# ICME 11, DG 13

Challenges and possibilities posed by different perspectives and theoretical approaches in mathematics education research

ICME, the International Congress on Mathematical Education, in its eleventh edition, is to be held in Monterrey, Mexico from July 6th to 13th, 2008. The DG (Discussion Group) 13 is dedicated to challenges posed by different perspectives, positions, and approaches in mathematics education research.

As the name suggests, a Discussion Group (DG) is designed to gather congress participants who are interested in discussing, in a genuinely interactive way, certain challenging or controversial issues and dilemmas –of a substantial, non-rhetorical nature– pertaining to the theme of the DG. The team will identify more specific issues and questions for the DG, and participants in the group will be invited to propose responses to the issues thus raised, including answers to specific questions and possibly recommendations to relevant categories of policy or decision makers. There will be no oral presentations in a DG, except introductions by the organizers of the group to provide the background and framework for the discussion. Information or position papers are expected to be made available to group participants electronically through a web site forming part of the congress web site, in due time before the congress.

This webpage is dedicated to DG 13, which could also be found on the general ICME website.

The team of DG 13 is composed of:

Luc Trouche (France) – co-chair

Susanne Prediger (Germany) - co-chair

Patricia Camarena Gallardo (Mexico)

Jana Trgalova (France)

Soledad Ulep (Philippines)

The International Program Comittee (IPC) has appointed Richard Noss as IPC liaison for DG 13.

### Challenges and possibilities posed by different perspectives and theoretical approaches in mathematics education research. Aims and focus of DG13

The large diversity of different perspectives and theoretical approaches in the mathematics education research community has often been described. It poses challenges for international communication, for the integration of empirical results into a bigger corpus of scientific knowledge and in the long run for a cumulative and joint progress in the research field.

On the other hand, the diversity of different perspectives is unavoidable due to very different traditions and focuses. Even stronger, some researchers (including the organizing team) consider it to be a rich resource that enables researchers to grasp the complexity of the research objects.

The discussion group focuses not only on challenges, but also on possible strategies for connecting perspectives for a better understanding of complex phenomena in mathematics teaching and learning. In order to avoid a too abstract discussion without concrete basis, we will refer to concrete research projects in the field of mathematics teaching and learning (with most prefarably, but not only, a special attention paid to ICT, i.e. Information and Communications Technology, its design and use):

- How are different perspectives and theoretical approaches reflected in concrete research practices? (exemplify the expression of different theories on research practices)
- Where has the diversity of perspectives and theoretical approaches been used as a resource for a better understanding of complex phenomena? (Exemplify the use different perspectives for understanding empirical phenomena)
- What strategies can be used to connect different perspectives and approaches in concrete research projects? (connect different theories for their development)? How can different theoretical approaches learn from each other?

### How to participate

Discussion Groups are expressly not about the presentation of research papers. How-ever, as starting points for discussions, the OT is calling for written contributions that bring forward examples of research projects and their general reflections due to one or more of the questions listed above.

Accepted contributions will be posted on the web page of DG13 and will be referred to during the discussions at ICME11. Contributors are expected to participate in all the sessions of the Discussion Group, together with others who are interested in the theme but have not entered a written contribution. The written contributions should include the following information:

- The author(s) names and location (town and country, school, or establishment) and contact details (if possible the address of the personal webpage);
- Explicit references to the addressed questions;
- A general reflection on the issues of the Discussion Group, rather than a classical research report;
- A concrete example from research practices.

The written preliminary contribution should be 2 to maximally 4 pages (A4 please) in length using 12-point font, single spaced, with 2 cm margins, and should be submitted electronically to both DG13 co-chairs Luc Trouche and Susanne Prediger.

### Timeline

The deadline for submission of the preliminary short versions is January 20th 2008.

- By the end of February 2008 – Information about acceptance will be sent to contributors who will be asked to prepare an extended version of their text (6 pages maximum) by the end of March 2008.
- By the April 20th 2008 – The OT will prepare and publish an organising framework for the operation of DG13 based on the extended contributions received.

### Preliminary plan for the sessions

- At this stage we envision devoting the first 2 sessions (of 2 hours each) to small group discussions after a brief introduction to the key issues by the members of the OT.
- For the final session (1 hour only) the OT is considering the possibility of a common report back session to share general observations and issues and possibly plan further cooperation.

## Patricia Camarena Gallardo

Patricia Camarena is researcher professor at Mechanical and Electrical Engineering University School in National Politechnical Institute. My research lines are mathematical education and applied mathematics to engineering.

My interest in the DG 13 is to discus about the challenges posed on how to change teachers mind and role with the uses of ICT in classroom, knowing how and when to use it; knowledge about how to help to the student to become autonomous.

