Preservice Elementary Teacher's Thinking about Algebraic Reasoning

Walter F. Castro Gordillo; Juan D. Godino

doi:10.29333/iejme/287

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Walter F. Castro Gordillo, Juan D. Godino

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Abstract

This paper reports on the results of a research about the comprehension of elementary algebraic reasoning exhibited by preservice elementary teachers. Such comprehension is studied by means of the elementary algebraic tasks that preservice teachers propose to be worked out by primary pupils, as well as their conceptions about including elementary algebraic reasoning in primary school curriculum. Based on the analysis of the tasks it stands out the procedural and numerical conception exhibited by the preservice teachers. Additionally they conceded little importance to the relational aspect of the equal sign. Some implications for teacher education are discussed.

Keywords

Early Algebra
Preservice Teachers
Algebraic Reasoning
Conceptions on Early Algebra

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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

INT ELECT J MATH ED, 2014, Volume 9, Issue 2, 147-162

https://doi.org/10.29333/iejme/287

Publication date: 12 Dec 2014

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References

Anderson, J. R. (2002). Spanning seven orders of magnitude: A challenge for cognitive modeling. Cognitive Science, 26, 85-112.
Asquith, P., Stephens, A., Knuth, E., & Alibali, M. (2007). Middle school mathematics teachers´ knowledge of students´ understanding of core algebraic concepts: equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272.
Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(4), 430-511.
Bastable, V., & Schifter, D. (2007). Classroom stories: Examples of elementary students engaged in early algebra. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades. Mahwah, NJ: Lawrence Erlbaum Associates.
Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem solving tool:continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to algebra (pp. 115-136). Dordrecht, The Netherlands: Kluwer Academics.
Bell, A. W. (1996). Problem-solving approaches to Algebra:Two aspects. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to Algebra. Perspectives for Research and Teaching (pp. 167-186). Dordrecht/Boston/London: Kluwer Academic Press.
Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equal sign. Mathematics Teaching, 92, 13-15.
Blanton, M. L., & Kaput, J. J. (2001). Algebrafying the elementary mathematics experience Part II. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), The future of the teaching and learning of algebra (Vol. 1, pp. 87-102). Melbourne: University of Melbourne, Australia.
Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers´ "algebra eyes and ears.". Teaching Children Mathematics, 10, 70-77.
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446.
Booth, L. R. (1984). Algebra: Children´s strategies and errors. A report of the strategies and errors in secondary mathematics project. Windsor, Berkshire: Nfer-Nelson.
Borko, H., Pittman, M., Eiteljorg, E., Nelson, M., & Jacobs, J. (2005). Preparing teachers to foster algebraic thinking. ZDM The International Journal on Mathematics Education, 37(1), 43-52.
Borko, H., & Shavelson, R. (1990). Teacher decision making. In B. F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 311-346). Hillsdale,NJ: Lawrence Erlbaum.
Boulton-Lewis, G. M., Cooper, T. J., Atweh, H. P., Wilss, L., & Mutch, S. (1997). The transition from arithmetic to algebra: a cognitive perspective. In E. Pehkonen (Ed.), 21th International Group for the Psychology of Mathematics Education (Vol. 2, pp. 185- 192). Latki, Finland.
Burkhardt, H. (2001). Algebra for all: What does it mean? How are we doing? In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), The future of the teaching and learning of algebra (Vol. 1, pp. 140-146). Melbourne: University of Melbourne, Australia.
Carpenter, T. P., & Franke, M. L. (2001). Developing algebraic reasoning in the elementary school: generalization and proof. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceeding of the 12th ICMI Study Conference. The Future of the Teaching and Learning of Algebra (Vol. 1, pp. 155-162). Melbourne: University of Melbourne.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH:Heinemann
Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades Retrieved February 24, 2006, from http://www.mth.msu.edu/cmp.
Carraher, D. W., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization. ZDM The International Journal on Mathematics Education, 40(1), 3- 22.
Carraher, D. W., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (Vol. 2, pp. 669-705). Reston: NCTM.
Chevallard, Y. (1989). Le passage de l´arithmétique à l´algébrique dans l´enseignement des mathématiques au collége. Petit x, 5, 51-94.
Clement, J. (1982). Algebra word problem solution through process underlying a common misconception. Journal for Research in Mathematics Education, 3(16- 30).
Cooney, T. J. (1985). A beginning teacher´s view of problem solving. Journal for Research in Mathematics Education, 16, 324-336.
Cooper, T. J., Williams, A. M., & Baturo, A. R. (1999). Simplifying algebraic expressions, Proceedings of the 1999 annual conference of the Australian Association for the Research in Education. Melbourne.
English, L., & Warren, E. (1998). Introducing the variable through pattern exploration. The Mathematics Teacher, 91(2), 166-171.
Ferrero, L; I, Gaztelou; Luelmo, M.J; Martin, P y Martinez, L.(2007) Matemáticas para sexto grado. Editorial Anaya.
Filloy, E., & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19-25.
