Abstract
The purpose of this chapter is to highlight the importance of “thinking ahead” in mathematical problem solving. This process, though seemingly central to the work of mathematicians, seems to be largely overlooked in the mathematics education literature. This chapter presents my recent attempts to characterize future-oriented processes in mathematical work and summarizes evidence of mathematicians engaging in such processes. The main new results presented here concern students’ future thinking in mathematical situations. Student participants’ work in problem situations was analysed through the lens of mathematical foresight. This analysis serves to deepen the mathematical foresight model and opens up a number of directions for future research.
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References
Atance, C.M., & O’Neill, D.K. (2001). Episodic future thinking. Trends in Cognitive Science, 5(12), 533–539.
Buckner, R. L., & Carroll, D. C. (2007). Self-projection and the brain. Trends in Cognitive Science, 11, 49–57.
English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In L. English & B. Sriraman (Eds.), Theories of mathematics education (pp. 263–290). Heidelberg: Springer.
Galbraith, P., Stillman, G., & Brown, J. (2015). The primacy of ‘noticing’: A key to successful modelling. In G. A. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modelling and applications (pp. 83–94). New York: Springer.
Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton: Princeton University Press.
Hamilton, E. (2007). What changes are needed in the kind of problem solving situations where mathematical thinking is needed beyond school? In R. Lesh, E. Hamilton, & J. Kaput (Eds.), Foundations for the future in mathematics education (pp. 1–6). Mahwah, NJ: Lawrence Erlbaum.
Kilpatrick, J. (1985). A retrospective account of the past 25 years of research on teaching mathematical problem solving. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives. Hillsdale, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modelling. In F. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte, NC: Information Age Publishing.
Lester, F. K., & Cai, J. (2017). Can mathematical problem solving be taught? Preliminary answers from 30 years of research. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems: Advances and new perspectives (pp. 117–135). Switzerland: Springer.
Lester, F. K., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. A. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 501–518). Mahwah, NJ: Lawrence Erlbaum Associates.
Maciejewski, W. (2012). Resistance and relatedness on an evolutionary graph. Journal of the Royal Society, Interface, 9(68), 511–517.
Maciejewski, W. (2017). Mathematical knowledge and memories of mathematics. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 209–216). Singapore: PME.
Maciejewski, W., & Barton, B. (2016). Mathematical foresight: Thinking in the future to work in the present. For the Learning of Mathematics, 47(3), 31–37.
Maciejewski, W., Roberts, R., & Addis, D. R. (2016). Episodic future thinking in mathematical situations. Episodic future thinking in mathematical situations. In C. Csikos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 227–234). Szeged, Hungary: PME.
Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically. Essex, UK: Pearson.
Mayer, R. (1982). The psychology of mathematical problem solving. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research. The Franklin Institute: Philadelphia, PA.
Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical competencies (pp. 43–59). New York: Springer.
Nowak, M. (2006). Evolutionary dynamics: Exploring the equations of life. Cambridge, MA: Harvard University Press.
Poincaré, H. (1910). Mathematical creation. The Monist, 20(3), 321–335.
Pólya, G. (1945). How to solve it. Garden City, NY: Doubleday.
Raichle, M. E., MacLeod, A. M., Snyder, A. Z., Powers, W. J., Gusnard, D. A., & Shulman, G. L. (2001). A default mode of brain function. Proceedings of the National Academy of Sciences, 98, 676–682.
Schacter, D. L. (2012). Adaptive constructive processes and the future of memory. American Psychologist, 67(8), 603–613.
Schacter, D. L., & Addis, D. R. (2007). The cognitive neuroscience of constructive memory: Remembering the past and imagining the future. Philosophical Transactions of the Royal Society B, 362, 773–786.
Schacter, D. L., Addis, D. R., & Buckner, R. L. (2007). Remembering the past to imagine the future: The prospective brain. Nature Reviews Neuroscience, 8, 657–661.
Schacter, D. L., Addis, D. R., & Buckner, R. L. (2008). Episodic simulation of future events: Concepts, data, and applications. Annals of the New York Academy of Science, 1124, 39–60.
Schacter, D. L., Addis, D. R., Hassabis, D., Martin, V. C., Spreng, R. N., & Szpunar, K. K. (2012). The future of memory: Remembering, imagining, and the brain. Cell: Neuron Review, 76(4), 677–694.
Schoenfeld, A. (1985). Mathematical problem solving. New York, NY: Academic Press.
Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 334–370). New York, NY: Macmillan.
Silver, E. (1985). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.
Taylor, S. E., Pham, L. B., Rivkin, I. D., & Armor, D. A. (1998). Harnessing the imagination. Mental simulation, self-regulation, and coping. American Psychologist, 53(4), 429–439.
Tulving, E. (1983). Elements of episodic memory. Oxford: Oxford University Press.
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Maciejewski, W. (2019). Future-Oriented Thinking and Activity in Mathematical Problem Solving. In: Liljedahl, P., Santos-Trigo, M. (eds) Mathematical Problem Solving. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-030-10472-6_2
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