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Richard P.
R. (2000)
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© Philippe R. Richard |
IntroductionFor lack of specific training, the secondary school student (12-16 years of age) is unfamiliar with the procedures of proof recognized by the mathematical community. To the point that carrying out of authentic proofs, possibly articulated by deductive reasoning, appears to be exceptional. Yet, even at the beginning of this stage, the problem of the degree of certainty in the notions under consideration occurs at many moments, whether in order to ascertain that a property is well-founded or to validate a conjecture. The student develops discursive habits that are rooted as much in the history of the class as in daily life, thereby increasing the attention to reasoning that he/she knows how to use in a validation situation. Although social interaction and the teacher's role as tutor continue to be determinants in the formation of proof procedures, the study based on the state of habits acquired by the analysis of written proofs -- and not on the process of acquisition as a whole -- permits the decomposition of the discursive structure of the semiotics of texts. Thus, from geometric situations/problems, we have identified a step of reasoning, employed effectively by the student, based on the intrinsic and extrinsic features of the geometric figure. Appearance of the figurative inferenceFrom the formulation of a conjecture to the realization of a proof, the processes of figuration, signification and reasoning interpenetrate to such a degree that it becomes difficult to discern when and how they intervene. On the one hand, it is well known that reasoning in geometry is not only based on words or on symbols, but also on drawings and visual images (mental pictures). In a situation of validation, reasoning controls the action (construction of lines or handling of drawing tools) as much as the reflection (development of visual images and of their relationship to the ideal). On the other hand, even when one admits a distinction between verbal (linguistic - symbolic) and nonverbal (nonlinguistic - symbolic) reasoning, one doesn't generally consider the possibility that the latter could be associated with the forms of reasoning other than in intuitive thought or in thought that turns exclusively toward action. However, on posing a diagnosis on the discursive structure of concrete proofs produced by students of 14-15 years of age (Richard, 1999), we showed that they use a type of inference from the semiotic register of a geometric drawing. To bring about or to justify certain steps in reasoning, the student incorporates a drawing -- or a comic strip -- in the discursive structure of the proof for the same purpose as a verbal proposition, by addition of the graphic reasoning to the discursive reasoning. The identification of a figurative inference occurs inasmuch as it is impossible to capture the reasoning locally while concealing the drawing, even if one attempts simultaneously to bring the inferred verbal proposition closer to the thematic continuity developed in the proof. Semiotic register of a drawingExcept in a situation of contemplation, the notion of a geometric figure does not coincide with a petrified picture, frozen in a same state or merely involutive, without being conditioned by a demand or without resulting from an operation. In terms of visualization, interpretation or validation, the geometric figure is composed of significant units (points, groups of points that share a same characteristic or that ensue from a transformation) that are combined according to the functions that they perform. As to material representation of the figure, the drawing can be cast in turn in significant and functional units, even though the significance of these units doesn't refer to the theoretical ideal but to a personal, certainly implicit model (such as one based on confusion between the ideal and its tangible representation). Thus, in a drawing, the significant units are the elementary signs (e.g. the tracing of a segment to represent a straight line) that combine first in graphic syntagmas (e.g. the small square placed at the intersection of two segments to denote two perpendicular lines), and then inside a graphic proposition (e.g. the drawing of a trapezium rectangle) or an assembly of graphic propositions (e.g. a complete drawing). The function of every significant unit relates to the meaning produced by the unit itself or by combining with other significant units (production of an unit of meaning or of a complete meaning relative to a model). Structure and quality of a figurative inferenceAs for all types of inference, first the question of structural recognition is posed, then the question of quality. In the first instance, in accordance with the semiotic register of the drawing, we recognized two types of structure in the students' proofs:
where the graphic proposition born of the drawing plays simultaneous roles as antecedent and as justification within the inference, in the manner of a semantic inference (see Duval, 1995);
Verbal Proposition ==> Verbal Proposition where the graphic proposition born of the drawing
sustains the justification of the inference, in the manner
of a discursive inference, without all semantic
compatibility between the verbal propositions or all
accommodation to the thematic continuity proving to be
insufficient.
Regardless of the type of structure, the meaning of the verbal proposition inferred depends on whether it refers to the domain of interpretation or the domain of function attached to the graphic propositions of the justification --- as in the distinction between the drawing and the figure, Laborde and Capponi (1994). However, even when the justification invokes spatial properties of the drawing, the figurative inference remains structurally and functionally valid. Because therefore, given that the drawing doesn't have objective significance and that it is interpreted at the least in relation to the implicit model of the student, the comparison with the geometric model guarantees the validity of the graphic proposition that underlies the justification and assures, for the possible reader, a degree of acceptance of the proposition inferred. The fact that the meaning of the graphic proposition adheres to a spatial property or a visual property generated by a highly personal model is integrated more in the question of the quality of the inference, a problem that turns more toward:
Advantages and disadvantages of the figurative inferenceFrom a theoretical point of view, the figurative
inference constitutes recognition of an existing bridge
between the geometric figure and the process of proof.
Within the cooperation between the role of the drawing and
the role of the reasoning, if Presmeg (1986) deduces the
importance of reasoning in the development of visual images,
the figurative inference grants to the drawing an authentic
role of justification in a step of reasoning. This is why,
in adding this type of inference to the forms of discursive
reasoning, we complete the set of inferences of Duval (1995)
by a consistent extension that respects the foundation of
the functional definition of reasoning. We even propose to
substitute the semantic unit "the form of discursive
expansion" for "the form of discursive-graphic expansion",
which broadens the scope of the definition, without changing
its spirit or its intention.
In a civilization of instant and visual gratification, it is necessary nevertheless to take cover from the abusive use of the figurative inference. Although its application encourages the development of intuitive reasoning, so important in problem solving and the formulation of conjectures, it risks, in the long run, annihilating the very essence of the process of proof while causing problems of structure (e. g. managing to infer graphic propositions); and the same is true if the student is no longer being forced to begin to construct the reasoning from the beginning, or has the tendency to start out from an earlier admitted statement. Figurative inference is useful insofar as understanding it is a question of decoding signs related to the domain of geometric interpretation, and not images. If one can determine its value as mentioned earlier, its usefulness remains subordinate to, at the least, the quality of the rest of the verbal reasoning. ConclusionJust as the expert uses visual support as a means of introduction to a complex mathematical abstraction (Bishop, 1996), the use of the figurative inference in teaching/learning expands the didactic space available around situations in which reasoning is likely to intervene. Already, the advantages enumerated earlier evoke a natural adaptation of the student during a situation of validation, in a state of equilibrium between what he/she knows and what he/she can do. Verbal (of the lexicon) and symbolic (of the mathematical register) rigidity as much as the obvious linearity of a text, the intrinsic complication that arises in reasoning in geometry without deductive expertise and the difficulty of concentrating simultaneously on several properties obliges the student to use the means that bring him a certain flexibility, whether in action or in reflection. However, contrary to purely intuitive thought, the figurative inference shows itself to be more structured as it connects graphic propositions to at least one verbal proposition (the antecedent or the consequence of the inference), especially when it is incorporated into the thematic continuity of a text or when it emerges in an oral debate. BibliographyBalacheff N. (1987) Processus de preuve et
situations de validation. Educational Studies in
Mathematics 18(2), 147-176. |