The international groups which I am linked are working Mathematics in Science Context, which is a theoretical position supported by Ausubel, Vygotsky and Piaget works. Their main development point is the link between Mathematics and interest students areas, such as subject engineering carriers, everyday problems and professional work problems. This means, mathematical knowledge construction as a social mathematics or mathematics for life, that is to say, from mathematics knowledge to the abilities the mathematics develops in the pupils.

We work with context events, like problems and projects in science context, everyday problems and professional and technical work problems, the central point in didactics of mathematic is the mathematical modeling, as an interdisciplinary point of view. So I propose to discuss mathematics modeling as an integrative element to knowledge.

This theoretical position has some of the next references:

Ausubel, D. P., Novak, J. D. y Hanesian, H. (1990). PsicologÃa educativa, un punto de vista cognoscitivo. Editorial Trillas.

Bucciarelli, L. (1996). Designing Engineers. The MIT Press, Cambridge, London, England.

Camarena, G. P. (2001). Reporte de investigaciÃ³n intitulado: Los modelos matemÃ¡ticos como etapa de la matemÃ¡tica en el contexto de la ingenierÃa, con No. de registro: CGPI 2000731, IPN, MÃ©xico

Camarena, G. P. (2004). La formaciÃ³n de los profesores de las ciencias bÃ¡sicas en el nivel superior. CientÃfica, The Mexican Journal of Electromechanical Engineering, Vol 8. No.1.

OlazÃ¡bal, A. M. (2004). CategorÃas en la traducciÃ³n del lenguaje natural al lenguaje algebraico de la matemÃ¡tica en contexto. Tercer Congreso Internacional “Retos y Expectativas de la Universidad”, MÃ©xico.

Muro, U. C. (2004). AnÃ¡lisis del conocimiento del estudiante relativo al campo conceptual de la serie de Fourier en el contexto de la transferencia de masa. Tesis de Doctorado. IPN, MÃ©xico.

Polya, G. (1976). CÃ³mo plantear y resolver problemas. Editorial Trillas.

Salett, B. M. • Santos, T. L. M. (1997). Principios y mÃ©todos de la resoluciÃ³n de problemas en el aprendizaje de las matemÃ¡ticas. Grupo Editorial IberoamÃ©rica SA de CV.

Schoenfield, A. (1992). Learning to think mathematically: problem solving, metacognition and sense making in mathematics. In Handbook for Research on Matematics Teaching and Learning. New York: Macmillan.

SchÃ¶n, D. (1998). El profesional reflexivo. Temas de EducaciÃ³n PaidÃ³s

## Susanne Prediger

Susanne Prediger is full professor at the Institute for Development and Research in Mathematics Education in Dortmund, Germany.

My interest in the challenges posed by different theoretical perspectives has evolved in re-search projects on the development of students’ thinking from everyday conceptions to mathe-matical conceptions (e.g. in the domain of fractions, probability etc.) when I tried to coordinate different theoretical frameworks in order to make sense of my empirical data.

The question how to deal with different theoretical frameworks was focused in the CERME working group 2005 and 2007. To me, some key points are most crucial:

• Of course, diversity of theoretical approaches is a challenge (for communication, for building upon existing empirical results etc.), but it is even more important to mention that diversity is an important resource

• Hence, search for good ways of dealing with diversity does (for me) not mean aiming at a grand unification, but at exploiting the diversity as a resource for a multi-facetted under-standing of the complex phenomena in teaching and learning

• We can only make use of diversity as a resource when we start connecting theoretical ap-proaches. Connecting ca mean comparing and contrasting, but also combing or synthesiz-ing theories, this depends on the concrete situations.

• When comparing theoretical approaches, the focus in on the impact of the theoretical choices on the research practices.

My tentative first reactions to the grand and difficult questions:

• Is it possible to distinguish today main trends in mathematics education research? Are there divergent, or convergent processes? I am not sure if we should try to answer this question that!! This is a very huge question we could definitely not handle in our discussion group. My personal impression: the process is not convergent since the field is more and more dif-ferentiated and interconnected,

• What are the roots of these different perspectives, positions, and approaches (Cul-tural? Technological? Political? Epistemological?…). I see the roots in cultural differences, but culture is a very huge word, including for exam-ple: the institutional situation of mathematics educators in different countries, typical phi-losophical and disciplinary traditions , different research practices and teaching cultures in schools, … Emphasizing the cultural roots of differences is operative in so far as I emphasize the exis-tence of a huge part of implicit factors.

The last two questions seem to be more instructive to answer:

• What are the relationships between theoretical frameworks in mathematics, and more generally, science education? This is an important question which sometimes gets lost in all talking on differences! It would be worth to gather commonalities since it helps for the long term process of estab-lishing an identity as a scientific discipline. I would like to work on this point at the discus-sion group.

