
NOMAD 24(34), 2019
Algebra teachers’ questions and quandaries – Swedish and Finnish algebra teachers discussing practice
Cecilia Kilhamn and AnnSofi RöjLindberg
Abstract
Taking the teachers’ own practices as a point of departure, this study investigates what areas of mathematical knowledge algebra teachers brought up in collegial discussions and how they used their knowledge in acts of decompressing, trimming and bridging. The discussions centered around aspects of teaching and learning school algebra previously shown to be problematic, but gave rise to mathematical quandaries, revealing gaps in the teachers’ own understanding of the mathematical content. The study implies that the ability to unpack a mathematical concept is essential in algebra teaching and that teachers may need external input concerning mathematical knowledge to enable development in pedagogical content knowledge.Cecilia Kilhamn
Cecilia Kilhamn is a doctor in Mathematics Education and works at the University of Gothenburg. She has done research in algebra teaching and learning in the VIDEOMAT project in Gothenburg and as a guest researcher at Uppsala University. Currently she studying the intersection of algebraic thinking and computational thinking brought on by the implementation of programming in school curricula. She works as an editor for the journal Nämnaren at the national center for mathematics education (NCM) in Gothenburg.AnnSofi RöjLindberg
AnnSofi RöjLindberg is a university teacher in the didactics of mathematics at Åbo Akademi University in Vasa, Finland. Before working in teacher education she taught mathematics, physics and chemistry in grades 7 to 9. Her PhD has a focus on school mathematical practices at the lower secondary level from the perspectives of students and teachers. Her resent scientific effort is to contribute to the to development of primary school teacher education and the professional development of teachers.Skapad: 20191127 kl. 16:33

NOMAD – 24(34), 2019
Volume 24, No 34, November 2019
eNOMAD
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Thomas Kaas
Tilgange til tidlig algebra
[PDF]Peter Hästö and Riikka Palkki
Finnish students’ flexibility and its relation to speed and accuracy in equation solving
[PDF]AnnaMaija Partanen and Pieti Tolvanen
Developing a frame for analysing different meanings of the concept of variable mediated by tasks in elementaryschool mathematics textbooks
[PDF]Inger Eriksson, Sanna Wettergren, Jenny Fred, AnnaKarin Nordin, Martin Nyman och Torbjörn Tambour
Materialisering av algebraiska uttryck i helklassdiskussioner med lärandemodeller som medierande redskap i årskurs 1 och 5
[PDF]Jenny Fred
Att designa för elevers deltagande i ett algebraiskt arbete – elever i årskurs 2 och 3 utforskar visuellt växande mönster
[PDF]Helena Eriksson
Algebraic thinking and level of generalisation: students’ experiencing of comparisons of quantities
[PDF]Cecilia Kilhamn and AnnSofi RöjLindberg
Algebra teachers’ questions and quandaries – Swedish and Finnish algebra teachers discussing practice
[PDF]Skapad: 20191127 kl. 16:33

NOMAD 24(34), 2019
Algebraic thinking and level of generalisation: students’ experiencing of comparisons of quantities
Helena Eriksson
Abstract
This article explores grade 1 students’ different ways of experiencing quantity comparisons after participating in teaching designed as a learning activity using tasks from the Davydov curriculum. A phenomenographic analysis generated three hierarchical ways of experiencing comparisons: counting numerically, relating quantities, and conserving relationships. The first category comprises arithmetic ways of thinking, whereas the second and third categories comprise algebraic ways of thinking. Algebraic thinking was identified as reflections on relationships between quantities at different levels of generalisation. The implications of these results in relation learning activity theory are discussed.Helena Eriksson
Helena Eriksson is a PhD student at the Department of Mathematics and Science Education, Stockholm university. Her PhDproject concerns learning activity as a framework for designing teaching as well as a framework for research. Helena is engaged in teacher education at Dalarna University and is a teacher for special needs education in the municipality of Borlänge.Skapad: 20191127 kl. 16:27

NOMAD 24(34), 2019
Att designa för elevers deltagande i ett algebraiskt arbete – elever i årskurs 2 och 3 utforskar visuellt växande mönster
Jenny Fred
Sammandrag
Artikelns syfte är att beskriva och analysera vad i olika lektionssekvenser som skapar förutsättningar för att elever ska engageras i ett algebraiskt arbete och därmed urskiljer kritiska aspekter. Artikeln bygger på data från tre forskningslektioner i vilka lärandeverksamhet (learning activity) tillsammans med Radfords arbete om mönstergeneraliseringar har utgjort teoretiska utgångspunkter. I analysen har didaktiska principer från lärandeverksamhet samt kritiska aspekter gällande att uttrycka och argumentera för mönstergeneraliseringar fungerat som analysredskap. Resultatet kan bidra till att fördjupa förståelsen gällande på vilka sätt principerna från lärandeverksamhet kan stödja ett etablerande och upprätthållande av ett algebraiskt arbete och därmed möjliggöra för elevers urskiljande av kritiska aspekter.Abstract
The aim of the article is to describe and analyze what in different lesson sequences that creates the conditions for students to be involved in algebraic work and thereby distinguish critical aspects. The article is based on data from three research lessons in which Learning activity together with Radford’s work on pattern generalizations were theoretical starting points. In the analysis, didactic principles of Learning activity along with a few identified critical aspects regarding the ability to express and justify algebraic generalizations served as analytical tools. The result can contribute to deepened understanding of the ways the principles can support the establishment and maintenance of algebraic work enabling students to distinguish critical aspects.Jenny Fred
Jenny Fred är fil lic i matematikämnets didaktik och lärare åk F–6. Hennes forskningsintresse handlar främst om den tidiga algebraundervisningen samt hur och vad i undervisningen som skapar förutsättningar för elevers lärande.Skapad: 20191127 kl. 16:21

NOMAD 24(34), 2019
Materialisering av algebraiska uttryck i helklassdiskussioner med lärandemodeller som medierande redskap i årskurs 1 och 5
Inger Eriksson, Sanna Wettergren, Jenny Fred, AnnaKarin Nordin, Martin Nyman och Torbjörn Tambour
Sammandrag
Syftet med denna artikel är att beskriva och diskutera vilka funktioner lärandemodeller kan ha för att främja yngre elevers kollektiva diskussioner om algebraiska uttryck. Artikeln bygger på data från ett designforskningsprojekt baserat på Davydovs principer för lärandeverksamhet, bestående av videofilmade forskningslektioner i årskurs 1 och 5. Analysen fokuserar på vad som skapar förutsättningar för helklassdiskussioner om algebraiska uttryck, hur de drivs framåt och kvalificeras samt vilka funktioner lärandemodeller kan ha för elevernas utforskande av matematiska strukturer och relationer i algebraiska uttryck. Resultatet indikerar att lärandemodeller som medierande redskap gör det möjligt för eleverna att föra kreativa och reflekterande diskussioner om algebraiska uttryck och deras komponenter.Abstract
The aim for this article, which draws upon on data from a design research project based on Davydov’s principles of learning activity, is to discuss which functions learning models can have to promote students’ collective discussions on algebraic expressions. The data is comprised of videotaped lessons in Grade 1 and 5 respectively. The analysis focuses on conditions for qualifying wholeclass discussions and the functions learning models can have for the students’ collective exploration of mathematical structures and relationships in algebraic expressions. The result indicates that learning models as mediating tools enable the students to conduct creative and reflective discussions on algebraic expressions and their components.Inger Eriksson
Inger Eriksson är professor i pedagogik vid Stockholms universitet. Hennes forskningsintresse är elevers kunskapsutveckling och förutsättningar för lärande, speciellt i matematik och kemi, i ett verksamhetsteoretiskt perspektiv.Sanna Wettergren
Sanna Wettergren är fil lic i didaktik. Hennes forskningsintresse är kvalificering av matematikundervisning i grundskolan, speciellt algebra och resonemang, samt olika aspekter av formativ bedömning.Jenny Fred
Jenny Fred är fil lic i matematikämnets didaktik. Hennes forskningsintresse handlar främst om den tidiga algebraundervisningen samt hur och vad i undervisningen som skapar förutsättningar för elevers lärande.AnnaKarin Nordin
AnnaKarin Nordin är fil lic i matematikämnets didaktik och arbetar vid Stockholms universitet. Hennes forskningsintresse är skapandet av matematiska argument och resonemang i helklassdiskussioner.Martin Nyman
Martin Nyman är doktorand i matematikämnets didaktik. Hans forskningsintresse rör modeller för utformandet av en undervisning där elever ges möjligheter att engagera sig i ett aktivt undersökande av matematikens abstrakta dimensioner.Torbjörn Tambour
Torbjörn Tambour är docent i matematik vid Stockholms universitet. Hans forskningsintresse är algebra och lärande av algebra.Skapad: 20191127 kl. 16:14

