# STUDENTS’ WAYS OF REASONING ABOUT NONLINEAR FUNCTIONS IN GUESS-MY-RULE GAMES

- 396 Downloads
- 1 Citations

## Abstract

There is a growing movement toward the introduction of algebra in early grades. This is supported by an increasing number of research studies that have reported success in getting young students to “do” mathematics considered beyond their reach. Yet, the consensus is that more research is needed to provide insights into the processes by which students make sense of algebraic ideas. In particular, more studies are needed on how nonlinear functions can be introduced in early grades. This article reports an algebra research strand that introduced seventh grade students to quadratic functions using Guess-My-Rule games. The article describes several instances of the students engaging successfully with ideas and forms of reasoning involving quadratic functions. The purpose is to contribute to the debate on the introduction of algebra in early grades by providing further evidence of young students’ ability to engage with algebraic ideas usually considered to be beyond their reach.

## Key words

algebraic reasoning generalization Guess-My-Rule games nonlinear functions quadratic functions## Preview

Unable to display preview. Download preview PDF.

## Supplementary material

## References

- Alston, A. & Davis, R. B. (1996).
*The development of algebraic ideas. Report of mathematics education seminar for teachers and educators*. New Brunswick, NJ: Rutgers University.Google Scholar - 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—The International Journal on Mathematics Education, 40*(1), 111–129.CrossRefGoogle Scholar - Anthony, G. & Hunter, J. (2008). Developing algebraic generalization strategies. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.),
*Proceedings of the joint meeting of PME32 and PME-NA XXX*(Vol. 2, pp. 65–72). Mexico: Cinvestav-UMSNH.Google Scholar - Bass, H., Howe, R. & Milgram, R. J. (2005). Analysis of NAEP items classified under the algebra and function content strand.
*Discussion panel at the conference on algebraic reasoning: Developmental, cognitive, and disciplinary foundations for instruction*.Google Scholar - Carpenter, T. P. & Frank, L. M. (2001). Developing algebraic reasoning in elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.),
*Proceedings of the twelfth ICMI study conference: The future of the teaching and learning of algebra*(Vol. 1, pp. 163–170). Melbourne, Australia: University of Melbourne Press.Google Scholar - Carpenter, T. P. & Levi, L. (2000).
*Developing conceptions of algebraic reasoning in the primary grades. (Res. Rep. 00-2)*. Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science.Google Scholar - Carraher, D. W., Brizuela, B. M. & Earnest, D. (2001a). The reification of additive differences in early algebra. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.),
*Proceedings of the twelfth ICMI study conference: The future of the teaching and learning of algebra*(Vol. 1, pp. 163–170). Melbourne: University of Melbourne Press.Google Scholar - Carraher, D. W. & Earnest, D. (2003). Guess my rule revisited. In N. A. Pateman, B. J. Dougherty & J. Zilliox (Eds.),
*Proceedings of the 2003 joint meeting of PME and PMENA*(Vol. 2, pp. 13–18). Honolulu, HI: College of Education, University of Hawaii.Google Scholar - Carraher, D. W., Martinez, M. & Schliemann, D. (2008). Early algebra and mathematical generalization.
*ZDM—The International Journal on Mathematics Education, 40*(1), 3–22.CrossRefGoogle Scholar - Carraher, D. W. & Schliemann, D. (2007). Early algebra and algebraic reasoning. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics*(pp. 669–705). Charlotte, NC: National Council of Teachers of Mathematics, Information Age.Google Scholar - Carraher, D. W., Schliemann, D., & Brizuela, M. B. (2000). Early algebra, early arithmetic: Treating operations as functions.
*Plenary address presented at the twenty-second annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*. Tucson, ArizonaGoogle Scholar - Carraher, D. W., Schliemann, D. & Brizuela, M. B. (2001b). Can students operate on unknowns? In M. V. D. Heuvel-Panhuizen (Ed.),
*Proceedings of the twenty-fifth conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 130–140). Utrecht, the Netherlands: Freudenthal Institute.Google Scholar - Carraher, D., Schliemann, A.D. & Brizuela, B. (2005). Treating operations as functions. In D. W. Carraher & R. Nemirovsky (Eds.),
*Monographs of the Journal for Research in Mathematics Education*, XIII, CD-Rom only issue.Google Scholar - Carraher, D. W., Schliemann, D., Brizuela, M. B. & Earnest, D. (2006). Arithmetic and algebra in early education.
*Journal for Research in Mathematics Education, 37*(2), 87–115.Google Scholar - Chazan, D. & Yerushalmy, M. (2003). On appreciating the cognitive simplicity of school algebra: Research on algebra learning and directions of curriculum change. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 123–135). Reston: National Council of Teachers of Mathematics, VA.Google Scholar - Cooper, T. & Warren, E. (2008). The effect of different representations on years 3 to 5 students’ ability to generalize.