Filloy, E., Rojano, T., & Rubio, G. (2001). Propositions concerning the resolution of arithmetic-algebraic problems. In R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds.), Perspectives on School Algebra (pp. 155-176). Dordrecht, The Netherlands: Kluwer Academic.
Freiman, V., & Lee, L. (2004). Tracking primary student´s understanding of the equality sign. In H. M. J. & A. B. Fuglestad (Eds.), Proceedings fo the International Group for the Psychology of Mathematics Education. Bergen, Norway.
Godino, J. D., Batanero, C., & Font, V. (2007). The ontos-semiotic approach to research in mathematics education. ZDM The International Journal on Mathematics Education, 19(1-2), 127-135.
Herscovics, N., & Kieran, C. (1980). Constructing meaning for the concept of equation. Mathematics Teacher, 73(8), 572-580.
Jacobs, V., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children´s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258-288.
Kaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by `algebrafying´ the K12 curriculum. Paper presented at the The nature and role of algebra in the K-14 curriculum, Washington, D.C.
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317-326.
Kieran, C., & Chalouh, L. (1993). The transition from arithmetic to algebra. In T. D. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 179-198). New York: MacMillan
Knuth, E., Stephens, A., McNeil, N., & Alibali, M. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297-312.
Kücheman, D. (1981). Algebra. In K. Hart (Ed.), Children´s understanding of mathematics:11- 16 (pp. 102-119). London: John Murray.
Linchevski, L., & Livneh, D. (1999). Structure sense: the relation between algebraic and numerical contexts. Educational Studies in Mathematics, 40(173-196).
MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebra learning. Journal for Research in Mathematics Education, 30(4), 449-467.
Mason, J., Pimm, D., Graham, A., & Gower, N. (1985). Routes to/Roots of Algebra: Milton Keynes:Open University.
McGowen, M. A., & Davis, G. (2001). Changing preservice elementary teachers´ attitudes to algebra. In H. Chick, Stacey, J. Vincent & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (Vol. 2, pp. 438-446). University of Melbourne, Australia.
McNeil, N., & Alibali, M. (2002). A strong schema can interfere with learning: The case of children´s typical addition schema. In C. D. Schunn & W. Gray (Eds.), Proceedings of the Twenty-Fourth Annual Conference of the Cognitive Science Society (pp. 661-666): Mahwah, NJ: Erlbaum.
McNeil, N., & Alibali, M. (2004). You´ll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28, 451-466.
McNeil, N., & Alibali, M. (2005). Why Won´t You Change Your Mind? Knowledge of Operational Patterns Hinders Learning and Performance on Equations. Child Development, 76(4), 883-899.
McNeil, N., Grandau, L., Knut, E., Alibali, M., Stephens, A., Hattikudur, S., et al. (2006). Middle-School Students´Understanding of the Equal Sign: The Books They Read Can´t Help. Cognition and instruction, 24(3), 367- 385.
Nathan, M., & Koedinger, K. (2000). An investigation of teachers´ beliefs of students´ algebra development. Cognition and Instruction, 18(2), 209-237.
MEC (2006). Real Decreto 1513/2006, BOE number 293
Nesher, P., & Teubal, E. (1975). Verbal cues as an interfering factor in verbal problem solving. Educational Studies in Mathematics, 6(1), 41-51.
Puig, L., & Cerdan, F. (1990). Acerca del caracter aritmético o algebraico de los problemas verbales. In E. Filloy & T. Rojano (Eds.), Simposio Internacional sobre Investigación en Educación Matemática (pp. 35-48). Cuernava: Morelos:PNFAPM.
Rojano, T. (1988). Teaching equation solving: A teaching experiment with a concretesyntactic approach. In M. Behr, C. Lacampagne & M. Wheeler (Eds.), Tenth International Group for the Psychology of Mathematics Education. North American Chapter (pp. 150-156). Dekalb, Illinois.
Schifter, D. (1999). Reasoning about operations: Early Algebraic Thinking, Grades K through 6. In L. V. Stiff & F. Curio (Eds.), Mathematical Reasoning, K-12:1999 NCTM Yearbook (pp. 62-81). Reston, VA: National Council of Teachers of Mathematics.
Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.
Simon, M. A., & Blume, G. W. (1994). Building and Understanding Multiplicative Relationships: A Study of Preservice Elementary Teachers. Journal for Research in Mathematics Education, 25(5), 472-494.
Sinha, S. M. (2006). Mathematical Programming:Theory and methods: Elsevier Inc.
Stephens, A. (2008). What "counts" as algebra in the eyes of preservice elementary teachers? Journal of Mathematical Behavior, 27, 33-47.
Stump, S. L., & Bishop, J. (2002). Preservice elementary and middle school teachers´ conceptions of algebra revealed throuhg the use of exemplary curriculum materials. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant & K. Nooney (Eds.), Proccedings of the Twenty-Fourth annual meeting of the International group for the psychology of mathematics education (pp. 1903-1914). Columbus,OH.
Subramaniam, K., & Banerjee, R. (2004). Teaching arithmetic and algebraic expressions. In M. Johnsen & A. Berit (Eds.), Proceedings of the 28th International Group for the Psychology of Mathematics Education (Vol. 3, pp. 121-128). Bergen, Noruega.
Van Dooren, W., Vershaffel, L., & Onghema, P. (2003). Preservice teachers´s preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6(27-52).
Warren, E. (2005). Young children´s ability to generalize the pattern rule for growing patterns. In H. Chick & J. Vincent (Eds.), 29 th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 305-312). Melbourne:University of Melbourne.
Williams, A. M., & Cooper, T. J. (2001). Moving from arithmetic to algebra under the time pressures of real classrooms. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceedings fo the 12th ICMI Study Conference: The future of the Teaching and Learning of Algebra (Vol. 2, pp. 665-662). Melbourne: University of Melbourne.