NOMAD 24(34), 2019
Developing a frame for analysing different meanings of the concept of variable mediated by tasks in elementaryschool mathematics textbooks
AnnaMaija Partanen and Pieti Tolvanen
Abstract
Pupils’ studies in arithmetic can support the development of their algebraic thinking if arithmetic is taken as a starting point for generalising in sensemaking discussions. One of the most prominent concepts in algebra is that of the variable, which can have many different meanings, depending on its context. In this paper, we develop a frame for content analysis of tasks in elementaryschool mathematics textbooks. New categories for the meaning of variable are added to previous summaries, based on the literature review and the analysis. The developed frame can be used for analysing curricular materials, especially at the elementaryschool level.AnnaMaija Partanen
AnnaMaija Partanen is a senior university lecturer in mathematics education at the University of Lapland in Finland. She is also the director of LUMA centre Lapland. Her research interests are the discussion culture of mathematics classroom and earlyalgebraic approaches to teaching in elementery school.Pieti Tolvanen
Pieti Tolvanen is a lecturer in didactics of mathematics, physics and chemistry for preservice class teachers and a PhD candidate in Faculty of Education at University of Lapland in Finland. His research interest is the development of elementary pupils’ algebraic thinking.Skapad: 20191127 kl. 16:04

NOMAD 24(34), 2019
Finnish students’ flexibility and its relation to speed and accuracy in equation solving
Peter Hästö and Riikka Palkki
Abstract
A total of 266 Finnish students participated in a flexible equation solving test. By flexibility we understand the knowledge of multiple strategies and ability to choose the most mathematically appropriate strategy for a given task. Here we focus on the first aspect, namely knowledge of appropriate alternative, socalled innovative strategies. The test measured students’ capacity and inclination for producing innovative strategies. We consider the relationship between these measures and students’ speed and accuracy in solving equations. We find that students with high capacity for innovation have high speed and accuracy. On the other hand, some low capacity students had high speed or accuracy whereas others had low. Inclination for innovation is not related to speed or accuracy.Peter Hästö
Peter Hästö is professor of mathematics at University of Turku and University of Oulu, with responsibility for mathematics teacher education.Riikka Palkki
Riikka Palkki is PhD student at University of Oulu, Finland. She has taught mathematics, physics and chemistry in secondary school. She has been working on a mathematics teaching development project since 2014. She is especially interested in flexibility and intentional errors in teaching mathematics.Skapad: 20191127 kl. 15:58

NOMAD 24(34), 2019
Tilgange til tidlig algebra
Thomas Kaas
Sammendrag
Undervisning af 6–12 årige i algebra og algebraisk tænkning har under betegnelsen ”tidlig algebra” gradvist etableret sig som forskningsområde og undervisningspraksis i et stigende antal lande. På basis af et hermeneutisk inspireret litteraturstudie karakteriserer denne artikel de indholdsmæssige tilgange til undervisning i tidlig algebra, som forskningslitteratur i perioden 1995–2017 giver. Analysen har resulteret i et rammeværk for tilgange til tidlig algebraundervisning, som præsenteres og diskuteres. Artiklen konkluderer bl.a., at undervisning i tidlig algebra typisk tager afsæt i elevers arbejde med tal, kvantiteter og/eller funktionelle sammenhænge, og at elevers algebraiske tænkning søges udviklet gennem aktiviteter, der enten har opdagelser af generelle egenskaber og egenskaber eller ræsonnementer vedrørende ukendte talstørrelser som fokus. Forskelle mellem de forskellige tilgange vedrører desuden den rolle, som repræsentationer og kontekster har i undervisningen.Abstract
Teaching of 6–12 year old students in algebra and algebraic thinking has gradually become established as a research area and teaching practice in an increasing number of countries, under the term early algebra. This article presents a hermeneutically inspired review that was conducted to characterize and discus content approaches to early algebra teaching provided by the research literature in the period 1995–2017. The analysis of the included literature has resulted in a framework for approaches to early algebra teaching, which is presented and discussed.
The article concludes that early algebra teaching typically takes its starting point in students’ work on numbers, quantities and / or functional contexts, and that students’ algebraic thinking is taught through activities that have either identification of general relations and properties or reasoning with unknown quantities as focus. Differences between the different approaches relate also to the role of representations and contexts in the teaching.Thomas Kaas
Thomas Kaas er ph.d.studerende ved Aarhus Universitet og Københavns Professionshøjskole. Han har tidligere arbejdet som folkeskolelærer, læreruddanner og lærebogsforfatter. Hans primære forskningsinteresser er algebraisk tænkning og evaluering i matematikundervisning.Skapad: 20191127 kl. 15:49

NOMAD 24(2), 2019
University students’ general and specific beliefs about infinity, division by zero and denseness of the number line
Kristina Juter
Abstract
A study of university students’ beliefs about infinity and related concepts, e.g. division by zero and denseness of the number line, was conducted. The concepts were chosen for the students’ proven cognitive challenge in coping with them, and part of the study was to analyze individual beliefs of the different concepts in relation to each other. A questionnaire was designed to discover relationships between pre service teachers’ and technology students’ beliefs. Particular foci in the study were general and specific perspectives of the concepts and admission requirements for the programs. The results show incoherence with respect to general and specific representations of aspects concerning denseness of the number line, and also show that admission requirements are significant when it comes to validity of beliefs about division by zero.Kristina Juter
Kristina Juter is Professor of Mathematics education at Kristianstad Uni versity in Sweden. Her current research interests are students’ understandings of concepts related to calculus, preservice mathematics teacher development and upper secondary school physics teachers’ use of mathematics in physics teaching.Skapad: 20190620 kl. 08:52

NOMAD – 24(2), 2019
Volume 24, No 2, June 2019
eNOMAD
[PDF] displays the full text pdf. The two most recent volumes are password protected. Use ”Open access” in the menu for full text of older articles.
Trond Stølen Gustavsen and Olav Gravir Imenes
Investigating the fit of a model for students’ understanding of fractions in a Norwegian context
[PDF]Abdel Seidouvy, Ola Helenius and Maike Schindler
Authority in students’ peer collaboration in statistics: an empirical study based on inferentialism
[PDF] OPEN ACCESSSvanhild Breive
Engaging children in mathematical discourse: a kindergarten teacher’s multimodal participation
[PDF]Kristina Juter
University students’ general and specific beliefs about infinity, division by zero and denseness of the number line
[PDF]Skapad: 20190619 kl. 16:09

NOMAD 24(2), 2019
University students’ general and specific beliefs about infinity, division by zero and denseness of the number line
Kristina Juter
Abstract
A study of university students’ beliefs about infinity and related concepts, e.g. division by zero and denseness of the number line, was conducted. The concepts were chosen for the students’ proven cognitive challenge in coping with them, and part of the study was to analyze individual beliefs of the different concepts in relation to each other. A questionnaire was designed to discover relationships between pre service teachers’ and technology students’ beliefs. Particular foci in the study were general and specific perspectives of the concepts and admission requirements for the programs. The results show incoherence with respect to general and specific representations of aspects concerning denseness of the number line, and also show that admission requirements are significant when it comes to validity of beliefs about division by zero.Kristina Juter
Kristina Juter is Professor of Mathematics education at Kristianstad Uni versity in Sweden. Her current research interests are students’ understandings of concepts related to calculus, preservice mathematics teacher development and upper secondary school physics teachers’ use of mathematics in physics teaching.Skapad: 20190619 kl. 16:08

NOMAD 24(2), 2019
Engaging children in mathematical discourse: a kindergarten teacher’s multimodal participation
Svanhild Breive
Abstract
This article reports from a case study which investigates a kindergarten teacher’s multimodal participation in a teachinglearning activity involving addition and counting. By multimodal participation the kindergarten teacher engages nine children (age 4.9–5.9) in mathematical discourse and supports their opportunities for learning. Implications for practice are that kindergarten teachers (and school teachers) can benefit from being consciously aware of the affects their bodily actions have on children’s mathematical reasoning and how they can engage children in mathematical discourse without having to ”teach” (i.e., tell) children mathematical concepts and relations. The article also considers how kindergarten teachers can prepare for a smooth transition to school by introducing children to mathematics through semistructured activities.Svanhild Breive
Svanhild Breive defended her PhD thesis at University of Agder in June 2019. She has then been employed as Assistant Professor in Mathematics Education at University of SouthEastern Norway. Her key research interests are mathematics teaching and learning in kindergarten.Skapad: 20190619 kl. 16:04

NOMAD 24(2), 2019
Authority in students’ peer collaboration in statistics: an empirical study based on inferentialism
Abdel Seidouvy, Ola Helenius and Maike Schindler
Abstract
Students’ peer collaboration efforts in mathematics and statistics is a topic that has increasingly gained attention in research. In any collaboration, authority relations play a role for how meaning is constituted: Whenever things are discussed and decisions are made, authority is involved in a sense that some arguments or persons may be more convincing and powerful than others. In this article, we investigate how authority changes dynamically in type and in distribution as groups of fifth grade students collaborate in data generation processes. We identify and categorize authority using an epistemological framework, which is based on the philosophical theory of inferentialism. The results show that the three different types of authority described in inferentialism are all identifiable in students’ collaborative work. We also find and categorize further types of authority connected to the statistics group work, some of which are hardly addressed in previous research.Abdel Seidouvy
Abdel Seidouvy is PhD student at Örebro University, Sweden. His main research interest concerns statistics education, student collaboration, and inferentialism in statistics education.Ola Helenius
Ola Helenius is a researcher and a designer of teaching sequences and professional development programs at the National Centre for Mathematics Education at University of Gothenburg. His main research interests concerns the epistemology, psychology and neuropsychology of elementary mathematics, and professional development of mathematics teachers and preschool teachers.Maike Schindler
Maike Schindler has a PhD in mathematics education from TU Dortmund University, Germany. After her postdoc at Örebro University, Sweden, she became professor at the University of Cologne, Germany. Her main research interests relate to theories in mathematics education, learning difficulties and special education in mathematics, creativity and giftedness in mathematics, inclusive teaching and learning, and – methodically – the use of eye tracking in mathematics education.Skapad: 20190619 kl. 15:59