*ZDM—The International Journal on Mathematics Education, 40*(1), 23–37.CrossRefGoogle Scholar - Davydov, V. (Ed.). (1969/1991).
*Soviet studies in mathematics education*:*Vol*.*6*.*Psychological abilities of primary school children in learning mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Ebersbach, M. & Wilkening, F. (2007). Children’s intuitive mathematics: The development of knowledge about nonlinear growth.
*Child Development, 78*(1), 296–308.CrossRefGoogle Scholar - Fey, J. T. & Good, R. A. (1985). Rethinking the sequence and priorities of high school mathematics curricula. In C. R. Hirsch (Ed.),
*Research issues in the learning and teaching of algebra*(Vol. 4 of Research agenda for mathematics education, pp. 199–213). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as tool.
*Journal for Research in Mathematics Education, 31*, 396–428.Google Scholar - Kaput, J. (1998). Transforming algebra from engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum.
*Paper presented at the 1995 National Council of Teachers of Mathematics meeting*.Google Scholar - Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics*(pp. 707–762). Charlotte, NC: National Council of Teachers of Mathematics, Information Age.Google Scholar - Lannin, J., Barker, D. & Townsend, B. (2006a). Recursive and explicit rules: How can we build student algebraic understanding?
*The Journal of Mathematical Behavior, 25*, 299–317.CrossRefGoogle Scholar - Lannin, J., Barker, D. & Townsend, B. (2006b). Algebraic generalization strategies: Factors influencing student strategy selection.
*Mathematics Education Research Journal, 18*(3), 3–28.CrossRefGoogle Scholar - Lee, L. (1996). An initiation into algebra culture through generalization. In N. Bednarz, C. Kieran & L. Lee (Eds.),
*Approaches to algebra: Perspectives for research and teaching*(pp. 87–106). Dordrecht, The Netherlands: Kluwer.Google Scholar - Lin, F.-L., Yang, K.-L. & Chen, C.-Y. (2004). The features and relationships of reasoning, proving and understanding proof in number patterns.
*International Journal of Science and Mathematics Education, 2*, 227–256.CrossRefGoogle Scholar - Martinez, M. & Brizuela, B. M. (2006). A third grader’s way of thinking about linear function tables.
*The Journal of Mathematical Behavior, 25*, 285–298.CrossRefGoogle Scholar - RAND Mathematics Study Panel (2003).
*Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education*. Santa Monica, CA: RAND Mathematics Study Panel.Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Pimm, D. (1995).
*Symbols and meanings in school mathematics*. London: Routledge.CrossRefGoogle Scholar - Powell, A. B., Francisco, J. M. & Maher, C. A. (2003). An evolving analytical model for understanding the development of mathematical thinking using videotape data.
*The Journal of Mathematical Behavior, 22*(4), 405–435.CrossRefGoogle Scholar - Radford, L. (2010). Layers of generality and types of generalization in pattern activities.
*PNA, 4*(2), 37–62.Google Scholar - Radford, L. & Demers, S. (2004).
*Communication et apprentissage. Repères conceptuels et practiques pour la sale de classe mathématiques*. Ottawa, ON, Canada: Centre Franco-Ontarien des Ressources Pédagogiques.Google Scholar - Rivera, F. & Becker, J. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns.
*ZDM—The International Journal on Mathematics Education, 40*(1), 65–82.CrossRefGoogle Scholar - Saul, M. W. (1998). Algebra, technology, and a remark of I. M. Guelfand. In the nature and role of algebra in the K-14 curriculum.
*Proceedings of a national symposium organized by the National Council of Teachers of Mathematics Board*,*and the National Research Council*(pp. 137–144). Washington, DC: National Academy Press.Google Scholar - Schliemann, A. D., Carraher, D. W. & Brizuela, B. M. (2001). When tables become function tables. In M. V. D. Heuvel-Panhuizen (Ed.),
*Proceedings of the twenty-fifth conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 145–152). Utrecht, The Netherlands: Freudenthal Institute.Google Scholar - Schwartz, J. & Yerushalmy, M. (1992). Getting students to function in and with algebra. In E. Dubisnky & G. Harel (Eds.),
*The concept of function: Aspects of epistemology and pedagogy (MAA notes)*(Vol. 25, pp. 261–289). Washington, DC: Mathematical Association of America.Google Scholar - Sfard, A. & Linchevsky, L. (1994). The gains and the pitfalls of reification.
*The case of algebra. Educational Studies in Mathematics, 26*, 191–228.Google Scholar - Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
*Journal for Research in Mathematics Education, 26*, 114–145.CrossRefGoogle Scholar - Swafford, J. & Langrall, C. (2000). Grade 6 students' preinstructional use of equations to describe and represent problem situations.
*Journal for Research in Mathematics Education, 31*(1), 89–112.CrossRefGoogle Scholar - Warren, E., Cooper, T. & Lamb, J. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning.
*The Journal of Mathematical Behavior, 25*, 208–223.CrossRefGoogle Scholar - Wheeler, D. (1996). Backwards and forwards: Reflections on different approaches to algebra. In N. Bednarz, C. Kieran & L. Lee (Eds.),
*Approaches to algebra: Perspectives for research and teaching*(pp. 317–325). Dordrecht, The Netherlands: Kluwer.Google Scholar