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Balancing competing demands: Enhancing the mathematical problem posing skills of prospective teachers through a mathematical letter writing initiative

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Abstract

Responding to mathematical problems is a core activity in classrooms. The problems that teachers select determine the mathematical content, processes and nature of mathematical inquiry occurring in classrooms and thereby contribute to the development of mathematical skills and dispositions. Selecting, designing or reformulating mathematical problems is a critical skill, then, for prospective and practising teachers. This study explores the influence of a mathematical letter writing initiative in developing the problem posing skills of 28 prospective primary teachers. We examine the characteristics of mathematical problems designed by prospective teachers, and their understandings of what constitutes a good mathematical problem, prior to and following completion of a 12-week letter writing initiative with 10–11-year-old children. Analysis of the data reveals the benefits of engaging in the initiative as evidenced in improvements in several problem characteristics. There was an increase in the number of multiple approach and multiple solution problems and in the level of cognitive demand of problems posed. The challenge of posing non-traditional problems, alongside the competing demands of building in opportunities for success, may have diminished participants’ ability to evaluate and attend to the cognitive demand of problems.

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References

  • Barab, S., & Squire, K. (2004). Design-Based Research: Putting a Stake in the Ground. The Journal of the Learning Sciences, 13(1), 1–14.

    Article  Google Scholar 

  • Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62.

    Article  Google Scholar 

  • Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37–55.

    Article  Google Scholar 

  • Chapman, O. (2008). Helping pre-service elementary teachers develop flexibility in using word problems in their teaching. Paper presented at the Annual meeting of the North American Chapter of the International group for the Psychology of Mathematics Education, Toronto.

  • Chapman, O. (2004). Instructional practices to facilitate prospective mathematics teachers’ learning of problem solving for teaching (pp. 158–167). Paper presented at ICME, Monterrey, Mexico. Accessed at http://www.matedu.cinvestav.mx/~rptec/Sitio_web/Documentos_files/tsg19icme11.pdf

  • Chapman, O. (2012). Prospective elementary school teachers’ ways of making sense of mathematical problem posing. PNA, 6(4), 135–146.

    Article  Google Scholar 

  • Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., & Whitenack, J. (1997). Mathematizing and symbolizing: The emergence of chains of signification on one first-grade classroom. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: Social, semiotic, and psychological perspectives (pp. 151–233). Lawrence Erlbaum.

  • Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in education research. Educational Researcher, 32(1), 9–13.

    Article  Google Scholar 

  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum.

  • Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52, 243–270.

    Article  Google Scholar 

  • Crespo, S. (2003). Using math penpal letters to promote mathematical communication. Teaching Children Mathematics, 10(1), 34–39.

    Article  Google Scholar 

  • Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415.

    Article  Google Scholar 

  • Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5), i–113.

    Article  Google Scholar 

  • Einstein, A., & Infeld, L. (1938). The evolution of physics from early concepts to relativity and quanta. Simon and Schuster.

  • Ellerton, N. F. (1988). Exploring children's perception of mathematics through letters and problems written by children. In A. Borbas (Ed.) Proceedings of the Twelfth International Conference for the Psychology of Mathematics Education. Veszprém (Hungary): International Group for the Psychology of Mathematics Education. (Vol. 1, pp. 280-287).

  • Ellerton, N. F. (2013). Engaging pre-service middle-school teacher-education students in mathematical problem posing: Development of an active learning framework. Educational Studies in Mathematics, 83, 87–101.

    Article  Google Scholar 

  • Freudenthal, H. (1973). Mathematics as an educational task. Reidel.

  • Geeslin, W. E. (1977). Using writing about mathematics as a teaching technique. Arithmetic Teacher, 70, 112–115.

    Google Scholar 

  • Guberman, R., & Leikin, R. (2013). Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teacher Education, 16, 33–56.

    Article  Google Scholar 

  • Kelly, A. E., Lesh, R. A., & Baek, J. Y. (Eds.). (2008). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching. Routledge.