NOMAD 24(2), 2019
Investigating the fit of a model for students’ understanding of fractions in a Norwegian context
Trond Stølen Gustavsen and Olav Gravir Imenes
Abstract
To capture the complexity of students’ understanding of fractions, a model linking partwhole to the subconstructs ratio, operator, quotient and measure has been proposed. We ask if this model is compatible with students’ achievements in a Norwegian context. Responses from 638 students were analysed using structural equation modelling (SEM), and a good fit of the model was obtained after removing the ratio subconstruct. In particular, partwhole is seen to be important for operator, quotient and measure. Using qualitative analysis of interviews, we found reasoning associated with ratio, with a weak link to the partwhole subconstruct.Trond Stølen Gustavsen
Trond Stølen Gustavsen is Professor of mathematics at the University of SouthEastern Norway and Professor of mathematics/mathematics education at the University of Bergen. He holds a Dr. Scient. degree from the University of Oslo and has done research in pure mathematics. Gustavsen is editor and coauthor of textbooks for mathematics teacher education and his research interests include the teaching and learning of fractions and argumentation and proof.Olav Gravir Imenes
Olav Gravir Imenes is Associate Professor of mathematics/mathematics education at Oslo Metropolitan University. He has a Ph.D. in mathematics from the University of Oslo in the subject of noncommutative algebraic geometry and has published research in mathematics education and contributed to textbooks for teacher education in mathematics. His research interests include noncommutative algebraic geometry and the teaching and learning of fractions.Skapad: 20190619 kl. 15:47

NOMAD – 24(1), 2019
Volume 24, No 1, March 2019
eNOMAD
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Annika Pettersson, Yvonne Liljekvist and Jorryt van Bommel
Studying concept elements as a way to trace students’ conceptual understanding
[PDF]Kajsa Bråting, Lars Madej and Kirsti Hemmi
Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks
[PDF]Johan Sidenvall
Literature review of mathematics teaching design for problem solving and reasoning
[PDF]Janne Fauskanger
Ambisiøse undervisningspraksiser i Teacher time out
[PDF]Skapad: 20190215 kl. 14:20

NOMAD 24(1), 2019
Ambisiøse undervisningspraksiser i Teacher time out
Janne Fauskanger
Sammanfattning
Denne studien undersøker ambisiøse undervisningspraksiser lærere får muligheter til å øve på å utføre gjennom rutinen Teacher time out (TTO) i et etterutdanningsforløp. Datamaterialet analysert er fra prosjektet Mestre ambisiøs matematikkundervisning, hvor lærere arbeider med bestemte matematiske aktiviteter i sykluser av utforsking og utprøving. Analysene av 139 TTO viser at deltakerne får øve på følgende undervisningspraksiser: 1) å få frem elevers matematiske ideer, 2) å orientere elevene mot hverandres ideer, 3) å respondere på elevenes matematiske ideer, 4) å vurdere elevenes matematiske forståelse, samt til utvikling av mer generell undervisningskompetanse. Implikasjoner for fremtidig etterutdanning og for fremtidig forskning diskuteres.Abstract
This study investigates ambitious teaching practices teachers have an opportunity to practice through the routine Teacher time out (TTO). The data material analyzed is taken from the project Mastering ambitious mathematics teaching, wherein teachers in their professional development work on given teaching activities in cycles of enactment and investigation. 139 TTOs have been analyzed. The analyses indicate that the teachers in TTOs have an opportunity to practice the following teaching practices: 1) eliciting students’ mathematical ideas, 2) orienting students towards each other’s ideas, 3) responding to students’ mathematical ideas, 4) evaluating students’ mathematical understanding, and in addition developing their general teaching competence. Implications for future professional development and research are discussed.Janne Fauskanger
Janne Fauskanger er førsteamanuensis i matematikkdidaktikk ved Universitetet i Stavanger. Hennes forskningsinteresser knyttes hovedsakelig til matematikklæreres kunnskap og praksis, samt til utvikling av læreres kunnskap og praksis.Skapad: 20190215 kl. 14:09

NOMAD 24(1), 2019
Literature review of mathematics teaching design for problem solving and reasoning
Johan Sidenvall
Abstract
To characterize teaching designs intended to enhance students’ problem solving and reasoning skills or to develop other mathematical competencies via problem solving and reasoning, a literature review was conducted of 26 articles published in seven topranked journals on mathematics education from 2000 to 2016. Teaching designs were characterized by a) the educational goals of the designs, b) the claims about how to reach these goals, and c) the empirical and theoretical arguments underlying these claims. Thematic analysis was used to analyze the retrieved articles. All but two studies had goals concerned with developing students’ mathematical competencies. The overarching ideas of the identified emergent claims regarding the achievement of stipulated goals, concerned scaffolding students’ learning and letting students construct their own mathematics. Four recurring theoretical arguments were found to support emergent claims: hypothetical learning trajectories, realistic mathematics education, theory of didactical situations and zone of proximal development.Johan Sidenvall
Johan Sidenvall, postgraduate student at the Department of Science and Mathematics Education, and Umeå Mathematics Education Research Centre (UMERC), Umeå University, Sweden. His research interest is how and under what conditions mathematical teaching, aimed at supporting students’ own construction of solutions via reasoning, may lead to more e ective learning.Skapad: 20190215 kl. 14:03

NOMAD 24(1), 2019
Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks
Kajsa Bråting, Lars Madej and Kirsti Hemmi
Abstract
In search of the reasons for Swedish students’ low achievement in algebra in international and national evaluations, we investigate how the development of algebraic thinking is addressed in the Swedish national mathematics curriculum and two widely used mathematics textbook series for grades 1–6 in Sweden. The analytical tool used is based on the classification of ”big ideas” which research has shown as important for developing pupils’ algebraic understanding in early school grades. The results show that functional thinking, expressions, and equations are well represented topics both in the curriculum and the textbooks; however generalized arithmetic is a topic that is poorly developed in both the curriculum and the textbooks.Kajsa Bråting
Kajsa Bråting is Associate professor in mathematics education at the Department of education, Uppsala university. Currently she leads the VRfunded research project Integrating programming in school mathematics – exploring the intersection of algebraic and computational thinking and participates in a research project aiming at characterizing the algebraic content in the Swedish mathematics curriculum and textbooks for grades 1–9. She writes textbooks in mathematics for upper secondary school level and for teacher education.Lars Madej
Lars Madej is a doctoral student and university teacher in mathematics education at the Department of education, Uppsala university. He is also fil. lic. in mathematics. Currently he participates in the research project Towards researchbased teaching of algebra – diachronic and synchronic analyses of steering documents, curriculum materials and teachers, funded by the Swedish research council. He writes textbooks in mathematics for intermediate school.Kirsti Hemmi
Kirsti Hemmi is Professor in mathematics and science education at Åbo akademi university in Finland and Guest professor at Uppsala university in Sweden. She has led research projects comprising cultural aspects of mathematics education, curricula, textbooks and teachers’ interaction with materials. She is currently leading a fouryear research project at Uppsala university focusing on the progression of algebra in the Swedish mathematics curriculum.Skapad: 20190215 kl. 13:59

NOMAD 24(1), 2019
Studying concept elements as a way to trace students’ conceptual understanding
Annika Pettersson, Yvonne Liljekvist and Jorryt van Bommel
Abstract
The understanding of mathematical concepts has been described in terms of concept definition and concept image. We suggest an elaboration of these constructs, the concept element, to find a way to theoretically describe students’ understanding. The concept element construct was tested in a setting with students working with linear functions at the secondary school level. Our empirical findings reveal traces of students’ concept elements regarding linear functions. Some concept elements appeared early in the process while others appeared after a cognitive conflict (e.g. evoked by the task construction and setting). The detailed grid on which concept elements are defined was a useful tool, yielding new insights into students’ knowledge and understanding.Annika Pettersson
Annika Pettersson is licentiate in Mathematics and lecturer at Kristine hamns kommun, Sweden. She works in upper secondary school for adults (Komvux). Her research interests are students’ learning, teaching development and the possibilities for teachers to research base their teaching.Yvonne Liljekvist
Yvonne Liljekvist is senior lecturer in Mathematics Education at the Department of Mathematics and Computer Science at Karlstad University, Sweden. One of her research interests is mathematics teachers’ professional development. One of her research interests is mathematics teachers’ professional development, focusing on the relations between the subject and processes of teaching – studying – learning.Jorryt van Bommel
Jorryt van Bommel is senior lecturer in Mathematics Education at the Department of Mathematics and Computer Science at Karlstad University, Sweden. Her research focusses on teachers’ professional development as well as the teaching and learning of mathematics in preschool class, primary and secondary school.Skapad: 20190215 kl. 13:53