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APA

Gordillo, W. F. C., & Godino, J. D. (2014). Preservice Elementary Teacher's Thinking about Algebraic Reasoning. International Electronic Journal of Mathematics Education, 9(2), 147-162. https://doi.org/10.29333/iejme/287

Vancouver

Gordillo WFC, Godino JD. Preservice Elementary Teacher's Thinking about Algebraic Reasoning. INT ELECT J MATH ED. 2014;9(2):147-62. https://doi.org/10.29333/iejme/287

AMA

Gordillo WFC, Godino JD. Preservice Elementary Teacher's Thinking about Algebraic Reasoning. INT ELECT J MATH ED. 2014;9(2), 147-162. https://doi.org/10.29333/iejme/287

Chicago

Gordillo, Walter F. Castro, and Juan D. Godino. "Preservice Elementary Teacher's Thinking about Algebraic Reasoning". International Electronic Journal of Mathematics Education 2014 9 no. 2 (2014): 147-162. https://doi.org/10.29333/iejme/287

Harvard

Gordillo, W. F. C., and Godino, J. D. (2014). Preservice Elementary Teacher's Thinking about Algebraic Reasoning. International Electronic Journal of Mathematics Education, 9(2), pp. 147-162. https://doi.org/10.29333/iejme/287

MLA

Gordillo, Walter F. Castro et al. "Preservice Elementary Teacher's Thinking about Algebraic Reasoning". International Electronic Journal of Mathematics Education, vol. 9, no. 2, 2014, pp. 147-162. https://doi.org/10.29333/iejme/287