Pre-service Elementary Teachers’ Reasoning Types of Generalization and Justification on a Figural Pattern Task

Year 2021, Volume: 8 Issue: 3, 422 - 440, 01.08.2021

Sümeyra Doğan Coşkun

https://doi.org/10.17275/per.21.74.8.3

Abstract

The purpose of this study is to examine how pre-service elementary teachers generalize a non-linear figural pattern task and justify their generalizations. More specifically, this study focuses on strategies and reasoning types employed by pre-service elementary teachers throughout generalization and justification processes. Data were collected from 32 pre-service elementary teachers who were enrolled in the Elementary Teacher Education program of a university, Turkey. During the data collection process, these pre-service teachers were first asked to generalize a non-linear figural pattern task and were then asked to justify their generalizations. To analyze the pre-service elementary teachers’ written answers for the task considering reasoning types for both generalization and justification, data reduction and constant comparative methodologies were used (Miles, Huberman, & Saldana, 2014). The findings indicated that the pre-service teachers were better able to find a rule for the pattern using the explicit strategy. It was also found that although these pre-service teachers used different types of reasoning which were numerical reasoning, figural reasoning, and pragmatic reasoning, figural reasoning was the most frequent one throughout the generalization process. Reasoning types for justification by the pre-service teachers fell into two categories: inductive and deductive. Most pre-service teachers resorted to inductive reasoning; however, there were a few pre-service teachers who referred to deductive reasoning. In addition, the pre-service teachers who articulated figural reasoning to generalize appeared to be more successful in justifying their developed rules deductively.

Keywords

figural pattern task, generalization, justification, reasoning

References

• Barbosa, A., & Vale, I. (2015). Visualization in pattern generalization: Potential and Challenges. Journal of the European Teacher Education Network, 10, 57-70.
• Becker, J. R., & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121-128). Melbourne, Australia: University of Melbourne.
• Becker, J. R., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 465-472). Prague, Czech Republic: Charles University.
• Becker, J. R., & Rivera, F. (2007). Factors affecting seventh graders’ cognitive perceptions of patterns involving constructive and deconstructive generalization. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 129-136). Seoul, Korea: PME
• Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization (pp. 5-23). Heidelberg, Germany: Springer Berlin Heidelberg.
• Chua, B. L., & Hoyles, C. (2012). The effect of different pattern formats on secondary two students’ ability to generalize. In T. Y. Tso (Ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 155-162). Taipei, Taiwan: PME.
• Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM, 40(1), 143-160.
• Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194-229.
• Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DRN based instruction. In S. Campbell & R. Zaskis (Eds.), Learning and teaching number theory, journal of mathematical behavior (pp. 185-212). New Jersey: Albex.
• Harel, G., & Tall, D. (1989). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38-42.
• Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805-842). Charlotte, NC: Information Age Publishing.
• Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennama & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp.133-155). Mahwah, NJ: Erlbaum.
• Kaput, J. (2000). Teaching and learning a new algebra with understanding. National Center for Improving Student Learning & Achievement in Mathematics & Science.
• Kaput, J., Blanton, M., & Moreno, L. (2008). Algebra from the symbolization point of view. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 19-56). Mahwah, NJ: Lawrence Erlbaum Associates/Taylor & Francis Group.
• Kilpatrick, J., & Izsák, A. (2008). A history of algebra in the school curriculum. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics: Seventieth Yearbook (pp. 3-18). Reston, VA: National Council of Teachers of Mathematics.
• Kinach, B. M. (2014). Generalizing: The core of algebraic thinking. The Mathematics Teacher, 107(6), 432-439.
• Kirwan, J. V. (2017). Using Visualization to Generalize on Quadratic Patterning Tasks. Mathematics Teacher, 110(8), 588-593.
• Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258.
• Lannin, J., Barker, D., Townsend, B. (2006). Algebraic generalization strategies: Factors influencing student strategy selection. Mathematics Education Research Journal, 18(3), 3-28.
• Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Lo, J. J., Grant, T. J., & Flowers, J. (2008). Challenges in deepening prospective teachers’ understanding of multiplication through justification. Journal of Mathematics Teacher Education, 11(1), 5-22.
• Markworth, K. (2012). Growing patterns: seeing beyond counting. Teaching Children Mathematics, 19(4), 254-262.
• Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Miles, M., Huberman, A., & Saldaña, J. (2014). Qualitative Data Analysis: A Methods Sourcebook. Thousand Oaks, CA: Sage.
• National Council of Teachers of Mathematics (NCTM). (2000). Learning mathematics for a new century (2000 Yearbook). Reston, VA: Author.
• Patton, M. Q. (2002). Qualitative evaluation and research methods (3rd Ed.). Thousand Oaks, CA: Sage.
• Radford, L. (1996). Some reflections on teaching algebra through generalization. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 107-111). Dordrecht, The Netherlands: Kluwer.
• Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37-70.
• Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM, 40(1), 83-96.
• Richardson, K., Berenson, S., & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2-3), 188-199.
• Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297-328.
• Rivera, F. D., & Becker, J. R. (2003). The effects of numerical and figural cues on the induction processes of preservice elementary teachers. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 4, pp. 63-70). Honolulu, HI: University of Hawaii.
• Rivera, F. D., & Becker, J. R. (2005). Figural and numerical modes of generalizing in algebra. Mathematics Teaching in the Middle School, 11(4), 198-203.
• Rivera, F. D., & Becker, J. (2009). Algebraic reasoning through patterns. Mathematics Teaching in the Middle School, 15(4), 212-221.
• Smith, E. (2003). Stasis and change: Integrating patterns, functions and algebra throughout the k-12 curriculum. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 136-150). Reston, VA: National Council of Teachers of Mathematics.
• Warren, E. (2005). Young children’s ability to generalize the pattern rule for growing patterns. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 305-312). Melbourne: PME.
• Warren & Cooper (2008). Patterns that support early algebraic thinking in elementary school. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (pp. 113-126). Reston, VA: National Council of Teachers of Mathematics.
• Whitin, D. J., & Whitin, P. (2014) Building squares and discovering patterns. Teaching Children Mathematics, 21(4), 210-219.
• Wilkie, K. J. (2014). Upper primary school teachers‟ mathematical knowledge for teaching functional thinking in algebra. Journal of Mathematics Teacher Education, 17(5), 397-428.
• Wilkie, K. J. & Clarke, D. M. (2014). Developing students’ functional thinking in algebra through different visualizations of a growing pattern’s structure. Mathematics Education Research Journal, 28(2), 223-243.
• Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379-402.