Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking

Samuel Obara

doi:10.29333/ejmste/109262

Full Text (PDF)

Samuel Obara ¹ ^*

More Detail

¹ Texas State University, San Marcos, Texas, USA^* Corresponding Author

Abstract

This paper explores how a group of pre-service elementary school teachers training to become mathematics teachers for elementary schools arrived at generalizations based on patterns. Two representative problems were investigated with these preservice teachers. The focus of this study was how these preservice teachers analyze and symbolize algebraically their generalizations during a problem-solving process. The results indicate that the preservice teachers had difficulty making use of input-output (having two variables in the table) relationships in a generalization process associated with developing symbolic functions. This study identifies the crucial need for introducing students to pattern activities early on in their lives.

Keywords

algebraic
generalization
patterns
proof
reasoning
variable

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2019, Volume 15, Issue 11, Article No: em1763

https://doi.org/10.29333/ejmste/109262

Publication date: 09 May 2019

Article Views: 2031

Article Downloads: 1311

Open Access References How to cite this article

References

Allen, J. (2013). Updating the ACT College Readiness Benchmarks. ACT Research Report Series 2013 (6). ACT, Inc.
Amit, M., & Neria, D. (2008). “Rising to the challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM, 40(1), 111-129. https://doi.org/10.1007/s11858-007-0069-5.
Arzarello, F. (1992). Praic pre-algebraic problem solving In J. P. Ponte, J. F. Matos, J. M. Matos & D. Fernandes (Eds.), Mathematical problem solving and new information technologies (Vol. 89, pp. 155-166). Berlin: Springer-Verlag. https://doi.org/2010.1007/978-3-642-58142-7.
Bay-Williams, J. M., Skipper, E. M., & Eddins, S. K. (2004). Developing a well-articulated algebra curriculum: Examples from the NCTM academy for professional development. In R. N. Rubenstein & G. W. Bright (Eds.), Perspective on the teaching of mathematics: Sixty-sixth yearbook of the National Council of Teachers of Mathematics yearbook (pp. 15-26). Reston, VA: National Council of Teachers of Mathematics.
Billings, E. (2008). Exploring generalization through growth patterns. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 279-293). Reston, VA: National Council of Teachers of Mathematics.
Billstein, R., Libeskind, S., & Lott, J. W. (2007). Problems solving approach to mathematics for elementary school teachers (9th ed.). Boston: Pearson Eductional, Inc.
Brown, T. (2001). Mathematics education and language: Interpreting hermeneutics and post-structuralism. Dordrecht, The Netherlands: Kluwer Academic. https://doi.org/10.1007/978-94-010-0726-9.
Burgis, K., & Morford, J. (2008). Investigating college algebra with technology. Oakland, CA: Key curriculum press.
Chazan, D. (2008). The shifting landscape of school algebra in the United States. In C. Greenes & R. N. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 19-25). Reston, VA: National Council of Teachers of Mathematics.
Cuevas, G. J., & Yeatts, K. (2001). Navigating through algebra in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Dienes, Z. P. (1961). On abstraction and generalization. Harvard Educational Review, 3, 289-301.
Dreyfus, T. (1991). Advanced mathematical thinking process. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25-41). Dordrecht, The Netherland: Kluwer. https://doi.org/10.1017/CBO9781139013499.008.
Driscoll, M. J. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10. Portsmouth, N.H.: Heinemann.
Greenberg, J., & Walsh, K. (2008). No common denominator: The preparation of elementary teachers in mathematics by America’s education schools. Retrieved from http://nctq.org.
Healy, L., & Hoyles, C. (1999). Visual and symbolic reasoning in mathematics: Making connections with computers. Mathematical Thinking and Learning, 1, 59-84. https://doi.org/10.1207/s15327833mtl0101_3.
Kaput, J. (2000). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. Dartmouth, MA: National Center for Improving Student Learning and Achievement in Mathematics and Science.
Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-18). New York: Lawrence Erlbaum Associates.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390 - 419). New York: Macmillan.
