How do mathematicians learn math?: resources and acts for constructing and understanding mathematics
 Michelle H. WilkersonJerde,
 Uri J. Wilensky
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a network of mathematical resources by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics graduate students that were asked to read and make sense of an unfamiliar, but accessible, mathematical proof in the domain of geometric topology. We find that experts are more likely to refer to definitions when questioning or explaining some aspect of the focal mathematical idea and more likely to refer to specific examples or instantiations when making sense of an unknown aspect of that idea. However, in general, they employ a variety of types of mathematical resources simultaneously. Often, these combinations are used to deconstruct the mathematical idea in order to isolate, identify, and explore its subcomponents. Some common patterns in the ways experts combined these resources are presented, and we consider implications for education.
Inside
Within this Article
 Introduction
 Theoretical framework
 Methods and analytic framework
 The mathematical research paper
 Coding framework
 Results
 Discussion
 References
 References
Other actions
 Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69, 111–129. CrossRef
 Ball, D. L., Hoyles, C., Jahnke, H. N., & MovshovitzHadar, N. (2002). The teaching of proof. In Proceedings of the International Congress of Mathematicians (Vol. 3, pp. 907–920).
 Bell, A. W. (1976). A study of pupils’ proofexplanations in mathematical situations. Educational Studies in Mathematics, 7(1), 23–40. CrossRef
 Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
 Burton, L. (1999). The practices of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37, 121–143. CrossRef
 Chi, M. (1997). Quantifying qualitative analyses of verbal data: A practical guide. Journal of the Learning Sciences, 6(3), 271–315. CrossRef
 Clement, J. (2000). Analysis of clinical interviews: Foundations & model viability. In R. Lesh (Ed.), Handbook of research methodologies for science and mathematics education (pp. 341–385). Hillsdale, NJ: Lawrence Erlbaum.
 Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15, 375–402. CrossRef
 de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24(1), 17–24.
 Duffin, J., & Simpson, A. (2000). A search for understanding. The Journal of Mathematical Behavior, 18(4), 415–427. CrossRef
 Ericsson, K. A., & Simon, H. A. (1984). Protocol analysis: Verbal reports as data. Cambridge, MA: MIT Press.
 Glaser, B. G., & Strauss, A. L. (1977). The discovery of grounded theory: Strategies for qualitative research. London: Aldine.
 Gray, E., Pinto, M. M. F., Pitta, D., & Tall, D. (1999). Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, 38(1), 111–133. CrossRef
 Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM, 40(3), 345–353. CrossRef
 Hanna, G., & de Villiers, M. (2008). ICMI study 19: Proof and proving in mathematics education. ZDM, 40, 329–336. CrossRef
 Hatano, G., & Inagaki, K. (1986). Two courses of expertise. In H. William Stevenson, & K. H. Hiroshi Azuma (Eds.), Child development and education in Japan (pp. 262–272). San Francisco, CA: W. H. Freeman.
 Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399. CrossRef
 Inglis, M., MejiaRamos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66, 3–21. CrossRef
 Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press.
 MejiaRamos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the international commission on mathematical instruction study 19, proof and proving in mathematics education (Vol. 2, pp. 88–93). Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
 Michener, E. R. (1978). Understanding understanding mathematics. Cognitive Science, 2, 361–383. CrossRef
 NCTM (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics.
 Papert, S. (1971). On making a theorem for a child. Paper presented at the ACM Annual Conference, Boston, MA.
 Papert, S. (1993). The children’s machine: Rethinking school in the age of the computer. New York: Basic Books.
 Patel, V. L., & Groen, G. (1991). The specific and general nature of medical expertise: A critical look. In K. A. Ericsson, & J. Smith (Eds.), Toward a general theory of expertise: Prospects and limits. Cambridge, UK: Cambridge University Press.
 Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.
 Roth, W. M., & Bowen, G. M. (2003). When are graphs worth ten thousand words? An expert–expert study. Cognition and Instruction, 21(4), 429–473. CrossRef
 Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
 Sierpinska, A. (1994). Understanding in mathematics. Bristol, PA: The Falmer Press, Taylor & Francis Inc.
 Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.
 Stanford, T. (1998). Found observations of ntriviality and Brunnian links. http://arxivorg/abs/math/9807161.
 Stylianou, D. A., & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expertnovice similarities and differences. Mathematical Thinking and Learning, 6(4), 353–387. CrossRef
 Tall, D. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In M. van den HeuvelPanhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 65–72). Utrecht, The Netherlands: PME.
 Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. Advanced Mathematical Thinking, 11, 65–81.
 Watson, A., & Mason, J. (2002). Studentgenerated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237–249. CrossRef
 Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2/3), 209–234. CrossRef
 Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel, & S. Papert (Eds.), Constructionism (pp. 193–203). Norwood, NJ: Ablex.
 Wilensky, U. (1993). Connected mathematics: Building concrete relationships with mathematical knowledge. Ph.D. thesis, MIT.
 Wineburg, S. (1997). Reading Abraham Lincoln: An expert/expert study in the interpretation of historical texts. Cognitive Science, 22(3), 319–346. CrossRef
 Title
 How do mathematicians learn math?: resources and acts for constructing and understanding mathematics
 Journal

Educational Studies in Mathematics
Volume 78, Issue 1 , pp 2143
 Cover Date
 20110901
 DOI
 10.1007/s1064901193065
 Print ISSN
 00131954
 Online ISSN
 15730816
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Expert mathematicians
 Topology
 Proof
 Reasoning
 Knowledge resources
 Industry Sectors
 Authors

 Michelle H. WilkersonJerde ^{(1)} ^{(2)}
 Uri J. Wilensky ^{(1)} ^{(2)} ^{(3)} ^{(4)}
 Author Affiliations

 1. Center for Connected Learning, Northwestern University, 2120 Campus Drive, Evanston, IL, USA
 2. Learning Sciences Program, Northwestern University, 2120 Campus Drive, Evanston, IL, USA
 3. Computer Science and Electrical Engineering, Northwestern University, 2120 Campus Drive, Evanston, IL, USA
 4. Northwestern Institute on Complex Systems, Northwestern University, 2120 Campus Drive, Evanston, IL, USA