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Godino J.
D., Recio A. M. (1997)
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Abstract |
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1. IntroductionGrowing interest in the problems of the teaching and learning of proof is presently to be found within Mathematics Education (Hanna, 1995; 1996). Research publications on this subject have increased over the last five years (see the special issue in Educational Studies in Mathematics, 1993), though we also find earlier relevant contributions (Lester, 1975; Bell, 1976; Fischbein, 1982; Balacheff, 1987; etc.) This interest is justified by the essential role of validation situations within mathematics, and by the students' poor level in understanding and building mathematical proofs (Senk, 1985; Recio and Godino, 1996; Harel and Sowder, in press). In spite of the aforementioned research, there is still room for research into clarifying the meaning of mathematical proof, its different types and mutual relationships. In particular, the idea of demonstration ,which is understood in a rigid and absolute way by the mathematical community, seems to be the sole, valid conception. We consider it necessary to carry out a systematic study of the various meanings of proof, not just from the subjective point of view, but also in the different institutional contexts. This study would allow for a comparison of the different research contributions, posing new investigation problems, alternative interpretations of students' difficulties and elaborating new didactic proposals. In this research report, we analyze the different meanings that the idea of proof takes in different institutional contexts, using the theoretical framework by Godino and Batanero (1994) and Godino (1996), concerning mathematical objects and their meanings. Here, this implies taking the validation situations and the corresponding argumentative practices as primitive notions. Proof notions emerge from argumentative practice systems. We, furthermore, distinguish between personal and institutional dimensions thereof. 2. Situations of validation and argumentative practicesThe word 'proof' is used with various senses in different contexts. Sometimes these various senses are recognized through terms such as 'explanation', 'argumentation', 'demostration', etc. Though in all theses cases there is a common idea, - that of justifying or validating a statement (thesis) by providing reasons or arguments -, in fact, the differences in the types of situations in which they are used, their characteristic features and the expressive resources used in each case can be different. These changes in situations and argumentative practices suggest different senses of the concept of proof, e.g., various "object proof" according to our ontosemantic model. In this paper we shall use the term 'proof' to refer to the objects emerging from argumentative practices (or arguments) systems accepted at the heart of a community, or by a person, in validation and decision situations. That is to say, situations that require justifying the truth of a statement, or the efficacy of an action. An important distinction between analytical and substantial arguments is made by Krummheuer (1995), following Toulmin. Analytical arguments, characteristic of correct logic deductions are tautological. That is, a latent aspect of the premises is visibly elaborated, but they add to the conclusion nothing more than what is already a potential part of the premises. Substantial arguments, on the contrary, expand the meaning of the propositions to the extent to which they adequately relate a specific case by means of updating, modification, and/or application. From a cognitive viewpoint, we consider that the relationships between reasoning and argumentative practices are those established between a construct and its empirical indicators. Balacheff (1987, p. 148) defines the reasoning as the "very often not explicit intellectual activity of manipulating information to produce new information from data". From our perspective, this intellectual activity gives rise to personal or institutional argumentative practices, which constitutes its ostensive dimension. Simultaneously, reasoning is developed by means of such practices, so that the study of reasoning is intrinsically linked to the study of argumentation. In the next section we shall show that a proof is a contextual and pragmatic attribute of a discursive practice. 3. Meanings of proof in different institutional contextsFrom a cultural viewpoint, Wilder (1981) wote that, "we must not forget that what constitutes 'proof' varies from culture to culture, as well as from age to age" (p. 346). We are trying to show that this relativity must be widened to different institutional contexts, when we are interested in psychological and didactic problems involved in the teaching of proof. We consider that a context or institutional framework is a local viewpoint or perspective concerning a given 'problÈmatique', characterized by using expressive resources and specific tools, as well as habits and specific behavior procedures. In this section, we shall study the diversity of proofs according to the following institutional contexts: logic and foundations of mathematics, professional mathematics, daily life, empirical sciences and the teaching of elementary mathematics (including primary, secondary and university levels). We have to recognize that in each of these contexts it is also possible to identify more local viewpoints in which the problem of truth and proof takes on specific connotations. However, we consider that the level of analysis adopted in this paper is sufficient to show the diversity of identificable 'object proof', and in particular that there is no uniform theory and practice firmly established about mathematical proof. 