# How do mathematicians learn math?: resources and acts for constructing and understanding mathematics

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## Abstract

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.

## Keywords

Expert mathematicians Topology Proof Reasoning Knowledge resources## References

- Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures.
*Educational Studies in Mathematics, 69*, 111–129.CrossRefGoogle Scholar - Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In
*Proceedings of the International Congress of Mathematicians*(Vol. 3, pp. 907–920).Google Scholar - Bell, A. W. (1976). A study of pupils’ proof-explanations in mathematical situations.
*Educational Studies in Mathematics, 7*(1), 23–40.CrossRefGoogle Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999).
*How people learn: Brain, mind, experience, and school*. Washington, DC: National Academy Press.Google Scholar - Burton, L. (1999). The practices of mathematicians: What do they tell us about coming to know mathematics?
*Educational Studies in Mathematics, 37*, 121–143.CrossRefGoogle Scholar - Chi, M. (1997). Quantifying qualitative analyses of verbal data: A practical guide.
*Journal of the Learning Sciences, 6*(3), 271–315.CrossRefGoogle Scholar - 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.Google Scholar - Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula.
*Journal of Mathematical Behavior, 15*, 375–402.CrossRefGoogle Scholar - de Villiers, M. (1990). The role and function of proof in mathematics.
*Pythagoras, 24*(1), 17–24.Google Scholar - Duffin, J., & Simpson, A. (2000). A search for understanding.
*The Journal of Mathematical Behavior, 18*(4), 415–427.CrossRefGoogle Scholar - Ericsson, K. A., & Simon, H. A. (1984).
*Protocol analysis: Verbal reports as data*. Cambridge, MA: MIT Press.Google Scholar - Glaser, B. G., & Strauss, A. L. (1977).
*The discovery of grounded theory: Strategies for qualitative research*. London: Aldine.Google Scholar - 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.CrossRefGoogle Scholar - Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge.
*ZDM, 40*(3), 345–353.CrossRefGoogle Scholar - Hanna, G., & de Villiers, M. (2008). ICMI study 19: Proof and proving in mathematics education.
*ZDM, 40*, 329–336.CrossRefGoogle Scholar - 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.Google Scholar - Hersh, R. (1993). Proving is convincing and explaining.
*Educational Studies in Mathematics, 24*(4), 389–399.CrossRefGoogle Scholar - Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification.
*Educational Studies in Mathematics, 66*, 3–21.CrossRefGoogle Scholar - Lakatos, I. (1976).
*Proofs and refutations: The logic of mathematical discovery*. Cambridge, UK: Cambridge University Press.Google Scholar - Mejia-Ramos, 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.Google Scholar - Michener, E. R. (1978). Understanding understanding mathematics.
*Cognitive Science, 2*, 361–383.CrossRefGoogle Scholar - NCTM (1989).
*Curriculum and evaluation standards for school mathematics*. Reston, VA: The National Council of Teachers of Mathematics.Google Scholar - Papert, S. (1971). On making a theorem for a child.
*Paper presented at the ACM Annual Conference, Boston, MA*.Google Scholar - Papert, S. (1993).
*The children’s machine: Rethinking school in the age of the computer*. New York: Basic Books.Google Scholar - 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.Google Scholar - Rav, Y. (1999). Why do we prove theorems?
*Philosophia Mathematica, 7*(1), 5–41.Google Scholar - 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.CrossRefGoogle Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. Orlando, FL: Academic Press.Google Scholar - Sierpinska, A. (1994).
*Understanding in mathematics*. Bristol, PA: The Falmer Press, Taylor & Francis Inc.Google Scholar - Skemp, R. R. (1976). Relational understanding and instrumental understanding.
*Mathematics Teaching, 77*, 20–26.Google Scholar - Stanford, T. (1998).
*Found observations of n-triviality 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 expert-novice similarities and differences.
*Mathematical Thinking and Learning, 6*(4), 353–387.CrossRefGoogle Scholar - Tall, D. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In M. van den Heuvel-Panhuizen (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.Google Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics.
*Advanced Mathematical Thinking, 11*, 65–81.Google Scholar - Watson, A., & Mason, J. (2002). Student-generated examples in the learning of mathematics.
*Canadian Journal of Science, Mathematics and Technology Education, 2*(2), 237–249.CrossRefGoogle Scholar - Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions.
*Educational Studies in Mathematics, 56*(2/3), 209–234.CrossRefGoogle Scholar - 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.Google Scholar - Wilensky, U. (1993).
*Connected mathematics: Building concrete relationships with mathematical knowledge*. Ph.D. thesis, MIT.Google Scholar - Wineburg, S. (1997). Reading Abraham Lincoln: An expert/expert study in the interpretation of historical texts.
*Cognitive Science, 22*(3), 319–346.CrossRefGoogle Scholar