• How can different ‘schools of thought’ in mathematics education learn from one another? Here, I could bring in some answers from the CERME working group, which are basically systematized propositions for local answers, no general procedures. I would be interested in discussing different strategies of “learning from one another".

Prediger S., Ruthven K. (in press): From teaching problems to research problems. Proposing a way of comparing theoretical approaches, to appear in the Proceedings of CERME 5 (Larnaca, Zypern)

Arzarello F., Bosch M., Lenfant A., Prediger S. (2007): Different theoretical perspectives in research, to appear in the Proceedings of CERME 5 (2007 in Larnaca, Cyprus).

Bikner-Ahsbahs A., Prediger S. (2006): Diversity of Theories in Mathematics Education – How can we deal with it?, in: Zentralblatt fÃ¼r Didaktik der Mathematik 38(1), S. 52-57.

Dreyfus T., Artigue A., Bartolini-Bussi M., Gray E., Prediger S. (2006): Different theoretical perspectives and approaches in research in mathematics education, in: M. Bosch et al. (eds.): Proceedings of the 4th CERME, Sant Feliu de Guixols 2005, published at Fundemi IQS – Universitat 2006, S. 1239-1244. ( ISBN 84-611-3282-3

Prediger S. (2004): Intercultural Perspectives on Mathematics Learning – Developing a Theoretical Framework, in: International Journal of Science and Mathematics Education 2 (2004) 3, S. 377-406.

## Jana Trgalova

Jana Trgalova is a researcher in the French National Institute for Pedagogical ResearchMy interest in the DG13 comes mainly from my participation to the European project TELMA (Technology Enhanced Learning of Mathematics) addressing the issue of design and use of ICT tools for the teaching and learning mathematics.

Within this project, 6 European teams started working towards understanding the role played by theoretical frames in design and research in this area, and building tools to improve communication between researchers from different cultures (TELMA ERT 2006). This collaborative work began by an identification of the main theoretical frames used by TELMA teams in their research: Theory of didactic situations (Brousseau 1997), Activity theory (Cole and EngestrÃ¶m 1993; EngestrÃ¶m 1987; Vygotsky 1978), Situated abstraction (Noss & Hoyles 1996), Constructionism (Harel & Papert 1991). A first attempt at understanding how these theoretical frames influence the visions they develop as regards technology enhanced learning in mathematics was done in this first phase. A specific methodology has been developed in order to deepen the reflection and to better understand the exact role played by the theoretical frames they use in their research: a cross-experimentation, where each team experimented an ICT tool developed by another team (Cerulli & al. 2007). A methodological tool for the systematic exploration of the theoretical frames used in technology enhanced learning in mathematics has been built, with the aim to support the analysis of the cross-experimentation, but also the understanding of the role played by theoretical frames in the design and analysis of uses of ICT tools and the search for interesting connections and complementarities between such frames (Cerulli & al., submitted).

DG13 offers an opportunity to share this experience with other participants and to learn about and discuss other approaches towards a comparing and networking of theoretical frameworks.

Brousseau, G. (1997). The Theory of Didactical Situations in Mathematics, N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield (Trans. & Eds.). Dordrecht: Kluwer Academic Publishers.

Cerulli, M., Georget, J.-P., Maracci, M., Psycharis, G., Trgalova, J. (2007). TELMA’s approach to integration of research teams. 5e CongrÃ¨s de ERME (European Society for Research in Mathematics Education), Larnaca (Chypre), 22-26 fÃ©vrier 2007.

Cerulli, M., Georget, J.-P., Maracci, M., Psycharis, G., Trgalova, J. (submitted). Comparing theoretical frameworks enacted in experimental research: TELMA experience.

Cole, M., & EngestrÃ¶m, Y. (1993). A cultural-historical approach to distributed cognition. In G. Salomon (Ed.), Distributed cognitions: psychological and educational considerations (pp. 1-47). Cambridge: Cambridge University Press.

EngestrÃ¶m, Y. (1987). Learning by expanding: An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit.

Harel, I., & Papert, S. (1991). Constructionism. Norwood, NJ: Ablex Publishing Corporation.

Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings. Dordrecht: Kluwer Academic Publishers.

TELMA ERT (2006). Developing a joint methodology for comparing the influence of different theoretical frameworks in technology enhanced learning in mathematics: the TELMA approach. In Le Hung Son, N. Sinclair, J.-B. Lagrange, C. Hoyles (eds.) Proceedings of the ICMI 17 Study Conference: background papers for the ICMI 17 Study. HanoÃ¯ University of Technology, December 3-8 2006.

Vygotsky, L. S. (1978). Mind and Society. The development of higher psychological processes. Harvard University Press.