3.1. Logic and foundations of mathematicsIn these contexts, the veracity of a theorem rests on the validity of the logic rules used in the proof; the theorem appears as a logical and necessary consequence of the premises, through the corresponding deductive inference. A statement (or theorem) accepted as true has a universal and intemporal validity. It is also important to emphasize the nature of the problematic situations that are faced in these contexts. The aim of the validation process is to justify, with the maximum guarantees, the truth of the system of mathematical propositions, or at least part thereof. This implies looking for the minimal independent, non contradictory and complete system of axioms (selfevident truths), such that the other mathematical propositions may be derived by applying the inference logic rules. Hence, it deals with the theoretical problem of organizing and structuring the system of mathematical knowledge. The use of formal languages is required to achieve the greatest guaranties and rigor in this work. The 'object proof' in these institutional contexts may be synthetically described as emerging from the system of analytical argumentative formal practices, and its meaning is given by the intensional, extensional and representational characteristics described. Nevertheless, we recognize that substantial argumentations are also used to justify some statements even in these institutional contexts. In any mathematical system, the acceptance of axioms or postulates is necessarily reached through intrinsically inductive arguments. Let's remember what PoincarÈ (1902)wrote : "What is the nature of mathematical reasoning? Is it actually deductive as it is ordinarily believed? A deep analysis shows us that it is not so; it participates to some extent in the nature of inductive reasoning, and that is why it is productive" (p. 15) 3.2. Professional mathematicsAs regards to the real practice of mathematics, the notion of proof clearly differs from formal logic and foundation studies in mathematics . Formal proofs become extraordinarily complex, which in practice makes the complete formalization of proofs in many mathematic investigations impossible, even when it would be feasible, in principle. "They may require time, patience, and interest beyond the capacity of any human mathematician. Indeed, they can exceed the capacity of any available or foreseeable computing system" (Hersh, 1993, p. 390). As asserted by Resnick (1992), this makes contemporary mathematics full of "working proofs", i.e., informal and non axiomatized proofs. In the field of professional mathematic, proofs are deductive but not formal. They are expressed through ordinary language completed with symbolic expressions. There is no generally accepted standard of rigor for systemizing mathematical proof. In this way, mathematical theorems in fact lose their character of absolute and necessary truths. Real mathematics acquires a falibilist, social, conventional, and temporary character. This situation induces us, in real mathematical practice, to describe proof as a 'convincing argument, as judged by qualified judges' Hersh (1993, p. 389). The problem faced by professional mathematicians is to solve new problems, to increase the knowledge body, and, secondarily, to organize and found the whole system of mathematics. The highest degree of safety of the work carried out by people interested in the foundations of mathematics is not required. 3.3. Experimental sciences and daily lifeProof, in these contexts, is mainly based on substantial arguments (empirical inductive, analogical, etc. ) from which we conclude that what is true for some individual in one class is true for all the members of that class, or that what is sometimes true, will be true in similar circuntances, or with a given probability. The simultaneous use of deductive arguments, in particular statistical inferences, is not discarded:
Proof uses the expressive resources of ordinary language, symbols and any type of concrete devices. In the experimental sciences, experiments or observations are made with maximum care, controlling all possible factors that might influence the results. They also use symbolic resources. Reasoning by analogy plays an important role in natural reasoning showed in our daily inferences. All analogical inferences start from the similarity of two or more things, concerning one or more aspects, concluding with the similarity of those things in another aspect. 3.4. Proof in the mathematics classroomAs a rule, mathematical theorems are necessarily true for secondary and university level curricula, textbooks and mathematics teachers. But arguments establishing their truth are frequently informal-deductive, not deductive, or they are even based on external authority criteria. Elementary mathematics -including mathematics at university courses - is a knowledge whose truth is considered to be completely certain. There are some proofs for theorems accepted by the generality of the professional mathematicians. Therefore, this knowledge has not the falibilist character attributed to advanced mathematics, or at least, is presented in this way in textbooks and in mathematics classrooms. In these institutional contexts, particularly at the higher levels, students are expected to acquire the capacity of understanding and carrying out mathematical proofs, to establish the truth of theorems with absolute safety, and to convince themselves and any person of such unquestionable truth. This is an idiosyncratic use of proof, different from what is done by professional mathematicians. Mathematicians must develop proofs to convince referees for journals; mathematics students must convince themselves, and convince the teacher of the necessary and universal truth of theorems. 