NOMAD – 23(34), 2018
Volume 23, No 34, November 2018
eNOMAD
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Tamsin Meaney and Toril Eskeland Rangnes
Language diversity in mathematics education in the Nordic countries 2008–2018
[PDF]Eva Norén and Petra Svensson Källberg
Fabrication of newlyarrived students as mathematical learners
[PDF]Petra Svensson Källberg
Identity formations as mathematical learners in the context of transition
[PDF]Marie Sjöblom
Developing mathematical reasoning by using questions in a multilingual mathematics classroom
[PDF]Maria Ahlholm and Päivi PortaankorvaKoivisto
The language factor – what exactly is it? Bilingual speakers of Russian and Finnish solving mathematical tasks
[PDF]Jöran Petersson
Newly and earlyimmigrated secondlanguage students’ knowledge of arithmetic syntax
[PDF]Hilja L. Huru, AnnaKaisa Räisänen and Anita Movik Simensen
Culturally based mathematics tasks: a framework for designing tasks from traditional Kven artefacts and knowledge
[PDF]Mette Hjelmborg and Ane Fleischer
En registeranalyse af centrale matematiske begreber i en grønlandsk kontekst
[PDF]Anne Birgitte Fyhn, Ellen J. Sara Eira, Ole Einar Hætta, Inga Anne Marit Juuso, Siv Ingrid Nordkild og Ellen Margrethe Skum
Bishop Sámegillii – utfordringer ved oversetting av matematikkdidaktisk fagterminologi
[PDF]Dorota Lembrér
Polish parents’ views on mathematics activities at home and in Swedish preschools
[PDF]Troels Lange and Tamsin Meaney
Talking about mathematics in two languages: Can parental views inform the development of digital games for young children?
[PDF] OPEN ACCESSAndrea Eikset and Tamsin Meaney
When does a difference make a difference? Teaching about language diversity in mathematics teacher education
[PDF] OPEN ACCESSSusanne Prediger
Multilingual issues in Nordic mathematics education – What is achieved and where to go next?
[PDF]Skapad: 20181106 kl. 11:33

NOMAD 23(34), 2018
Multilingual issues in Nordic mathematics education – What is achieved and where to go next?
Susanne Prediger
Abstract
This Nomad special issue provides eleven highly interesting insights into current research and development projects in mathematics education on multilingual and multicultural issues. It shows the diversity of approaches currently adopted in the Nordic Countries with an impressing richness of perspectives and ideas. I am grateful to have had the opportunity to read and discuss the papers carefully as I learned a lot. In this commentary, I compare and connect the papers with each other and the international state of research and to suggest some directions for further research and development. The commentary is structured in the following steps: Di erent implicit and explicit conceptualization of languages are identi ed in the articles ( rst section); di erent research approaches are summarized with a need to strengthen Design research (second section); and di erent instructional approaches for activating multiple language resources for mathematics learning and further enhancing both languages (third section).Susanne Prediger
Susanne Prediger is full professor for mathematics education research at TU Dortmund University in Germany and currently the vicedirector of the German Center for Mathematics Teacher Education. She leads a big research group on language diversity in mathematics education and is interested in fostering language learners’ mathematics learning.Skapad: 20181106 kl. 11:22

NOMAD 23(34), 2018
When does a difference make a difference? Teaching about language diversity in mathematics teacher education
Andrea Eikset and Tamsin Meaney
Abstract
There has been little attention in mathematics education research about how to include issues to do with language diversity in teacher education. This paper describes the process used by two teacher educators to examine their own practices of linking multilingual perspectives to mathematics education in their work with preservice teachers. By systematically analysing their discussion about a threehour, mathematicsteachereducation workshop on proportional thinking, the teacher educators were able to identify a series of Discourses. They considered that these Discourses underlay their decision making about how language diversity could be raised with preservice teachers. The results highlight the complexity connected to raising language diversity issues in mathematics teacher education. For example, deciding what challenging content should be provided to preservice teachers is a ected by the need to develop relationships with them as well as managing their learning. Joint re ection by the teacher educators was needed to ensure that the aim of challenging preservice teachers about how to deal with language diversity issues in mathematics classrooms could be achieved.Andrea Eikset
Andrea Eikset is an assistant professor working at Western Norway University of Applied Sciences in Bergen. Her academic background is in mathematics didactics, and her main research interests are mathematics in early childhood education, diversity and sustainable development. This is also connected to teacher educators own professional development.Tamsin Meaney
Tamsin Meaney is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. She has written journal articles on language diversity in mathematics education for twenty years.Skapad: 20181106 kl. 11:20

NOMAD 23(34), 2018
Talking about mathematics in two languages: Can parental views inform the development of digital games for young children?
Troels Lange and Tamsin Meaney
Abstract
In this article, the results are presented from a survey of parents’ views about the digital games that their young multilingual children play. Previous research has indicated that parents struggled to describe how their children were learning from playing digital games. The results from this study indicate that parents could provide information about the digital games and the mathematical language they invoked. This information could be useful in developing playful, digital games that support multilingual children to talk about mathematics. The survey also provides insights into the followup qualitative research studies that are needed to support the development of new digital games.Troels Lange
Troels Lange is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. He has a long standing interest in children ́s perceptions of mathematics education and societal issues that in uence the views of parents, teachers and politicians about mathematics education for young children.Tamsin Meaney
Tamsin Meaney is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. She has written journal articles on language diversity in mathematics education for twenty years.Skapad: 20181106 kl. 11:18

NOMAD 23(34), 2018
Polish parents’ views on mathematics activities at home and in Swedish preschools
Dorota Lembrér
Abstract
This article describes the results of a digital survey of 41 Polish immigrant parents’ views on mathematics activities at home and at preschool as parents’ views potentially provide a range of perspectives on mathematics activities for young children. Parents were asked to describe and justify their views about how children engage with mathematical ideas and nominate activities that children engage in at home and at preschool. When parents justi ed their views about young children and mathematics, they tended to align themselves with the norms and values of the Swedish preschool curriculum. The ndings suggest that parents, like children, are socialised into Swedish preschools. However, this alignment could limit possibilities for broadening perspectives about mathematics education in preschool, which could be available by incorporating input from immigrant parents’ di erent cultural and linguistic backgrounds.Dorota Lembrér
Dorota Lembrér is a doctoral student at Western Norway University of Applied Sciences in Bergen, Norway. She has a background as a preschool teacher and a lecturer at Malmö University in Sweden. Her main research interests are early childhood mathematics education and aspects of mathematics activities in preschool and home environments.Skapad: 20181106 kl. 11:16

NOMAD 23(34), 2018
Bishop Sámegillii – utfordringer ved oversetting av matematikkdidaktisk fagterminologi
Anne Birgitte Fyhn, Ellen J. Sara Eira, Ole Einar Hætta, Inga Anne Marit Juuso, Siv Ingrid Nordkild og Ellen Margrethe Skum
Sammenfatning
I artikkelen belyser vi kulturelle og språklige utfordringer som kan oppstå når teoretiske begreper fra matematikkdidaktisk forskning oversettes til et språk og en kultur som er ernt fra språket og kulturen der begrepene ble utviklet. I samarbeid med forskere oversatte lærere ved Guovdageainnu nuoraidskuvla/Kautokeino ungdomsskole et matematikkdidaktisk rammeverk fra engelsk til nordsamisk. Vi fokuserer på oversetting av aktiviteten locating som inngår i rammeverket. Locating omfatter grunnlaget for elevers utvikling av romlig forståelse. Vi bidrar også til diskusjonen om hvorvidt hovedområdene i matematikklæreplanen skal utformes som verb eller substantiver.Abstract
In this article we enlighten cultural and linguistic challenges that may occur when concepts from mathematics education are translated into a language and a culture that is far from the language and the culture where the concepts were developed. In cooperation with researchers, teachers at Guovdageainnu nuoraidskuvla/Kautokeino lower secondary school translated a mathematics education framework into North Sámi. We focus on translations of the activity locating, which is part of this framework. Locating constitutes the basis for students’ development of spatial understanding. We also contribute to the discussion about whether the national mathematics curriculum’s content areas should be verbs or nouns.Anne Birgitte Fyhn
Anne Birgitte Fyhn is professor in mathematics education at UiT, the Arctic University of Norway, Tromsø. She holds a professor II position at Sámi Allaskuvla/ Sámi University of Applied Sciences. Her main research interests are relations between mathematics education and culture.Ellen J. Sara Eira
Ellen J. Sara Eira is principal of Guovdageainnu nuoraidskuvla. She has taught mathematics in Sámi for 35 years and she taught Sámi at Sámi Allaskuvla/Sámi University of Applied Sciences for two years. Since 2010, she is mathematics sensor for the national compulsory school exam. From 1983–1998 she was member of the national exam group in Sámi at upper and lower secondary level.Ole Einar Hætta
Ole Einar Hætta is mathematics teacher at Guovdageainnu nuoraidskuvla. He has been sensor at the Sámi mathematics exam for the compulsory school for ten years. He was member of the 2017 national curriculum group that worked out core elements in mathematics. He is master student in mathematics education at UiT, the Arctic University of Norway.Inga Anne Marit Juuso
Inga Anne Marit Juuso is mathematics teacher at Guovdageainnu nuoraidskuvla. She also teaches duodji (Sámi handicraft). She is master student in mathematics education at UiT, the Arctic University of Norway.Siv Ingrid Nordkild
Siv Ingrid Nordkild is Ph.D. student at UiTThe Arctic University of Norway, Campus Tromsø. Her Ph.D. is based in mathematics education in the North Sámi area in Norway. Her research interests are culture based mathematics education and indigenous related issues in education. She is educated as an electrical engineer with Master’s degree in pedagogy.Ellen Margrethe Skum
Ellen Margrethe Skum is vice principal of Guovdageainnu nuoraidskuvla. She teaches Sámi language and duodji (Sámi handicraft). She uses her competencies in Sámi tradition and culture, among others from reindeer husbandry, in her teaching.Skapad: 20181106 kl. 11:12

