Eva Riesbeck
Sammanfattning
Denna studie är ett exempel på hur vi bygger meningsmönster i vår omvärld. Vårt lärande och vår kunskap om omvärlden börjar i den värld som vi kan kalla vår livsvärld. Dissonansen mellan miljön i och utanför skolan visar sig tydligt inom skolmatematiken. Elevers tolkningar av vardagsuppgifter i matematik kan vara ett hinder för deras lärande. Fokus för studien är hur språket i den kommunikativa situationen eller aktiviteten kan förstås som en hjälp för elevers meningsbyggande. Kontextens och kommunikationens betydelse för matematik-undervisningen visas med hjälp av en textuppgift, som behandlar decimaler. Genom delaktighet och interaktion i dialogen finns det möjlighet för elever att utveckla nya meningshorisonter i matematiken.

Abstract
This study is about how we build patterns of meaning in the world around us. Our learning and knowledge about the surrounding world begins in what we may call the life world. The dissonance between the environment in and outside school is very distinct in mathematics as a school subject. Students’ interpretations of real life problems may become an obstacle for their learning in mathematics. The focus of the study is how the language in the communicative situation or activity could be understood as a means of construction of meaning. The importance of context and communication in the teaching of mathematics is visualized through analyses of 5 and 6 grade students’ work with a real life task, which involves decimals. Through participation and interaction, in the dialogue, the students develop new patterns of meaning that may facilitate their learning of mathematics.

EVA RIESBECK
Eva Riesbeck works as a lecturer at the University of Linköping. Her research interests include problemsolving, language and interaction in mathematics education.

Danmarks Matematiklærerforening planlægger i samarbejde med matematiklærerforeningerne i Norden at afholde en nordisk matematikkonference i Roskilde.

Mandag den 4. september – onsdag den 6. september 2006

Temaet for konferencen er:
Undervisning og læring Hvordan underviser vi alle elever i grundskolen, så de ud fra egne forudsætninger lærer optimalt?

Konferencen henvender sig til personer med indblik i og erfaring vedrørende matematik i grundskolen – specielt set i forhold til undervisning og læring.
Formålet er at give undervisere i matematik i Norden mulighed for at udveksle erfaringer og at lære om hinandens praksis inden for matematikundervisning i grundskolen.
Der sigtes mod, at konferencens oplæg og konklusioner efterfølgende bliver offentliggjorte med henblik på videndeling.
Antal deltagere vil blive ca. 80 nogenlunde ligeligt fordelt mellem landene.

Yderligere oplysninger kan ses på Danmarks Matematiklærerforenings hjemmeside: www.matematik.ffw.dk

Perspectives on Identity in Learning and Education Research

PhD course, 14-17 November 2006
Aalborg University, Denmark
The concept of identity has featured strongly in many research fields dealing with learning and education. Mathematics and science education research have not been an exception. Recently, learning has been understood as a process of becoming, in which individuals construct a sense of who they are and what they are demanded to do in a given environment. The way people learn mathematics and science in different educational (formal and informal) environments has been interpreted as a process of becoming a self in which mathematics and science play a central role. Concepts such as school students' mathematical (science) identity or science (mathematics) teachers' identity have provided alternative ways of understanding learning processes in these subject areas.
This PhD course is open to all research students who have an interest in the notion of identity. The course will address questions such as:
Why and how is the concept of identity relevant and interesting in educational research?
What are the various perspectives opened up by different concepts of identity?
What are the methodological challenges in investigating identity?
What are the implications for research in learning and education in the future?

The setting up of the course is interdisciplinary, however the specificity of science and mathematics education will be dealt with by the presence of Anna Sfard (University of Haifa, Israel) and of other researchers and students in the field. The perspective of mathematics and science education into researching learning and identity will be complemented by two other international guests, namely Etienne Wenger, (North San Juan, California, USA) and Carl Rhodes (University of Technology, Sydney, Australia).

For further information visit:
http://auaw2.aua.auc.dk/fak–tekn/phd/kurser/s10_1.htm
or contact Paola Valero, Aalborg University
Phone: +45 9635 9782
E-mail: paola@learning.aau.dk.

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As I wrote in the issue 3-4 of Nomad in 2005, the year 2006 will be exceptional because of the number of doctoral students who will finish their research education in mathematics didactics and defend their dissertations. Already in the first quarter of the year there have been at least five dissertations printed and made public. A short presentation of them will be made here, as they are likely to be of interest to the whole Nordic Graduate School and many Nordic researchers.

Five new dissertations in mathematics didactics
Morten Misfeldt defended his dissertation in January at the Danish University of Education. The title is Mathematical writing and it consists of a monograph, where all chapters except two build on earlier published work or work in progress (17 titles), such as one journal paper, 5 reviewed conference presentations, 3 book chapters and some other publications. The thesis reports research into the ways various technologies support mathematical writing. It comprises empirical and theoretical parts. One of the main theoretical outcomes is the attempt to view mathematical writing simultaneously as a creative writing process and as a mathematical problem-solving process.

Kristin Bjarnardottir defended her thesis at Roskilde University Center in February. Her thesis carries the title Mathematical education in Iceland in a historical context of socio-economic demands and influences. The data consists of historical documents from the development of school mathematics in Iceland and her dissertation is a monograph. The thesis consists of three parts and examines the history of mathematical education in Iceland and its position in comparison to its neighbouring countries. Mathematics education in Iceland differs primarily from its neighbours in the absence of demand for furthering higher mathematical education, nearly total dominance of a few institutions, and initiatives of individuals.

In March at the University of Helsinki, Iiris Attorps defended her dissertation Mathematics teachersí conceptions about equations. Her work is a monograph (a substantial volume with 231 pages). The aim of the study is to describe and clarify the mathematics teachersí subject matter and pedagogical content conceptions about equations. The research shows that some of the teachers included in the study do not have clear conceptions about what the pupils should learn in algebra at compulsory school. Both expert and novice teachers have various apprehensions of the pupilsí difficulties concerning equations.

Kristina Juter defended her dissertation in the beginning of April at Kristianstad University. The thesis comprises six published papers and an extended summary binding them together (referred to as ëkappaí in Swedish). The title of her work is Limits of functions ñ University students concept development. She followed university students during their first course of tertiary mathematics and inquired into their development of the limit concept. One of her results is that many students complete basic courses in calculus without ever understanding the notion of limits, or even understanding that they do not understand limits. Kristina Juter interprets her findings to argue that connectedness and continuity are essential features of teaching and learning limits to prevent students from failing.

Lil Engström carried out the defence of the thesis later in April and the work is called Möjligheter till lärande i matematik (Opportunities to learn mathematics). It is a substantial (238 pages) monograph, written in Swedish. The study examines how teachers formulate mathematical problems, how they use the experiences students have gained and what use they make of the potential of computer software. One result is that the teachersí ability to pose thought-provoking open-ended problems is the most important factor as it significantly influences what the students learn. Lil followed three teachers (in Sweden and Switzerland) and a group of student teachers and the software used was CabriñGeometrie.

It is not possible in the limited space here to do real justice to the content of the work of these five research students. We congratulate the new doctors in mathematics didactics and the research community in the Nordic countries for the new knowledge gained. I hope to have provoked the curiosity of the reader to explore the publications in more depth ñ they are certainly worth the effort.

The language and format of disserations
The dissertations can be used to illustrate some of the questions that are constantly discussed by doctoral students in the Nordic Graduate School. One question is what language to use in your dissertation. Here we see four dissertations in English and one in Swedish. Each doctoral student must create an overview of the pros and cons of writing in his or her own language or in English. This is one question that has been discussed also in the supervisorsí seminars arranged by NoGSME. It is a complex issue and there is no general answer. Each individual must carefully consider her or his situation and how to best meet different needs.

Another question concerns if it is better to write a monograph or use a collection of published papers. In the five examples above we see one collection of published papers, three monographs and then the interesting case where Morten Misfeldt builds on his papers of different kinds but using the material by rewriting it in the chapters of the dissertation. This is a challenging hybrid of a collection of papers. He thereby avoids the hard question of what kind of journals are at an acceptable level for papers in a collection. Which journals are of acceptable scientific quality and which are not? As in the question of language my advice would be that the issue is best dealt with by the doctoral student and the supervisor together.

The next seminar for supervisors in NoGSME will investigate different doctoral programmes and look at differences and similarities between them. One obvious difference at the moment is the length of the programmes. Denmark and Norway have three-year (fulltime) doctoral programmes while Sweden has four years. The prerequisites are different at the moment. For example in Norway the doctoral students must have a mastersí degree when they start but in Sweden a teacher education qualification can be accepted. Might all this be changed when all countries have taken the Bologna agreements into consideration?

To publish or not to publish the thesis?
Another visible difference between theses is whether they are printed as proper books or not. It appears that in Denmark the custom is to make compendiums, A4-format copies of the manuscript with no proper copy-page (with ISBN but not ISSN). The other countries appear to publish dissertations as books, in book-format and with both ISBN and ISSN. Are they all considered to be publications or is it as in the UK or the US that they are called unpublished dissertations? In Sweden there is a tradition that copies of the thesis are delivered to university libraries automatically and thus they are available all over the country.

As we know there are many more dissertations that will be completed during 2006, it will be interesting to see at the end of the year how many are in English and how many are monographs. The demand for researchers to publish regularly in journals is emphasised today so maybe students will consider it fruitful to start to publish in journals already during the research studies. Those who publish monographs will hopefully soon try to write papers for journals based on their work. Nomad is a good journal to start with for the Nordic doctoral students as we saw in the latest issue, nr 3-4 for 2005, with four contributions from doctoral students.

A good forum for discussion
It is to be hoped that the discussion among doctoral students and supervisors about choice of language and format for the dissertation will continue so that all arguments are clear for everyone. The summer school for doctoral students in D¯mmesmoen in June will offer a good forum for such a discussion, as will the seminar for supervisors in May, which will focus on different doctoral programmes in mathematics didactics.

The board of the Nordic Graduate School welcomes suggestions for future doctoral courses, seminars and workshops covering different themes and we look forward to offers from different universities to host such events. It only requires an email to start a planning process.

Barbro Grevholm

Director of the Nordic Graduate School
Agder University College
Serviceboks 422
N-4604 Kristiansand
Norway
barbro.grevholm@hia.no

Markus Hähkiöniemi

Abstract
This paper is a report of a study on how a less successful student perceives the derivative from the graph of a function. A task-based interview of a grade 11 student was analyzed to find how she perceived the derivative from a graph of a function and what kind of representations she used for this. The results show how she used representations of the increase, the steepness, and the horizontalness of the graph to perceive the derivative. Gestures were an integral part of her thinking. This case shows that with appropriate representations students can perceive essential aspects of the derivative from the graph of the function, and that students can consider the derivative as an object at the very beginning of the acquisition process.

Yhteenveto
Tässä tutkimuksessa tarkastellaan miten hieman heikommin matematiikassa menestynyt opiskelija havaitsee derivaatan funktion kuvaajasta. Lukion toisen luokan opiskelijan tehtäväpohjaisesta haastattelusta analysoitiin miten hän havaitsi derivaatan funktion kuvaajasta ja millaisia representaatioita hän tähän käytti. Hänen huomattiin käyttävän kuvaajan kasvamisen, jyrkkyyden ja vaakasuoruuden representaatioita. Eleet olivat olennainen osa hänen ajatteluaan. Tulosten perusteella opiskelijat voivat tarkoituksenmukaisia representaatioita käyttäen tehdä olennaisia havaintoja derivaatasta ja tarkastella derivaattaa objektina jo oppimis prosessin alkuvaiheessa.

MARKUS HÄHKIÖNIEMI
Markus Hähkiöniemi, M.Sc., is a researcher in the Finnish Graduate School of Mathematics, Physics, and Chemistry Education. He is working in the Department of Mathematics and Statistics in the University of Jyväskylä and writing his doctoral thesis on the role of different representations in learning the derivative.

As announced in the previous number, it is time for the editorship to move to another country, namely Denmark. From the beginning of 2006, Morten Blomhøj, IMFUFA, Roskilde University, and Paola Valero, Department of Education, Learning and Philosophy, Aalborg University, are the new chief editors. During the next four years the editorship of NOMAD will be a cooperative enterprise between two persons. We will work hard to strengthen NOMAD's profile and its position on the Nordic scene of mathematics education research both in terms of the number of subscribers and in terms of the scientific quality of the journal.

One of the first tasks we face is raising the number of submitted manuscripts for publication. During the last years research in the field of mathematics education has significantly grown in the Nordic region, not only because of the expansion of the work of experienced researchers, but also because of the increasing interest in doctoral studies. The political emphasis on the necessity of improving mathematical instruction in the Nordic countries has also meant public investment in a variety of developmental projects in which researchers and teachers cooperate in bringing together theory and practice, and producing knowledge about changing processes in educational institutions. From these different sources there is a production of knowledge concerning the phenomena of teaching and learning of mathematics. NOMAD's role is making public this knowledge, as a basis for an academic debate and an improvement of praxis. We know that there are many exciting research projects going on in our field. Therefore, we encourage all members of the mathematics education community in the region to participate actively in preparing manuscripts and submitting them for publication.

The issue of language also needs to be reconsidered. It has been clear from NOMAD's very beginning the necessity of building scholarship in mathematics education in the Scandinavian languages. As a result of this the journal has published three numbers per year with most papers written in the Scandinavian languages (Danish, Norwegian and Swedish), and one number in English. Although we still believe in the strength of publication in these languages, we can also confirm that there are many Nordic authors who prefer English as their working language. We also need to add the fact that NOMAD has also become an attractive journal for publication of mathematics education research produced by authors from other countries in the world. It is clear that mathematics education research in our countries is responsive to globalisation and internationalisation. This fact makes it difficult to have a clear-cut language policy. More than suggesting a particular principle for a language policy, we encourage an opening of this debate in our community. We continue considering manuscripts for publication in all the four languages mentioned above, and authors outside the Nordic region are also welcomed to publish research of relevance for the region in NOMAD.

A new initiative the editors will engage with is the production of one special thematic issue per volume, normally the fourth issue. The editors will send out both special invitations and a general call for contributions for authors to submit manuscripts for the thematic issue. Suffice to say that all papers will go through the usual review procedure.
The theme proposed for the special issue in 2006 is Learning difficulties in/with mathematics. In the Nordic countries there is a strong tradition of building inclusive educational systems. In the field of mathematics education many recent initiatives have chosen to teach mathematics for all, also for those who experience great difficulties in/with the learning of the subject. These efforts have faced great obstacles and research on these issues have raised political and scientific debates about, for example, the meaning of the very same term "difficulties in/with learning mathematics", the theoretical perspectives that are suitable to give an account of these problems, and the methodological challenges for carrying out research on the topic. It is clear that a deeper understanding of all these issues is a first necessary step in tackling the problems. We invite contributions within this topic for the thematic issue of NOMAD 2006. The deadline for submission of manuscripts is September 1st, 2006.

About this number

This number collects an interesting series of papers, which represents the variety of research going on in the field in the Nordic region. In accordance with our language policy this issue contains articles both in English and Swedish. In their article "Accountability affects the use of small group learning in school mathematics", John Berry and Pasi Salhberg raise a discussion about how teachers in England and Finland view the role of small group work in the learning of mathematics, and point to how teachers' views do not only have to do with their understanding of the teaching/learning activity, but also with how they interpret and learn to live in the contextual constraints of the school. The paper in Swedish "Det hänger på decimalen! Om hur vi formar och bygger meningsmönster i vår omvärld" by Eva Riesbeck analyses in-depth how students make sense of a real world situation and a related problem. The problem, which is about the ordering of four decimal numbers in the context of a 60 meters running competition, is an excellent basis for analysing the tremendous complexity involved in students' sense making processes. Through analysing the students' communication in different types of interview situations, Riesbeck shows that the students need to be actively involved in the communication in order to develop and maintain a mathematical perspective on the real life situation. Hence the study underlines the importance of inclusive dialogue in mathematics teaching. Markus Hähkiöniemi in his paper "Perceiving the derivative: the case of Susanna" presents a case study on a 11th grade student's perceptions of the derivative of a function in its different forms of representations. Carefully designed task-based interviews and connected analyses allow the author to draw a both detailed and comprehensive picture of Susanna's perception. The findings are interpreted from the theoretical standpoint learning in the embodied and in the symbolic world. Hereby the paper contributes to the ongoing theoretical discussion about learners' formation of mathematical concepts.

Morten Blomhøj and Paola Valero
NOMAD Editors

John Berry and Pasi Sahlberg
Abstract
This study investigates the perspectives of a sample of teachers on the use of cooperative small groups in the teaching and learning of mathematics. We asked teachers (N = 18) in England and Finland about their experiences and ideas of small group learning in mathematics. The research tool used the ordering by each teacher of eight mathematics tasks into a hierarchy from those tasks that are best for small group working to those tasks that are best for individual working as a frame for in-depth interviews. We conclude that the role of small group learning as seen by most of the teachers is for doing mathematics, introducing social skills and discussion rather than learning mathematical knowledge and skills. Furthermore we report on the barriers to using small group learning caused by the accountability structures inherent in the educational systems of both countries.

Sammanfattning
I den föreliggande studien undersöktes en grupp lärares uppfattningarom användning av elevsamarbete i smågrupper vid undervisning i matematik. Vi frågade 18 lärare i England och Finland om deras erfarenheter och idéer gällande lärande av matematik i smågrupper. Lärarna fick i uppgift att ordna åtta matematikuppgifter hierarkiskt, från de uppgifter som lämpar sig bäst för arbete i smågrupper till uppgifter som lämpar sig bäst för individuellt arbete. Undersöknings-instrumentet utnyttjade de ordnade uppgifterna som ett ramverk för djupintervjuer. Vår konklusion är att användningen av smågrupps-inlärning, enligt de flesta lärarna, bäst lämpar sig för att lösa matematiska uppgifter, introducera sociala färdigheter och diskussion framom att lära sig matematiska kunskaper och färdigheter. Vi belyser också svårigheter, då det gäller användningen av smågruppsinlärning, som sammanhänger med underliggande strukturer i de båda ländernas undervisningssystem.

JOHN BERRY
Professor John Berry is Professor of Mathematics Education at The University of Plymouth, is Mathematics Professor in Residence at Wells cathedral School, Somerset, is a consultant to the National Academy of Gifted and Talented Youth delivering workshops to members of the Academy and to primary pupils and delivers professional development courses for teachers nationally and internationally. His research interests are in mathematics education, particularly students’ development and understanding of key concepts, the use of hand-held technology and symbolic algebra in learning and teaching mathematics.

PASI SAHLBERG
Dr. Pasi Sahlberg is a Senior Education Specialist in the World Bank, Washington, DC. He is a former Counsellor of Education in the National Board of Education (Finland) and director of the Centre for School Development in the University of Helsinki. He has worked as a mathematics teacher, teacher educator and researcher. He has also advised several governments in their mathematics education reforms. His main fields of research interests are school improvement, global education policies and mathematics education.

Assessment of university students’ understanding of abstract binary operations

Timo Ehmke, Martti E. Pesonen and Lenni Haapasalo

Abstract

This paper presents the results of a study on the use of interactive online tasks to assess students’ conceptual understanding of abstract binary operations in a first-year linear algebra course. The assessment consists of recognition, identification and production tasks and uses verbal, graphic and symbolic representations of binary operations in numerous point set contexts. The aim of the study is to directly assess the students’ understanding of binary operations and – more indirectly – to identify different profiles for the students’ procedural and conceptual knowledge levels. A latent class analysis revealed different levels in students’ conceptual understanding. Implications will be drawn for teaching abstract binary operations – and other similar concepts. Finally, some suggestions about conceptual qualifications for mathematics teacher education will be discussed.

Timo Ehmke

Timo Ehmke, Dr. habil., works as a research scientist at the Leibniz Institute for Science Education (IPN) in Kiel, Germany. He studied Mathematics, Technology and Education at the Universities of Kiel and Flensburg in Germany. He received the state examination in Mathematics and Technology Education in 1997 and a Ph.D. in Education in 2001. Since 2001 he has been working at the Department of Education at the IPN Kiel in the field of Educational Research. In 2007, he finished his habilitation (postdoctoral lecture qualification) in Educational Research at the University of Kiel.

Martti E. Pesonens

Martti E. Pesonen, Ph.D., acts as senior lecturer at the University of Eastern Finland (Joensuu campus). He studied mathematics and physics in Joensuu and Paris and got the doctor’s degree in mathematics in 1986. He achieved the pedagogical competence of mathematics and physics subject teacher in 1997. He teaches mostly the first year university mathematics courses including basic relational concepts, discrete mathematics and axiomatic approach to linear algebra.

Lenni Haapasalo

Prof. Dr. Lenni Haapasalo worked at the University of Jyväskylä, at first 9 years as mathematician, and after that18 years as Senior Lecturer/Associate Professor in Mathematics Education. Since 1999 he is full-time Professor of Education at the University of Eastern Finland. His research interest is to develop practical theories based on modern socio-constructivist views, emphasizing technology-based self-determined learning environments, links between conceptual and procedural knowledge, and the genesis of sustainable heuristic processes.

Identity development in limbo: teacher transition from education to teaching

Hanna Palmér

Abstract

The theories and results discussed in this article are from a study investigating the identity development of novice primary mathematics teachers. The article has two aims: first, to elaborate the notion of beliefs in relation to the notions of identity and identity development, with the purpose of developing a framework to investigate the process of becoming and being a teacher of mathematics; and second, to offer an example of the use of this framework in a study of novice primary mathematics teachers. The core of the example is the case of Jenny, a Swedish novice primary mathematics teacher. Jenny’s case, however, is not simply about her but also about identity development when the formal aspect of employment is missing, a case not rare in Sweden.

Hanna Palmér

Hanna Palmér is a postgraduate student in mathematics education at Linnaues University, Sweden. Her main research interest is mathematics teacher education, and the becoming, being and mathematics teaching of primary teachers after graduation.

Affektive sider ved lærerstudenters arbeid med matematikk

Leif Kværnes

Sammenfattning

Formålet med artikkelen er å belyse og drøfte sider ved allmennlærerstudenters utvikling av lærerkompetanse i matematikk. I empiriske analyser har jeg har valgt å fokusere på studenters affektive eller følelsesmessige forhold til læring av/arbeid med matematikk; som er sett som et delaspekt ved lærerkompetansen. I første del av artikkelen redegjør jeg for sentrale teoretiske utgangspunkt; et triadisk syn på læring og en kommunikativ tilnærming til analyser og beskrivelser av læring og undervisning. Andre delen av artikkelen starter med analyser og beskrivelser av affektive sider gjennom utvalgte eksempler fra studenters arbeid med matematikk. Disse beskrivelsene er utgangspunkt for en avsluttende problematisering og drøfting av hvordan affektive sider kan influere på studenters utvikling av lærerkompetanse i faget.

Abstract

My intention with this article is to discuss some aspects of teacher student’s development towards becoming mathematics teachers. My main focus is on the relations between affect and cognition. First part of the article will be theoretical. I will here outline how this relation is seen, and I also describe what may be called a communicative or discursive approach to this relation. In the second part I use this approach on student’s utterances while working with mathematics. My intentions are not to make representative or broad descriptions of relations between affect and cognition. The descriptions will be used as points of departure for discussing student’s development towards becoming mathematics teachers.

Leif Kværnes

Leif Kværnes er høgskolelektor i matematikk ved Høgskolen i Oslo, avdeling for lærerutdanning, hvor han underviser både grunnutdanningsstudenter, masterstudenter og lærere som søker etter- eller videreutdanning. Hans forskningsinteresse er knyttet til lærerutdanning og til utvikling av lærerkompetanse for undervisning i matematikk.

JESSICA CARTER

Abstract

This paper poses two problems for von Glasersfeld's Radical Constructivism. The first problem concerns the rejection of the idea that it is possible to share meanings. The second problem is that Radical Constructivism rejects the notion of an objectively existing reality of which we can have objective knowledge. Yet with respect to mathematics, von Glasersfeld seems to claim that it is possible to obtain objective knowledge. We propose an alternative position - Constructive Realism - that gives a description of what mathematical objects are and gives an account of why knowledge in mathematics is objective. Furthermore, we argue that some of the assumptions, used in von Glasersfeld's description of how numbers are formed, support the claim that some meanings are objective and that communication is possible. Finally, we consider some of the implications this position has for mathematics education.

Sammendrag

Artiklen formulerer to problemer for von Glaersfeds Radikale Konstruktivisme. Det f¯rste problem vedr¯rer forkastningen af ideen om mulighed for delagtigg¯relse af mening. Det andet problem er, at den Radikale Konstruktivisme benÊgter eksistensen af en objektiv ydre virkelighed, som vi kan have objektiv viden om. Dog ser det ud til, at von Glasersfeld mener, at vi kan have objektivt gyldig viden i matematik. Vi prÊsenterer en alternativ position - Konstruktiv Realisme - som beskriver hvad matematikkens objekter er og redeg¯r for hvorfor viden i matematik er objektiv. Desuden argumenterer vi for, at nogle af de antagelser som von Glasersfeld bruger i beskrivelsen af hvordan tallene dannes, st¯tter pÂstanden om at nogle meninger er objective og at kommunikation er mulig. Til sidst giver vi nogle forslag til hvilke implikationer denne position har for undervisning i matematik.

JESSICA CARTER
Jessica received her Ph.d. at the Center for Educational Development in University Science, University of Southern Denmark (SDU) in 2002. She works mainly in Philosophy and History of Mathematics, where she tries to give a description of mathematical objects that agrees with mathematical practice. She is currently assistant professor at the Department of Mathematics and Computer Science and member of the Science and Mathematics Education Research Group at SDU.

ATHANASIOS GAGATSIS, BHARATH SRIRAMAN, ILIADA ELIA & MODESTINA MODESTOU

Abstract

This study explores young children's strategies while transforming polygons, through the use of geometrical models. Data were collected from 291 children ranging from 4 to 8 years of age in Cyprus. Children were asked to draw a stairway of specific polygons, with each shape being bigger or smaller than its preceding one. Relationships between children's responses in the transformation tasks, their ability to recognize geometric shapes and their IQ level were investigated. Results showed that children used three alternative strategies in the transformation tasks. Children's IQ score was directly associated with their transformation strategies, while only a low recognition ability was associated with the use of a defective strategy.

Sammendrag

I dette studium undersøges 4-8 årige børns strategier til transformation af trekanter, kvadrater og rektangler. Datamaterialet omfatter tegninger fra 291 cypriotiske børn, der har løst hver seks transformationsopgaver. Børnene blev bedt om at tegne serier af specifikke ligedannede polygoner af henholdsvis stigende og faldende størrelse. Sammenhænge mellem børenes transformationsstrategier, deres evne til genkendelse af de geo-metriske figurer og deres scorer i en IQ-test blev undersøgt statistisk. Resultaterne viser, at børnene bruger tre alternative strategier i transformationsopgaverne, at børnenes strategier er direkte forbundet med deres IQ-scorer, og at dårlig genkendelse af geometriske figurer har en sammenhæng med brugen af utilstrækkelige transformationsstrategier.

ATHANASIOS GAGATSIS
Athanasios Gagatsis is Professor of the Department of Education and Dean of the Faculty of Social Sciences and Education at the University of Cyprus. He received his Ph.D. in the Didactics of Mathematics from the Department of Mathematics at the University of Strasbourg, France. His research focuses on the cognitive development of mathematical concepts, with a particular emphasis on representations and on the History of Mathematics Education. He is the Chief Editor of the Mediterranean Journal for Research in Mathematics Education, and serves on the editorial board of several international scientific journals.

BHARATH SRIRAMAN
Bharath Sriraman is Associate Professor of Mathematics at the University of Montana, with an eclectic range of research interests. He works in the domains of Cognitive Science; Gifted and Talented Education; History and Philosophy of Mathematics and Science; Mathematics Education and Elementary Ergodic Theory. He received his PhD in Mathematical Sciences from the Department of Mathematics at Northern Illinois University, USA. Bharath is the Chief Editor of The Montana Mathematics Enthusiast and he also holds positions on the editorial boards of several international scientific journals.

ILIADA ELIA
Iliada Elia recently completed her Ph.D. on Mathematics Education at the University of Cyprus under the supervision of Professor Athanasios Gagatsis. She is currently a researcher and educational personnel of Mathematics Education at the University of Cyprus. Her research focuses on the role of representations in problem solving and the cognitive development of mathematical concepts, particularly the function concept.

MODESTINA MODESTOU
Modestina Modestou is a Ph.D. candidate on Mathematics Education at the University of Cyprus currently working with Professor Athanasios Gagatsis. Her research focuses on proportional reasoning and the role of representations in the learning of mathematics.

In this issue you will find three very different papers. A paper about the philosophy of mathematics, a paper analysing quantitative data about young children's geometrical transformations of polygons, and a paper based on a developmental project at lower secondary level. The project aims are to develop the students' methods for metal calculations and their reflections about choices of methods for simple calculations.

This variation reflects the ambition of NOMAD to present a broad coverage of the field of mathematics education and to channel both research papers and reports from developmental projects. The editorial process gives priorities to authors from the Nordic countries. Joined papers, by Nordic and non-Nordic authors, are of course most appreciated, and we also intend to continue to publish papers by non-Nordic authors addressing issues of interest to the Nordic community.

About this issue

Jessica Carter has written a paper challenging two of the basic claims in von Glasersfeld's radical constructivism, namely that there are no such thing as objective knowledge and that exact knowledge about another person's knowledge is an illusion. Her point of departure is the position that a philosophy of mathematics and mathematical learning, which is indeed included in the domain of radical constructivism, should be in accordance with the practice of mathematical research and the practice of mathematics teaching, respectively. Illustrated with examples of the development of advanced mathematical concepts, Carter argues that even though concepts are constructed by human beings in the practice of mathematical research, mathematical knowledge about these constructions and their representations can still be seen as objective knowledge.

With learning still seen as a personal construction of meaning, this view on the ontology of mathematical concepts and the epistemology of mathematical knowledge allows for inter-subjectivity in a community of mathematics learners. Inter-subjectivity is normally rejected by radical constructivism despite the fact that inter-subjectivity is evidently present in the practice of mathematics teaching. Carter names her position Constructive Realism and ends up by drawing a few basic implications for mathematics education. The genesis and the motivation for the introduction of the mathematical concepts should be made subject to the teaching, and the process of assigning meaning to symbolic representations through which the students may gain access to the mathematical concepts should be in focus in mathematics teaching.

The paper "Exploring young children's geometrical strategies" reports on a large quantitative research project involving 291 children form Cyprus. One of the very interesting findings in this project is that in all cases more than 40% of the 4-6 year old children are able to efficiently transform simple polygons and draw series of similar triangles, squares and rectangles with increasing or decreasing sizes. Moreover, it is documented that there is a tendency that older children (6-8 years), who have already attained one or two years of mathematics teaching, more frequently use inadequate one-dimensional strategies when transforming simple polygons. Such findings raise interesting and challenging questions about how to identify and recognise young children's geometrical knowledge and intuition when they enter the school system. More research and developmental projects are needed in order to find ways to use young children's knowledge and intuition as a basis for mathematics teaching. This is indeed most relevant in relation to the ongoing political discussion about the subject matter content in pre- and early school activities.

However, the main focus of the paper is to investigate the relationship between the children's strategies working with geometrical transformation tasks, the children's IQ-level, and their ability to recognise the relevant geometrical shapes. An extensive statistical analysis shows that the IQ-level is directly associated with the children's use of transformation strategies and that children with low recognition of the geometrical shapes tend to have defective transformation strategies. Such results are in themselves of scientific interest, but moreover they give rise to an important discussion about the role of mathematics in pre- and early schooling. Can mathematics teaching play a progressive role in the educational system by identifying and helping children with special needs? The problematique concerning students with special needs in mathematics is precisely the theme for the coming thematic issue of Nomad (no. 4, 2006), and this specific issue will most certainly be discussed here.

The last paper, by Frode Olav Haara, reports and analyses a developmental project, which aims at making lower secondary students more reflective about their choice of method for elementary arithmetic calculations. In this particular case, mental calculations were observed to be disregarded by the students when they had a calculator at their disposal - even for very simple calculations. Research literature confirm this to be a general tendency and expose concerns about the possible consequences for the students' conceptions of the base 10 number system and their number sense in general. In the developmental project, a special didactical intervention is designed and carried through. The intervention includes the students' work with traditional drill exercises under time limits but also a strong emphasis on the teacher's dialogue with the entire class and with individual students about their choice of method. In what sense and what degree the intervention can be considered in accordance with a social constructivist view on learning and with a critical perspective on mathematics teaching is interestingly discussed in the paper.

The intervention, which is quite easily implemented in practise, seems to make the students use mental calculations more frequently in their work with mathematical problems. In addition to this, all students improved their mental calculations during the intervention. The author's analysis of the class dialogue indicates that the intervention also made the students more aware of their choices of calculation methods. The paper is rounded off with considerations on important frame conditions concerning the practical implementation of the intervention.

In addition to the three papers you will find information on the activities of The Nordic Graduate School in Mathematics Education. This time, Barbro Grevholm, director of the graduate school, reports on six new dissertations, which all have been successfully defended. Added to the five dissertations presented in the previous issue of NOMAD, we are witnessing a historical growth rate in the Nordic community of researchers in mathematics education, and in the very near further, more are coming. From the perspective of NOMAD the future looks bright in terms of excepted subscriptions and submitted papers.

We wish all readers a pleasant summer.

Morten Blomhøj and Paola Valero
NOMAD Editors

Summer school of 2006

The NoGSME summer school for doctoral students took place in June. Twenty participants from the Nordic and Baltic countries met for a week at the campus of Agder University College in Dömmesmoen, Norway. Most of the time was spent in working groups of 6-8 persons with an experienced group leader. The work focussed on the studies of the participants and dealt with their research questions, choice of theoretical framework, methods, data collection, analysis, results, discussion and implications for teaching mathematics. Many hard questions were raised and everyone had to justify their research design and decisions taken. The group leaders were Kath Hart, Ole Björkqvist, Christer Bergsten and Trygve Breiteig. Lectures were given by Kath Hart and Simon Goodchild. A series of workshops dealt with questions such as How to read a scientific paper productively, The role of theory in scientific papers and How to write a scientific paper. All participants produced a draft or sketch of their next paper for a scientific journal. NOMAD certainly can look forward to good access to Nordic and Baltic research papers.
The next summer school will take place in Iceland in the beginning of June 2007. Suggested themes are how to design a classroom study and how to collect and analyse classroom data.

Dissertations in mathematics education

In the last issue of NOMAD five Nordic dissertations were presented. We can now take part of another six dissertations treating questions on teaching and learning mathematics.
Mette Andresen defended her dissertation, Taking advantage of computer use for increased flexibility of mathematical conceptions, in May at the Danish University of Pedagogy. Mette Andresen's study is part of a larger project in Denmark called World Class Math & Science. In a sub project each student had a laptop, with computer algebra software, at their disposal. Participants gained the experience that computer use in upper secondary school mathematics has a potential. Mette studied: "How could these potentials be captured and conceptualised?" Later, her questions became: "Is flexibility a supportive construct for articulation of experiences of teaching and learning within a modelling approach? Is it useful for realisation of the learning potential of students' concept formation?" The methods can be described as an interpretative approach to experimental teaching design. Data were collected through classroom observations with field notes and video recording, and through interviews with students and teachers at four different schools all involved in the same project.

Gunnar Sjöberg defended his dissertation at Umeå University in the Graduate School of Pedagogical work. The title is Om det inte är dyskal-kyli - vad är det då? (If it is not dyscalculia - what is it then?). He investigated the concept of dyscalculia in the research literature and found that it is an ill-defined or not defined concept. The pupils he followed from grade five in school to upper secondary school were said to be in mathematics difficulties but many of them later succeded in the subject. One crucial factor seems to be the short time these pupils spend on mathematics learning, often less that half an hour per school week. Compulsory school mathematics teachers will find interesting interviews as well as noteworthy pupils' comments in this work.

Monica Johansson wrote about Teaching mathematics with textbooks - a classroom and curricular perspective. She defended her work in June at Luleå University of Technology. The dissertation consists of four papers and a preamble and the focus of all parts is the relationship between the textbook and the curriculum. She shows that the textbook influences not only what kind of tasks students are working with during the lessons, but also the examples the teacher presents on the board, what kind of concepts are introduced and how they are introduced. The teacher can get into problems because of too much reliance on the textbook. The study shows the relative autonomy of the mathematics teacher in relation to the most common teaching tool in Swedish classrooms - the textbook.

Örjan Hansson is, as is Monica, a member of the Swedish Graduate School in Mathematics Education. He defended his dissertation the day after Monica at the same university. The work carries the title Studying the views of pre-service teachers on the concept of function. His work consists of five papers and an overview that binds the work together. Three different groups of pre-service mathematics and science teachers for grade 4-9 were his informants. He used questionnaires, concept maps, and interviews in order to understand and analyse how they perceive the concept of function. The concept of function is rarely a well integrated concept and the pre-service teachers view of the concept is represented by a less developed knowledge structure than one could wish for. Thus there are many implications for the teaching of pre-service teachers.

On the same day as Örjan, Maria Bjerneby-Häll defended her thesis at Linköping University. The title is Allt har förändrats och allt är sig likt: en longitudinell studie av argument för grundskolans matematikundervisning. The aim of this thesis is to describe and analyse arguments for mathematics in compulsory school and to understand why and how the official arguments change. The point of departure is that the conditions and the reality for school mathematics can be understood through an analysis of official arguments and of personal arguments given by teacher students and teachers. She followed a group of teacher students through their education and their first three years of teaching. The result shows that during their education the teacher students develop a view on mathematics and mathematics education harmonizing with the goals of mathematics in the national syllabus. The novice teachers experience quite different conditions when they start to work as teachers. Preparing their pupils for the national test becomes the most important goal. A factor influencing the mathematics teacher is the qualification requirement in mathematics from compulsory school to go into the national programs in the upper secondary school. The novice teachers experience a conflict between different goals in the national curriculum and course syllabus for mathematics.

Andreas Ryve's dissertation took place at Mälardalen University in the end of June with Anna Sfard as opponent. The title is Approching mathematical discourse. Two analytical frameworks and their relation to problem solving interactions. His main aim is to investigate how conceptual understanding and problem solving can become a natural part of mathematics teaching and thus of students' mathematical knowledge construction. He wants to characterize the classroom discourse in two different problem solving courses in teacher education and also to investigate and further develop two analytical frameworks - a communicational approach and a dialogical approach used to study mathematical discourses. He shows that the classroom discourse can be characterized in terms of subject oriented, didactically oriented and problem solving oriented discourses. The analytical frameworks are further developed in his study.
Readers are recommended to examine all these dissertations closer. They are all available in electronic form from the authors. Already in September still a number of dissertations will be presented.

Workshop on mathematics textbooks and curricula

The Nordic Graduate School organised a workshop on textbooks and curriculum studies in the end of May this year. This area of research seems to have been neglected for at least two decades in the Nordic countries but a new interest has grown and a number of doctoral students are now working in this area. Birgit Pepin from Manchester University and Linda Haggarty from Open University, well known international researchers, were invited. They have both published studies on textbooks and are often quoted in recent works in the Nordic countries. Both doctoral students working in the area and more senior researchers took part. Others interested in being part of this new network are welcome to contact us. The first follow-up will be a discussion group on the issue at PME30 in Prague, were many of the participants will meet again and continue to develop the ideas from the workshop. The presentations from the workshop will be documented and published. We hope that this network will support the development of research on textbooks and curricula as well as comparative studies in the Nordic and Baltic countries.

A similar workshop will take place during the autumn, probably on the use of technology in mathematics teaching and learning. Information will be distributed as soon as the programme is settled. All ideas and suggestions for content of seminars or workshops are welcome.

Seminar for supervisors in NoGSME
The fifth seminar for supervisors organised by NoGSME took place in Vasa in Finland in the beginning of May. At several universities in the Nordic countries there is a wish to build research education in mathematics education. The process often starts with the creation of a doctoral stipend for a student who is then enrolled in an existing programme at another university. The hope seems to be that this student after finishing doctoral studies will work in the home university and be one of the staff members in a new programme. Thus the seminar this time investigated different existing research programmes in mathematics education and we discussed some characteristics and crucial questions to be aware of in the creation of new programmes (and continuation of existing ones). The sixth seminar will take place during autumn 2006 and suggestions for themes to deal with are welcome to the NoGSME board.

Half way through the life of NoGSME

NoGSME has now existed and acted for two and a half years and the same amount of time is left for our joint work. We have offered two summer schools, many doctoral courses in all of the Nordic countries, five seminars for supervisors, two workshops, five mobility stipends for doctoral students, and travel support to courses for a number of doctoral students. As NoGSME started it was clearly said by the funding organisers, NordForsk, that this is a one time commitment. It is not possible to get a prolongation of the grant. Thus we ask you all to think carefully about how to use the rest of the time and money accessible in line with the application from 2003.

As said before, the board of the Nordic Graduate School welcomes suggestions for future doctoral courses, seminars and workshops and we look forward to offers from different universities to host such events. Just send an email in order to start a planning process.

BARBRO GREVHOLM
Director of the Nordic Graduate School
barbro.grevholm@hia.no

FRODE OLAV HAARA

Stadig oftere opplever vi at elever bruker kalkulator til å gjøre utregninger som må karakteriseres som enkel hoderegning. Artikkelen omhandler et prosjekt gjennomført på ungdomstrinnet med fokus på å bevisstgjøre elevene i valget av arbeids-metode, og øke bruken av hoderegning. Sentralt står bruken av et tradisjonelt virkemiddel i en matematikkundervisning preget av lærerens tro på aktive og læringskritiske elever.

Abstract

Constantly we experience the fact that pupils use calculator to do what we must refer to as elementary mental calculations. This article is based on a project which was carried through in secondary school, focusing on making the pupils more aware when choosing a manner of calculation, and increasing their use of mental arithmetic. The use of a traditional approach in an environment of teaching mathematics featured by faith in active and critical pupils is vital for the project.

FRODE OLAV HAARA

Frode Olav Haara er Cand. Scient. med hovedfag i matematikkdidaktikk fra Høgskolen i Agder, og ansatt som høgskolelektor i matematikkseksjonen ved Høgskulen i Sogn og Fjordane - avdeling for lærarutdanning og idrett. For tiden er han i en fireårig åremålsstilling som studieleder for allmennlærerutdanningen ved Høgskulen i Sogn og Fjordane.
Forskningsinteresser: Bruk av hoderegning og kalkulator, Valg av arbeidsmetoder ved utregninger, Problemløsningens posisjon i grunn-skolen, Praktiske aktiviteters betydning for læring av matematikk.

The sixth seminar for supervisors and new doctors

During two days in the beginning of October, twenty-two supervisors and new doctors were gathered in Magleås in Denmark for a seminar. Participants from seven countries took part. The theme this time was how to develop oneself to become a good supervisor. The role of NOMAD as a support for new doctors in their academic career was discussed. Some of the questions that were analysed were: How can NOMAD contribute to the fostering of new researchers in mathematics education in the Nordic countries? Which types of activities could be fruitful to support the interplay between NOMAD and doctoral programmes in mathematics education? What can be done to ensure the scientific quality and relevance of NOMAD? What could be interesting thematic issues in NOMAD? One suggestion was that a future seminar could offer a programme, where authentic review reports of journal papers were analysed and discussed. Some international journals share the review reports of a paper with all the reviewers of the paper and this was indicated to be a strong learning process in developing a constructive reviewer of papers. Such a competence of a supervisor was considered fruitful for the doctoral students. This suggestion can be used at one of the seminars in spring 2007. Another issue at the seminar was courses for supervisors at different universities. As examples were used the courses in Linköping University and Luleå University of Technology. They both have similarities in taking up questions as the regulations for doctoral education, financing of research, equity issues, ethics in research, demands and levels for doctoral examination, models of supervision, and the relation between supervisor and doctoral student.

Episodes in supervision

All participants were invited to write down and share with others a crucial episode in supervision, which one had experienced as supervisor or doctoral student. Starting points were two episodes from the book Hand-ledning av doktorander by Jitka Lindén at Lund University. There were in all nine different episodes to analyse and discuss. Many different questions of general character were raised based on the episodes. The discussion touched upon for example if the supervisor should publish together with the doctoral student. Some of the arguments in favour of this viewpoint were that it could actually help to raise the willingness to read the paper, and it can be an appropriate way to give credit to the supervisor for the work of supervision, which is otherwise not visible as an academic reward. If a doctoral student has difficulties with the writing process, a co-writing experience with the supervisor might help. The case must always be that the supervisor is a real co-writer of the paper. Other interesting issues were how to manage the delicate balance between patient waiting and setting demands, how to assist in the writing process, and how to support the doctoral student in taking responsibility for her/his own work. The discussions were based on five ethical principles presented in the book by Jitka Lindén, and which she claims are part of the ethical code for many professions.

Three new dissertations in mathematics education

At Ålborg University Lene Östergaard Johansen defended her thesis "Hvorfor skal voksne tilbydes undervisning i matematik? - en diskursanalytisk tilgang til begrundelsesproblemet" (Why should adults be offered teaching in mathematics? - a discourse-analytical approach to the problem of justification). She claims that justification for mathematics teaching is rarely explicit and only on specific events can we get access to the real reasons for why a group of students is offered mathematics. She has chosen to answer the question by analysing the development of preparatory mathematics teaching for adults (FVU), which was introduced through a Danish reform in 1999. Lene differs analytically between three discourses: The political discourse, The planning of teaching persons' discourse and The mathematics teachers' discourse. In her thesis, she develops a framework for analysing the explicit and implicit justifications. Although there are common explicit justifications in the system, her analysis shows that there are conflicting implicit justifications. These conflicting justifications build upon different view of human beings, different views of mathematics knowledge and skills and of mathematics learning.

Kirsti Hemmi at Stockholm University defended her thesis "Approaching proof in a community of mathematical practice". Her aim is to describe how students encounter proof in a community of mathematical practice at a mathematics department and how they are drawn to share mathematicians' views and knowledge of proof. She tries to combine socio-cultural theories, social practice theories and theories about proof into a framework for understanding and describing the diversity of the culture which involves the complex notion of proof. Students felt that they were confronted with proofs from the beginning of their studies. Proof was there as a mysterious artefact and many aspects of proof remained invisible for the students when they struggled to find out what a proof is and to understand its role and meaning in the practice. The first oral examination in proof seems to be significant in drawing students to the practice of proof.

At Umeå University Jesper Boesen defended his dissertation "Assessing mathematical creativity." The thesis consists of four papers and a preamble. Jesper claims that use of superficial reasoning seems to be a main reason for learning difficulties in mathematics. Therefore he finds it important to investigate reasons for this use and the components that may affect students' mathematical reasoning development. Assessments have been claimed to be one such component that significantly influences students' learning. This study shows that a majority of the tasks in the teacher-made assessment could be solved by successfully using only imitative reasoning. The national tests however required creative mathematically founded reasoning to a much higher extent. He also investigates which kind of reasoning the students really use, why teacher made tests emphasise low-quality reasoning and if national tests influence teachers in their development of classroom tests. This impact seems to have a limited effect.

Readers are invited to study these dissertations more closely. They are all available via the authors. We now have fourteen in all by adding the mentioned three dissertations to the eleven that have been presented in earlier issues of NOMAD during 2006. And more are coming later this year and in the beginning of 2007.

Future events in the Nordic Graduate School

In November, there will be a workshop about research on the use of ICT in mathematics teaching and learning at Agder University College. Invited speakers are Luc Trouche and John Monaghan and the focus will be on which theoretical frameworks can be of use in this kind of research. Nordic and Baltic doctoral students, recently educated doctors and more experienced researchers will work together for two days. All will contribute with presentations of their own studies. The outcomes of the workshop will be made available for all interested.

The NoGSME-programme of 2007

The board has planned activities for 2007 and it might be wise to book these dates already now if you want to participate. The workshops or seminars will take place on February 8-9, April 26-27, October 11-12 and November 22-23. The workshop in April will be in cooperation with The Swedish Society for Research in mathematics Education (SMDF) and deal with Mathematics and language. The summer school in Iceland will take place in the southern parts and the dates are 4th to10th of June. The invitation to the summer-school will be sent out in November and we expect great interest in this event. Among the group-leaders we will have Abraham Arcavi from Israel, Mariana Bosch from Spain, and Marcelo Borba from Brazil.

Mathematics didactics courses for doctoral students

During autumn 2006, Nordic students are participating in the Theory of science from a mathematics didactics perspective and Theories of teaching and learning mathematics at Agder University College. The course Perspectives on identity in learning and education research will take place at Ålborg University in 14-17 November 2006. During spring 2007 we offer the course MA607 Methods in mathematics education (15 ECTS) and this course will take place in weeks 4, 7 or 9, 12 and 16. We also hope to be able to offer courses in Sweden, Finland and Estonia. Doctoral students in mathematics education can get travel costs covered from NoGSME in the same way as before, by sending an application with an estimated budget to the director.

Mobility stipends for visits at other Nordic or Baltic universities

NoGSME offers mobility stipends for five students each year. They cover travel expenses and lodging for one month at another Nordic or Baltic university. Thus students can work with colleagues or other supervisors for some time with very low costs for the stay. We encourage all students to consider this option and to apply for a stipend (see web-page for more info, www.hia.no/realfag/forskerskolen).

Contact the board members

The board consists of one member from each of the Nordic countries, one representative from the Baltic countries and the director. You are most welcome to contact any of us if you have suggestions for activities that match our funding. On the web-page you find names of all board members. Just send an email and let us know what you would like to see among our activities.

On behalf of the board of the Nordic Graduate School in Mathematics Education

Barbro Grevholm, Director

JILL BROWN and GLORIA STILLMAN

Abstract

A case study investigated cognitive, mathematical, and technological processes undertaken by senior secondary students as they searched for a complete graph of a difficult cubic function using a graphing calculator. Intensive qualitative macroanalysis identified several defining moments in the solution process. Those related to use of scale marks and identification of key function features are presented. Students' understanding of scale marks varied and this impacted on the efficiency and elegance of their solution. A range of calculator features was used in identifying key feature coordinates. These were not always used successfully or with an understanding of the mathematics underpinning their operation.

Sammenfatning

Denne artikel rapporter et case-studie til undersøgelse af kognitive, matematiske og teknologiske processer hos gymnasieelever (11. og 12. klassetrin), der arbejder med at tegne en fuldstændig graf for et vanske-ligt 3. gradspolynomium ved hjælp af en grafregner. Intensive analyser af elevernes virksomhed har identificeret en række "defining moments" (afgørende momenter) i elevernes løsningsprocess, der er bestemmende for forløbet af deres virksomhed og for deres brug af grafregneren. I artiklen præsenteres og analyseres "defining moments" knyttet til elevernes brug af skalering og enheder ved tegning af funktionens graf samt til elevernes udnyttelse af grafregnerens faciliteter til bestemmelse af koordinater for funktionsgrafens karakteristiske punkter. Der var stor variation i elevernes forståelse af skalering og dette havde indvirkning på effektiviteten og grad af elegance i elevernes løsningsstrategier. Eleverne brugte en række af grafregnerens forskellige faciliteter til bestemmelse af koordinater for grafens karakteristiske punkter, men ikke altid på en succesfuld måde og heller ikke altid baseret på forståelse af den matematik der ligger til grund for deres operationer på grafregneren.

