Early Algebraization pp 239-258 | Cite as

# Representational Competence and Algebraic Modeling

## Abstract

This chapter reviews some key empirical results and theoretical perspectives found in the past three decades of research on students’ capacities to reason with algebraic and graphical representations of functions. It then discusses two recent advances in our understanding of students’ developing capacities to use inscriptions for representing situations and solving problems. The first advance is the insight that students have criteria that they use for evaluating external representations commonly found in algebra, such as algebraic and graphical representations. Such criteria are important because they play a central role in learning. The second advance has to do with recognizing the importance of adaptive interpretation, which refers to ways in which students must coordinate shifts in their perspective on external representations with corresponding shifts in their perspective on problem situations. The term adaptive highlights the context sensitive ways in which students must learn to interpret external representations. The chapter concludes with implications of these two advances for future research and algebra instruction.

## Keywords

Word Problem Algebraic Representation Algebraic Modeling Equal Sign External Representation## Preview

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## References

- Booth, L. R. (1981). Child-methods in secondary mathematics.
*Educational Studies in Mathematics*,*12*, 29–41. CrossRefGoogle Scholar - Carraher, D., Schliemann, A., Brizuela, B., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education.
*Journal for Research in Mathematics Education*,*37*(2), 87–115. Google Scholar - Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization.
*ZDM—The International Journal of Mathematics Education*,*40*(1), 3–22. CrossRefGoogle Scholar - Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception.
*Journal for Research in Mathematics Education*,*13*(1), 16–30. CrossRefGoogle Scholar - Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing.
*Focus on Learning Problems in Mathematics*,*11*(2), 77–87. Google Scholar - Coxford, A., Fey, J., Hirsch, C., Schoen, H., Burrill, G., Hart, E., & Watkins, A. (with Messenger, M., & Ritsema, B.). (1998).
*Contemporary Mathematics in Context: A Unified Approach*. Chicago: Everyday Learning. Google Scholar - diSessa, A. A. (2002). Students’ criteria for representational adequacy. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.),
*Symbolizing, Modeling and Tool Use in Mathematics Education*(pp. 105–129). Dordrecht, The Netherlands: Kluwer Academic. Google Scholar - diSessa, A. A., & Sherin, B. (2000). Meta-representation: An introduction.
*Journal of Mathematical Behavior*,*19*, 385–398. CrossRefGoogle Scholar - diSessa, A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children.
*Journal of Mathematical Behavior*,*10*(2), 117–160. Google Scholar - Hall, R. (1990).
*Making mathematics on paper: Constructing representations of stories about related linear functions*(Technical Report No. 90-17). University of California, Irvine. Google Scholar - Hall, R., Kibler, D., Wenger, E., & Truxaw, C. (1989). Exploring the episodic structure of algebra story problem solving.
*Cognition and Instruction*,*6*(3), 223–283. CrossRefGoogle Scholar - Harper, E. (1987). Ghosts of diophantus.
*Educational Studies in Mathematics*,*18*, 75–90. CrossRefGoogle Scholar - Izsák, A. (2000). Inscribing the winch: Mechanisms by which students develop knowledge structures for representing the physical world with algebra.
*The Journal of the Learning Sciences*,*9*(1), 31–74. CrossRefGoogle Scholar - Izsák, A. (2003). “We want a statement that is always true”: Criteria for good algebraic representations and the development of modeling knowledge.
*Journal for Research in Mathematics Education*,*34*(3), 191–227. CrossRefGoogle Scholar - Izsák, A. (2004). Students’ coordination of knowledge when learning to model physical situations.
*Cognition and Instruction*,*22*(1), 81–128. CrossRefGoogle Scholar - Izsák, A., & Findell, B. (2005). Adaptive interpretation: Building continuity between students’ experiences solving problems in arithmetic and in algebra.
*Zentralblatt für Didaktik der Mathematik*,*37*(1), 60–67. CrossRefGoogle Scholar - Izsák, A., Çağlayan, G., & Olive, J. (2009). Teaching and learning to model word problems with algebraic equations.
*The Journal of the Learning Sciences*,*18*, 1–39. CrossRefGoogle Scholar - Janvier, C. (Ed.) (1987a).
*Problems of Representation in the Teaching and Learning of Mathematics*. Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Janvier, C. (1987b). Translation processes in mathematics education. In C. Janvier (Ed.),
*Problems of Representation in the Teaching and Learning of Mathematics*(pp. 27–32). Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Kaput, J. (1987). Towards a theory of symbol use in mathematics. In C. Janvier (Ed.),
*Problems of Representation in the Teaching and Learning of Mathematics*(pp. 159–195). Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Kaput, 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). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum. Google Scholar - Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glasersfeld (Ed.),