NOMAD 23(34), 2018
En registeranalyse af centrale matematiske begreber i en grønlandsk kontekst
Mette Hjelmborg and Ane Fleischer
Sammenfatning
Denne artikel beskriver resultaterne af en registeranalyse af grønlandske matematiske begreber. Fokus for analysen er de karakteristiske træk, der opstår når man genererer et matematisk register i et polysyntetisk sprog. Vores teoretiske ramme er faglige registre og registerkontinuum i et systemisk funktionelt lingvistisk perspektiv. Metodisk foretages dokumentanalyser, der mættes af interviewsvar fra semistrukturerede interviews. Vores analyse indikerer, at der som oftest benyttes samtlige tre teknikker når grønlandske matematiktermer genereres: omskrivninger, nominaliseringer og metaforer. Det matematiske register knytter sig dermed i høj grad og på mange og varierede måder til det grønlandske dagligdagssprog.Abstract
This article describes the results of an analysis of the mathematical register in Greenland concerning the characteristics for developing terms for mathematical concepts in a polysynthetic language. The theoretical frame is registers, register continuum with respect to systemic functional linguistic. The analysis indicates that three techniques (gerunding, circumlocutions, and metaphors) are used simultaneously when generating new terms in Greenlandic. The mathematical concepts in Greenlandic thereby have a direct link to everyday meanings of similar concepts.Mette Hjelmborg
Mette Hjelmborg er lektor på UCL Erhvervsakademi og Professionshøjskole, læreruddannelsen på Fyn. De vigtigste interesser er: sprog og begrebsdannelse, Dansk som andetsprog, elever med særlige behov, læremiddelanalyse.Ane Fleischer
Ane Fleischer underviser på læreruddannelsen i Grønland, efteruddanner, lærebogsforfatter. De vigtigste interesser er: matematiklæring og især sprog og matematikbegreber på grønlandsk, elever med særlige behov.Skapad: 20181106 kl. 11:04

NOMAD 23(34), 2018
Culturally based mathematics tasks: a framework for designing tasks from traditional Kven artefacts and knowledge
Hilja L. Huru, AnnaKaisa Räisänen and Anita Movik Simensen
Abstract
This article discusses mathematical and cultural task design to support minority and endangered languages and cultures. More precisely, we propose a theoretical framework to design mathematical tasks for language immersion in mathematics for Kven students. Drawing on previous studies, we suggest that traditional tools have the potential to support the learning of mathematics, language, and culture. One challenge for endangered languages and cultures is that the younger generations may have lost connections with their traditional language and culture. We argue that the older generations can mediate authentic aspects of Kven culture to students, which then become historicalcultural authentic (HiCuA) aspects.Hilja L. Huru
Hilja L. Huru is a professor in mathematics at OsloMet – Oslo Metropolitan Univerity and UiT – The Arctic University og Norway. Her background is in pure mathematics with a PhD in noncommutative algebra connected to mathematical physics. Her research interest also includes multicultural classrooms with a focus on pressured minorities and indegenous mathematics.AnnaKaisa Räisänen
AnnaKaisa Räisänen is a language advisor at the Kainun institute – Kvensk institutt, Norway and a doctoral student at the University of Oulu, Finland. She is involved with language education programs for Kven language and leads Kven language nests projects in northern Norway. Her main research interests are sociolinguistics and language policy as well as language revitalization and language vitality.Anita Movik Simensen
Anita Movik Simensen is assistant professor of mathematics education at UiT – The Arctic University of Norway. Her research interests include mathematics education in inclusive classrooms, indigenous mathematics, and the use of natural outdoor learning environments as setting for young children’s (age 1–6) mathematical experiences.Skapad: 20181106 kl. 10:55

NOMAD 23(34), 2018
Newly and earlyimmigrated secondlanguage students’ knowledge of arithmetic syntax
Jöran Petersson
Abstract
The present study investigated how 259 Swedish, grade 9 students, of whom 90 had an immigrant background, achieved on twelve written test items in the content area of number. Four of the twelve test items required good knowledge of arithmetic syntax, such as when it was appropriate to apply orderofoperation rules and the associative and distributive laws of arithmetic operations. On these four test items, the mostrecently arrived students showed on average signi cantly more knowledge than the students who had immigrated when they were younger and had participated in Swedish schools for longer periods of time. The outcome suggests that these two groups of immigrant students in later school years should be considered as separate subcategories of secondlanguage students when it comes to teaching, assessment and research.Jöran Petersson
Jöran Petersson is a senior lecturer at Stockholm University, Sweden. He wrote his PhDdissertation on test achievement of second language students in the last year of compulsory school. Presently he is doing postdoctoral research on how foundational number sense appears in textbooks and homework in the rst year of compulsory school. Jöran also has an interest in mathematical modelling.Skapad: 20181106 kl. 10:52

NOMAD 23(34), 2018
The language factor – what exactly is it? Bilingual speakers of Russian and Finnish solving mathematical tasks
Maria Ahlholm and Päivi PortaankorvaKoivisto
Abstract
A lack of knowledge of the language of instruction is often believed to be the main reason for low achievement among students with an immigrant background. We regard language as a tripartite unit comprising aspects of concept formation, pragmatic language usage and the linguistic form. In this theoretical framework, we report two case studies of bilingual, Russian and Finnish speaking students’ explanations of their procedures while solving mathematical tasks. The students’ linguistic processing varied in terms of conceptualization, pragmatic meaningmaking and grammatical form. In a bilingual context, the labelling of concepts and meaningmaking through argumentation are simultaneously processed in two languages.Maria Ahlholm
Maria Ahlholm is a University lecturer (PhD) in the Faculty of Educational Sciences at the University of Helsinki, and an adjunct professor in Finnish language, especially applied linguistics. Ahlholm is interested in the multilingual socialisation process of emerging second language.Päivi PortaankorvaKoivisto
Päivi PortaankorvaKoivisto is a University lecturer (PhD) of mathematics didactics in the Faculty of Educational Sciences at the University of Helsinki. PortaankorvaKoivisto is interested in developing mathematics teacher education and teachers’ awareness of the language factor in teaching and learning mathematics.Skapad: 20181106 kl. 10:49

NOMAD 23(34), 2018
Developing mathematical reasoning by using questions in a multilingual mathematics classroom
Marie Sjöblom
Abstract
In this paper, students’ questions while working in small groups on mathematical problemsolving tasks are investigated. In order to improve students’ reasoning and communication abilities in mathematics, an intervention study was designed in a multilingual upper secondary mathematics classroom in Sweden. In their discussions students used Swedish, which was their second language and also the language of instruction. The changes in students’ ways of using questions across the three cycles of the intervention were analysed. The results showed how students over the cycles changed their ways of framing questions from looking for the correct answer towards clarifying other students’ meaning in order to understand each other’s reasoning. The implication from the study is that it is important to promote interactions between students rather than focusing on students’ need to develop their second language competencies.Marie Sjöblom
Marie Sjöblom is a PhDstudent in mathematics education at Malmö University. She is also a mathematics teacher, and work as senior lecturer with school development in Malmö, supporting teachers and school leaders on collegial learning processes. Key research interests are interaction in multilingual mathematics classrooms and educational design research.Skapad: 20181106 kl. 10:46

NOMAD 23(34), 2018
Identity formations as mathematical learners in the context of transition
Petra Svensson Källberg
Abstract
This paper explores the relation between discourses and identity formations as mathematical learners in a context of transition. The data consists of an interview with two 16 yearold immigrant girls, who were relocated when their school, in a multicultural and socioeconomically disadvantaged area in Sweden, was closed. The girls showed dynamic and unstable identities by drawing on di erent discourses. Social relational discourses, more than mathematical pedagogical discourses, governed their actions as learners of mathematics; enabling identities as noisy, unengaged, but able students in the old school, and as engaged and accepted, but also as strangers, in the new school.Petra Svensson Källberg
Petra Svensson Källberg has a doctoral degree in mathematics education from Stockholm University, the department of mathematics and science education. She works at Pedagogisk Inspiration, a department which works with school development and research in the muncipality of Malmö, Sweden. Her main research interests concern sociopolitical issues in mathematics education and are related to multilingual and multicultural issues in mathematics education.Skapad: 20181106 kl. 10:44