JILL BROWN
Jill Brown was a secondary mathematics teacher for over two decades and has recently joined the staff at Australian Catholic University where she is a lecturer in mathematics education. Research interests include use of graphing calculators in the teaching of function at the secondary level. Jill is currently undertaking her doctoral studies within the field of technology-rich teaching and learning environments at the University of Melbourne. The research presented in this article was undertaken for Jill's research masters thesis at the University of Melbourne.

GLORIA STILLMAN
Gloria Stillman is a senior lecturer of mathematics education in the science and mathematics education cluster within the faculty of education at the University of Melbourne, Victoria, Australia. Research interests include teaching and assessing higher-order thinking through applications and mathematical modelling, metacognition, technology use in mathematics teaching at the secondary level, curriculum change, and ethnomathematics.

In recent months NOMAD has received many new submissions from Nordic researchers. This is a very promising development that leaves us with good prospects for promoting the main aim of NOMAD, namely to stimulate, support and foster the work of Nordic researchers and post-graduate students in mathematics education, and to develop mathematics education and teacher education in theory and practice at all levels of the educational system. Most important is of course that NOMAD is a channel for Nordic researchers, in a broad sense, for publishing research and development papers in a scholarly scientific journal. The review process is another way which NOMAD can contribute to the development of mathematics education in the Nordic countries, in particular as regards supporting new researchers entering our field.

When a paper has been submitted to NOMAD (via nomad@ncm.gu.se) the editors have a first reading of it in order to decide whether or not the paper is relevant for NOMAD and of sufficient quality to go through the review process. If so, the editors choose two researchers with expertise within the area of the paper to make reviews. The guidelines for the reviews can be found at our web site: www.ncm.gu.se. The reviews are produced anonymously. Based on the two reviews the editors decide which of the following four categories the paper should be placed in:

• The paper can be published as it is (which is very rarely the case).
• The paper can be published after it has been improved in accordance with the editors' recommendations in the review report given to the author(s). During the process of revision the author(s) correspond(s) with one of the editors in order to improve the paper. Often several versions of the paper are sent back and forth.
• The paper has to be revised according to the recommendations in the review report, and the new version of the paper has to be submitted through a new review process.
• The paper is rejected and the author(s) receive(s) no recommendation to send in a revised version of the paper (this, too, is very rarely the case).

This review process, which is very similar to what can be found in other research journals, is an extensive process, often running between half a year and one year. The quality of this process is of course strongly depending on the work of the reviewers and the editors. During our short period as editors we have already experienced a great willingness in the Nordic community to act as a reviewer for NOMAD. In addition to the group of more experienced researchers some of the newly graduated doctors have acted as reviewers, and others have expressed their interest in making reviews. Allowing young researchers to gain experience as a reviewer is another way in which NOMAD can contribute to developing the Nordic communities. We are very happy about this evolution and we encourage young researchers to contact the editors, if they would like to write reviews for NOMAD. Occasionally we will bring the list of reviewers in NOMAD, thus acknowledging their important contribution to the journal.

In the report in this issue of the activities in the Nordic Graduate School of Mathematics Education (NoGSME) you will find a short account of the discussion on NOMAD's review process at a recently held seminar for supervisors. Based on this discussion we have decided that the editors' final review report, sent to the author(s), should also be sent to the two reviewers. In this way the reviewers will learn about the editors' decision and get some feed back on their work. Besides, this opens for critique of the editors' work. At the same seminar it was also decided that NoGSME should organise a seminar for reviewers with the aim of discussing and furthering the quality of the review process.

Of course a sufficient inflow of submissions is a prerequisite for securing the quality of the review process and of the journal at the end of the day. Therefore, we are very pleased with the latest increase in the number of submissions. This enables us to operate with a longer turn-over time in the review process, thus giving better opportunities to secure the quality of the process and eventually in the papers published. However, regularity still has to be given top priority by the editorship. Since also this issue is behind schedule (even though only by one month) regularity remains something we have to strive to achieve.

About this issue

In this issue we publish three quite large papers focusing on different aspects of secondary mathematics teaching and learning. The first paper by Monica Johansson is based on the author's recently defended doctoral thesis: Teaching mathematics with textbooks - A classroom and curricular perspective. Starting from the solid findings in many studies that the textbook plays a key role in determining both the content and the organisation of mathematics teaching, Monica Johansson investigates how the textbook is seen and used by three teachers as an instrument for their teaching of mathematics for eight graders. It is a qualitative study with an extensive empirical basis consisting of teacher interviews and questionnaires, and video-recorded classroom observations. The analyses allow specifying in great details to which degree, and how, the three teachers rely on the textbook. It is argued that the textbook is an indispensable means for the teaching of mathematics - at least for most teachers. These findings call for further research on the potentials and limitations of textbook presentation of different mathematical topics.

The second paper by Eugenia Koleza and Elisabeth Kabani investigates lower secondary students' reasoning in relation to geometrical problem solving involving constructions of isosceles triangles. The empirical basis consists of extensive qualitative studies of twenty 10th grade students' work on geometrical construction for one year! Based on analyses of the students' problem solving activities, of which the paper gives clear accounts, three forms of reasoning are identified and described, namely: visual, heuristic and theoretical reasoning. Together with a distinction between the type of evidence used as ground for the reasoning and the reasoning process itself; this categorization captures the students' reasoning in relation to geometrical problem solving. From the point of view of developing teaching practice the findings may be helpful for teachers in recognising their students' modes of reasoning and in finding ways to support and challenge the students as they develop their competence.

The last paper by Jill Brown and Gloria Stillman investigates upper secondary students' use of graphing calculators in relation to the problem of drawing a complete graph of a complicated cubic function. The empirical basis of this study is qualitative analyses of five pairs of students' working on such a problem. In this context the graphing calculator can both be an effective instrument for the students and an artefact which distracts their attention away from the mathematical features of the problem. The paper describes how the analysis has led to the proposal of the concept of a defining moment as a way to describe and understand how the students' activity develops on the basis of how they use the calculator. A number of defining moments are identified in the students' work and two of them, "Use of scale marks" and "Identification of key function features" are analysed in great detail. It is very interesting to see how the students' use of the different features of the calculator and their interpretations of its output is interwoven with their conceptual understanding of the underlying mathematics.

Morten Blomhøj and Paola Valero
NOMAD editors

MONICA JOHANSSON

Abstract

This paper reports a study of three teachers' way to organize their lessons and how textbooks are incorporated in their work. Despite the differences between the teachers, it is noticeable that in these three classrooms, the textbooks, to a large degree, guide the teaching. The textbooks are present: (a) in the students' indivi-dual work, (b) in many of the examples presented on the board, (c) as a source for background and motivational discussions, (d) in how mathematics is presented, and (e) for homework.

Sammandrag

Den här artikeln handlar om hur tre lärare organiserar sina lektioner och på vilket sätt läroboken är integrerad i deras arbete. Trots skillnader när det gäller lärarnas undervisningserfarenhet och elevgruppernas sammansättning visar det sig att läroboken bestämmer undervisningens innehåll i mycket stor utsträckning. Läroboken används som källa: (a) då eleverna arbetar individuellt, (b) i många exempel som läraren visar på tavlan, (c) för diskussioner om bakgrund och syfte, (d) för hur mate-matiken presenteras och (e) för hemläxa. Huruvida lärobokens styrande roll i matematikundervisningen ska betraktas som ett problem eller inte diskuteras. Å ena sidan är det inte rimligt att förvänta sig att lärare ska frångå boken utan goda skäl - läroboken är ett hjälpmedel som förenklar deras dagliga arbete. Dessutom kan den ses som ett stöd för progressionen och likformigheten i elevers väg genom skolsystemet. Å andra sidan kan innehållet i läroböckerna diskuteras och ifrågasättas - till exempel hur rika och verklighetsnära uppgifterna är samt hur väl de är anpassade till varje elevs behov. Viktiga frågor i detta sammanhang är om (a) läro-boken erbjuder utmaningar för varje elevs förmåga och behov, (b) elever kan lära sig matematik på egen hand (med viss hjälp från läraren) och (c) om det är möjligt att uppnå styrdokumentens strävansmål när det gäller matematik via en lärobok.

MONICA JOHANSSON
Monica Johansson received her PhD in mathematics education (Matematik och lärande) this year at Luleå University of Technology, Sweden. She is now working as an associate professor at the Department of Mathematics and at the local school government as a supervisor for developmental works in mathematics in the schools. Her main research interest concerns textbooks and how they are used in the mathematics classroom.

EUGENIA KOLEZA and ELISABETH KABANI

Abstract

The cognitive processes of 15-year-old students, when they solve geometrical problems involving the construction of isosceles triangles, and the different forms of reasoning which they use, are investigated in this paper. First we explore the large variety of reasoning processes which appear, categorize them in three approaches (visual, heuristic and theoretical) and look at the language which is used by each one. Then we focus on the weaknesses of students' reasoning and examine their reasons. The analysis of the data intends to support teachers to recognize and understand the relationship between students' reasoning (nature of justification) and their
geometrical thought.

Sammenfatning

Artiklen undersøger kognitive processer hos 15-årige elever under deres arbejde med geometriske problemer i forbindelse med konstruktion af ligebenede trekanter. Tre forskellige ræsonnementsformer er udviklet til kategorisering af elevernes problemløsning; nemlig visuel, heuristisk og teoretisk ræsonneren. Inden for hver ræsonnementsform analyseres elevernes sprogbrug, grundlaget for og svaghederne ved deres ræsonnementer. Analyserne har som intention at støtte lærere i at genkende og forstå relationen mellem elevernes ræesonneren og deres geometriske tænkning og forståelse. En sådan viden kan anvendes som grundlag for at støtte og udfordre elevernes problemløsning inden for geometri på en relevant måde.

EUGENIA KOLEZA
Eugenia Koleza is Associate Professor and Ph.D. in Mathematics education in the Department of Primary Education, University of Ioannina, Ioannina, Greece. She is the Director of the Laboratory of Research in Mathematics Education of the University of Ioannina. She has served the editorial boards of Hellenic and international journals and has published books for elementary mathematics teachers and over 60 articles. Her main research interests are focused on the epistemological and sociological aspects of Mathematics Education.

ELISABETH KABANI
Elisabeth Kabani, who has been a mathematics teacher for 26 years, is the headmistress of the 5th High School of Alimos. She has a Master and Ph.D. in didactics of mathematics. She has participated in the group of implementation, supervision and support of innovative educational programs in Greek High Schools, and in many European Programs (Lingua, Comenius, Arion).

ANN AHLBERG

Abstract
This research investigates children's understanding of numbers when solving addition and subtraction word problems. A comparative approach - studying children who are blind, children with a hearing impairment and children without these impairments aims at illuminating and describing the differences and similarities between the three different groups. Thereby, the investigation contributes to the under-standing of critical aspects characterising development of numerical competence in each of the three groups. The design of the study also makes it possible to describe children's development of number concepts in the context of solving addition and subtraction word problems on a more general level.

Sammanfattning
Den redovisade forskningen handlar om hur barn förstår tal när de löser enkla additions- och subtraktionsproblem. I studien deltar barn som är blinda, barn med hörselnedsättning och barn utan dessa funktionshinder. Studiens syfte är att beskriva skillnader och likheter mellan hur de tre barngrupperna hanterar och förstår tal. Den komperativa ansatsen bidrar till förståelsen av kritiska aspekter som karakteriserar utvecklingen av den aritmetiska förmågan för varje barngrupp. Studiens design möjliggör också en beskrivning av barnens förståelse av tal, när de löser aritmetiska textproblem, på en mer generell nivå.

ANN AHLBERG
Ann Ahlberg is professor in Special Education at Göteborg University. Her research interest is directed towards inclusion and exclusion processes in school focusing on participation, communication and learning. A special interest is directed towards mathematics for younger children. In this research she has studied children's different ways of experiencing and learning mathematics as well as the organisation and content of teaching.

OLOF MAGNE

Abstract

Research on failure to master mathematics (often used term: disability) is a modest speciality, compared with related domains. Research on low attaining persons appears to be too much directed towards a small number of topics en vogue while other issues, often more impressive ones, are mainly left unnoticed. Not the least disquieting is the excessive concentration on computation with small natural numbers in a setting of formalism. This presentation has the aim to demonstrate that the number of parameters (factors, vectors, dimensions) is great in the field of research on education and learning mathematics and that research is a problematic matter, due to the complex relations between mathematics, individual and environment (MIE).

Sammandrag

Forskning om lågprestationer i matematik (ofta använd term matematiksvårigheter) är en blygsam specialitet i jämförelse med besläktade vetenskapsområden. Studier om låga prestationer i matematik - både i skola och utanför skola - är få, ofta godtyckligt utvalda och ibland subjektivt evaluerade. De kan stå för lättillgängliga smakriktningar snarare än vardagligt anspråkslösa teman med vetenskaplig tyngd och djup. Ett annat oroande drag är en överdriven koncentration på enkel aritmetik med små naturliga tal i en formalistisk tankekostym. Komplexiteten i matematikens struktur går förlorad. Vardagsmatematik försummas trots dess betydelse för den lågpresterande räknarens livskvalitet.
Denna framställning ägnas främst åt (1) att visa och klarlägga mäktigheten av parametrar (faktorer, vektorer, dimensioner) i vetenskapen om matematikens undervisning och inlärande samt (2) att exemplifiera hur dess utforskande har fyllts av motsägelsefulla objekt till följd av de komplexa relationerna mellan matematik, individ och omgivning (MIO) - engelska: mathematics, individual and environment (MIE). Forsknings- och utvecklingsarbetet är mera energiskt i projekt för att förbättra undervisning och inlärning för elever med läs- och skrivsvårigheter.

OLOF MAGNE
In 1952 Olof Magne was appointdocent at the Göteborg College, from 1954 Göteborg University, after his doctoral thesis and was 1952 to 1960 responsible for the Department of Education at the College (from 1954 University). Ped. Dr h.c. (Åbo Akademi). County School Director (Karlskrona) 1961 to 1971. Assistant Professor and Professor at the Malmö School of Education 1971 to 1983. After his retirement Olof Magne has continuously acted as a consultant, researcher and author. Among his chief research interests the following ones could be mentioned: mathematics education, learning and memory, special education. Magne has been engaged with missions in various parts of the world and entrusted with national and international assignments. In Norway the Olof Magne Foundation has been created in order to support teachers' special education studies in mathematics.

Vanskeligheder i/med matematiklæring - behov for øget forskning

Med dette temanummer om vanskeligheder i/med matematiklæring sætter NOMAD fokus på et forskningsfelt inden for matematikkens didaktik, der fortjener øget opmærksomhed. Pt. er det ikke særligt udbredt hverken på nordisk eller internationalt plan og ofte bliver det opfattet som marginalt placeret i det matematikdidaktiske forskningsfelt. Det er på trods af, at forskningsfeltet faktisk behandler problemstillinger, der har stor økonomisk og samfundsmæssig og dermed også politisk betydning, og at forskning med fokus på vanskeligheder i/med matematiklæring kan give nye frugtbare perspektiver til forskning i matematikdidaktik generelt.
De danske resultater i sammenlignende internationale studier som SIALS og PISA kan - uanset hvad man ellers måtte mene om relevansen og kvaliteten af dem - tjene som illustration af den samfundsmæssige betydning af læringsvanskeligheder i matematik. De viser, at omkring 15% af danske voksne og unge ikke præsterer tilfredsstillende i test af grundlæggende matematiske kompetencer. Det er oplagt, at denne store gruppe af befolkningen, i meget varierende grad, vil opleve deres kompetencer i matematik som begrænsende ved valg og gennemførelse af uddannelse, i deres arbejdsliv og i livet i øvrigt. Der er under alle omstændig-heder tale om dokumentation af et problem af stor samfundsmæssig betydning. Tilsvarende foruroligende resultater rapporteres fra Norge og Sverige i artiklerne i dette nummer. Problemstillingen omkring læringsvanskeligheder i matematik har da også en vis politisk bevågenhed, men de konkrete initiativer går - trods flotte formuleringer i læseplaner og politiske programerklæringer - ofte kun i retning af flere obligatoriske prøver i skolesystemet, hvorimod det kniber gevaldigt med flere ressourcer til undervisning, videre- og efteruddannelse af lærere, forsøgs- og udviklingsarbejde samt egentlig forskning og forskningsformidling.
Som det vil fremgå af artiklerne i dette nummer, er diskussionen om terminologi og afgrænsning af elevgrupper med læringsvanskeligheder stadig særdeles påtrængende inden for forskningsfeltet. Det er en bemærkelsesværdigt kompleks diskussion. Dels handler den om hvor årsagerne til læringsvanskelighederne eftersøges: i elevens hjerne, hos eleven og dennes forhold til matematikken, i elevens sociale eller kulturelle sammenhæng, i lærernes holdninger og uddannelsesbaggrund, i matematikundervisningens indhold og form, eller i matematikken. Dels drejer det sig om den tæt forbundne diskussion om, hvordan man afgrænser gruppen af elever eller mere generelt gruppen af mennesker, der er i fokus for forskningen om læringsvanskeligheder i matematik. Med overskriften for temanumret, Vanskelighed i/med matematiklæring har vi netop søgt at annoncere en åbenhed over for denne diskussion.
Diskussionen har afgørende betydning ikke bare internt i forskningsfeltet, men i høj grad også i forhold til hvordan uddannelsessystemet og i sidste ende den enkelte lærer forholder sig til elever, der oplever alvorlige vanskelighed i deres matematiklæring. Hvilke elever skal for eksempel have tilbud om specialundervisning, og hvilken type specialundervisning er mest hensigtsmæssig? Hvornår udløses der ekstra lærerressourcer til inkluderende klasseundervisning? Hvordan praktiserer man undervis-ningsdifferentiering over for elever med særlige vanskeligheder og behov? Hvordan taler man som lærer med sine elever om læringsvanskeligheder og læring generelt i matematik?
Det er oplagt, at svarerne på sådanne spørgsmål har klare økonomiske og organisatoriske konsekvenser, og at de kun kan tilvejebringes gennem dybdegående forskningsmæssige undersøgelser af problemfeltet. Artiklerne i dette temanummer viser, at der er forskning at bygge på i de nordiske lande, men samtidig også at der er stort behov for mere systematisk forskning inden for området. Som Olof Magne påpeger i den første artikel savnes forskning, der rækker udover læringsvanskeligheder i den tidlige matematikundervisning og som behandler læringsvanskeligheder som en integreret del af studiet af læring og undervisning i matematik.
I de nordiske lande har vi tradition for at udvikle inkluderende uddannelsessystemer og for at overlade det metodemæssige ansvar for undervisningen til den enkelte lærer. Derfor ville det være naturlig, at der netop i vore lande blev iværksat forskellige former for forsøgs- og udviklingsvirksomhed som led i udforskning af, hvordan man kan imødegå nogle af de hyppigt forekommende vanskeligheder med matematiklæring. Vi kan blot håbe, at dette teamnummer kan bidrage til en sådan udvikling.
Som forskningsfelt i matematikkens didaktik har Vanskeligheder i/med matematiklæring væsentlige bidrag og nye perspektiver at tilbyde netop fordi feltet interesserer sig specifikt for særlige typer af "ekstreme tilfælde" inden for matematiklæring. Det gælder forbindelsen mellem matematiklæring, kognitiv psykologi og en neurologisk forståelse af hjernens opbygning og funktion. Indsigt i hjernens neurologiske processer vundet gennem avancerede hjernescanningsteknikker kan måske på sigt bidrage til forståelse af principielle vanskeligheder knyttet til matematiklæring. Herved kan forskningen bidrage til at klarlægge, hvad der kan opfattes som en biologisk begrundet naturlig variation i oplevede vanskeligheder ved matematiklæring. Forskningsfeltet sætter i stigende grad fokus på, hvad de enkelte elever kan i forskellige undervisningssammenhænge og bidrager hermed til udvikling af inkluderende undervisningsformer, der rumme relevante faglige udfordring for alle elever. Hvad angår betydningen af sociale og kulturelle forhold for matematikundervisningen og de enkelt elevers udbytte heraf, kan studiet af elever med særlige vanskeligheder og behov i høj grad afdække fænomener og mekanismer, der er generelle for matematiklæring. Det gælder ikke mindst dannelsen af stereotype opfattelser af egne evner for matematiklæring og negative feedbackvirkninger heraf. Endelig sætter forskningen i matematikvanskeligheder begrundelsesproblemet i et nyt perspektiv, det vil sige spørgsmålet om hvorfor og i hvilket omfang givne grupper af børn og voksne skal udsættes for matematikundervis-ning. Hvordan skal en almen matematikundervisning vægte hensynet til det enkelte menneskes personlige udvikling og kommende medleven i et moderne højteknologisk demokrati samfund overfor arbejdsmarkedets behov for uddannelse af tilstrækkelig kvalificeret arbejdskraft?

Om artiklerne i dette nummer

Temanummeret omfatter fire fyldige artikler, der belyser problemfeltet hver på deres måde. Der er to artikler på nordiske sprog, dansk og norsk og to artikler på engelsk af svenske forfattere.
I den første artikel giver Olof Magne et historisk overblik over udvik-lingen inden for specialundervisning i matematik. Det er en udvikling som Olof Magne som forsker og debattør selv har været en markant deltager i både på nordisk og internationalt plan i hen ved 60 år. Som det fremgår, er han fortsat forskningsaktiv. Vi er således vidne til endnu et resultat af en imponerende livslang virksomhed inden for forskningsfeltet læringsvanskeligheder og specialundervisning i matematik, og det er ikke uden stolthed, at vi kan publicere Olof Magnes artikel i dette temanummer. Artiklen gennemgår den tidlige udvikling i første halvdel af det 20. århundrede af begreber om læringsvanskeligheder i matematik og viser hvordan disse har været tæt forbundet med forskning i hjernens funktion og specielt effekten af forskellige typer af hjerneskader. Sådan studier af vanskeligheder med matematik har bidraget til forståelse af hjernens funktion, men ikke i afgørende grad til forståelse af årsagerne til og udvikling af mulige pædagogiske indsatser over for læringsvanskeligheder i matematik som de optræder i uddannelsessystemet. Dette er måske en af forklaringerne på den manglende interesse for læringsvanskeligheder i matematik inden for den matematikdidaktiske forskning som Magne så klart påviser. Nyere forskning har bidraget til udvikling af begreber, der er mere pædagogisk orienteret som f.eks. elever med særlige behov i matematikundervisning. Med reference til mange store undersøgelser - ikke mindst forfatterens egne - dokumenteres forekomsten af elever med særlig behov i skolens tidlige matematikundervisning at være betydeligt (10%-15%) og samtidig dokumenteres omfanget af disse elevers læringsvanskeligheder at være betragtelige (4. klasses niveau efter 9 års grundskole undervisning). Det burde således være krystalklart, at vi har at gøre med et særdeles omfattende problem af samfundsmæssig betydning, og som kalder på en genuin forskningsmæssig belysning. Magne kritiserer den eksisterende forskning for at være for specialiseret og diagnoseorienteret og uden tilstrækkelig forbindelse til det matematikdidaktiske forskningsfelt, og peger på behovet for at anskue feltet ud fra en forståelse af den komplekse relation mellem matematik, individ og omgivelser herunder læringsmiljøet. Magne lægger således op til at inddrage Guy Brousseau's teori for didaktiske situationer i forståelsen af specifikke læringsvanskeligheder i matematik.
Udgangspunktet for artiklen af Tone Dalvang og Olav Lunde er en påpegning af diskrepansen mellem indførelse af klare kompetencemål i den norske læreplan som eleverne skal opnå i matematik ved fire forskellige årstrin og de faktiske forhold angående elevers og lærestuderendes matematikkompetencer, som de er dokumenteret i forskellige norske undersøgelser. Dette aktualiserer ifølge forfatterne behovet for et skifte i forskning og praksis angående læringsvanskeligheder i matematik væk fra en diagnose- og specialundervisningsorienteret tilgang mod en tilgang, der i højere grad fokuserer på elevernes læringspotentiale og på udvikling af inkluderende undervisningsformer. Forfatterne redegør for fire forskellige forklaringsmodeller til forståelse af læringsvanskeligheder i matematik; den medicinske/neurologiske, den psykologiske, den sociologiske og den didaktiske forklaringsmodel. I artiklen fokuseres på den didaktiske forklaringsmodel og forfatterne opstiller en model, der kan bruges som "kompas" både til forståelse og afhjælpning af specifikke læringsvanskeligheder i matematik. Modellen integrerer elevens læringsforudsætninger,
undervisningens indhold og form og de otte matematikkompetencer fra det danske KOM-projekt. Modellens forudsætninger, praktiske anvendelse og didaktiske konsekvenser diskuteres mod baggrund af forfatternes mangeårige erfaring fra specialpædagogisk praksis.
Den tredje artikel af Lena Lindenskov identificerer to - til dels modsatrettede - internationale tendenser i udviklingen og diskussion af læreplaner i matematik. Den ene tendens går i retning af at opfatte menneskers matematikholdige kompetence som nøglekompetence for samfund og for uddannelsen af kvalificeret arbejdskraft til arbejdsmarked. Den anden er en uddannelsespolitisk tendens i retning af en inkluderende skole med et nyt syn på afvigelser og normalitet og med ændret organisation, der anses for afgørende for fastholdelse og udvikling af demokrati. Tenden-serne anses i artiklen for at være afgørende, både for hvordan matematikvanskeligheder kan forstås, og for hvordan der kan handles i forhold til dem: Begrebet nøglekompetence gør matematikvanskeligheder til et fokuspunkt for matematikkens didaktik, samtidig med at det udfordrer intentioner og læreplaner. Begreber om skolens rummelighed, integration og inklusion på et generelt plan udfordrer gængse forestillinger om normalitet og om undervisning og læring, men mangler en diskussion af, hvordan begreberne kan få en matematikfaglig dimension. Dermed peges på behovet for en ny begrebssætning af matematikvanskeligheder og der gives et forslag hertil, som er i harmoni i syn og i praksis med intentioner i en inkluderende skole.
Den sidste artikel af Ann Ahlberg præsenterer et sammenlignende studium af talforståelse hos tre forskellige grupper af 6-10årige elever, der er henholdsvis blinde, høre-handicappede og uden disse handicap. Ud fra en fænomenografisk tilgang analyseres elevernes arbejde med additions- og subtraktionsproblemer i interviewsituationer. Elevernes måder at håndtere tallene i deres problemløsning karakteriseres og elevernes bagvedliggende talopfattelser afdækkes i artiklen. Det viser sig, at elever kan have de samme opfattelser af tal selvom de håndterer de selv samme talstørrelser forskelligt i arbejdet med de forelagte problemer, og her er der interessante forskelle mellem de tre grupper. Undersøgelsen giver også grundlag for at belyse udviklingen af elevernes talopfattelse i hver af de tre grupper. Selvom studiet angår læringsmæssige udfordringer forårsaget af fysiske sansemæssige handicap er det i høj grad relevant i forhold til læringsvanskeligheder generelt, dels ved at afdække principielle vanskeligheder i børns udvikling af talforståelse, dels ved at den anvendte metodo-logi er særdeles relevant i forhold til afdækning af andre elevgruppers
talforståelse og matematiklæring i øvrigt.

Temanumret i 2007

Temaet for NOMAD, no.4, 2007 bliver Samspil mellem matematikdidaktisk forskning og udvikling af undervisningspraksis. Redaktionen inviterer hermed til indsendelse af artikler om forskningsprojekter, der har et eksplicit fokus på udvikling af en bestemt undervisningspraksis og som indeholder dokumentation af og refleksion over en sådan udvikling. Vi er specielt interesseret i artikler, der er skrevet i, eller som beskriver, samarbejde mellem forsker(e) og lærer(e). Begrundelsen for temaet er, at vi gerne vil bidrage til at tydeliggøre, at NOMAD som forskningstidsskrift kan give inspiration til udvikling af undervisningspraksis. Det er samtidig vores vurdering, at der rundt om i de nordiske miljøer foregå spændende og interessante udviklingsprojekter med et forskningssigte, men som ikke normalt publiceres som forskning. NOMADs næste temanummer kan være en publiceringsmulighed for sådanne projekter.
Alle modtagne manuskripter vil naturligvis gennemgå NOMAD's sædvanlige review-proces. Fristen for indsendelse af manuskripter er den 15. juni, 2007, men interesserede forfattere bedes snarest muligt indsende titel og abstrakt for det påtænkte bidrag. Det giver redaktørerne mulighed for at være opsøgende, hvis det skulle vise sig nødvendigt. Forfattere er også velkommen til at indsende en foreløbig version af en artikel med henblik på at få kommentarer om egnethed og forslag til forbedringer fra redaktørerne inden indsendelse af manus.

Til sidst vil vi gerne benytte lejligheden til med lidt forsinkelse at ønske NOMAD's læsere et godt og lykkebringende 2007.

Morten Blomhøj and Paola Valero
NOMAD editors

LENA LINDENSKOV

Sammendrag
To internationale tendenser af afgørende betydning for, hvordan matematikvanskeligheder kan forstås og imødegås, identificeres i artiklen: Matematikholdig kompetence som nøglekompetence og inkluderende skole. På denne baggrund og ud fra empiriske undersøgelser argumenteres for en ny og pragmatisk begrebssætning af matematikvanskeligheder. Et overordnet begreb regnehuller til støtte for forståelse af det faglige indhold i vanskelighederne og for inspiration til videre matematiklæring. Desuden en model til forståelse af elevers oplevelser. Modellens elementer er matematik som system, mening og forståelsesbehov, blokeringer og modstand, undervisning og eleven i klassen/på holdet. De behandlede spørgsmål og udviklingstendenser er generelle, men danske resultater og dokumenter bruges i artiklen som case.

Abstract
Two international trends are presented by some central documents as they constitute important general frames for understanding and coping with students in mathematics difficulty: a trend towards a political focus on math-containing competence for all, which stresses the need to further work in research and practice, and a trend towards political and economical interest in inclusive schools, which challenges thinking on normality and school organisation and where concept of inclusion needs to be given a mathematical dimension. The article suggests a new pragmatic concept of mathematics difficulty - in Danish termed regnehuller - in order to support individual and group of teachers to understand, focus and support individual students in their learning mathematics and to develop mathematics education for all. A model with five elements is then suggested to further the understanding of how students experience mathematics difficulties. Elements are termed math as a system, meaning and need for understanding, blockage and resistance, teaching, and student in the class. Danish results serve as a case to illustrate general questions, trends and concepts.

LENA LINDENSKOV
Lena Lindenskov, kandidat i matematik og samfundsfag, ph.d. i matematikkens didaktik. Lena har arbejdet som gymnasie- og HFlærer i mate-matik og samfundsfag og har udført mange konsulent- og udviklings-opgaver, blandt andet inden for matematik på voksenområdet. Forsker inden for matematikkens didaktik i matematiklæring i uddannelses-institutioner, kompetencer, hverdagsmatematik, matematikvanskelig-heder, blandt andet i EU-projekterne MiA og ALMAB om voksne og matematik. Ansat som lektor på Danmarks Pædagogiske Universitet, Institut for curriculumforskning, i Forskningsenhed for matematikkens og naturfagenes didaktik.

TONE DALVANG og OLAV LUNDE

Sammendrag

Artikkelen belyser matematikkvansker som et sammensatt fenomen. Ulike årsaksforklaringer fremføres, der forfatterne legger hovedvekten på didaktiske forhold. En modell presenteres som et redskap til å samtale, drøfte og forstå matematikkvansker i et sosio-kulturelt perspektiv. Ulike forklaringer og forståelser veves sammen i lys av konteksten. Modellen er en syntese som bygger på forfatternes kjennskap til praksisfeltet og teoretiske antagelser om for eksempel betydning av virksomhet og praksisfellesskap for læring. Artikkelen drøfter hvordan modellen kan brukes i praktisk arbeid ut fra en didaktisk vinkling med vekt på at eleven mestrer matematikken.

Abstract

The article enlightens learning difficulties in mathematics as a complex phenomenon. Different explanations to the problems are put forward, of which the authors emphasize didactical relations. A model is presented as a tool for dialogue, reasoning and understanding of the difficulties in mathematics in a socio-cultural perspective. Varied explanation and understandings are woven together in the context. The model is a synthesis of the authors' experiences and knowledge about many years of practice in the field and theoretical reflections about learning, as for instance the importance of activity and communities of practice. The article discusses how the model can be useful from a didactical point of view when emphasizing the students' mastering of mathematics.

TONE DALVANG
Tone Dalvang har undervist i grunnskole, videregående skole og lærerutdanning, og har mastergrad i pedagogikk. Hun har deltatt i utredning av matematikksituasjonen i Norge, MiSS-utvalget, og i oppstart og ledelse av Landslaget for matematikk i skolen - Lamis. Tone Dalvang er rådgivere i Forum for matematikkmestring ved Sørlandet kompetansesenter og har skrevet artikler om matematikk-læring og -vansker og holdt en rekke kurs/foredrag i Norge og i Norden.

OLAV LUNDE
Olav Lunde har arbeidet i en årrekke som pedagogisk-psykologisk rådgiver og leder i PPT. Han har magistergrad i pedagogikk fra Universitetet i Oslo og er godkjent spesialist i ped.-psyk. rådgivning. Olav Lunde er rådgivere i Forum for matematikkmestring ved Sørlandet kompetansesenter og har skrevet artikler om matematikk-læring og -vansker og holdt en rekke kurs/foredrag i Norge og i Norden.

The Nordic Graduate School into its fourth year of activities

The year 2007 is the fourth year of the Nordic Graduate School. The funding from NordForsk will be given for five years and then it is expected that the community of doctoral students and supervisors that we have created will survive on its own. Will it do so? The group of persons involved is growing and new institutions giving doctoral education enter into the work. In the end of 2006 letters were sent out to all environments participating in the Nordic Graduate School in order to get updated information on the students and supervisors that are active in mathematics education. The updated lists are available on the NoGSME webpage and now contain names of 106 supervisors (current or prospective) and 87 doctoral students. As many doctoral students have taken their degree during 2006 the list of doctoral students looses names but they enter into the list of prospective supervisors instead. It is valuable if information about changes can be sent to us regularly.

The seventh seminar for supervisors in Trondheim

The seventh seminar will focus on review reports for scientific journals. How can such a report be designed in order to be valuable for both the author of the paper and for the editor of the journal? How can supervisors support their doctoral students both in the use of a received report and in learning how to write such a report? The editors of NOMAD and the editorial board will be part of the programme and discuss the review process and how decisions are taken when reviewers have different opinions. Two editors in chief will be interviewed about the policy of a scientific journal and its function. There will be work in small groups where participants will address authentic review reports. Different review guidelines will be investigated and discussed.

The fourth workshop will be about mathematics and language

In April 26-27 NoGSME will organise a research workshop on mathematics and language in Sweden, near Stockholm. It will be done in cooperation with the Swedish Society for Research in Mathematics Education, SMDF, and build on an earlier workshop on the same theme that SMDF arranged in the spring of 2005. Workshops by NoGSME are for both doctoral students and supervisors who are doing research related to the theme.

The third summer school by NoGSME will be in Iceland

As announced earlier, the summer school in 2006 will be in Laugarvatn in Iceland. Many doctoral students have already given us notice of their interest to participate. As before, the group leaders will be excellent inter-national researchers in mathematics education. The evaluation of the summer school in Dømmesmoen in 2006 convinced us that a summer school is a valuable and important event for doctoral students. We welcome all doctoral students in mathematics education from the Nordic and Baltic countries. The second announcement for the summer school will soon be available.

Seven new dissertations in mathematics education

In NOMAD, volume 11 numbers 1-3, I have written about fourteen dissertations from 2006. Here I can report on still seven more and maybe there are still more that I have not been told about. Please inform us if you know of such theses in mathematics education.

Ole Einar Torkildsen defended his thesis for the academic degree of doctor philos at Oslo University. It has the title Mathematical archaeology on pupils' mathematical texts. Un-earthing of mathematical structures. The data basis for his work is the pupils' solutions for six tasks given in a competition in Tangenten in the early 1990's. They make up 23 mathematical texts. The purpose of his analysis method is to make explicit the mathematical structures that are inherent in the pupils' solutions. Thus he is intending to uncover the mathematical structures by analysing pupils' solutions through the glasses of a mathematician. The analysis revealed that the pupils in their solving process heavily relied on some fundamental mathematical structures. The relationships in the answers can be identified as functions, in some instances functions with more than one independent variable. The mathematical structures are localised at two levels and factors influencing the solution process are studied. Finally Torkildsen argues that mathematical archaeology is a suitable tool for increasing the knowledge about pupils' mathematical activity.

Åse Streitlien's thesis has the title Room for participation - a study of inter-action and communication in mathematics classrooms. It is written in Norwegian and makes up an extensive text of 350 pages and was defended at Oslo University. The aim of her work is to study interaction and communication between the teacher and her students in mathematics classrooms. The research focus is concerned with the opportunities young students have for participating in the discourse of mathematics and how the dynamics of reasoning and discussion gives rise to mathematical meaning. Taped lessons from two classrooms were analysed using discourse analysis. Focus was on how discourse patterns influence what counts as mathematics knowledge and what communicative competences the students need for participating in the classroom discourse. Streitlien suggests that what students learn in mathematics depends on how their teacher responds to their responses and the opportunities there are given them in the negotiation of mathematical meaning.

Ewa Bergqvist calls her thesis Mathematics and mathematics education - two sides of the same coin: some results on positive currents related to polynomial convexity and creative reasoning in university exams in mathematics. It was defended at Umeå University in Sweden. The dissertation consists of two different but connected parts. Part A is based on two papers in mathematics and part B on two papers in mathematics didactics. In part B the focus is on what kind of reasoning university students in mathematics use in courses and exams. Bergqvist differs between imitative reasoning and creative reasoning. About 70% of the tasks in exams can be solved by imitative reasoning. The teachers constructing the exams are pleased with this situation. They claim that otherwise the exams would be too difficult and lead to too low passing rates.

Sharada Gade was the first to present her thesis in the doctoral programme of Agder University College in Norway. The title is The micro-culture of a mathematics classroom. Artefacts and activity in meaning making and problem solving. The work is based on a yearlong classroom study in the first grade in upper secondary school in Norway. The thesis points to the centrality of both meaning-making in teaching-learning and goal-directedness in problem-solving, as important parts of the instruction in a mathematics classroom. The classroom was bilingual with emphasis on cooperation or group-learning by students. The thesis offers a synthesis based on socio-cultural perspectives of the micro-culture of teaching-learning of mathematics established and situated in the classroom.

Magnus Österholm defended his thesis at Linköping University. The thesis has the title Cognitive and metacognitive perspectives on reading comprehension in mathematics. The purpose of the dissertation is to examine whether a reader needs special types of knowledge or abilities in order to read mathematical texts. The reading of mathematical texts is studied from a cognitive perspective and from a meta-cognitive perspective. In the first case reading abilities and content knowledge are studied in relation to reading comprehension. In the second case the focus is on beliefs and how a reader determines whether a text has been understood or not. The results show that courses at upper secondary level and at university level do not affect the special reading ability. There is a need to focus on reading but it does not need to be about learning to read mathematical texts but to use existing, more general reading ability also for mathematical texts.

Markus Hähkiöniemi at the University of Jyväskylä has written and defended the thesis The role of representations in learning the derivative. The aim of the study is to find out how students may use different kinds of representations for thinking about the derivative in a specific approach. To achieve this, the author designed and implemented a five-hour teaching-learning sequence introducing the derivative concept in a Finnish high school (grade 11). Five students were selected to take part in carefully designed task-based interviews. He found that the embodied world offered powerful thinking tools for the students. They used increase, steepness, horizontalness and tangent of the graph for thinking about the derivative qualitatively without calculating anything. On the basis of the analysis of the students' use of representations a hypothetical learning path to the derivative was constructed.

Per Nilsson's thesis presented at Växjö University has the title Exploring probabilistic reasoning. A study of how students contextualise compound chance encounters in explorative settings. The focus is on what learners with little experience of formal theories of probability do and can do then they are dealing with compound random situations in which they are offered opportunities to integrate different probabilistic lines of reasoning. Two part studies have been done on 12 to 13 year old students and one study on 14 to 16 year old students. The younger students acted within a dice-game setting and the older with ICT-versions of compound, independent events. Prior to instruction students were able to devise ideas of under-lying probability distributions in the case of compound random phenomena. The students brought into the discussions geometrical and numerical considerations as well as arguments reflecting principles of the law of large numbers.

How will these theses influence classroom practice?

The twenty-one theses in mathematics education that have been defen-ded during 2006 are probably pointing to the highest number of theses in the area during one year in the Nordic countries. The year 2006 has been an exceptional year for mathematics didactics in the Nordic countries. Will the dissertations have an effect, in the long run, on mathematics classroom practice? What will the effects be, if any? Is it possible for mathematics teachers to learn about the research results and try to interpret them and implement some outcome of them in the classroom? These are questions that could be interesting to try to answer through future research studies. We will have to show some patience before we can answer the questions.

Contact the NoGSME board if you have suggestions

The work of the Nordic Graduate School is intended to support and develop research education in mathematics education in the Nordic and Baltic countries. If you have any suggestions for activities that are of value for doctoral students or supervisors and follow the intentions of our sponsors NordForsk, please contact us and let us know about your ideas.
The NoGSME board is looking forward to another year with many exciting activities and collaboration with colleagues in all the Nordic and Baltic countries.

Barbro Grevholm
Director of NoGSME
Agder University College

Mathematical competitions and classroom collaboration: antonyms or a new direction for research on teacher beliefs?

Tine Wedege and Jeppe Skott (2006), Changing views and practices: a study of the KappAbel mathematics competition. Research report, Norwegian Center for Mathematics Education & Norwegian University of Science and Technology, Trondheim. ISBN 82-471-6040-4
 

BHARATH SRIRAMAN, The University of Montana
OLOF STEINTHORSDOTTIR, University of North Carolina at Chapel Hill

GAIL E. FITZSIMONS and TINE WEDEGE

Abstract

Internationally, adult literacy and numeracy are in general recognized as cultural techniques. However, the two competences and their development are contested among politicians and researchers. Numeracy is often subsumed under literacy and/or described in isolation from the situational context. Adult numeracy at work is often described unproblematically as transfer from school to workplace. With reference to Bernstein’s theoretical framework, we claim that adult numeracy on the labour market is a horizontal discourse, in contrast to the vertical discourse of mathematics. This article draws on the findings from an Australian study into numeracy in the context of chemical spraying and handling, utilising a methodology based on activity theory. The main findings are that mathematically straightforward skills become transformed into workplace numeracy competence, when the complexities associated with successful task completion as well as the supportive role of mediating artefacts and the workplace community of practice are taken into account.

Sammendrag

Internationalt er voksnes ”literacy” og ”numeracy” anerkendt som kulturteknikker. Men blandt politikere, uddannelsesbureaukrater og forskere er der langtfra enighed om indholdet i de to kompetencer, og om hvordan de udvikles. Numeracy bliver ofte underordnet literacy og/eller beskrevet isoleret fra den situationelle kontekst. Voksnes numeracy i arbejdet beskrives ofte som en helt uproblematisk overførsel (transfer) af matematiske kundskaber og færdigheder fra skole til arbejdsplads. Med reference til Bernstein’s teoretiske ramme påstår forfatterne at voksnes numeracy på arbejdsmarkedet er en horisontal diskurs i modsætning til matematikkens vertikale diskurs. Artiklen er baseret på resultater fra et australsk studie af numeracy inden for kemisk sprøjtning og håndtering. Heri er metodologien baseret på virksomhedsteori. Hovedresultaterne er at rene matematiske færdigheder bliver transformeret til numeracy som arbejdspladskompetence, når der tages højde for kompleksiteten i den succesfulde opgaveudførelse ligesom det medierende artefakt og arbejdspladsens praksisfællesskab.

GAIL E. FITZSIMONS
Gail FitzSimons was a teacher of mathematics, statistics, and numeracy subjects to adult students of further and vocational education in community, industry, and institutional settings for 20 years. She was awarded an Australian Research Council Post-Doctoral Research Fellowship, 2003–2006, for a project entitled: Adult numeracy and new learning technologies: an evaluative framework.

TINE WEDEGE
Tine Wedege is associate professor at the School of Teacher Education, Malmö University, Sweden, and professor II at the Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim. Her research interests include social and affective dimensions of people’s relationship with mathematics, adult’s mathematics in work and identity of mathematics education as a research domain.

NICHOLAS MOUSOULIDES, BHARATH SRIRAMAN and CONSTANTINOS CHRISTOU

Abstract

More than 25 years ago, a research project in the U.S investigated the question: ”What is needed by students, beyond having a mathematical idea, that enables students to use the mathematical idea in everyday problem solving situations? (Lesh, Landau & Hamilton, 1983). The answer to this question has begun to emerge after 25 years of systemic work in the domain of modeling. In this paper, we chronicle the emergence of models and modeling perspectives (MMP) from the genre of problem solving research via a synthesis of the major strands in the extant literature.

Sammanfattning

Ett forskningsprojekt i USA undersökte för mer än 25 år sedan följande fråga: ”What is needed by students, beyond having a mathematical idea that enables students to use the mathematical idea in everyday problem solving situations? (Lesh, Landau & Hamilton, 1983). Efter 25 års systematiskt arbete inom området ”modellering” har svaret på frågan börjat framträda. I artikeln beskriver vi hur utvecklingen av ”models and modeling perspectives” (MMP) har skett utifrån forskning om problemlösning. Beskrivningen sker med hjälp av en syntes av de viktigaste linjerna i befintlig forskningslitteratur.

NICHOLAS MOUSOULIDES
Nicholas Mousoulides is a Ph.D. candidate, researcher and educational personnel of mathematics education at the University of Cyprus. His research focuses on the implementation of ICT in mathematics education and the effect of integrating modeling in the learning and teaching of mathematics. He participated as a researcher in seven European and national research projects. His main research interests are modeling and applications, curriculum development in computer based environments, and student cognitive and mathematical development.

BHARATH SRIRAMAN
Bharath Sriraman is associate professor of mathematics at the University of Montana, with a wide range of eclectic research interests including mathematical modeling. He received his PhD from the department of mathematics at Northern Illinois University, USA. Bharath is the editor-in-chief of the Montana Mathematics Enthusiast, , associate editor of ZDM – the International Journal on Mathematics Education (formerly known as Zentralblatt für Didaktik der Mathematik), consulting editor of Interchange: A Quarterly Review of Education, and reviews editor of Mathematical Thinking & Learning as well as ZDM. He holds very active research ties with researchers working in his domains of interest in Australia, Canada, Cyprus, Denmark, Germany, Iceland, India, Turkey and USA.

CONSTANTINOS CHRISTOU
Constantinos Christou received his PhD from the University of Toledo, Ohio, USA. He is professor of mathematics education at the University of Cyprus. His research focuses on the cognitive development of mathematical concepts. He is in the editorial board of the journals Mediterranean Journal of Mathematics Education and the Montana Mathematics Enthusiast and participated as coordinator and partner in ten European and national research projects. Currently he studies mathematical modeling and applications, student spatial reasoning, the reasoning of students in mathematical tasks including their intuitive knowledge, and the effects of integrating technology in the teaching of mathematics on the cognitive development of students.

The first year of the Danish editorial team

One year ago, when we took over the role of Nomad editors, we met with enthusiasm the challenging job of taking care of and helping nourishing the job that colleagues in other countries started fifteen years ago. The beginning was not easy since we had to learn how to manage the whole editorial process. The support from Johan Häggström and Ole Björkqvist was fundamental in setting in place new routines that would make more efficient the processing of manuscripts, from their submission to their publication. The support of the Science and Mathematics Education Research Group at the Department of Education, Learning and Philosophy at Aalborg University made possible that Hanne Lützow Kirk gave us a helping hand as editorial assistant during the first year.
As the year passed, both the Nordic and the international mathematics education community showed a rising interest in Nomad, reflected in the increase in submissions from authors inside and outside the region. With a growing number of papers in the review process our job got more complicated, but at the same time more exciting. The support of colleagues who diligently reviewed manuscripts was fundamental for succeeding in producing three regular issues and a special issue on difficulties in/with mathematics, all of which have received good comments from Nomad readers. We thank all authors and reviewers for their efforts in bringing papers of a good quality to be part of Nomad. The following persons reviewed papers processed during 2006:

Mette Andresen, Christer Bergsten, Morten Blomhøj, Trygve Breiteig, Tone Dalvang, Hans Christian Hansen, Marit Johnsen Høines, Simon Goodchild, Gunnar Gjone, Pedro Gómez, Barbro Grevholm, Lisen Häggblom, Lena Lindenskov, Thomas Lingefjärd, Olav Lunde, Candia Morgan, Mogens Niss, Torulf Palm, Jeppe Skott, Ole Skovsmose, Paola Valero, Tine Wedege, Carl Winsløw.

We would also like to acknowledge the generous support of the Swedish National Center for Mathematics Education (NCM) in Göteborg, whose funding has been fundamental in the administration, production and distribution of the journal. In particular we thank our managing editor Johan Häggström, NCM, for his great job producing the issues. NCM’s support, however, needs to be replaced by Nomad’s own finances. Therefore, one of the greatest challenges this year is to balance financially though the increase in the number of subscribers. We appreciate all the help that we can get from the readers in promoting Nomad as the Nordic journal for mathematical education research and developmental work.
This year we have also experienced the importance of a strong Editorial Committee whose members can provide a working hand and wise advice in running the journal. Therefore, it has been decided to enlarge this committee to include three members from each country – though two from Iceland. We are in the process of renewing and welcoming the new members. We thank Ole Skovsmose for the many years he served the committee; he has decided to open space for “fresher blood” to join the committee. We welcome then Tine Wedege and Jeppe Skott from Denmark, Elin Reikerås from Norway and Johan Lithner from Sweden.
We would also like to highlight the special help from the Nordic Graduate School of Mathematics Education, NoGSME, led by Barbro Grevholm, for allowing the creation of strong connections between the education of doctoral students, the qualification of doctoral supervisors and the strengthening of Nomad. Last year there was a fertile year for research in the region due to the very many research students who finished their dissertations. Nomad has been a journal in which they have decided to publish parts of their doctoral work. At the same time, Nomad has been used in several supervisor seminars to illustrate different issues related to writing, reviewing and publishing of research papers. These discussion have not only enriched the participants’ understanding of the processes of publication of academic work, but also has allowed Nomad editors to bring up discussions about the possibilities for the journal within the mathematics education community in the region. A new support action of NoGSME is announced at the end of this issue. In connection to the conference Teaching mathematics: Retrospectives and perspectives, which is to be held in Riga in May, NoGSME has arranged a one day seminar for supervisors from the Baltic countries on the writing of papers for scientific journals in mathematics education. As editors we will be given the opportunity to talk about Nomad and encourage submissions of papers to Nomad from the Baltic research environments. As a start in the strengthening of the link with Baltic countries, in this issue we publish a paper by two Estonian researchers.

