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Grenier D.
(2000)
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I. Elements of ecology of proof practice |
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We will give here some elements about the ecology of proof practice in recent French official programs and textbooks corresponding to school levels where proof is a " sensible object " in France - 4th, 3rd and 2nd classes (13-16 year old pupils). I.1. In the official French programsThe excerpt below of some directives concerning French programs for 4th and 3rd classes (still in effect today) shows the intention to grant a genuine place to proof practice as well as to the activity of writing proofs. " The deepening of the already acquired concepts, the training to deductive reasoning are led in the spirit of the former classes,without systematic rebuilding and in connection with new situations, in order to develop the abilities of discovery and conjecture as much as proving. The pupils will be involved to exercise their ability to write, but the premature requirement of reaching a formalization will be avoided [...] " This point of view is confirmed in the comments of the programs of 2nd class (BO n° 20, May 90), under the heading " Scientific education " : " The capacities of experimenting and reasoning, the imagination and critical analysis, far from being incompatible, must be developed on : to formulate a problem, to conjecture a result, to test on examples, to construct a proof, to implement theoretical tools, to put a solution in shape, to control the results obtained , to evaluate their relevance according to the problem raised, are just different steps of a single mathematical activity. In this context, the clarity and accuracy of the reasoning, the quality of the written and oral expression constitute significant objectives. However, acquiring the expertise of reasoning and of mathematical language must be placed in a perspective of progress; one will thus keep from any premature requirement of formalization, either for statements or for proofs. In particular, the vocabulary and the notation are not imposed a priori; they are introduced in the course of the teaching process according to a criterion of utility. " The official programs clearly offer a real possibility for a scientific work with the students on the objects related to proof practice : conjecture, modeling, refutation. We will examine now how these recommendations are transposed in textbooks which correspond to these programs. I. 2. The status of proof in the textbooksA study of some textbooks for 4th to 2nd classes leads us to make two remarks. 1. The proof issues are primarily (not to say exclusively) concentrated in geometry. This state of things is well known and has been largely considered in didactic articles on proof practice. We can reasonably make the assumption that a consequence of this fact is that the proof is built with very particular characteristics related to the choice of geometry as a subject. Let us justify thess assertions, by describing the forms under which proof seems to exist in these textbooks. We classify them in three types. 1st type : observation --> proof --> validation (?) by observation First of all, the property to be proven or shown either
is explicitly given, or is to find from an activity of
observation of figures. This scheme is traditional : pupils
are asked to construct lines, points, circles etc..., then
to observe the figure, to locate certain objects and the
properties of these objects. Those are obvious, even on a
quickly constructed figure. " a. Observe the figure; find parallel lines, middles, parallelograms. or " a. Observe the figure and state some visible properties. It is thus a question of proving something which is sufficiently visible so that one is assured that any construction made by pupils " shows it ". Moreover, when the proof has been produced, pupils are invited to control the result of his proof through a visit back to the figure. Let us quote " Maths 4ème, Collection M classique Hachette 1979 ", in the Teacher's book, Chapter 3, p.36-37 : " [...] " To draw a figure " means : " Reflect by a drawing the situation considered ". This figure is only a visual aid which helps you to argue, but not to establish what you are asked to prove. You can then check on this figure if the conclusion of your proof is in good conformity with the graphic observation. " One can legitimately wonder about the validity of this check : if the property is conjectured from the figure that has been drawn, any conclusion is necessarily in conformity with the figure ! 2nd type : proof = " formal " proof In glossaries which can be found at the end of some textbooks, the term " proof " sometimes appears. Here an example of a mini-dictionary (Deledicq, 3ème, 1993) : " A proof is a succession of sentences showing how starting from assumptions (regarded as true) one can arrive logically to a conclusion based on properties on which everyone would then agree. In mathematics, at the Collège level, these properties are those which are written in mathematics textbooks. " The proof is often the object of discourses intended to introduce or explain its utility or its need. The activity of proving is in general reduced to a work of formulation and to the implementation of writing techniques. Thus, for example, one reads, in the collection " Perspectives Mathématiques, 2nde, Hachette " 1990, p.10 : "Proving: These two examples are remarkable by their presentation : the words in boldface characters in the two proofs (10 lines each) are exclusively " note ", " by definition ", " since ", " it follows ", " i.e. ", " consequently ", " one obtains ", " one recognizes ". Even if these words of course play a significant role in the structure of a proof, the fact that they are the only one to be highlighted illustrates quite well the distance between the status of the proof in school programs and its status in the textbooks. The proof practice tends to be reduced to its aspect of writing details, the explicit Didactical Contract for pupils merely consists of recognizing and use formulations and words which will be regarded as significant in proof training. 3rd type : for each property, a specific method of proof The " methods to prove " or show often constitute titles of paragraphs or headings, just as for calculation, construction, reasoning, etc. They seem to be contextualized techniques, closely related to concepts or properties, and are almost all geometrical. This choice, which consists in associating with each task a particular technique, produces an extreme effect of dividing knowledge into sparse parts. Here are some illustrations. In " Maths Magnard, 3ème, 1989 " in an index of methods which recapitulates the methods seen in the textbook, one finds in particular : " " How to " :- show that a triangle is a rectangular triangle [... ] The textbook produced by IREM Strasbourg for class of 4ème suggests in several places scattered regularly through the book, " keys to show that... ". These keys all are connected to very specialized geometric or algebraic properties. For example, here is the exhaustive list of keys proposed p.244 and following -- on which we will refrain from making any additional comment : " three points are aligned / points are on a same circle / a point is the middle of a segment / two segments have equal lengths / two lines are perpendicular / two lines are parallel / three lines are concurrent / two angles have the same value / a line is the mediatrix of a segment / a line is the bisecting line of an angle / a triangle is isosceles / a triangle is equilateral / a triangle is rectangular / a quadrilateral is a parallelogram / a quadrilateral is a rectangle / a quadrilateral is a rhombus / a quadrilateral is a square. " In addition, the 4th chapter " rules of the game for
deduction in geometry " specifies in a clear way the proof
contract to be used in class. The particular case of the Deledicq textbook, " Faire des mathématiques " (Hachette 1993, 3rd class) which attempts to approach conjectures and argumentation, only confirms in fact the difficulty of implementing proof practice and keeping it alive. Indeed, the headings " justify and argue ", which contain the subentries : " show an equality " (p.90) are all written in such a way that the real stakes are
found in the Didactical Contract, finally very far from the
activities related to conjectures or proofs. " The problem of searching an optimal path in Geometry is quite difficult. Such problems are generally solved in a simpler way by using results of Analysis. But with the indications given here, the problems which are posed are feasible. " Follows the text of the problem : " D is a line, A and B are two points located outside D and in the same half-plane delimited by D. A' is the reflection of A with respect to D. Find a point M located on D so that the sum AM+BM is minimal. " We will make only one comment : the difficult part of the problem of minimality is evacuated by the (obvious) indication that the reflection A' of A with respect to D is to be used. The mathematical problem is closed ! In conclusion, the proof practice - for 13-16 years students - seems to live primarily in the field of geometry and under two forms : - simple deductive reasoning (with a small number of steps), based on the use of statements or theorems from the theoretical part of the course, Pupils are supposed to learn how to pass from hypotheses
which are stated and recognized as true to a conclusion
which is either stated, or more or less obvious and to be
found (from experimentation and observation).
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