How do mathematicians learn math?: resources and acts for constructing and understanding mathematics
 Michelle H. WilkersonJerde,
 Uri J. Wilensky
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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.
 Alcock, L, Inglis, M (2008) Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics 69: pp. 111129 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, AW (1976) A study of pupils’ proofexplanations in mathematical situations. Educational Studies in Mathematics 7: pp. 2340 CrossRef
 Bransford, JD, Brown, AL, Cocking, RR eds. (1999) How people learn: Brain, mind, experience, and school. National Academy Press, Washington, DC
 Burton, L (1999) The practices of mathematicians: What do they tell us about coming to know mathematics?. Educational Studies in Mathematics 37: pp. 121143 CrossRef
 Chi, M (1997) Quantifying qualitative analyses of verbal data: A practical guide. Journal of the Learning Sciences 6: pp. 271315 CrossRef
 Clement, J Analysis of clinical interviews: Foundations & model viability. In: Lesh, R eds. (2000) Handbook of research methodologies for science and mathematics education. Lawrence Erlbaum, Hillsdale, NJ, pp. 341385
 Cuoco, A, Goldenberg, EP, Mark, J (1996) Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior 15: pp. 375402 CrossRef
 Villiers, M (1990) The role and function of proof in mathematics. Pythagoras 24: pp. 1724
 Duffin, J, Simpson, A (2000) A search for understanding. The Journal of Mathematical Behavior 18: pp. 415427 CrossRef
 Ericsson, KA, Simon, HA (1984) Protocol analysis: Verbal reports as data. MIT Press, Cambridge, MA
 Glaser, BG, Strauss, AL (1977) The discovery of grounded theory: Strategies for qualitative research. Aldine, London
 Gray, E, Pinto, MMF, Pitta, D, Tall, D (1999) Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics 38: pp. 111133 CrossRef
 Hanna, G, Barbeau, E (2008) Proofs as bearers of mathematical knowledge. ZDM 40: pp. 345353 CrossRef
 Hanna, G, Villiers, M (2008) ICMI study 19: Proof and proving in mathematics education. ZDM 40: pp. 329336 CrossRef
 Hatano, G, Inagaki, K Two courses of expertise. In: William, H, Hiroshi Azuma, KH eds. (1986) Child development and education in Japan. W. H. Freeman, San Francisco, CA, pp. 262272
 Hersh, R (1993) Proving is convincing and explaining. Educational Studies in Mathematics 24: pp. 389399 CrossRef
 Inglis, M, MejiaRamos, JP, Simpson, A (2007) Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics 66: pp. 321 CrossRef
 Lakatos, I (1976) Proofs and refutations: The logic of mathematical discovery. Cambridge University Press, Cambridge, UK
 MejiaRamos, JP, Inglis, M Argumentative and proving activities in mathematics education research. In: Lin, FL, Hsieh, FJ, Hanna, G, Villiers, M eds. (2009) Proceedings of the international commission on mathematical instruction study 19, proof and proving in mathematics education (Vol. 2). The Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, pp. 8893
 Michener, ER (1978) Understanding understanding mathematics. Cognitive Science 2: pp. 361383 CrossRef
 Curriculum and evaluation standards for school mathematics. The National Council of Teachers of Mathematics, Reston, VA
 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. Basic Books, New York
 Patel, VL, Groen, G The specific and general nature of medical expertise: A critical look. In: Ericsson, KA, Smith, J eds. (1991) Toward a general theory of expertise: Prospects and limits. Cambridge University Press, Cambridge, UK
 Rav, Y (1999) Why do we prove theorems?. Philosophia Mathematica 7: pp. 541
 Roth, WM, Bowen, GM (2003) When are graphs worth ten thousand words? An expert–expert study. Cognition and Instruction 21: pp. 429473 CrossRef
 Schoenfeld, AH (1985) Mathematical problem solving. Academic Press, Orlando, FL
 Sierpinska, A (1994) Understanding in mathematics. The Falmer Press, Taylor & Francis Inc., Bristol, PA
 Skemp, RR (1976) Relational understanding and instrumental understanding. Mathematics Teaching 77: pp. 2026
 Stanford, T. (1998). Found observations of ntriviality and Brunnian links. http://arxivorg/abs/math/9807161.
 Stylianou, DA, Silver, EA (2004) The role of visual representations in advanced mathematical problem solving: An examination of expertnovice similarities and differences. Mathematical Thinking and Learning 6: pp. 353387 CrossRef
 Tall, D Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In: HeuvelPanhuizen, M eds. (2001) Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 3). PME, Utrecht, The Netherlands, pp. 6572
 Vinner, S (1991) The role of definitions in the teaching and learning of mathematics. Advanced Mathematical Thinking 11: pp. 6581
 Watson, A, Mason, J (2002) Studentgenerated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education 2: pp. 237249 CrossRef
 Weber, K, Alcock, L (2004) Semantic and syntactic proof productions. Educational Studies in Mathematics 56: pp. 209234 CrossRef
 Wilensky, U Abstract meditations on the concrete and concrete implications for mathematics education. In: Harel, I, Papert, S eds. (1991) Constructionism. Norwood, NJ: Ablex, pp. 193203
 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: pp. 319346 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
 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