About this issue

In this number we have gathered three papers, which address three different sites and aspects of mathematics education. Anu Palu and Eve Kikas in their paper “Primary school teachers’ beliefs about teaching mathematics” report on a study that investigated primary teachers’ beliefs about the purposes and methods of teaching mathematics in primary school. The sample consisted of 103 practicing teachers and 26 pre-service teachers in Estonia. Teachers with similar teaching experience agree in their evaluations of the purposes of teaching mathematics. Experienced teachers give priority to the purpose of acquiring knowledge before the purpose of developing the pupils’ personality. All teachers valued formalist teaching methods the least. However, teachers with different teaching experience held different beliefs about using traditional, formalist and social teaching methods. The results open for interesting discussions and further research on how teachers’ beliefs are formed.
In the paper “From problem solving to modeling” Nicholas Mousoulides, Bharath Sriraman and Constantinos Christou analyse and discuss a fair amount of the literature on mathematical modelling. The authors have identified three major strands in the literature, which are used to structure the paper, namely: (1) mathematical modelling as a problem solving activity, (2) basic principles for designing modelling activities and (3) benefits for students and teachers working with revealing modelling activities. Throughout the paper, the North American perspective on mathematical modelling is deliberately dominating. As editors we see this as a quality of the paper. The Nordic research on mathematical modelling may benefit from considering these perspectives and the paper may even raise some debate about the relation between problem solving and modelling.
Gail FitzSimons and Tine Wedege in their paper “Developing numeracy in the workplace” present an overview of the main discussions related to adult literacy and numeracy. They explore the meaning of the concept of numeracy and engage in clarifying how adults’ mathematical skills become transformed into workplace numeracy competence. Using the result of the authors’ research in Denmark and particularly in Australia, the authors show the way in which completing a complex task successfully puts in place a series of relationships among workers where mathematical competence is being created. This paper presents a contribution not only in delineating some of the main issues in the field of research on adults’ mathematical learning, but also bringing forwards a strong case to feed the public debate about mathematical competencies outside school and in the workplace.
In this issue we are also happy to publish a book review. Bharath Sriraman and Olof Steinthorsdottir have reviewed the research report of the study on KappAbel 2005–06: “Mathematical competitions and classroom collaboration: antonyms or a new direction for research on teachers’ beliefs?” by Tine Wedege and Jeppe Skott. We are especially happy since the research under review is the fruit of a truly Nordic collaboration. The Norwegian KappAbel in its present form as a competition for classes is developed by Ingvill Stedøy, leader of the Norwegian Center for Mathematics Education. During the ICME-10 years 2000–2004 the Nordic Contact Committee together with enthusiastic collaborators in each of the Nordic countries managed to extend KappAbel to become a Nordic competition and the Nordic final in 2004 was held during ICME-10 in Copenhagen. These activities were financed by a generous grant from the Nordic Council of Ministers and together with support from the Norwegian Center for Mathematics Education, the research project on KappAbel conducted by Tine Wedege and Jeppe Skott was also funded.

Morten Blomhøj and Paola Valero
Editors

ANU PALU and EVE KIKAS

Abstract

The main aim of the study was to investigate the beliefs about the purposes and methods of teaching mathematics in primary school teachers with different teaching experience. The sample consisted of 103 practicing teachers and 26 pre-service teachers. It was shown that teachers with different teaching experience were concordant in their evaluations of the purposes of teaching mathematics – they evaluated the purpose of acquiring knowledge higher than the purpose of the development of personality. Also, all groups of teachers valued formalist teaching methods the least. However, teachers with different teaching experience held different beliefs about using traditional, formalist and social teaching methods.

Sammanfattning

Huvudsyftet med studien var att undersöka uppfattningar (beliefs) om syftet med och metoder för matematikundervisning hos grundskollärare (primary school teachers) med olika undervisningserfarenhet. Undersökningsgruppen bestod av 103 verksamma lärare och 26 lärarstuderande. Resultatet gav att lärarna, trots olika undervisningserfarenhet, var samstämmiga beträffande syftet med att undervisa i matematik – de värderade syftet att utveckla kunskaper högre än den personliga utvecklingen. Dessutom värderades formalistiska undervisningsmetoder lägst av samtliga. Däremot visade det sig att lärare med olika undervisningserfarenhet hade skilda uppfattningar beträffande traditionella, formalistiska och sociala undervisningsmetoder.

ANU PALU
Anu Palu (PhD student) is an assistant of the methodology of teaching mathematics. Main research interests: the primary school pupils’ knowledge in mathematics and its development.

EVE KIKAS
Eve Kikas (PhD in psychology) is a professor of pre- and primary school education. Main research interests: the influence of school education on the development of thinking; development of everyday, synthetic, and scientific concepts (basing on Vygotskian approach); adults’ (including teachers) thinking.

The third summer school of NoGSME

In June the third summer school of the Nordic Graduate School in mathematics education will take place in Laugarvatn in Iceland. Laugarvatn is a small school community 93 km from Reykjavík. The sports and health science faculty of Iceland University of Education is located there. The interest for this summer school is high and we expect about 40 participants from all the Nordic and Baltic countries. A few participants come from other countries. The scientific leaders in the working groups during the summer school will be Abraham Arcavi from Israel, Marcelo Borba from Brazil, Marianna Bosch from Spain and Eva Jablonka from Germany/Sweden. Participants will spend most of the time in working groups, discussing research questions, theoretical frameworks, relevant literature, data collection and analysis, results and conclusions of the students’ projects. But there will also be a series of workshops, lectures by the group-leaders followed by discussions and a social programme. Opportunities to discuss one’s research project individually with the scientists or other doctoral students will be offered in the free time. An excursion to interesting places nearby will be offered. The programme is guided by the experiences from the earlier summer schools and the organisers have listened carefully to students suggestions.

The fourth workshop of NoGSME

Mathematics and language is the theme of the fourth workshop, which will take place in Uppsala on April, 26–27, 2007. The workshop is a joint enterprise of NoGSME and the Swedish Society for Research in Mathematics Education, SMDF, and has attracted 25 participants, both doctoral students and more experienced researchers. The workshop starts with two lectures by invited speakers. Heinz Steinbring will talk about Central issues of mathematical communication in the classroom. Another lecture will be given by Candia Morgan and she has chosen the title Discourse theoretic approaches to studying mathematics education. After this hopefully inspiring and challenging introduction participants of the workshop will be invited to present their own projects and discuss them with colleagues. There will be a panel discussion on themes related to mathematics and language and the different presentations.

Doctoral courses in the Nordic countries during 2007/2008

We hope that doctoral students have started to plan for the courses they will take during the next academic year. Added to the local courses there is a choice of course through the Nordic Graduate School. As in earlier years Agder University College will offer Theory of science from a mathematics didactics perspective, Theories of teaching and learning mathematics in the autumn, and in the spring probably either Problem solving, Meta perspectives in mathematics and mathematics learning in a technological environment, or Methodology in mathematics education. The later course has 16 participants at the moment of writing, which is the most ever.
We are also preparing a course on Research on assessment in mathematics education, which will be given by Umeå University in collaboration with NoGSME. In the autumn we hope to offer in Denmark as alternatives a doctoral course and workshop on the theme Justification of findings in mathematics and science education research, with particular regard to the role of theory in such justification. It can be used as a course by doctoral students and as a workshop by the supervisors. Planned dates are 21–23 of November, 2007. If you are interested in taking part of any of these courses we would like to know that as soon as possible. Most of the courses are given as distance courses with physical meetings at a few occasions during the semester. The courses will of course be given only if there are enough students interested in taking part. As usual doctoral students in NoGSME can get travel support to go to courses outside their own university.

A half way evaluation of NoGSME

NordForsk, our funding partner, is organising a half way evaluation of all the Graduate Schools, including NoGSME. Thus some of the doctoral students and supervisors in NoGSME will soon be asked to answer a self evaluation. We hope that you will be willing to do this. The self evaluations will be read by a committee of experts, who will judge the value of our activities. The director of NoGSME is then invited to meet this committee and discuss the outcome. The outcome of this evaluation is depending on honest contributions from participants in the activities of NoGSME.

Norma08 in spring 2008

The fifth Nordic conference on mathematics education is hosted in Denmark in April, 21–25, 2008. As in 2005 NoGSME will offer some arrangements in connection to this conference. It could be a seminar for supervisors maybe just before the conference and/or a workshop during the conference.

Mobility stipends for doctoral students

We would like to remind doctoral students of the opportunity to apply for mobility stipends to go to another Nordic university. The stipend can cover travels and accommodation for a one to two month stay. Those students who have used this opportunity found it highly rewarding. NoGSME can offer five such stipends each year. You can for example stay with another group of researchers and doctoral students, who are working with areas of research that are of interest to you. Or you can take especially relevant courses and get external supervision if agreed with the ordinary supervisors. An added value of such stays is that you get acquainted to other traditions of work and organisation than you are used to in your home university. It offers a fruitful experience for you future research career and might even open for a later position as post doctoral student or research assistant.

Other future plans and the new web address www.nogsme.no

In May 10–11, 2007, a Baltic conference on mathematics education will take place in Riga: Teaching mathematics: Retrospective and perspectives, the 8th international conference. Following that on the 13th of May NoGSME organises a one day seminar for supervisors in the Baltic countries with the theme concerning writing of scientific papers in mathematics education. NoGSME will also offer a seminar for supervisors on October 10–12, 2007. More information about that seminar will be given later. We have succeeded in getting an easier address for the NoSGME web-pages. Try out www.nogsme.no
Agder University College will soon introduce a new web portal and after that we hope that the NoGSME web pages will be more accessible. So you can try to follow announcements for future events on the new web pages.
As always any suggestions for themes to treat in seminars or workshops are welcome. Just contact anyone among the board members.

Barbro Grevholm
Director of the Nordic Graduate School
barbro.grevholm@hia.no

Anmeldelse af Didaktiske elementer – en indføring i matematikkens og naturfagenes didaktik

Carl Winsløw (2006). Didaktiske elementer – en indføring i matematikkens og natur-fagenes didaktik. Frederiksberg: Bifolia. (254 sider) ISBN: 87-9131-933-1
 

MOGENS NISS, Roskilde Universitet

CARL WINSLØW and VIVIANE DURAND-GUERRIER

Abstract

This paper presents a comparative study of two surprisingly different systems of preparing teachers for lower secondary level teaching of mathematics, namely those of Denmark and France. We first describe these systems differences succinctly. The main part of the paper reports on a qualitative study of how final year teacher students in the two countries handle two hypothetical situations of mathematics
teaching. Then we discuss how the findings could be related to the systems of
formation described first.

Sammendrag

Denne artikel præsenterer et komparativt studium af to overraskende forskellige matematiklæreruddannelser til det indledende sekundære niveau (12–15 årige), nemlig den danske og det franske. Vi giver først en kort beskrivelse af de to uddannelser. Hovedparten af artiklen præsenterer et kvalitativt studium af hvordan lærerstuderende i det afsluttende studieår håndterer to hypotetiske situationer i matematikundervisning. Dernæst diskuterer vi hvordan resultaterne af dette studium kan relateres til de to uddannelsers indretning, som først beskrevet.

CARL WINSLØW
Carl Winsløw is professor and vice-chair for research at the Department of Science Education, University of Copenhagen. His research in didactics of mathematics deals mainly with tertiary education and epistemological aspects. He teaches the didactics of mathematics and science to future high school teachers in the preservice education at the faculty of science. He is currently involved in the organisation of the NORMA conference in 2008 as chair of the scientific programme committee.

VIVIANE DURAND-GUERRIER
Viviane Durand-Guerrier is associate professor at the IUFM (university institute for teacher education) in Lyon and member of the research laboratory LIRDHIST (history, epistemology and didactics of the sciences and technology) at the University of Lyon 1. Her research is mainly on the epistemological role of logic and language within secondary and tertiary mathematics education. She is currently the president of ARDM (the French association for research on the didactics of mathematics). Her teaching addresses future primary and secondary mathematics teachers at the IUFM as well as masters students at the university.

The important and difficult task of improving mathematics teacher education

In 2003 an international symposium on mathematics teacher education was held in Malmö. The symposium was planed in connection with the preparation of the ICME-10 congress in Copenhagen in 2004, where mathematics teacher education also was one of the main themes. The Nordic mathematics education community took advantage of the symposium to raise and discuss issues and problems related to mathematics teacher education in the Nordic countries in dialogue with many prominent international researchers in the field. The book Educating for the future – proceedings of an international symposium on mathematics teacher education (Strässer et al., 2004) contains a wide range of reports about research related to mathematics teacher education.

In connection with recent reforms in the structure of teacher education in Norway and Sweden, a team of researchers-teacher educators produced a background document focusing on the current state of affairs in teacher education in these two countries. The main intention of the document was setting the scene for relating theoretical discussions emerging from research in different countries in the world, to the particularities of the educational systems in the Nordic countries (Bergsten et al., 2004). The background document highlighted five issues that needed academic and political attention: (1) The recruitment of teachers, (2) the contents of initial teacher education, (3) the relation between initial education and school practice, (4) the demands of a changing society and (5) the need for research on teacher education.

We believe that these points are crucial in the development of mathematics teacher education at all levels of the educational systems in the Nordic countries. Raising the quantity and quality of mathematics teacher education is becoming an economic and democratic imperative in the development of modern societies. At the same time research in mathematics education has documented that the practice of mathematics teaching is very stable and resistant to reform processes. Changes leading to improvement in the practice of mathematics teaching are more likely to occur gradually through changes in initial teacher education and in-service professional development, where teachers’ mathematical knowledge gets closely related to didactical knowledge in contexts of teaching experience or concrete teaching situations.

The papers in this issue address all, in one way or another, different aspects of teacher education and teacher practice. In the paper Education of lower secondary mathematics teachers in Denmark and France, Carl Winsløw and Viviane Durand-Guerrier present a comparative study of teacher education in these two countries. Besides describing the general characteristics and structure of teacher education in the two countries, they show the results of a qualitative study in which a group of teacher students in their last year, in each country, was asked to handle two hypothetical teaching situations that the researchers have designed. The researchers interpret the differences in students’ way of engaging in the situations in relation to the emphasis that each one of the educational systems places on the different aspects of mathematics teacher education. While French students seem to be more clearly competent in the handling of the mathematical content involved in the situations, their sense of how to bring that knowledge in an educational situation is less clear. On the contrary, Danish students have a better understanding of the pedagogical dynamics of a classroom and can easily imagine concrete teaching situations; however, their possibilities for acting as a teacher seem to be constrained by a not so rich mathematical understanding. Winsløw and Durand-Guerrier conclude that student teachers could gain in qualifications if teacher education emphasized

an earlier and more comprehensive work with didactics and pedagogy in France, paying attention to the potential of advanced mathematical knowledge; and a considerably broader basis in both mathematics and its didactics in Denmark, while maintaining the attention to students’ awareness and knowledge about pupil perspectives.

This paper, we think, illustrates clearly some of the predicaments related to the contents of initial teacher education and the relation between initial education and school practice, identified in Bergsten et al. (2004).

Even though international research has been addressing the issue of what the contents of initial teacher education should be in order to equip student teachers with tools that could help becoming an effective teacher, the issue is far from being resolved. As more research allows having an insight into the complexities of teacher education, the question of what should be the core of teacher education is transformed in many directions, among those what are the competencies and qualifications that may help a teacher when entering the teaching profession. Raymond Bjuland, in his paper Mathematically productive discourses among student teachers, focuses on the characteristics of this type of discourse in a teacher education setting. Building on the assumption that it is important for student teachers to engage in collaborative reasoning processes where rich mathematical discourses can emerge, he studies concrete cases in which the conversation among teacher students when solving geometry problems develop the special characteristics of being mathematically productive. Bjuland presents the set of theoretical constructs that supports the idea of learning being a discursive and communicative activity and identifies the features that research has identified to be associated with productive mathematical discourse – in opposition to ineffective mathematical communication. Using few cases of a larger study, he exemplifies how such mathematically effective communication among teacher students can emerge. Besides providing insight into communication processes among students when working in groups, we think that this paper indirectly opens the discussion of what it takes to educate good mathematics teachers: more than a matter of the right contents and the right organisation of activities, teacher education could be thought of as a communicative space where teacher students should experience productive mathematical discourse with the hope that such communicative competence can be a solid basis for entering the practice of teaching in schools.

In Bergsten et al. (2004) the need to think about mathematics teacher education in a broader social perspective was emphasised. The preparation of teachers should take into account the complexity of the teacher profession, since qualifications and competencies gained in initial education are always at stake when entering teacher students graduate, get their first job and enter the institutional settings that shape the life of mathematics teaching in schools. Teacher education and its potential to bring change in the teaching and learning of mathematics in schools are always under examination. Laila Pehkonen in her paper To change or not to change – how primary school teachers speak about stability and change invites into the world of in-service teachers and how they perceive their work and, in particular, the demands on changing their practice. In contrast to many studies that ask teachers directly on how they see transformation, Pehkonen addressed the issue of change through seeing how teachers view stability, since change and stability and continuity are complementary notions. In her case study in Finland, the teachers interviewed expressed a series of dilemmas related to their practice. Teachers have to operate in the midst of several demands and, before engaging in changing something as policy makers, teacher educators or researchers suggest, they need to feel confident with the possible results of such changes. The need of stability and continuity is not a simple desire of preserving tradition; it is rather part of a professional critical stance in search of good reasons for change. We see Pehkonen’s paper as a contribution to understanding teachers’ practice and teachers’ good professional reasons for action in a turbulent time of increasing demands to the work of mathematics teachers.

The book review presented in this issue has clearly relevance for the education of mathematics teachers. The book in Danish Didaktiske elementer – en indføring i matematikkens og naturfagenes didaktik by Carl Winsløw is meant to be a textbook for teacher education on the didactics of mathematics and the natural sciences. In his review Mogens Niss highlights both the achievements and shortcomings of the book and certainly welcomes this contribution within the community of teachers and researchers in mathematics and science education.

We hope that the reader finds in this number of nomad good sources of reflection on the difficult, yet absolutely important task of understanding
teachers’ work and improving teacher education.

Morten Blomhøj and Paola Valero
Editors

References
Bergsten, C., Botten, G., Fulgestad, A.-B., Grevholm, B., Holden, I. & Lingefjärd, T. (2004). Background document. In R. Strässer, G. Brandell, B. Grevholm & O. Helenius (Eds.), Educating for the future – proceedings of an international symposium on mathematics teacher education (pp. 9–38). Göteborg: NCM, National Centre for Mathematics Education.
Strässer, R., Brandell, G., Grevholm, B. & Helenius, O. (Eds.) (2004). Educating for the future – proceedings of an international symposium on mathematics teacher education. Göteborg: NCM, National Centre for Mathematics Education.onal Centre for Mathematics Education.

RAYMOND BJULAND

Abstract

This article reports research that focuses on the characteristics of mathematically productive discourses (MPD) while student teachers are working collaboratively on a geometry problem in a problem-solving context. Analyses from the discourses of two groups of students are presented in order to illustrate mathematically non-productive and productive discourses respectively. A definition of MPD is presented and used as an analytical tool to identify critical characteristics of sequences of productive discourses. This definition involves the following five criteria: 1) Student utterances, stimulating a monitoring utterance, 2) the monitoring utterance, 3) student responses, stimulating a second monitoring utterance, 4) the second monitoring utterance, 5) further elaborations, advancing the mathematical discussion among the students. The article also discusses the difficulty of concluding when a discourse is productive or not, especially when students are challenged to work on complex problems in which a solution is not usually reached within a school lesson.

Sammendrag

Målet med artikkelen er å identifisere viktige kjennetegn på en produktiv, matematisk diskurs når lærerstudenter samarbeider i smågrupper om å løse en geometrioppgave i en problemløsningskontekst. Analyser av diskurssekvenser fra den matematiske diskusjonen i to studentgrupper blir presentert for å identifisere både ikke-produktive og produktive diskurssekvenser. En definisjon på en matematisk produktiv diskurs (MPD) blir presentert og brukt som et analytisk redskap i analysen. Definisjonen omfatter følgende fem kriterier: 1) Ytringer som stimulerer en monito-rerende ytring, 2) den monitorerende ytringen, 3) responsytringer som stimulerer en ny monitorerende ytring, 4) den nye monitorerende ytringen, 5) responsytringer som fører kommunikasjonen matematisk videre mellom studentene. Artikkelen diskuterer også hvor vanskelig det kan være å konkludere når en matematisk diskurs er produktiv eller ikke, særlig når studenter blir utfordret til å arbeide med kompliserte problemer der en løsning vanligvis ikke blir funnet i løpet av en skoletime.

RAYMOND BJULAND
Raymond Bjuland is Associate Professor in mathematics education at Agder University College. At the moment he has a post. doc. position in the LCM project, Learning Communities in Mathematics at AUC. This project is based on the idea of creating ”communities of inquiry” between teachers and didacticians to question existing practices and design new approaches to teaching and learning mathematics. He is interested in classroom research and in problem solving with a focus on students working in collaborative small groups.

Supervisors' seminar in Riga

The Nordic Graduate School in Mathematics Education has as one of its aims to support and develop the competence of supervisors in mathematics education. In connection to the 8th international conference on mathematics education that took place in Riga on May 10–11, the Latvian University offered an opportunity to NoGSME to organise a supervisors' seminar there. Thus on May 12, fifteen supervisors were gathered and listened to the presentation by one of the editors in chief of nomad, Morten Blomhøj. He discussed the mission of nomad, the policy and review process, the editors’ work and guidelines for authors and reviewers. After a presentation about the writing process the participants worked in groups on drafts or ideas for papers suitable for nomad. A number of realistic ideas were elaborated and we hope to see these papers take form and be published in nomad later on. This was the first time NoGSME could carry out an activity in one of the Baltic countries and we hope to be able to do so on later occasions also.

European educationalists introduced to NoGSME

The National Centre for Mathematics Education in Göteborg organised a four-day programme for government representatives from ten European countries in May. This was done on the initiative of the Swedish Government. One of the sessions were about the Nordic Graduate School in Mathematics Education and the interested participants asked a number of questions. They were eager to hear how the researcher education could complement teacher education and offer qualified persons to the profession of mathematics teaching. The labour market for new doctors was also discussed. In some of the countries there was an interest to get qualified mathematics teachers, who can take on more advanced and developed tasks in school as responsible for the mathematics teaching. Curriculum development and implementations are in focus in many countries and a search for new ways to work with this is going on.

New doctors in mathematics education in the Nordic countries

In February Elin K. Lie Reikerrås defended her thesis Aspects of arithmetical performance related to reading performance: a comparison of children with different levels of achievement in mathematics and reading at different age levels. Her dissertation was the first one in a new doctoral program in special education at Stavanger University in Norway. The performance of pupils on different kinds of tasks was studied: word problems, counting facts, multi-step tasks and mental calculation tasks. Three studies were carried out using quantitative methods with 941 pupils in ages 8 to 15. Results show that the ability to solve word problems is not related so strongly to reading abilities as to general level of mathematics. In tasks with counting facts the level of reading ability did not influence mathematics performance. In tasks with several steps the findings show that weak reading ability influences in early ages. In mental calculation when pupils had no visual support the level of reading ability was strongly related to the level of mathematics performance, while general level of mathematics was not of crucial importance.
Bodil Kleve’s dissertation Mathematics Teachers’ interpretation of the curriculum reform, L97, in Norway was presented at Agder University College in May. It is a qualitative study of mathematics teachers in compulsory school and how they interpret and implement the curriculum reform, L97. In the case study both the individual and the social perspectives in teaching and learning are illuminated. Data consist of classroom observations, focus group interviews, conversation with teachers, self-estimation and questionnaires. The results indicate that teachers have interpreted the plan very differently and had different teaching practices. The study shows that the introduction of a new curriculum does not necessarily lead to change in teaching practices in mathematics and it takes a long time to carry out reforms in school. It is teachers’ mathematical and didactical competence that is decisive for what kind of mathematics teaching pupils will meet in school.
At Ålborg University Sikunder Ali Baber defended his thesis on May 3: Interplay of citizenship, education and mathematics: Formation of foregrounds of Pakistani immigrants in Denmark. The project aims at making a contribution to the debate around multiculturalism while bringing attention to the foregrounds and backgrounds of immigrants. The research study presents an investigation into the foregrounds and backgrounds of three Pakistani immigrant families in Denmark. How have they been engaged in perceiving and making their living within the Danish nation-state context and how are their lives being transformed as an effect of 9/11 scenario and the globalization processes? Attention is given to factors that are responsible for the formation of the foregrounds of Pakistani immigrants as part of conditions of their citizenship in the Danish welfare state. Here immigrants’ access to and experiences with education and mathematics have been recognized as ways to secure their rightful place within Danish welfare society.
The importance of mathemacy has been recognised as a tool to deal with the complex interplay of numbers within the transformation of modern society.
A few more dissertations are going to take place in June both in Norway and Sweden and we hope that thesis from other places will also be made known to us so they can be included among these short reports here and visible to all Nordic doctoral students and supervisors. It is a lively activity going on at the moment and more doctoral students are taken up in the programmes.

Is there a future for Nordic cooperation after NoGSME in 2009?

One of the questions that NordForsk is posing in the evaluation of NoGSME is what is going to happen after the funding from NordForsk finishes in 2009. In the conference Norma05 we started to discuss the creation of a Nordic umbrella organisation, the Nordic Society in Mathematics Education, NSME. Such an organisation could assist in coordinating the activities that are going on in all the Nordic countries. It could also be the host organisation of the Norma-conferences that at the moment do not have a real home.
Even nomad could profit from an active, formal organisation that could help in searching for funding for the journal. A group was formed in the Norma05-conference to prepare a document about a Nordic Society that can be presented at Norma08 and discussed there. This group has investigated the national societies and their statutes in order to get some inspiration for the new society.
Now the question is of course if such a society will be enough to carry on the activities from NoGSME. It is probably also necessary that the institutions that are involved at the moment have a wish to share activities with participants from other universities and offer courses, seminars and workshops to people from all participating departments in NoGSME. So far we have experienced a great willingness to take on the organisation of such activities locally and we can hope that this will continue. This question about the future of the Nordic collaboration in mathematics education should be a concern of all involved and we hope it will be discussed widely and that all creative solutions will be shared. The Nordic cooperation builds on long-standing traditions from common conferences both for mathematics teachers and researchers and common projects. As a region in EU the Nordic and Baltic countries can profit much from collaboration and bridging activities.

Please contact us if you have wishes for the activities or suggestions for themes for workshops or seminars organised by NoGSME.

Barbro Grevholm
Director of the Nordic Graduate School
barbro.grevholm@hia.no

LEILA PEHKONEN

Abstract

Changing mathematics education and teachers has proved to be difficult. In this case study the matter has been turned upside down. The purpose is to approach the question of change from the perspective of stability. In this paper I will discuss how primary school teachers speak about stability, and – in this connection – about changes in mathematics teaching. The data consists of semi-structured interviews of nine primary school teachers and is analyzed qualitatively. The findings suggest complex relationships between the need for continuity and the desire for changes.

Sammandrag

Att ändra matematikundervisning och matematiklärare har visat sig vara svårt. I den här fallstudien har saken vändts upp och ner. Syftet är att närma sig frågan om förändring utgående från stablitetens perspektiv. Detta arbete handlar om hur klasslärare talar om stabilitet och – i detta sammanhang – om förändringar i matematikundervisningen. Data består av halv-strukturerade intervjuer med nio klasslärare och har analyserats kvalitativt. Resultaten tyder på komplexa samband mellan behovet av stabilitet och viljan till förändringar.

LEILA PEHKONEN
Leila Pehkonen is university lecturer in Education at the University of Helsinki. Her research interests include mathematics education, education for gifted students, and teaching and learning in higher education.

KEIKO YASUKAWA

Abstract

The United Nations declared 2005 to 2014 as the Decade of education for sustainable development. This presents an opportune moment for mathematics educators and mathematics education researchers to reflect about the effectiveness that mathematics education has had in creating citizens for a sustainable future. There is an important distinction between education about sustainable development, and education for sustainable development; the latter is the more important, but also the more difficult and challenging. The paper examines some of the obstacles that mathematics educators face in educating for sustainable development, and identifies the need for some radical alternatives. These alternatives will need to challenge the dominant discourses that shape identities of both learners and teachers.

Sammendrag

De Forenede Nationer lancerede 2005–2014 som Årtiet for Uddannelse for BæredygtigUdviklingen. Det er en anledning for matematiklærere – ikke mindst i læreruddannelser – og for matematikdidaktikere til at reflektere over, hvordan matematikundervisning aktuelt bidrager og i fremtiden kan bidrage til dannelse af borgere, der kan udvikle og opretholde bæredygtige samfund. Det er her vigtigt at skelne mellem uddannelse om holdbar udvikling, og uddannelse for holdbar udvikling. Den sidste udfordring er den vigtigste men samtidig også den sværeste og pædagogisk-didaktisk mest udfordrende. Artiklen undersøger nogle barriere som matematiklærere møder, når de forsøger at praktisere matematikundervisning for holdbar udvikling. Artiklen identificerer et behov for radikale alternativer til traditionel matematikundervisning, når det gælder bidrag til holdbar samfundsudvikling. Det kræver imidlertid at de dominerende diskurser i matematikundervisningen, der er med til at forme identiteten hos elever og lærere udfordres.

KEIKO YASUKAWA
Keiko Yasukawa is a lecturer in adult education and in engineering at the University of Technology. Keiko has been researching the mutual shaping of mathematics and social change, drawing on recent theoretical developments in Science and Technology Studies (STS). She is exploring questions around the meanings of mathematics educators as activist educators, and mathematics education for social action.

OLOF BJORG STEINTHORSDOTTIR AND BHARATH SRIRAMAN

Abstract

This study was conducted to investigate the influence of contextual and number structures on individuals’ use of strategies in solving missing value proportion problems, and to examine gender differences in strategy use. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males were individually interviewed as they solved sixteen missing value proportion problems. The problems represented four contextual structures. No gender differences were identified in the overall success rate. However, girls were more successful than boys in handling associated sets and symbolic problems, and boys were more successful than girls in part-part-whole problems. Moreover, the data suggest that the contextual structures influence females’ choice of strategy more than that of males.

Sammendrag

Denne artikel handler om strategier ved løsning af proportionalitets-problemer og det undersøges hvordan opgavernes kontekstuelle og talmæssige strukturer influerer på henholdsvis drenges og pigers valg af løsningsstrategier.
Der kunne ikke påvises kønsforskelle i elevernes generelle succesrater. Pigerne var dog mere succesfulde end drengene ved A–S og S–P opgaver mens drengene klarede P–P–W opgaver bedre end pigerne. Endvidere indikerer data, at den kontekstuelle struktur i højere grad påvirker pigernes valg af strategi end drengenes.

OLOF BJORG STEINTHORSDOTTIR
A former mathematics classroom teacher in her native country of Iceland, assistant professor Olof Bjorg Steinthorsdottir teaches mathematics education courses. Her scholarly interests include the teaching and learning of mathematics among students in pre-kindergarten through middle school, specifically students' understanding of mathematics and how teachers can use that knowledge to make instructional decisions. Her interests also include gender and mathematics. Steinthorsdottir has been active member within PME and has in resent years co-organized the working/discussion group about gender and mathematics. She received her PhD in Mathematics Education from the University of Wisconsin in Madison.

BHARATH SRIRAMAN
Bharath Sriraman is associate professor of mathematics at the University of Montana, with a wide range of eclectic research interests including mathematical cognition and gender issues. Bharath is the editor-in-chief of the Montana Mathematics Enthusiast, associate editor of ZDM – the International Journal on Mathematics Education, consulting editor of Interchange: A Quarterly Review of Education, and reviews editor of Mathematical Thinking & Learning as well as ZDM. He holds very active research ties with researchers in many countries including Iceland.

Mathematics education: a key for success in a globalised world?

In 2006 the Danish government set up a commission with the task of analysing Denmark’s situation in the global scene, identifying current problems and suggesting a course of action for securing Denmark’s growth, competitiveness and security in a global economy. The report Progress, Innovation and Cohesion. Strategy for Denmark in the Global Economy [ 1] by the Globalisation Council emphasises the need of strengthening education in general and in particular citizens’ mathematical and scientific competencies and education. The report has placed mathematics education high in the political agenda. A series of initiatives with accompanying resources have followed and are still under design now and in the years to come. Hopefully, this political initiative will open new possibilities for practitioners and researchers to engage in activities aiming at the betterment of mathematics education in the country.

The focus on mathematics education in 2007 and in the future, as a result of the recommendations of the Globalisation Council, seem to be in conflict with the results of the implementation of previous, yet very recent reforms in two main sectors of the Danish educational system. The reform of the structure of high schools (gymnasium) intended, among others, increasing the amount of students who will choose the mathematical and scientific string by offering a compulsory, interdisciplinary basic course where all students had the opportunity to do mathematics and science. The result until now has been a clear decrease in the numbers of students who choose the mathematical and scientific path in high school; what is quite worrying given the already low numbers of students who are qualified for and who actually decide to follow mathematics and science-based studies at university levels. The reform of teacher education intended, among others, to extend the actual time that primary and lower secondary teachers dedicated to their preparation for teaching mathematics, science and Danish. More focus on the subject areas (instead of on general pedagogical processes in school) was a central interest of the reform. The results until now of the brand-new implemented structure show an alarming low amount of students having choosen mathematics as a central area in their teaching education. The numbers are even worse for science. These trends are highly problematic in a context with a shortage of interested youngsters, qualified mathematics teachers in schools, and a population of school and high school teachers close to retirement age. If these tendencies continue to be the case, they can easily threaten the intention of strengthening mathematics education as a key for improving citizens’ competencies for action and participation in a globalised Denmark.

Are these happenings in Denmark of relevance for the Nordic Region? All Nordic countries have been engaged recently in educational
reforms impacting on the actual teaching and learning of mathematics in different levels of schooling. Despite of differences among countries, policies and structures, similarities can be fund in these reforms. The overall political goal has been clearly stated: we need to increase the percentages of the youth cohorts that enrol in mathematics, science, engineering and the technical branches at a higher educational level. This goal has primarily been pushed through structural changes in the educational systems and/or through the installation of new assessment systems. Discussions of how to improve the quality in the teaching and learning of mathematics at the different levels of the educational system typically play a very limited role in the reform process. Accordingly research findings and experiences from developmental projects are seldom used to inform the political decisions.

The state of affairs poses a double challenge for the research community in mathematics education. We need to improve our communication with politicians and policy makers about the type of research findings that can be useful in a reform process, and we need to investigate more directly how to develop mathematics teaching and learning at different school levels in order to empower the students to meet the mathematical challenges of a gobalised world. It remains to be seen whether the initiatives that are being launched nowadays will involve research for the development of practice, and whether they will be able open possibilities for mathematics and science to play a decisive role in the formation of not only economically active citizens but also democracy-aware people under globalisation.

A new member at the editorial board

This year NOMAD started a process of renewal and expansion of the editorial committee. It is now a pleasure for us to inform that Kristín Bjarnadóttir, from Iceland University of Education, has joined the editorial committee. We would also like to express our gratitude to Anna Kristjánsdóttir who for many years has served NOMADs editorial committee.

About this issue

This issue brings together four quite different papers. The first paper by Keiko Yasukawa, rather than a research report, is an essay of relevance for the continuous discussion about what is the role of mathematics and mathematics education in society. It provides a meta-discussion that invites researchers and practitioners to question the intentions behind mathematical instruction in a time in history where sustainability has become more important than ever. The other two papers are research reports of two studies with two quite different methodological designs and areas of study. While the paper by Olof Bjorg Steinthorsdóttir and Bharath Sriraman presents a screening study on a group of 53 students’ proportional thinking, the paper by Lil Engström and Thomas Lingefjärd presents a study on how dynamic geometry systems are used in three high school classes in two different countries. The fourth paper by Uffe Jankvist is something between a book review and review paper on empirical investigations of the effects of including the history of mathematics as part of the teaching of the subject. All papers highlight important issues about the practices of teaching and learning of mathematics in the Nordic region.

At a time where discussions about the state of our planet are part of daily news, Yasukawa’s paper An agenda for mathematics education in the decade of education for sustainable development presents an important challenge for mathematics educators and mathematics education researchers: Does mathematics education have something to do with education citizens for sustainable development? That the mathematical education of students at different levels of schooling fulfils important functions in society is a recognized fact of policy makers, educators, researchers, the labour market and the public in general. However, which kinds of functions is a less clear issue. Yasukawa argues that mathematics, being a powerful tool to describe, model and act in the world, can be connected to the development of a critical capacity in students for judging the effects of development on the world. The arguments of critical mathematics education, partly emerging from Scandinavian contexts, are brought in relation to the perception that youngsters and, in general, people have about themselves and their world currently. One of the main points in an attempt of relating mathematics education and the possibilities of contributing to sustainable development is the need to concentrate on how students construct their identities and how they adopt different values about themselves and the world.

In the paper Gender and strategy use in proportional situations: an Icelandic study by Steinthorsdottir and Sriraman, we find a detailed report on an empirical study on 53 Icelandic eight graders’ strategies for solving missing value proportion problems of the mathematical form: . The students have been interviewed while solving 16 different problems spanning a variation of four categories of the contextual structure in the formulation of the problems and four categories of the numerical structure of the problem according to the integer/non-integer nature of the involved proportions and the answer. The authors’ analyses show that both types of structures influence the students’ choice of strategy and their success rates. The gender differences found in the study are quite subtle but still interesting. In accordance with other findings in the literature, it seems that girls have stronger tendency to make use of a context familiar to them than boys. In general the results of the study are an argument in favour of systematic variation in context and numerical structure when teaching proportional reasoning. The design of the study invites to carry out follow up studies where negative integers, rational numbers and more than one variable are introduced in proportional problems.

The paper Posing problems using Cabri by Engström and Lingefjärd presents some of the major findings from the first author’s PhD dissertation. The teaching of geometry with dynamical geometry software (DGS) in three upper secondary classes – two from Sweden and one from Switzerland – is studied in detail. Based on extensive classroom observations, the students’ problem solving activities using DGS and the related dialogues between the teacher and the students are analysed. The teachers’ objectives for using DSG and their general beliefs about teaching and learning of mathematics are explored with a questionnaire and used in the analysis. The main result in the study is that the way in which the teachers formulate the tasks for students and ask questions during their problem solving activities are the most important factors for the potential learning outcome of using DGS. Quite small differences in the questions asked during the students’ computer-based activities can make a difference between the students’ instrumental use of DGS for finding right answers to unrelated tasks or the students’ engagement in mathematical investigations using DGS as an instrument for learning mathematics.

As a non-standard type of contribution this issue also includes the paper Empirical research in the field of using history in the teaching of mathematics by Jankvist. The paper is something between a book review and a literature review of the reasons for including the history of mathematics in the teaching in the subject, and of empirical investigation of its effects. The point of departure is the author’s reading of the extensive proceedings from the International study group on the relations between the History and Pedagogy of Mathematics (HPM) 2004 and the European Summer University on the history and epistemology in mathematics education (ESU) 4, and in particular his review of the four papers reporting empirical studies in these volumes. The purposes for including the history of mathematics in the four papers are discussed in relation to the guidelines for teaching history of mathematics in the Danish gymnasium. The paper concludes with an argumentation for the need of more empirical research on the possible effects of including the history of mathematics as an integral part of the teaching of mathematics.

Morten Blomhøj and Paola Valero
Editors

Notes
1 Available at http://www.globalisering.dk/

LIL ENGSTRÖM AND THOMAS LINGEFJÄRD

Abstract

The purpose of Engström’s research was to investigate in which ways and to what extent three different teachers, one in Switzerland and two in Sweden, used a specific dynamical geometry software, Cabri Géomètre, in upper secondary school.
The method consisted mainly of field notes and audio recording during observations in classrooms. The field research was then analysed and evaluated according to the following questions: a) How do teachers pose problems and b) how do teachers make students’ experiences useful when Cabri Géomètre is accessible?
The result showed that one important factor enabling to challenge the pupils in learning mathematics when using Cabri Géomètre is the way the teacher poses the problems or the questions. The word ’challenging’ means among other things, that the pupils continue asking themselves questions so that there will be continuous learning, not limited to finding just one correct answer.

Sammanfattning

Artikeln presenterar en studie av hur tre gymnasielärare, en i Schweiz och två i Sverige, använder dynamisk programvara, Cabri Géomètre, i sin matematikundervisning. Empirin består av ljudupptagningar och fältanteckningar från lektionsobservationer. Anlysen genomfördes med avseende på lärarens problemformuleringar samt hur läraren tar vara på elevernas erfarenheter när Cabri Géomètre används.
Resultatet visar på betydelsen av hur läraren formulerar problemen eller uppgifterna för att utmana eleverna, samt för att utnyttja programvarans särskilda förutsättningar. I en laborativ undervisningsmiljö i matematik kan ett dynamiskt datorprogram skapa möjligheter för elever att lära sig längs ej förväntade vägar.

LIL ENGSTRÖM
Lil Engström is assistant professor in mathematics education at Stockholm Institute of Education. She has an extensive experience of teaching in lower and upper secondary school. For 14 years she has educated prospective matehmatics teachers both for primary, lower and upper secondary school. She has been involved in different projects about using computers in the classroom and also been a speaker in a number of international conferences about Cabri Géomètre.

THOMAS LINGEFJÄRD
Thomas Lingefjärd is associated professor (docent) in mathematics education at Gothenburg university and has long experience of teaching mathematics and mathematics education with or without the aid of technology. He share his time between teaching, supervising and research. Most recently Thomas Lingefjärd was a member of the International program Committee for the 14th ICMI Study: Modelling and applications in mathematics education and he is also a member of the recently funded European Comenius project DQME II.

Review of empirical studies in HPM2004 & ESU4: Empirical research in the field of using history in mathematics education

UFFE THOMAS JANKVIST

Abstract

This paper discusses empirical studies in the proceedings HPM2004 & ESU4. More precisely the paper deals with four of the more clear-cut empirical studies. These are the contributions by B. Smestad, C. Tzanakis & M. Kourkoulos, W.-S. Horng, and Y.-W. Su. These contributions are first presented and then later discussed in the context of whether their purpose of involving the history of mathematics in mathematics education is to promote the learning of mathematics or if it is to bring about aspects of mathematics which are not normally part of the teaching and learning agenda, e.g. cultural or social aspects of mathematics and its history – ’history as a tool’ or ’history as a goal’. The papers and their purposes for involving history are then related to a Nordic case, namely the official regulations for the Danish upper secondary mathematics programme for involving the history of mathematics. In the end the need for empirical research studies in the field of using history in mathematics education is discussed as well as further perspectives for the community regarding such studies or the lack of them.

UFFE THOMAS JANKVIST
Roskilde University

The Nordic graduate school in mathematics education

The summer school for doctoral students in Iceland took place in rather ice-cold weather with storm and rain but inside the temperature was good and everything happened in good mood and spirit. The campus at Laugarvatn showed to be excellent for our work and both students and group leaders were highly satisfied with the development that took place in the groups during the week. We were 47 participants in all including the two students from institutions outside the Nordic countries. One of the students has suggested that in 2008 the summer school should become a winter school. The argument is that so many activities are going on next year. And indeed there are many opportunities for doctoral students to present their work in 2008.
Already in January 30 the research seminar of the Swedish society for research in mathematics education welcomes doctoral students and supervisors to present research studies in an international context. Keynote speakers will be Eva Jablonka and Rosamund Sutherland and the theme of the seminar is Mathematical knowledge. Immediately after that we have Matematikbiennalen, where there is room for many presentations of different kinds. In April doctoral students and supervisors can take part in Norma08, the fifth Nordic conference in mathematics education, which takes place in Denmark in April 21–25 and has several interesting themes for the work. The themes are Didactical design in mathematics education, Education and identity of mathematics teachers, Technology in mathematics education and Mathematics for all: Why? What? When?
In March 2008 the centennial jubilee of ICMI is celebrated in Rome with a symposium, with investigations into how ICMI and the ICME-conferences have interrelated with the development of research in mathematics education.
In summer 2008 it is time again for the great international conference ICME and now it is the 11th in order. Number 10 took place in Denmark, organised by the Nordic countries together in 2004. Normally there is only one ICME-conference during one's doctoral study time so for those who experience this now it is time to go. For all these events there are excellent web pages to visit when you plan your participation. ICME-11 will be in Mexico from July 6 to 13. The scientific programme is as rich as ever.

The eighth NoGSME seminar for supervisors

In Lund 20 members of the NoGSME network will be gathered on October 11–12 for a seminar with the theme Outcomes of research. What is it we are producing with research in mathematics education? What are the results? Does the outcome have any implications? How is the new knowledge from research disseminated to mathematics teachers in school, to teacher educators, to academic teachers and to society in large? Where do the research questions come from? Are there significant areas where nothing is going on? Many questions can be raised and we want to open for a broad discussion about at least some of them in this seminar. Mogens Niss will start the seminar by discussing Outcome of research in mathematics education, followed by Ole Björkqvist who will present about evaluations of such research. Participant will give their view about where we stand currently in the areas they are specialised in.

Three NoGSME courses are going on in autumn 2007

The great interest from doctoral students in taking part in the courses offered via NoGSME is very encouraging. The course in University of Agder (new name since September 1, 2007) has more than ten participants from outside the university. This course named Theories of teaching and learning mathematics is now given for the fifth time and Simon Goodchild and Maria Luiza Cestari are teachers in the course together with some doctoral students, who have just finished writing their theses.
Another course is on Research on assessment in mathematics education, and it is given by Torulf Palm and Peter Nyström at Umeå University in collaboration with NoGSME. In the autumn we also offer in Denmark as alternatives a doctoral course/workshop on the theme Justification of findings in mathematics and science education research, with particular regard to the role of theory in such justification. It can be used as a course by doctoral students and as a workshop by the supervisors. The course runs already and it will end with the workshop in November 22–24 in Nyborg, Denmark. In the board meeting for NoGSME in October plans will be made for all the activities during 2008 and then announced in the next issue of NOMAD.

New doctoral dissertations since June 2007

In June Anne Birgitte Fyhn defended her doctoral work Angles as tool for grasping space: Teaching angles based on students’ experiences with physical activities and body movement at the University of Tromsö in Norway. Her work consists of four papers and two DVDs held together by a preamble, which includes the theoretical framework and the methodology chapter. She asks ’How can the teaching of angles be based on the students’ experiences with physical activity and body movement?’ Another question dealt with is how students describe and explain angles in drawings and written text when they mathematise climbing with respect to angles. She also investigates how teachers do attain students’ mathematising of climbing as approach to their teaching of angles. The papers illustrate tries to use compass and climbing as a tool together with analytical drawings as alternative ways for students to learn about geometry and angles. Teachers need to get acquainted with inductive enactive mathematics teaching before they are able to grasp the students’ mathematising of climbing. In the papers and DVDs teachers can get new ideas about how to introduce pupils to different parts of geometry by using physical activities and body movement.
Johan Prytz at Uppsala University in Sweden defended his dissertation also in June 2007. The title is Speaking of geometry: a study of geometry textbooks and literature on geometry instruction for elementary and lower secondary levels in Sweden, 1905–1962, with a special focus on professional debates. His purpose is to investigate textbooks and literature, related to instruction of geometry, used by teachers in elementary schools (ES) and lower secondary schools (LSS) . Attention is given to debates about why a course should be taught and how the content should be communicated. In the period 1905–1962, the Swedish school system changed greatly but it is not really known how the teaching of mathematics changed in Sweden. The time before 1950 is often described as traditional, static or isolated. Geometry instruction in Sweden did change in the period 1905–1962 and geometry instruction in LSS was discussed. Two major issues were the axiomatic method and spatial intuition. Textbooks for LSS not following Euclid were produced also, but the axiomatic method was kept. By 1930, these alternative textbooks were the most popular. The textbooks in ES also changed. Visualizability was a central concept in the debate. Some features did not change. Throughout the period, the rationale for keeping axiomatic geometry in LSS was to offer training in reasoning. The axiomatic method was the dominating theme in ongoing discussions but not heuristics. Discussion on heuristics would have been relevant considering the final exams in the LSS. A skilled problem solver had better chances to succeed than a master of proof.
In August Tomas Højgaard Jensen discussed his thesis with the opponents at Roskilde University Centre. He wrote about Utvikling af matematisk modelleringskompetence som matematikundervisningens omdrejningspunkt – hvorfor ikke? (Developing mathematical modelling competency as the hub of mathematics education – why not?). He wants to investigate if he, based on analyses from the perspective of mathematics as a teaching subject and cognitive psychology, can argue for potentials of working with the analysis and construction of mathematical models in general education with a mathematical content. He also inquires into what meaning he can ascribe to the concepts mathematical modelling, competency, technological competency and democratic competency to make them a constructive tool with respect to the identified potentials in relation to thinking about and plan, carry out and evaluate general education with a mathematical content. Further Tomas wants to reply to what organizational characteristics of the way mathematical modelling can potentially be integrated into the teaching he can defend as being central based on the theoretical analyses, if the goal is to develop pupils’ mathematical modelling competence as much as possible. Finally a fourth question in the work is what the nature is of the hindrances that in a specific case stand in the way of the Utopia of a complete realization of the good practice in accordance with the central organizational characteristics. A number of potentials can be found but there are also hindrances in this work, such as when the examination conditions aren't fully in accordance with the teaching goals and methods used.
Still a couple of theses are coming up to defence during autumn 2007, but there will not be as many this year as last year when we had 21.

