NOMAD – 10(2), 2005

NOMAD 10(2), 2005. Conceptual change in mathematics



In traditional educational contexts, mathematics is considered a hierarchical structure in which new concepts logically follow from prior ones. From the viewpoint of the theories of conceptual change, however, the learning of mathematics is characterized more by discontinuity than gradual and continuous enrichment. These theories stress the crucial role of prior knowledge in learning. According to these theories, prior knowledge does promote learning, but it can also restrict it and lead to misconceptions. This is the case especially with those kinds of concepts where learning demands a radical change in prior knowledge, which is typical of mathematics and science. One example of these kinds of changes in mathematics is the enlargement of number concept from natural to rational numbers. In this article, three different theories of conceptual change are presented and the perspectives of these theories on the difficulty of the above-mentioned enlargement are discussed. Results of empirical research and some implications for teaching mathematics from the viewpoint of theories of conceptual change are also dealt with.

Perinteisessä matematiikan opettamisen ajattelussa matematiikka näyttäytyy hierarkkisena käsitejärjestelmänä, jossa uudet käsitteet seuraavat johdonmukaisesti aikaisemmista. Käsitteellisen muutoksen teorioiden lähtökohdista matematiikan oppiminen näyttää kuitenkin edistyvän todennäköisemmin epäjatkuvana tapahtumasarjana kuin jatkuvana käsitteiden vähittäisenä rikastumisena. Käsitteellisen muutoksen teoreettisessa ajattelussa painotetaan aikaisemman tietämyksen keskeistä roolia uuden oppimisessa. Näistä teoreettisista lähtökohdista tehdyt empiiriset tutkimukset osoittavat, että vaikka aikaisempi tietämys edistää uuden oppimista, se voi myös rajoittaa sitä ja johtaa väärinkäsityksiin. Näin käy todennäköisesti sellaisten käsitteiden oppimisessa, jotka vaativat oppijalta radikaalia muutosta aikaisempaan ajatteluun. Yksi esimerkki tällaisesta muutosvaatimuksesta on lukualueen laajentaminen luonnollisten lukujen alueelta rationaalilukujen alueelle. Tässä artikkelissa esitellään kolme erilaista käsitteellisen muutoksen teoreettista suuntaa ja tarkastellaan empiirisen tutkimuksen valossa lukualueen laajennuksen problematiikkaa näistä näkökulmista.

Kaarina Merenluoto, MSc (majoring in mathematics), PhD in education, is an Adjunct professor in science education. She works as a senior researcher in the Department of Teacher Education at the University of Turku. Her research interests are focused on problems of learning and conceptual understanding in mathematics at the secondary level, and on the dynamics of cognitive and motivational processes in conceptual change.

NOMAD 10(2), 2005. Secondary mathematics teachers' beliefs about mathematics assessment

Secondary mathematics teachers' beliefs about mathematics assessment and components that influence these beliefs


The espoused beliefs of 465 secondary mathematics teachers regarding mathematics assessment are the focus of this study. The data for this investigation were collected using a 19 items questionnaire. There is evidence from this study that there are teachers who espouse a 'socio-constructivist' orientation to mathematics assessment, teachers who espouse a 'problem solving' orientation to mathematics assessment and teachers who espouse an 'accountability' orientation to mathematics assessment.

Fokus för denna studie utgör de uttalade uppfattningar (beliefs) om utvärdering i matematik som innehas av 465 grekiska matematiklärare på de stadier som motsvarar grundskolans högre årskurser och gymnasiet. Data insamlades med en enkät som omfattade 19 frågor. Studien ger belägg för att det finns lärare som ger uttryck för en socio-konstruktivistisk orientering, lärare som ger uttryck för en problemlösnings -orientering och lärare som ger uttryck för en orientering som betonar ansvarighet i fråga om utvärdering i matematik.

Adjunct Professor, National and Kapodistrian University of Athens, Greece and Head of the Mathematics Faculty, St Joseph's College, Melbourne, Australia

Professor of Mathematics Education, Curtin University of Technology, Australia

NOMAD 10(2), 2005. The Nordic graduate school in mathematics education - a growing network


Let us know about you
When the Nordic Graduate School in Mathematics Education, NoGSME, started in January 2004 it included 37 research environments, about 45 supervisors and about 80 doctoral students in the Nordic and Baltic countries. Since then some of the students have graduated and will sooner or later belong to the group of supervisors instead. New doctoral students are being enrolled in research education programmes all the time. We ask you to remember to report to the Nordic Graduate School all such changes, so that we can update our sending lists. Sending information via email is one of our opportunities to inform about resources that are offered by NoGSME.

