Algebra at the Meta and the Object Level

Year 2015, , 366 - 379, 10.12.2015

Reinhard Oldenburg

https://doi.org/10.16949/turcomat.18001

Abstract

Two different interpretations of algebra that differ in the ontological status assigned to variables are distinguished. Variables may either be viewed as meta-mathematical tools to express generality or as objects similar to numbers and other members of the mathematical ontology. Both interpretations are detailed and linked with the literature and the use of variables in computer programming. Furthermore, it is analyzed how these two conceptualizations lead to two different understandings of the process of change of values. Some evidence from algebra assessment on the understanding of change by students is given that that illustrate that the theory is useful in analyzing students work.

Keywords

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References

• Bednarz, N., Kieran, C., & Lee, L. (1996). Approaches to algebra: Perspectives for research and teaching (pp. 3-12). Springer Netherlands.
• Chiappini, G.P. (2011). The role of technology in developing principles of symbolical algebra. Pytlak M., Rowland T., Swoboda E. (eds), Proceedings of CERME 7, (pp. 429-439). University of Rzeszow, Poland.
• Dougherty, B. (2008). Measure Up: A quantitative view of early algebra. In J. Kaput, D. Carraher & M. Blanton (Eds.), Algebra in the early grades (pp. 389-412). New York: Erlbaum.
• Drouhard, J.-Ph., & Teppo, A. R. (2004) Symbols and language. In Stacey, K., Chick, H., Kendal, M. (Eds). The Future of the Teaching and Learning of Algebra The 12 th ICMI Study (pp. 225-264) Springer Netherlands.
• Euler, L. (1810) Elements of Algebra (2nd edition). Johnson & Co.
• Felleisen, M., Findler, R. B., Flatt, M., & Krishnamurthi, S. (2001). How to design programs. MIT Press.
• Filloy, E., Rojano, T., & Puig, L. (2008) Educational Algebra. Springer, New York.
• Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
• Küchemann, D. (1978). Children’s understanding of numerical variables. Math. School, 7(4), 23-26.
• Li, W. (2010) Mathematical Logic. Berlin: Birkhäuser.
• Linchevski, L. (2001) Operating on the unknown: What does it really mean? Proceedings of PME25, 141-144.
• MacGregor, M., & Price, E. (1999) An exploration of aspects of language proficiency and algebra. Journal for Research in Mathematics Education, 30(4), 449-467.
• Mendelson, E. (1997): Introduction to mathematical Logic. Monterey.
• Menghini M. (1994) Form in algebra: Reflecting, with peacock, on upper secondary school teaching. For the learning of Mathematics, 14, 9-14.
• MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebra learning. Journal for Research in Mathematics Education, 30(4), 449-467.
• Nemirowski, R., Tierney, C., & Ogonowski, M. (1993). Children, additive change and calculus. TERC, Cambridge.
• Nicaud, J.-F., Bouhineau, D., & Gélis, J.-M. (2001). Syntax and semantics in algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 475–486). Melbourne, Australia: The University of Melbourne.
• -
• Nie, B., Cai, J., & Moyer, J. C. (2009). How a standards-based mathematics curriculum differs from a traditional curriculum: With a focus on intended treatments of the ideas of variable. ZDM, 41(6), 777-792.
• Oldenburg, R. (2015). Reflections on the importance of reference for understanding. International Journal of Research in Education and Science (IJRES), 1(1), 1-6.
• Peacock G. (2004) Treatise on Algebra. Volume I Arithmetical Algebra. Volume II Symbolical Algebra. Dover Publications, Inc.
• Quine, W.v.O. (1960). Variables explained away. Proc. of the American Philosophical Society, 104(3) 343-347.
• Schwank, I. (1999) On predicative versus functional cognitive structures. European research in mathematics education, 1(2), 85-97.
• Sfard, A.(1991) On the dual nature of mathematical conceptions. Educational Studies in Mathematics, 22, 1-26.
• Tourlakis, G. (2003) Lectures in logic and set theory vol. 1: Mathematical logic. Cambridge University Press, Cambridge.
• Usiskin, Z. (1988) Conceptions of school algebra and uses of variable. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 8-19). Reston, VA: National Council of Teachers of Mathematics.
• White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1), 79-95.