An American conference on doctoral programmes

In 1999 the first national conference on doctoral programmes in mathematics education was organised in the United States. In September 23–26 a follow-up conference was organised to investigate progress in the past decade. Bob Reys at university of Missouri Columbia was the main responsible for this event. A number of important questions were raised: What is happening with doctoral programs in mathematics education? What core knowledge do doctoral students in mathematics education need to know? Do we need accreditation of doctoral programs? What about program delivery, issues, challenges and opportunities? In the international panel visitors from Brazil, Japan, Spain and Norway were invited to present about the doctoral programs in their countries. I was invited to talk about the doctoral programs in the Nordic countries and the interest was great. Participants envy us for the collaboration we can have in the Nordic Graduate School and students' opportunities to take courses in other universities, to go to summer schools, to get mobility stipends and so on. I will try to return to the content of the conference later.
All kinds of suggestions for content of program for the Nordic Graduate School are welcome. Just contact any of us in the board. See www.NoGSME.no

Barbro Grevholm, Director of NoGSME
University of Agder, Norway

ULLA RUNESSON

Abstract

This paper describes an approach to teaching that enhances pupils’ learning in mathematics. The model described – Learning study – involves teachers and researchers cooperating in an iterative process, gathering data about teaching and pupils’ learning, analysing the data, planning and revising their teaching. A particular theoretical framework was used as a guiding principle when designing and analysing learning. The goal was to identify aspects critical for learning the angle concept. It is demonstrated how the teachers were able to identify the critical aspects and change the teaching in a way that promoted pupils’ learning. What these critical aspects may entail and what teachers and researchers can learn from a Learning study is discussed.

Sammanfattning

I denna artikel beskrivs en modell för hur lärare kan samarbeta för att förbättra elevernas lärande. Modellen – Learning study – grundar sig på forskning om lärande och har en teoretisk grund. Det är en modell för samarbete mellan lärare i lärarlag, där lärare tillsammans utvecklar en gemensam kompetens kring frågor som: Hur kan man på bästa sätt lära ut något som är svårt? Vad gör skillnad mellan olika möjligheter att lära? I den aktuella studien undervisade man om vinklar och avsikten var att undersöka vad som var nödvändigt för att eleverna skulle lära sig begreppet. I artikeln beskrivs hur lärarna lyckades komma underfund med vad detta var samt hur de lyckades förändra sin undervisning så att fler elever lärde sig.

ULLA RUNESSON
Ulla Runesson is associate professor at the Department of Education, Göteborg University. Research interest: learning and teaching mathematics and the teaching profession in general. Ulla Runesson has been involved in international research projects studying and comparing features of mathematics classrooms in different countries. Several of her publications are based on data collected mainly in Hong Kong, but also in Australia. She is also engaged in the development of variation theory; an educational approach with practical applications.

Researching, teaching and the practice of mathematical modelling and applications

Christopher Haines, Peter Galbraith, Werner Blum, and Sanowar Khan (Eds.) (2007), Modelling ICTMA12: Education, Engineering and Economics. Chichester, UK: Horwood Publishing. ISBN-13: 978-1-904275-20-6

GLORIA STILLMAN
The University of Melbourne

THOMAS VILS PEDERSEN

Abstract

We describe how we used Brousseau’s theories of didactic  situations and didactic engineering as a framework for the development of an exam project in a first year mathematics course at a life science university. The main learning goals of the project were to (re)discover eigenvalues and eigenvectors partly by studying the asymptotic behaviour of matrix models for population growth and to understand the role eigenvalues play in such models. Moreover, the students would gain experience with mathematical experiments with the use of computers and with drawing conclusions from such experiments.

Sammendrag

Vi beskriver hvordan vi benyttede Brousseaus teorier om didaktiske situationer og didaktisk ingeniørarbejde som en ramme for udviklingen af et eksamensprojekt i et førsteårs matematikkursus på et biovidenskabeligt universitet. De vigtigste læringsmål i projektet var at (gen)opdage egenværdier og egenvektorer bl. a. ved at studere den asymptotiske opførsel af matrixmodeller for populationsvækst samt at forstå den rolle egenværdier spiller i sådanne modeller. Derudover skulle de studerende opnå erfaring med matematiske eksperimenter med brug af computer og med at drage konklusioner fra sådanne eksperimenter.

THOMAS VILS PEDERSEN
Thomas Vils Pedersen is associate professor of mathematics at the Faculty of Life Sciences, University of Copenhagen. He comes from a research background in pure mathematics, but his present research interests also include mathematics education, in particular course development and the teaching of mathematics as a service subject at university level.

Developing mathematics teaching and learning through research

Following the idea that started in the previous volume, of having a number per year with a special topic, it is our pleasure to present this number on the relationship between research in mathematics education and the development and improvement of teaching and learning practices. When we decided about the relevance of this topic for NOMAD, we considered, on the one hand, the importance of promoting actively one of the aims of NOMAD, namely, developing mathematics teaching and teacher education in theory and practice at all levels of the educational system in the Nordic region. On the other hand, we took into account our knowledge about the state of affairs in the mathematics education community in the region and wanted to open a publication space for the very many projects existing at the moment, where teachers and researchers collaborate in order to provide well-reflected and solid educational alternatives for the improvement of mathematics teaching and learning.

The issue of the relation between the development of practice and research has been well discussed in the international community of mathematics education. A central discussion related to this has to do with different views about the main aim of mathematics education research. Some people (e.g., Hart, 1998) argue that mathematics education research emerged from the interest of mathematics educators to intervene in practice in order to better it. Therefore, an essential feature of research in the field is a close connection with the work of teachers for devising teaching methods leading to effective mathematics learning in students. This driving aim has been at the heart of, for example, design-research (e.g., the work of Paul Cobb and collaborators in the USA), the realistic mathematics approach in the Netherlands, and the French didactical engineering approach. As the discipline has advanced, other people have argued that research, in the consolidation of a field of study, does not only have the intention of improving practice, but rather of explaining, understanding and theorizing it (e.g., Ernest, 1998). The improvement of practice, then, is a secondary aim subordinated to the main goal of explaining, understanding and theorizing. Thus, the collaboration between researchers and teachers is not a necessary condition for the realization of research. The formulation of theoretical propositions about mathematics teaching and learning is seen as belonging to the realm of the intangible world of research with little contact to the concrete world of practice. In the case of research for practice, the connection between research and improvement of practice is strong and evident. In the case of research about practice, the connection may not be immediate. Thus, the connection between the two is often formulated in terms of the divide between theory and practice, and many people have written about it. For a comprehensive and updated view on this discussion we recommend to see the results of the survey team ”The relation between research and practice in mathematics education” at http://www.icme10.dk/ .

We prefer to define mathematics education research as a field of study that has the double aim of explaining, understanding and theorizing mathematics teaching and learning, as well as improving it. From this perspective both aims are equally important. If this is the case, several forms of systematic inquiry are needed: the work of teachers-as-researchers (e.g., Zack, Mousley & Breen, 1997) doing inquiry in their own practice for changing it; and the collaboration between researchers and teachers in setting up environments for action-research (e.g., Atweh, 2004) and for collective inquiry (e.g., Jaworski, 2006). From projects of this nature important understandings about practice are generated, and innovative alternatives to mathematics education practices are effected.

The papers in this issue represent different approaches to and aspects of developing the practices of mathematics teaching through research and reflections. The paper by Ulla Runesson, A collective enquiry into critical aspects of teaching the concept of angles, reports a learning study involving three classes of forth and fifth graders. A learning study is a method for systematic collaboration between a researcher or a group of researchers and a group of teachers with a common intention of studying the teaching and learning of a particular concept or piece of knowledge. The method is closely related to the Japanese lesson studies, which is an institutionalized format for in-service teacher training in Japan. The learning study, however, puts more emphasis on studying the learning effect of variations in the teaching. Through a cyclic process of developing, testing and analyzing variations in the teaching of the concept of angles, the study group (the researcher and three teachers) developed knowledge about the pupils’ learning difficulties and about how to deal with them in the teaching situation.

In the second paper, Design of a didactic situation – mathematical experiments in linear algebra, by Thomas Vils Pedersen we are presented to the development of a course in linear algebra for life-science, university students. The course includes an exam project which is designed explicitly on the basis of the French Theory of didactical situations (developed by Guy Brousseau) and Didactical engineering (developed by Michelle Artigue), with the purpose of creating situations enabling the students to experiment with Leslie matrix population models and eventually (re)discover that the limit of iterating multiplication with the matrix can be expressed by the dominant eigenvalue and the corresponding eigenvector. The paper is rounded off with a section of reflections on the usefulness of the two theories from the perspective of a course designer. It is argued that, in the case at hand, the theories where helpful both on an operational level in designing the course and as a basis for understanding and analyzing the students’ learning difficulties.

In the third paper, Some aspects of web-courses in mathematics based on PC screen recorded video lectures, by Dag Lukkassen, Lars Erik Persson and Anna Sierpinska, an innovative and comprehensive design for web-based mathematics teaching is described and analyzed. Two master degree courses, one in complex analysis and one in partial differential equations, were designed and subsequently further developed since 2001. More than 200 students have taken the two web-based courses so far, and the format has now been institutionalized at Narvik University College as the only way of teaching these courses. Furthermore, the design has been used extensively in Ph.D. courses in applied mathematics. Students generally welcome the web-based format, which gives them a lot of flexibility in their study process. However, it is argued by the authors that the effects that the web-based format may have on the students’ learning need to be researched more closely. They call for collaboration between mathematicians teaching and designing web-based university courses and researchers in mathematics education in order to develop the quality of mathematics teaching using modern information technology.

The three papers offer examples of various forms of relationship between teaching practice and research, in developing possibilities for improving teachers’ practice. While the first paper addresses collaboration in the basic school, the other two papers show university mathematics teachers engage in design and reflection on their own design and implementation of innovative teaching strategies. Their contact with the tools of research has been a fundamental part in their advance.

New member of the editorial committee

During 2007 we have engaged in engaging new members in the editorial committee of NOMAD. The last new member is Guðný Helga Gunnarsdóttir, assistant professor at the Iceland University of Education. Guðný is also a member of the board of the Nordic graduate school in mathematics education. With Guðný we complete the enlargement of the editorial committee, which gives support and advice to our work as editors.

(Multi)culturality and diversity in mathematics education

The next thematic issue, planned for December 2008, will be addressing the challenges of (multi)culturality and diversity in mathematics education. In this issue we will be dealing with the challenges that an increase in diversity of students from different cultures and backgrounds posed to the teaching and learning of mathematics in educational institutions in the Nordic region. While for some decades ago it was able to consider the population in most of the region as homogeneous and mono-national, the increase in migration of peoples in the world has changed the composition of the student body. Students from different nationalities, ethnicities, languages and religions meet in mathematics classrooms. Such diversity has implication for the work of teachers, in particular for how individual students are met by existing, dominant teaching and learning practices which have been based on an assumption of homogeneity.
We invite submissions of research papers addressing these challenges, providing understanding and illuminating practice. The deadline for submission is the 15th of August 2008.

Thanks to authors and reviewers

Finally, we would like to thank the authors and reviewers of the papers that were published this year, and those that have been processed in our review system. We thank all of them.

Andreas Ryve
Anna Sierpinska
Anne Berit Fuglestad
Anu Palu
Barbro Grevholm
Bharath Sriraman
Carl Winsløw
Christer Bergsten
Claus Michelsen
Constantinos Christou
Dag Lukkassen
David Wagner
Elin Reikerås
Erkki Pehkonen
Eve Kikas
Frode Haara
Gail Fitzsimmons
Gunnar Gjone
Inge Henningsen
Jeppe Skott
Johan Lithner
Keiko Yasukawa
Lars Burman
Lars Erik Persson
Leila Pehkonen
Lil Engström
Markku Hannula
Mette Andresen
Mogens Niss
Nicholas Mousoulides
Ola Helenius
Olof Steinthorsdottir
Pasi Sahlberg
Pedro Gómez
Pekka Kupari
Raymond Bjuland
Simon Goodchild
Thomas Lingefjärd
Thomas Vils Pedersen
Tine Wedege
Tomas Bergqvist
Uffe Thomas Jankvist
Ulla Runesson
Viviane Durand-Guerrier

Morten Blomhøj and Paola Valero
Editors

References

Atweh, B. (2004). Understanding for changing and changing for understanding. Praxis between practice and theory through action research in mathematics education. In P. Valero & R. Zevenbergen (Eds.), Researching the socio-political dimensions of mathematics education: issues of power in theory and methodology (pp. 187–206). Boston: Kluwer Academic Publishers.

Ernest, P. (1998). A postmodern perspective on research in mathematic seducation. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: a search for identity (p. 71–86). Dordrecht: Kluwer.

Hart, K. (1998). Basic criteria for research in mathematics education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: a search for identity (p. 409–413). Dordrecht: Kluwer.

Jaworski, B. (2006). Theory and practice in mathematics teaching development: critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education (Special issue: relations between theory and practice in mathematics teacher education), 9 (2), 187–211.

Zack, V., Mousley, J. & Breen, C. (Eds.). (1997). Developing practice: teachers’ inquiry and educational change. Geelong: Centre for Studies in Mathematics, Science and Environmental Education, Deakin University.

DAG LUKKASSEN, LARS ERIK PERSSON and ANNA SIERPINSKA

Abstract

In this paper we will describe and discuss two graduate level web-based mathematics courses based on PC screen recorded video lectures. These courses have been developed by the first author of this paper and followed by master’s students at the Narvik University College since 2001. Comparison to other available courses is made. We argue that the technical solutions, our personal positive experiences and didactical ideas (e.g. concerning the physical presence of students and teachers in addition to instruction by video) in relation to our concept constitute an interesting field for mathematical education research

Sammendrag

I denne artikkelen beskrives og diskuteres to master-kurser som er basert på videoopptak av PC-skjerm. Disse kursene har blitt fulgt av sivilingeniørstudenter ved Høgskolen i Narvik siden 2001. Ett antall andre kurser, som finns tilgjengelig, blir diskutert og sammenliknes med vårt konsept. Kurserne evalueres ut i fra en matematikkdidaktisk synsvinkel. Vi mener at vårt konsept inneholder både tekniske løsninger, personliga positive erfarenheter og didaktiske ideer (for eksempel vedrørende fysisk nærvær av lærere/veiledere i tillegg til videoundervisning) som utgjør et interessant emne for matematikkdidaktisk forskning.

DAG LUKKASSEN
Dag Lukkassen is professor of mathematics at Narvik University College (HiN) and part-time professor in mathematics at the Department of Mathematics, Luleå University of Technology. He has been the chairman of the R&D-board and later Doctoral Education-board at HiN. Lukkassen leads the research group within Homogenization theory at the same institution and is a member of the editorial boards of 4 international scientific journals in pure mathematics, applied mathematics and mechanics.

LARS ERIK PERSSON
Lars-Erik Persson is chairprofessor of mathematics at the Department of Mathematics, Luleå University of Technology, but also part-time professor in mathematics at Narvik University College and Uppsala University. He has been supervisor of 32 PhD exams (three of them in mathematics education) and he has been president of the Swedish Mathematical Society. He is editor-in-chief of Journal of Function Spaces and Applications and a member of the editorial boards of 9 international scientific journals in pure mathematics, applied mathematics and engineering.

ANNA SIERPINSKA
Anna Sierpinska is professor of mathematics and mathematics education in the Department of Mathematics and Statistics of Concordia University in Montreal. Her research is related to epistemological, cognitive, affective and institutional aspects of mathematics learning. In the years 1990–1998, she was a member and then a vice-president of the Executive Committee of ICMI, and in 2001–2005 she served as the editor-in-chief of Educational Studies in Mathematics.

Programme for 2008

The board of NoGSME had a meeting in October and made plans for the activities in 2008. NoGSME will offer a course on Methodology in mathematics education given at the University of Agder in Norway (15 ECTS). The course Problemsolving will also be offered (10 ETCS) there and run parallel to the other course in the same course weeks. If you are interested please contact Elna.Svege@uia.no and let us know about it. The course-weeks will be weeks 3, 7, 10, and 15/16. We will also offer a shorter course for doctoral students in Helsinki, 11–13 April on conceptions of mathematics organised by Erkki Pehkonen and with Fulvia Furinghetti as one of the course teachers. More information will be sent out by email.

The ninth seminar for supervisors will be in Tallin and Helsinki on April 10–11 (partly overlapping with the short course in Helsinki). In addition there will be a workshop for doctoral students and supervisors during the Norma08-conference, 22–25 April in Copenhagen. A seminar especially for the Baltic supervisors will be organised in connection to the Baltic conference, around May 16 in Vilnius, Lithuania.

An international seminar for supervisors together with our international contacts will take place in October, 8–11 outside Copenhagen with the theme: Internationalism versus localism as a challenge in our field.

As there are so many different activities going on during summer 2008, for example the ICME11-conference, we have decided to transform the traditional summer school into a winter school for doctoral students in week 48 or week 49. It is planned to take place in Sigtuna near Arlanda airport in Sweden. Please follow further information about the programme in our email letters and on the web at www.NoGSME.no

Report from the evaluation of NoGSME by NordForsk

The halfway evaluation of NoGSME done by the funders NordForsk has been reported back to us recently. The evaluation of NoGSME is very positive and we have been promised to get the funding for the final year 2008 according to our application, one million NOK. Here are some quotations of what the expert panel writes:

When comparing scientific quality and relevance in relation to programme guidelines and NoGSME’s initial plan for objectives and strategy with the actual report, presented after a few years of cooperative work, the impression is overwhelming: It shows the success of strong, innovative, intensive and effective collaborative endeavour in creating a well functioning Nordic research community in mathematics education able to successfully implement various initiatives and working plans for capacity building in mathematics education research.

In another section the evaluation talks about what NoGSME has achieved in internationalisation:

International orientation
The International orientation is very strong and of high quality: When reading the names of international lecturers and seminar leaders, they make up a list of the most famous who’s who on the world scene of mathematics education research and guidance around the world: From Paul Cobb (USA) – the recent winner of the Freudenthal-Award and leading cognitive scientist and mathematics educator starting social constructivism, to Hyman Bass (USA) and Michele Artigue (F), the former and the actual president of ICMI and both active mathematicians and mathematics educators, to philosophers and historians like Paul Ernest (UK), and Frank Lester (USA), teacher education and classroom researchers like Kathleen Hart, Simon Goodchild, Barbara Jaworski (all UK) to well known social cognitivists and epistemologists and critical researchers like Anna Sfard, Abraham Arcavi and Uri Leron (IS), Etienne Wenger (USA) and Marcello Borba (BR) and others. The support of such a fine selection of the international community is of very high quality. All of them named here as lecturer, seminar or workshop leaders or just consultant and reviewer are from the group of the world’s best researchers in mathematics education, and were mostly paired with much appreciated colleagues from Nordic countries, a very good and often complementary mix in collaboration and view, all very well known and highly appreciated, leading people in ICME or PME and in their own home country.

The evaluating panel is also expressing some concern about the future of NoGSME after the funding period and here we all have a challenge to work with in coming years:

Plans for continuing the co-operation/activities after the end of the funding period
It was emphasized and clearly outlined that there is still a need for training researchers in the field of mathematics education research, and the success of the project in its obvious impact on research training within is taken as an argument to continue the successful collaborative work in the Nordic community at least. Concrete plans for continuing the co-operation/activities after the end of the funding period were not quite clear despite of the desire to be able to continue that which has been so successfully started. Some concrete measures mentioned were the idea to create a Nordic society to continue the work of the Board of NoGSME and to host the NOMAD journal and the regular Norma conferences. A planning group has been elected to investigate national societies, and will propose a Nordic umbrella organisation. A willingness to secure the openness of all activities to Nordic participation in the future was also expressed. Finally, an application had been put forward for a Nordic Master programme in mathematics education. It remained clear that these measures do not solve all and there was concern for how to secure the incentive for cooperation in the future without the Nordic frame grant.

Four new dissertations during autumn 2007

Eva Taflin at Dalarna University College defended her thesis at Umeå University in June. The title is Matematikproblem i skolan: för att skapa lärande (Mathematical problems in school: in order to create learning). The purpose of her work was to define and explore what mathematical problem solving entails. The first part of the dissertation explores a sketch of what mathematical problem solving can offer in the teaching and learning processes. The second part of her work presents and analyses two so called rich problems. With rich problem she means problems which are especially constructed for mathematics education in a school context. Rich problems enable pupils with different capacity for mathematics to work with the same problem and solve it with different mathematical ideas. A set of criteria for rich problems is presented. The methods used are video- and audio-recordings, stimulated recall with pupils and teachers, interviews and pupils’ drawings. In the analysis indications are given of where concepts, procedures, conventions, strategies and formulae appear in the problem solving process. Examples from pupils’ and teachers’ work with problems are exposed.

Camilla Björklund at Åbo Akademi University has defended her thesis with the title Critical conditions for learning – toddlers encountering mathematics. The aim of her study is to discern how toddlers experience and learn mathematics in a day-care environment. Twenty-three children were observed and video-recorded during everyday activities. The methodology aims to describe and interpret human actions in natural settings. The analysis was based on phenomenography in order to discover how the children come to understand the different aspects of mathematics they encounter. The results show that toddlers encounter various mathematical concepts, similarities and differences and the relationship between part and whole. For a certain type of learning to occur, some critical conditions must exist. They are variation, simultaneity, reasonableness and fixed points. Adults working with young children play an important role for children’s experiences and opportunities to explore mathematical concepts and phenomena.

Constanta Olteanu took part in the graduate school in pedagogical work in Umeå and defended her thesis at Kristianstad University in October. The title is "Vad skulle x kunna vara?" Andragradsekvation och andragradsfunktion som objekt för lärande ("What could x be?” Second degree equation and second degree function as objects of learning). The aim of the thesis is to analyse, understand and explain the relation between the handled and learnt content, which are second degree equations and quadratic functions. The study includes two teachers and 45 students in two different classes. The data consists of video-recordings of lessons, individual sessions, interviews and the teachers’ and researcher’s review of individual sessions. The students’ tests were also an important part of the data collection. Variation theory was used for the analysis. The results imply that there is convergent variation leading the students to improve their learning. The variation leads students to make generalisations in each object of learning and between those objects (equations and functions).

Lisser Rye Ejersbo is situated at Denmark Pedagogical University in Copenhagen and her thesis, defended in December, has the title Design and redesign of an in-service course: the interplay of theory and practice in learning to teach mathematics with open problems. The question she asks is to what extent a meta-didactical transposition for mathematics educational research concepts can be incorporated into successive stages of redesigned courses and how effective are these redesigns measured by the participating teachers’ reactions during the course. Four theoretical concepts were selected to assist in the redesign of the course: the interactive flowchart, the epistemological triangle, the virtual monologue and the socio-mathematical norms. Using design research methods she has gone through several cycles of redesign of the course, each time sharpening her tools for working with teachers on the skills of communication and reflection. The careful interweaving of theory and practice in the study should be of value for both practitioners and researchers.
With the report of these four theses we have presented in all ten dissertations in mathematics education during 2007 compared to twenty-one in 2006, which was an exceptional year in mathematics education. There might be more, but these are the ones we have been told about. Please let us know if there are more to inform about. Are ten theses what we can expect in 2008 or will the normal level be higher in the future?

Nordic collaboration in math education after NoGSME-period

During 2008 steps must be taken to prepare for coming work when the funding of NoGSME is over in mid-2009. There are several opportunities to get funding from NordForsk, such as for Nordic networks, for Nordic doctoral courses, for competence development of supervisors and mobility stipends. But in each case an application must be written so we will have to work more for comparable funding to what we have had over recent years. Preparations are being made for the creation of a Nordic ‘umbrella’ society for research in mathematics education. More information about that will be available in 2008.

Barbro Grevholm, Director of NoGSME
University of Agder, Norway

The International Commission on Mathematical Instruction: a centenary of history and a future to construct

Between the 5 th and the 9 th of March 2008 there was held a very special symposium in Rome: Reflecting and shaping the world of mathematics education. The first century of the International Commission on Mathematical Instruction (1908–2008). Organised in collaboration between ICMI and an Italian committee with Ferdinando Arzarello on the lead, the symposium took place at the Accademia Nazionale dei Lincei, the very same place where hundred years ago a selected group of members of the International Mathematical Union decided to engage in the creation of ICMI.
The symposium gathered around 200 invited participants from 43 countries around the world. It intended to present a historical look at the formation and evolution of ICMI during the first century of its existence, and reflect about the status of mathematics education research and envision its future. The symposium had a series of activities: eight plenary addresses with a reaction, eight parallel short talks, and five working groups with a series of short papers as a basis for discussion. For more details about the symposium, program and papers consult http://www.unige.ch/math/EnsMath/Rome2008/.
Participating in this event was an enriching opportunity and we would like to share with nomad readers some of the main points that, in our perspective, were prominent issues of this important symposium. We will shortly touch upon what is ICMI, some of the characteristics in its first hundred years, the role of the Nordic countries in ICMI, and some of the challenges for the future.

What is ICMI?

ICMI is an official commission of the International Mathematical Union, IMU. ICMI was first established at the International Congress of Mathematicians held in Rome in 1908, on the suggestion of David Eugene Smith, American mathematician and historian of mathematics. The first President of ICMI was Felix Klein, and Henri Fehr was the first Secretary-General.
ICMI members are both the countries belonging to IMU, and some countries which do not belong to IMU but who show special national interest in the organisation of mathematics education. Member countries have national ICMI commissions with at least an appointed national representative.
ICMI has played an important role in being an international organisation for promoting initiatives for the improvement of mathematics education in its member countries, through the maintenance of a series of activities (for further details about the following activities consult http://www.mathunion.org/ICMI ):
– L’ Enseignement Mathématique, the official organ of ICMI.
– ICMI bulletin and the new electronic ICMI News.
– ICMI Awards, through the Felix Klein Award which honours a lifetime achievement, and the Hans Freudenthal Award which recognises a major cumulative program of research.
– ICMI Studies aiming at fostering efforts around the world to improve the quality of mathematics teaching and learning, though the growth, synthesis, and dissemination of new knowledge (research) and of resources for instruction (curricular materials, pedagogical methods, technology, etc.).
– The International Congress on Mathematics Education (ICME), a quadrennial event gathering the international community of mathematics education around the world.
– ICMI five affiliated permanent study groups: The International Study Group on the Relations between the History and Pedagogy of Mathematics (1976), The International Group for the Psychology of Mathematics Education (1976), The International Organization of Women and Mathematics Education (1987), The World Federation of National Mathematics Competitions (1994), and The International Study Group for Mathematical Modelling and Applications (2003).
– Regional conferences series such as the AFRICME and SEAME.

Some highlights on ICMI’s history

At the symposium there was a series of plenary lectures aiming at presenting the history of ICMI from different perspectives and based on original documentation. Hyman Bass (the former president of ICMI) presented an overview of the century of ICMI. The first fifty years were characterised by the initiatives of the founding fathers, a reduced group of mathematicians who had a commitment to the improvement of mathematical instruction. At the beginning of the 20 th century schooling was only provided for a selected social and economic elite, and mathematics education was mainly discussed in terms of the content and organisation of the curriculum relevant for the education of that minority. A main figure in this period was Felix Klein who was a strong advocator for a renewing of mathematics curricula with more emphasis on the process of doing mathematics in the classroom. Bass named this period the ”Klein era”. In his plenary talk, Gert Schubring provided a detailed account of this period, of the types of collaboration among the first interested actors and of the conflicts emerging as a result of participants’ engagement in World War I and II, leading to the dissolution of the commission in between wars.
The following 50 years staring from the rebirth of ICMI in 1952, began thanks to the initiative of Hans Freudenthal (president of ICMI 1967–1971). With the expansion of education to the masses and the influence of Freudenthal’s thinking, the didactical aspects of the teaching and learning of mathematics started to be central concerns of ICMI. The beginning of the ”Freudenthal era”, as named by Bass, was heavily influenced by the New Math reform of which Freudenthal was a very strong opponent. In their plenary talk, Fulvia Furinghetti and Martha Menghini presented details of the renaissance in the Freudenthal era, highlighting the important new organisational initiatives in this period such as the formation of the international journal Educational Studies in Mathematics and the launching of the ICMEs, the affiliated study groups (HPM and PME in 1976) and regional conferences.
Through out the history of ICMI the relationship to IMU on the organisational level and in general the relations to the mathematics community have played a significant role, sometimes of positive synergy and some other times of conflict. Until recently, the IMU general assembly elected the president and executive committee of ICMI. The presidents in the past have all been well respected male mathematicians and ICMI has never had a president with mathematics education as his main research area. However, at the election in 2007, Michèle Artigue became president of ICMI. With the election of a woman mathematics educator these traditions had been changed and a new era – the ”Artigue era”, in Bass’ words – is beginning. Moreover, due to the persistent work of Hyman Bass during his presidency, the election regulations were revised so that ICMI’s general assembly is now the legitimate voting body for the election of all ICMI officers.
In her closing address Michèle Artigue spoke about ICMI as an interface between mathematics education and mathematics or mathematics educators and mathematicians and expressed her visions for a fruitful interplay between mathematics and research in mathematics education for the mutual benefit. Mathematics education research has actually a much more to offer to the mathematics community, other than providing conditions for students to appreciate mathematics, developing adequate competencies for further studies in mathematics, and supporting the development of mathematics teaching at university level. Mathematics education research is contributing to the pool of mathematical knowledge with research on the nature and functioning of mathematics in education and in society.

The Nordic participation in ICMI

Nordic mathematics education research was well represented at the conference both in terms of number of participants (Denmark 5, Finland 1, Norway 2 and Sweden 3) and in terms of involvement in the scientific program. Mogens Niss was member of the International Program Committee for the symposium and he gave a plenary lecture on the balance between application and modeling, and ’pure’ mathematics in the teaching and learning of mathematics. Barbro Grevholm was co-chairing the working group on the professional formation of teachers together with Deborah Ball. Morten Blomhøj contributed to a closing plenary panel on challenges for ICMI in the future. All the other Nordic participants presented papers in their working groups. The papers can be found at the conference web-site.
In the account for and discussion of the history of ICMI, Nordic researchers where mentioned prominently in two different contexts. Bent Christiansen (1921–1996), the first professor in mathematics education in Denmark and in the Nordic countries, was especially mentioned for his commitment to mathematics education research and for his work as a vice-president for ICMI for more than ten years in the period 1976–1988. Also Mogens Niss was acknowledged for his achievements during his two periods as the Secretary-General for ICMI, in particular for having being a key figure in the publication of the ICMI-studies reports as a collection at Springer Publishers.
The socio-political dimension in mathematics education research was pinpointed by Jeremy Kilpatrick in his plenary talk as one the most important recent developments in our field of research. Stieg Mellin-Olsen (1939–1995), who was one of the two founding editors of nomad (together with Göran Emanuelsson) was recognised as a pioneer in the area, and the work of Ole Skovsmose on critical mathematics education was pointed out to be the break through for the recognition of the socio-political dimension in mathematics education research.

The challenges towards the future of ICMI

In the closing panel on challenges for ICMI in the future, a number of pending problems for mathematics education research were mentioned and it was briefly discussed what ICMI could do in relation to these challenges. These are some of the major challenges discussed:
– the need of keeping the meta-reflections on the nature and status of our research field alive,
– the need of developing and maintaining good relations with mathematics,
– the need of improving teacher education and teachers’ professional development,
– the need of strengthening the interplay between research and the development of mathematics teaching practices, and
– the need of working actively for reaching mathematics education for all.
We see that these global challenges are also challenges for the Nordic region. A testimony to that is the fact that, in one way or another, these challenges have been addressed either explicitly in our editorial comments, in papers and in the special issues of nomad in the last three years. The issue of mathematics education for all opens for the need of thinking about new research approaches and the concern for new topics and areas of research. We would like to remind our readers about the coming thematic issue for the present volume, which will address exactly the challenges of (multi)culturality and diversity in mathematics education, an important aspect of mathematics education for all. We encourage
submissions to this issue and remind you the deadline of August 15 th.

About this number

In this issue we publish three papers, all reporting on empirical inve-stigations but from different perspective and for different groups of students.
The paper Word problems in upper secondary algebra in Sweden over the years 1960–2000 by Teresia Jakobsson-Åhl reports results from a study on school algebra at the upper secondary level. The study considered changes in the algebraic content as it was presented in mathematics textbooks in the second half of the twentieth century in Sweden. The paper describes the changes of word problems in terms of the way they are used for developing algebraic skills. It is shown in what ways the place of the problems, the context and the use of mathematical models changed in the period in question. The use of word problems is related to the curricular documents of the time and to different views on school algebra.
In her paper Växelverkan mellan intuitiva idéer och formella resonemang Kerstin Pettersson analyses the work of a group of first year university students’ formulating and proving a conjecture concerning the relation between the n-derivative of a function being without zeros and the largest possible number of zeros for the function itself. The students were prompt to use an induction proof to prove their conjecture and this seems to have facilitated a rich interplay in the students’ work between intuitive ideas and formal reasoning. By means of intentional analysis of a nearly two-hour video recording of the students’ work, which is documented in the paper through rich transcripts, Kerstin Pettersson shows how various forms of interplay between intuitive understanding and formal reasoning serve as resources in the students’ problem solving process. The learning potentials of the interplay between intuition and formal reasoning are analysed in relation to the particular mathematical content. As a case study the paper illustrates the importance of intuition for the learning of mathematics and it rise interesting questions concerning how to facilitate the students’ interplay between intuitive ideas and formal reasoning when teaching mathematics.
Joakim Samuelsson, in his paper Classroom settings, self-regulated learning skills and grades in mathematics, addresses the issue of the plausible influences on students’ self-regulated learning skills. This study concentrates on the contextual aspects of the educational preconditions. The group climate seems to be the most influential factor to students’ self-regulated learning skills in mathematics. A supportive climate is related to the view of mathematics as something important, while a non-supportive climate is related to difficulties in mathematics. Students with difficulties in mathematics are affected by classroom settings that they perceive as demanding in terms of objectives, and in teacher centred instructions. To some students, high demands, distinct information and invitations to participate can result in positive relationships with mathematics. However, the same conditions can create difficulties in
mathematics among other students.

The editors

ANTTI VIHOLAINEN

Abstract

In this study, formal and informal reasoning skills of 146 Finnish subject-teacher students in mathematics are investigated. The students participated in a test in which they were asked to argue two claims concerning derivative both informally and formally. The results show that the success in the formal tasks and the success in the informal tasks were dependent. However, there were several students who did well in the formal tasks despite succeeding poorly in the informal tasks. The success both in the formal tasks and in the informal tasks was dependent also on the amount of passed studies in mathematics and on the success in these studies. Moreover, these factors could have a stronger effect on the formal than on the informal reasoning skills.

Sammanfattning

I denna studie undersöktes den formella och informella argumentationsfärdigheten hos 146 finska blivande ämneslärare i matematik. Studenterna deltog i ett test, där de skulle argumentera både formellt och informellt för två påståenden gällande derivatan. Resultaten visar, att framgången i formella uppgifter och i informella uppgifter beror av varandra. Studenterna kunde emellertid klara sig bra i formella uppgifter utan att ha likadan framgång i informella.
Resultaten i båda uppgiftstyperna var beroende både av mängden av och framgången i tidigare matematikstudier. Dessutom kan de här faktorerna ha haft en starkare effekt på de formella argumentationsfärdigheterna.

ANTTI VIHOLAINEN
Antti Viholainen is a PhD candidate at University of Jyväskylä, Finland. Since the year 2003, he has worked as a researcher in the area of mathematics education. His main interests are the nature of mathematical knowledge, informal and formal reasoning and the coherence of mental structures. In his studies, he has especially concentrated on the informal and formal representations regarding the concept of derivative.

TORULF PALM

Abstract

Mathematical communication, oral and written, is generally regarded as an important aspect of mathematics and mathematics education. This implies that oral mathematical communication also should play a part in various kinds of assessments. But oral assessments of subject matter knowledge or communication abilities, in education and elsewhere, often display reliability problems, which render difficulties with their use. In mathematics education, research about the reliability of oral assessments is comparably uncommon and this lack of research is particularly striking when it comes to the assessment of mathematical communication abilities. This study analyses the interrater reliability of the assessment of oral mathematical communication in a Swedish national test for upper secondary level. The results show that the assessment does suffer from interrater reliability problems. In addition, the difficulties to assess this construct reliably do not seem to mainly come from the communication aspect in itself, but from insufficiencies in the model employed to assess the construct.

Sammanfattning

Matematisk kommunikation, muntlig och skriftlig, ses i allmänhet som en viktig aspekt av matematik och matematikutbildning. Detta medför att muntlig matematisk kommunikation också bör bedömas i olika sorters prov. I både utbildningssammanhang och inom andra områden är dock muntliga prov av ämneskunskap eller kommunikationsförmåga behäftade med reliabilitetsproblem vilket orsaker svårigheter med dess användning. När det gäller matematikutbildning är reliabilitets­studier om muntliga prov dock relativt ovanliga, och detta gäller speciellt vid bedömning av matematisk kommunikationsförmåga. Denna studie analyserar interbedömarreliabiliteten för bedömningen av muntlig matematisk kommunikation i ett svenskt nationellt matematikprov för gymnasial nivå. Resultaten visar att provet var behäftat med reliabilitets­problem. Det verkar dock inte vara kommunikationsaspekten i sig själv som gör att denna förmåga var svår att bedöma reliabelt utan
otillräckligheter i den då använda bedömningsmodellen.

TORULF PALM
Torulf Palm, PhD, is a member of Umeå Mathematics Education Research Centre (UMERC) at Umeå University, Sweden and works at the Department of Mathematics, Technology and Science Education. His special research interests are assessment, mathematical reasoning, mathematical modelling and the authenticity of word problems.

Strengthening NOMAD as a means of research communication in the Nordic region

One of the aims of NOMAD is to contribute to the consolidation of a community of research in mathematics education in the Nordic region. As editors of the journal we have not only been engaged in estabilising the publication in terms of its regularity, its contents and quality, and its visibility among researchers, students and practitioners in mathematics education. We have also tried to come closer to many of those people who can contribute with submitting papers for publication in the journal. In this editorial we have decided to highlight some important points that emerged from a special workshop session run at Norma08. We hope that some of these comments can be of help for those readers who would like to send a contribution to the journal. We also present the contents of this issue with some of the reflections that the papers have inspired in us.

NOMAD in Norma08

The Nordic conference on mathematics education research (Norma) was held in Copenhagen, from the 21st to the 25th of April 2008. This conference was a meeting space for discussion about current research in mathematics education in the region, but also for connection with other research groups in countries such as France, Hong Kong, the Netherlands and Spain. The conference was organised around four key topics, each one of which was explored by a plenary talk and by a series of papers, short presentations and discussion groups:

• Didactical design in mathematics education: This includes all types of ’controlled intervention’ research into the processes of planning, delivering and evaluating concrete mathematics education. It also includes the problem of reproducibility of results from such interventions.
• Education and identity of mathematics teachers: This includes research into teacher education programmes, teacher educators’ practices, and the relation between teacher education and the formation of teachers’ professional identity and competence as mathematics teachers.
• Technology in mathematics education: This includes studies of the rationales, modes and effects of technology use in mathematics teaching and learning at all levels.
• Mathematics for all: why?, what? and when?: This includes studies of mathematical literacy, rationales for ’general’ mathematics education, and the challenges of socio-cultural diversity in mathematics education.

A full version of the programme and most contributions are posted in the conference website and will also be available soon in printed proceedings. For further details about this event we suggest consulting the homepage: http://www.dpu.dk/site.aspx?p=10797.

One special activity at Norma08 was a workshop entitled Publishing in Nordic Studies in Mathematics Education (NOMAD). We proposed this workshop as an activity that could inform Norma08 participants about the journal, its aims and purposes, and about the review and publication process. We considered that many of the contributions to Norma could be a very good basis for the elaboration of a journal paper presenting research results of current projects being carried out in the region.

The workshop had a good attendance. Around twenty-five participants – mostly novice researchers engaged in doctoral studies or researchers that have defended their doctoral thesis recently, but also some more experienced researchers – were present in the session. At the workshop we discussed a few important points that we also would like to address here, since we consider them to be useful for the NOMAD audience and, in particular, for potential authors.

What types of papers are appropriate for NOMAD?
NOMAD publishes research papers. Many people interpret this statement focusing on the report of empirical studies. However, we do not limit our understanding of ’research’ to empirical studies. For NOMAD the term also encompasses theoretical research where notions are developed, literature review papers where an overview of an topic of relevance for Nordic research is explored through a critically organised discussion of existing research results in a particular area, methodological papers exploring issues of the research process, and research-based essays discussing critical issues in a particular topic. Independently of the type of research, texts will have to follow standard criteria of quality for communicating academic work.

Does a paper need to be original?
In many journals there is a demand that submitted papers have not been published in any form before, and that the material presented, therefore, needs to be original. In NOMAD, we interpret this demand on originality in the following terms: the paper submitted should not be published previously in its present or similar form. However, we think that it is consistent with one of the purposes of the journal – namely, contributing to strengthening research in mathematics education in the region – to accept that the papers are substantial elaborations of previous conference papers, and that previous background material is acknowledged.

A slightly different aspect of the discussion about originality is whether the paper presents new findings or insights to the regional and/or international field of mathematics education. In this respect we are modest and demand authors to relate their paper to existing research, emphasise what they consider to be new, and declare how they think their research contributes to existing knowledge in the region.

In what language do papers need to be written?
Although NOMAD’s policy had been to publish papers in Danish, Norwegian, Swedish and English, the dominant practice in the recent years has been the preference of English as a language of publication. In fact, few papers have been published in one of the three Nordic languages. We encourage authors to submit papers in the language that they feel most comfortable with, to express efficiently their ideas.

We can see that this activity in fact had an impact: We have already received two submissions based on Norma08 papers. In general, the situation concerning the number of submissions for publication has improved and we look forwards to keep on receiving new interesting papers.

About this number

In this issue we present two papers that are addressing research questions related to the reform of mathematics teaching and the assessment of oral mathematics communication at the upper secondary level.

Uffe Jankvist’s paper Matematikopfattelser hos 2g’ere: fokus på de ’tre aspekter’ is addressing problems with implementing curriculum reforms in the Danish gymnasium (a three year programme aiming at preparing the students for further theoretical studies). The Danish high school has, since the late eighties, strived to broaden the view on and the beliefs about mathematics communicated through the mathematics teaching. These changes have been included in the curriculum through the formulation of three essential aspects of mathematics that should be communicated and discussed with the students as an integrated part of the teaching of mathematics. The three aspects are the historical evolution of mathematics, the application of mathematics in society, and the inner structures and nature of mathematics as a scientific discipline. In the paper Uffe Jankvist relates the three aspects and their role and placement in the curriculum, to the Danish Competence Project and in the newly launched mathematics programme from 2007. The paper also reports results from a qualitative investigation of the students’ compliance with the intentions of the three aspects in a particular 2.g class. These results are compared with results from a similar investigation from 1980, before the first reform introducing the three aspects. The author concludes that the students’ beliefs about mathematics in the relation to the three aspects are very similar in the two samples and that the implementation of the three aspects in the matter taught and learned still have a long way to go. The paper ends with a discussion of the possible reasons for the difficulties with the realisation of the curriculum intentions connected to the three aspects. The treatment of the three aspects in the mathematics textbooks, the teachers’ educational background and beliefs about the importance of the three aspects, and the role of the aspects in the final written and oral examinations are pinpointed as crucial for the advancement of the implementation process.

The current regulations from 2007 for mathematics in the Danish gymnasium allow a form of oral examinations where the individual student is examined for 25 minutes with point of departure in a report written by the student on a project or a theme that he or she has made as an integrated part of his or her mathematics classes. In our opinion, this form of assessment provides a unique opportunity for assessing more advanced competencies and reflections such as those connected to the three aspects investigated by Uffe Jankvist. One of the recent changes in examinations allowed teachers the possibility to choose between this new particular form of examination and an oral examination with randomly chosen questions referring to well defined elements of the mathematical curriculum, which has been a dominant practice in Denmark. So far, less than two percent of the oral examinations have had the new form. Many teachers argue that this form of examination is too demanding for many students, and that it requires quite dramatic changes in the normal teaching to prepare the students for such an examination. A recent evaluation of the mathematics B level curriculum in the 2007 reform has analysed the situation and recommended that there should only be one form of oral examination including both questions of the traditional form and questions that relates to projects covering the three aspects (Danmarks Evalueringsinstitut, 2008, p. 35). This story illustrates very clearly the crucial importance of assessment in the implementation of reforms in mathematics.

This connects to the theme of the second paper in this issue, Interrater reliability in a national assessment of oral mathematical communication where Torulf Palm reports on an investigation of the reliability of the assessment of oral mathematical communication in a Swedish national test for the upper secondary level. Attempts to assess oral communication in mathematics are often met with scepticism concerning their reliability. Therefore, oral communication competence is often not assessed formally in mathematics programmes. A first concern assessing oral mathematical communication is the extent to which it is possible for different assessors (raters) to agree on the assessment of an oral mathematical performance – this is called the interrater reliability. Palm’s investigation shows that the assessment of oral mathematical communication in term of the two constructs ”Line of thought” and ”Mathematical terminology” in the Swedish national test is suffering from very low interrater reliability, even though the grading system only operates with four grade levels. This very distinctive result is argued by the author to have more to do deficits of the assessment model employed that with the communication of mathematics in itself. The main deficits seem to be related to the ambiguity of the two constructs assessed and the practical organisation of the assessment. The duration of the students oral performance assessed (only 5 minutes), and the limited resources for educating the teachers for the intended assessment are pinpointed as two important organisational constrains.

Together the two papers indicate a need for developing and researching the reliability and validity of oral assessment of high level mathematical competencies such as mathematical modelling and communication, as well as of higher order reflections connected to the three aspects included in the curriculum for the Danish gymnasium.
The third paper in this issue addresses teacher students’ informal and formal reasoning. The paper is based on a comprehensive empirical material consisting of the test results of 146 Finnish subject-teacher students reacting to informal and formal arguments concerning the derivative concept. The results indicate that the students’ formal and informal reasoning are dependent, and that the students’ ability to argue informally about the derivative does not follow automatically from being able to argue formally with this concept. The amount of mathematics training is shown to have a stronger effect on the teacher students’ formal reasoning than on their informal reasoning. Since, for coming teachers it is very important to be able to support and challenge the students’ formal reasoning as well as their informal reasoning, these findings emphasise the need for addressing students’ informal reasoning explicitly in teacher education and maybe especially to include the nature of and the relation between these two forms of reasoning in the education of coming mathematics teachers.

The editors

References
Danmarks Evalueringsinstitut (2008). Matematik B på hhx og stx. Fag­eva­lue­ringer 2007. København: Danmarks Evalueringsinstitut.

UFFE THOMAS JANKVIST

Sammendrag

Med udgangspunkt i de såkaldte ‘tre aspekter‘ fra den danske gymnasiale bekendt­gørelse for matematik anno 1987 analyserer artiklen 2g’eres (17–18 årige) opfattelser af faget matematik. Mere præcist er der tale om faget matematiks historiske udvikling, dets anvendelser i samfundet og dets indre strukturer. Det undersøges i artiklen hvor de ‘tre aspekter‘ oprindelig stammer fra og hvordan de går igen i såvel KOM-rapporten fra 2002 som i 2007-bekendtgørelsen. Endvidere undersøges det med udgangspunkt i en enkelt adspurgt 2g-klasse, hvordan denne klasses elever synes at indfri den ny bekendtgørelses mål svarende til de ‘tre aspekter’. Resultaterne af denne undersøgelse sammenlignes med en ældre og i nogen grad tilsvarende undersøgelse fra 1980 og forskelle og ligheder i resultaterne diskuteres.

Abstract

Based on the so-called ‘three aspects‘ from the 1987-regulations for the Danish upper secondary mathematics programme this article discusses second-year upper secondary students’ beliefs about the nature of mathematics. That is to say, it investigates the students’ beliefs concerning the historical evolution of mathematics, the application of mathematics in society, and the inner structures of mathematics as a scientific discipline. Firstly, the article examines the origin of the ‘three aspects‘ as well as the role they play in both the KOM-project of 2002 and the new regulations for the Danish upper secondary mathematics programme of 2007. Secondly, it discusses how the students in a concrete second-year class of upper secondary level seem to fulfil the goals of the ‘three aspects’. Thirdly, the results of this study are compared to a similar study from 1980 and differences and similarities between the two are discussed. It is concluded that there still is room for improvement concerning the fulfilment of the three aspects, and that the students’ beliefs in the 1980-study and in the 2007-study are very similar. In the end, the article speculates upon why the ‘three aspects’ do not seem to have had a larger impact on the mathematics teaching on upper secondary level when they have been in the regulations for twenty years now.

UFFE THOMAS JANKVIST
Uffe Thomas Jankvist is a third year Ph.D. student at Roskilde University. His Ph.D. study concerns the use of history of mathematics in mathematics education. As part of his Ph.D., he has designed and implemented two historical teaching modules in an upper secondary mathematics class with the purpose of seeing how the intentions of the KOM-project and the new Danish regulations for the upper secondary mathematics programme concerning the inclusion of history may be fulfilled. It is this work that has led him to consider students’ beliefs about the nature of mathematics.

The ninth seminar for supervisors of doctoral students took place in Tallin in April and the invited international guest was professor Gabriele Kaiser, who is the editor in chief of Zentralblatt für Didaktik der Mathematik. She talked about the publishing policies of international journals in mathematics education and the scientific profile of different journals. The participating supervisors from six countries worked in groups with copies of journals, descriptions of aims and goals for different journals and discussed issues in supervision related to choice of journal for papers. Questions were discussed, such as how insight into publishing policy can serve as support for doctoral students, and what kind of communication with the editors would be helpful. Should they be contacted with a pre-submission or rather after the reviews have been received? The review process was discussed and Gabriele Kaiser spoke about different criteria and principles used in the review process and by editors in their communication with authors. How early in the studies should doctoral students be encouraged to send papers to journals? The characteristics of a number of important journals in mathematics education were listed and discussed.

A NoGSME workshop took place on April 22 during Norma08. The theme of the workshop was The use of ICT in mathematics education – neither salvation nor catastrophe? What can we learn from research and what are our conclusions? Three short lectures focusing on research results served as an introduction to the work in smaller groups. Mette Andresen talked about Use of ICT in School mathematics, Per-Eskil Persson about Use of graphic calculators in school mathematics, and Christer Bergsten about Teacher education and use of ICT in mathematics learning. Intense discussions followed in seven groups led by Mette Andresen, Paul Drijvers, Per-Eskil Persson, Christer Bergsten, Guðný Gunnarsdóttir, Mary Billington, Ingvald Erfjord, and Hildegunn Espeland. The presentations and the reports from groups will be made available in the proceedings of Norma08. Some of the questions raised in the groups were:

• Pre-school children use computers today without any problems. How will this influence the learning of mathematics in schools?
• Development work is going on where mathematics classes in upper secondary school only use one tool during lessons: the computer. What could be the consequences of that?
• Teachers need competence development in the area of use of ICT-tools in teaching and learning mathematics. How could this issue be resolved? And what kind of competence is needed?
• What are the characteristics of use of ICT in teaching and learning mathematics at university level?
• What are the most strengthening features and most threatening features in use of ICT in school mathematics learning?

