Skip to main content
SpringerLink
Log in

Mathematical Beliefs — A Search for a Common Ground: Some Theoretical Considerations on Structuring Beliefs, Some Research Questions, and Some Phenomenological Observations

  • Chapter
Beliefs: A Hidden Variable in Mathematics Education?

Part of the book series: Mathematics Education Library ((MELI,volume 31))

Abstract

A range of research and theory from different sources is reviewed in this chapter, in an attempt to understand better the construct of mathematical beliefs. Definitions of mathematical beliefs in the literature are not consistent and thus working out the core elements of a definition is one aspect of the chapter. Specifically, a four-component definition of beliefs is presented. The model focuses on belief object, range and content of mental associations, activation level or strength of each association, and some associated evaluation maps. This framework is not empirically derived but is based on common characteristics of the literature on didactics, particularly mathematics didactics. This effort towards achieving a precise definition can provide new understandings of fundamental issues in research on mathematical beliefs and give rise to new research questions. In particular, it allows description of the term “belief systems” allowing clustering of individual beliefs into a system across each of the four components. Furthermore, it makes sense to distinguish between global beliefs, domain-specific beliefs and subject-matter beliefs. The question immediately arises as to what interdependencies exist between the individual beliefs. Some observations from a survey of mathematical beliefs of students studying calculus are also included.

So, my hypothesis is: whatever the notion of belief is, it may solve our problem. (Bogdan,1986, p. 2)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abelson, R. (1979). Differences between belief systems and knowledge systems. Cognitive Science, 3, 355–366.

    Article  Google Scholar 

  • Alexander, P.A., & Dochy, F.J.R.C. (1995). Conceptions of knowledge and beliefs: A comparison across varying cultural and educational communities. American Educational Research Journal, 32(2), 413–442.

    Google Scholar 

  • Amit, M., & Vinner, S. (1990). Some misconceptions in calculus. Anecdotes or the tip of an iceberg? In G. Booker, P. Cobb & T.N. de Mendicuti (Eds.), Proceedings of the 14th Conference of the International Group For the Psychology of Mathematics Education (PME) with the North American Chapter 12th PME-NA Conference Vol. 1 (pp. 3–10). México.

    Google Scholar 

  • Azcarate, C. (1991). Instantaneous speed: Concept images at college students level and its evolution in a learning experience. In F. Furinghetti (Ed.), Proceedings of the 15th International Conference of the International Group for the Psychology of Mathematics Education (PME). Vol. 1 (pp. 96–103). Genoa (Italy): Dipt. di Matematica.

    Google Scholar 

  • Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(3), 132–144.

    Google Scholar 

  • Ball, D. L. (1991). Research on teaching mathematics: Making subject-matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching Vol. 2. Teacher’s knowledge of subject matter as it relates to their teaching practice. A research annual (pp. 1–48). Greenwich, CT: Jai Press.

    Google Scholar 

  • Bauersfeld, H. (1983). Subjektive Erfahrungsbereiche als Grundlage einer Interaktionstheorie des Mathematiklernens und-lehrens. In H. Bauersfeld, H. Bussmann, G. Krummheuer, J. H. Lorenz, & J. Voigt (Eds.), Lernen und Lehren von Mathematik. Untersuchungen zum Mathematikunterricht (pp. 1–56). Köln: Aulis.

    Google Scholar 

  • Baroody, A. J. (1987). Children’s mathematical thinking. A developmental framework for preschool, primary, and special education teachers. New York: Columbia University, Teachers College Press.

    Google Scholar 

  • Berger, P. (2001). Computer und Weltbild. Habitualisierte Konzeptionen von Lehrern im Kontext von Informatik, Mathematik und Computerkultur. Dissertation. Wiesbaden: Westdeutscher Verlag.

    Google Scholar 

  • Berliner, D. C., & Calfee, R. (Eds.). (1996). Handbook of Educational Psychology. New York: Macmillan Library Reference USA.

    Google Scholar 

  • Bogdan, R. J. (1986). The importance of belief. In R. J. Bogdan (Ed.), Belief: Form, content, and function (pp. 1–16). New York: Oxford University Press.

    Google Scholar 

  • Calderhead, J. (1996). Teachers: Beliefs and knowledge. In: D. C. Berliner. & R. Calfee, R. (Eds.), Handbook of Educational Psychology (pp. 709–725). New York: Simon & Schuster Macmillan.

