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Effects of variations in task design on mathematics teachers’ learning experiences: a case of a sorting task

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Abstract

The goal of the study presented in this article was to examine how variations in task design may affect mathematics teachers’ learning experiences. The study focuses on sorting tasks, i.e., learning tasks that require grouping a given set of mathematical items, in as many ways as possible, according to different criteria suggested by the learners. We present an example of a sorting task for which the items to be grouped are related to basic concepts of analytical geometry that are connected to the notion of loci of points. Based on a design experiment of three iterations with practicing secondary school mathematics teachers, we report on intended and enacted objects of learning inherent in three versions of the task. Empirically based suggestions are drawn about design of sorting tasks that potentially evoke desirable learning experiences.

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Notes

  1. A short version of this article was presented at ICMI Study 22 Conference “Task design in mathematics education,” Oxford, UK, July 2013 (Koichu et al. 2013b).

  2. Note that, according to Goldin (1998), representation and representational system are conventional rather than fully formalized constructs and, as such, they unavoidably bear some extent of ambiguity.

  3. The term “pathological item” is used in the meaning compatible with the meaning assigned to the term “pathological example,” an example specifically designed to violate certain almost universal properties (e.g., see WolframMathWorld website http://mathworld.wolfram.com/Pathological.html).

References

  • Askew, M., Brown, M., Rhodes, V., Wiliam, D., & Johnson, D. (1997). Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency. London: King’s College, University of London.

    Google Scholar 

  • Bell, A. (1993). Principles for the design of teaching. Educational Studies in Mathematics, 24, 5–34.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Cooney, T. J., & Krainer, K. (1996). Inservice mathematics teacher education: The importance of listening. In A. J. Bishop, et al. (Eds.), International handbook of mathematics education (pp. 1155–1185). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Dolev, L. (2012). Considerations in the design of learning tasks for teachers: The case of classification of mathematical items. Unpublished Ph.D. dissertation. Technion – Israel Institute of Technology.

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 101–131.

    Article  Google Scholar 

  • English, L. D., & Sharry, P. V. (1996). Analogical reasoning and the development of algebraic abstraction. Educational Studies in Mathematics, 30, 135–157.

    Article  Google Scholar 

  • Friedlander, A., & Dreyfus, T. (1991). Is the graph of y = kx straight? The Mathematics Teacher, 84(7), 526–531.

    Google Scholar 

  • Glaister, E. M., & Glaister, P. (2006). Introducing conics without eccentricity. International Journal of Mathematical Education in Science and Technology, 37(2), 235–245.

    Article  Google Scholar 

  • Goldin, G. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.

    Article  Google Scholar 

  • Goren, B. (2006). Mathematics (for 5 credit point curriculum). Part 7 (questionnaire 035007). Tel Aviv: Author (in Hebrew).

  • Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders. Singapore: World Scientific Publishing.

    Google Scholar 

  • Howard, A. (1988). Calculus: With analytic geometry. New York, NY: Wiley.

    Google Scholar 

  • Koichu, B., Berman, A., & Moore, M. (2007). Heuristic literacy development and its relation to mathematical achievements of middle school students. Instructional Science, 35, 99–139.

    Article  Google Scholar 

  • Koichu, B., & Kontorovich, I. (2013). Dissecting success stories on mathematical problem posing: A case of the Billiard Task. Educational Studies in Mathematics, 83(1), 71–86.

    Article  Google Scholar 

  • Koichu, B., Harel, G., & Manaster, A. (2013a). 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 

  • Koichu, B., Zaslavsky, O., & Dolev, L. (2013b). Effects of variations in task design using different representations of mathematical objects on learning: A case of a sorting task. In C. Margolinas, J. Ainley, J. B. Frant, M. Doorman, C. Kieran, A. Leung, M. Ohtani, P. Sullivan, D. Thompson, A. Watson, & Y. Yang (Eds.), Proceedings of ICMI Study 22 Task Design in Mathematics Education (pp. 463–472). Oxford: Oxford University Press.

    Google Scholar 

  • Lai, M. Y., & Murray, S. (2012). Teaching with procedural variation: A Chinese way of promoting deep understanding of mathematics. International Journal for Mathematics Teaching and Learning. http://www.cimt.plymouth.ac.uk/journal/lai.pdf. Retrieved 18 Nov 2013.

  • Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task. Journal of Mathematics Teacher Education, 10, 239–249.

    Article  Google Scholar 

  • Ling, L. M., & Marton, F. (2012). Towards a science of the art of teaching: Using variation theory as a guiding principle of pedagogical design. International Journal for Lesson and Learning Studies, 1(1), 7–22.

    Article  Google Scholar 

  • Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.

    Article  Google Scholar 

  • NCTM (National Council of Teachers of Mathematics). (2000). Principles and standards for teaching mathematics. Reston, VA: NCTM.

    Google Scholar 

  • Runesson, U. (2005). Beyond discourse and interaction. Variation: A critical aspect for teaching and learning mathematics. The Cambridge Journal of Education, 35(1), 69–87.

    Article  Google Scholar 

  • Santos-Trigo, M. (2007). Mathematical problem solving: An evolving research and practice domain. ZDM, 39, 523–536.

    Article  Google Scholar 

  • Simon, M., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91–104.

    Article  Google Scholar 

  • Sun, X. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76, 65–85.

    Article  Google Scholar 

  • Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: A design research study. Journal of Mathematics Teacher Education, 10, 217–237.

    Article  Google Scholar 

  • Swan, M. (2011). Designing tasks that challenge values, beliefs and practices: A model for the professional development of practicing teachers. In O. Zaslavsky & P. Sullivan (Eds.), Constructing knowledge for teaching secondary mathematics. New York: Springer.

    Google Scholar 

  • Thanheiser, E. (2014). Developing prospective teachers’ conceptions with well-designed tasks: Explaining successes and analyzing conceptual difficulties. Journal of Mathematics Teacher Education,. doi:10.1007/s10853-014-9272-9.

    Google Scholar 

  • Tirosh, D., & Wood, T. (Eds.). (2008). The international handbook of mathematics teacher education (Vol. 2). Rotterdam: Sense Publishers.

    Google Scholar 

  • Tzur, R. (2007). Fine grain assessment of students’ mathematical understanding: Participatory and anticipatory stages in learning a new mathematical conception. Educational Studies in Mathematics, 66, 273–291.

    Article  Google Scholar 

  • Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.

    Article  Google Scholar 

  • Wiersma, W. (1995). Research methods in education (6th ed.). Massachusetts: Allyn and Baker.

    Google Scholar 

  • Zaslavsky, O. (2008a). Attention to similarities and differences: A fundamental principle for task design and implementation in mathematics education. Presented at the Topic Study Group on Task Design and Analysis (TSG34) at ICME-11. http://tsg.icme11.org/document/get/290. Retrieved 19 July 2012.

  • Zaslavsky, O. (2008b). Meeting the challenges of mathematics teacher education through design and use of tasks that facilitate teacher learning. In T. Wood, P. Sullivan, D. Tirosh, K. Krainer, & B. Jaworski (Eds.), The international handbook on mathematics teacher education, 4 (pp. 93–114). The Netherlands: Sense Publishers.

    Google Scholar 

  • Zaslavsky, O., & Leikin, R. (2004). Professional development of mathematics teacher educators: Growth through practice. Journal of Mathematics Teacher Education, 7(1), 5–32.

    Article  Google Scholar 

  • Zaslavsky, O., & Sullivan, P. (Eds.). (2011). Constructing knowledge for teaching: Secondary mathematics tasks to enhance prospective and practicing teacher learning. New York: Springer.

    Google Scholar 

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Authors and Affiliations

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

    Boris Koichu, Orit Zaslavsky & Lea Dolev

  2. New York University, New York City, NY, USA

    Orit Zaslavsky

  3. ORT Israel, Tel Aviv, Israel

    Lea Dolev

Authors
  1. Boris Koichu
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  2. Orit Zaslavsky
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  3. Lea Dolev
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Correspondence to Boris Koichu.

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Koichu, B., Zaslavsky, O. & Dolev, L. Effects of variations in task design on mathematics teachers’ learning experiences: a case of a sorting task. J Math Teacher Educ 19, 349–370 (2016). https://doi.org/10.1007/s10857-015-9302-2

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