Doctoral courses in Helsinki and Malmö

In April Erkki Pehkonen at University of Helsinki organised a course on Conceptions in mathematics. Fulvia Furinghetti from University of Genova acted as one of the teachers during the course. For the autumn 2008 a doctoral course in Malmö University College has been announced with the title Mathematical literacies: construction of concepts and the implication for design of research and education. The course teachers will be Tine Wedege, Eva Jablonka and Ole Skovsmose. For spring 2009 Jo Boaler, who is currently a Marie Curie professor at University of Sussex, is preparing a course which will be offered to the students in NoGSME. It will be a course on Research into effective mathematics teaching. The content will be a combination of findings of studies of teaching (Jo Boaler’s and others) and issues related to the ways teaching and classrooms can be researched.

Furthermore, in the autumn of 2008 students can take the courses given at University of Agder. The courses are Theory of science from a perspective of didactics of mathematics and Theories of teaching and learning mathematics. All courses given at Nordic universities entitle students to apply for travel stipends from NoGSME.

Also mobility stipends for doctoral students are available. They open the opportunity to visit another Nordic or Baltic university for a month and get both travel expenses and lodging covered by the stipend. Application can be sent to NoGSME at any time and information about the mobi­lity stipend is given on the web page of NoGSME (see www.nogsme.no).

Coming events in NoGSME

An international seminar for supervisors of doctoral students will take place in October 8–11, 2008 at Schæffergården in Gentofte just outside Copenhagen. NoGSME has received a generous grant from The Danish­-Norwegian Collaboration Foundation, which enable us to host the seminar at this very nice place. The international contact persons of NoGSME have been invited and we will at least have the following international guests: Michele Artigue, Hyman Bass, Willibald Dörfler, and Jeremy Kilpatrick. The theme for the seminar will be Local, global and international perspectives in mathematics education research and we will look both into the future and into the past as we have now almost five years of experience from work in NoGSME and ask ourselves where do we stand and where are we going.

The winter school 2008 for doctoral students will take place in Sigtuna in November 24–29 and the first announcement has been sent our in May. The second announcement will be sent in June to all who have indicated an interest in coming. There will be a limit of 28 participants this time, for financial reasons.

The updating of the web page of NoGSME is going on continuously so it is worth checking now and then the information available. Recently the self-evaluation from 2007 and the evaluation from NordForsk have been made accessible and also a paper about doctoral programmes in the Nordic and Baltic countries (see www.nogsme.no).

A Nordic Society for Research in Mathematics Education

During Norma08 a meeting took place to follow up an earlier meeting in Norma05 and an initiative to create a Nordic Society for Research in Mathematics Education (NoRME). In order to make it easier to access funding from NordForsk and EU-funds a joint Nordic society was considered useful. Some of the arguments for the society were presented in Nomad nr 1, 2008. In the meeting the decision was taken to establish such a society and current members are the national societies in Denmark, Finland, Norway and Sweden and the Nomad-society. The aim of the society is given in the constitution:

The aim of the society is to support and raise the quality of Nordic and Baltic research in mathematics education, and especially through the collaboration among researchers in the Nordic and Baltic countries. An important aim is to achieve funding for the activities among members in NoRME by initiating applications to funding bodies. The member societies are autonomous and NoRME will not interfere with their internal affairs. The aim of the society is also to support the aims of the member societies concerning research in mathematics education and to ensure the continuation of Nomad and Norma-conferences, as well as to create forums for discussions and constructive meetings for researchers in mathematics education. The aim is also to ensure the collaborative continuation of activities such as those carried out by the Nordic Graduate School of Mathematics Education, NoGSME.

The constitution has been distributed to all individuals in the societies that are members in NoRME. For NoGSME this is one way of assuring that the activities created by NoGSME will continue after the funding of NordForsk has ended in 2009. A board of NoRME was elected. Frode Rønning, Sør-Trøndelag University College, was elected to be the chair and the four members of the Board of NoRME are Christer Bergsten, Linköping University, Morten Blomhøj, Roskilde University, Markku Hannula, Helsinki University, and Tine Wedege, Malmö University College. Two substitute board members were elected, namely Guðný Gunnarsdóttir, Iceland University of Education, and Madis Lepik, Tallin University. The new board will work in close connection with NoGSME initially in order to take over some of the traditions and experiences from NoGSME.

Recent Nordic dissertations in didactics of mathematics

Currently there seems to be a continuous stream of doctoral theses coming up in the different countries in the Nordic area. New students are taken up in doctoral programmes, thus the situation seems to have come into a stable phase of development. Recently new professorships in didactics of mathematics have been advertised in Trondheim, Bergen and Stavanger following earlier ones in for example Växjö and Turku. Research and doctoral education in mathematics education have developed into a viable state and seem to survive, although many universities struggle with recruiting new students in mathematics and science.

Martin Carlsen at University of Agder in Norway has defended his thesis Appropriating mathematical tools through problem solving in collaborative small-group settings. The aim of his thesis was to give both empirical and theoretical contributions to the understanding of appropriation processes in mathematics learning of students at upper secondary school, 17–18 years of age. In particular he investigated how the students appropriate the concepts of dot product and geometric series. Through analyses of student collaborative problem solving in small-group settings Martin was able to present the following research findings. The students steadily improved their accuracy in using mathematical, concept related terminology. The students were able to explain mathematical components of the concepts to their fellow students. Opportunities for learning to occur were created by student questioning, externalisation of thinking, and calling for help. The issue of resistance in the appropriation process was documented. The students made use of semiotic means of objectification throughout their participation in joint activity. Finally, the process of appropriating mathematical concepts is a time-consuming enterprise.

Eva Riesbeck at Linköping University in Sweden presented and defended her thesis På tal om matematik: matematiken, vardagen och den matematikdidaktiska diskursen (Talking about mathematics: mathematics, the every day and the mathematics didactical discourse). The aim of the dissertation is to describe and analyze how discourse as a theoretical and didactical concept can help in advancing knowledge about the teaching of mathematics in school. The author takes a socio-cultural perspective, where active participation and support from artefacts and mediation are viewed as important contributions to the development of understanding. In order to grasp language use, knowledge construction and mathematical content in the teaching practises discourse analysis was used as a theoretical point of departure. The empirical data consisted of video and audio tape recordings of the interaction of teachers and pupils in mathematics classrooms and of discussions between student teachers. The results of her studies demonstrate that often discussions are located in either a mathematical or in an every-day discourse. The results also demonstrate how change between every-day and mathematical language often takes place unknowingly. Further the results underline that a specific and precise dialogue can contribute towards teachers’ and pupils’ conscious participation in the learning process. Translated into common vocabulary such as speak, think, write, listen and read teachers and pupils would be able to interact over concepts, signs, words, symbols, situations and phenomena in every-day discourse and its mathematical counterpart. When teachers and pupils become aware of crossing the discursive boundaries in mathematics an understanding of mathematical phenomena can start to develop. Teachers and pupils can construct a meta-language leading to new knowledge and new learning in mathematics.

Johan Häggström defended his dissertation with the title Teaching systems of linear equations in Sweden and China: What is made possible to learn? at Gothenburg University in Sweden. A starting point for his study is the aim to better understand the relation between teaching and learning of mathematics. The assumption is that what is possible for students to learn about mathematics must be related to how they experience the mathematical content. This in turn must be related to how the content is handled during the mathematics lesson. Mathematics teachers must always make decisions about the handling of the content – what examples to use, what aspects of concepts and methods to emphasise, what exercises students should work on etc. Häggström intends to produce results that can inform practice on the classroom level, as well as teacher education, and to contribute to the development of methods for analysing teaching that focus on the specific content of instruction. Sixteen lessons from six classes in Sweden and China – video recorded within the Learners’ perspective study – were analysed, based on variation theory and with focus on differences in how the same mathematical content was taught. The concept ‘object of learning’ was used to denote what teachers try to teach and what students are supposed to learn. Three objects of learning were analysed from the perspective of a student: systems of linear equations in two unknowns, solutions to systems of linear equations in two unknowns and the method of substitution. An aspect was considered made possible to experience if the corresponding ‘dimension of variation’ was opened, and not taken for granted during teaching and kept invariant. The analytical approach employed made it possible to detect even subtle differences in how the teachers handled the content and made it possible to learn for the students. The description of these differences points out several aspects that could be so familiar to many teachers that they face the risk of being taken for granted in teaching.

A common feature of these three studies is the use of video recordings for the data collection. The two first studies use socio cultural theory and the third one is based on variation theory as the theoretical framework. The discourse, the communication between learners and between learners and teacher is crucial in all three studies. Although there are different foci in the three studies, appropriation of concepts in collaborative settings, discourse as a theoretical and didactical concept for advancing knowledge about teaching, and differences in how the same mathematical content was taught, respectively, the studies are using similar methods for data creation and analyses. In all three cases there is a clear aim to inform practices in the classroom and the authors should be encouraged to work closely with teachers to make their findings become tools for teachers in their everyday work with pupils or students in mathematics.

Any suggestions for activities for NoGSME or themes for courses, workshops or seminars are as always welcome. Please contact any member of the board or the director of NoGSME. Well met in the upcoming events of NoGSME.

Barbro Grevholm
Director of NoGSME
University of Agder
Norway

ANDREAS RYVE

Abstract

The dialogical approach has been introduced for studying mathematical classroom discourse in a growing body of studies conducted by researchers from the Nordic countries. However, since it is developed for analyzing human action, communication, and cognition in general, it is important to explicitly discuss how it could be developed and complemented for serving the purposes of mathematics education research. In this article I initiate such a discussion by drawing on theoretical analysis as well as my own experiences of using the dialogical approach. By relating it to a framework of criteria for research in mathematics education it is shown that the dialogical approach could be a useful tool for fulfilling several aspects of relevance for mathematics education research. The article concludes by suggesting further aspects that need to be discussed and elaborated on in the project of making it even more useful for understanding mathematical teaching and learning.

Sammanfattning

Den dialogiska approachen har introducerats för att studera matematiska klassrumssamtal i ett växande antal publikationer genomförda och skrivna av nordiska forskare. Eftersom den dialogiska approachen är utvecklad för att studera mänskligt handlande, kommunikation och kognition i allmänhet är det viktigt att explicit diskutera hur den kan utvecklas och kompletteras för att uppfylla de ändamål som matematikdidaktisk forskning kräver. I denna artikel initierar jag en sådan diskussion med utgångspunkt i teoretiska analyser och empiriska exempel från min egen forskning. Genom att relatera den till ett ramverk för kvalité inom matematikdidaktisk forskning visas att den dialogiska approachen är ett verktyg som kan användas för att uppfylla många av dessa kriterier. Artikeln avslutas med förslag på aspekter som behöver diskuteras och utvecklas för att göra den dialogiska approachen ännu mer användbar för att förstå lärande och undervisning inom matematik.

ANDREAS RYVE
Andreas Ryve is a research fellow at Stockholm University and a senior lecturer at Mälardalen University. His research interest includes theoretical and methodological issues of how to analyze mathematical classroom discourse, conceptualizations of teacher practice, and teaching mathematics through problem solving.

Omtale av Matematik for lærerstuderende – Ypsilon, basisbog

Hansen, H. C., Skott, J. & Jess, K. (2008), Matematik for lærerstuderende. Ypsilon, basisbog. Forlaget Samfundslitteratur. ISBN 978-87-593-1302-2 (Bind1), ISBN 978-87-593-1346-6 (Bind2)

FRODE RØNNING
Høgskolen i Sør-Trøndelag, Trondheim

MAGNUS ÖSTERHOLM

Abstract

The main question discussed in this paper is whether students need to learn how to read mathematical texts. I describe and analyze the results from different types of studies about mathematical texts; studies about properties of mathematical texts, about the reading of mathematical tasks, and about the reading of mathematical expository texts. These studies show that students seem to develop special reading strategies for mathematical texts that are not desirable. It has not been possible to find clear evidence for the need of a specific ”mathematical reading ability”. However, there is still a need to focus more on reading in mathematics teaching since students seem to develop the non-desirable reading strategies.

Sammanfattning

Huvudfrågan som diskuteras i denna artikel är om studerande behöver lära sig att läsa matematiska texter. För att besvara denna fråga beskriver och analyserar jag resultat från olika typer av studier kring matematiska texter; studier om egenskaper hos matematiska texter, om läsning av uppgiftstexter och om läsning av förklarande texter. Studierna visar att studerande verkar utveckla speciella lässtrategier för matematiska texter som inte är önskvärda. Det finns inga tydliga bevis för att man behöver utveckla någon sorts ”matematisk läsförmåga”. Dock är det i alla fall nödvändigt att behandla läsning i matematikundervisning eftersom studerande verkar utveckla de icke önskvärda lässtrategierna.

MAGNUS ÖSTERHOLM
Magnus Österholm has a PhD in mathematics education from Linköping University and now works as a research fellow at the Department of Mathematics, Technology and Science Education at Umeå University. He is also a member of Umeå Mathematics Education Research Centre (UMERC). His research interests deal with mathematics education at the upper secondary and university levels, where cognitive and metacognitive perspectives are of special interest, together with studying the language in the learning and teaching of mathematics.

OLE SKOVSMOSE & ROGER SÄLJÖ

Abstract

The role of inquiry in teaching and learning has been discussed for a long time and by many leading educational philosophers and analysts. The purpose of this article is to analyse the assumptions and some of the outcomes of two interrelated and extensive developmental projects in mathematics teaching and learning in Norway. The projects – referred to as the KUL  projects (Knowledge, Instruction, Learning) – aimed at introducing the notion of communities of inquiry as a basis for developing mathematics teaching and learning in participating schools, and as a model for organizing developmental work in cooperation between teachers and researchers. In several respects, it seems as if the projects have been successful in the sense that they were accepted by the teachers (especially at lower levels) as a productive mode of engaging in developmental work. In the article, the interpretation of the concept of inquiry in the projects is scrutinized. It is argued that in order to develop our understanding of inquiry processes, detailed analyses of the nature of inquiry in interactional activities in mathematics learning is necessary. It is also argued that the notion of inquiry adopted by the projects is based on a conception where inquiry is seen as a means of learning mathematics better. An alternative conception is to see inquiry as a means of promoting critical thinking in which understanding of mathematics is at the core of the development of more general reasoning skills that play a central role in a democratic society.

Sammanfattning

Begreppet inquiry har spelat, och spelar ännu, en framträdande roll i den pedagogiska diskussionen och i forskares och didaktikers försök att utveckla undervisning. Syftet med artikeln är att diskutera en del av de antaganden om inquiry som utgjort utgångspunkt för två omfattande forsknings- och utvecklingsprojekt inom matematikundervisning som bedrivits vid Universitetet i Agder. Dessa projekt – som lokalt går under beteckningen KUL-projekten (Kunskap, Undervisning, Lärande) – syftade till att introducera en förnyelse av matematikundervisning som bygger på idéer om att etablera “communities of inquiry/learning” mellan universitet (forskare/didaktiker och doktorander) och skolor (lärare, skolledning, elever). Projekten är ovanliga i den mening att de har en väl artikulerad teoretisk bas för utvecklingsarbetet grundad i community begreppet och i ett medvetet försök att arbeta inom ramen för den modell som numera går under beteckningen design research. Intresset var dessutom inriktat både mot att utveckla matematikundervisningen och samtidigt att lära sig om hur utvecklingsarbete mellan universitet och skolar kan utformas. En preliminär utvärdering av de inledande resultaten av projekten, utförd av författarna till denna artikel, visar att arbetet på många sätt varit framgångsrikt. Bland annat framgår att idéerna bakom sätten att utveckla undervisningen, liksom samarbetsformerna mellan forskare och lärare, uppfattades som mycket givande för deltagarna på skolorna (särskilt bland lärare i grundskolan). Projekten har också avkastat många intressanta publikationer och publiceringen pågår alltjämt.
I föreliggande artikel diskuteras den tolkning av begreppet inquiry som de båda projekten bygger på. Det påpekas att för att vår förståelse av vad inquiry innebär som beståndsdel i pedagogisk praktik skall utvecklas, så är det nödvändigt att ingående analysera interaktion i undervisningen. Analysens mål måste vara att klarlägga vad inquiry innebär som kommunikativ praktik, och att urskilja vilka slags interaktiva mönster som karaktäriserar det slags aktivitet som kan kallas inquiry. Inquiry är mer än att personer samtalar med varandra. Det påpekas också att den föreställning om inquiry som projekten bygger på innebär att man ser inquiry som ett sätt att förbättra inlärningen av matematik. En alternativ ansats är att se inquiry som en aktivitet som utvecklar människors förmåga till kritiskt tänkande och där förståelse av matematik blir kärnan i utvecklingen av mer generella analytiska förmågor som spelar en central roll i ett demokratiskt kunskapsbegrepp och i ett demokratiskt samhälle.

OLE SKOVSMOSE
Ole Skovsmose has a special interest in critical mathematics education. He has investigated the notions of mathematics in action, students’ foreground, globalisation, ghettoising with particular reference to mathematics education. He is professor at Department of Education, Learning and Philosophy, Aalborg University, Denmark. He has been author or co-author of many books including Towards a Philosophy of Critical Mathematics Education (1994); Educação Matemática Crítica: A Questão da Democracia (2001); Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique (together with Helle Alrø, 2002) and Travelling Through Education: Uncertainty, Mathematics, Responsibility (2005). He serves in the editorial board of several scientific journals. He has been co-director of The Centre for Research of Learning Mathematics, a co-operative project between different universities.

ROGER SÄLJÖ
Roger Säljö is professor of educational psychology at the University of Gothenburg, Sweden. His research centres on issues of learning, development and comunication in a sociocultural perspective, and in particular how people learn to think and reason with technologies. Within the field of mathematics learning, he has studied the issue of how children learn to move between everyday (written) language and mathematical modeling, especially in the context of what in the literature is referred to as ‘word problems’. Recent publications include What’s the problem? Meaning making and learning to do mathematical word problems in the context of digital tools, Instructional Science (co-authors A. Lantz-Andersson and J. Linderoth, in press) and Learning to model: co-ordinating linguistic openness and mathematical precision when solving word problems (in B. Greer, L. Verschaffel et al. (Eds.), Title unknown, Sense publishers, in press, co-authors E. Riesbeck and J. Wyndhamn). Roger Säljö is director of LinCS – The Linnaeus Centre for Research on Learning, Interaction and Mediated Communication in Contemporary Society – which is a national Centre of Excellence funded by the Swedish Research Council since 2006.

Nordic mathematics education research in the world and in the region

The period of 2008, since the publication of the previous number of NOMAD, has been a very agitated period for mathematics education in the world, and also in the region. In this editorial we would like to continue a line of reflection that we started from our editorial in the first volume of this year, namely the prominent role that Nordic mathematics educators have played internationally, regionally and nationally in each of the countries in the region. We will bring some comments about this topic focusing on the realisation of ICME 11 in Monterrey, Mexico, and of the big international seminar for PhD supervisors organised by NoGSME.

The eleventh International Congress on Mathematics Education

ICME is always a big event: the conference for the large community of mathematics educators in the world. After the remarkable work of the Nordic organisation in planning and holding ICME 10 in Copenhagen, it was the turn of the Mexican mathematics education community to host this congress. It was the first time an ICME took place in a Latin American country and there were many expectations about what the country and the continent as a whole had to offer to the international community of mathematics education. Many histories can be told about ICME 11 (see, for example ICMI News 5, at http://www.mathunion.org/pipermail/icmi-news/2008-September.txt). In this occasion we would like to highlight some points of this meeting through a Nordic view, focusing on the participation of our research environment in the conference.

A first noteworthy fact is the number of participants from the region in the congress. In a counting of the countries with the biggest delegations, Norway, Denmark and Sweden were well ranked with 20, 35 and 64 participants respectively. It was not possible for us to get the official figures about the amount of participants from Iceland and Finland, but from having seen many colleagues in Monterrey, we suppose that these countries also had good numbers of participants. We find these numbers to be significant if we compare the size in the population in our countries with the size of the delegations from other strongly represented countries. In fact, even compared to the participation of many Latin American countries, we can say that the Nordic representation was significant.

A second remarkable fact was the engagement of these delegates in both organisational tasks and in presentations in the different activities in the congress. Serious and committed work was carried out by Nordic colleagues in the top organisation of the programme, in subplenary lectures, in the different topic study and discussion groups, as well as in poster presentations. Again such active participation has impacted the overall functioning of the conference, as well as it has been set research and practice in our region on the world map of mathematics education.
Finally, following the practice initiated by ICME 10, ICME 11 has gathered an impressive resource in mathematics education by keeping the conference website open to the public. In the website it is possible to have access to many of the activities, papers and products of this important worldwide activity. We encourage colleagues who did not participate in ICME 11 to use this resource of value for practice and research.

Nordic (and Baltic) collaboration in mathematics education

Just recently The Nordic Graduate School in Mathematics Education (NoGSME) held its last seminar for researchers and doctoral supervisors from Nordic and Baltic countries. This time four internationally highly recognised researchers where invited as plenary speakers to give their views on the theme of the seminar: Local, global and international perspectives in mathematics education. Details about this very interesting seminar can be found in the report from NoGSME in the back of this issue. The seminar was a natural occasion for looking back on the recent developments in the Nordic collaboration in mathematics education research, to evaluate the contribution made by NoGSME and to look forward to see future challenges and opportunities for strengthening the collaboration in the field in the Baltic and Nordic regions.

Mogens Niss gave in his address a historical overview on the Nordic collaboration in mathematics education as a field of research. The field is quite young. Not much research was done in the Nordic countries in this field before 1970 and it was not until the late 1980’s that the first signs of Nordic collaboration in research activities emerged. The organisation of Nordic research symposia and conferences such as the symposium on the criteria for scientific quality and relevance in the didactics of mathematics under the Danish initiative Mathematics teaching and democracy 1988–1992 (see Nissen & Blomhøj, 1993), and the first NORMA conference in Finland in 1994 where some of the first Nordic activities. It is also worth mentioning the Nordic collaboration driven by Göran Emanuelsson and Stieg Mellin-Olsen, which resulted in the foundation of a Nordic journal for research in mathematics education. As a result, the first volume of Nomad was published in 1993.

In its entire history of twenty years the Nordic collaboration in the field has been held up by synergetic effects between the growing Nordic network and national or often local research initiatives by various research groups or institutions in the Nordic countries. Whenever possible local research grants have been used to strengthening the Nordic collaboration and local environments have take advantage of the Nordic network and its connections to international research groups to strengthen their own development. The Nordic collaboration has been particular important in relation to the foundation and development of doctoral programmes at various institutions around the Nordic and also Baltic countries. With a few exceptions, the research groups in mathematics education in the Nordic and Baltic area are very small – typically two or three researchers with permanent positions and less than five doctoral students at the time. Nordic and international connections are therefore crucial in order to secure the quality of research education in the field.

Since 2004, the activities of NoGSME have raised tremendously the quality and quantity of Nordic and Baltic collaboration in the field. This fact was recognised by all participants in the seminar and was also highlighted by the international guest speakers. Formally NoGSME finishes in 2008, and therefore the seminar was a natural forum for discussing the future challenges and possible forms of organisation for the continuation of the Nordic and Baltic collaboration.

A new umbrella organisation, Nordic Society for Research in Mathematics Education (NORME), was founded at NORMA08 in Copenhagen (see our editorial in Nomad 13(2)). This society gathers as its (potential) members national societies of research in mathematics education in all the Nordic and Baltic countries. We see this organization as one new important actor to support the future development of the field. As an important part of its constitution, it aims at supporting Nomad and the continuous holding of the NORMA conferences every third year. In addition, it is meant to take initiatives to continue some of the activities related to research education initiated and co-ordinated by NoGSME. A crucial issue is of course how to raise economic support for such activities. Therefore, one of the first tasks of NORME is to inquire the possible sources for financing activities in the Nordic and Baltic regions.

We see that early in the development of the Nordic collaboration Nomad has played a prominent role as a means of communication in the region about the research being produced, and also as a showcase of the region to the world. The journal is still considered to be a central element in collaboration since it is the only journal for research in mathematic education in which research papers in the Scandinavian languages can be published. Moreover for many doctoral students and young researchers NOMAD is a natural choice for submitting a paper to a research journal for the first time. But the journal is not only a publication exercise for novice researchers, but it also collects the work of well-established researchers. It is this mixture of authors and different levels of research expertise a characteristic that we see to be promising in any effort to develop NOMAD as a well recognised scientific journal.

During the operation of NoGSME the number of papers submitted to NOMAD every year has more than doubled and we are sure that the doctoral school is one of the most important factors behind this development. Several of its activities have involved Nomad directly in form of seminars for reviewers and authors, and the doctoral courses have raised awareness on young research students’ possibility of making the journal an important tool of information and of publication. At the same time NOMAD has served as a means of communication for NoGSME.

Unfortunately we still wait to see a substantial increase in the number of subscribers to NOMAD. The number is growing but very slowly. For the time being there are nearly 250 individuals or institutions subscribing. This is still way below the 400 during the first years of the journal. Together with NCM in Göteborg, we strive to find ways of increasing subscriptions and of cofinancing the expenses associated with keeping publication. Hopefully, with the support of NORME, we will be able to create a better financial basis for maintaining the journal in the future and hereby also providing a basis for the continuous development of the quality of NOMAD as a research journal.

About this number

Andreas Ryve in Analyzing mathematical classroom discourse. Initiating elaborations on the usefulness of the dialogical approach presents a discussion of the use of the dialogical approach to research mathematics education classrooms. In mathematics education the focus on interactions as an important dimension of learning has led researchers to recontextualise different analytical models of human communication, dialogue and interaction into the specificities of the mathematics conversations among students and teachers and students in classrooms. Drawing mainly on the work of the Swedish linguist Per Linell, Ryve discusses how Linell’s dialogical approach could be developed and complemented for serving the purposes of mathematics education research. Ryve builds on his research experience using this approach, as well as on the experience of other Scandinavian researchers such as Maria Luiza Cestari and Raymond Bjuland. He also engages in the exercise of contrasting the insights that are possible to gain from researching from within the approach, against criteria for the quality of mathematics education research. He concludes by suggesting further aspects that need to be discussed and elaborated on in the project of making it even more useful for understanding mathematical teaching and learning.

The paper Learning mathematics through inquiry by Ole Skovsmose and Roger Säljö bring the very important notion of inquiry in learning to discussion. The paper uses the Knowledge, Instruction, Learning Projects (KUL-projects), two very large and significant projects in Norway and in general in the Nordic region, as an entry point to examine the notion of inquiry in relation to mathematical learning. Using their own research as a background for reading the use of the notion of inquiry in the KUL-projects, the authors argue that in order to develop an understanding of inquiry processes, detailed analyses of the nature of inquiry in interactional activities in mathematics learning is necessary. It is also argued that the notion of inquiry adopted by the projects is based on a conception where inquiry is seen as a means of learning mathematics better; while an alternative and complementary view on inquiry could emphasise the promotion of critical thinking in understanding the functioning of mathematics in democratic societies.

In the paper Do students need to learn how to use their mathematics textbooks? The case of reading comprehension by Magnus Österholm we learn about three different studies addressing receptively the properties of mathematical texts, students reading of mathematical tasks and of mathematical expository texts. The two first studies are literature surveys while the third one is an empirical study of upper secondary and university students’ reading and comprehension of expository mathematical texts with different degrees of use of mathematical symbols. The findings from this complex of studies are analysed and distilled to form the basis for a discussion of the research question in the title of the paper. Generally students do develop special strategies for reading mathematical tasks and texts. Parts of these strategies are based on superficial aspects of the texts and are not desirable in relation to the students’ learning processes. In order to challenge and develop the students reading strategies it is argued that in general students need more varied experiences with reading mathematical texts and support to reflect about their reading and comprehension of different types of mathematical texts. Students cannot be expected by themselves to develop forms of reading strategies for mathematical texts and to see the benefits of using general literacy skills also for mathematical texts. In order for students to be active and competent users of mathematics textbooks in full, mathematics teaching needs to focus more on reading and reading comprehension and to make sure that the students are exposed to varied experience with reading different types of mathematical texts. And not surprisingly, we need more research on how to support the students’ development of strong and effective reading strategies for mathematical texts.
Finally, Frode Rønning presents a review of the book Matematik for lærerstuderende. Ypsilon, basisbog, the first book in one of the newest and most comprehensive textbook system for mathematics teacher education produced in Denmark by Hans Christian Hansen, Kristine Jess, Jeppe Skott and John Schou. Rønning gives and overview and critical reading of the content of the book.

The editors

References

Nissen, G. & Blomhøj, M. (Eds) (1993). Criteria for the scientific quality and relevance in the didactics of mathematics. Roskilde: RUC – Danish Research Council for the Humanities.

Graduate school plans for 2009

The funding of NoGSME from NordForsk will run out during 2009 but the board has planned to use the final resources for a summer school for doctoral students in August 2009, either in Estonia or in Denmark and for one more supervisors’ seminar in the spring. The preliminary date for the seminar is April 23–24 and it will take place in Kristiansand in Norway, at University of Agder, the home institution of NoGSME. The theme of the seminar is suggested to be Critical review of research methodologies in mathematics education.
The expectation from NordForsk is that after the 5 Million NOK have been spent by NoGSME during 2004–2009, there will be strength enough in the participating units to carry on the activities and the networking. To support this to happen the Nordic Society for Research in Mathematics Education (NoRME) was created this year and it is the hope that NoRME will be able to inspire groups of Nordic and Baltic universities in the NoGSME network to apply for funding for common doctoral courses, summer schools, seminars and research workshops. In order to be eligible to apply from NordForsk there has to be at least three different Nordic countries involved in the application. Deadline for most of the offers from NordForsk is in March and April so the time for starting to prepare applications is already approaching. The NoRME board had a meeting in Denmark on October 8 during the NoGSME seminar and has taken several initiatives for applications.

The tenth NoGSME seminar – an important international event

The greatest and most spectacular supervisors’ seminar in NoGSME took place in Schæffergården, north of Copenhagen on October 8–11. The theme for the seminar was Local, global and international perspectives in mathematics education research. The number of participants was 36 and the size and length of the seminar, four days, was possible thanks to a generous grant from the Danish-Norwegian Collaboration Foundation. The international collaborating centres of NoGSME generously contributed through the presence of some of their most outstanding researchers in mathematics education.
As a background to the work during the seminar an introductory lecture was given by Mogens Niss about the development of the Nordic collaboration in mathematics education research and the sociology of that development, and by Barbro Grevholm about NoGSME, the idea, its development and life – a historical sketch. In the historical overview Mogens took us all back to the 1950's and Nordic LMFK-congresses, the Nordic committee for mathematics teaching in school, the isolated early individual researchers and up to more recent initiatives like the Danish Mathematics and democracy in the beginning of the 90's, the start of Nomad in 1993 and the Norma-conference in 1994, initiated by Erkki Pehkonen.
The first international guest in the seminar was Jeremy Kilpatrick, asking critical questions to a panel consisting of the members in the NoGSME board. The main questions were: What have we learnt from the Nordic collaboration? What is the current situation in mathematics education research in the Nordic countries?
Jeremy Kilpatrick gave a lecture on Mathematics education research in the Nordic countries – trends and development seen from an international perspective, where he drew from his longstanding collaboration with Nordic universities and the NoGSME activities. He contrasted and compared the research situation in the Nordic countries to the situation in the Iberian countries and pointed to a number of interesting aspects. Questions from the audience and a lively discussion followed. Jeremy related research to what he called the instructional triangle with corners the student, the teacher and mathematics. From his sources, which were overviews of Nordic research and NOMAD, he found that most of the studies were in the corner the student and with some links from student to teacher and to mathematics. For the Iberian research he found focus on students but also much on teachers and gave examples of such studies. Jeremy then raised the question of the impact of research on ordinary classrooms. He emphasised the following criteria for impact:

He then asked us: How are the discourses of mathematics education in Nordic countries affected by your research?

The lecture by Hyman Bass was about Scientific challenges in mathematics education as a discipline and research domain. Three challenges were carefully explored in the talk:

From the discussion about these challenges Hyman Bass drew conclusions for doctoral education. The implications for doctoral training demands from us that we attend to the following three challenges:

The next international speaker was Willibald Dörfler and he presented on The place of mathematics education research in the academic system – links between mathematics, mathematics education and other disciplines. One important message from him was that we have neglected to transform the results of research to products useful in the classroom. He asked how people in our field are finding their theoretical frameworks. The tension created by specialisation was discussed. Willi claimed the there is no relation to mathematics and referred back to the time when mathematics education research in Bielefeld was flourishing. Finally he talked about semiotics and diagrammatic thinking, which caused a heated scientific debate among all participants in the session after the presentation.
Between the lectures participants were active in group work and groups were asked to compare and contrast some of the traditions in research education in the Nordic and Baltic countries and internationally. The theme Structure and organisational forms – how do they influence the content of our work? evolved around the following questions:

• What can be noticed about traditions for the form of the thesis – monograph or collection of papers with ”kappa”?

• What about the traditions for supervision? Are there hidden traditions in supervision? How many and what kind of supervisors?

• What qualifications for supervisors? What forms for supervision? How is the work structured?

• What about the way to evaluate the thesis? How is it done? Is it the supervisors who decide when it is time to hand in the thesis, and how is it done? Is there a committee to evaluate before decision is taken? Are international evaluators engaged? How and when? Is it possible to fail?

• What criteria are used to evaluate the thesis? Are they local or national? Are they explicit, published? What instructions are there for the evaluators?

• What traditions are there concerning choice of language for the thesis? What are our experiences from that? What are the pros and cons for the choices?

• What do we know about the supervision for the writing process? How is the need to develop academic writing abilities met in the ph d education? What is the experience from courses on academic writing?

• What traditions are there for the dissertation day? How is this influencing quality of the thesis? What about the public character of the scientific discussion of theses? Has that anything to say about quality issues?

• What traditions exist for the follow up publications of a thesis? Which theses are made visible after the dissertation in scientific journals or other media? Are theses published by publishing houses or official channels?

• Can we learn something from this contrasting and comparing of traditions? What?

In one of the groups narratives were created based on the questions posed. Here is one of them (written one sentence after the other by different participants):

Finally Michele Artigue talked about Balancing national and international experiences and perspectives in the training and supervision of researchers in mathematics education. She gave a careful and detailed insight into doctoral education in France and its development since 1975. The doctoral programmes in France seem to be much more homogenous than the different programmes in the Nordic countries. She ended her talk by presenting some evident challenges of being a supervisor.

Each supervision has been and still is, for me, a human adventure and something unique. Helping an individual to become a researcher is a particular challenging task:

helping the doctorate student transform rather vague or too much ambitious ideas into something accessible to research, compatible with the doctorate constraints, the human and material means accessible, seeing and keeping in mind how it can be inserted into more global perspectives;

orientating and often re-orientating because things do not work as expected, supporting, leading, without imposing your views;

pushing, stimulating without generating too much anxiety;

facing the psychological fragility revealed by engagement in research work.

The success of the enterprise does not only depend on your scientific expertise, and you learn from it as much as your student. But it is a so rewarding task and you are so proud when your student for the first time speaks of his (her) thesis better than you could do.

The concluding activity of the seminar was a panel debate and discussion with Jeremy Kilpatrick, Michele Artigue, Willibald Dørfler, Hyman Bass, Ole Björkqvist and Mogens Niss over the theme The future of mathematics education research in the Nordic area and internationally. During this panel discussion many important suggestions for future work were made. Here is just the place to mention a few of them: There is a need to accumulate, organise, systematise and criticise what we have achieved so far. We should look critically to what we call theories in our field. We are just able to make contributions to research. Research results can only be provisional. It is important to recognise the value of our research for decision makers and be sensitive to that. Concepts that are closer to everyday notions should be preferred, if possible, and we should not use different concepts for similar phenomena. The dominance of qualitative research should be complemented by quantitative studies and casual effect could be estimated. We need to provide a rational framework for decision-making and educate practitioners and decisionmakers. We have been doing the right thing, when we coordinated actions, but we should now try to make the totality more than the pieces. A number of small studies with different methods at hand can offer a collection of results that indicate directions. Survey such work and search for different aspects of a common problem. We are facing a collective enterprise and it can not be done by individual researchers. There is an inner diversity in the Nordic community which makes us stronger from some aspects than for example the research environments in France. There is a necessity of doing some deep evaluations and negotiate what we consider to be good research.
Participants in the seminar probably listened to different parts of the suggestions and advice and other narratives can be told, but the hope is that all were inspired for the future work by this final panel from the international guests and some Nordic voices.

New Nordic dissertations in mathematics education

In June Tone Bulien defended her thesis at Tromsø University. The title is Matematikkopplevelser i lærerutdanningen: en fenomenologisk orientert narrativ analyse av studenttekster (Mathematical experiences in teacher education: a phenomenologically oriented analysis of students’ texts). The thesis, in the form of a monograph, is a study of texts from and interviews with six Norwegian student teachers in a compulsory course in mathematics. The aim was to listen to the students, sharing their experiences while studying mathematics, through the author’s critical constructive descriptive investigation. The work is a contribution towards defining the didactic challenges teacher training is faced with. The thesis is written from a phenomenological perspective, using narratives as an important feature in both the analysis itself and the presentation of the results. A description of the students’ perceptions of teaching and learning mathematics, both prior to and in the course of the compulsory course, is made visible through narratives. The methodology is narrative analysis. The students’ experiences are divided into four main areas of beliefs: beliefs about mathematics in general, beliefs about themselves as practitioners of mathematics, beliefs about teaching mathematics, and beliefs about how mathematics is learnt. One indication is that students’ experience of the compulsory course in mathematics did not depend on their previously held beliefs on mathematics education or their attitudes towards mathematics in general. About half of the students had higher expectations about their grade at the beginning of the semester than what they actually ended up with. It is likely that the way mathematics is taught in a teacher education program differs from the students’ previous experiences in how to learn mathematics. The author suggests that this should be taken into consideration in prospective mathematics programs, for instance by supervising the students about their own beliefs in a meta-perspective by analyzing their own narratives and how they are subject to alterations during the course.
In September Antti Viholainen defended his thesis at University of Jyväskylä in Finland. The title of the dissertation is Prospective mathematics teachers’ informal and formal reasoning about the concepts of derivative and differentiability. His study, which is a collection of six papers and an extended summary, examined informal and formal understanding of the concepts of derivative and differentiability and the use of informal and formal reasoning in problem solving situations, where these concepts were needed. The subjects of the study were mathematics education students in the middle or in the final phase of their studies. The data were based on a written test given at six Finnish universities and on some oral interviews. The methods used could be called an explanatory mixed method design and the sample included 146 student teachers. One outcome was that connecting informal and formal reasoning was often difficult for the students. In particular, the students seemed to have a tendency to avoid using the definition of the derivative in problem solving situations. This was a considerable obstacle in problem solving processes and in some cases led to erroneous conclusions. Inability to use the definition is not a sufficient reason to explain this tendency, as several students were able to use the definition when they were asked to do so. The author recommends that the teaching of mathematics should support the development of coherence of students’ knowledge structure. It should also strengthen the understanding of connections between informal and formal representations.
The common characteristics of these two theses are that they study teacher education and they focus on student teachers. Bulien’s thesis explores student teachers’ beliefs about aspects of mathematics and its teaching and learning, while Viholainen’s thesis investigates student teachers’ reasoning and understanding in relation to two central mathematical concepts, derivative and differentiability. It is reported internationally that research on mathematics teacher education is increasing and these two Nordic theses seem to align the Nordic trend of research interests with the international. Nordic studies on mathematics teacher education have not been so common earlier, although there are a few such studies.

Winterschool for doctoral students in Sigtuna in November

The summerschool is going to be a winterschool for 2008 in order to avoid collision with ICME11 in Mexico. There will be 29 participants from all the Nordic countries and one Baltic country and two students from Freudenthal Institute in the Netherlands. Groupleaders will be Morten Blomhøj from Roskilde University, Cyril Julie from University of Agder and Joao Pedro da Ponte from University of Lisboa in Portugal.
We look forward to the cooperation while we make use of the final parts of the funding from NordForsk and welcome doctoral students as always to apply for travel stipends and mobility stipend. They have become increasingly popular.

Barbro Grevholm
Director of NoGSME
University of Agder, Norway

Seminar for supervisors

The 11th seminar for supervisors will take place in April 23–24 in Kristiansand in Norway, at University of Agder, the home institution of NoGSME. The theme of the seminar will be Critical review of research methodologies in mathematics education. The aim of this seminar, as for the 10 previous, is to offer competence development to current and prospective supervisors of doctoral students in mathematics education in the Nordic and Baltic countries. Many different themes have been on the agenda of the seminars before but it is the first time methodologies in mathematics education will be discussed. Experts in the area will contribute to the seminar.

Mobility stipends and travel support for students

As previously, students can apply to NoGSME for a mobility stipend to stay at another Nordic or Baltic university or for support for travel costs to doctoral courses at other universities. The budget for such costs is smaller in 2009 than before and several applications have already been accepted and support promised. Thus this opportunity may very soon come to an end as the funding from NordForsk will not be renewed for 2009. There will probably not be any chance to support applications after the spring semester.

Summer school for doctoral students in 2009

The NoGSME board is planning a summerschool for doctoral students to take place September 21–26, 2009 at Søminestationen in Holbæck, Denmark. This summer school will be offered by Roskilde University in cooperation with NoGSME in the form of a doctoral course but with the same structure as earlier summer schools. The evaluation from the winter school in Sigtuna shows that the participants want NoGSME to keep the same format for the summer school as before. International scholars will be invited to function as group leaders during the summer school.

Preparations for a summer school in 2010

As the funding from NordForsk to NoGSME will end in 2009 the new organisation NoRME, Nordic Society for Research in Mathematics Education, has initiated work to prepare the NoGSME summer school in 2010. This means that mathematics educators at three different Nordic Universities will collaborate in preparing an application to NordForsk for a summer school. The didacticians at University of Agder will take the lead in this application and seek support from colleagues in Finland and Denmark.

New dissertations in the Nordic countries

Ingvald Erfjord defended his thesis at University of Agder in Norway on November 29, 2008. The title is Teachers’ implementation and orchestration of Cabri-use in mathematics teaching. Ingvald Erfjord reports from a study of three teachers’ first ever use of a particular software tool in teaching at two lower secondary schools in Norway and gives a characterisation of teachers’ progression through a development process in which they implemented and orchestrated Cabri-use in their teaching. The study was situated within two developmental projects run by didacticians (including the author) at the University of Agder and the teachers in the study participated in these projects. Data were collected in sessions within this frame in many different ways. Teachers’ motives and goals for implementation of Cabri were analysed by utilising activity theory. During the implementation process, the teachers worked in teams with other teachers and didacticians and raised many issues. Two teachers at one of the schools had a focus on institutional school related issues while the teacher at the other school had a focus on personal issues, indicating a difference between the schools. From an activity theory perspective, the kinds of issues and teachers’ ways of coping with them are seen to illuminate teachers’ motives and indicate their goals for implementation of Cabri. Although issues raised in the study were particular to these teachers, the issues are argued to be relevant to teachers and educators more widely. Analysis of teachers’ orchestration of students’ work with Cabri is also guided by the instrumental approach. The term instrumental orchestration accounts for the role of the teacher when software tools are used in mathematics teaching. Teachers’ emphases and ways of accomplishing their Cabri-teaching as well as how they arranged these lessons are considered as being part of teachers’ orchestration of Cabri-use. Two kinds of orchestration are illuminated and their consequences for students’ work and achievements with Cabri are discussed.
The thesis suggests implications for mathematics teachers considering implementation and orchestration of software tool-use in teaching, indicating that the established and evolving collaboration among mathematics teachers in schools influences to a great extent teachers’ implementation of new tools and the sustainability of development in teaching. Conclusions are presented indicating that implementation of a new computer software tool can offer teachers a medium to develop new styles of mathematics teaching. Implications are also suggested concerning future developmental projects aiming to support teachers’ development in mathematics teaching.
On December 15 Sverker Lundin defended his thesis in educational sociology at Uppsala University in Sweden. The title is The mathematics of schooling. A critical analysis of the prehistory, birth and development of Swedish mathematics education. He claims that common beliefs about mathematics come from the practices of elementary mathematics instruction rather than science. In school pupils come to believe that mathematics has a set of properties in itself. For example, pupils believe that mathematics is useful in everyday life, even if it is not necessarily the case. He labels this object of belief the mathematics of schooling (skolans matematik), while the system of practices by which the belief is created is called mathematics education (skolmatematik). He uses terminology inspired by psychoanalytic theory to describe the specific properties of the mathematics of schooling and suggests that it can be understood as an object of an ideology spread by the system of education. He gives an overview of the history of mathematics education in Sweden, based on curricula, textbooks, discussions in teacher magazines, and other published material mainly from the eighteenth to the twentieth century. The story intertwines social factors determining the practices of mathematics education, the changing ideas about mathematics, and the interplay between external social factors and internal ideological meaning. His conclusion is that while elementary arithmetic is part of common knowledge, the mathematics of schooling is something different. Sverker Lundin claims that this object is thoroughly ideological and plays a central part in society mainly by making the social effects of mathematics education – keeping children away from production while sorting them – to appear as something else, namely as most often failed attempts to give children a necessary knowledge of mathematics.
Both of these theses are extensive monographs, the first one in English and the second one in Swedish with more the 300 and 400 pages, respectively. There is a clear tendency that theses in the form of monographs take up many more pages than the theses that are in the form of a selection of published papers. We can compare for example by the one presented in NOMAD no 3, 2008, by Viholainen, containing 6 papers and 137 pages. Quality of scientific work can of course not be measured by the size of the theses. However, keeping in mind that the education to become a doctor of philosophy in mathematics education in Norway and Finland is a three year full time programme (mandated in all those countries that have signed the Bologna agreement), it is relevant to discuss what is reasonable and relevant for the size of such theses. Are some supervisors demanding too much of the doctoral students and are some of the doctoral candidates too ambitious? We certainly know and we are concerned that it is common that students do not finish their dissertation within the financed time for their doctoral studies.
The two theses presented here are different in the fact that one is produced within a programme for educational sociology (Lundin) and the other in a programme for mathematics education (Erfjord). The two theses not only have very different aims and foci, approaches, theoretical frameworks and kinds of empirical data used. They also present very diverging images of mathematics in school. Erfjord’s thesis is situated inside a developmental research project, where the aim is collaboration between teachers and researchers to create opportunities for development of mathematics teaching. It represents a kind of optimistic view on mathematics in school with a hope that it can be improved. Lundin’s thesis deals with old school mathematics documents, providing arguments to claim that mathematics in school has been used to keep the students away from production in society and to sort them and invite them to fail in their learning. In Lundin’s thesis the image of school mathematics is presented as something very negative with highly despicable aims. This huge difference in perspectives between the theses points to the scope and complexity of mathematics education, and raises general issues on how to interpret, relate, and evaluate research in the field. These are difficult but important tasks challenging our scientific community.

A book about some of the recent doctoral studies in Sweden

The Swedish Graduate School in Mathematics with direction of subject didactics that existed between 2000 and 2006 has produced a book with contributions from nine of the doctors in that school (Brandell, Grevholm, Wallby & Wallin, 2009). This graduate school was financed by the Bank of Sweden Tercentenary Foundation and the Swedish Research Council over 5 years but has not yet received any continuation of the funding (Leder, Brandell & Grevholm, 2004). The school had in all 24 participating doctoral students and so far 12 of them have been awarded a doctoral degree and another 4 have finished with a licentiate degree. A report of the outcome from the graduate school will soon be available. In the book, which has the title Matematikdidaktiska frågor – resultat från en forskarskola (Issues in the didactics of mathematics – results from a graduate school), each of the nine new doctors writes a chapter about her/his study. The book is published by SMDF, The Swedish Society for Research in Mathematics Education in cooperation with NCM, the National Center for Mathematics Education. The research areas that are dealt with in the chapters are diverse and presented in a way that should be inviting for student teachers, as well as teachers and teacher educators. The theses of the nine authors have been presented in earlier issues of NOMAD.
The studies reported in this book mainly concern teaching and learning mathematics at university level. That is the case for the studies about the function concept, the limit concept, about proofs and proving, about reading ability and the interplay between different aspects of conceptions. Some of these studies also touch upon learning at upper secondary school level. That is the case for the study about proof and proving and the study on reading ability. One study about assessment in mathematics concerns upper secondary school level. Two of the studies investigate teachers and pupils, respectively, in compulsory school. Those studies deal with the concept of probability and the use of mathematics textbooks, respectively. One study is historical and investigates geometry teaching in a form of schooling that no longer exists, the realskola. Thus the studies deal with teaching and learning mathematics for older pupils and students than has normally been the case in earlier research in Sweden.
Both the number of recent papers in NOMAD by new doctors in mathematics education and a book such as the one presented above show that the area of research in mathematics education in the Nordic countries is in good shape and very much alive. There is hope for a continuation of this situation.

Barbro Grevholm
Director of NoGSME
University of Agder, Norway

References

Brandell, G., Grevholm, B., Wallby, K. & Wallin, H (red). (2009). Matematik-didaktiska frågor – resultat från en forskarskola. NCM, Göteborgs universitet.
Leder, G. C., Brandell, G. & Grevholm, B. (2004). The Swedish graduate school in mathematics education: conception, birth and development of a new doctoral programme. Nordic Studies in Mathematics Education, 9 (2), 165–182.

EVA NORÉN

Abstract

This article presents some of the main results of a bilingual mathematics teaching project, which run in five multicultural schools in Sweden. The main research question was: How do mathematical practices emerge in bilingual mathematics classrooms? In the project bilingual mathematics teachers seemed to promote mathematical learning and engagement in the classroom by using two languages in mathematical discourses. Pupils and teachers communicated mathematically in different ways, and the interplay between mathematics and language often became obvious. Bilingual pupils participating in the project expressed that they were able to learn more and they felt secure with the ways of using languages and learning mathematics. Participating in the project gave many of the pupils’ confidence in their mathematics learning competence.

Sammanfattning

Artikeln presenterar några av resultaten från ett tvåspråkigt matematikundervisningsprojekt, vilket bedrevs i fem mångkulturella skolor i Sverige. Den övergripande forskningsfrågan var: Hur utformas matematikpraktiker i tvåspråkiga matematikklassrum? Genom möjligheten att använda två språk verkade elevernas engagemang i klassrummet öka. Därigenom stärktes deras potential att lära matematik.
Elever och lärare kommunicerade matematik på olika sätt, och samverkan mellan matematik och språk blev ofta uppenbar. Tvåspråkiga elever som deltog i projektet uttryckte att de hade möjlighet att lära sig mer, och de kände sig mer säkra, när de kunde använda båda sina språk på matematiklektionerna. Genom att delta i projektet ökade många elevers tilltro till sin egen förmåga i matematik.

EVA NORÉN
Eva Norén is lecturer and doctoral student at the Department of Mathematics and Science Education, University of Stockholm. Research interest: multicultural and multilingual issues in mathematics education, power relations, disciplining and consequences for students learning of mathematics. Eva Norén has been a mathematics teacher in lower primary school for more than 20 years and a teacher trainer the last six years. She is also interested in the development of mathematics teacher education.

