Abstract
This paper examines the role and function of experimentation in mathematics with reference to some historical examples and some of my own, in order to provide a conceptual frame of reference for educational practise. I identify, illustrate, and discuss the following functions: conjecturing, verification, global refutation, heuristic refutation, and understanding. After pointing out some fundamental limitations of experimentation, I argue that in genuine mathematical practise experimentation and more logically rigorous methods complement each other. The challenge for curriculum designers is therefore to develop meaningful activities that not only illustrate the above functions of experimentation but also accurately reflect the complex, interrelated nature of experimentation and deductive reasoning.
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- 1.
In fact, it holds for a tetrahedron with equi-areal faces, and for equilateral or equi-angled polygons.
- 2.
Of course, sometimes a combination of reflective thought and experimentation is needed. For example, from \( {3^2} + {4^2} = {5^2} \)and \( {5^2} + {12^2} = {13^2} \), we can see that perhaps \( {7^2} + {24^2} = {25^2} \) and guess that similar equations might hold for \( {9^2},\;{11^2},\;{13^2},\;\ldots \)Indeed, noting the structure – that, say \( {13^2} - {12^2} \) = (13 − 12)(13 + 12) = 25 = \( {5^2} \) – gives us a clue for constructing and checking, with a minimum of pain, other instances.
- 3.
It would obviously be instructive for students to further examine this conjecture and to identify the additional key property that one of the perpendicular diagonals should bisect the other. The initial inarticulation of hypotheses happens quite regularly with inexperienced students, and necessitating the fostering of a state of mind characterised by acute analysis and thoroughness.
- 4.
As before, it might be a valuable learning experience to guide students to identify the additional property that the equal diagonals need to cut each other in the same ratio.
- 5.
However, not all counter-examples are constructed by experimentation or quasi-empirical testing. For example, since 41 is clearly a factor of \( {n^2} - n + 41 \) when n= 41, one might easily notice without any quasi-empirical substitution that it provides an immediate counter-example to the conjecture that \( {n^2} - n + 41 \) is always prime for n= 1, 2, 3, etc.
- 6.
Though Descartes already in 1639 knew of the invariance of the so-called “total angle deficiency” of polyhedra and Euler’s formula can be derived from this, there is – according to Grünbaum and Shephard (1994:122) – no historical evidence that Descartes actually saw the connection.
- 7.
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 \( {x^2} - d{y^2} = 1 \) 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).
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Acknowledgment
Reprinted adaptation of article by permission from CJSMTE, 4(3), July 2004, pp. 397–418, http://www.utpjournals.com/cjsmte , © 2004 Canadian Journal of Science, Mathematics and Technology Education (CJSMTE).
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de Villiers, M. (2010). Experimentation and Proof in Mathematics. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_14
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