Abstract
In the course of the last few years, we have investigated shifts of knowledge among different settings in inquiry-based mathematics classrooms: the individual, the small group and the whole class community. The different theoretical perspectives we used for analysing group work and for analysing whole class discussions, and the empirical data, led us to hypothesize links between shifts of knowledge and students’ creative reasoning. Therefore, the goal of the current study is to investigate creative reasoning within the shifts of knowledge in an inquiry-based classroom. Specifically, we ask: What is the role of creative mathematical reasoning in the shifts of knowledge between the knowledge agents and their followers in the classroom? To this end, we analysed a whole class discussion and the subsequent work of a small group. Our findings show that creative reasoning has a role in researchers’ characterization of shifts of knowledge in the classroom. In particular, we found that the students who expressed creative reasoning all had followers and thus became knowledge agents, while students’ contributions that were not characterized as creative were not always followed up. Finally, in cases where both, the knowledge agent and the follower expressed creative ideas, we named these ideas milestones.
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Notes
As a reminder, by downloading of knowledge, we mean shifting of knowledge from a whole class discussion into a group discussion.
References
Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83, 37–55.
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of Learning Sciences, 10(1&2), 113–163.
Dreyfus, T., Hershkowitz, R., & Schwarz, B. (2015). The nested epistemic actions model for abstraction in context—theory as methodological tool and methodological tool as theory. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 185–217). Dordrecht: Springer, Advances in Mathematics Education series.
Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behavior, 37, 48–62.
Hadas, N., Hershkowitz, R., & Ron, G. (2008). Instructional design and research-design principles in probability. In M. Kourkoulos & C. Tzanakis (Eds.), Proceedings of the 5th international colloquium on the didactics of mathematics (pp. 249–260). Rethymnon, Crete, Greece: The University of Crete.
Hayne, N., & Tabach, M. (2014). When stories and multiple solution tasks meet in a technological environment: The case of equality. Mediterranean Journal for Research in Mathematics Education, 12, 23–37.
Hershkowitz, R., & Schwarz, B. B. (1999). Reflective processes in a technology-based mathematics classroom. Cognition and Instruction, 17, 65–91.
Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32, 195–222.
Hershkowitz, R., Tabach, M., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts in a probability classroom: a case study coordinating two methodologies. ZDM—The International Journal on Mathematics Education, 46, 363–387.
Kattou, M., Kontoyianni, K., Pitta-Pantazi, D., & Christou, C. (2013). Connecting mathematical creativity to mathematical ability. ZDM—The International Journal on Mathematics Education, 45, 167–181.
Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Rotterdam, the Netherlands: Sense Publisher.
Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st international conference for the psychology of mathematics education (Vol. 3, pp. 161–168). Korea: The Korea Society of Educational Studies in Mathematics.
Leikin, R., & Lev, M. (2013). On the connections between mathematical creativity and mathematical giftedness in high school students. ZDM—The International Journal on Mathematics Education, 45, 183–197.
Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: The state of the art. ZDM—The International Journal on Mathematics Education, 45, 159–166.
Levenson, E. (2011). Exploring collective mathematical creativity in elementary school. Journal of Creative Behavior, 45, 215–234.
Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67, 255–276.
Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM—The International Journal on Mathematics Education, 40, 165–178.
Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 195–215). London: Taylor & Francis.
Saxe, G. B., Gearhart, M., Shaughnessy, M., Earnest, D., Cremer, S., Sitabkhan, Y., et al. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 203–222). London: Routledge.
Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: new questions and directions. Educational Studies in Mathematics, 83, 1–7. doi:10.1007/s10649-013-9478-2.
Tabach, M., & Friedlander, A. (2013). School mathematics and creativity at the elementary and middle grade level: How are they related? ZDM—The International Journal on Mathematics Education, 45, 227–238. doi:10.1007/s11858-012-0471-5.
Tabach, M., Hershkowitz, R., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts and knowledge agents in the classroom. Journal of Mathematical Behavior, 33, 192–208.
Torrance, E. P. (1974). Torrance tests of creative thinking. Lexington: Ginn.
Tsamir, P., Tirosh, D., Tabach, M., & Levenson, E. (2010). Multiple solution methods and multiple outcomes: Is it a task for kindergarten children? Educational Studies in Mathematics, 73, 217–231.
Van Harpen, X. Y., & Presmeg, N. (2013). An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge. Educational Studies in Mathematics, 83, 117–132.
Voica, C., & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. ZDM—The International Journal on Mathematics Education, 45, 267–279.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press.
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This study was partially supported by the Israel Science Foundation (Grants Nos. 1057/12 and 438/15).
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Hershkowitz, R., Tabach, M. & Dreyfus, T. Creative reasoning and shifts of knowledge in the mathematics classroom. ZDM Mathematics Education 49, 25–36 (2017). https://doi.org/10.1007/s11858-016-0816-6
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DOI: https://doi.org/10.1007/s11858-016-0816-6