KAY OWENS

Abstract

The purpose of this paper is to draw upon extensive multicultural experiences and research to present some key aspects of effective multicultural teaching and learning. With an emphasis on cultural context and language, the discussion uses examples from Sweden and Australia, although experience and research in other cultures have informed the perspectives. Aspects of culture that are relevant to mathematics and approaches that maintain culture are emphasised. An emphasis on national values and language is also influencing schooling but strategies that take account of diversity of language and culture are provided.

KAY OWENS
Kay Owens has been a primary, secondary and tertiary teacher of mathematics and health education. Her current focus in lecturing is on social justice and sustainability in addition to mathematics education. Her research is in ethnomathematics, space and geometry, and measurement.

TROELS LANGE

Abstract

In this paper, I contrast an immigrant 10 years old girl’s perception of her home support and her mathematics teacher’s rather different perception. I show how the girl tries to align her perception of her home support with middle class Danish family values, and how the public discourse about immigrants apparently frames the teacher’s perception of the resources that are available or not available to the girl. The analysis becomes an example of how mathematics teaching and learning are embedded in a wider socio-political field. It suggests that sometimes resources could be available that schools do not see because students are constructed as disadvantaged.

Sammendrag

I artiklen modstilles en 10-årig piges opfattelse af den støtte hun får fra sit hjem og hendes matematiklæreren noget anderledes opfattelse. Pigen tilhører en minoritet i Danmark idet hendes forældre er indvandret fra Mellemøsten. I artiklen vises dels hvordan pigen prøver af tilpasse sin opfattelse af sit hjems støtte til den indfødte danske majoritets normalitetsforestillinger, og dels hvordan den offentlige diskurs om immigranter tilsyneladende former lærerens opfattelse af de ressourcer pigens familie råder over. Analysen er et eksempel på hvorledes matematikundervisning er indlejret i et socio-politisk felt. Den antyder at der kan være ressourcer til stede i minoritetsbørns familier som skolen ikke har øje for, fordi den offentlige diskurs konstruerer minoritetselever elever som underprivilegerede.

TROELS LANGE
Troels Lange is a senior lecturer in mathematics at VIA College of Education and a Ph.D. student at Aalborg University. His research interest are children’s perspectives on mathematics teaching and learning, especially on learning difficulties in mathematics.

Bringing focus to mathematics education in multicultural and multilingual settings

During the last three years the last number of a year volume in NOMAD has been dedicated to particular research topics which are of relevance in the Nordic region. This number is dedicated to mathematics education in multicultural and multilingual settings. We present some general reflections about this topic as well as introduce the papers in this thematic issue.
Asserting that mathematics education is connected to culture should not surprise many nowadays. Since the 1980’s international research in mathematics education has shown that all human groups have developed practices, thinking and techniques that can be called mathematical, even though they are not similar or have not been formulated in the language and form of the Western, European mathematics. Alan Bishop’s own research and his review of existing anthropological studies in different aboriginal groups have been one of the first systematic research work to argue for the emergence of mathematical practices in all human cultures (Bishop, 1988). The ethnomathematical program proposed by Ubiratan D’Ambrosio (D’Ambrosio, 1994) has also furthered our understanding of the way in which different human groups, from any aboriginal community to any professional community, produce mathematical thinking and knowledge. D’Ambrosio’s defines the term emphasizing its three components: ethno-mathema-tics. Mathema- refers to a cultural group’s understanding and coping with reality; -tics refers to the techniques and arts developed in the understanding and coping with reality; and together with the prefix ethno- it comes to mean the culturally embedded techniques of understanding developed by a cultural group. More recently the adoption of socio-cultural and anthropological theories for the study of mathematical thinking and learning have pointed to the fact that thinking and cognition in mathematics ”are forms of reflective, mediated social praxis where the organization of individuals’ sensuous cognitive processes are related to the meaning of things as they become objectified in practical and theoretical activity.” (Radford, 2008, p. 440) According to Luis Radford, meaning construction always happens in cultural, rational contexts, where the word ”rational” refers to the particular rationality determined in and by the practices of the human group where the thinking is situated. All this research has contributed to acknowledge the embededness of mathematics, mathematics thinking and mathematics education in culture and, more broadly in society. The fact that Ubiratan D’Ambrosio was awarded the ICMI Felix Klein Medal for 2005, announced at ICME-11 in Monterray 2008, illustrates the international recognition of the importance of the socio-cultural perspective in mathematics education research.
Simultaneously with this trend, there has been a growing amount of research focusing on the challenges that students’ diversity pose for the teaching and learning of mathematics. As the world gets more globalised and migration of peoples around the world becomes a spread phenomenon, many classrooms have turned evidently diverse. This does not mean that classrooms have ever been homogeneous! Cultural differences always exist among people according to a series of factors. It suffices to look at any classroom in any big Nordic city to find differences among students according to their gender, family background, social and economic status, regional belonging and even dialect spoken. However, the meeting with others who are really different in ethnic group, language or religion have made diversity more visible. While in countries with a longer tradition of immigration research has for a long time investigated these differences mainly in relation to the challenges of multilingualism, for many European countries and not leas for the Nordic countries the attention to cultural and linguistic diversity has been a recent phenomenon (Abreu, César, Gorgorió, & Valero, 2007). Nevertheless in the recent years some attention has been paid to how to understand the complexity of a learning and teaching situation when students and teachers seem to have little in common and, still, have to work together in schools and mathematics classrooms.
International research in multilingual and multicultural situations has illuminated aspects of such complexity. A first trend of studies has focused on the challenges posed to teaching and learning when there are differences between the language of instruction and students’ mother tongue or home languages. Part of this trend has identified that it is students’ lack of fluency in the main language what obstructs instruction and, thereby, mathematical learning. Critique has been raised to research studying bilingualism in mathematical learning and reinforcing a cognitive deficit perspective. Other research agendas have tried to focus on the dilemmas that bi/multilingualism poses to teachers (Adler, 2001), and some others have also tried to explore the broader network of social, cultural and political practices in schools and mathematics classrooms that contribute to the construction of bi/multilingualism as problematic.
In particular, the focus on the mathematical underachievement of minority groups in many societies has raised awareness on the necessity of expanding research views on this matter. Multiple examples of this type of research can be found in, for example, the work of the group on multiculturalism and mathematics education in the last four CERME conferences (see for the proceedings of conferences at http://ermeweb.free.fr).
In this volume we have collected a series of four papers around this issue, representing both research done mainly in Sweden and Denmark, as well as international research particularly from Australia, thanks to collaborative research enterprises. The papers address the issue of multiculturalism and multilingualism from perspectives that highlight the complexity of the challenges. All papers go beyond deficiency discourses of minority students.
The involvement of Kay Owens, Charles Sturt University (Australia), with Swedish colleagues in researching mathematics education for Sámi people in Sweden, led her to propose to the editors of Nomad the idea of having a special number of the journal dedicated to culture, ethnomathematics and mathematics education in the Nordic region. Her proposal resonated well with our own idea for a thematic issue. Building on her long personal experience and research work in different cultures, Kay Owens, in the article Culturality in mathematics education: a comparative study, extracts and discusses six key issues that are important to consider when dealing with mathematics education in multicultural and multilingual settings.
In the article Bilingual students’ mother tongue: a resource for teaching and learning mathematics, Eva Norén presents some of the main results of a project promoting the teaching of mathematics in the mother tongue of the students and not only in Swedish. The project run in five multicultural schools in the Stockholm area ad involved a number of bilingual teachers and students. Bilingual mathematics teachers seemed to promote mathematical learning and engagement in the classroom by using two languages in mathematical discourses. Pupils and teachers communicated mathematically in different ways, and the interplay between mathematics and language often became obvious. Bilingual students participating in the project expressed that they were able to learn more than what they normally did in a Swedish-only mathematics classroom, and they felt secure with the ways of using languages and learning mathematics. Participating in the project contributed to the students’ confidence in their mathematics learning competence.
Troels Lange’s article Homework and minority students in difficulty with learning mathematics: the influence of public discourse, raises the issue of the connection between minority children’s experiences in school mathematics and the public images about how minority parents provide parental support for their children’s schooling, particularly in the case of doing homework. He contrasts an immigrant 10 years old girl’s perception of her home support and her mathematics teacher’s rather different perception. He shows how the girl tries to align her perception of her home support with middle class Danish family values, and how the public discourse about immigrants apparently frames the teacher’s perception of the resources that are available or not available to the girl. The analysis provided in this article is an example of how mathematics teaching and learning are embedded in a wider socio-political field. For the case of minority students, the analysis suggests that sometimes resources could be available that schools do not see because students are constructed as disadvantaged.
The article School mathematical discourse in a learning landscape: understanding mathematics education in multicultural settings is a collaborative venture between Helle Alrø, Ole Skovsmose and Paola Valero who have been researching mathematics learning in multicultural settings in Denmark for the last 5 years, and Tamsin Meaney, Uenuku Fairhall and Tony Trinick, who have been researching mathematics education in multilingual settings in New Zealand. By bringing their conceptual tools together they discuss the potential of combining two particular notions: the learning landscape and school mathematical discourse. They aim at formulating concepts and methodological tools to challenge the simplification of issues in regard to mathematics learning in multicultural settings, when adopting restricted perspectives on issues of bilingualism. In the paper they propose how these two notions relate and offer a framework for the analysis of the complexity of mathematics education practices in these settings. They present two cases from their empirical material and analyze it with the proposed model. They conclude by pointing to the potentialities and limitations of such framework.

New structure of the editorship

From January 2009 Johan Häggström (NCM) enters the chief editorial team. Johan has been for many years the managing editor of NOMAD and now he will also be serving as a chief editor together with Morten Blomhøj and Paola Valero. This is the first step in a process of changing the structure of the editorship for NOMAD from shifting among the Nordic countries in periods of four years to an editorship that represents more than one country at the time.
During 2008 we have had the help of Anita Lindkvist Pedersen as an editorial assistant. Unfortunately due to lack of founding she can not continue working with us. We would like to thank her for having been the contact between editors, authors and reviewers, and for administering the review process.

Thanks to authors and reviewers

Finally, we would like to thank all the authors that have submitted papers and book reviews to NOMAD in 2008. We have experienced an increasing number of submissions and, for the first time in the history of NOMAD, we have papers queuing for publication. Accordingly we have been drawing on a large number of reviewers, and we are very keen on expressing our gratitude for the hard and important work done by all of them in 2008. We sincerely thank their engagement with the journal. Below the list of reviewers for the papers processed in 2008.

Andreas Ryve
Antti Viholainen
Bharath Sriraman
Barbro Grevholm
Bettina Dahl Søndergaard
Bodil Kleve
Carl Winsløw
Christer Bergsten
Elin Reikerås
Erkki Pehkonen
Dave Wagner
Frode Rønning
Guðný Gunnarsdóttir
Gunnar Gjone
Jeppe Skott
Johan Häggström
Johan Lithner
Juha Oikkonen
Kaarina Merenluoto
Kay Owens
Kjersti Wæge
Kristin Bjarnadóttir
Lena Lindenskov
Lil Engström
Lisser Rye Ejersbo
Markku Hannula
Marta Civil
Marit Johnsen Høines
Mette Andresen
Mogens Niss
Morten Blomhøj
Núria Planas
Nora Lindén
Ola Helenius
Paola Valero
Raymond Bjuland
Stine Timmermann Ottesen
Tine Wedege
Thomas Lingefjärd
Tomas Bergqvist
Tomas Højgaard Jensen
Troels Lange
Trygve Breteig
Uffe Jankvist
Ulla Runesson

References

Abreu, G. de, César, M., Gorgorió, N. & Valero, P. (2007). Issues and challenges in researching mathematics education in multicultural settings. In M. Bosch (Ed.), Proceedings of the IV Congress of the European Society for Research in Mathematics Education (pp. 1125–1130). Barcelona: Universitat Ramon Llull – ERME.
Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht: Kluwer.
Bishop, A. J. J. (1988). Mathematical enculturation: a cultural perspective on mathematics education. Dordrecht: Kluwer.
D’Ambrosio, U. (1994). Cultural framing of mathematics teaching and learning. In R. Biehler, R. W. Scholz, R. Strässer & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 443–455). Dordrecht: Kluwer.
Radford, L. (2008). Culture and cognition: towards and anthropology of mathematical thinking. In L. D. English & M. G. Bartolini Bussi (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 439–464). New York: Routledge.

PAOLA VALERO, TAMSIN MEANEY, HELLE ALRØ, UENUKU FAIRHALL, OLE SKOVSMOSE and TONY TRINICK

Abstract

By bringing our research work together, we are able to discuss the potential of combining the notions of the learning landscape and school mathematical discourse. We do so in a search for concepts and methodological tools to challenge the simplification of issues in regard to mathematics learning in multicultural settings, when adopting restricted perspectives on issues of bilingualism. In the paper we discuss the relationship between the learning landscape and school mathematical discourse. We then use these notions to analyse two case studies in Danish and New Zealand schools. Our conclusion raises possibilities about how these notions can be used when researching mathematics education in multicultural settings.

Sammendrag

I denne artikel bringer vi vore respektive forskningsperspektiver i
spil ved at diskutere forholdet mellem begreberne læringslandskab og skolematematisk diskurs. Formålet er at udvikle teoretiske begreber og metodologiske redskaber, som kan udfordre faren for en forenkling af de forhold, som har indflydelse på matematiklæring i multikulturelle kontekster. En sådan forenkling kan f.eks. ses i begrebet om ”de tosprogede”. Vi diskuterer læringslandskab og skolematematisk diskurs i forbindelse med en analyse af to case studier fra hhv. Danmark og New Zealand. I konklusionen peger vi på de muligheder, som disse begreber kan tilføre forskningen inden for matematikundervisning i det multikulturelle klasseværelse.

PAOLA VALERO
Paola Valero is associate professor in mathematics education at the Department of Education, Learning and Philosophy, Aalborg University, Denmark. Her research interests are, among others, the political dimension of mathematics education in areas such as school reform processes, curricular innovation, and multiculturalism and mathematics learning.

TAMSIN MEANEY
Tamsin Meaney has worked as a teacher in many situations which have made her consider the relationship between language and mathematics learning. She currently lectures in mathematics education at Charles Sturt University, Wagga Wagga, Australia.

HELLE ALRØ
Helle Alrø has a research interest in interpersonal communication and learning in helping relationships: (mathematics) education, coaching, supervision, conflict mediation etc. She is professor in Interpersonal Communication at Department of Communication and Psychology, Aalborg University, Denmark; and professor II at Bergen University College, Department of Teacher Education, Bergen, Norway. She is leader of the ’Centre for Interpersonal Communication in Organizations’
situated at Aalborg University.

UENUKU FAIRHALL
Uenuku Fairhall, Te Arawa: Ngäti Rangiwewehi, Waitaha and Ngäi Te Rangi: Ngäti Hë is the principal of Kura Kaupapa Mäori o te Koutu. He is one of the most experienced teachers of mathematics in te reo Mäori in New Zealand.

OLE SKOVSMOSE
Ole Skovsmose has a special interest in critical mathematics education. He has investigated the notions of mathematics in action, students’ foreground, globalisation, ghettoising with particular reference to mathematics education. He is professor at Department of Education, Learning and Philosophy, Aalborg University, Denmark.

TONY TRINICK
Tony Trinick has many years of experience in Mäori medium primary, secondary and tertiary mathematics education, pre-service teacher and teacher professional development. He is currently Associate Dean Mäori at the University of Auckland’s Faculty of Education.

Several Nordic research initiatives to follow up the funding of the Nordic Graduate School Mathematics Education

Planning for future NoGSME-activities

Much activity has been going on in March and April to secure further funding of the Nordic network that NoGSME has created. At the moment NoGSME consists of over 40 participating institutions, over 120 supervisors and about 90 doctoral students. Almost one hundred new Nordic doctors with theses in the area of didactics of mathematics have graduated during the period of NoGSME (2004–2009). The funding for NoGSME from NordForsk will end after the summer school in September 2009. There is no new opportunity to apply for Nordic graduate schools but NordForsk expects the graduate school, which have been funded over five years, to be able to continue to exist by the help of the participating institutions. Thus we are now approaching the minute of truth, where we will find out if the NoGSME-activities will continue after 2009.
The viability of NoGSME is at least strong enough to produce a number of promising applications for future activities. Thus Frode Rønning, as the chair of the Nordic Society for Research in Mathematics Education, NoRME, has handed in to NordForsk an application for funding of the network NoRME over three years. This is meant to give NoRME some basic resources to plan and act for joint Nordic activities in mathematics education. Another initiative was taken by Morten Misfelt and colleagues in Denmark to apply for a researcher network with focus on studies of use of ICT in mathematics education. Contacts have been made with colleagues in other Nordic countries, as an application to NordForsk has to be supported by institutions in at least three Nordic countries. A third application for resources for researcher networks has been constructed by Madis Lepik at Tallin University, who is planning for networking among Baltic and Nordic researchers in order to carry out some comparative studies in three different areas.
NoRME has asked didacticians at University of Agder to initiate an application for a summer school (doctoral course) in 2010. Thus
contacts have been made with Åbo Academy University, Roskilde University, Iceland University and Linköping University. A joint application for a summer course in May 2010 has been sent to NordForsk. Additionally we are considering repeating the application for a Nordic Master programme, which was sent in 2007. NordForsk has opened a second opportunity to apply for Nordic Master programmes and there seems to be a wish to try again with this application. In 2007 there were extremely many applications for such master programmes, but maybe it will be more reasonable this time.
We can only hope that all or at least some of these applications will be successful. There are also other kinds of contacts going on in order to ensure continued Nordic collaboration in mathematics and science education. We thank all the applicants for the good initiatives and the work done and wish them luck with the continued work.

The eleventh seminar for supervisors of doctoral students

On April 23–24 the final and eleventh seminar for supervisors organised by NoGSME will take place at University of Agder. There will be 35 participants from all the Nordic countries, which is the most ever in the seminars. The theme of the seminar is Critical reviewing of research methodologies in mathematics education. Guest lecturer will be professor emeritus Frank Lester from University of Indiana. He will give two main presentations during the seminar. The first one will be about Making mathematics education research more effective: philosophical, theoretical & methodological considerations. Here is Frank’s description of what he intends to illustrate in the lecture:

The basic argument I will present in this presentation is that the choice of research methods must be made in view of: (1) the researcher’s philosophical and theoretical stance, and (2) what the research community considers the purpose of doing research to be. I will also argue that the purpose of our research should be to make lasting changes in both practice and policy; that is, it should be transformative in nature. Furthermore, I will argue that there often are forces – both political and ideological – at work that influence the methods we use and, consequently, the sorts of questions we seek to answer. Finally, I will argue that some research methods are more likely to lead to the sorts of transformative change that I think are needed.

The second lecture by Frank will have the title Preparing future researchers: what methods should they learn and how should they learn them? This is his abstract for that lecture:

Over the past 40 years, numerous research methods, both qualitative and quantitative in nature, have been developed having particular promise for studying questions related to teaching and learning. Many of these have been ”borrowed” from other fields – anthropology, sociology, and psychology come readily to mind. Indeed, there are so many research methods to choose from that it can seem nearly impossible to decide which ones to emphasize and which to ignore. Then, there is the matter of how to go about developing expertise in the use of the methods that are selected. In this presentation I will address these issues with the intent of generating discussion that may be useful in the continuing development of graduate programs in mathematics education.

The question about how to develop expertise in the use of research methods is of course crucial for supervisors and doctoral students. We can expect a lively debate in the seminar about that question. Frank’s lectures will be followed by discussions in groups and three of our Nordic experts will give shorter presentations preparing that discussion. These presentations will be given by Simon Goodchild, Mogens Niss and Eva Jablonka. In a final panel the members of the NoGSME board will discuss The way forward for research in mathematics education? What are the challenges for the Nordic mathematics education research community?

The NoGSME summer school 2009

In September 21–26 the summer school will take place in Søminestationen in Holbæck, Denmark. This year’s summer school will be organised according to the traditional content and layout, but will also be a doctoral course run by Roskilde University (giving 7.5 ECTS for those who pass the exam). Søminestationen is a course house owned by Roskilde University and it has excellent opportunities for the summer school and beautiful surroundings near the Isefjord. The second announcement has been sent out in the beginning of April and doctoral students are asked to apply to the summer school before April 30. We expect around 30 participants, as in earlier summer schools.

Two recent doctoral theses in mathematics education

At Lund University Ingrid Dash defended her thesis in social sciences on January 30, 2009. The title in English is Flexibility in knowing school mathematics in the contexts of a Swedish and an Indian school class. The main objective of her thesis was to obtain insights into flexible modes of knowing in school mathematics in two school class contexts. Dash wanted to find out how the contexts relate to modes of being a learner in them, with specific focus on learners’ flexible ways of discerning parts and delimiting wholes, and how they understand part- and whole-relationships while doing mathematics. The theoretical exploration was informed by perspectives from phenomenography, variation theory and by constructionist theoretical standpoints. Empirical material was collected from a school class in Southern Sweden and a school class in Orissa, Central-Eastern India. Using contextual analysis, the meaning of the learners was analysed, as it was expressed during interviews and observations, verbally or with the help of mathematics.
The main results of the thesis are different categories of description. The claim is that three modes of knowing emerged from the empirical material. These were according to the author: associative flexible experiencing; compositional flexible experiencing and contextual flexible experiencing. These modes of knowing feature distinct differences: in the depth of understanding mathematics, in how learners use variation when dealing with an object of knowledge, and in learner identity. The associative mode of knowing involved the learner in arbitrary ways of making sense of the material s/he was working with, with a focus on arbitrarily discerned aspects in chains of associations. The compositional mode of knowing meant that the learner made an effort to understand, keeping a focus on compositions, such as number-relations or formulas. The contextual mode of knowing engaged the learners in ways of understanding the context from which critical aspects were to be discerned.
The contexts gave meanings to the content. The knowledge about the context, mathematical and also reality-based, gave meaning to the theoretical constructs. The logic of the mathematical context and content was understood in a more differentiated way than within the two other modes of knowing. The compositional flexible mode of knowing predominated in all parts of the empirical material. The dominant mode of being a learner in the Swedish school class context was simultaneously independent and collaborative, as well as creative and productive. In the Indian school class context, the dominating mode of being a learner was autonomous and committed. Dash claims that in mathematics education there is a need to give pupils tasks containing possibilities both for experiencing variation and for authorship. This also demands of the teacher to observe and evaluate the individual pupil’s understanding and use of the possibilities offered.
On April 2 Unni Wathne defended her thesis at University of Agder, Norway. The thesis is written in Norwegian and the title is Barns tilnærming til analogiske og kombinatoriske resonnement. En longitudinell studie av samspill i smågrupper (Children’s approaches to analogical and combinatorial reasoning. A longitudinal study of collaboration in small groups). The aim of the study was to make empirical and theoretical contributions to the understanding of how children in the lower grades in primary school (grades 1–4) appropriate tools in mathematics. The study focused on children’s analogical and combinatoric reasoning and Wathne studied the appropriation of these tools through collaborative problem solving in four small groups. The study also investigated children’s approaches in solving combinatoric problems.
A sociocultural perspective on learning and development was used. Thirteen children aged between 8–10 years were observed while they worked on combinatoric problems in small groups in a natural school setting. The data material consists of videotapes of group sessions, field notes, and copies of the childrens’ inscriptions.
The analysis is divided into three distinct parts. In the first part of the analysis, three strategies were identified that the children used to solve combinatoric problems: counting all, grouping and learned product. The study also identified instances when children used combinations (hybrids) of these strategies in order to find a solution of the problem at hand.
The second part of the analysis elucidated children’s appropriation of combinatoric problems. Important results are that the children were engaged in common activity, they achieved a common attention by focusing on understanding the problem, on understanding the structure of the problem, and by utilising inscriptions in their problem solving process. Further, the children developed shared meanings for concepts and utterances, they identified connections between their earlier experience with combinatoric problems and their engaging in similar problems. The children were involved in a transformation process where they appropriated tools, actions and remarks from other children through collaboration and used these as tools in solving the problem or other problems encountered later.
The third part of the analysis identified indications of analogical reasoning through the children’s appropriation. When children made an analogy on the background of contextual properties identified in the problem, they were aware that a problem encountered earlier could be helpful in solving the new problem. They were unable to make use of the analogy. When the children made an analogy on the background of structural properties identified in the problem, they were able to identify corresponding relationships between the problem encountered earlier and the new problem. The children realised the connection between earlier experience with such problems and their work with this tool in small groups. The children were involved in a process where they were appropriating the tool and the structure.
The two theses are examples of studies, where the doctoral students have decided to go in the footsteps of the supervisor, when it comes to choice of theoretical framework. Another common feature is that the supervisors are rather general educationalist than researchers in the didactics of mathematics. Do such facts have an influence on the results of the studies and on the impact of them? An open debate among supervisors on such questions would be refreshing. To find a forum for such debates could be one important issue for future work in doctoral programmes.

Future plans for doctoral courses in mathematics education

At the University of Agder two doctoral courses will be given in the autumn of 2009. Although NoGSME cannot offer travel support for Nordic doctoral students any more, the courses are open to all students in the Nordic and Baltic area. The courses are Theory of science from a mathematics education perspective (5 ECTS) and Theories of teaching and learning mathematics (10 ECTS). Students who are interested in these courses can contact Elna Svege at the university (elna.svege@uia.no).
We hope that in the spirit of NoGSME doctoral courses given at other universities will also be open to all students in NoGSME. Announcement of such courses can still be done via the NoGSME emailing list. Material can be sent to me for further information. For information about NoGSME and activities see www.nogsme.no.
NoGSME is also grateful to Jo Boaler at Sussex University and her colleagues in UK for offering a workshop on classroom research in mathematics education, which has attracted interest from many Nordic doctoral students. The workshop runs over a couple of days rather late in spring. Another upcoming yearly event is the 10th conference Teaching mathematics: retrospectives and perspectives that will be held May 14–16, 2009 at Tallinn University, Institute of Mathematics and Sciences. This year the conference will be followed by a one day research seminar on May 16 with participants from the Nordic countries. Information regarding the Conference will be updated on the conference web-page http://www.tlu.ee/bcmath2009.

Barbro Grevholm
Director of NoGSME
University of Agder

Omtale av Identitet og forskning

Wedege, T. (red) (2008). Identitet og forskning: ni essays om at blive matematikdidaktisk forsker. København: Navimat. ISBN 978-87-92397-00-3

GUNNAR GJONE
Universitetet i Oslo

Dialogical inquiry in practice teaching
MARIT JOHNSEN-HØINES

Abstract

This article refers to a project in which the preconditions for a subject-oriented 1, reflective approach towards mathematics and mathematics education in the context of practice teaching were investigated. Student teachers, their tutors and teacher educators participated in the investigation. The article elaborates on their participation as ”co-researchers” in developing the methodological approach and analyses. In addition, the article explores how the analyses provide insight into the didactical conditions for including a subject-oriented approach in practice teaching. More specifically, it was found that practice teaching communication bears the imprint of an evaluative approach that restrains the development of a subject-oriented reflective approach. The conflicting processes characterising these approaches are highlighted.

Sammendrag

Artikkelen utgår fra et samarbeid der lærerstudenter, øvingslærere og matematikkdidaktikere utforsker praksissamtalen slik den foregår i lærer-studiets praksisopplæring. Fokus rettes mot hvordan det etablereres arbeidsformer som stimulerer til at studenter og øvingslærere deltar i samarbeidende forskning. De studerer hvordan samtaler med matematisk og matematikkdidaktisk innretning kan utvikles som del av praksissamtalene. Den faglig forstsettende samtalen løftes fram som didaktisk begrep. Det dokumenteres at praksissamtalen i stor grad er preget som evaluerende samtale, og at dette kan ses i motstrid til og kan virke hemmende i forhold til å utvikle en faglig fortsettende samtale. Artikkelen utdyper et perspektiv der samtaler består av flere samtaler, og der samtaler utvikles i bevegelse mellom, og i konflikt mellom, samtaler.

MARIT JOHNSEN-HØINES
Marit Johnsen-Høines, Professor, Dr. philos ved Høgskolen i Bergen. Hun har lang praksis som grunnskolelærer og lærerutdanner i matematikk, er forfatter av matematikkdidaktisk litteratur og ledet konferansen PME28 (the international group for the Psychology of Mathematics Education) i Bergen 2004. Hun leder forskningsprosjektet ”Læringssamtalen i matematikkfagets praksis”(LIMP), støttet av Norges Forskningsråd.

Quality criteria in mathematics education research

The nature and status of mathematics education as a research discipline has been a subject of reflection in the international research community during the latest three decades. As time passes and research activity in the area proliferates, a meta-reflection on the discipline or the research area becomes more central. One of those meta-reflections concerns what is taken as quality of research and its results. Already in 1994 the ICMI study What is research in mathematics education and what are its results? (Sierpinska & Kilpatrick, 1998), a subgroup within the study had as a task to discuss what criteria should be used to evaluate the results of research in mathematics education. Starting from the idea that, as a scientific field becomes a discipline, there is a need of reaching agreements about what counts as quality in research, the group concluded that it is not possible to present one unified set of criteria of quality, but that it was important to take into account the ”considerations that inform judgements about the quality of research in mathematics education” (p. 29). Therefore, the group proposed exploring how the different elements that constitute mathematics education as a field could generate different sets of criteria. For example, is it possible to generate criteria for quality of research when considering the relationship between research and educational practice? Or its relationship to mathematics? Or to other foundational disciplines? (p. 30–31). Each one of these sets of relationship will lead to considering different definitions of quality.
More recently, the issue was addressed at the ICMI-2008 symposium in Rome celebrating the centennial of the foundation of ICMI. In the proceedings from the symposium, subtitled Reflecting and reshaping the world of mathematics education (Menghini et al., 2008), there is a chapter by Jeremy Kilpatrick, with the title The development of mathematics education as an academic field. Here the scientific status of mathematics education research is discussed and it is questioned to what extent the research in the field has developed scientific theories. From the reaction to the lecture given by Jean Luc Dorier and from the debate that followed, it was clear that there where different opinions on how to judge the nature and scientific status of different research trends, of the results achieved and of the theories developed in the field. Such differences may be explained by differences in research traditions that have developed differently in various research milieus and countries. This type of discussion could suggest the idea that even though in a community there is a need for having criteria of quality, it may be almost impossible to think that there can be one and only one set of criteria that would be equally applicable and relevant to all research carried out in the field. In fact, many authors argue for the need of reconstructing classic research criteria such as validity, reliability and generality in relation to the emergence of new research methodologies and new research problems (e.g., Lesh et al., 2000; Vithal & Valero, 2003).
In the Nordic region mathematics education research as an academic field has been undergoing a rapid development during the last two decays. In 1992, at the time that the symposium with the title Criteria for quality and relevance in mathematics education (Nissen & Blomhøj, 1993) was held, there were only few professors and less than 10 Ph.D. students in the whole Nordic region. This symposium was one of the importing starting points for the Nordic collaboration in the field. The plans of the publishing nomad were finalised at a meeting during the symposium, and the first issue of the journal was published the year after. Internationally prominent researchers – among them Jeremy Kilpatrick – gave presentations and commenting on ongoing Nordic research projects. Since that time, the discussion of criteria of quality has occupied an important space in the conversations among researchers. The need for a broader and updated discussion on this issue is even more actual now than ever since we are experiencing a boost in the growth of the community. Nowadays, we have a Nordic Graduate School in Mathematics Education, several national doctoral programs and growing research milieus with professorships in many places in the Nordic region. However, mathematics education is still in the process of establishing itself as an academic discipline institutionally. Reflecting on the characteristics and the scientific status of research and research results in mathematics education is of extreme importance in relation to the interdisciplinary relationship with supporting sciences such as mathematics, pedagogy, psychology and sociology. What is specific for the forms of research that mathematics educators develop, and why are results relevant or even necessary? Moreover, we need to keep the meta-reflection on our discipline alive as a basis for prioritising the research effort and resources between different possible research programmes.
For these reasons we have decided to make Quality criteria in mathematic education research the topic for the next thematic issue of nomad. The theme can be addressed from many different perspectives: normatively, philosophically, historically, by meta-analysis of research papers or through case studies, and all such approaches are of interest. Of course, we prefer contributions with a distinct Nordic perspective. Papers for the thematic issue should be submitted no later than August 15, 2009.

In this issue

Two of the three papers presented in this issue are researching student teachers’ beliefs and didactical reflections concerning the teaching and learning of mathematics. In both cases the underlying purpose is to provide scientific knowledge for raising the quality of teacher education in mathematics. The third paper reports on a theoretical and empirical investigation of how to operationalize the concept of contextualisation in a teaching experiment challenging and supporting the students’ probabilistic reasoning.
The paper by Lisen Häggblom Lärarstuderandes syn på lärande i matematik reports a phenomenological study on prospective teachers’ attitudes and beliefs in relation to the learning of mathematics. The study includes 77 Finnish prospective teachers, and their attitudes and beliefs are uncovered by means of qualitative text analysis of essays produced on the encouragement: Describe your view of students’ learning in mathematics and your role as a mathematics teacher. In the first part of the paper the author presents an overview of part of the research literature on student teachers’ and teachers’ beliefs and reflections on the learning of mathematics. The analysis of the essays identifies respectively four and six general positions concerning the students’ beliefs about pupils’ learning of mathematics and about their own role as a mathematics teacher. Beliefs about the importance of pupils’ motivation for learning mathematics and its connection to affective factors and the important role of pupils’ activity and communication in the learning process are among the most dominant. On the other hand, addressing diversity among pupils, showing the relevance of mathematics in connection to the daily life of the pupils, and organising the learning environment for activity and communication are some of the student teachers’ beliefs concerning their role as mathematics teachers. In addition, the analysis shows that the students express clear general positions about mathematics as a school subject, and that these positions are connected to their general beliefs about the learning and teaching of mathematics. In general, the analysis documents that the students have a holistic perspective on the learning and teaching of mathematics, and hence the research supports that didactical reflections from a holistic point of view is given appropriate attention in teacher education programmes.
In the paper Dialogical inquiry in practice teaching, Marit Johnsen-Høines reports from a developmental project in which researchers/teacher educators, student teachers and tutor teachers collaborated in subject-oriented conversations within the framework of teaching practice. The paper is written in a way that invites the reader to have a sense of how the process of collaboration between these three groups of people developed along the project. Starting from the mismatch between teaching practice being recognised as a central element in teacher education and teaching practice not being perceived by teacher students as an opportunity to connect theory with practice, the teacher educators – who also were the researchers – invited tutor teachers and student teachers to engage in a process of improving the types of conversations that they hold in the evaluative meetings connected to teaching practice. The openness of the invitation made by the teacher educators contributed in the creation of a sense of ownership to all participants in the project. The main aim of developing subject-oriented conversations, that is, conversations around mathematical and mathematics education issues emerging in the context of teaching practice, became a common goal. The subject-oriented conversations also permitted to alter the characteristics of the conversations that normally take place in teaching practice, making the new space of conversation a richer space for the professional qualification of student teachers.
The third paper by Per Nilsson Operationalizing the analytical construct of contextualization aims at developing and testing a set of analytic tools for organizing our thinking about teaching and learning mathematics. The theoretical frame concerns the process to assimilate new elements of knowledge into a conceptual network and it relates explicitly to constructivism. The theory includes thus the process to contextualize experiences by doing investigations. The theoretical issues are investigated in the paper by means of analysing the students activity in a very carefully designed teaching experiment which challenge and support lower secondary pupils’ probabilistic reasoning. In the experiment eight students are engaged with a structured series of tasks with two dices, designed as a set of games. The overall purpose has been to challenge the students to base their probability reasoning on the structure of the sample space. For example, with rolling two dices marked (222244) and (333355), the students are challenged to reason on the outcome space for the sum, namely [5, 7, 9] and in the next turn to recognise the different probability of the outcomes. The author analyses the mathematical activities involved in a contextualization process using four sets of categories, identified from the literature as crucial in contextualization. These four categories, which are (1) the context and mathematical potential of the problems, (2) issues of familiarity, (3) context variation and (4) reflection on validity, explain very well the difficulties that the students experienced in the experiment.

References

Lesh, R., Lovitts, B. & Kelly, A. E. (2000). Purposes and assumptions of this book. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 17–34). Hillsdale: Lawrence Erlbaum Associates.
Menghini, M., Furinghetti, F., Giacardi, L. & Arzarello, F. (Eds.). (2008). The first century of the International Commission of Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education. Roma: Istituto della Enciclopedia Italiana fondata da Giovanni Treccani.
Nissen, G. & Blomhøj, M. (Eds.). (1993). Criteria for scientific quality and relevance in the didactics of mathematics. Roskilde: Danish Research Council for the Humanities.
Sierpinska, A. & Kilpatrick, J. (1998). Mathematics education as a research domain: a search for identity. Dordrecht: Kluwer.
Vithal, R. & Valero, P. (2003). Researching mathematics education in situations of social and political conflict. In A. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.), Second international handbook of mathematics education (Vol. 2, pp. 545–592). Dordrecht: Kluwer.

Lärarstuderandes syn på lärande i matematik
LISEN HÄGGBLOM

Sammanfattning

Föreliggande studie syftar till att belysa lärarstuderandes syn på lärande i matematik. Ur metodologisk synvinkel har studien en fenomenologisk ansats som karakteriseras genom analyser av individers åsikter och uppfattningar om matematiklärande. Studien återspeglar inte bara studerandes åsikter utan synliggör även aspekter av ett komplext matematiklärande. Utgående från ett skriftligt datamaterial på 77 essäer utformades en analysmodell där de studerande uttrycker sina åsikter om elevers lärande i matematik och om den egna lärarrollen. Åsikterna har sammanställts till kvalitativa uppfattningar om matematiklärandet och resultatet visar på ett holistiskt förhållningssätt till undervisningen i matematik. Studien är ett bidrag till forskning om lärarstuderandes didaktiska reflektioner.

Summary

The aim of the study is to illustrate the beliefs of teacher students considering learning mathematics. The study has been carried out using a qualitative research approach where 77 written documents have been analysed to show the beliefs of the students concerning pupils’ learning, the teacher role and the view of mathematics as a subject. During the analysis some shared points of view that were central and some that were more peripheral surfaced. The shared points of view considering pupils’ learning are expressed as the pupils being motivated and acquiring a feeling of success, pupils being different and learning in different ways, pupils learning by being active, and pupils learning through linguistic and oral communication. The beliefs describing the teacher’s role as a mathematics teacher consist of the teacher being able to plan and organize teaching, the teacher understanding and considering the differences in pupils, the importance of the teacher’s own subject knowledge, the importance of the teacher in relation to the learning environment, the teacher’s interest and devotion igniting the flame and the teacher as the captain of a ship.
The beliefs considering mathematics are described as visualizing a reality-based content, mathematics presupposing good knowledge, mathematics as an important school subject with a wide potential for development and as mathematics triggering emotions. The described points of view often reflect a holistic perspective of the learning process by arguing that a teacher should motivate pupils and consider different possibilities of learning, as well as making the mathematical content reality-based and meaningful. The study is a contribution to research on the didactic reflections of teacher students.

LISEN HÄGGBLOM
Lisen Häggblom är ped dr och verksam som lektor i matematikens didaktik vid Pedagogiska fakulteten, Åbo Akademi i Vasa. Forsknings-området är elevers kunskapsutveckling i matematik, språkliga dimensioner i elevers matematiklärande samt lärarstuderandes syn på lärande i matematik.

Operationalizing the analytical construct of contextualization
PER NILSSON

Abstract

This article elaborates on the construct of contextualization, which constitutes a constructivist contextual view on learning. Principles of constructivism and contextualization are operationalized into a set of four analytical categories that teachers and researchers can use in organizing their thinking about teaching and learning mathe­matics. The categories are discussed and verified throughout the design and analysis of a classroom compatible learning activity, which is thought to promote probabilistic reasoning. The article discusses suggestions for developing the operationalization and, thus, encourages future efforts that further explore the explanatory power of contextualization and its analytical categories.

Sammanfattning

Denna artikel elaborerar kontextualiseringsmodellen, som är en konstruktivistisk kontextuell modell för lärande. Principer avseende konstruktivism och kontextualisering operationaliseras i fyra analytiska kategorier, som lärare och forskare kan använda för att organisera undervisning och lärande i matematik. Kategorierna är diskuterade och verifierade genom utformningen och analysen av en klassrumsliknande aktivitet, som syftar till att stimulera resonemang om sannolikhet.
Förslag på hur operationaliseringen kan utvecklas diskuteras och artikeln inbjuder, i anslutning till sådana förslag, till framtida insatser, där kontextualisering och de analytiska kategoriernas förklaringsvärde ytterligare utforskas.

PER NILSSON
Per Nilsson is PhD in mathematics education. His main research interest is on students’ learning in the domain of probability.

Gard Olaf Brekke in memoriam

Tidligere redaktør av Nomad høgskoledosent Gard Olaf Brekke døde den 18. mars 2009 nær 66 år gammel. Norge har mistet en av sine sentrale matematikkdidaktikere som også hadde stor betydning for lærerutdanningen i matematikk i Norge.

Gard Brekke var født 21. september 1943. Han vokste opp på Voss i nærheten av Bergen. Etter examen artium tok han til med studier i matematikk og fysikk ved Universitetet i Bergen. Han tok hovedfag i matematikk i 1969 ved Universitetet i Bergen.
Fra 1971 var han tilsatt som høgskolelektor i matematikk ved Notodden lærarskole (senere Telemark lærarhøgskole og Høgskolen i Telemark). De tre første årene hadde han redusert stilling for å undervise i ungdomsskolen, dette for å få bedre kjennskap til skoleslaget.
Da Gard Brekke ble tilsatt ved Notodden lærerhøgskole var Norge inne i en omfattende reform av skoleverket – både grunnskole, videregående skole og lærerutdanning. Den obligatoriske skolegangen skulle utvides fra 7 til 9 år, og strukturen i videregående skole skulle endres. Som en konsekvens måtte Norge også få en ny lærerutdanning. Spesielt for lærerutdanningen spilte miljøet ved lærerskolen på Notodden en sentral rolle. Gard Brekke ble tidlig trukket inn i arbeid for de sentrale skolemyndigheter, både for grunnskolen og lærerutdanningen. Han deltok i læreplanarbeid for lærerutdanning, ikke bare for matematikk – men også for ”datalære” i lærerutdanningen (før en betegnet dette som informatikk). Han deltok videre i utvikling av planer for fjernundervisning i matematikk, og var med å lage eksamensoppgaver for avgangsprøva i matematikk i grunnskolen. I perioden 1980 til 1984 var han fagrettleder i matematikk i Lærerutdanningsrådet, noe som ga han stor innflytelse på utformingen av lærerutdanningen.
Gard Brekke var hele sitt yrkesaktive liv knyttet til det som etter hvert ble Høgskolen i Telemark. Sammen med andre realister ved høgskolen var han med å etablere institusjonen som et sentralt miljø for realfag i lærerutdanningen i Norge. I 1990 ble han førsteamanuensis og fra 1994 var han høgskoledosent i matematikkdidaktikk. Gard Brekke hadde også undervisnings- og veiledningsoppgaver ved andre institusjoner, som Høgskolen i Agder – der han var Høgskoledosent II fra 1998 – og Universitetet i Oslo. Han var med i flere lærebokprosjekter, både for grunnskolen og for lærerutdanning.
I skoleåret 1984–1985 hadde han et studieopphold ved Shell Centre for Mathematical Education ved University of Nottingham. Her kom han i kontakt med miljøet rundt Alan Bell, som blant annet hadde fokus på såkalt diagnostisk undervisning. Oppholdet ledet til en doktorgrad i Mathematics Education ved University of Nottingham i 1991. Tittelen på doktorgradsarbeidet var: Multiplicative structures at ages seven to eleven. Studies of children’s conceptual development and diagnostic teaching experiments. Arbeidet førte til en utvidet kontakt med miljøet ved Shell Centre for norske matematikkdidaktikkmiljøer. Flere norske matematikkdidaktikere har hatt kortere eller lengre opphold ved institusjonen, noe som klart har beriket den matematikkdidaktiske forskningen i Norge.
Fra 1992 var han også forsker ved Telemarksforsking-Notodden. Dette kombinerte han med undervisning ved høgskolen. Det som vi vil spesielt trekke fram er Gard Brekkes engasjement innenfor ulike forskningsprosjekter, både nasjonale og internasjonale.
I perioden 1991 til 1997 var han medlem i den nasjonale ledelsen av TIMSS prosjektet (Third International Mathematics and Science Study). Dette prosjektet innledet Norges deltaking i internasjonale sammenlikningsstudier, noe som har fått betydning for utviklingen av matematikkunder-visningen videre i Norge.
Et internasjonalt engasjement som må trekkes fram er arbeidet innenfor The International Group for the Psychology of Mathematics Education (PME). Han var medlem av ”eksekutiv” komiteen fra 1997–2001, og var med i programkomiteer for PME konferansene i 1997 og 2001. Senere har også andre norske matematikkdidaktikere engasjert seg innenfor PME, og det er lett å trekke forbindelseslinjene tilbake til det engasjementet Gard Brekke hadde i organisasjonen. Han var også engasjert i de nordiske konferansene for matematikkundervisning (NORMA), der han var med i programkomiteene for konferansene i 1994 og 1998.
Et annet forhold som var sentralt i nordisk matematikkdidaktisk forsk-ning var arbeidet som redaktør i dette tidsskriftet – NOMAD. Fra 1997 til 2000 var vi begge redaktører, med Gard som ansvarlig redaktør. Her la han ned et betydelig arbeid, slik at vi etter 4 år kunne sende redaktørarbeidet videre til Finland. Det var en rekke praktiske forhold som måtte avklares da tidsskriftet flyttet til Norge, og Gard måtte bruke mye tid på å få tidsskriftet trykket og publisert. Det er vel riktig å si at NOMAD i perio-den etablerte seg som et internasjonalt forskningstidsskrift. Forskere utenfor Norden ble oppmerksomme på tidsskriftet og det kom inn flere bidrag fra slike forskere. Kontaktene som Gard Brekke hadde i andre land – spesielt England – kom her til god nytte. Gard Brekke har vært medlem av redaksjonsrådet for NOMAD siden han sluttet som redaktør.
Gard Brekke og undertegnede hadde i denne perioden samarbeid på en rekke områder. I 1992 ble KIM-prosjektet etablert, som et samarbeid mellom Telemarksforsking-Notodden og Institutt for lærerutdanning og skoleutvikling (ILS) ved Universitetet i Oslo (etter noen drøftinger sto forkortelsen KIM for Kvalitet I Matematikkundervisningen). KIM-prosjektet bygget på ideene om såkalt diagnostisk undervisning som Gard Brekke hadde kommet i kontakt med ved Shell Centre. Prosjektets ide var å utvikle oppgaver for å kart-legge elevenes alternative oppfatninger av matematiske størrelser og begreper, såkalte ”misoppfatninger”. Mange lærere i Norge har brukt materialet som ble utviklet, prosjektet har utviklet seg til et sentralt element innenfor matematikkundervisningen i Norge. Mange lærere har brukt det i undervisningen, og det har også fått en plass innenfor lærerutdanningen ved flere institusjoner. KIM-prosjektet ga også data til en rekke hovedfagsoppgaver i matematikk-didaktikk. Prosjektet er nå i ferd med å gå over i en ny fase – oppgavene skal digitaliseres og gjøres tilgjengelig for nedlasting fra internett. Gard Brekke rakk å være med i forberedelsene til dette arbeidet.
Ved oppstarten av de nasjonale prøvene i regning, var Gard Brekke en selvskreven deltaker i utvalget for prøvene. I dette arbeidet kom han med viktige bidrag, der han brukte sine erfaringer fra konstruksjoner av diagnostiske oppgaver.
Fra slutten av 1990-årene har det i Norge vært satt i gang en rekke utredninger, etablert ulike typer nettverk, og det har blitt foretatt en rekke vurderinger av forhold ved matematikkundervisningen i Norge. Gard Brekke deltok i flere slike aktiviteter. Her skal bare kort nevnes at han var nettverkskoordinator for matematikk i regi av departementet og han hadde en sentral posisjon – som prosjektleder – i vurderingen matematikkfaget i læreplanen av 1997. Gard Brekke holdt en lang rekke kurs for lærere. Gjennom deltakingen i prosjektene hadde han en omfattende erfaring å bygge på.
Gard Brekke påvirket matematikkundervisningen og lærerutdanningen i Norge gjennom flere tiår. Han var videre en inspirator for mange matematikkdidaktikere. I den perioden da Gard Brekke var yrkesaktiv ble matematikkdidaktikken etablert som et eget vitenskapsområde i Norge. I denne etableringen spilte han en betydelig rolle. Mange yngre forskere hadde et opphold ved høgskolen på Notodden, der de ble kjent med arbeidene til Gard Brekke. Han var også et bindeledd til det internasjonale matematikkdidaktikkmiljøet.

Gunnar Gjone
Universitetet i Oslo

Research education activities in the Nordic graduate school in mathematics education

The 11th seminar for supervisors organised by NoGSME attracted 35 participants and became a lively activity at University of Agder. Frank Lester, in his introductory plenary presentation, began with some philosophical, theoretical and methodological considerations about how to make mathematics education research more effective. Frank Lester referred to what Paul Cobb writes in the Second handbook of research on mathematics teaching and learning. Paul Cobb mentions many of the theoretical perspectives used in mathematics education, such as radical constructivism, sociocultural theory, symbolic interactionism, distributed cognition, information-processing psychology, situated cognition, critical theory, critical race theory, and discourse theory. In this rather bewildering array of theoretical perspectives Cobb seeks to address how researchers might make and justify their decisions to adopt one theoretical perspective rather than another. Frank Lester also quoted the National Research Council about scientific research in education, saying ”at its core scientific inquiry is the same in all fields”.
They present six scientific principles, namely, to pose significant questions and study them empirically, to link research to theory, to study questions directly, to create a coherent and explicit chain of inferential reasoning, to replicate and generalise across studies and to disclose research to allow for scrutiny and critique. Frank Lester asked how we are to pursue both our desire for fundamental understanding and the need to put our results into practice.
The American Statistical Association report presents components of a research programme in the following way: Generate ideas about a phenomenon, clarify goals and define concepts and constructs, develop concepts and constructs, make assessment of measurement and feasibility, implement small scale studies, confirm and generalise, and extend and create long-term community follow up. Finally Frank Lester discussed some design experiment research like Gravemeijer’s teacher-developer-researcher collaboration, Battista and Clement’s curriculum research as design, Lesh and Kelly’s design research and teaching experiments as described by Coob and Steffe and Thompson. Design research could be seen as one of the recent trends in development of methodology.
A lively discussion was followed by a panel debate about methodological issues. In the panel Mogens Niss, Eva Jablonka and Simon Goodchild all presented their personal preferences and views about methodologies and argued against each others views.
In his second presentation Frank Lester asked what methods doctoral students should learn and how they should learn them. He claimed that from the 1960's to the present the nature of doctoral programmes changed from highly structured studies with little or no actual experience in doing research to apprenticeship training. Earlier, students were trained primarily in mathematics but today they come from a wide variety of degrees. Research used to focus on teaching but today there are more wide ranging interests, today there are also more interest in continuing doing research. Frank Lester presented four questions that were discussed during the seminar: What ”core” knowledge should all doctoral students have? What research training should they receive? Who should be responsible for preparing doctoral students? What should be the purpose of a doctoral programme? These are all vital questions to handle for the doctoral programmes in mathematics education in the Nordic countries. Other questions that created lively discussion in the small groups during the seminar were: Should research in mathematics education be transformative in nature? How do you distinguish between methods and methodology? What other ways of categorising research methods in addition to the distinction between qualitative and quantitative could be useful. What methods for triangulation are used in research in mathematics education? What can we learn from these examples?
The seminar concluded with a discussion among NoGSME board members concerning the way forward for research in mathematics education and what challenges there are for the mathematics education community. Wishes were expressed that the new organisation NoRME will secure continued activities in the spirit of NoGSME and that the collaboration built so far in the NoGSME network can survive and continue to support both doctoral students and supervisors in the Nordic and Baltic countries.

