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Dissecting success stories on mathematical problem posing: a case of the Billiard Task

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Abstract

“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.

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Notes

  1. This section is an abridged and modified version of a section in Kontorovich, Koichu, Leikin, and Berman (2012).

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Acknowledgments

The research of the first-named author was supported by the Loewengart Research Fund. The research of the second-named author was supported by the Technion graduate school. The authors are grateful to the participants in our study for their time, creativity, and readiness to collaborate.

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Authors and Affiliations

  1. Department of Education in Technology and Science, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Boris Koichu & Igor Kontorovich

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  1. Boris Koichu
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  2. Igor Kontorovich
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Correspondence to Boris Koichu.

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Koichu, B., Kontorovich, I. Dissecting success stories on mathematical problem posing: a case of the Billiard Task. Educ Stud Math 83, 71–86 (2013). https://doi.org/10.1007/s10649-012-9431-9

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