Abstract
Developing abilities to create, inquire into, qualify, and choose among mathematical problems is an important educational goal. In this paper, we elucidate how mathematicians work with mathematical problems in order to understand this mathematical process. More specifically, we investigate how mathematicians select and pose problems and discuss to what extent our results can be used to inform, criticize, and develop educational practice at various levels. Selecting and posing problems is far from simple. In fact, it is considered hard, complex, and of crucial importance. A number of criteria concerning personal interest, continuity with previous work, the danger of getting stuck, and how fellow mathematicians will respond to the findings are considered when mathematicians think about whether to approach a specific problem. These results add to previous investigations of mathematicians’ practice and suggest that mathematics education research could further investigate how students select and develop problems, work with multiple problems over a longer period of time, and use the solutions to problems to support the development of new problems. Furthermore, the negative emotional aspects of being stuck in problem solving and students’ conceptions of solvability and relevance of or interest in a mathematical problem are areas of research suggested by our study.
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Notes
A sampling strategy considering that pure but not applied mathematics can of course be contested, because applied mathematics—with its attention to models and real-life situations—could be more suitable as inspiration to and mirror of the teaching and learning of mathematics. However, there are a multitude of practices in society that can serve as such inspiration (engineering, science, and economy to mention a few). Furthermore, in this investigation, we have chosen a narrow focus and aim only at relating the selection of research problem in the area of pure mathematics, to problem posing and selection when teaching and learning mathematics.
Burton quotes an earlier version (from 1982—in this edition, the quotation is on page 49) of Mason et al. (2010), cited in this article. Leone Burton is a co-author of the book.
To secure the anonymity of the interviewees, we will refer to them using a sequence of random numbers (obtained from www.random.org) throughout the paper. Thus, the order of the numbers might not reflect the order in which the interviews were made.
References
Appelbaum, P. (1999). The stench of perception and the cacophony of mediation. For the Learning of Mathematics, 19(2), 11–18. doi:10.2307/40248294
Artigue, M., & Baptist, P. (2012). Inquiry in mathematics education (Resource booklet). Retrieved from The Fibonacci Project website: http://www.fibonacci-project.eu
Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathematiques, 1970–1990. Dordrecht, Holland: Kluwer Academic Publishers.
Brown, S. I. (1984). The logic of problem generation: From morality and solving to de-posing and rebellion. For the Learning of Mathematics, 4(1), 9–20.
Brown, S. & Walter, M. (2005). The art of problem posing. Mahwah, NJ: Lawrence Erlbaum. Retrieved from http://public.eblib.com/EBLPublic/PublicView.do?ptiID=227452
Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Boston: Kluwer Academic Publishers.
Cai, J., Moyer, J. C., & Wang, N. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83(1), 57–69. doi:10.1007/s10649-012-9429-3
Charmaz, K. (2006). Constructing grounded theory. London; Thousand Oaks, CA: Sage Publications.
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. Journal of the Learning Sciences, 10(2), 113–163.
d’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the learning of Mathematics, 5(1), 44–48.
Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.
English, L. D. (1997). The development of fifth-grade children’s problem-posing abilities. Educational Studies in Mathematics, 34(3), 183–217.
English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–106.
European Commission. (2007). Science education now: A renewed pedagogy for the future of Europe. Luxembourg: Office for Official Publications of the European Communities.
Johansen, M. W., & Misfeldt, M. (n.d.). An empirical approach to the mathematical values of problem choice and argumentation. Unpublished manuscript.
Johansen, M. W., & Misfeldt, M. (2014). Når matematikere undersøger matematik - og hvilken betydning det har for undersøgende matematikundervisning. MONA, 2014(4), 42–59.
Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale, NJ: Erlbaum.
Kirshner, D. (2002). Untangling teachers’ diverse aspirations for student learning: A crossdisciplinary strategy for relating psychological theory to pedagogical practice. Journal for Research in Mathematics Education, 33(1), 46–58.
Kvale, S. (1996). InterViews: An introduction to qualitative research interviewing. Thousand Oaks, CA: Sage Publications.
Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically. Harlow; Munchen: Pearson Education Limited.
Misfeldt, M. (2005). Media in mathematical writing: Can teaching learn from research practice? For the Learning of Mathematics: An International Journal of Mathematics Education., 25(2), 36–42.
National Research Council (U.S.). (1996). National Science Education Standards: Observe, interact, change, learn. Washington, DC: National Academy Press.
Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.
Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
Silver, E. A. (2013). Problem-posing research in mathematics education: Looking back, looking around, and looking ahead. Educational Studies in Mathematics, 83(1), 157–162. doi:10.1007/s10649-013-9477-3
Strauss, A. L., & Corbin, J. M. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage Publications.
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–178.
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Misfeldt, M., Johansen, M.W. Research mathematicians’ practices in selecting mathematical problems. Educ Stud Math 89, 357–373 (2015). https://doi.org/10.1007/s10649-015-9605-3
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DOI: https://doi.org/10.1007/s10649-015-9605-3