Summer school 2009 in Denmark

The NoGSME summerschool in 2009 will, as announced before, take place in Søminestationen in Holbæck in Denmark. Almost all accessible places for participants are taken by the 30 who have registered by the end of April. There are possibilities for a few more to be accepted. Contact the director of NoGSME at once if you have missed the chance.
This year the summer school is formally a doctoral course organised by Roskilde University. Thus the demand on participants will be somewhat different from earlier summer schools. The paper prepared by students in advance will be more elaborated and students will also have to prepare questions to fellow students about their research studies. After the summer school there will be a formal examination in the form of an essay to write.
Group leaders will be Mogens Niss, Morten Blomhøj from Roskilde University and Marta Menghini from La Sapienza in Rome. A fourth group leader will take part if the number of students exceeds 32.

Summer school in Norway in 2010?

An application has been sent to NordForsk for a summer school in May 2010. NordForsk will make the decision in June 2009 and if the summer school is supported financially it will be announced as soon as possible in order to ease students planning. It is planned to take place in Dömmesmoen, which is part of the campus of University of Agder, and beautifully situated in Grimstad, on the south coast of Norway.

The 10 th international conference in the Baltic countries

This year is the 25th anniversary of the Baltic yearly conference and it took place in Tallin University in May 14–16. There were four themes during the conference: Teaching and learning mathematics, Extracurricular activities in mathematics education, Education and professional development of mathematics teachers, and Technology in mathematics education. The first plenary session was about didactics of mathematics as a research discipline and speakers were Markku Hannula and Barbro Grevholm. The second plenary session dealt with mathematics education research and researcher education in Baltic countries. In the third plenary session Erkki Pehkonen spoke about how Finns learn mathematics: What is the influence of 25 years of research in mathematics education? According to him there is no easy explanation to the good results in international comparisons by Finnish students. In the final plenary session the focus was on Developments of school mathematics in Baltic countries.
In addition to the regular conference a research seminar was organised on May 16. The aim of the seminar was to initiate joint Nordic-Baltic comparative research projects in different areas of mathematics education. The proposed topics for the projects were:
– proof and proving in school mathematics,
– mathematics teachers’ educational beliefs, and
– mathematics textbooks.
Experienced researchers from Nordic countries acted as project leaders, and they will also be responsible for methodology and general planning of the project. The leaders were Markku Hannula, Kirsti Hemmi and Barbro Grevholm. Researchers from Baltic countries interested in these topics were invited to participate and the projects were started. Funding has been applied for over a period of three years. A web page will be created to make the projects accessible for interested colleagues.

Two new doctoral dissertations in the Nordic countries

Stine Timmermann Ottesen defended her thesis on April 17 at Roskilde University. Relating University mathematics teaching practices and students’ solution processes is the title of her work. The teaching practices were examined via observations, according to a new instrument developed by her, building on earlier studies. The students’ solution pro­cesses were examined through a specifically developed research design. In a pilot study a hypothesis was presented and then tested in the main study. The findings show that it is difficult for students to find a proof strategy, which could provide them with a proof structure. For many of the students signs can be found indicating that a sociomathematical norm of proof production has been established among them. This norm sometimes says that a proof can be constructed just by combining the wordings of some well chosen theorems that combine and include words appearing in the tasks. That norm could be related to the norm that proofs are constructed through the use of tricks. Students hesitate to search for the formal definitions of the concepts involved and prefer to base their reasoning on the concept images they have developed. The students find it difficult to understand the explanations of details in the proofs and this might be because the structure is unclear to them. A factor that sustains the misconceived sociomathematical norms could be the lack of attention given to the connection between the structure and the details in the proofs.
Ole Kristian Bergem defended his thesis at University of Oslo on May 19 with the long title: En analytisk redegjørelse for relasjonen mellom allmenn­didaktikk, realfagsdidaktikk og matematikkdidaktikk, med særlig henblikk på en belysning av sentrale forskningsmessige bidrag fra de respektive feltene til forståelsen av matematikklasserommet (An analytical account of the relation between general didactics, natural science didactics and mathematics didactics, with special focus on enlightenment of central research contributions from the fields to the understanding of the mathematics classroom). The empirical material in the study is observations in grade 9 classrooms and analyses of the video recordings and interviews with pupils and teachers. Focus of the study is challenges related to implementation and use of new learning tools in mathematics. From observations and interview data is shown that use of work plans mediates pupils’ learning work in mathematics. This new learning tool is discussed from a perspective of activity theory. Work plans seem to be a reply to the demand for individually designed learning, but the author argues that these plans generate new pedagogical and didactical challenges for mathematics teachers. The didactical contract in the classroom seems to change and the new role- and responsibility distribution seems unclear to both pupils and teachers. For example, work plans allow pupils to work with mathematics only one or two days per plan period of two to three weeks. This might reduce the learning opportunities of the pupils. The use of work plans also leads to much individual written work, which can be seen as problematic in relation to sociocultural theories of learning.
Another part of the thesis investigates the relation between tasks from daily life and different discourses in the classroom. The analyses show that when pupils discuss among themselves they have serious problems to relate the mathematical knowledge to the authentic tasks. They end up in an everyday discourse with little mathematical relevance. In the whole class discussion the teacher holds the discourse in a mathematically relevant track through his balancing interference. With empirical data from classrooms and practice oriented questions these analyses and findings should be useful and relevant both in school and teacher education.
The two theses investigate teaching practices in relation to the learning opportunities that pupils/students are offered. The different theore­tical approaches allow authors to find answers to different kinds of questions. But both studies offer insights that can be used by mathematics teachers in their development of teaching and reflections on their own professional identity.

Already now we see more theses coming up before summer

It has been announced that Lovisa Sumpter in Umeå University, Kajsa Bråting in Uppsala University, Claire Berg at University of Agder, and Liisa Näveri at University of Helsinki will defend their theses in the beginning of June. We will return to these dissertations in the next issue of Nomad. Thus, in the first half of 2009 there seems to be at least 8 dissertations in mathematics education. The year 2009 might be as productive in number of dissertations as 2006, with 21 theses.

Barbro Grevholm
Director of NoGSME
University of Agder

An anthropological approach to a transitional issue. Analysis of the autonomy required from mathematics students in the French lycee

CORINE CASTELA

Abstract

This paper intends to contribute to the process of theoretical networking within the mathematics education research community. Some key elements of the Anthropological Theory of Didactics are recalled and used to deal with the issue of French students’ transitional difficulties in mathematics between Collège (lower secondary school) and Lycée (upper secondary school). The intention is showing how this theoretical framework, in contrast with a theoretical framework of Advanced Mathematical Thinking, provides tools to analyse the changes between these two institutions and thus supports the following assumption: An increasing autonomy as problem solvers as well as mathematics learners is required from the upper secondary school students. This hypothesis led to a clinical investigation on high school students’ homework. This paper addresses the hypothesis by drawing on the case of three high-achieving students.

CORINE CASTELA
Corine Castela teaches mathematics and didactics to secondary school student teachers. As a member of the André Revuz Laboratory (previously DIDIREM-Paris), her research topics are: knowledge involved in mathematics problem solving, secondary school and university students’ homework, international curricular comparison, mathematics in workplace and professional education, and developing and connecting theoretical approaches. She is specially concerned with the issue of the variety of research working languages. She holds regular collaboration in Spanish with South American researchers in countries such as Mexico and Chile.

Bokanmälan. Relating practices and research in mathematics education

C. Bergsten, B. Grevholm, H. Strømskag Måsøval & F. Rønning (red.) (2007). Relating practices and research in mathematics education. Proceedings of NORMA 05, fourth Nordic conference on mathematics education. Trondheim: Tapir Academic Press. ISBN 978-82-519-2212-8

OLA HELENIUS
Örebro universitet och NCM, Göteborgs universitet

Handheld calculators as tools for students’ learning of algebra

PER-ESKIL PERSSON

Abstract

What evidence can be found in recent research literature of the potential positive or negative effects of using graphic calculators (GC) and symbolic calculators (CAS) in mathematics education? The focus of this literature review is the use of handheld calculators and their effect on algebra learning, with theoretical backgrounds for the use of this type of technology in classroom practice. Special attention is given to three areas: students’ conceptions of literal symbols and of algebraic expressions, fundamental for their ability to work with algebra; functional and modelling approaches, both important for students’ view of algebra as a useful tool in problem solving; and approaches within CAS, which put special demands for changes in educational methods. Results of some recent meta-studies, based on a relatively large number of research papers and reports, are also discussed, as well as the importance of students’ and teacher’s beliefs. Common results are compiled and synthesised for a formulation of some important implications for teaching and pre-service teacher education.

Sammanfattning

Vilka belägg kan man finna i nyare forskningslitteratur för potentiellt positiva eller negativa effekter av att utnyttja grafräknare (GC) och symbolhanterande räknare (CAS) i matematikundervisningen? Fokus för denna litteraturgenomgång är användningen av räknare och deras effekt på algebralärande, inklusive teoretisk bakgrund för användningen av denna typ av teknologi i klassrumsarbetet. Särskild uppmärksamhet ägnas tre områden: elevers uppfattningar om bokstavssymboler och algebraiska uttryck, fundamentala för deras förmåga att arbeta med algebra; funktions- och modelleringsansatser, båda viktiga för elevers syn på algebra som ett användbart verktyg i problemlösning; samt ansatser med CAS, som ställer särskilda krav på förändringar i undervisningsmetoderna. Resultat från några nyare metastudier, baserade på ett relativt stort antal forskningsrapporter, diskuteras såväl som betydelsen av elevers och lärares uppfattningar om räknare. En sammanställning och syntes av vanligt förekommande resultat görs för en formulering av några viktiga implikationer för undervisning och lärarutbildning.

PER-ESKIL PERSSON
Per-Eskil Persson has a long background as a teacher in mathematics and physics at upper secondary level, and is presently working as lecturer at the School of Teacher Education at Malmö University. He holds a licentiate in Mathematics and learning and is also a doctoral student at Luleå University of Technology. His main research interests are different aspects of algebra learning, especially at upper secondary level, and the use of technology in mathematics education.

The role of overview papers in mathematics education research

The explosion of mathematics education research has been an issue of concern for many researchers in the field. For some people the explosion in terms of the growing amount of research results and, particularly, theoretical and methodological orientations, is problematic. Some people argue that some kind of unification of theories and clarity about the concepts used in the field would be desirable. Some other argue that such aspiration of coherence and unification is impossible due to the nature of the objects of study of mathematics education and the historical time in which the field has developed. Independently of which position a researcher adopts in relation to this issue, it is evident that there is a need of much more reflexivity in the field. Reflexivity refers to efforts to meta-analyse the field of research, its theories, its methodologies, and its results. Reflexivity allows having better qualified understandings of the advancements in the field and of the blind spots.
In mathematics education research there has been a tradition for publication of research report papers, presenting the results of empirical studies or papers, which develop a particular theoretical perspective. Published papers presenting overviews, classifications of results and analysis of existing sub-domains have been rarer. The scarcity of this type of research in the field is quite interesting since in most research processes the production of research overviews – meant to place the significance of a particular project – is an important activity to which researchers devote time. In the case of doctoral studies, students use a great deal of their time producing comprehensive literature reviews which provide ideas about the trends in particular sub-domains, as well as a pinpointing of the areas that could be further explored. Nevertheless, it is seldom to find published papers that capitalize on this important research activity to inform the community about what is being researched in the field.
We have seen that the type of overview papers mentioned above can be of importance as the platform for generating more reflexivity and awareness of the field about itself. Therefore, we would like to invite the NOMAD audience to consider submitting solid, well-structured overview papers that address some of the debates in particular research areas. We find this type of papers to be very informative for the Nordic community. It is also an important source for generating the possibility of crossing the specificities of particular research projects and building a basis for examining the contributions of the field in terms of results and possible fruitful interpretation of and for practice.
In this issue we have one of the few theoretical overview papers that NOMAD has ever published, namely the paper by Per-Eskil Persson in which key issues of the use of ICT in mathematics teaching and learning are presented and discussed. We encourage readers to study this paper and be stimulated to submit articles of this type.

In this issue

In this issue we are publishing three research papers. The paper by Corine Castela, An anthropological approach to a transitional issue, analysis of the autonomy required from mathematics students in the French lycee, intends to contribute to the process of theoretical networking within the mathematics education research community by presenting to the Nordic community some key elements of the Anthropological Theory of Didactics. The intention is showing how this theoretical framework, in contrast to a theoretical framework of Advanced Mathematical Thinking, provides tools to analyse the changes when students move from one educational level to a higher level requiring different types of mathematical praxeologies.
The paper The case of Brandon: The dual nature of key ideas in the classroom, by Manya Raman and Michelle Zandieh, look at proof production in the midst of classroom interaction. The setting is a college level geometry course in which students are working on the following task: Prove that two parallel transported lines in the plane are parallel in the sense that they do not intersect. A proof of this statement is traced from a student’s idea, through a small group discussion, to a large class discussion moderated by a teacher. As the proof emerges through a series of increasingly public settings we see ways in which the key idea of the proof serves to both open and close class discussion. The authors look at several examples of opening and closing, and hereby showing how not only the key idea, but also the warrants and justifications connected to it, play an important role in the proof development.
Per-Eskil Persson, in his paper Handheld calculators as tools for students’ learning of algebra, presents a comprehensive literature review on the use and effects of two forms of ICTs in mathematics education. The focus of this literature review is the use of handheld calculators and their effect on algebra learning, with theoretical backgrounds for the use of this type of technology in classroom practice. Special attention is given to three areas: students’ conceptions of literal symbols and of algebraic expressions, fundamental for their ability to work with algebra; functional and modelling approaches, both important for students’ view of algebra as a useful tool in problem solving; and approaches within CAS, which put special demands for changes in educational methods. Results of some recent meta-studies, based on a relatively large number of research papers and reports, are also discussed, as well as the importance of students’ and teacher’s beliefs. Common results are compiled and synthesized for a formulation of some important implications for teaching and pre-service teacher education.
In addition to the research papers this issue contains a review of Relating practices and research in mathematics education – Proceedings of NORMA 05, fourth Nordic conference on mathematics education by Ola Helenius, in which he also reflects on the role of conferences and conference proceedings in the Nordic mathematics education research community. In Gard Olaf Brekke in memoriam Gunnar Gjone commemorates the former Editor of NOMAD. Gard Brekke passed away in the spring of 2009.

The case of Brandon: the dual nature of key ideas in the classroom

MANYA RAMAN SUNDSTRÖM & MICHELLE ZANDIEH

Abstract

This paper looks at proof production in the midst of classroom interaction. The setting is a college level geometry course in which students are working on the following task: Prove that two parallel transported lines in the plane are parallel in the sense that they do not intersect. A proof of this statement is traced from a student’s idea, through a small group discussion, to a large class discussion moderated by a teacher. As the proof emerges through a series of increasingly public settings we see ways in which the key idea of the proof serves to both open and close class discussion. We look at several examples of opening and closing, showing how not only the key idea, but also the warrants and justifications connected to it, play an important role in the proof development.

Sammanfattning

Artikeln beskriver en studie av hur bevis konstrueras i interaktion mellan elever. Under en geometrilektion på ett amerikanskt college arbetar eleverna med följande uppgift: Bevisa att två parallellförskjutna linjer i planet är parallella i meningen att de inte skär varandra. Formuleringen av beviset följs från en idé från en av eleverna, via diskussion i en mindre grupp till en lärarledd diskussion i helklass. Allteftersom beviset utvecklas genom en följd av diskussioner i allt större grupper finner vi olika sätt varpå bevisets "key idea" bidrar till att både öppna och sluta diskussionen. Vi beskriver flera exempel på öppnande och slutande, och visar hur inte bara nyckelidén, utan även de rättfärdiganden och motiveringar som är knutna till den, spelar en viktig roll i utvecklingen av beviset.

MANYA RAMAN SUNDSTRÖM
Manya Sundström is Docent at Umeå University, and a member of Umeå Mathematics Education Research Centre. She comes to Sweden from the USA, where she worked as an Assistant Professor in mathematics and mathematics education at Rutgers University. Her main research area is mathematical proof.

MICHELLE ZANDIEH
Michelle Zandieh is an Associate Professor in mathematics and mathematics education at Arizona State University in the USA.  Her research focuses on student mathematical reasoning at the university level, especially the transitions students make from less formal to more formal ways of reasoning and how teachers may foster this transition.

Book Review. Matematik for lærerstuderende.

Jeppe Skott, Hans Christian Hansen og Kristine Jess (2008). Matematik for lærerstuderende. Delta: fagdidaktik. Samfundslitteratur, Frederiksberg. ISBN 978-87-593-1340-4

ANDREAS RYVE
Mälardalen University, Stockholm University and University of Oxford

Future activities in the Nordic graduate school in mathematics education network?

Another summer school

The sixth summer-school organised by the Nordic graduate school in mathematics education took place in Søminestationen, Holbæk, in Denmark in the third week of September. Thirty doctoral students participated and four professors acted as their supervisors in the working groups. The group leaders were Morten Blomhøj, Barbro Grevholm, Marta Menghini and Mogens Niss. The students came from Finland (4), Denmark (5), Norway (10) and Sweden (11). This year no students from Iceland or any of the Baltic countries took part, but among the students we still find several other national roots, such as Albanian, Bosnian, Dutch, Mexican, and Kenyan. The main part of the programme in the summer school consists of 13 hours of working group activities. Here students present and discuss their research questions and the background for them, the theory they use, the methods and methodology, the data collection and analysis, the results of their study and the implications of them. Each group had 7–8 members so there was plenty of time to have deep conversations about all these issues. Many of the students witnessed how important the suggestions from the group leader and the fellow students were to them.

The two workshops

Two workshops were offered. One was about how to read a scientific paper and how to write one. A paper had been sent out in advance for all to read and in the session the participants could discuss what they experienced when reading the paper, what they discerned during this reading and how they interpreted the paper. A number of different ways of putting emphasis during the reading emerged and some interesting comparisons took place. Few of the readers observed structure during the first reading but, when discussing how to write a paper, issues of structure became visible. The participants discussed the intention of the abstract, the introduction and background, the theory section, how to present methods and methodology. The selection of what data to present is crucial, and so is the issue of how to be transparent about the analysis of data. Presentation of results and the implications are of course highly important. After this work through a typical structure for a paper in a scientific journal all participants got the task to sketch their next paper and a number of interesting drafts emerged from this work.
The other workshop dealt with actual data analysis of authentic video material from a doctoral study. This workshop was led by Uffe Jankvist and built on his experiences from video analysis. Here participants could work hands on with a realistic situation. Another point of this workshop was to illustrate to the doctoral students how one can be inspired by and apply (elements of) a theory in mathematics education (in this case Sfard’s theory of commognition) to analyze the actual data.

The four lectures

Each morning students could enjoy a lecture by one of the group leaders or an invited researcher. Mogens Niss was first talking about What is quality in a PhD dissertation?. He structured his discussion in two parts: The quality of the underlying research and the quality of the dissertation as a report of the research conducted, and of its results. For the first part he emphasised the quality of the reserch questions, being clear, deep, of scholarly interest, original and researchable. Further he mentioned the quality of the research design adopted, and the corresponding methods of investigation and discussed this in depth. Focus was then on the quality of the results obtained, mentioning for example their range, applicability and generalisability. Next Mogens turned to discuss some frequently encountered problems with quality of research and finally two aspects of the report of the work done: scientific/scholarly quality and communicative quality. The concluding discussion about what could or should lead to a rejection of a dissertation might have been scaring to listen to for some of the participants. The final remarks placed the responsibility on the student by observing that every PhD project is unique and puts high demands on the student’s independence and originality and every PhD student must ”think from scratch”, even though supervised by the most experienced and respected of supervisors. Mogens has promised to write a paper for Nomad based on his presentation so doctoral students and supervisors can look forward to a more elaborated text on this important issue.
The second lecture was held by Morten Blomhøj about Five major challenges in mathematics education research and how to relate to them in a PhD project?. The five were the challenge of keeping the meta-reflections on mathematics education research alive, supporting the interplay between research and the development of teaching practices, accumulating theoretical knowledge and avoiding isolation among sub-paradigms, defining and strengthening the relations to the supporting sciences, and integrating mathematics education in general liberal education. Morten elaborated on what these challenges could mean and why he claims that this is important. He then discussed what to do in a PhD project concerning each of these five challenges and gave concrete suggestions of possible actions and why they could be important. The interested reader can find more about the challenges in the proceedings from the ICMI Jubilee conference in Rome in 2008 documented in the book: The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education.
The third lecturer was Marta Menghini presenting From geometrical figures to definitional rigor: Teachers’ analysis of teaching units mediated through van Hiele’s theory. She sees this as a curriculum design project and the work is based in the fact that teachers realize that pupils entering upper secondary school, even if they know names and shapes of many geometric figures, are not familiar with their properties and are not always able to point out specific differences expressed in the definitions. In the project they used a conceptual framework for teacher enhancement, a didactical framework called From perception to definitions and a mathematical framework: Exclusive and inclusive definitions. The design of the project and the methods were presented and illustrated by exemplars of pupils and teachers work. Among the conclusions Marta Menghini mentioned that making the first levels of van Hiele explicit allowed teachers to clarify their teaching aims. The development of the passage from exclusive to inclusive definitions, and the role of the latter in starting deduction, became clear to them. van Hiele’s theory is a framework for different experiences, such as ”a pupil’s knowledge acquisition is local and temporary”, ”definitions may be inclusive for some figures and exclusive for others”, and ”a pupil may be at a certain level for some points and at a different level for others”. In the discussion she commented on the Italian school of mathematics education research and its tradition to work together with teachers in mathematics classrooms, as illustrated in her presentation.
Mette Andresen from the NAVIMAT (national science centre for didactics of mathematics) in Copenhagen gave the fourth lecture about Recent developments in school mathematics roles and relations, Mathematics in multi-disciplinary teaching projects, Danish science gymnasiums (DASG) and NAVIMAT. The centre is rather new and a presentation can be found at www.navimat.dk. The research perspective is ”Potentials for mathematics education of multi-disciplinarity”. The reasons for this work of multidisciplinarity is that it is prescribed in the Danish upper secondary schools’ curriculum since 2005 in mathematics. The curriculum demands support of students’ knowledge about ”important aspects of the interplay between mathematics and culture, science and technology” and students are supposed to know how mathematics adds to understanding, formulating and treating problems in different subject areas”. Further students must know about mathematical reasoning and the aim is to enable students to competently take a position on the applications of mathematics and to pass further education. Mette then described a number of collaboration projects with teachers in this spirit of multidisciplinarity and the outcome of them. Finally she talked about how the centre initiates professional development for teachers instead of giving courses to them. The main reasons for this part of the work are to locate the teachers’ learning in the institutional setting, to account for the collective learning of the teacher group, and to relate teachers’ activity in professional development sessions and in the classroom.

The networking

Another important feature of the summer schools is the networking that takes place. Doctoral students have time to get to know each other and discuss research projects, and as the local conditions were excellent this could happen both in the social activities, like the excursion, walks in the forest and meals and at late evening dances. Students have during the time slots in the programme for informal conversations the opportunities to meet individually with the group leaders and discuss their projects. This was used extensively and thus a network is built of doctoral students and more experienced researchers. Now that the funding from NordForsk to NoGSME has come to an end the hope is that this network will be strong enough to survive from its own power. One initiative came already during the summer school, a Swedish subgroup of students decided to organise a local conference in Karlstad to unite forces and learn with help from each other.

Seventh summer school in 2010

The application sent in spring 2009 to NordForsk for support to a summer school in May 2010 was granted and thus there will be another chance for new doctoral students to experience such an event. It will take place in Dømmesmoen in May 25–30, 2010. The first announcement was sent out in October 2009. We already have a promise from Professor Jo Boaler in Sussex that she will be one of the group leaders and half promises from some other outstanding researchers to come. Dømmesmoen is a course- and conference centre, part of the University of Agder campus beautifully situated in Grimstad, on the south coast of Norway. Doctoral students are welcome to register for the summer course (7.5 ECTS).

The 11th international conference in the Baltic countries

Peteris Daugulis, who is chairing the Organizing committee of TM2010, has announced that the 11th international conference Teaching mathematics: restrospective and perspectives will take place on May 6–7 at Daugavpils University, Daugavpils, Latvia. More information about this yearly Baltic conference that circulates in the Baltic countries is posted on the web at http://www.de.dau.lv/tm2010

Four new doctoral dissertations in the Nordic countries

Lovisa Sumpter presented her thesis On aspects of mathematical reasoning. Affect and gender at Umeå university on June 1. She explores two aspects of mathematical reasoning: affect and gender. Studying upper secondary students’ reasoning, when solving tasks, she revealed that, when not guided by an interviewer, algorithmic reasoning was predominant. The reasoning was based on memorising algorithms, which may or may not be appropriate for the task. She then continued to investigate students’ different strategy choices and conclusions. Beliefs about safety, expectation and motivation were important in the decisions made during task solving. Her third study investigated upper secondary teachers’ conceptions about gender and mathematical reasoning. Findings indicate that the teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies, guessing and chance-taking to boys. Results from the final study show that students share the teachers’ rather traditional view on femininities and masculinities. A surprising result was that, when students were asked to reflect on their own behaviour, this result was not repeated. The result implies that girls and boys share many of the same core beliefs about mathematics. Sumpter concludes that still much work is needed to create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning.
Kajsa Bråting’s dissertation took place at Uppsala university on June 5. The title of the thesis is Studies in the conceptual development of mathematical analysis. In the thesis the development of certain mathematical concepts are considered from a historical and didactical point of view. In particular, Bråting has studied the conceptual development in analysis during the mid-19th century, for instance, concepts such as functions, continuity, convergence and infinite series have been investigated. The use of basic concepts was studied in connection with two important theorems: Cauchy’s sum theorem from 1821 and Cauchy’s theorem on power series expansions of complex valued functions from 1840. In the thesis she also investigates the role of visualizations in mathematics from a historical and didactical perspective. The visualizations have been considered on the basis of historical examples as well as on her own empirical studies. The thesis consists of a ”kappa” and three papers, two of which are published in scientific journals. The first paper deals with a new look at E. G. Björling and the Cauchy sum theorem and the second with visualisations in mathematics. The third manuscript is about E. G. Björling’s view of power series expansions of complex valued functions.
Claire Berg’s thesis Developing algebraic thinking in a community of inquiry: collaboration between three teachers and a didactician was defended at university of Agder on June 12. The study includes three teachers from lower secondary school and a didactician from a university in Norway (Claire). The thesis offers an account of the relationship between the participants’ development of algebraic thinking and the processes related to the creation and development of a community of inquiry. In addition, the thesis presents elements of the relationship between the teachers’ development of algebraic thinking and their thinking in relation to their teaching practice. The theoretical framework was elaborated according to the criteria of relevance and coherence. In order to conceptualise the participants’ development of algebraic thinking within the community of inquiry, Berg started from Wenger’s theory of community of practice and expanded it in order to include both the dimension of inquiry and Vygotsky’s ideas of mediation and scientific concepts. Methodologically, she classifies the study as a case study, within a developmental research paradigm. The results of the study indicate that the participants’ development of algebraic thinking is deeply interwoven with the processes related to the creation and development of the community of inquiry. It seems that the participants’ confidence in the community was developing gradually while the confidence in the subject-matter was related to the nature of the mathematical tasks with which the participants engaged. In addition, the study shows how the teachers engaged in a process of both investigating critically their own teaching practice as a consequence of their collaborative engagement within the community of inquiry, and of envisaging possible implications for their future teaching practice. Furthermore, insights are offered into the doctoral student’s own development both as a didactician and as a researcher and how these relate to research outcomes. Berg claims that the thesis contributes to a better understanding of issues related to collaboration between in-service teachers and a didactician from a university, while focusing on the development of algebraic thinking. Implications are also suggested concerning the way algebra could be addressed in schools.
Uffe Thomas Jankvist defended his thesis Using history as a ’goal’ in mathematics education on August 28 at Roskilde University. The work is an analytical and empirical study of using history of mathematics in mathematics education. The analytical part consists in proposing two categorizations based on a literature survey, one for the arguments of using history (history as a tool and history as a goal) and one for the approaches to doing so (the illumination, the modules, and the history-based approaches), and then analyzing the interrelations between these ”whys” and ”hows” of using history. A modules approach is chosen to fulfil the purpose of using history as a goal in the new Danish upper secondary mathematics programme. Two historical modules are designed and implemented in a particular upper secondary class. The purpose of the empirical study is to see whether students at upper secondary level are (1) capable at engaging in meta-issue discussions and reflections of mathematics and its history, (2) if these discussions and reflections in any way are anchored in the taught and learned subject matter (in-issues) of the modules, and (3) if such modules in any way may give rise to changes in students’ beliefs about mathematics (as a discipline) or the development of new beliefs. Based on videos of the implementations, students’ essays, mathematical exercises, questionnaires, and follow up interviews, the conditions on and ways in which the students are able to carry out and engage in meta-issue discussions and reflections are analyzed and discussed and so are the levels of anchoring of these in the related in-issues. In particular, four different levels regarding the students’ discussions about meta-issues are identified: the non-anchored, anchored comments, anchored arguments, and anchored discussions. It is found that modules like the ones designed in the present study may cause some changes in students’ views of mathematics on a content specific level as well as in the way the students hold their beliefs. In particular it is found that the students’ beliefs seem to grow in consistency and that the students’ desire to justify and exemplify their beliefs increases over the one year period of the study.
The four theses are very different. Sumpter’s is the only one so far reported in the NoGSME network that relates to gender, Bråting’s is in history of mathematics, Berg’s is focused on algebraic thinking, and Jankvist’s is about use of history of mathematics in the teaching of mathematics. Two of them include historical aspects but in different ways. Maybe one common trait could be seen in the quest for conceptual learning and avoiding mere superficial memorization and use of mechanical algorithms? The three theses in didactics of mathematics are clearly aiming for implications of how to improve mathematics teaching and learning.
Meanwhile Erkki Pehkonen has reported about some Finnish dissertations defended lately that have not been mentioned here. We will try to present them in the next issue of Nomad.

The final meeting of the NoGSME board

On December 14 the NoGSME board will hold its final meeting and hand over responsibility for many of the NoGSME activities to the NoRME board. A final report will be sent to NordForsk and the final part of the funding will be concluded and reported. We are convinced that NoRME will take good care of the NoGSME spirit and that both participating doctoral students and supervisors are eager to initiate future cooperation within the NoGSME network. It has been a great adventure to run this Nordic graduate school in mathematics education and to get to know so many supervisors and doctoral students and help them with different kinds of support in their scientific work. Now all must share the
responsibility for the future activities.

Barbro Grevholm
Director of NoGSME
University of Agder

Identities-in-action. Exploring the fragility of discourse and identity in learning mathematics

DIANA STENTOFT and PAOLA VALERO

Abstract

The notion of identity is often used in mathematics education research in an attempt to link individual and social understandings of mathematical learning. In this paper we review existing research making use of the notion of identity, and we point to some of the strengths and weaknesses in the ways the notion of identity is being constructed. We propose a conceptualization of the notion which points to the fragility and instability of identification processes as embedded into discourse. We contend that a notion of identity formulated from a poststructuralist perspective and emphasising the dialectic relationship between identification and discourse offers interesting possibilities for interpretations of mathematical learning as a fragile process characterised more by discontinuities and disruptions than by continuity and stability. We further argue that a poststructuralist notion of fragile identities in action allows us to bring attention to what is normally considered as ”noise” or ”impossibilities” in our understandings of mathematics education and classroom interaction.

Sammendrag

I forskningen om matematiklæring bliver et begreb om identitet ofte anvendt i forsøget på at forbinde individuelle og sociale forståelser af matematiklæring. I denne artikel gennemgår vi eksisterende forskning der anvender identitetsbegrebet, og vi peger på styrker og svagheder i konstrueringen af dette begreb. Vi foreslår en konceptualisering af identi-tetsbegrebet, der peger mod det skrøbelige og ustabile i identifikations-processen som den er indlejret i den diskursive praksis. Vi argumenterer for, at et identitetsbegreb formuleret fra en poststrukturalistisk synsvinkel og som fremhæver det dialektiske forhold mellem identitet og diskurs kan bidrage til interessante fortolkninger af matematiklæring som en skrøbelig proces karakteriseret mere ved diskontinuitet og forstyrrelse end ved sammenhæng og stabilitet. Vi fremhæver endvidere hvordan en poststrukturalistisk konstruktion af skrøbelige identiteter i bevægelse åbner op mod muligheder for at rette blikket mod det der sædvanligvis betragtes som ’larm’ og udelades i vores forståelse af matematiklæring og interaktion i klasseværelset.

DIANA STENTOFT
Diana Stentoft has a PhD in mathematics education and currently teach and do research in the Department of Education, Learning and Philosophy at Aalborg University, Denmark. Her research is primarily concerned with bringing issues of complexity and instability into contact with education research and practice by critically addressing elements often left behind in research.

PAOLA VALERO
Paola Valero is associate professor in mathematics education at the Department of Education, Learning and Philosophy, Aalborg University, Denmark. Her research interests are, among others, the political dimension of mathematics education in areas such as school reform processes, curricular innovation, and multiculturalism and mathematics learning.

The role of research in mathematics education reform work

Mathematics curricula seem to be undergoing constant changes – in the Nordic countries no curriculum is older than 10 years. At the moment a new curriculum for the Swedish compulsory school is in progress and, just like the curricula in Denmark, Iceland and Norway, the structure and organisation is influenced by an international trend that separates the mathematical content from aims regarding general mathematical competences. The draft versions of the new Swedish curriculum published on the web by the Swedish national agency for education have three sections: Rationale and Aim of the subject, Central content (for grades 1–3, 4–6 and 7–9), and Required knowledge (for grades 3, 6 and 9). By structuring the curriculum in this way two dimensions – content and competences – become clearly separated. This can be viewed as a result of a long and gradual transformation from older curricula that in many cases were merely describing the content to be covered. With respect to curricular documents that only focused on the syllabus or list of contents, these newer formulations show a progress. The current documents are influenced by recent trends and discussions emerging in the mathematics education community. For example, the aims of mathematics instruction, as for instance in the first section of the new Swedish curriculum, are influenced by several, quite recent, reports where mathematical competencies are described and categorised. Sources of influence that can be detected include among others the Standards of the National Council of Teachers of Mathematics in the United States (see e.g. NCTM, 2000), and documents such as the Danish report Kompetence og Matematiklæring (Niss & Højgaard Jensen, 2002) and Adding it up (Kilpatrick, Swafford & Findell, 2001). Despite this influence, it is less clear to see in the documents a better integration and a substantial change of focus from a syllabus-oriented thinking to a competence based curricula. The introductory declarations that intend to guide the direction of the curricula very seldom infiltrate the lists of contents to learn. Since mathematics curricula in the Nordic countries are organised in similar ways, the general image that persists is that mathematical content and mathematical competences can be regarded separately. From a mathematics education research point of view such separation can be questioned. If the educational aim is to develop mathematical competencies then the choices of content need to be justified as a suitable means for developing the target competencies. Such analyses are crucial also in relation to the process of implementation. Teachers need to develop connections between the mathematical content and the mathematical competencies in order to teach in a way that actually supports the curriculum goals.
A common way to view the mathematics curriculum is the model adopted by the IEA studies (TIMSS), which is described in three levels: the intended, the implemented and the attained curriculum. The political documents described above constitute one part of the first level – the intended curriculum – in the sense of what is prescribed by the national authorities. In that capacity they reflect the curriculum that is intended on the national level. A naïve assumption is that what is prescribed will flow down in the system and be implemented in the classrooms. However, reforming curricula and reforming teaching involve complex processes. Another part of the intended curriculum is the intended curriculum of mathematics teachers and students. In a recent investigation of mathematics teaching in the compulsory school in Sweden, the Swedish schools inspectorate (Skolinspektionen) found that the intentions in terms of ”competences to strive for”, as described in the current curriculum, are quite blurred to many teachers (Skolinspektionen, 2009). This means that the teachers’ intended curriculum in many cases is different from the nationally intended curriculum. Such discrepancy naturally bears consequences for the implemented curriculum and the actual teaching taking place in the classrooms.
The recent reform work in the Nordic countries raise many questions regarding the role of research in mathematics education. Does it matter to what extent the basic ideas behind the present curricula reforms draw on research in mathematics education? Are the effects of the former curricula properly evaluated, and if so, in what ways are the results guiding the reform work? In what ways are the problematic implementation of a new curriculum considered in the reform work? In what ways are teachers, students and parents beliefs about mathematics, and mathematics learning and teaching considered?
The editors would like to call for research articles regarding the process of reforming mathematics education for publication in NOMAD. We see a need for deepening our understanding of processes of curricular change. Research can certainly contribute to this.

About this issue

Two of the three papers published in this issue actually connect to the problematique of implementing new forms of mathematics teaching, in both cases in a Norwegian context.
In his paper Nokre spesielle trekk ved arbeidet med matematikkfaget i begynnaropplæringa Leif Bjørn Skorpen reports on the findings from an investigation of the format of the teachers’ and the pupils’ activities in Norway in 27 classes at grade 1–4 (age 6–10 years). The background motivation for this investigation was that, in the latest decade, the curriculum and the guidelines for mathematics teaching in the lower grades have undergone a change in direction of placing more emphasis on the connections between mathematics and the pupils’ activities with games and practical investigations in the teaching situation as well as on the connection to pupils daily life experience. In this reform process the ideal is to place in the centre of mathematics teaching the pupils’ activities with solving meaningful problems using mathematics. A main responsibility for the teacher in the prescribed form of teaching is to help structuring and organising the pupils’ learning in order to form a common mathematical knowledge in the classroom. Against the background of this intended curriculum the findings in this paper are quite depressing. The author has found that, although there are large variations among classes and teachers and although some very nice and interesting pupil activities have been observed, the overall picture shows that more than 2/3 of the time in class are spent on the pupils’ individual work with textbook drill exercises or on the teacher’s presentations of how to do the exercises. These findings are in line with what was found in a similar investigation in 1997; thus the research documents a strong impact of the tradition of mathematics teaching on practice. The dominant form of mathematics teaching found in this investigation is not in agreement with the intended curriculum and in this way the paper pinpoints the need for research on teachers’ professional development in relation to curriculum reforms.
The second paper Practical activities in mathematics teaching – mathematics teachers’ knowledge based reasons by Frode Olav Haara and Kari Smith also addresses the state of affairs in Norwegian mathematics teaching in compulsory schooling. Here the focus is on the use of practical activities in mathematics and the teachers’ reasons for using such activities. Eight acknowledged mathematics teachers have been interviewed about their use of practical activities and these qualitative interviews have been analysed hermeneutically. From the interviews it appears that pupils’ practical activities do not play a very prominent or well integrated role in the teachers’ organisation of mathematics teaching. Less experienced teachers may use practical activities but they do not give very clear and specific reasons related to the intended learning for their use of practical activities. Their reasons are mostly of a general pedagogical nature such as positive affects or variation of the teaching format. More experienced teachers tend to give more specific and mathematical content related reasons for using practical activities, and some of the experienced teachers also express some doubts as to whether the time spent on practical activities is worthwhile.
The findings are related to the research literature in the field of mathematics teacher education and in particular to research on the interplay between different forms of teachers’ knowledge – i.e. mathematical, didactical or pedagogical knowledge – and their beliefs about mathematics and mathematics teaching and learning.
Taken together, the two papers concerning mathematics teaching in Norway indicate that there might be an imbalance between the intended curriculum and the actual mathematics teaching taking place. From a mathematics education research point of view the situation calls for further investigations of the reform process that has led to the current curriculum and of possible ways of supporting teachers’ professional development in the process of implementing the curriculum.
The third paper in this issue addresses a more general theoretical issue. Diana Stentoft and Paola Valero, in Identities-in-action. Exploring the fragility of discourse and identity in learning mathematics, examine the notion of identity, a concept that has been adopted recently in mathematics education research in relation to diverse socio-cultural and discursive readings of mathematical learning. The notion has gained acceptance as a good tool for linking individual and social understandings of mathematical learning. The authors review existing research using the notion of identity, and point to some of the strengths and weaknesses in the ways the notion of identity is being constructed. Based on observations in an empirical setting of initial teacher education in Denmark, the authors propose a conceptualization of the notion which points to the fragility and instability of identification processes. Drawing on post-structural theories, the contention they put forward is that a notion of identity emphasising the dialectic relationship between identification and discourse offers interesting possibilities for interpretations of mathematical learning as a fragile process characterised more by discontinuities and disruptions than by continuity and stability. Such discontinuities and disruptions seem to grasp the observation that people’s engagement in mathematical learning is a much more discontinuous process than what our theoretical lenses assume it to be. Therefore, the authors argue that a poststructuralist notion of fragile identities in action allows bringing to the fore what research in mathematics education normally constructs as ”noises” or ”impossibilities” in the analysis and understanding of
mathematics education and classroom interaction.

References

Kilpatrick, J., Swafford, J. & Findell, B. (Eds.) (2001). Adding it up: helping children learn mathematics. Washington, D.C.: National Academy Press
NCTM (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Niss, M. & Højgaard Jensen, T. (Eds.) (2002). Kompetencer og matematiklæring: ideer og inspiration til udvikling af matematikundervisning i Danmark. København: Undervisningsministeriets forlag. Retrieved Novemer 4, 2009 from http://pub.uvm.dk/2002/kom/index.html
Skolinspektionen (2009). Undervisningen i matematik – utbildningens innehåll och ändamålsenlighet (Rapport 2009:5). Retrieved November 2, 2009 from http://www.skolinspektionen.se/Documents/Kvalitetsgranskning/Matte/granskningsrapport-matematik.pdf?epslanguage=sv

Nokre spesielle trekk ved arbeidet med matematikkfaget i begynnaropplæringa
LEIF BJØRN SKORPEN

Sammanfattning

Denne artikkelen fokuserer på arbeidsformer i matematikk i begynnaropplæringa i grunnskulen, og byggjer på data innsamla gjennom klasseromsobservasjonar over ein treårsperiode (2002–2005). Funna som her blir presenterte viser stor variasjon i arbeidsmåtar, men tradisjonelle arbeidsmåtar har ei dominerande rolle. Lærebøkene styrer svært mykje av aktiviteten og taus oppgåveløysing er den mest typiske arbeidsforma. Temaarbeid utgjer ein betydeleg mindre del av undervisningstida enn det læreplanen føreskriv. Det er lite samarbeid og elevane får i liten grad uttrykke tankane sine i ein sosial samanheng. Elevane får i liten grad hjelp frå lærar til å strukturere kunnskapen og til å sette ny kunnskap inn i ein større samanheng. Det mest oppsiktsvekkjande einskildresultatet er at det vert brukt markert mindre tid til arbeid med matematikkfaget enn det læreplanen legg opp til.

Abstract

This article focuses on kinds of working methods in mathematics in grades 1–4. It is based on classroom observations as part of a research project at Volda University College called: ”Elementary teaching and adapted learning”. Though our findings show a great variation in the way to organise and work with mathematics, it is the traditional working methods that dominate. Much of the teaching is directed by the textbooks, where silent and individual work with exercises has a dominating role. The pupils have to a small extent been challenged to collaborate, to put their thoughts into words and to communicate their thoughts and opinions to others. The pupils get little help from the teacher to structure their knowledge, and to connect new knowledge with existing knowledge. The total time used for mathematic activities is far less than stated by the curriculum.

LEIF BJØRN SKORPEN
Leif Bjørn Skorpen er høgskulelektor i matematikkdidaktikk ved Høgskulen i Volda. Forskingsinteresser: Klasseromsforsking, matematikk i tverrfaglege samanhengar og haldningar til matematikk.

Practical activities in mathematics teaching – mathematics teachers’ knowledge based reasons

FRODE OLAV HAARA and KARI SMITH

Abstract

The current article assumes that mathematics teachers’ understanding of mathematics and professional beliefs are integrated into their professional knowledge. The focus of the article is on teachers’ knowledge based explanations and reasons for choosing practical activities in the teaching of mathematics. Based on interviews of eight mathematics teachers in Norwegian elementary school (where the pupils are 6 to 16 years old), the article analyses and discusses relations between mathematics teachers’ professional knowledge and choice of using practical activities. The findings give grounds for suggesting that both disciplinary and didactical knowledge have an impact on teachers’ choice, and that inexperienced teachers do not have clear knowledge based explanations or reasons for using practical activities at the level of experienced teachers. However, the inexperienced, yet acknowledged teacher will, regardless of high or minimal disciplinary knowledge in mathematics, develop a more thorough and clear opinion on about practical activities as the pedagogical content knowledge deepens through experience.

Sammendrag

I artikkelen antas det at matematikklæreres forståelse av matematikk og deres profesjonelle oppfatninger er integrert i deres profesjonelle kunnskap. Fokuset i artikkelen er på læreres kunnskapsbaserte forklaringer og grunner for å velge praktiske aktiviteter i matematikkundervisning. Basert på intervjuer med åtte matematikklærere i norsk grunnskole (hvor elevene er 6 til 16 år gamle), analyserer og diskuterer artikkelen sammen­henger mellom matematikklæreres profesjonelle kunnskap og valg av praktiske aktiviteter. Resultatene antyder at både faglig og didaktisk kunnskap påvirker lærerens valg, og at uerfarne lærere ikke har like tydelige kunnskapsbaserte forklaringer eller grunner for å bruke praktiske aktiviteter som erfarne lærere har. På den annen side vil den uerfarne men anerkjente lærer, uavhengig av faglig kunnskapsnivå i matematikk utvikle en grundigere og tydeligere mening om praktiske aktiviteter etter som den pedagogiske fagkompetansen utvikler seg gjennom økt erfaring.

FRODE OLAV HAARA
Frode Olav Haara is lecturer in mathematics education at the Faculty of teacher education and sports, Sogn & Fjordane University College, Norway. Haara’s research interests are on practical activities in mathematics, mental calculations and choice of method for calculation, and the relationship between mathematics and society.

KARI SMITH
Kari Smith is professor of education at the Faculty of psychology, University of Bergen, Norway. Smith’s main research interests lie in teacher education, professional development and assessment.

An investigation of Norwegian students’ affective domain in mathematics

KIRSTI KISLENKO

Abstract

After decades of research in the affective domain in mathematics education, and search for ways to enhance students’ positive attitudes towards the discipline, the perception that to be able to do mathematics is innate remains a widespread belief. Already twenty years ago the Fourth NAEP study concluded that students believe mathematics to be important, difficult and based on rules, and theses attributions also characterise the view of mathematics even two decades later. As a relationship exists between the claims ”mathematics is difficult” and ”mathematics is boring” one could assume that students lack interest towards mathematics. The previous conclusions about the present situation are made based on a study carried out in Norway in 2005 investigating six factors in relation to students’ affective domain in mathematics – interest, hard-working, self-confidence, usefulness, insecurity, and MAD (Mathematics as an Absolute Discipline).

Sammanfattning

Elevers uppfattningar om och attityder till matematik är viktiga eftersom de är relaterade till resultaten av lärandet i matematik. Utgående från en studie som genomfördes i Norge 2005 verkar det som om elevers syn på matematik skulle kunna relateras till sex faktorer: intresse, att arbeta hårt, självförtroende, användbarhet, ängslan och MAD (matematik som en absolut disciplin). Trots att eleverna saknar intresse för matematik kan de erkänna dess användbarhet, betydelse och att man måste arbeta hårt med matematik. Nästan hälften av eleverna tror på medfödd förmåga att lära matematik.

Appendix

KIRSTI KISLENKO
Kirsti Kislenko is a PhD student in Mathematics Education at the Faculty of Engineering and Science, University of Agder in Kristiansand, Norway. Main research interests are students’ beliefs and attitudes towards
mathematics teaching and learning.

Development of students’ concept images in analysis

KRISTINA JUTER

Abstract

Students’ pre-knowledge and conceptual development in analysis were investigated at a teacher education program to reveal what pre-knowledge endured and how the students perceived the concepts a year after the course had ended. Questionnaires and interviews were used to collect data. Two students’ results are presented in more detail in the article. The study was cognitively framed with the influence of situated theories to take as many aspects of concept development into account as possible. The students showed numerous connections between concepts, but they were often unable to discern valid links from invalid links. The perceived richness from many connections causes unjustifiably strong self-confidence which prevents further work with the concept. A tool for classification of the students’ connections between concepts resulted from the analysis.

Sammanfattning

Blivande gymnasielärares förkunskaper och begreppsliga utveckling i analys undersöktes i anslutning till deras första analyskurs. De fick en enkät i början av kursen och en intervju genomfördes ett år efter kursens slut för att spegla utvecklingen i deras begreppsbilder. Kopplingar mellan olika begreppsrepresentationer undersöktes och kategoriserades i en klassificeringsmodell som utvecklades efter analysen av studenternas hantering av begreppen. Den teoretiska ramen är kognitiv men har inslag av kontextuella teorier som komplement. Två studenter har valts ut för djupare beskrivningar i artikeln. Studenterna visade många kopplingar mellan begreppen, men också en oförmåga att avgöra om de är meningsfulla. Studenter upplever då att de har förmåga att koppla samman begrepp, som i sin tur ger ett obefogat starkt självförtroende. Delar av båda studenternas begreppsbilder var så pass utvecklade att studenterna kunde beskriva avgörande processer, men ingen av dem hade en formell begreppsbild.

KRISTINA JUTER
Kristina Juter works mainly in teacher education at Kristianstad University College but is currently involved in a research project about future mathematics teachers’ identities at Växjö University. She is particularly looking at students' conceptions of analysis.

