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)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abelson, R. (1979). Differences between belief systems and knowledge systems. Cognitive Science, 3, 355–366.
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.
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.
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.
Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(3), 132–144.
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.
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.
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.
Berger, P. (2001). Computer und Weltbild. Habitualisierte Konzeptionen von Lehrern im Kontext von Informatik, Mathematik und Computerkultur. Dissertation. Wiesbaden: Westdeutscher Verlag.
Berliner, D. C., & Calfee, R. (Eds.). (1996). Handbook of Educational Psychology. New York: Macmillan Library Reference USA.
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.
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.
Cooney, T. (1994). Research and teacher education: in search of common ground. Journal for Research in Mathematics Education, 25, 608–636.
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.
Dedekind, R. (1995). What are numbers and what should they be? Orono, ME: Research Institute for Mathematics (first published in 1888).
Dungan, J. F., & Thurlow, G. R. (1989). Students’ attitudes to mathematics: A review of the literature. The Australian Mathematics Teacher, 45(3), 8–11.
Eagly, A. H., & Chaiken, S. (1992). The psychology of attitudes. San Diego, CA: Harcourt Brace Janovich.
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.
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.
Ernest, P. (1989). The knowledge, beliefs, and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(10), 13–33.
Ernest, P. (1991). The philosophy of mathematics education. Hampshire (UK): The Falmer Press.
Erlwanger, S. (1975). Case studies of children’s conceptions of mathematics, Part 1. Journal of Children’s Mathematical Behavior, 1, 157–283.
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.
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.
Green, T. F. (1971). The Activities of Teaching. Tokyo: McGraw-Hill Kogakusha.
Jones, D. L. (1990). A study of the beliefs systems of two beginning middle school mathematics teachers. Dissertation. University of Athens, Georgia.
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.
Klaus, G., & Liebscher, H. (1979). Wörterbuch der kybernetik. Frankfurt: Fischer Taschenbuch Verlag.
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.
Kuhs, T.M., & Ball, D.L. (1986). Approaches to teaching mathematics, unpublished paper, National Center for Research on Teacher Education, Michigan State University.
Lawler, W. (1981). The progressive construction of mind. Cognitive Science, 5, 1–30.
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.
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.
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
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.
McLeod, D. B., & Adams, V. M. (Eds.). (1989). Affect and mathematical problem solving — A new perspective. New York: Springer.
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.
Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of Curriculum Studies, 19(4), 317–328.
Olson, J. M., & Zanna, M. P. (1993). Attitudes and attitude change. Annual Review of Psychology, 44, 117–154.
Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.
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.
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.
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.
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.
Perry, W. (1970). Forms of intellectual and ethical development in the college years: A scheme. New York: Holt, Rinehoart, & Wilson.
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.
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.
Rokeach, M. (1960). The organisation of belief-disbelief systems. In M. Rokeach (Ed.), The open and closed mind. New York: Basic Books.
Rokeach, M. (1968). Beliefs, attitudes, and values: A theory of organization and change. San Francisco: Jossey-Bass.
Ruffell, M., Mason, J., & Allen, B. (1998). Studying attitude to mathematics. Educational Studies in Mathematics, 35, 1–18.
Ryan, M. P. (1984). Monitoring text comprehension: Individual differences in epistemological standards. Journal of Epistemological Psychology, 76, 248–258.
Schoenfeld, A.H. (1985). Mathematical problem solving. Orlando (FL): Academic Press.
Schoenfeld, A.H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Tirosh, D., & Graeber, A.O. (1989). Preservice elementary teachers’ explicit beliefs about multiplication and division. Educational Studies in Mathematics, 20(1), 79–96.
Törner, G., & Pehkonen. (1996). On the structure of mathematical belief systems. International Reviews on Mathematical Education (ZDM), 28(4), 109–112.
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.
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.
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.
Vinner, S., & Dreyfus, T. (1989). Images and definition for the concept of function. Journal for Research in Mathematics Education, 20, 356–66.
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.
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.
Zimmermann, H. J. (1990). Fuzzy set theory and its applications. Boston: Kluwer.
Editor information
Editors and Affiliations
Rights 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