Kilpatrick, J., & Izsak, A. (2008). A history of algebra in the school curriculum. In C. Greenes (Ed.), Algebra and algebraic thinking in school mathematics: 2008 NCTM yearbook (pp. 3-18). Reston, VA: NCTM.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258. https://doi.org/10.1207/s15327833mtl0703_3.
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. 87-106). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-009-1732-3_6.
Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification (Research report). Montreal: Concordia University, Mathematics Department.
Malara, N., & Navarra, G. (2003). April Project: Arithmetic pathways towards favoring pre-algebraic thinking. Bologna: Pitagora Editrice.
Mason, J. (1996). Expressing generality and roots of algebra. In L. Lee (Ed.), Approaches to algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht, The Netherlands: Kluwer Academic. https://doi.org/10.1007/978-94-009-1732-3_5.
McCrory, R., Floden, R., Ferrini-Mundy, J., Reckase, M. D., & Senk, S. L. (2012). Knowledge of algebra for teaching: A framework of knowledge and practice. Journal for Research in Mathematics Education, 43(5), 584-615. https://doi.org/10.5951/jresematheduc.43.5.0584.
National Council of Teachers of Mathematics (2000). Principles and standard for school mathematics: Reston, VA: Author.
Patton, M. Q. (2015). Qualitative evaluation and research methods (p. 5321990). Newbury Park, CA: Sage.
Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237-268. https://doi.org/10.1023/A:1017530828058.
Smith, E. (2003). Stasis and change: Integrating pattern, functions, and algebra throughout the K-12 curriculum. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 135-150). Reston, VA: National Council of Teachers of Mathematics. https://doi.org/10.1002/jsc.645.
Smith, E. (2008). Representational thinking as a framework. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133-160). New York: Lawrence Erlbaum Associates.
Spradley, J. P. (1980). Participant Observation. Chicago: Holt, Rinehart, and Winston.
Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147-164. https://doi.org/10.1007/BF00579460.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-307). Mahwah, NJ: Erlbaum.
Stephens, A. (2008). What “counts” as algebra in the eyes of preservice elementary teachers? Mathematical Behavior, 27, 33-47. https://doi.org/10.1016/j.jmathb.2007.12.002.
Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equation to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112. https://doi.org/10.2307/749821.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press. https://doi.org/10.4324/9780203454220.
Warren, E. (2000). Learning comparative mathematical language in the elementary school: A longitudinal study. Educational Studies in Mathematics, 62(2), 169-189. https://doi.org/10.1007/s10649-006-4627-5.
Warren, E. (2006, July). Teacher actions that assist young students write generalizations in words and in symbols. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 377-384).
Yin, R. K. (2017). Case study research and applications: Design and methods. Sage publications.
Zazkis, R., & Liljedehl, P. (2002). Generalization of patterns: The tension between algebra thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402. https://doi.org/10.1023/A:1020291317178.

How to cite this article

APA

Obara, S. (2019). Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking. Eurasia Journal of Mathematics, Science and Technology Education, 15(11), em1763. https://doi.org/10.29333/ejmste/109262

Vancouver

Obara S. Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking. EURASIA J Math Sci Tech Ed. 2019;15(11):em1763. https://doi.org/10.29333/ejmste/109262

AMA

Obara S. Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking. EURASIA J Math Sci Tech Ed. 2019;15(11), em1763. https://doi.org/10.29333/ejmste/109262

Chicago

Obara, Samuel. "Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking". Eurasia Journal of Mathematics, Science and Technology Education 2019 15 no. 11 (2019): em1763. https://doi.org/10.29333/ejmste/109262

Harvard

Obara, S. (2019). Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking. Eurasia Journal of Mathematics, Science and Technology Education, 15(11), em1763. https://doi.org/10.29333/ejmste/109262

MLA

Obara, Samuel "Pre-service Teachers Exploring the Role of Pattern-based Reasoning in the Context of Algebraic Thinking". Eurasia Journal of Mathematics, Science and Technology Education, vol. 15, no. 11, 2019, em1763. https://doi.org/10.29333/ejmste/109262