Preuve Proof Prueba

Web Newsletter
Mai/Juin 1997


The theme of the newsletter

Proof and visualization

The goal of the Letter is the stimulation of exchanges on certain current questions about the learning of mathematical proof. In no case is it an advanced contribution; it remains modest in content and editorial position. It is the letters and contributions which will constitute the substance.

The idea that recourse to images can facilitate the presentation of mathematical proofs is an ancient one; basically it is the idea that the concrete is more accessible than the abstract. It is undergoing a return of interest in the world of mathematics teaching, particularly as a result of the development of virtual realities to which computer science gives access. And after all, isn't a picture worth a thousand words? This point of view is reinforced by the attention being given to ancient mathematics, especially oriental mathematics, in which drawing was permitted to play an essential role in the communication of knowledge. But are things really that simple? Is visualization really a facilitator?

A Cabri-Java illustration by
Jean-Philippe Ibarlucia and
Gilles Kuntz

A visual and animated proof of the Pythagorean Theorem:

In the figure to the left, grab the point M and move it around. To "see things well" it is best to move M clockwise along a large arc.

The equality in the famous theorem is borne out by the continuous deformation of the two small squares to produce two parallelograms which can be attached to each other and deformed yet again to produce the large square.
    The essential element, of course, is the observation that the areas are conserved. But the reason for this conservation is not "given" by the representation observed--it must be constructed by the person who is examining the animation, and it should therefore engage his/her knowledge to understand how it has to do with a proof, and what the sources of validity of that proof are: seeing is knowing...
    Thus a visual proof might be no simpler than a verbal one. If it seems to many to have more evocative power, one should not forget that it is rich in implicit content.

It is an essential function of language to pursue this implicit content and, in bringing them into view, to submit the articulations of reasoning to critical examination.

What place should be given to so-called visual proofs?

    David Tall, in his contribution to this question, suggests that they are of interest in the perspective of a progression in learning from the concrete, and manipulation, to the more abstract, and enunciation. He designates as the crucial issue, which can be regarded as the lever for progressing towards verbal proofs: "Difficulties occur when the enactive or visual form of the proof does not suggest an obvious sequence of deductions to use for a formal proof, so that the individual seems to 'know' that the theorem is true and yet has no method of proving it" (ibid.). The work of Claude Tisseron on third degree polynomial functions is such an example, probably reinforced in the case he presents by the students' confidence in the images produced by calculators and computers.
    If multi-media expression is to develop in mathematics, perhaps one should create tools for the analysis of images, a language to explore them and encode the exploration in the manner of the "window-shopping" which Joel Hillel suggests for the study of functions in interface with a computer.
    After a decade of discovering the potential of visual expression in mathematics, one might suggest for the next decade an increased attention to the complexity of this use and to the conditions in which it is put to use in teaching. Some authors are engaged in this task, such as Raymond Duval , who has examined in a very advanced way the semiotic aspects of proof and demonstration in mathematics.

Reactions? Remarks?

Contibutions to this theme will be published in the July/August Newsletter

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