Abstract
The learning of mathematics starts early but remains far from any theoretical considerations: pupils’ mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners’ understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners’ way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory.
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Notes
- 1.
Argumentation means here “verbal, social and rational activity aimed at convincing a reasonable critic of the acceptability of a standpoint by putting forward a constellation of one or more propositions to justify this standpoint” (van Eemeren et al., 2002, p.xii). “In argumentative discussion there is, by definition, an explicit or implicit appeal to reasonableness, but in practice the argumentation can, in all kinds of respects, be lacking of reasonableness. Certain moves can be made in the discussion that are not really helpful to resolving the difference of opinion concerned. Before a well-considered judgment can be given as to the quality of an argumentative discussion, a careful analysis as to be carried out that reveals those aspects of the discourse that are pertinent to making such a judgment concerning it reasonableness.” (ibid., p.4)
- 2.
See e.g. Piaget J. (1969) p. 239: “L’enfant n’est guère capable, avant 10-11 ans, de raisonnement formel, c’est-à-dire de déduction portant sur des données simplement assumées, et non pas sur de vérités observées.” More precisely, For more, c.f. Piaget J. (1967) Le jugement et le raisonnement chez l’enfant. Delachaux et Niestlé.
- 3.
Popper (1959) proposed falsification as the the empirical criterion of demarcation of knowledge, scientific theories or models.
- 4.
Or should not be...
- 5.
From Capponi (1995), Cabri-classe, sheet 4–10.
- 6.
E.g. Cabri-geometry (here used for the drawing), or Geometer Sketchpad; or Geogebra or one of the several others now available sometimes open access.
- 7.
- 8.
Another student’s search for an explanation illustrates well what is meant here by mechanical world: “So... I have said... But is not very clear... That when, for example, we put P to the left, then P3 compensates to the right. If it goes up, then the other goes down...” (Sébatien, [prot. 78–84]).
- 9.
See Claudi Alsina and Roger B. Nelsen (2006), Math Made Visual: Creating Images for Understanding Mathematics, published by MAA, and a good example in Roger B. Nelsen (1993), Proofs without words: exercises in visual thinking, published by MAA. See Hanna (2000, esp. pp.15–18) for an analysis.
- 10.
Considering the sequence of complex numbers zn+1 = z 2n + c, the Mandelbrot set (or set M) is obtained by fixing z0=0 and varying the complex parameter c.
- 11.
Quotation from p.250 of Mendelbrot (1980) Fractal aspects of the iteration of z→λz(1-z) for complex λ and z. Annals of the New York Academy of Sciences. 357 (1) 249 - 259
- 12.
Régine Douady remembers that Adrien had been quickly convinced of the connectivity of M, thanks to the theoretical argument which convinced him in an astonishingly “simple” way. However, to complete the explicit proof took some time (2008, personal communication).
- 13.
Personal communication
- 14.
For the convenience of the English-speaking reader, I take all the references to Brousseau’s contributions to mathematics education from Kluwer, 1997 but Brousseau’s work was primarily published between 1970 and 1990.
- 15.
This proposition should be understood in the light of the development of the “situated learning paradigm” of Jeane Lave and Etienne Wenger, whose work was published in the early 1990s.
- 16.
The letters cK¢ stand for : “conception,” “knowing” and “concept”; more about this model is presented and discussed on [http://ckc.imag.fr]
- 17.
Vergnaud in fact proposed this definition at the beginning of the 1980s.
- 18.
By extension, one can often refer to students’ conceptions as acceptable given that one can account precisely for the circumstances, which are the milieu and the constraints within which [S↔M] functioned.
- 19.
“Je le vois, mais je ne le crois pas,” wrote Cantor to Dedekind, in 1877, after having proved that for any integer n, there exists a bijection between the points on the unit line segment and all of the points in an n-dimensional space.
- 20.
figure 9.11 sketches the interactions between these three poles
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Balacheff, N. (2010). Bridging Knowing and Proving in Mathematics: A Didactical Perspective. 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_9
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