    Google Scholar 

  • Cooney, T. (1994). Research and teacher education: in search of common ground. Journal for Research in Mathematics Education, 25, 608–636.

    Google Scholar 

  • Cooney, T. J., Shealy, B. E., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for the Research in Mathematics Education, 29, 306–333.

    Google Scholar 

  • Dedekind, R. (1995). What are numbers and what should they be? Orono, ME: Research Institute for Mathematics (first published in 1888).

    Google Scholar 

  • Dungan, J. F., & Thurlow, G. R. (1989). Students’ attitudes to mathematics: A review of the literature. The Australian Mathematics Teacher, 45(3), 8–11.

    Google Scholar 

  • Eagly, A. H., & Chaiken, S. (1992). The psychology of attitudes. San Diego, CA: Harcourt Brace Janovich.

    Google Scholar 

  • Edwards, B. (1999). Revisiting the notion of concept image / concept definition. In F. Hitt & M. Santos, (Ed.), Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME) Vol. 1 (pp. 205–210). Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental.

    Google Scholar 

  • Eisenhart, M. A., Shrum, J. L., Harding, J. R., & Cuthbert, A. M. (1988). Teacher beliefs: Definitions, findings, and directions. Educational Policy, 2(1), 51–70.

    Google Scholar 

  • Ernest, P. (1989). The knowledge, beliefs, and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(10), 13–33.

    Google Scholar 

  • Ernest, P. (1991). The philosophy of mathematics education. Hampshire (UK): The Falmer Press.

    Google Scholar 

  • Erlwanger, S. (1975). Case studies of children’s conceptions of mathematics, Part 1. Journal of Children’s Mathematical Behavior, 1, 157–283.

    Google Scholar 

  • Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for the Research of Mathematics Education, 24, 94–116.

    Google Scholar 

  • Furinghetti, F., & Pehkonen, E. (1999). A virtual panel evaluating characterizations of beliefs. In E. Pehkonen & G. Törner (Eds.), Mathematical Beliefs and their Impact on Teaching and Learning of Mathematics. Proceedings of the Workshop in Oberwolfach, November 21–27, 1999 (pp. 24–30). Schriftenreihe des Fachbereichs Mathematik, No. 457. Duisburg: Universität Duisburg.

    Google Scholar 

  • Green, T. F. (1971). The Activities of Teaching. Tokyo: McGraw-Hill Kogakusha.

    Google Scholar 

  • Jones, D. L. (1990). A study of the beliefs systems of two beginning middle school mathematics teachers. Dissertation. University of Athens, Georgia.

    Google Scholar 

  • Kaldrimidou, M., Sakonidis, H., & Tzekaki, M. (2000). Epistemological features in the mathematics classroom: Algebra and geometry. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 3 (pp. 111–118). Hiroshima: Hiroshima University.

    Google Scholar 

  • Klaus, G., & Liebscher, H. (1979). Wörterbuch der kybernetik. Frankfurt: Fischer Taschenbuch Verlag.

    Google Scholar 

  • Köller, O., Baumert, J., & Neubrand, J. (2000). Epistemologische Überzeugungen und Fachverständnis im Mathematik-und Physikunterricht. In J. Baumert, W. Bos, & R. Lehmann, (Hrsg.), TIMSS / III — Dritte Internationale Mathematik — und Naturwissenschaftstudie — Mathematische und naturwissenschaftliche Bildung am Ende der Schullaufbahn (pp. 229–269). Band 2: Mathematische und physikalische Kompetenzen am Ende der gymnasialen Oberstufe. Opladen: Leske + Budrich.

    Google Scholar 

  • Kuhs, T.M., & Ball, D.L. (1986). Approaches to teaching mathematics, unpublished paper, National Center for Research on Teacher Education, Michigan State University.

    Google Scholar 

  • Lawler, W. (1981). The progressive construction of mind. Cognitive Science, 5, 1–30.

    Article  Google Scholar 

  • Lerman, S. (1983). Problem-solving or knowledge centered: The influence of philosophy on mathematics teaching. International Journal of Mathematical Education in Science and Technology, 14(1), 59–66.

    Google Scholar 

  • Lerman, S. (1997). The psychology of mathematics teachers’ learning in search of theory. In E. Pehkonen (Ed.), Proceedings of the 21st International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 3 (pp. 200–207). Helsinki: University of Helsinki; Lahti Research and Training Center.