NOMAD 23(34), 2018
Fabrication of newlyarrived students as mathematical learners
Eva Norén and Petra Svensson Källberg
Abstract
As a response to recent laws on how to support newlyarrived students’ schooling, new policy texts have been released in Sweden. By analyzing policy texts we show how a particular kind of human, ”the newlyarrived student as a mathematical learner” is fabricated through discursive processes. We show how the policy texts are framed within an including discourse that encourages multiculingualism and views students’ mother tongue and backgrounds as resources. However, simultaneously the newlyarrived student is thought of, in a more excluding discourse, as being in need of rescue and as lacking the most valuable asset, the Swedish language.Eva Norén
Eva Norén is senior lecturer in mathematics education at the department of mathematics and science education, Stockholm University. Her main research interest is multilingual students in mathematics classrooms. She defended her PhD thesis, Flerspråkiga matematikklassrum [Multilingual mathematics classrooms] in 2010. She has also researched gender issues related to mathematics teaching and learning. Since 2017 she is involved in a development and research project on programming in subject didactics.Petra Svensson Källberg
Petra Svensson Källberg has a doctoral degree in mathematics education from Stockholm University, the department of mathematics and science education. She works at Pedagogisk Inspiration, a department which works with school development and research in the muncipality of Malmö, Sweden. Her main research interests concern sociopolitical issues in mathematics education and are related to multilingual and multicultural issues in mathematics education.Skapad: 20181106 kl. 10:41

NOMAD 23(34), 2018
Language diversity in mathematics education in the Nordic countries 2008–2018
Tamsin Meaney and Toril Eskeland Rangnes
Abstract
A paper in which the guest editors introduce this thematic issue of NOMAD.
”The aim of the thematic issue is to provide an overview of what was being done and from this to determine what still needed to be done on language diversity in mathematics classrooms and early childhood centres in the Nordic countries.”Tamsin Meaney
Tamsin Meaney is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. She has written journal articles on language diversity in mathematics education for twenty years.Toril Eskeland Rangnes
Toril Eskeland Rangnes is associate professor in mathematics education at the Faculty of Education, Arts and Sports at Western Norway University of Applied Sciences, campus Bergen. Her main research interest are critical mathematics education, teacher professional development and language diversity in mathematics classrooms.Skapad: 20181106 kl. 10:35

NOMAD 23(2), 2018
Scrutinizing teacherlearner interactions on volume
Anita Tyskerud and Reidar Mosvold
Abstract
This study adds to research on volume and spatial reasoning by investigating teacherlearner interactions in the context of Lesson study. Our analysis illustrates how the mathematical object of volume is realized, and what metarules of discourse that can be observed over two iterations of a research lesson. The study unpacks the mathematical work of teaching volume in terms of discourse, and shows how an undesirable and unexpected result from the first research lesson can be attributed to the communicational work of teaching rather than to lack of skills among students.Anita Tyskerud
Anita Tyskerud is a PhD candidate in Educational Science, Department of Education and Sports Science, University of Stavanger, Norway. Her research interests are related to teachers’ professional development in mathematics and Lesson study.Reidar Mosvold
Reidar Mosvold is Professor of Mathematics Education at the Department of Education and Sports Science, University of Stavanger, Norway. His research interests are related to mathematics teaching and developing mathematics teachers.Skapad: 20180822 kl. 16:08

NOMAD 23(2), 2018
Læreres utbytte av kunnskap om hjernen
Jan Roksvold
Sammandrag
Konkrete klasseromsanvendelser av hjerneforskning har latt vente på seg. I denne oversiktsartikkelen undersøkes det potensielle utbyttet for lærerstudenter ved å kjenne til ulike temaer knytta til hjernens befatning med tall og aritmetikk – uavhengig av hvorvidt slike entilenanvendelser eksisterer eller kan eksistere. Av potensiell verdi for lærere framheves blant annet kunnskap om hvilke vanskeligheter et assosiativt minne forårsaker i forbindelse med aritmetiske tabeller. Med bakgrunn i moderne hjerneforskning belyses ei tallbehandling som kan deles opp i en medfødt ”tallsans” og et kultur og utdanningsavhengig eksakt tallsystem, hvordan ulike binæroperasjoner behandles på grunnleggende forskjellig vis av hjernen, og hvordan innlæringsstrategi kan påvirke lagringa av aritmetisk kunnskap. Temaer som barnets ”logaritmiske indre tallinje” og dyskalkuli blir også belyst. Jeg konkluderer med at denne typen kunnskap om hjernen vil utvide lærerstudentenes forståelse av det lærende barnet, og dermed kunne påvirke deres praksis.Abstract
Concrete applications of neuroscience to the classroom are yet to be confirmed. The topic of this research article is the potential gains to be had for trainee teachers in knowing about various topics concerning the brain’s processing of numbers and arithmetic – regardless of whether onetoone applications exist or can exist. Highlighted as potentially valuable to teachers, is knowledge about: the dichotomy between an inborn ”number sense” and a culturally and educationally dependant exact number system; how different binary operations are processed; how learning strategy can affect the encoding of arithmetic facts; the difficulties caused by an associative memory in relation to arithmetic tables; the child’s ”logarithmic inner number line”; dyscalculia, and the neuromyth pertaining to it. I conclude that this type of knowledge will expand trainee teachers’ understanding of the learning child, and thereby possibly in uence their practice.Jan Roksvold
Jan Roksvold er førsteamanuens i matematikkdidaktikk ved UiT Norges arktiske universitet. Hans forskningsinteresser omfatter anvendelse av funn fra kognitiv psykologi i matematikkundervisning, hjernens befatning med matematikk, samt bruk av matematikkhistorie og historiefortelling i undervisning.Skapad: 20180822 kl. 16:02

NOMAD 23(2), 2018
Disciplinary competence descriptions for external use
Jens Højgaard Jensen and Uffe Thomas Jankvist
Abstract
The article addresses the need for competence descriptions of disciplines as a means for fostering more productive communication between different disciplines and between the disciplines and their surroundings. It is argued that the usual competence descriptions devised for use within a discipline itself, e.g. in relation to teaching and learning of the discipline – socalled competence descriptions for internal use – are not the best means to achieve this. The same is true for the general, nondisciplinary competence descriptions. Instead, specially devised disciplinary competence descriptions for external use are called for. Our main illustration is a competence description of mathematics for external use devised so that it can support the dialogue about justi cation of mathematics education between the discipline’s practitioners and its recipients. This description for external use is counterposed with one for internal use i.e. that of the Danish KOM project. It is also counterposed with a competence description for external use for physics, taking into account the different justification problem of physics education. Together these two descriptions showcase how competence descriptions of disciplines for external use may support interdisciplinary collaboration and division of labor in the educational system.Jens Højgaard Jensen
Jens Højggard Jensen is associate professor in physics at Roskilde University. For many years he was involved in the governing of Roskilde University in various positions. Besides publications with more specific physics content he has published on university politics, general didactics, science didactics, and interdisciplinarity.Uffe Thomas Jankvist
Uffe Thomas Jankvist is professor with special responsibilities in mathematics education at the Danish School of Education, Aarhus University. He has published on the use of history of mathematics in mathematics education, technology in mathematics education, interdisciplinarity, and students’ learning difficulties in mathematics. Besides teaching and supervising future mathematics teacher educators at the Danish School of Education, he is also part of the Danish researchbased ”maths counsellor” upper secondary teacher program at Roskilde University.Skapad: 20180822 kl. 15:56

NOMAD – 23(2), 2018
Volume 23, No 2, June 2018
eNOMAD
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Jens Højgaard Jensen and Uffe Thomas Jankvist
Disciplinary competence descriptions for external use
[PDF]Jan Roksvold
Læreres utbytte av kunnskap om hjernen
[PDF]Anita Tyskerud and Reidar Mosvold
Scrutinizing teacherlearner interactions on volume
[PDF]Skapad: 20180822 kl. 15:48

NOMAD 23(1), 2018
The gap between school mathematics and university mathematics: prospective mathematics teachers’ conceptions and mathematical thinking
Jani Hannula
Abstract
In Finland, both prospective and inservice mathematics teachers report a discontinuity between universitylevel mathematics and mathematics taught at comprehensive and secondary school. In this study, ten prospective mathematics teachers (PMTs) were interviewed to examine their conceptions of the nature of this gap as well as their mathematical thinking. The study’s findings support research that has revealed difficulties experienced by PMTs in the secondary–tertiary transition and in connecting formal and informal components of mathematical thinking. Additionally, the study provides new insight into PMTs’ conceptions of teacher knowledge, such as the relationship between knowledge of advanced mathematics and the knowledge needed in teaching situations. The findings offer guidelines for further studies that could help the development of mathematics teacher education.Jani Hannula
Jani Hannula is a doctoral student at the University of Helsinki, Finland. He is involved with mathematics teacher education at the Department of Mathematics and Statistics. He has a background as a lecturer of mathematics and information technology at Helsinki Metropolia University of Applied Sciences. His main research interests are teacher knowledge and beliefs as well as cognitive aspects of mathematical thinking.Skapad: 20180320 kl. 15:50

NOMAD 23(1), 2018
Discourses in school algebra: the textbooks’ different views on algebra and the positioning of students
Kristina Palm Kaplan
Abstract
The purpose of this study is to understand the school algebra offered in Swedish mathematic textbooks for grade 8. Using a social semiotic perspective, textbook tasks are analysed with a method inspired by Systemic Functional Linguistics. Five school algebra discourses are identified: symbolic discourse, geometrical discourse, arithmetical discourse, (un)realistic discourse and the scientific discourse. It is argued that these offer different views on the nature of algebra and the positioning of students.Kristina Palm Kaplan
Kristina Palm Kaplan is a doctoral student at Uppsala University since September 2014. The main research interests are mathematics and language, especially algebra and social semiotics.Skapad: 20180320 kl. 15:47

NOMAD – 23(1), 2018
Volume 23, No 1, March 2018
eNOMAD
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Anna Ida Säfström
Preschoolers exercising mathematical competencies
[PDF]Magnus Fahlström and Lovisa Sumpter
A model for the role of the physical environment in mathematics education
[PDF]Kristina Palm Kaplan
Discourses in school algebra: the textbooks’ different views on algebra and the positioning of students
[PDF]Jani Hannula
The gap between school mathematics and university mathematics: prospective mathematics teachers’ conceptions and mathematical thinking
[PDF]Skapad: 20180320 kl. 12:27

NOMAD 23(1), 2018
Preschoolers exercising mathematical competencies
Anna Ida Säfström
Abstract
The mathematical ideas that emerge in children’s free and guided play can be both complex and sophisticated, and if they are linked to formal mathematics, they can be a powerful basis for mathematical development. To form such links, one needs knowledge of how children use and express these ideas. This is especially true in the intersection of arithmetic and geometry, where the intermingling of numerical and spatial concepts and skills is not yet fully understood. This study aims to gain understanding of children’s mathematical practices by describing the interplay of key mathematical ideas, and more specifically how young children exercise mathematical competencies in the intersection of early arithmetic and geometry. The results show that children can use spatial representations when reasoning about numbers, and that they are able to connect spatial and numerical structures. Furthermore, it is shown that children not only use and invent effective procedures, but also are able to explain, justify and evaluate such procedures.Anna Ida Säfström
Anna Ida Säfström is associate professor in mathematics education at Halmstad University. Her main research interests are mathematical competence, mathematics as conceptual fields, design research and teachers’ professional development.Skapad: 20180319 kl. 15:22

NOMAD 23(1), 2018
A model for the role of the physical environment in mathematics education
Magnus Fahlström and Lovisa Sumpter
Abstract
In this paper, we develop an analytical tool for the role of the physical environment in mathematics education. We do this by extending the didactical triangle with the physical environment as a fourth actor and test it in a review of literature concerning the physical environment and mathematics education. We find that one role played by the physical environment, in relation to mathematical content, is to portray the content in focus, such as geometry and scale. When focusing on teachers, students, and the interaction between them, the role of the physical environment appears to be a precondition, either positive (enabling) or negative (hindering). Many of the findings are valid for education in general as well, such as the importance of building status.Magnus Fahlström
Magnus Fahlström is PhDstudent in Microdata Analysis and a mathematics teacher educator at Dalarna University. Key research interests are physical school environment and mathematics education.Lovisa Sumpter
Lovisa Sumpter is Senior Lecturer and Associate Professor in Mathematics Education at Stockholm University. Key research interests are mathematical reasoning, affect and gender.Skapad: 20180319 kl. 15:16

NOMAD 22(4), 2017
Mathematics lecturers’ views on the teaching of mathematical modelling
Stephanie TreffertThomas, Olov Viirman, Paul HernandezMartinez and Yuriy Rogovchenko
Abstract
The paper reports on the views and use of mathematical modelling (MM) in university mathematics courses in Norway from the perspective of lecturers. Our analysis includes a characterisation of MM views based on the modelling perspectives developed by Kaiser and Sriraman (2006). Through an online survey we aimed to identify the main perspectives held in higher education by mathematics lecturers and the underlying rationale for integrating (or not) MM in university courses. The results indicated that most respondents displayed a realistic perspective on MM when it came to their professional practice. There was a more varied response when it came to their views on MM in teaching. Regarding conditions influencing the use or nonuse of MM in teaching, these mainly concerned the mathematical content and the institutional practices.Stephanie TreffertThomas
Stephanie TreffertThomas is a lecturer at Loughborough University (UK) with experience of teaching mathematics at school level, tertiary (college) level and at university, mainly to engineering students. Her research interests are in university level mathematics teaching and learning using sociocultural educational theories. She has a particular interest in the mathematical teaching practices of lecturers, including the use of mathematical modelling in teaching.Olov Viirman
When the research reported on in this paper was conducted, Olov Viirman was a postdoctoral researcher within the MatRIC centre at the University of Agder, Norway. He has recently taken up a position as senior lecturer at the University of Gävle, Sweden. His research is in university mathematics education, mainly focusing on the discursive practices of lecturers and students, and on the teaching and learning of mathematics, for instance mathematical modelling, in other academic disciplines.Paul HernandezMartinez
Paul HernandezMartinez is a senior lecturer in the Department of Mathematics at Swinburne University of Technology, Australia, and a visiting fellow in the Mathematics Education Centre at Loughborough University, UK. His research is in postcompulsory Mathematics Education, where he uses sociocultural educational theories to investigate teachinglearning practices (e.g. Mathematical Modelling) that have the potential to develop in students rich mathematical meanings while at the same time create in them positive dispositions towards the subject.Yuriy Rogovchenko
Yuriy Rogovchenko is a Professor of Mathematics at the University of Agder, Kristiansand, Norway. His research interests include qualitative theory of ordinary, functional and impulsive differential equations, mathematical modelling, and mathematics education related to teaching and learning of differential equations and mathematical modelling at university level.Skapad: 20180104 kl. 00:00