The summer school 2005
At the start of NoGSME the groups of students and of supervisors had limited knowledge about persons from environments in other countries. We can see now that networks are growing with students and supervisors. This summer NoGSME organised its first summer school at the University of Jyväskylä, Finland. Thirty doctoral students from Denmark, Estonia, Finland, Norway, Sweden, one from Germany and one from Turkey worked together for a week and learnt to know each other. The working groups that took up most of the time allowed each participant to present her/his research questions, theoretical framework, methods, data, analysis and results. The serious and deep discussions were led by three group leaders, experienced researchers in mathematics education, Gilah Leder from Melbourne, Tommy Dreyfus from Tel Aviv and Roger Säljö from Göteborg. The intensity of the discussions was high and the personal engagement that was created will probably lead to future cooperation in research between the participants. Two themes were discussed in additional discussion groups chaired by members of the NoGSME board. The themes were: How to read a scientific paper productively and How to write a scientific paper. Participants related to literature on these themes and actual sample papers were investigated. Plenary lectures were given by the group leaders and also by some members of the NoGSME board. For those doctoral students who could not participate this year there will be another opportunity in 2006.

Activities for supervisors
On September 1-2 the third seminar for supervisors of doctoral students in mathematics education took place in Trondheim, just before the fourth Nordic conference in mathematics education, Norma05. Professor Uri Leron from Haifa led the seminar and the focus was the actual supervision process, its joys and challenges. General issues related to supervision were treated, and discussions on specific cases took place as well. There will be a documentation based on the work during the seminar. Another seminar will take place later this year and focus on the process of reviewing papers for scientific journals. There is a need to provide competence development for future reviewers of, for example, NOMAD and other journals. Improving your ability to review a paper will probably also develop the ability to scrutinize your own papers and the papers of your doctoral students in a more constructive way.

Workshop on classroom research in mathematics education
One afternoon during the Norma05 conference was devoted to classroom research. This programme was initiated by NoGSME. Simon Goodchild led this workshop and used his own experience from a study where he followed a mathematics class during a whole year and investigated the students' goals. Participants in the workshop were invited to analyse transcriptions of student talk about their work during class and the goals of it. Simon indicated that what goes on in the classroom is mysterious and will never be fully uncovered as the activities are so complex. In the specific class studied by Simon there seemed to be little learning in mathematics taking place over the year. The workshop will be reported in the proceedings of the conference.

Use the offers of NoGSME
Students and supervisors are invited to use the offers of NoGMSE as much as possible. At the moment we offer travel support for doctoral students who take part in four different doctoral courses included in the NoGSME programme. For the spring 2006 a new set of courses will be available and every student taken up in a doctoral programme in mathematics education can apply for travel support to take part in them. NoGSME also offers five mobility stipends per year for students who want to go to another Nordic university to study or get supervision for a month. Expenses for travel and lodging will be covered by NoGSME. Instructions about how to apply are accessible on the web page. For further information see our web page For all kinds of questions about the Nordic Graduate School please contact the director.

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

Barbro Grevholm, Director
Faculty of Mathematics and Science
Agder University College
Serviceboks 422
N-4604 Kristiansand

NOMAD 10(2), 2005. The teaching of fractions – a comparative study of a Swedish and a Hong Kong classroom



The aim of this paper is to illustrate how the topic of fractions can be taught differently by making a comparison between two cultures. We have studied mathematics teaching in classrooms in Hong Kong and Sweden. One of our basic assumptions is that the way in which the content is taught in a classroom has an important implication for what students may possibly learn. With reference to the framework of Variation Theory, two different spaces of learning are delineated. The Hong Kong lesson demonstrated a pattern of many juxtaposed variations, whereas the Swedish lessons presented a pattern of sequential and wide spreading character.

Syftet med denna artikel är att visa hur jämförelse av oliknämniga bråk kan hanteras på olika sätt i undervisningen. Detta görs genom att jämfîra undervisning i Sverige och Hong Kong. Data har analyserats utifrån ett variationsteoretiskt perspektiv och utgångspunkten är ett antagande om att hur innehållet behandlas har betydelse för vad som är möjligt för eleverna att lära. Vi fann att i det svenska klassrummet behandlades innehållet så att det fick en sekventiell karaktär, medan det i Hong Kong gavs en mer komplex karaktär genom att flera aspekter behandlades samtidigt.

Dr Ulla Runesson is a Senior Lecturer and has currently a post doc position at the Department of Education at Göteborg University. Her research interest is in teaching and learning. She has been the scientific leader of a research project "The pedagogy of variation" and active in a number of projects researching the relation between teaching and learning, some in cooperation with Hong Kong University.

Dr Ida Ah Chee Mok is Assistant Professor in the Department of Curriculum and Educational Studies at the University of Hong Kong. Since 1990, she has worked in teacher education, specialized in mathematics education. She has been specialized in students' mathematics learning and active in a number of projects in the Hong Kong mathematics curriculum, teaching and learning of mathematics.