# Using Integer Manipulatives: Representational Determinism

## Abstract

Teachers and students commonly use various concrete representations during mathematical instruction. These representations can be utilized to help students understand mathematical concepts and processes, increase flexibility of thinking, facilitate problem solving, and reduce anxiety while doing mathematics. Unfortunately, the manner in which some instructionally employ representations potentially leaves students confounded and mathematical ideas unlearned. From the perspective of representational determinism, this paper explores the appropriate uses and misuses of concrete representations in respect to integer operations.## References

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McKendree, J., Small, C., & Stenning, K. (2002). The role of representation in teaching and learning critical thinking. Educational Review, 54, 57-67.

Merenluoto, K. (2003). Abstracting the density of numbers on the number line--A quasi-experimental study. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 joint meeting of PME and PMENA (Vol. 3, pp. 285-292). Honolulu, HI: CRDG, College of Education, University of Hawai’i.

Miles, M. B. & Huberman, M. N. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.

representational competence. Journal for Research in Mathematics Education, 30(5), 541-557.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-281.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification- The case of function. In E. Dubinsky, & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-85). Washington DC: Mathematical Association of America.

Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research. London, England: Sage Publications Ltd.

Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21, 555–570.

Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179-217.

Ainsworth, S. E., Bibby, P. A., & Wood, D. J. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. The Journal of the Learning Sciences, 11(1), 25-62.

Ball, D. L., Thames, M., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407.

Berenson, S. B., Valk, T. V. D., Oldham, E., Runesson, U., Moreira, C. Q., & Broekman, H. (1997). An international study to investigate prospective teachers’ content knowledge of the area concept. European Journal of Teacher Education, 20(2), 137–150.

Berman, B., & Friederwitzer, F. (1985). Color tile activities. NY: Cuisenaire Company of America, Inc.

Bogdan, R. C., & Biklen, S. K. (2003). Qualitative research for education: An introduction to theories and methods (4 ed.). Boston: Allyn and Bacon.

Brenner, M. E., Herman, S., Ho, H. -Z., & Zimmer, J. M. (1999). Cross-national comparison of representational competence. Journal for Research in Mathematics Education, 30(5), 541-557.

Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Durán, R., Reed, B. S., & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663-689.

Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Belknap Press.

Clements, D. H. (1999). “Concrete” manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45–60.

Creswell, W. J. (2003). Research Design: Qualitative, quantitative and mixed methods approaches (2 ed.). London: Sage Publications.

Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson Educational.

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.

Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics. Educational Studies in Mathematics, 61, 103-131.

Ernest, P. (1985). The numbers line as a teaching aid. Educational Studies in Mathematics, 16, 411-424.

Fuson, K., & Briars, D. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21, 180-206.

Goldin, G. (2000). A scientific perspective on structured, task-based, interviews in mathematics education research. In A. Kelly & R. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education. Mahwah, NJ: Lawrence Erlbaum Associates.

Goldin, G. A. (2002). Representation in mathematical learning and problem solving. Handbook of international research in mathematics education, 197-218.

Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. A research companion to principles and standards for school mathematics, 275-285.

Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. The roles of representation in school mathematics, 2001, 1-23.

Goldin, G.A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137-165.

Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 361-367.

Harris-Sharples, S. (1993). Introduction. In A. Hoffman, & A. Glannon (Eds.), Kits, games and manipulatives for the elementary school classroom: A source book. New York: Garland Publishing.

Herbst, P. (1997). The number-line metaphor in the discourse of a textbook series. For the Learning of Mathematics, 17(3), 36-45.

Janvier, C. (1983). The understanding of directed number. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the Fifth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 295-301). Montréal: Université de Montréal, Faculté de Sciences de l’Education.

Knuth, E. J. (2000). Understanding connections between equations and graphs. Mathematics Teacher, 93(1).

Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11, 65-99.

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.

McKendree, J., Small, C., & Stenning, K. (2002). The role of representation in teaching and learning critical thinking. Educational Review, 54, 57-67.

Merenluoto, K. (2003). Abstracting the density of numbers on the number line--A quasi-experimental study. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 joint meeting of PME and PMENA (Vol. 3, pp. 285-292). Honolulu, HI: CRDG, College of Education, University of Hawai’i.

Miles, M. B. & Huberman, M. N. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.

representational competence. Journal for Research in Mathematics Education, 30(5), 541-557.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-281.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification- The case of function. In E. Dubinsky, & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-85). Washington DC: Mathematical Association of America.

Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research. London, England: Sage Publications Ltd.

Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21, 555–570.

Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179-217.

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2016-11-04

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