## Luc Trouche

Luc Trouche is a researcher in the French National Institute for Pedagogical Research

Personal ideas come always from personal experience. My experience in the field is double:

- firstly as a French researcher in the field of didactics of mathematics, I am interested in comparing the theoretical frameworks in which French researchers use to situate their work, mainly the theory of didactic situations (Brousseau 1997) and the so called anthropological approach (Chevallard 2005)

- secondly as a researcher in the field of ICT integration in math education, I am interested in analysing how different theoretical frameworks help us catch a part of the reality, to understand particular aspects of learning processes.

This last experience was, for me, the most fruitful, because it is certainly when thinking about a particular problem that we can test the pertinence of a particular framework. Three pieces of research were particularly interesting for me:

- about Mind and Machine, a discussion at the third CAME symposium (Computer Algebra in Mathematics Education). It is possible to download my paper and Celia Hoyles's answer on the symposium webpage. The notions of orchestration and webbing were particularly discussed;

- about the integration of symbolic calculators (Guin et al. 2005): it was the occasion to study dialectic relationships between instrumentation and conceptualisation processes, in using frameworks originating from both cognitive ergonomics and didactics of mathematics;

- about an international survey of the research on ICT and maths education (Lagrange et al. 2003): it was the occasion to analyse the interest combining different dimensions (institutional, cognitive, instrumental…) to understand what is at stake in a complex environment.

My interest, in working in DG13, is to deepen these ideas, in more general contexts.

Brousseau, G. (1997). The Theory of Didactical Situations in Mathematics, N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield (Trans. & Eds.). Dordrecht: Kluwer Academic Publishers.

Chevallard, Y. (2005). Steps towards a new epistemology in mathematics education, proceedings of CERME 4, San Feliu de Guixols, Espagne.

Guin, D., Ruthven, K. and Trouche, L. (eds.) (2005). The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument, Springer, New York.

Lagrange, J.-B., Artigue, M., Laborde, C. and Trouche, L. (2003). Technology and Mathematics Education: a Multidimensional Study of the Evolution of Research and Innovation, in A. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick and F.K.S. Leung (eds.), Second International Handbook of Mathematics Education, Kluwer Academic Publishers, Dordrecht, pp. 239-271.

## Soledad Ulep

Soledad A. Ulep is a mathematics education specialist at the University of the Philippines National Institute for Science and Mathematics Education Development where she is involved in teacher training, curriculum development, extension work, and research.

My interest in DG 13 arose from my involvements in two international research studies. One is in the Learner’s Perspective Study which I used as an example below. The other is in the Collaborative Studies on Innovations for Teaching and Learning Mathematics in Different Cultures through Lesson Study. This is a project of the Asia-Pacific Economic Community participated by Japan, Thailand, Australia, Singapore, Malaysia, Philippines, Indonesia, Hong Kong, South Korea, Chile, USA, and Vietnam. Lesson study is a professional development activity of teachers that originated in Japan. Based on a long term goal (say develop students’ problem solving skills), teachers collaboratively plan lesson/s to address the goal. Later, one of them teaches the lesson while others carefully observe primarily how students think or learn it. Afterwards, they reflectively evaluate it. And then, they revise it accordingly. Another teacher can teach the revised lesson using another class. Thus, teachers research their own practices.

Both research studies dealt with classroom practices. I find it is interesting to note the diversity of theoretical perspectives and frameworks used to interpret or develop classroom practices. Some relate to the use of mathematical tasks (Marton’s theory of variation and Brosseau’s theory of didactical situations in mathematics), development of social and socio-mathematical norms (Cobb and Yackel’s emergent perspective), and mathematical thinking (Bishop’s socio-cultural framework). And I find that these are not only useful to research, but to teacher training and curriculum development, as well.

Clarke, D. (2001). Perspectives on Practice and Meaning in Mathematics and Science Classrooms. Dordrecht: Kluwer Academic Press.

Clarke, D., Keitel, C., & Shimizu, Y. (Eds.) (2006). Mathematics Classrooms in Twelve Countries: The Insider’s Perspective. The Netherlands: Sense Publishers.

Leung, F.K.S., Graf, K.D. & Lopez-Real, F. J. (Eds.) (2006). Mathematics Education in Different Cultural Traditions: A Comparative Study of East Asia and the West- The 13th ICMI Study. New York: Springer Science + Business Media, Inc.

Isoda, M, Shimizu, S. Miyakawa, T., Aoyama, K., & Chino, K. (Eds. ) (2006). Special Issue on the APEC-Tsukuba International Conference “Innovative Teaching of Mathematics through Lesson Study.” Tsukuba Journal of Educational Study in Mathematics (vol. 25). Tsukuba, Japan: Mathematics Education Division.

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