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The Intimate Interplay Between Experimentation and Deduction: Some Classroom Implications

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Handbook of the History and Philosophy of Mathematical Practice

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Abstract

We understand experimental mathematics as the systematic experimental investigation of concrete examples of a mathematical structure in the search for conjectures about its properties. Experiments might be done by pencil-and-paper work, building physical models, and, of course, by using available computer programs for doing time-consuming calculations, geometric constructions, and other types of visualizations. The guiding intention of the chapter is the study of the interplay of experimentation and deduction (proof) in the context of the teaching of mathematics at lower and upper secondary schools as well as the education of teachers of mathematics.

The didactical intentions of the chapter make it necessary to take a special philosophical perspective, which might be characterized as a combination of Lakatosian and non-Lakatosian viewpoints (Sect. 2).

In section 3 we shortly discuss three historical examples (Euler, Fourier series, Chinese Remainder Theorem) to give a more realistic image of the interplay between experimentation and deduction in contrast to the somewhat artificial character which didactical examples necessarily involve.

In the following Sects. 4, 5, 6, 7, and 8, we discuss a few classroom examples of mathematical experimentation. Section 4 is on checking, guessing, and finding a proof, and in Sect. 5 we put a particular emphasis on finding and handling counterexamples. These two sections can be characterized as the classical domain of Lakatosian ideas transferred to the classroom. Section 6 explores the potential of viewing elementary Euclidean geometry from the point of view of statics. This approach has had a long tradition in mathematics since ancient times and represents a transition to discussing experimentation with hypotheses in Sect. 7 on modelling. The latter is definitely a non-Lakatosian viewpoint.

Lastly, in Sect. 8 it is concluded that the interplay between experimentation and deduction needs to be incorporated in primary education and extended throughout for the long-term development of sound mathematical thinking and understanding of students.

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Notes

  1. 1.

    To fully understand the choice of the arbitrary seeming numbers 40, 45, and 36 requires some modular arithmetic and of congruence relations between integers. More information can be found in abstract algebra and/or number theoretic texts.

  2. 2.

    This is a specific case of a Pell equation, for which solutions were discovered as an offshoot of theoretical work rather than quasi-empirical testing. For example, one can see with a modest amount of experimentation that x2 − dy2 = 1 is solvable in positive integers when d is a small, positive nonsquare integer and infer (as Indian mathematicians did in the twelfth century) that it is probably solvable for more general d. This led to ad hoc algorithms that worked pretty well (Bhaskara managed the case d = 61), and finally, to a theory that produced the present continued-fraction treatment, which is guaranteed to churn out a solution (and will do so with d = 991 in fairly short order).

  3. 3.

    Although ideally students ought to do such a dynamic construction for themselves, the instructor could provide them with a ready-made dynamic sketch to save time.

  4. 4.

    In physics, a lamina is a two-dimensional object with uniform density and negligible thickness.

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Authors and Affiliations

  1. Stellenbosch University, Stellenbosch, South Africa

    Michael de Villiers

  2. Faculty of Mathematics, University of Duisberg-Essen, Essen, FR, Germany

    Hans Niels Jahnke

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  1. Michael de Villiers
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  2. Hans Niels Jahnke
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Correspondence to Michael de Villiers .

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Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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    Jessica Carter

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de Villiers, M., Jahnke, H.N. (2023). The Intimate Interplay Between Experimentation and Deduction: Some Classroom Implications. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_40-1

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