    Google Scholar 

  • Lim Chap Sam. (2000). A comparison between Malaysian and United Kingdom teachers’ and students’ images of mathematics. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 3 (pp. 323–330). Hiroshima: Hiroshima University

    Google Scholar 

  • Lloyd, G. M., & Wilson, M. R. (1998). Supporting innovation: The impact of a teacher’s conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29(3), 248–274.

    Google Scholar 

  • McLeod, D. B., & Adams, V. M. (Eds.). (1989). Affect and mathematical problem solving — A new perspective. New York: Springer.

    Google Scholar 

  • McLeod, D. B. (1989). Beliefs, attitudes, and emotions: new views of affect in mathematics education. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving. A new perspective (pp. 245–258). New York: Springer-Verlag.

    Google Scholar 

  • Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of Curriculum Studies, 19(4), 317–328.

    Google Scholar 

  • Olson, J. M., & Zanna, M. P. (1993). Attitudes and attitude change. Annual Review of Psychology, 44, 117–154.

    Article  Google Scholar 

  • Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.

    Google Scholar 

  • Pajares, M. F., & Miller, M. D. (1994). Role of self-efficacy and self-concept beliefs in mathematical problem solving: A path analysis. Journal of Educational Psychology, 86, 193–203.

    Article  Google Scholar 

  • Patronis. T. (1994). On students’ conceptions of axioms in school geometry. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 4 (pp. 9–16). Lisboa (Portugal): University of Lisboa.

    Google Scholar 

  • Pehkonen, E. (1988). Conceptions and images of mathematics professors on teaching mathematics in school. International Journal of Mathematical Education in Science and Technology, 30, 389–397.

    Google Scholar 

  • Pence, B. (1994). Teachers perceptions of algebra. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 4 (pp. 17–24). Lisboa (Portugal): University of Lisboa.

    Google Scholar 

  • Perry, W. (1970). Forms of intellectual and ethical development in the college years: A scheme. New York: Holt, Rinehoart, & Wilson.

    Google Scholar 

  • Rodd, M. M. (1997). Beliefs and their warrants in mathematics learning. In E. Pehkonen (Ed.), Proceedings of the 21st International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 4 (pp. 64–65). Helsinki: University of Helsinki; Lahti Research and Training Center.

    Google Scholar 

  • Rogers, L. (1992). Images, metaphors and intuitive ways of knowing: The contexts of learners, teachers and of mathematics. In F. Seeger & H. Steinbring (Eds.), The dialogue between theory and practice in mathematics education: Overcoming the broadcast metaphor. Proceedings of the Fourth Conference on Systematica Co-Operation between Theory and Practice in Mathematics Education (SCTP). Brakel.

    Google Scholar 

  • Rokeach, M. (1960). The organisation of belief-disbelief systems. In M. Rokeach (Ed.), The open and closed mind. New York: Basic Books.

    Google Scholar 

  • Rokeach, M. (1968). Beliefs, attitudes, and values: A theory of organization and change. San Francisco: Jossey-Bass.

    Google Scholar 

  • Ruffell, M., Mason, J., & Allen, B. (1998). Studying attitude to mathematics. Educational Studies in Mathematics, 35, 1–18.

    Article  Google Scholar 

  • Ryan, M. P. (1984). Monitoring text comprehension: Individual differences in epistemological standards. Journal of Epistemological Psychology, 76, 248–258.

    Google Scholar 

  • Schoenfeld, A.H. (1985). Mathematical problem solving. Orlando (FL): Academic Press.

    Google Scholar 

  • Schoenfeld, A.H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94.

    Google Scholar 

  • Shulman, L.S. (1986). Paradigms and research programs in the study of teaching: a contemporary perspective. In M. C. Wittrock (Ed.), Third Handbook of Research on Teaching (pp. 3–36). New York: Macmillan.

    Google Scholar 

  • Snow, R., Corno, L., & Jackson III, D. (1996). Individual Differences in affective and conative function. In: D. C. Berliner & R. Calfee (Eds.), Handbook of Educational Psychology (pp. 243–310). New York: Simon & Schuster Macmillan.