NOMAD – 22(4), 2017
Volume 22, No 4, December 2017
Volume 22, No 4, December 2017
eNOMAD
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Simon Goodchild and Barbara Jaworski
Developing practice through research into university mathematics education
Angeliki Mali and Georgia Petropoulou
Suela Kacerja, Toril Eskeland Rangnes, Rune Herheim, Meinrad Pohl, Inger Elin Lilland and Ragnhild Hansen
Mervi A. Asikainen, Antti Viholainen, Mika Koponen and Pekka E. Hirvonen
Sinéad Breen, Niclas Larson, Ann O’Shea and Kerstin Pettersson
A study of students’ concept images of inverse functions in Ireland and Sweden
Margrethe Naalsund and Joakim Skogholt
Oral presentations as a tool for promoting metacognitive regulation in real analysis
Stephanie TreffertThomas, Olov Viirman, Paul HernandezMartinez and Yuriy Rogovchenko
Mathematics lecturers’ views on the teaching of mathematical modelling
Ian Jones and David Sirl
Peer assessment of mathematical understanding using comparative judgement
Barbro Grevholm
Innehåll: JH
Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
Stimulating critical mathematical discussions in teacher education: use of indices such as the BMI as entry points
Suela Kacerja, Toril Eskeland Rangnes, Rune Herheim, Meinrad Pohl, Inger Elin Lilland and Ragnhild Hansen
Abstract
The main purpose of our research project is to gain insight into, and develop teaching on indices and their applications in society. In this paper, the focus is to present insights into teachers’ reflections when discussing the Body Mass Index (BMI). Skovsmose´s concept of mathemacy, and source criticism, are chosen as conceptual framework. The data analysed were collected in a numeracy across the curriculum class with practising teachers. The findings show that the practising teachers engaged in meaning making of the index formula, and they critically discussed how BMI is used in society and the role the BMI index can have in our lives. We gain insight into the potential of such an index for developing teachers’ awareness of the application of mathematics to the real world and the issues it raises, both for the teachers and for ourselves.Suela Kacerja
Suela Kacerja has a postdoc position in mathematics education at the Western Norway University of Applied Sciences. She has a background as mathematics teacher educator from Albania and Norway. Her research interests are: developing critical mathematics education possibilities in teacher education and in schools, in intersection with reallife contexts in learning mathematics, as well as preservice teachers’ reflections about their own practice.Toril Eskeland Rangnes
Toril Eskeland Rangnes is associate professor in mathematics education. She works at the Department of Teacher Education Study in Mathematics, Western Norway University of Applied Sciences. Rangnes has a background as primary school teacher, textbook author and editor for Tangenten. Her main research interests are critical mathematics education, teacher professional development and language diversity in mathematics classrooms.Rune Herheim
Rune Herheim is associate professor at Western Norway University of Applied Sciences. His research focuses on connections between communication qualities and learning in mathematics with a particular focus on argumentation and agency in reallife contexts and when students use digital learning tools. Herheim is the Editor in chief for Tangenten, a Norwegian journal on mathematics teaching.Meinrad Pohl
Meinrad Pohl is associate professor in history. He works at the Department of Social Science, Western Norway University of Applied Sciences, Bergen. His main research interests are early modern economic theory and economic policy, trade history and mining history.Inger Elin Lilland
Inger Elin Lilland is associate professor at the Western Norway University of Applied Sciences, where she works at the Department of Teacher Education Study in Mathematics. She has previous experience as mathematics teacher at the upper secondary school level. Her main research interests are critical mathematics education and mathematics teacher professional development.Ragnhild Hansen
Ragnhild Hansen is associate professor at the Department of Teacher Education Study in Mathematics at Western Norway University of Applied Sciences (HVL). She received her master and PhD degrees from the University of Bergen within applied mathematics. Hansen has a background in as a researcher in different modelling projects. Her main research interests are critical mathematics education and teacher professional development.Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
Peer assessment of mathematical understanding using comparative judgement
Ian Jones and David Sirl
Abstract
It is relatively straightforward to assess procedural knowledge and difficult to assess conceptual understanding in mathematics. One reason is that conceptual understanding is better assessed using openended test questions that invite an unpredictable variety of responses that are difficult to mark. Recently a technique, called comparative judgement, has been developed that enables the reliable and valid scoring of openended tests. We applied this technique to the peer assessment of calculus on a firstyear mathematics module. We explored the reliability and criterion validity of the outcomes using psychometric methods and a survey of participants. We report evidence that the assessment activity was reliable and valid, and discuss the strengths and limitations, as well as the practical implications, of our findings.Ian Jones
Ian Jones obtained a PhD in Mathematics Education from the University of Warwick and is a Senior Lecturer in the Mathematics Education Centre at Loughborough University, UK. Prior to this he was a Royal Society Shuttleworth Education Research Fellow and taught in primary and secondary schools for ten years. His research interests are in school children’s learning of algebra and the assessment of procedural and conceptual understanding of mathematics.David Sirl
David Sirl is a Lecturer in the School of Mathematical Sciences at the University of Nottingham. He is enjoying spending some time working with education researchers to explore new ways of improving teaching and learning.Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
Oral presentations as a tool for promoting metacognitive regulation in real analysis
Margrethe Naalsund and Joakim Skogholt
Abstract
Real Analysis is for many students their first proofbased mathematics course, and many find it challenging. This paper studies how oral presentations of mathematical problems for peers can contribute to students’ metacognitive reflections. The paper discusses several aspects tied to preparing for, and carrying out, oral presentations, that seem to spur important subcomponents of metacognitive regulation such as planning, monitoring, and evaluating. Thoughtful guidance from an expert encouraged the students to further monitor their cognition, and evaluate their arguments and cognitive processes when expressing their reasoning to their peers.Margrethe Naalsund
Margrethe Naalsund is associate professor in Mathematics Education. She works at Faculty of Science and Technology (Section for Learning and Teacher Education) at Norwegian University of Life Sciences (NMBU). Her main research interests are learning and teaching algebra at primary and secondary school, and learning and teaching real analysis at university level.Joakim Skogholt
Joakim Skogholt is PhDstudent in Mathematics. He works at Faculty of Science and Technology (Section for Applied Mathematics) at Norwegian University of Life Sciences (NMBU). His main research interests are applied linear algebra, and learning and teaching real analysis at university level.Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
A study of students’ concept images of inverse functions in Ireland and Sweden
Sinéad Breen, Niclas Larson, Ann O’Shea and Kerstin Pettersson
Abstract
In this paper we focus on firstyear university students’ conceptions of inverse function. We present results from two projects, conducted in Ireland and Sweden respectively. In both countries, data were collected through questionnaires, as well as through student interviews in Sweden. We draw on the notion of concept image and describe the components of students’ evoked concept images. The students’ responses involved e.g. ”reflection”, ”reverse”, and concrete ”examples”, while just a few students gave explanations relating to the definition of inverse functions. We found that the conceptions of inverses as reflections and reverse processes are important and relatively independent of local factors, and the data seemed to suggest that a ”reverse” conception is linked to an appreciation of injectivity more than a ”reflection” conception.Sinéad Breen
Sinéad Breen holds a PhD in Mathematics (on Asymptotic Analysis) from Dublin City University and has recently returned there as an Assistant Professor in the School of Mathematical Sciences. She conducts research in mathematics education, her main interest being in the teaching and learning of mathematics at undergraduate level.Niclas Larson
Niclas Larson is an associate professor at the Department of Mathematical Sciences, University of Agder, Kristiansand, Norway. His research interest lies in the teaching and learning of mathematics at secondary or university level. Current projects, both comparative, deal with students’ understanding of proof by mathematical induction and student teachers’ explanations of solutions to linear equations respectively. His methodological and theoretical standpoints are varied and driven by current research questions.Ann O’Shea
Ann O’Shea is a Senior Lecturer in the Department of Mathematics and Statistics at the Maynooth University in Ireland. She received a PhD in Mathematics from the University of Notre Dame, Indiana in 1991. Currently her research interests lie in Mathematics Education, especially at undergraduate level.Kerstin Pettersson
Kerstin Pettersson is an associate professor at the Department of Mathematics and Science Education, Stockholm University, Sweden. Her research interests concern university students’ conceptions of threshold concepts. Current projects deal with students’ learning in small groups teaching and students’ understanding of proof by mathematical induction.Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
Finnish entrylevel students’ views of teacher knowledge and the characteristics of a good mathematics teacher
Mervi A. Asikainen, Antti Viholainen, Mika Koponen and Pekka E. Hirvonen
Abstract
This paper reports a study of the views held by Finnish students at the start of their university studies concerning their understanding of the knowledge and characteristics of a good mathematics teacher. A total of 97 students following a basic university course responded to a questionnaire. The results showed that a knowledge of teaching mathematics was more often used to describe the good mathematics teacher than a knowledge of mathematics. According to the students’ views, mathematics teachers need to be able to take the level of understanding of individual students into account in their teaching. Good mathematics teachers were also considered to be skilled in explaining, simplifying and transforming mathematical contents for their students. A good mathematics teacher was often described by the respondents as a patient, clear, inspiring and consistent person. On the other hand, characteristics such as humorous, likeable, empathetic, or fair were seldom used in the students’ responses to describe a good mathematics teacher. Those respondents who planned to become teachers demonstrated a more learnercentred concept of a good mathematics teacher than did those who were aiming at some other subject specialist profession than that of teaching.Mervi A. Asikainen
Docent Mervi A. Asikainen is a senior lecturer at the UEF Department of Physics and Mathematics. Asikainen directs the UEF physics and mathematics education research group. Her current field of interest include teacher knowledge of mathematics and physics teachers, teaching and learning of physics in higher and secondary education, and researchbased development of STEM education.Antti Viholainen
Antti Viholainen is a senior lecturer in mathematics / mathematics education at University of Eastern Finland. His research areas are mathematical beliefs, mathematics teacher education, learning materials (textbooks etc.) in mathematics, and mathematical argumentation.Mika Koponen
Mika Koponen is a postdoctoral researcher at the University of Eastern Finland. He has used Mathematical Knowledge for Teaching (MKT) framework for evaluating and improving mathematics teacher education. In his dissertation study, he presented a novel approach for investigating teacher knowledge and its interconnections by making use of network analysis methods. His postdoctoral research continues from this work by focusing on how the components of teacher knowledge are interconnected.Pekka E. Hirvonen
Docent Pekka E. Hirvonen is a senior lecturer at the UEF Department of Physics and Mathematics. Hirvonen has published more than 30 peerreviewed articles in international journals, books, and proceedings.Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
Characterising undergraduate mathematics teaching across settings and countries: an analytical framework
Angeliki Mali and Georgia Petropoulou
Abstract
This paper explores the characteristics of teaching of a sample of university mathematics teachers in two countries, Greece and Great Britain, and in two settings, lectures and tutorials, seeking to identify a common ground for undergraduate mathematics teaching. Our observations of teaching and our sociocultural perspectives enabled us to develop a framework for a detailed description of the observed teaching. The description reveals categories of teaching actions, and the associated tools teachers use in selecting tasks for their students, providing comprehensive explanations, extending students’ mathematical thinking, or evaluating students’ mathematical meaning. The findings are across settings and countries in the direction of a profound understanding of undergraduate mathematics teaching.
Angeliki Mali
Angeliki Mali is a Postdoctoral Research Fellow in the School of Education at the University of Michigan. Prior to her fellowship, she was member of the Culture, Pedagogy and Identity group in the Mathematics Education Centre at Loughborough University, where she was awarded her PhD. She holds a BSc in Mathematics, and an MSc in Didactics and Methodology of Mathematics from the University of Athens in Greece. Her research focuses on university mathematics education. She has experience in teaching mathematics to students attending STEM programmes at university level.Georgia Petropoulou
Georgia Petropoulou is finishing her PhD in the Mathematics Department at the University of Athens, Greece. Her PhD is in mathematics education, focusing on university mathematics teaching for students’ learning needs. She has an MSc in Didactics and Methodology of Mathematics and a BSc in Mathematics, both awarded by the University of Athens. Her research interests are in university mathematics teaching practice and its development to meet students’ learning needs.Skapad: 20180104 kl. 00:00