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

Articles

- 321 Downloads
- 12 Citations

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## 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

## Copyright information

© Mathematics Education Research Group of Australasia Inc. 2008