Reflections on the Importance of Reference for Understanding Thinking

Abstract views: 35 / PDF downloads: 24


  • Reinhard Oldenburg


Semantics, algebra, variables


An intentional form of the semantics of algebraic expressions in the tradition starting with Frege is popular in mathematics education. On the other hand, mathematical logic including predicate calculus and lambda calculus is dominated for more than 50 years by referential semantics. A third field of investigation is the semantics of mathematics as realized in programming languages and computer algebra systems. The paper explores the tension between these approaches and tries to clarify the role of reference both in he developed mathematics as well as in the learning process. Referential semantics simplifies theories but requires mental objects to be constructed to be useful. This links the topic to reification theory. A small collection of observations of learners’ behavior adds support to the claim that reference is of some importance in the learning process.

Anahtar Kelimeler: Semantics, algebra, variables.

Author Biography

Reinhard Oldenburg

Corresponding Author: Reinhard Oldenburg,, Augsburg University, Germany.


Abelson, H., Dybvig, R. K., …., Wand, M. (1998). The revised5 report on the algorithmic language Scheme. MIT 1998.

Arzarello, F., Bazzini, L., Chiappini, G. (1994). Intensional semantics as a tool to analyze algebraic thinking. Rend. Sem. Mat. Univ. Poi. Torino, Vol. 52, 2.

Arzarello F., Bazzini, L., Chiappini, G. (2001). A model for analyzing algebraic processes of thinking. In R. Sutherland et al. (Eds.), Perspectives on School Algebra, pp. 61-81. Dordrecht: Kluwer Academic Publishers.

Davenport, J., Siret, Y., Tournier, E. (1988). Computer Algebra: Systems and Algorithms for Algebraic Computation. London: Academic Press.

DeMarois, P. (1998) Facets and Layers of the Function Concept: The Case of College Algebra. PhD thesis, University of Warwick, England.

Dörfler, W.(2005): Diagrammatic Thinking. In: M. Hoffmann (Ed.) Activity and sign – Grounding mathematics education. Berlin: Springer.

Drouhard, J.-P. & Teppo, A. (2004). Symbols and Language. In K. Stacey, H. Chick, M. Kendal (Eds.) The

Future of the Teaching and Learning of Algebra: The 12th ICMI Study. (Pp 226 - 264), Dordrecht:


Li, W. (2010). Mathematical Logic. Berlin: Birkhäuser.

Meyer, A. (2013). Algebraic thinking and formalized mathematics – formal reasoning and the contextual. Paper presented at Cerme 2013.

Michaelson, G. (2011). An Introduction to Functional Programming through Lambda Calculus. New York: Dover.

Oldenburg, R. (2011). Reification and symbolization. Proceeding of Koli Calling 2011. ACM Digital library.

Quine, W. v. O. (1960). Word and Object. Cambridge: MIT Press.

Sfard, A., Linchevsky, L. (1994). Gains and pitfalls of reification. Educational Studies in Mathematics, 26: 191-228.

Tall, D. (2012). How Humans learn to think Mathematically. Cambridge: Cambridge.

Vollrath, H.J. (1989). Funktionales Denken. Journal für Mathematikdidaktik (JMD), 10, 3-37.




How to Cite

Oldenburg, R. (2015). Reflections on the Importance of Reference for Understanding Thinking. International Journal of Contemporary Educational Research, 2(1), 54–60. Retrieved from