# How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra

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## Abstract

Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense components, and that several components of university algebra structure sense are analogies of high school algebra structure sense components. We present a theoretical argument for these hypotheses, with some examples. We recommend emphasizing structure sense in high school algebra in the hope of easing students’ paths in university algebra.

## Keywords

Binary Operation Identity Element Mathematical Thinking Algebraic Expression Future Teacher
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## References

- Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics.
*For the Learning of Mathematics, 14*(*3*), 24–35.Google Scholar - Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.),
*Perspectives on School Algebra*(pp. 99–119). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar - Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.),
*The nature of mathematical thinking*(pp. 253–284). Mahwah, NJ, USA: Lawrence Erlbaum Associates.Google Scholar - Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory.
*Educational Studies in Mathematics, 27*, 267–305.CrossRefGoogle Scholar - Esty, W. W. (1992). Language concepts of mathematics.
*Focus on Learning Problems in Mathematics, 14*(4), 31–53.Google Scholar - Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain.
*Journal for Research in Mathematics Education, 22*(3), 170–218.CrossRefGoogle Scholar - Harel, G., & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics.
*For the Learning of Mathematics, 11*(1), 38–42.Google Scholar - Hoch, M. (2003). Structure sense. In M. A. Mariotti (Ed.),
*Proceedings of the 3rd Conference for European Research in Mathematics Education*(CD). Bellaria, Italy: CERME.Google Scholar - Hoch, M. (2007).
*Structure sense in high school algebra*. Unpublished doctoral dissertation, Tel Aviv University, Israel.Google Scholar - Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effect of brackets. In M. J. Høines & A. B. Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 49–56). Bergen, Norway: PME.Google Scholar - Hoch, M., & Dreyfus, T. (2005). Students’ difficulties with applying a familiar formula in an unfamiliar context. In H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 145–152). Melbourne, Australia: PME.Google Scholar - Hoch, M., & Dreyfus, T. (2006). Structure sense versus manipulation skills: An unexpected result. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.),
*Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 305–312). Prague, Czech Republic: PME.Google Scholar - Kieran, C. (1992). The learning and teaching of algebra. In D. A. Grouws (Ed.),
*Handbook of Research on Mathematics Teaching and Learning*(pp. 390–419). New York: MacMillan.Google Scholar - Kirshner, D., & Awtry, T. (2004). Visual salience of algebraic transformations.
*Journal for Research in Mathematics Education, 35*(4), 224–257.CrossRefGoogle Scholar - Linchevski, L., & Livneh, D. (1999). Structure sense: the relationship between algebraic and numerical contexts.
*Educational Studies in Mathematics, 40*(2), 173–196.CrossRefGoogle Scholar - Linchevski, L., & Vinner, S. (1990). Embedded figures and structures of algebraic expressions. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.),
*Proceedings of the 14th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 85–92). Oaxtepec, Mexico: PME.Google Scholar - Novotná, J. (2000). Teacher in the role of a student — A component of teacher training. In J. Kohnova (Ed.),
*Proceedings of the International Conference of Teachers and Their University Education at the Turn of the Millennium*(pp. 28–32). Praha: UK PedF.Google Scholar - Novotná, J., Stehlíková, N., & Hoch, M. (2006). Structure sense for university algebra. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.),
*Proceedings of the 30*^{th}*Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 249–256). Prague, Czech Republic: PME.Google Scholar - Novotná, J., & Trch, M. (1993).
*Algebra and theoretical arithmetics*. Volume 3. Introduction to Algebra. Praha. [Textbook.] (In Czech.)Google Scholar - Pierce, R., & Stacey, K. (2001). A framework for algebraic insight. In J. Bobis, B. Perry, & M. Mitchelmore (Eds.),
*Numeracy and Beyond. Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia*(Vol. 2, pp. 418–425). Sydney, Australia: MERGA.Google Scholar - Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification — The case of algebra.
*Educational Studies in Mathematics, 26*, 191–228.CrossRefGoogle Scholar - Simpson, A., & Stehlíková, N. (2006). Apprehending mathematical structure: A case study of coming to understand a commutative ring.
*Educational Studies in Mathematics, 61*(3), 347–371.CrossRefGoogle Scholar - Stehlíková, N. (2004).
*Structural understanding in advanced mathematical thinking*. Praha: Univerzita Karlova v Praze — Pedagogická fakulta.Google Scholar - Tall, D. O. (2007). Embodiment, symbolism and formalism in undergraduate mathematics education, Plenary at
*10*^{th}*Conference of the Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education*, Feb 22–27, 2007, San Diego, California, USA. [Available from electronic proceedings http://cresmet.asu.edu/crume2007/eproc.html. Downloaded 30 July, 2008].Google Scholar - Tall, D., & Thomas, M. O. J. (1991). Encouraging versatile thinking in algebra using the computer.
*Educational Studies in Mathematics, 22*, 125–147.CrossRefGoogle Scholar - Zorn, P. (2002). Algebra, computer algebra, and mathematical thinking. In I. Vakalis, D. H. Hallett, C. Kourouniotis, D. Quinney, & C. Tzanakis (Eds.),
*Proceedings of the 2nd International Conference on the Teaching of Mathematics at the Undergraduate Level*(on CD). Hersonissos, Crete, Greece: University of Crete.Google Scholar

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© Mathematics Education Research Group of Australasia Inc. 2008