  • Kennedy, W. (1985). Writing letters to learn math. Learning (Feb.), 59-60.

  • Koichu, B., Harel, G., & Manaster, A. (2013). Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions. Instructional Science, 41(4), 681–698.

    Article  Google Scholar 

  • Leavy, A. M., & Hourigan, M. (2020). Posing mathematically worthwhile problems: developing the problem-posing skills of prospective teachers. Journal of Mathematics Teacher Education, 23(4), 341–361.

    Article  Google Scholar 

  • Lee, J. E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15, 429–452.

    Article  Google Scholar 

  • Lesh, R., & Doerr, H. (Eds.). (2003). Beyond constructivism: models and modeling perspectives on mathematics problem solving learning and teaching. Lawrence Erlbaum Associates.

  • Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Information Age Publishing.

  • Leung, S. S. (1997). On the role of creative thinking in problem posing. Zentralbatt für Didaktik der Mathematik (ZDM), 29, 81–85.

    Article  Google Scholar 

  • Norton, A., & Kastberg, S. (2012). Learning to pose cognitively demanding tasks through letter writing. Journal of Mathematics Teacher Education, 15(2), 109–130.

    Article  Google Scholar 

  • Phillips, E., & Crespo, S. (1996). Developing Written communication in mathematics through math penpals. For the Learning of Mathematics, 16(1), 15–22.

    Google Scholar 

  • Polya, G. (1954). Mathematics and plausible reasoning. Princeton, New Jersey: Princeton University Press.

    Book  Google Scholar 

  • Polya, G. (1973). How to solve it: A new aspect of mathematics method. Princeton University Press.

  • Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.

  • Schoenfeld, A. H. (1983). Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography. The Mathematical Association of America.

  • Schoenfeld, A. H. (1989). Exploration of students’ mathematical beliefs and behaviours. Journal of Research in Mathematics Education, 20(4), 338–355.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: MacMillan.

    Google Scholar 

  • Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.

    Google Scholar 

  • Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75–80.

    Google Scholar 

  • Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.

    Article  Google Scholar 

  • Silver, E. A., Mamona-Downs, J., Leung, S., & Kenny, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.

    Article  Google Scholar 

  • Singer, F. M., & Voica, C. (2013). A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educational Studies in Mathematics, 83(1), 9–26.

    Article  Google Scholar 

  • Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM, 41, 557–567.

    Article  Google Scholar 

  • Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building Student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455–488.

    Article  Google Scholar 

  • Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275.

    Article  Google Scholar 

  • Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143.

    Article  Google Scholar 

  • Torrance, E. P. (1974). Torrance tests of creative thinking: Norms, technical manual. Ginn.

  • Van den Akker, J., Gravemeijer, K., McKenney, S., & Nieveen, N. (2006). Introduction to Educational design research. In J. Van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research. London: Routledge.

    Chapter  Google Scholar 

  • Van Harpen, X. Y., & Sriraman, B. (2013). Creativity and mathematical problem posing: An analysis of high school students’ mathematical problem posing in China and USA. Educational Studies in Mathematics, 82(2), 201–221.

    Article  Google Scholar 

  • Voica, C., & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. Zentralbatt für Didaktik der Mathematik (ZDM), 45, 267–279.

    Article  Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.

    Article  Google Scholar 

Download references

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  1. Department of STEM Education, Mary Immaculate College, South Circular Road, Limerick, Ireland

    Aisling Leavy & Mairéad Hourigan

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  1. Aisling Leavy
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  2. Mairéad Hourigan
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Correspondence to Aisling Leavy.

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Leavy, A., Hourigan, M. Balancing competing demands: Enhancing the mathematical problem posing skills of prospective teachers through a mathematical letter writing initiative. J Math Teacher Educ 25, 293–320 (2022). https://doi.org/10.1007/s10857-021-09490-8

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