*Radical Constructivism in Mathematics Education*(pp. 53–74). Dordrecht: Kluwer Academic. Google Scholar - Kerslake, D. (1981). Graphs. In K. M. Hart (Ed.),
*Children’s Understanding of Mathematics: 11–16*(pp. 120–136). London: John Murray. Google Scholar - Kieran, C. (1981). Concepts associated with the equality symbol.
*Educational Studies in Mathematics*,*12*(3), 317–326. CrossRefGoogle Scholar - 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. Google Scholar - Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. Lester (Ed.),
*Second Handbook of Research on Mathematics Teaching and Learning*(pp. 707–762). Charlotte, NC: Information Age Publishing. Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001).
*Adding It Up: Helping Children Learn Mathematics*. Washington, DC: National Academy Press. Google Scholar - Knuth, E. (2000). Student understanding of the Cartesian connection: An exploratory study.
*Journal for Research in Mathematics Education*,*31*(4), 500–507. CrossRefGoogle Scholar - 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. Google Scholar - Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.),
*Children’s Understanding of Mathematics: 11–16*(pp. 102–119). London: John Murray. Google Scholar - Lannin, J. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities.
*Mathematical Thinking and Learning*,*7*(3), 231–258. CrossRefGoogle Scholar - Lappan, G., Fey, J. T., Fitzgerald, W., Friel, S. N., & Phillips, E. D. (2002).
*Connected Mathematics Series*. Glenview, IL: Prentice Hall. Google Scholar - Lappan, G., Fey, J. T., Fitzgerald, W., Friel, S. N., & Phillips, E. D. (2006).
*Connected Mathematics 2*. Boston, MA: Prentice Hall. Google Scholar - Leinhardt, G., Zaslavsky, O., & Stein, M. K. (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 Representation in the Teaching and Learning of Mathematics*(pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman & J. S. Brown (Eds.),
*Intelligent Tutoring Systems*(pp. 25–50). New York: Academic Press. Google Scholar - Meira, L. (1995). The microevolution of mathematical representations in children’s activities.
*Cognition and Instruction*,*13*, 269–313. CrossRefGoogle Scholar - Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematical activity.
*Journal for Research in Mathematics Education*,*29*, 121–142. CrossRefGoogle Scholar - Mokros, J. R., & Tinker, R. F. (1987). The impact of microcomputer-based labs on children’s ability to interpret graphs.
*Journal of Research in Science Teaching*,*24*(4), 369–383. CrossRefGoogle Scholar - Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In A. Schoenfeld, E. Dubinsky, & J. Kaput (Eds.),
*Research in Collegiate Mathematics Education*(Vol. 1, pp. 139–168). Washington, DC: American Mathematics Association. Google Scholar - Moschkovich, J. (1998). Resources for refining mathematical conceptions: Case studies in learning about linear functions.
*The Journal of the Learning Sciences*,*7*(2), 209–237. CrossRefGoogle Scholar - National Council of Teachers of Mathematics (1989).
*Curriculum and Evaluation Standards for School Mathematics*. Reston, VA: Author. Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and Standards for School Mathematics*. Reston, VA: Author. Google Scholar - Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and the velocity sign.
*Journal of Mathematical Behavior*,*13*, 389–422. CrossRefGoogle Scholar - Radford, L. (2000). Signs and meaning in students’ emergent algebraic thinking: A semiotic analysis.
*Educational Studies in Mathematics*,*42*(3), 237–268. CrossRefGoogle Scholar - 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. CrossRefGoogle Scholar - Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts.
*The International Journal of Mathematics Education*,*40*(1), 82–96. Google Scholar - Rivera, F. D., & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns.
*ZDM—The International Journal of Mathematics Education*,*40*(1), 65–82. CrossRefGoogle Scholar - Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.) (1993).
*Integrating Research on the Graphical Representation of Functions*. Hillsdale, NJ: Erlbaum. Google Scholar - Sallee, T., Kysh, J., Kasimatis, E., & Hoey, B. (2002).
*College Preparatory Mathematics 1 (Algebra 1)*(2nd ed.). Sacramento, CA: CPM Educational Program. Teacher Edition: Version 6.1. Google Scholar - Schoenfeld, A. H., Smith, J., & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.),
*Advances in Instructional Psychology*(Vol. 4, pp. 55–175). Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Sherin, B. (2000). How students invent representations of motion: A genetic account.
*Journal of Mathematical Behavior*,*19*(4), 399–441. CrossRefGoogle Scholar - Sherin, S. (2001). How students understand physics equations.
*Cognition and Instruction*,*19*(4), 479–541. CrossRefGoogle Scholar - Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition.
*The Journal of the Learning Sciences*,*3*(2), 115–163. CrossRefGoogle Scholar - TERC (2008).
*Investigations in Number, Data, and Space*(2nd ed.) Glenview, IL: Pearson Scott Foresman. Google Scholar - Warren, E., & Cooper, T. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking.
*Educational Studies in Mathematics*,*67*(2), 171–185. CrossRefGoogle Scholar - Wisconsin Center for Educational Research & Freudenthal Institute (Eds.) (2006).
*Mathematics in Context*. Chicago: Encyclopedia Britannica, Inc. Google Scholar