Translating test items into Norwegian – without getting lost in translation?

REIDAR MOSVOLD, JANNE FAUSKANGER, ARNE JAKOBSEN and KJERSTI MELHUS

Abstract

In relation to the Learning Mathematics for Teaching (LMT) project, sets of measures were created in order to analyse teachers Mathematical Knowledge for Teaching (MKT). This article presents some of the challenges and complexities involved in an attempt to translate and adapt these measures for use with Norwegian teachers. The measures were originally created for use in a U.S. context only, and a number of differences between the two countries contribute to increase the difficulty of doing this. Our study builds upon a similar Irish study, and this article points to some similar and several additional issues that arise when attempting to translate and adapt the measures for use in Norway.

Sammendrag

I forbindelse med prosjektet: Learning Mathematics for Teaching (LMT) ble det utviklet måleinstrumenter for å analysere læreres matematiske undervisningskunnskap (MKT). Denne artikkelen presenterer noen av utfordringene som var involvert i et forsøk på å oversette og tilpasse disse målingene for bruk blant norske lærere. Instrumentet ble opprinnelig laget kun med tanke på å bli brukt i en amerikansk kontekst, og en rekke forskjeller mellom de to landene er med på å gjøre dette vanskelig. Vår studie bygger på en tilsvarende irsk studie. I vår studie støtte vi på flere problemstillinger som var tilsvarende de som ble funnet i Irland, men der var også flere nye utfordringer som oppsto når vi forsøkte å oversette og tilpasse måleinstrumentet for bruk i Norge.

REIDAR MOSVOLD
Reidar Mosvold is Associate Professor in mathematics education at the University of Stavanger, Norway. His main interest is related to teachers’ beliefs and knowledge of mathematics, and their influence on practice.

JANNE FAUSKANGER
Janne Fauskanger is Assistant Professor in mathematics education at the University of Stavanger, Norway. Her main interest is related to pre-school teachers’ mathematical knowledge for teaching and their practice.

ARNE JAKOBSEN
Arne Jakobsen is Associate Professor in mathematics at the University of Stavanger, Norway. His interests are mathematics, mathematical knowledge for teaching, and quantitative studies in mathematics education.

KJERSTI MELHUS
Kjersti Melhus is Assistant Professor in mathematics education at the University of Stavanger, Norway. She is especially interested in how to teach mathematics for understanding.

Commentary on Theorizing in mathematics education research: differences in modes and quality

BHARATH SRIRAMAN

The authors confront a major and troubling issue for the field of mathe-matics education, namely the ”bewildering array of theories, theoretical models, or theoretical frameworks” abundantly found in the literature that characterizes research today. This commentary is spurred by the provocative nature of the said article and having recently compiled and edited a major book on theories of mathematics education (Sriraman & English, 2010) whose research and development brought to the foreground many of the core issues eloquently and critically addressed by Jablonka and Bergsten. I will briefly spell out the salient points made by the authors in need of attention and consideration by the community within the larger framework of post-modernism.

BHARATH SRIRAMAN
Bharath Sriraman is Full Professor of Mathematics at The University of Montana with an honorary appointment in the Faculty of Central/Southwest Asian Studies. He has published 250+ journal articles, books, book chapters, reviews and commentaries in mathematics education, educational philosophy and mathematics. Recent major works include editing The first sourcebook on Nordic research in mathematics education (Information Age Publishing, 2010).

Connecting theories in mathematics education: from bricolage to professionalism

TINE WEDEGE

Abstract

Connecting theories is a normal activity in the practice of mathematics education researchers and the theories come from within the field of mathematics education (”home-brewed” theories) or from outside (psychological, sociological, anthropological; philosophical, linguistic etc. theories). Thus, the researcher needs methods and strategies for connecting theories; e.g. comparing/contrasting and integrating/synthesizing. I argue that a meta-language is also needed in order to move from bricolage to professionalism in the work of theory connection. Drawing on Radford’s morphology of theories as triplets of principles, methodologies and research questions, I suggest a set of quality criteria for research papers and reports which focuses on the explicitness in reporting theory connection.

TINE WEDEGE
Tine Wedege is professor in mathematics education at the School of Teacher Education, Malmö University, Sweden and professor II at the Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway. Her main research interests are people’s motivation/resistance to learn mathematics, mathematics in the workplace and the research field as such.

Developing mathematics teaching through inquiry – a response to Skovsmose and Säljö

BARBARA JAWORSKI & ANNE BERIT FUGLESTAD

Abstract

This paper constitutes a response to the article by Skovsmose and Säljö (2008) in Nomad. We focus on the concept of inquiry as used in the KUL projects at the University of Agder, Norway, 2004–2007, from which Skovsmose and Säljö offered an evaluation and critique. We begin by clarifying certain aspects of the two KUL projects, Learning communities in mathematics and ICT in mathematics learning. In doing so, we agree substantially with several of the points made by Skovsmose and Säljö. We go on to address their two main criticisms: that research in the KUL projects shows little documentation of inquiry processes or patterns of classroom interaction between teachers and students, or among students; and that the KUL projects demonstrate few attempts to use real life environments as a basis for establishing inquiry processes. Finally we come back to significant issues related to inquiry and the main focus of the two projects, further research questions and relations between the micro and the macro in mathematics education research.

BARBARA JAWORSKI
Barbara Jaworski is professor of mathematics education in the Mathematics education centre at Loughborough University, UK where she teaches in mathematics and mathematics education. Before this she worked at the University of Agder, Norway in doctoral education and at the University of Oxford in teacher education. She edited the international Journal of Mathematics Teacher Education for 6 years, was one of the editors of the first International handbook of mathematics teacher education (2008), and was President of ERME (European society for research in mathematics education) from 2005–2009. Her research principally is into the teaching of mathematics at all levels (currently at university level) and in the development of teaching in which research is a developmental tool. The use of ”inquiry” in collaborative processes between teachers and researchers is central to her work in exploring approaches to teaching and their contribution to students’ learning of mathematics.

ANNE BERIT FUGLESTAD
Anne Berit Fuglestad (PhD) is a professor (høgskoledosent) at University of Agder, Kristiansand, Norway. She has extensive experience in teaching and supervision of mathematics education in teacher education, master and PhD level. Fuglestad was project leader of the KUL project ICT and mathematics learning (ICTML) and is currently leading Teaching better mathematics.
Her research interests in mathematics education are in developmental research in collaboration with teachers, with an emphasis on inquiry approach to mathematics and mathematics teaching in general and with the use of ICT as a tool for mathematics teaching and learning.

Theorising in mathematics education research: differences in modes and quality

EVA JABLONKA & CHRISTER BERGSTEN

Abstract

In mathematics education research reports, we find a bewildering array of ”theories”, ”theoretical models” or ”theoretical frameworks”. The key notions and principles as well as the intellectual roots of these constructions are made more or less explicit, and the relations of theoretical entities to the empirical field under study are established in different ways. These differences imply discrepancies in quality. In this contribution we touch upon some of these issues. We attempt to show that an investigation of the relations between key concepts might help to read and evaluate theoretical underpinnings of research studies, and we argue that not all constructions that are labelled ”theoretical” meet the criteria we consider essential for productive theorising. We also allude to different modes of engaging with empirical material and different ways in which theories are used in research studies. The main part of our discussion is limited to examples of ”home-grown” theorising. The examples we have chosen to illustrate our points necessarily represent a biased selection.

EVA JABLONKA
Eva Jablonka is professor of mathematics education at Luleå University of Technology. Her research activities comprise studies of curriculum conceptions (mathematical modelling and mathematical literacy), international comparative classroom research, studies of the emergence of disparity in mathematics achievement and sociological theorizing in mathematics education.

CHRISTER BERGSTEN
Christer Bergsten is professor of mathematics education at Linköping University. His research has focused on mathematics teacher education, undergraduate mathematics education, semiotics, and the role of theory in mathematics education research.

What is quality in a PhD dissertation in mathematics education?

MOGENS NISS

Abstract

The present paper discusses the issue of quality in PhD dissertations in mathematics education on the basis of the author’s reflections, observations and experiences as a supervisor and as an assessor of PhD dissertations in several countries during the last three decades. Thus, the paper represents the personal stances and views of the author and does not claim to be written on behalf of any segment of the community of researchers in mathematics education. Two major components of quality in a PhD dissertation are being dealt with, quality of the underlying research, and quality of the dissertation as a reflective report of this research and its outcomes. Particular attention is being paid to the issue of what should cause a dissertation to be rejected. The paper emphasises, at the end, that because of the multiplicity of research paradigms and philosophies in research in mathematics education there is no royal road to quality in a PhD dissertation. Therefore, the student cannot avoid involving him- or herself in independent in-depth thinking.

MOGENS NISS
Mogens Niss is a professor of mathematics and mathematics education at Roskilde university, Denmark, where he has been working since 1972, the year the university was founded. He was a member of the Executive committee of ICMI, 1987–1991, the last eight years as the Secretary general of the commission. He is currently the chair of the ICMI Awards committee, a member of the Education committee of the European mathematical society, and a member of the board of National centre for mathematics education (NCM) in Sweden. His research interests and many publications focus on mathematics education, in particular mathematical competencies, mathematical modelling, the nature of mathematics education as a research discipline, and the justification problem of mathematics education in society.

Affektive sider ved lærerstudenters arbeid med matematikk

Leif Kværnes

Sammenfattning

Formålet med artikkelen er å belyse og drøfte sider ved allmennlærerstudenters utvikling av lærerkompetanse i matematikk. I empiriske analyser har jeg har valgt å fokusere på studenters affektive eller følelsesmessige forhold til læring av/arbeid med matematikk; som er sett som et delaspekt ved lærerkompetansen. I første del av artikkelen redegjør jeg for sentrale teoretiske utgangspunkt; et triadisk syn på læring og en kommunikativ tilnærming til analyser og beskrivelser av læring og undervisning. Andre delen av artikkelen starter med analyser og beskrivelser av affektive sider gjennom utvalgte eksempler fra studenters arbeid med matematikk. Disse beskrivelsene er utgangspunkt for en avsluttende problematisering og drøfting av hvordan affektive sider kan influere på studenters utvikling av lærerkompetanse i faget.

Abstract

My intention with this article is to discuss some aspects of teacher student’s development towards becoming mathematics teachers. My main focus is on the relations between affect and cognition. First part of the article will be theoretical. I will here outline how this relation is seen, and I also describe what may be called a communicative or discursive approach to this relation. In the second part I use this approach on student’s utterances while working with mathematics. My intentions are not to make representative or broad descriptions of relations between affect and cognition. The descriptions will be used as points of departure for discussing student’s development towards becoming mathematics teachers.

Leif Kværnes

Leif Kværnes er høgskolelektor i matematikk ved Høgskolen i Oslo, avdeling for lærerutdanning, hvor han underviser både grunnutdanningsstudenter, masterstudenter og lærere som søker etter- eller videreutdanning. Hans forskningsinteresse er knyttet til lærerutdanning og til utvikling av lærerkompetanse for undervisning i matematikk.

Communication and learning at computers: an overview

Rune Herheim

Abstract

The article highlights key findings from a research literature overview within the field of learning and communication, for face-to-face small group settings in which pupils use a computer. The overview surveys articles with a general learning approach and articles from the field of mathematics education. The purpose of the overview is to locate the most significant literature of the field and qualitatively summarize these articles by identifying the issues that are their focus. In addition, the article presents some of the sceptical arguments presented in the literature, and finally some important issues for future work are singled out.

Rune Herheim

Rune Herheim is a PhD-candidate at the Department of Education, University of Bergen. Before this he has worked as a lecturer in mathematics education at Bergen University College and as a teacher in primary school. Herheim’s main research interests lie in ICT and education, collaborative learning, and the relationship between pupils’ communication and pupils’ learning.

Cooperation and collaboration as zones of proximal development within the mathematics classroom

Sharada Gade

Abstract

Beyond understanding the Vygotskian construct of zone of proximal development or ZPD with reference to an individual student, this paper explores the formation of ZPD within the pedagogical constructs of cooperation, wherein students cooperate with each other within their groups; as well as collaboration, wherein students from different groups that constitute the classroom collaborate with each other. Identified on the basis of functions that are in the process of maturing, the formation of either ZPD is exemplified from a socio-cultural-historical study at an upper secondary mathematics classroom in Norway. An emphasis on what distinguishes events in instruction that are educational from those that are not is also explored, before illustrating what nature of ZPD is established. The role of guidance received, imitation and cultural resources in the development of higher mental functions is understood as these ZPD are formed, enabling students to act independently within the classroom teaching-learning of mathematics.

Sharada Gade

Sharada Gade combined her long teaching experience at Vidyaranya High School, Hyderabad, India with writing for teachers and popularisation. At University of Agder, Norway she pursued her doctorate in mathematics education with a classroom based thesis drawing upon sociocultural and activity theory perspectives. As Visiting fellow at Homi Bhabha Centre for Science Education, Mumbai, she taught master and doctoral courses and conducted narrative inquiry with middle school teachers about their experiences of classroom teaching-learning. Currently as postdoctoral researcher at Umeå University, Sweden, Sharada continues her pursuit of understanding classrooms as productive teaching-learning environments for students, their teachers and research.

Learning opportunities offered by a classical calculus textbooky

Mira Randahl and Barbro Grevholm

Abstract

In this paper we present results of an analysis of what the textbook used by the first year engineering students offers the students, when they take a basic calculus course. The aim of this analysis is to examine as an entirety what students are offered by the book to learn about the concept of derivative. The results show that the presentation of the concept is formal and depends on students’ previous knowledge. The treatment of the concept emphasises procedural knowledge. It is not easy for students using the book to make connections between conceptual and procedural knowledge of the concept of derivative.

Mira Randahl

Mira Randahl has worked at Narvik University College and is enrolled in the doctoral programme at University of Agder. Earlier she has been mathematics teacher in compulsory school, upper secondary school and in teacher education in Norway.

Barbro Grevholm

Barbro Grevholm is professor of mathematics education at University of Agder and leader of the doctoral programme there. She is also professor II at Narvik University College.

Boganmeldelse: Forskeres og studerendes mulige udbytte af at læse Springes nye bog om teorier i matematikkens didaktik

Uffe Thomas Jankvist

Head teachers’ conception of gifted students in mathematics in Swedish
upper secondary school

Linda Mattsson

Abstract

The article presents a study of how Swedish upper secondary head teachers, working within mathematically intensive study programs, conceptualize giftedness in mathematics. The study is based on a survey of 34 randomly selected head teachers, in a population of about 400, who have answered questions about how they characterize and detect gifted mathematics students. The results show that teachers characterize such students as creative, strong in logical ability and keen in their motivation for mathematics. The teachers detect such students by the students’ own initiative for engaging in mathematics, their inclination to orally reason about mathematics and their successfulness on tests. The findings, which are in accordance with results from internationally published studies, are of importance to the current discussion on special provision for gifted students in Sweden.

Linda Mattsson

Linda Mattsson is PhD student at the Department of Mathematical Sciences, University of Gothenburg and Chalmers University of Technology. Her research focus lies within the field of gifted education in upper
secondary school and on-going studies concern issues of nurturing gifted students in mathematics.

Orchestrating mathematical activities in the kindergarten: the role of
inquiry

Martin Carlsen

Abstract

The aim of my study is to address the role played by inquiry in orchestrating a mathematical activity in the kindergarten. The study takes the inquiry cycle as a methodological departure, i.e. the whole process of designing, acting, observing, reflecting, and feeding back. The subtleties of how the kindergarten teachers and a didactician reason in order to orchestrate the mathematical activity and the role played by inquiry in this orchestration are hence analysed. The analyses show how inquiry in every phase of the inquiry cycle, plays a significant role in the orchestration of the mathematical activity. The participants become co-learners in these processes, and are involved in appropriating the mathematical tools and concepts as well as didactical issues involved in orchestrating a mathematical activity in the kindergarten.

Martin Carlsen

The author’s research interests are within the scope of this paper, mathematical inquiry in the kindergarten. He is also particularly interested in mathematical appropriation through problem solving in collaborative small groups. He identifies with a sociocultural perspective on learning and development, and his research is taking place within this theoretical stance.

Understanding and solving multistep arithmetic word problems

Guri A. Nortvedt

Abstract

This article discusses the findings of a study in which the interplay between reading, numeracy, and strategies for working on multistep arithmetic word problems was researched through two approaches. The first approach involved analysing results on national tests in reading and numeracy for a representative sample of 1,264 grade 8 (13-years-old) students. A scale of ten multistep arithmetic word problems was identified in the numeracy test. Proficiency in reading explained 44 % of the variance in scores on this scale, indicating a positive relationship between reading comprehension and success in word problem solving. The second approach involved analysing verbal protocol data for 19 grade 8 students who worked on a collection of multistep arithmetic word problems. Protocols consisted of both independent and scaffolded work. Interpretive analysis of student work on one of the eight word problems given in the protocol sessions revealed three main areas of difficulties: representing quantities in the word problem text, retrieving number facts from memory, and performing basic operations. Difficulties within more than one area were frequent. To students with below-average numeracy skills, executing the basic operations was the main obstacle for this particular word problem.

Guri A. Nortvedt

Guri A. Nortvedt works at the University of Oslo. Her main research interests are 1) relationships between language and (learning and doing mathematics) and 2) assessment studies within mathematics education.
The research reported in this article was carried out at the Department of Special Needs Eduaction, University of Oslo.

Different views – some Swedish mathematics students’ concept images of the function concept

Olov Viirman, Iiris Attorps and Timo Tossavainen

Abstract

This study analyses what kind of concept images a group of engineering and teacher students have of the function concept, and how these concept images are related to the historical development of this concept. The study was conducted using questionnaires, and 34 students at a Swedish university participated. It is found that the students primarily rely on operational conceptions of the function concept, with only a minority of students possessing structural conceptions. The definitions given by the students mostly resemble an 18th or 19th century view of functions. The study also indicates that the character of the definitions given in the textbooks used by the students affect their concept images.

Olov Viirman

Olov Viirman is a graduate student in mathematics education at the department of Electronics, Mathematics and Science, University of Gävle, and the department of Mathematics, Karlstad University. His research interests are the teaching and learning of mathematical concepts, particularly at the university level.

Iiris Attorps

Iiris Attorps is senior lecturer in mathematics education at the Department of Electronics, Mathematics and Science, University of Gävle. Research interest: learning and teaching mathematical concepts from preschool to university level.

Timo Tossavainen

Timo Tossavainen is senior lecturer in mathematics at the University of Eastern Finland and docent of mathematics education at the University of Tampere. He has got a Ph.D. in mathematics and his research interests cover function theory and mathematics education at secondary and tertiary levels. He is a co-author of several mathematics textbooks for upper secondary schools and universities.

From beliefs to patterns of participation – shifting the research perspective on teachers

Jeppe Skott, Dorte Moeskær Larsen and Camilla Hellsten Østergaard

Abstract

Belief research was introduced to mathematics education in the early 1980s. It challenged the primarily cognitive and mathematical agenda of the time by investigating the character and significance of mental meta-constructs called beliefs. Particular attention has ever since been paid to teachers’ beliefs and their role in instruction.
Belief research has been troubled by conceptual and methodological problems since its early beginnings, and most of these are still unresolved. This indicates that it may be time to adopt a different perspective, if we are to understand the role of the teacher for the practices of the mathematics classroom.
Elsewhere we have discussed the problems of belief research at some length and suggested an alternative that we call patterns-of-participation research (e.g. Skott, 2009, 2010). In the present article we briefly recapitulate some of the arguments underlying this suggestion, but our main interest is to use the patterns-of-participation approach for empirical purposes. Consequently the article consists of two main sections. First we summarise some of the problems of belief research and present the contours of our alternative, patterns-of-participation research. Second, we in a much longer section present and analyse data on the case of a teacher, Susanne, whom we follow prior to and after her graduation from college. The overall intention is to suggest a change of research perspective from beliefs to patterns of participation.

Jeppe Skott

Jeppe Skott, PhD, is a professor of mathematics education at the Linnaeus University, Sweden, and an associate professor in the same field at the Danish School of Education. His main research interest is in the teacher’s role for classroom interaction, but he has also written on theory-practice relationships in mathematics education. Besides he has been involved in educational development in different countries in Europe, Africa, and Asia.

Dorte Moeskær Larsen

Dorte M. Larsen, MA, is a lecturer at University College Capital, Copenhagen. She teaches mathematics and mathematics education to prospective and practising teachers. Her main research interest is teacher development during their pre-service education and immediately after their graduation, especially the role of the practicum for the students’ professional development.

Camilla Hellsten Østergaard

Camilla Hellsten Østergaard, MA, is a lecturer at University College Capital, Copenhagen. She teaches mathematics and mathematics education to prospective teachers. Her main research interest is teacher development during their pre-service education and especially how they change their participation in the social practices at their schools in the first few year after their graduation.

Structure of students’ view of mathematics in the Estonian Business School

Indrek Kaldo

Abstract

Students’ mathematics-related beliefs are a decisive parameter for engagement and success in school. In the present research the students’ attitudes, beliefs and motivations regarding mathematics at an Estonian university was explored. The paper focuses on describing such a view of mathematics. By means of a confirmatory factor analysis, seven factors were confirmed. The data were collected from 93 first-year mathematics course students in the Estonian Business School through a questionnaire using a Likert-type scale. The study confirmed most of the components identified in earlier studies. It validates the use of the instrument in further studies of beliefs, attitudes and motivation at the university level in Estonia.

Indrek Kaldo

Indrek Kaldo is a PhD student in mathematics education at the Institute of Educational Sciences, Tallinn University. His research interests are students’ beliefs, attitudes and motivation towards the teaching and learning of mathematics, particularly at the university level. He has also 11 years experiences as lecturer in mathematics at university level.

Students’ mathematical identity formations in a Swedish multilingual mathematics classroom

Eva Norén

Abstract

In this article I explore how students’ mathematical identities are formatted in a multilingual mathematics classroom. The study has been conducted in a group of ten multilingual Arabic and Swedish speaking students in grade eight and nine. In the article the focus is on two of the students. Students’ mathematical identity formations are effects of exercise of a variety of discourses available in the mathematics classroom. In discourses promoting multilingualism and social relations students’ possibilities to positively build upon opportunities in the mathematics classroom seem to enhance and identity formations as engaged mathematics learners is not an obstacle.

Eva Norén

Eva Norén is PhD in mathematics education at Stockholm University, the Department of mathematics and science education, where she also teaches. Her research interest is in mathematics education related to multilingual and multicultural issues. She has been a teacher in primary school for many years and her practice as a teacher has inspired her to use ethnographic research methods, spending time in classrooms with students and teachers.

The theory of conceptual change as a theory for changing conceptionst

Peter Liljedahl

Abstract

It has become widely accepted that what and how mathematics teachers teach is linked to what it is they believe. What teachers believe, however, is not always in alignment with contemporary notions of mathematics and the teaching and learning of mathematics. As such, it is important for teacher educators to help facilitate changes in teachers’ beliefs in ways that will enable them to become more effective teachers of mathematics. In this article I present the results of a research project designed to examine the feasibility of using the theory of conceptual change as a theory for changing mathematics teachers’ conceptions about key aspects of mathematics and the teaching and learning of mathematics. The results indicate both that the theory of conceptual change is a viable theory for designing interventions for the purpose of changing beliefs, and that the implementation of these aforementioned interventions resulted in the rejection of participants’ a priori beliefs.

Peter Liljedahl

Dr. Peter Liljedahl is an Associate Professor of Mathematics Education in the Faculty of Education and an associate member in the Department of Mathematics at Simon Fraser University in Vancouver, Canada. He is a co-director of the David Wheeler Institute for Research in Mathematics Education. His research interests are creativity, insight, and discovery in mathematics teaching and learning; the role of the affective domain on the teaching and learning of mathematics; the professional growth of mathematics teachers; mathematical problem solving; and numeracy.

To translate between different perspectives in belief research: a comparison between two studies

Magnus Österholm

Abstract

A common problem in belief research seems to be a missing link between aspects of theory and empirical analyses and results. This issue highlights a question of how dependent empirical studies about beliefs actually are on the theoretical perspective described in the study. In this paper, I examine relationships between two different perspectives. One perspective focuses on belief change, and seems to rely on a type of cognitive perspective, where beliefs can be characterized as mental objects. The other perspective argues for moving away from such cognitive perspective and instead to adopt a participatory perspective in the analysis of mathematics teaching. The results show that the study about belief change is not dependent on seeing beliefs as mental objects, but that this study could as well have been located within a participatory perspective.

Magnus Österholm

Magnus Österholm has a PhD in mathematics education from Linköping University and now works as a research fellow at the Department of Science and Mathematics Education at Umeå University. He is also a member of Umeå Mathematics Education Research Centre (UMERC). During 2011 and 2012 he is a visiting scholar at Monash University in Melbourne, Australia. His research interests deal primarily with mathematics education at the upper secondary and university levels, where cognitive and metacognitive perspectives are of special interest, together with studying language and communication in the learning and teaching of mathematics.

Development of self-regulated learning skills in mathematics in lower secondary school in Sweden

Joakim Samuelsson

Abstract

In this study, the development of 219 students’ self-regulated learning skills in lower secondary school across ability groups were investigated and related to measures of students’ performance in mathematics. Self-regulated learning skills were assessed with a questionnaire originally designed and used in PISA 2003. Pre-testing was performed during the first two weeks in school in seventh grade. The first post-test was performed after one term in eighth grade, in January 2008. The second post-test was performed during the last two weeks in grade 9, in June 2009. All testing was performed by the class teacher. However, the result states that internal motivation, instrumental motivation as well as self-concept decline across year in lower secondary school. The development of interest and enjoyment of mathematics, self-concept in mathematics and anxiety in mathematics was similar in each ability group. No interaction effects across groups were significant in the study. This study highlights the importance of taking affective factors into account in discussions about the results of mathematics teaching and learning. The strong correlation between affective factors and achievement in mathematics helps us to identify some weaknesses in the Swedish education system.

Joakim Samuelsson

Joakim Samuelsson is associated professor at the Department of Behavioural Sciences and Learning at Linköping University. His research focuses on issues related mathematics teaching and learning in compulsory school.

Farmers do use mathematics: the case of animal feeding

Laia Saló i Nevado, Gunilla Holm and Leila Pehkonen

Abstract

This article presents findings from a study on the use of mathematics in the context of a farm. Ethnographic methods were used for the data collection and ethnomathematics provides the theoretical framework guiding the analysis. We present two different situations, as examples of ethnomathematics, in which the farmers make use of mathematics in daily life situations on a farm. The first situation has to do with how one of the farmers dealt with a barn as a space for feeding calves. The second situation is about the use of different objects as measuring tools.

Laia Saló i Nevado

Laia Saló i Nevado is doctoral student in the Institute of Behavioural Sciences at the University of Helsinki. Her research interests are focused on the everyday uses of mathematics and adults learning mathematics.

Gunilla Holm

Gunilla Holm is professor of education in the Institute of Behavioural Sciences at the University of Helsinki. Her research interests are focused on photography as a data collection method as well as issues in education related to race, ethnicity, class, and gender. She has published widely on multicultural education and on schooling in popular culture and has co-edited several books.

Leila Pehkonen

Leila Pehkonen is senior lecturer of education in the Institute of Behavioural Sciences at the University of Helsinki. Her current research interests include teaching and learning in higher education, mathematics education and teachers’ agency in vocational education.

Introduksjon til vektorer i norske lærebøker og i en undervisningsfilm

Anne Birgitte Fyhn

Sammendrag

Vektorer introduseres andre år på videregående skole i Norge. Denne teksten undersøker hvordan læreverkene og en klatrefilm introduserer dette emnet og hvorvidt filmen kan supplere bøkene. Film og lærebøker undersøkes ut fra samme kriterier. Fordi bøkene bygger på læreplanen, presenteres først en oversikt over vektorers plass i norske læreplaner. Analysene viser at filmen kan supplere lærebøkene ved å utfordre elevenes matematiske tenking i forhold til relasjoner mellom vektor og vinkel. Analysene indikerer også en svakhet ved læreplanen: Læreplanens kompetansemål med hensyn på vektorer fokuserer kun på regning og prosedyrer uten at disse eksplisitt inngår i en sammenheng. Kompetansemålene sier heller ikke noe om forståelse.

Anne Birgitte Fyhn

Anne Birgitte Fyhn har PhD i matematikkdidaktikk fra det matematisk-
naturvitenskapelige fakultet (nå: NT-fakultetet) ved Universitetet i Tromsø, Norge. Hun er tilsatt som førsteamanuensis i matematikkdidaktikk ved Institutt for lærerutdanning og pedagogikk ved Universitetet i Tromsø. Hun leder forskningsprosjektet Strukturer og mønstre i samisk ornamentikk som basis for undervisning i matematikk på ungdomstrinnet. Prosjektet er finansiert av Norges Forskningsråd.

Comparing perceptions of mathematics: Norwegian and Finnish university students‘ definitions of mathematics

Miika Vähämaa and Kennet Härmälä

Abstract

The article presents a comparison between Norwegian and Finnish university students’ perceptions of what mathematics is. To carry out the comparison, a mix of qualitative - the creation of abstract and concrete categories for mathematics representations - and quantitative (regression modeling) methods was used in the study. The main result of the study is that Norwegian students were more homogenous in their responses and the vast majority perceived mathematics in concrete terms. The Finnish students, on the contrary, showed greater variety in their responses. There are not many comparative studies among Nordic countries regarding students’ perceptions of mathematics. Therefore this study contributes to improving our knowledge about the possible differences and similarities on students’ perceptions of mathematics among Nordic students. A total of 239 students were asked how they perceive mathematics, numbers and personal applicability of mathematics via an open questionnaire. We propose that the divergent perceptions of mathematics stem from different types of communication cultures that surround mathematics. The argument is made that perceptions of mathematics should be treated as a type of mathematical knowledge that is valuable whenever mathematics is communicated.

Miika Vähämaa

Miika Vähämaa is a doctoral student in social psychology at the Department of Social Research at the University of Helsinki, Finland. Vähämaa’s research interests include group epistemologies, heuristics and decision making in groups.

Kennet Härmälä

Kennet Härmälä is a project researcher at the Aalto University School of Arts, Design and Architecture, Finland. Härmälä’s research interests include group epistemologies and precarious employment.

Does the format matter? How the multiple-choice format might complicate the MKT items

Janne Fauskanger, Reidar Mosvold, Raymond Bjuland and Arne Jakobsen

Abstract

In order to design appropriate professional development programs for teachers, an instrument has been developed in the U.S. to measure teachers’ mathematical knowledge for teaching. The process of translating and adapting these measures for use in other countries involves several challenges. This article focuses on issues related to the multiple-choice format of the items. Analyses of focus-group interviews reveal that the multiple-choice format may complicate the items. The teachers’ reflections about the format in this Norwegian case contribute to the understanding of this important challenge.

Janne Fauskanger

Janne Fauskanger is Assistant Professor in mathematics education at the University of Stavanger, Norway. Her main interest is related to primary school teachers’ mathematical knowledge for teaching and their practice.

Reidar Mosvold

Reidar Mosvold is Associate Professor in mathematics education at the University of Stavanger, Norway. His interests are related to teachers’ beliefs and knowledge of mathematics, and their influence on practice.

Raymond Bjuland

Raymond Bjuland is Associate Professor in mathematics education at the University of Stavanger, Norway. His interests are related to students’ collaborative problem solving in small groups, the use of gestures in teacher-student dialogues, and mathematical knowledge for teaching.

Arne Jakobsen

Arne Jakobsen is Associate Professor in mathematics at the University of Stavanger, Norway. His interests are mathematics, mathematical knowledge for teaching, and quantitative studies in mathematics education.

What characterises the heuristic approaches in mathematics textbooks used in lower secondary schools in Norway?

Tom Rune Kongelf

Abstract

In this paper I present findings of an analysis of how mathematics textbooks treat heuristic approaches. The aim of this analysis is to give a characterisation of the occurrence of nine well-known heuristic approaches by analysing 740 examples presented in six ninth grade textbook series. The findings show that many of the problems in the examples are being solved by using one or more heuristic approaches, but the characteristics of the examples and the textbooks’ lack of discussion of the approaches themselves make it challenging to teach and learn these in school. The heuristic approaches seem to be used rather incidentally, which is supported by the fact that none of the textbooks explicitly treat or mention problem solving.

Tom Rune Kongelf

Tom Rune Kongelf is a Ph. D. student in the doctoral program at University of Agder in Kristiansand and works as lecturer at Sogn og Fjordane University College in Sogndal. His main research interests concerns textbooks in mathematics in lower secondary school, with an emphasis on heuristic approaches, algebra and tasks.

Interference of subtraction strategies

Per-Olof Bentley

Abstract

This study concerns a particular kind of mistake that a number of pupils made when subtracting two positive whole numbers. The aim was to analyse the cause behind this particular mistake. According to the pupils, the difference was equal to the subtrahend. It was found that the pupils counted down to the subtrahend. But instead of finding the answer as the number of steps between the two terms, the pupils applied the last-number-word rule and gave the subtrahend, which was the last mentioned number word, as the result. When seeing subtraction as a concept, it could be assumed that the lack of experience of subtraction as a comparison and as equalization played a decisive role for this mistake. A comparable mistake described in previous research is also analysed.

Per-Olof Bentley

Per-Olof Bentley is senior lecturer at Gothenburg University. He has a Ph D degree in mathematics education and has been responsible for the TIMSS project in Sweden from 2004 to 2008. He has made three indepth analyses of the Swedish pupils mathematical knowledge exposed in TIMSS 2007 and TIMSS Advanced 2008. During three and a half years he had also carried out a large interview study in which about 500 pupils were in-depth interviewed.

Methodological issues when studying the relationship between reading and solving mathematical tasks

Magnus Österholm and Ewa Bergqvist

Abstract

In this paper we examine four statistical methods used for characterizing mathematical test items regarding their demands of reading ability. These methods rely on data of students’ performance on test items regarding mathematics and reading and include the use of regression analysis, principal component analysis, and different uses of correlation coefficients. Our investigation of these methods focuses on aspects of validity and reliability, using data from PISA 2003 and 2006. The results show that the method using principal component analysis has the best properties when taking into account aspects of both validity and reliability.

Magnus Österholm

Magnus Österholm has a PhD in mathematics education from Linköping University and now works as a research fellow at the Department of Science and Mathematics Education at Umeå University. He is also a member of Umeå Mathematics Education Research Centre (UMERC). During 2011 and 2012 he is a visiting scholar at Monash University in Melbourne, Australia. His research interests deal primarily with mathematics education at the upper secondary and university levels, where cognitive and metacognitive perspectives are of special interest, together with studying language and communication in the learning and teaching of mathematics.

Ewa Bergqvist

Ewa Bergqvist has a PhD in mathematics education from Umeå University and is an assistant professor at the Department of Science and Mathematics Education at Umeå University. She is a member of Umeå Mathematics Education Research Centre (UMERC) and a teacher in mathematics education for pre-service mathematics teachers. Her research focuses mainly on language, competencies, and reasoning in upper secondary and university level mathematics.

Using strands of tasks to promote growth of students’ mathematical understanding

John Francisco and Gunnar Gjone

Abstract

This article reports on the mathematical activity of a group of five high school students (15–16 year olds) who worked together on a series of challenging task in combinatorics and probability. The students were participants in an after-school, classroom-based, longitudinal research on students’ development of mathematical ideas and different forms of reasoning in several mathematical content strands. The purpose of the article is to contribute insights into how to promote growth of students’ mathematical understanding through problem-solving activities. In particular, the article shows that problem-solving activities involving strands of challenging tasks have the potential to promote growth of students’ mathematical understanding by providing opportunities for students to engage in reasoning by isomorphism. This is a type of reasoning whereby students rely on structural similarities, i.e., isomorphism among mathematical tasks to solve or deepen their understanding of the tasks. Implications for classroom teaching, and environmental conditions that promote reasoning by isomorphism are also discussed.

John Francisco

John Francisco is assistant professor in mathematics education in the School of Education at the University of Massachusetts at Amherst, USA. His research interests include students’ development of mathematical ideas and reasoning, personal epistemological beliefs and teacher learning.

Gunnar Gjone

Gunnar Gjone is professor of mathematics education at the University of Oslo, Norway, and guest professor at Karlstad University Sweden. His research interests are mathematics teaching and learning in lower and upper secondary school, as well as teacher training. He has special interest in the use of ICT in students’ learning of mathematical concepts.

Innovative approaches to teaching mathematics in higher education: a review and critique

Mahmoud Abdulwahed, Barbara Jaworski and Adam R. Crawford

Abstract

This paper provides a snapshot of emerging trends in mathematics teaching in higher education for STEM subjects (Science, Technology, Engineering and Mathematics). Overwhelmingly, papers identify a focus on conceptual understandings of mathematics in comparison to understanding that is instrumental or procedural. Calls for reform of mathematics teaching have been the basis for a range of studies; responses to these calls have embraced innovative methods for implementing changes in learning and teaching of mathematics, sometimes rooted in constructivist ideology. Observed trends have been categorised in six groups. In many studies, technology is being used as an enabler of reforms. Constraints to implementing new approaches in mathematics teaching are indicated. Discussion of contemporary research questions that could be asked as a result of the shift towards teaching mathematics in innovative ways is provided and is followed by a critique of the underlying theoretical positions, essentially that of constructivism.

Mahmoud Abdulwahed

Mahmoud Abdulwahed is Assistant Professor and Acting Head of the College Requirement Unit at the College of Engineering, Qatar University, Qatar. He was formerly a researcher and developer in ICT, STEM Education, and Innovation at Loughborough University, where he earned his PhD.

Barbara Jaworski

Barbara Jaworski is a Professor of Mathematics Education and Director of Research at the Mathematics Education Centre, Loughborough University, UK, formerly a Professor of Mathematics Education at the University of Agder, Norway, where she recently was awarded an Honorary Doctorate.

Adam Crawford

Adam Crawford is based within the School of Civil and Building Engineering at Loughborough University, UK, formerly Manager of the Engineering Centre for Excellence in Teaching and Learning at the University.

Structure of university students’ view of mathematics in Estonia

Indrek Kaldo and Markku S. Hannula

Abstract

This study reports on first-year Estonian university students’ view of mathematics. The data was collected from 970 university students of different disciplines. The participants filled out a Likert-type questionnaire that was developed using previously published instruments. The study confirmed that several different attitudes, beliefs, and motivational orientations can be identified and validly measured as separate components of Estonian university students’ view of mathematics. However, the low reliability of some scales highlights the necessity for careful testing of instruments in any new population.

Indrek Kaldo

Indrek Kaldo is a PhD student in mathematics education at the Institute of Educational Sciences, Tallinn University. His research interests are students’ beliefs, attitudes and motivation towards the teaching and learning of mathematics, particularly at the university level. He has also 11 years experiences as lecturer in mathematics at university level.

Markku S. Hannula

Markku S. Hannula is professor in mathematics education at the Department of Teacher Education at the University of Helsinki in Finland. His research interests include motivation, beliefs, emotions, problem solving, and gender in mathematics education.

Upper secondary school students’ gendered conceptions about affect in mathematics

Lovisa Sumpter

Abstract

This study explores upper secondary school students’ conceptions about gender and affect in mathematics. Two groups of students from Swedish Natural Science Programme each answered a questionnaire; the first with a focus on boys and girls in general and the other with a focus on individuals themselves. The results from two questionnaires were compared. The first questionnaire revealed a view of rather traditional femininities and masculinities, a result that did not repeat itself in the second questionnaire. There was a discrepancy between traits students ascribed as gender different and traits students ascribed to themselves.

Lovisa Sumpter

Lovisa Sumpter is lecturer and researcher at Dalarna University, Falun, Sweden. Her research interests are mathematical reasoning, affect and gender.

Making sense of a ”misleading” graph

Oduor Olande

Abstract

Given the importance of a critical-analytical disposition in the case of graphical artefacts, this paper explores graphicacy based on students’ answers to an item from PISA survey test. Primarily, results from the written test were analyzed using PISA’s doubledigit rubrics or coding. In evaluating these categories, it is observed that just a small percentage of students are able to produce answers that reflect a critical-analytical approach with respect to the use of statistical/mathematical operators and forms of expressions. Secondly, video observation shows that students tend to employ what is perceived as an ”identification approach” while discussing the task. Whereas elements of mathematical and statistical ideas can be identified in the students’ discussion, these are not explicitly stated and are largely submerged in everyday concerns and forms of expression.

Oduor Olande

Oduor Olande is a Ph. D. candidate in the didactics of mathematics at the Department of Applied Science and Design, Mid Sweden University (Mittuniversitetet), Sweden. His special research interest is in interactions (sense making and analysis) with graphical artefacts. He is also interested in issues concerning stochastic thinking.

Student teachers’ work on instructional explanations in multiplication – representations and conversions between them

Anita Valenta and Ole Enge

Abstract

In this study we are analysing student teachers’ instructional explanations. The study is based on student teachers’ written work on two different tasks about different strategies and properties in multiplication and explaining these. Our research questions concern the type of representation registers student teachers use in their explanations. In explanations where several representation registers are used, we analyse what can be challenges in conversions between representations. Data is analysed using the framework of Duval’s cognitive analysis, and analyses and discussions are related to development of mathematical knowledge for teaching.

Anita Valenta

Anita Valenta is Associate Professor in mathematics education at Sør-Trøndelag University College in Trondheim, Norway. Her main interest is in developing a knowledge base for mathematics teacher education, research in student teachers’ mathematical knowledge for teaching and algebra in the early grades.

Ole Enge

Ole Enge is Associate Professor in mathematics education at Sør-Trøndelag University College in Trondheim, Norway. He is interested in developing a knowledge base for mathematics teacher education, research on instructional routines in mathematics teaching and students’ proportional reasoning.

I need advanced mathematics to pursue the career of my choice”. Norwegian students’ motivations for enrolling in mathematics and plans for post-secondary studies

Ida Friestad Pedersen

Abstract

Participation in advanced science and mathematics courses in upper secondary school is a gateway to tertiary education and career opportunities in the STEM fields (Science, Technology, Engineering, and Mathematics). The purpose of this study is to investigate Norwegian final-year upper secondary school students’ motivations for choosing to enroll in the most advanced mathematics course offered (3MX), as well as to ascertain their plans for post-secondary education. Since females are underrepresented in mathematics and mathematics-related fields of study, special attention is paid to the gender perspective. The analyses are based on questionnaire data from the large-scale international achievement study TIMSS Advanced, and are framed by the expectancy-value model developed by Eccles and her colleagues (1985). Results show that the subject’s utility value was the primary reason for students’ enrollment in mathematics; that interest in mathematics as a school subject is somewhat more important to the girls than to the boys; and that there are some gender differences in students’ plans for post-secondary studies in the STEM fields.

Ida Friestad Pedersen

Ida Friestad Pedersen is a PhD-student in the doctoral program at the University of Oslo, and works as a lecturer at the University of Tromsø. Her research interests include students’ mathematical competence, with an emphasis on algebra, and their motivations for choosing mathematics in upper secondary school and beyond.

Assessing authentic tasks: alternatives to mark-schemes

DYLAN WILIAM

Abstract

The kinds of authentic tasks that have been used in national assessments in England and Wales over the last thirty years - typically open-ended, 'pure' investigative tasks - are described, and the marking schemes used for their assessment are classified as either task-specific or generic. Generic schemes are further classified according to whether the 'degree of difficulty' of the task or the 'extent of progress' through the task is given most emphasis. A view of validation is presented that requires consideration of the value implications and social consequences of implementing assessment procedures, and it is argued that both task-specific and generic schemes will have the effect of stereotyping student approaches to these tasks. An alternative paradigm to norm-referenced and criterion-referenced interpretations of assessments, entitled 'construct-referenced' assessment, is proposed as being more consistent with the rationale behind such authentic assessments. Suggestions for the implementation of such a system are made and indices derived from signal-detection theory are suggested as appropriate measures for the evaluation of the accuracy of such assessments.

DYLAN WILIAM
Dylan Wiliam är universitetslektor i matematikämnets didaktik vid Centre for Educational Studies, Kings College, University of London, Great Britain.

Seventh-graders 'experiences and wishes about mathematics teaching in Finland

ERKKI PEHKONEN

Abstract

The experiences and wishes of about five hundred Finnish seventh-graders towards mathematics teaching are surveyed using a postal questionnaire. The pupils' responses to three open-ended questions in the questionnaire are classified into six categories: Teacher/teaching, Mathematical topics, Learning control, Pupil, Interaction and working forms, and Resources. Most of the responses (65-70 %) are in the first two categories. In addition, a significant percentage (more than 10 %) of the responses are in the class "Pupil" for experiences and in "Interaction and working forms" for wishes. Differences in responses given by boys and girls are discussed and some general suggestions for change in mathematics instruction, based on pupils' experiences and wishes, are put forward.

ERKKI PEHKONEN
Erkki Pehkonen är docent i matematikdidaktik vid Institutet för lärarutbildning vid Helsingfors universitet.

Standardized mathematics testing in Sweden: the legacy of Frits Wigforss

JEREMY KILPATRICK & BENGT JOHANSSON

Abstract

Developed more than 50 years ago, the Swedish system of standardized testing as a means of moderating marks (or grades) is about to be replaced by a criterion-referenced measurement scheme. The principal developer of the original system, Frits Wigforss, was a psychologist and mathematics educator who understood the complex issues raised by any marking system and who attempted to use testing not to replace but to improve teachers' judgment A close examination of the history of standardized mathematics testing in Sweden reveals the magnitude of Wigforss's contribution as well as its subsequent eclipse by the elevation of measurement technique over mathematical substance and a serious absence of attention to the educational and social consequences of changes in the system.

JEREMY KILPATRICK
Jeremy Kilpatrick är professor i matematikämnets didaktik vid University of Georgia, Athens, USA.

BENGT JOHANSSON
Bengt Johansson är universitetslektor i matematikämnets didaktik vid Göteborgs universitet, Sverige.

What is good research?

GÖRAN WALLÉN

Review of Criteria for Scientific Quality and Relevance in the Didactics of Mathematics. Report from a
symposium held in Gilleleje, Denmark, 1992. Editors: Gunhild Nissen and Morten Blomhøj. Danish Research Council for the Humanities. Roskilde University, IMFUFA. 1993. ISBN 87-7349-178-0, ISSN 0906-0103.

GÖRAN WALLÉN
Göran Wallén is associate professor at the Department for the theory of science and research, Göteborg University, Sweden

Mathematics education as theoretical knowledge

VICTOR FIRSOV

Abstract

Didactics of mathematics (DM) is the theoretical part of our knowledge in mathematics education. Its connections with the mathematical sciences and with school mathematics determine the independent character of DM as a scientific discipline. Features of school mathematics such as its unique aims, the highly abstract nature and hierarchical construction of the material to be studied, and the varied kinds of educational activities lead to its specific character, situating it among the school subjects and making accepted theoretical conclusions exclusively applicable to mathematics education. The social character of DM generates the approaches to constructing categories within the discipline (inexact and "inaccurate" conceptions, exclusion principles in formulating absolute statements, plausible reasoning, diversity of proofs) and promotes the selection of adequate methods of research that are not typical of positivistic science (expert review, discussion, pedagogical experiment). The applied character of DM determines an appropriate methodology of research and efficient ways of overcoming contradictions. The concrete practice of education (partly in the form of experimental tests) gives teachers an opportunity to use "inaccurate" empirical methods, reasonable considerations, intuitive choice, and so forth. Some practical advice is given for discussing and conducting research projects.

VICTOR FIRSOV
Dr. Victor Firsov är chef för den fristående forskningsorganisationen "Education for AU" i Moskva, Ryssland. Han är ledare för projektet "Educational Standards" vid Moskvas utbildningsministerium och biträdande redaktör för tidskriften Nämnaren. Han var tidigare under många år chef för the Mathematics Teaching Laboratory vid Institute for Scientific Research into Content and Methods of Instruction, Academy of Pedagogical Sciences i Sovjetunionen.

On pupils' reactions to the use of open-ended problems in mathematics

ERKKI PEHKONEN

Abstract

This article will focus on the preliminary results of one experimental class from the three-year research project "Open tasks in mathematics", which has been carried out in junior high schools during the years 1989-92 in Helsinki (Finland). The experiment groups used open-ended problems (problem fields) regularly, i.e. once a month, within their normal teaching. The main results from one class (N = 18) were as follows: The pupils liked most of the problem fields used. Their mathematical views did not change statistically significantly, but the nonsignificant changes in questionnaire ratings, classroom observations and the teacher's evaluations indicate that there was a change, and the change was in most cases positive.

ERKKI PEHKONEN
Erkki Pehkonen är docent i matematik och universitetslektor vid Department of Teacher Education, University of Helsinki.

Staking claims

JEREMY KILPATRICK

Abstract

The field of mathematics education has both scholarly and professional aspects. On the scholarly side, the question of what counts as research is still being debated. An examination of two proposed sets of criteria for evaluating the quality of research in mathematics education reveals that, interpreted appropriately, criteria borrowed from the natural and social sciences are relevant to a field that is attempting to be scientific. On the professional side, mathematics education must inevitably be concerned with the application of specialized knowledge to assist the students and teachers who are its clients. Teacher education remains a major function of mathematics education, parallel to the search for reliable knowledge to be applied. University mathematics educators need to work closely with mathematicians and with classroom teachers in developing both theory and practice. Mathematics education has flourished in countries in which institutional structures have supported it as an identifiable academic field.

JEREMY KILPATRICK
Jeremy Kilpatrick är Regents professor i Mathematics Education vid Collegue of Education, University of Georgia, Athens, USA och gästprofessor vid Matematikavdelningen, Institutionen för ämnesdidaktik, Göteborgs universitet, där han den 21 oktober 1995 utnämndes till hedersdoktor.

GLORIA STILLMAN

Abstract

This study investigated the impact of engagement with the task context by upper secondary students on their performance on applications tasks and teacher beliefs about the effects of students' engagement with task context. Moderate to high engagement with a task context was not often associated with poor performance which was more likely to be associated with no to low engagement. High engagement with task context was not a necessary condition for success as the degree of engagement necessary for success may be task specific. Engagement with task context alone was not of sufficient explanatory power to account for all the patterns in the data and it is acknowledged other factors need to be considered. Students identified a sense of realism and having an objective to work towards as facilitators of their engaging with task context. Amongst the teachers interviewed, there was support for the following beliefs: (a) students' preferred degree of contextualisation determines whether success is accompanied by engagement with the task context; (b) if the mathematics is not integrated with the task context, students will not engage with the context and will develop the habit of ignoring it; (c) if the two are integrated, students will engage with the task context; (d) the setting of tasks where the context transcends reality is problematic. The last of these was not supported by all teachers, however.

Gloria Stillman, James Cook University, Australia