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Mathematics in Physics Education pp 53–74Cite as

Students’ Understanding of Algebraic Concepts

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Abstract

Redish and Kuo gave in 2015 in a paper on the language of physics and mathematics for higher physics education the following take-away message: “How mathematical formalism is used in the discipline of mathematics is fundamentally different from how mathematics is used in the discipline of physics—and this difference is often not obvious to students. For many of our students, it is important to explicitly help them learn to blend physical meaning with mathematical formalism.” We argue that this message already holds in lower secondary physics education and that students at this level are not to blame for being confused and failing to transfer between mathematics and physics. We discuss the challenges that these students face in algebra when using and building mathematical models of physical systems and on efforts to help them overcome their difficulties. We focus on the different ways of using variables, equations, and formulas in mathematics and physics.

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Notes

  1. 1.

    Although the term “polyvalent name” suggests that the variable represents more than one value, this is not necessary. Often, one does not know in advance how many values are possible: for example, when asked to find the real roots of a third-degree polynomial with real coefficients, one does not easily whether the answers will consist of one or three solutions.

  2. 2.

    Note that arithmetic computations like \(4+2\frac {1}{2}=6\frac {1}{2}\) and \(4+2\frac {1}{3}=6\frac {1}{3}\) may seduce a learner to generalize to 4 + 2x = 6x. Similarly, an arithmetic computation like \(3\frac {1}{2}-\frac {1}{2}=3\) may explain the mistake 3x − x = 3, and the calculation \(2+\frac {1}{2}=2\frac {1}{2}\) may make a learner believe that 2 + x is equal to 2x.

  3. 3.

    Karam, Uhden and Höttecke (this volume) use the same example to illustrate that in physics and mathematics, different meanings are assigned to multiplication.

  4. 4.

    Procept, a contamination of process and concept, is a term introduced for the combination of symbol, process, and concept, to make clear that a mathematical object never completely loses its process nature.

  5. 5.

    Arcavi (2005) described symbol sense as the ability to give meaning to symbols, expressions, and formulas and to have a feeling for their structure. Drijvers (2011, p. 22) confined the interpretation of symbol sense to the understanding of the meaning and structure of algebraic formulas and expressions, which involves (1) the strategic abilities to arrive at a problem approach and to maintain an overview of this process, (2) the capacity to view symbolic expressions globally, and (3) the capacity of algebraic reasoning.

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Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands

    André Heck

  2. Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Onne van Buuren

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  1. André Heck
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  2. Onne van Buuren
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Correspondence to André Heck .

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Editors and Affiliations

  1. Technische Universität Dresden, Dresden, Germany

    Gesche Pospiech

  2. DCFA - Section of Mathematics and Physics, University of Udine, Udine, Italy

    Marisa Michelini

  3. The Weizmann Institute of Science Rehovot, Israel

    Bat-Sheva Eylon

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Heck, A., Buuren, O.v. (2019). Students’ Understanding of Algebraic Concepts. In: Pospiech, G., Michelini, M., Eylon, BS. (eds) Mathematics in Physics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-04627-9_3

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