# PROCESSES AND REASONING IN REPRESENTATIONS OF LINEAR FUNCTIONS

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## ABSTRACT

This study examined student actions, interpretations, and language in respect to questions raised regarding tabular, graphical, and algebraic representations in the context of functions. The purpose was to investigate students’ interpretations and specific ways of working within table, graph, and the algebraic on notions fundamental to a conceptualization of linear functions. Through a case study method which investigated individual representations and student articulations within them, the study revealed that students can make a transition from a given representation of linear function to another and yet demonstrate limited understanding of linear functions.

## KEY WORDS

correspondence functions linear functions rate of change representations## Preview

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## References

- Ainsworth, S. (1999). The functions of multiple representations.
*Computers in Education, 33*(2–3), 131–152.CrossRefGoogle Scholar - Bogdan, R. C. & Biklen, S. K. (2003).
*Qualitative research for education: An introduction to theories and methods*(4th ed.). Boston: Allyn and Bacon.Google Scholar - Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education.
*Journal for Research in Mathematics Education, 23*(1), 2.CrossRefGoogle Scholar - Deloache, J. S. (1989). The development of representation in young children. In H. W. Reese (Ed.),
*Advances in child development and behavior*(Vol. 22, pp. 1–39). New York: Academic.Google Scholar - Dufour-Janvier, B., Bednarz, N. & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 109–122). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking: Basic issues for learning. In F. Hitt & M. Santos (Eds.),
*Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(pp. 3–26). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar - Duval, R. (2002). The cognitive analysis of problems of comprehension in the learning of mathematics.
*Mediterranean Journal for Research in Mathematics Education, 1*(2), 1–16.Google Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics.
*Educational Studies in Mathematics, 61*, 103–131.CrossRefGoogle Scholar - Eisenberg, T. (1992). On the development of a sense for functions. In G. Harel & E. Dubinsky (Eds.),
*The concept of function: Aspects of epistemology and pedagogy*(Vol. 25, pp. 153–174). Washington, DC: Mathematical Association of America.Google Scholar - Even, R. (1998). Factors involved in linking representations of functions.
*The Journal of Mathematical Behavior, 17*(1), 105–121.CrossRefGoogle Scholar - Freudenthal, H. (1983).
*Didactical phenomenology of mathematical structures*. Dordrecht, the Netherlands: D. Reidel.Google Scholar - Goldin, G. A. (1987). Cognitive representational systems for mathematical problem solving. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 125–145). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Janvier, C. (1987). Translation process in mathematics education. In C. Janvier (Ed.),
*Problems of representation in mathematics learning and problem solving*(pp. 27–31). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Kaput, J. (1987). Representation systems and mathematics. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 19–26). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.),
*Research issues in the learning and teaching of algebra*(pp. 167–194). Hillsdale, NJ: Erlbaum.Google Scholar - Keller, B. A. & Hirsch, C. R. (1998). Student preferences for representations of functions.
*International Journal of Mathematical Education in Science and Technology, 29*(1), 1–17.CrossRefGoogle Scholar - Leinhardt, G., Zaslavsky, O. & Stein, K. M. (1990). Functions, graphs and graphing: Tasks, learning and teaching.
*Review of Educational Research, 60*(1), 1–64.Google Scholar - Lesh, R., Post, T. & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.),
*Problems of representations in the teaching and learning of mathematics*(pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Miles, M. B. & Huberman, A. M. (1994).
*Qualitative data analysis*(2nd ed., pp. 10–12). Newbury Park, CA: Sage.Google Scholar - Monk, G. S. (1989). A Framework for describing student understanding of functions: Department of mathematics. Paper presented at the annual meeting of the American Educational Research Association. San Francisco, March 27–31, 1989.Google Scholar
- Porzio, D. T. (1999). Effects of differing emphasis on the use of multiple representations and technology on students’ understanding of calculus concepts.
*Focus on Learning Problems in Mathematics, 21*(3), 1–29.Google Scholar - Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification—the case of function. In G. Harel & E. Dubinsky (Eds.),
*The concept of function: Aspects of epistemology and pedagogy*(pp. 59–84). Washington, DC: Mathematical Association of America. MAA Notes 25.Google Scholar - Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky & G. Harel (Eds.),
*The concept of function: Aspects of epistemology and pedagogy*(Vol. 25, pp. 23–58). Washington, DC: Mathematical Association of America.Google Scholar - Steinbring, H. (2006). What makes a sign a mathematical sign?: An epistemological perspective on mathematical interaction.
*Educational Studies in Mathematics, 61*(1–2), 133–162.CrossRefGoogle Scholar - Sternberg, R. J. (1984). Mechanisms of cognitive development: A componential approach. In R. J. Stemberg (Ed.),
*Mechanisms of cognitive development*(pp. 163–186). New York: W. H. Freeman.Google Scholar - Tabachnek, H. J. M. & Simon, H. A. (1998). One person, multiple representations: An analysis of a simple, realistic multiple representation learning task. In M. van Someren, P. Reimann, H. A. P. Boshuizen & T. de Jong (Eds.),
*Learning with multiple representations*(pp. 197–236). Oxford: Pergamon.Google Scholar - Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function.
*Journal for Research in Mathematics Education, 20*(4), 356–366.CrossRefGoogle Scholar - Yerushalmy, M. (1991). Students perceptions of aspects of algebraic function using multiple representation software.
*Journal of Computer Assisted Learning, 7*, 42–57.CrossRefGoogle Scholar - Yerushalmy, M. (2000). Problem solving strategies and mathematical resources: A longitudinal view on problem solving in a function based approach to algebra.
*Educational Studies in Mathematics, 43*(2), 125–147.CrossRefGoogle Scholar - Zhang, J. (1997). The nature of external representations in problem solving.
*Cognitive Science, 21*(2), 179–217.CrossRefGoogle Scholar

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