4. Personal meanings of proofThe process employed by a person to suppress doubts about the truth of a conjecture is called proof scheme by Harel and Sowder (in press): "A person's proof scheme consists of what constitutes ascertaining and persuading for that person" (p. 12). The different categories of proof schemes they identify represent a cognitive stage, an intellectual ability in students'mathematical development, and are derived from the actions taken by the students in their process of proving. In the ontosemantic model developed by Godino and Batanero (1994), these proof schemes could be personal or mental objects, and their meanings are the systems of practices carried out by the person involved in decision and validation situations. Harel and Sowder distinguish three main proof scheme categories: based on external convictions (ritual, authoritarian and symbolic), empirical (inductive and perceptual) and analytical (transformational and axiomatic). For these authors, the high incidence of the three subtypes based on "external convictions" and of empirical-inductive proof schemes in the students could be explained by the influence of school habits, which reinforce such types of argumentative practices. The analysis presented in the previous sections suggests, indeed, that within elementary mathematics classes argumentative, not analytical practices, may prevail, mainly at primary and secondary school teaching levels. These arguments -unconsciously implemented by mathematics teachers - might be extrapolated from other institutional contexts, such as daily life or empirical sciences. Furthermore, the role played by substantial argumentation in the phases of searching and formulating conjectures in problem-solving should not be forgotten. Analytical arguments, characteristic of mathematical proof, are not the only argumentative practices of professional mathematicians to convince themself about the truth of their conjectures. This form of reasoning is frequently sterile, even an obstacle, in the phases of creation and discovery in problem-solving, where forms of substantial argumentation, in particular empirical induction and analogy, are allowed and even necessary. We may recall the words of Polya (1944, p. 116): "Mathematics presented with rigor is a systematical, deductive science, but mathematics at the embryo stage is an experimental, inductive science". 5. Conclusions and implications for research and teachingCertainly, we may appreciate some common features in the uses of the word 'proof' in the different institutional contexts described. This allows us to think about proof in a general sense. But this generic, abstract, metaphysical way of thinking, should not conceal the rich and complex variety of meanings acquired by the concept of proof, or, better, by the diversity of 'object proof' each one of them exists with a local meaning for the members of such institutions. We believe it is interesting to consider that there is not just a single concept of proof but several, depending on the subjective and epistemological viewpoint, when we are interested in the psychological and didactic problems involved in the processes of validating mathematical propositions (Godino and Batanero, 1994). By recognizing this diversity of objects and meanings, we shall be in a better position to study the components of meaning, the circumstances of their development, the roles performed in the different contexts. In fact, we would better understand the ecological relationships established between objects and the systemic nature of their meaning. This ontosemantic modelization can help to take into account the cognitive conflicts posed to each person forced to participate as a subject in different institutional contexts. Since students are simultaneously subjects of different institutions, at the heart of which different argumentative schemes are carried out, it seems reasonable that students may have difficulties in discriminating the respective use of each type of argumentation. Consequently, we consider that such institutional proof schemes might be explanatory factors for subjective schemes, and therefore they should be taken into account and investigated in depth. It is necessary to somehow articulate the different meanings of proof, at different teaching levels, thereby developing progressively among the students the knowledge, discriminative capacity and rationality required to apply them in each case. Informal proof schemes cannot just be considered to be incorrect, mistakes or deficiencies, but rather as stages in achieving and mastering argumentative mathematic practices. Understanding and mastering deductive argumentation by students require the development of a rationality and a specific state of knowledge. It demands "the adhesion to a problem that it is not that of the efficiency (exigency of practice) but rather that of rigor (theoretical exigency) (Balacheff, 1987, p. 170). But the construction of this rationality is a progressive process that requires time, as well as ecological adaptations of the 'object proof' (didactic transpositions) at different teaching levels. Acknowledgement : This report has been founded by DGES, MEC. Project PS93-0196. ReferencesBalacheff N. (1987). Processus de preuve et
situations de validation. Educational Studies in
Mathematics 18: 147-176. |