    Google Scholar 

  • Stacey, K. (1994). Algebraic sums and products: Students’ concepts and symbolism. In J.P. Ponte & J.F. Matos (Eds.), Proceedings of the 18th International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 4 (pp. 289–296). Lisboa (Portugal): University of Lisboa.

    Google Scholar 

  • Tall, D., & Vinner, S. (1981). Concept images and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.

    Article  Google Scholar 

  • Tall, D. (1987). Constructing the concept image of a tangent. In J.C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th International Conference of International Group for the Psychology of Mathematics Education (PME) Vol. 3 (pp. 69–75). Montreal.

    Google Scholar 

  • Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105–127.

    Google Scholar 

  • Thompson, A.G. (1989). Learning to teach mathematical problem solving: Changes in teachers’ conceptions and beliefs. In R. I. Charles & E. A. Silver (Eds.), The Teaching and Assessing of Mathematical Problem Solving Vol. 3 (pp. 232–243). Research Agenda for Mathematics Education. Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics.

    Google Scholar 

  • Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics learning and teaching (pp. 127–146). New York: Macmillan Publishing.

    Google Scholar 

  • Tierney, C., Boyd, C., & Davis, G. (1990). Prospective primary teachers’ conceptions of area. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.), Proceedings of the 14th Conference of the International Group For the Psychology of Mathematics Education (PME) with the North American Chapter 12th PME-NA Conference Vol. 2 (pp. 307–318). México.

    Google Scholar 

  • Tirosh, D., & Graeber, A.O. (1989). Preservice elementary teachers’ explicit beliefs about multiplication and division. Educational Studies in Mathematics, 20(1), 79–96.

    Article  Google Scholar 

  • Törner, G., & Pehkonen. (1996). On the structure of mathematical belief systems. International Reviews on Mathematical Education (ZDM), 28(4), 109–112.

    Google Scholar 

  • Törner, G. (2000). Domain specificbeliefs and calculus. Some theoretical remarks and phemonological observations. In E. Pehkonen & G. Törner (Eds.). Mathematical Beliefs and their Impact on Teaching and Learning of Mathematics. Proceedings of the Workshop in Oberwolfach, Nov. 21–27, 1999 (pp. 127–137). Schriftenreihe des Fachbereichs Mathematik, No. 457. Duisburg: Universität Duisburg.

    Google Scholar 

  • Uriza, R. C. (1989). Concept image in its origins with particular reference to Taylor’s series. In C. A. Maher, G. A. Goldin & R. B. Davis (Eds.), Proceedings of the 11th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME) Vol. 1 (pp. 55–60). New Brunswick (NJ): Rutgers-The State University of New Jersey.

    Google Scholar 

  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht: Kluwer.

    Google Scholar 

  • Vinner, S., & Dreyfus, T. (1989). Images and definition for the concept of function. Journal for Research in Mathematics Education, 20, 356–66.

    Google Scholar 

  • Vinner, S., & Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometrical concepts. In R. Karplus (Ed.), Proceedings of the 4th International Conference of the International Group for the Psychology of Mathematics Education (PME) Vol. 1 (pp. 177–184). Berkeley (CA): University of California, Lawrence Hall of science.

    Google Scholar 

  • Zandieh, M. J. (1998). The role of a formal definition in nine students’ concept image of derivative. In Berenson, S.B. et al. Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education Vol. 1 (pp. 136–141). Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education.

    Google Scholar 

  • Zimmermann, H. J. (1990). Fuzzy set theory and its applications. Boston: Kluwer.

    Google Scholar 

Download references

Authors
  1. GÜnter TÖrner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. La Trobe University, Australia

    Gilah C. Leder

  2. University of Turku, Finland

    Erkki Pehkonen

  3. University of Duisburg, Germany

    Günter Törner

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Kluwer Academic Publishers

About this chapter

Cite this chapter

TÖrner, G. (2002). Mathematical Beliefs — A Search for a Common Ground: Some Theoretical Considerations on Structuring Beliefs, Some Research Questions, and Some Phenomenological Observations. In: Leder, G.C., Pehkonen, E., Törner, G. (eds) Beliefs: A Hidden Variable in Mathematics Education?. Mathematics Education Library, vol 31. Springer, Dordrecht. https://doi.org/10.1007/0-306-47958-3_5

Download citation

  • DOI: https://doi.org/10.1007/0-306-47958-3_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-1057-6

  • Online ISBN: 978-0-306-47958-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation