### A Case Study of the Relationship Between Meaning And Formalism

#### Abstract

The purpose of this study was to explore the sources of mathematical ideas in terms of the relationships between meaning and formalism and their role in the transition between elementary mathematics and advanced mathematics. The two participants were high school mathematics teachers, who vary in their levels of experience. Two forms of data were collected to obtain more in-depth data about the transformations within among mathematical ideas: a questionnaire including 14 open-ended mathematical tasks and semistructured interviews. Results indicated that individuals had different ways in constructing mathematical ideas and that their mathematical ideas were derived from the transition between meaning and formalism.

**Keywords:** relations, meaning, formalism, advanced mathematics

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Bogdan, R., & Biklen, S. K. (2006). Qualitative research for education: An introduction to theories and methods (5th ed.). Boston: Allyn & Bacon.

Chin, E. T. (2002). Building and using concepts of equivalence class and partition. Unpublished PhD, University of Warwick.

Chin. E-T. & Tall. D. O. (2000). Making, having and compressing formal mathematical concepts. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference

of the International Group for the Psychology of Mathematics Education, 2, 177-184.

Chin. E-T., & Tall. D. O. (2001). Developing formal mathematical concepts over time. In M. van den Heuvel-Pabhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 2, 241-248.

Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among the five traditions. Thousand Oaks, CA: Sage.

Davis, R. B., & Maher, C. A. (1997). How students think: The role of representations. In L.D. English (Ed.), Mathematical reasoning: Analogies, metaphors,and images (pp.93-115). Mahwah, NJ: Erlbaum.

Davis, R. B., & Maher, C. A. (1990). What do we do when we ‘do mathematics’? In: N. Noddings (Ed.), Constructivist views of the teaching and learning of mathematics (Vol. Monograph No. 4). Reston, VA: National Council of Teachers of Mathematics.

Dörfler, W. (2000). Means for meaning. In: P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: perspectives on discourse, tools, and instructional design (pp. 99–131). Mahwah, NJ: Rulerence Erlbaum Associates.

Dörfler, W. (2003). Mathematics and Mathematics education: Content and people, relation and difference. Educational Studies in Mathematics, 54, 147-170.

Dubinsky, E. (2000). Meaning and formalism in mathematics. International Journal of Computers for Mathematical Learning, 5, 211-240.

Dubinsky, E. (1992). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.). Advanced mathematical thinking. (pp. 95-123). Dordrecht, The Netherlands: Kluwer.

Halmos. P. (1987). I want to he a mathematician. Washington. DC: The Mathematical Association of America.

Lakoff, G., & Nunéz, R. (2000). Where mathematics comes from. New York: Basic Books.

Maher, C. A. (2002). How students structure their own investigations and educate us: what we’ve learned from a fourteen year study. In: A. D. Cockburn, & E. Nardi (Eds.), Proceedings of the Twenty-Sixth Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 31–46). Norwich, UK: School of Education and Professional Development, University of East Anglia.

Maher, C. A., & Martino, A. M. (1996a). The development of the idea of mathematical proof: a 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.

Maher, C. A., & Speiser, R. (1997). How far can you go with block tower? Stephanie’s

intellectual development. Journal for Research in Mathematics Education, 16(2),

-132.

Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 22, 249–266.

Nunéz, R. E. (2000). Mathematical idea analysis: What embodied cognitive science can say about the human nature of Mathematics. In Proceedings of the Conference of the International Group for the Psychology of Mathematics Education, Japan, July,23.

Pinto, M., & Tall, D. O. (1999). Student constructions of formal theory: Giving and extracting meaning. In O. Zaslavsky (ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, 4, 65–72.

Sierpinska, A. (1992). Theoretical perspectives for development of the function concept. In G. Harel & E. Dubinsky (eds.), The concept of function: Aspects of epistemology and pedagogy, MAA Notes 25, (pp.23–58). Washington DC: MAA.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

Skemp. R. R. (1977). The psychology of learning mathematics. Penguin Books.

Skemp, R. R. (1979). Intelligence, Learning, and Action. New York: John Wiley & Sons.

Stylianou, D. A. (2000). Expert and novice use of visual representations in advanced mathematical problem solving. (Doctoral Dissertation, The University of Pittsburgh, 2000). Dissertation Abstracts International, 61(12), 9998584.

Tall, D., & Chin. E-T. (2002). University students embodiment of quantifier. In Anne D. Cockburn & Elena Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4, 273–280.

Tall, D. (1991). Intuition and rigor: The role of visualization in calculus. In W. Zimmerman and S. Cunningham (eds.) Visualization in teaching and learning mathematics, Mathematical Association of America, MAA Notes Series, 105–120.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press: Cambridge.

Vygosky, L. S. (1986). Thought and language. M.I.T. Press: Cambridge.

Wertsch, J. V. (1998). Mind as action. Oxford University Press: New York.