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Harel G.
(1997)
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Mariotti et al's paper is a report on three studies conducted by three teams in three different institutions. Although the report indicates the common features to these three studies (e.g., general goals, research methodology, epistemological analysis and cultural, cognitive, and educational hypotheses), I do not know the extent to which these studies were coordinated prior to and during the teaching experiments. I am mentioning this point because I found the consistency and complementarity among the findings remarkable. For example, in the three teaching experiments, which ranged from grade 5 to grade 10, students' processes of conjecturing and proving shared basic features which reassemble adult mathematicians' work. As the authors have indicated, this is an important observation, for it suggests that "the activities performed during [these] teaching experiments may represent an approach to mathematics theorems which is correct and meaningful from the cultural point of view". Another important example of the complementary and consistency among the studies in this collaborative research is the observation that "in all three fields of experience, the initial outcome is the generation of a space of possible configurations to be explored with different goals". In turn, "in the field of experience of representation of visible space, dynamic exploration contributes to the collective construction of the theory and to production of the conjecture." The authors focus on three major points: (1) The function of the different contexts in approaching geometry theorems These points are situated within and analyzed in terms of two basic constructs: Field of experience and mathematical discussion. Within the limits of time and space allocated to my reaction, I will be unable to fully address the richness of the theoretical framework of this research and the many facets of the findings. I will restrict my attention to a few isolated aspects of this research: I will begin with two interrelated issues: (1) The motivation to proof; and Based on their review of the literature, the authors have indicated that "motivation to proof can be expressed at different levels. At the first level the truth of the fact is central: Is a fact true? At the second level, truth may no longer be in question, but a foundation of truth is needed: Why is a fact true? ... In the first question, the truth of the fact is uncertain while in the second the truth of the fact may be certain." In their opinion, "the uncertainty status of the truth of a statement is crucial for the initial construction of the meaning of theorems and calls for the careful selection of problem-solving situations, where the production of a conjecture is required. A third level, which the authors do not consider in this report, concerns the communication of a validation in a form of formal proof. It is interesting to note that in the US students are first introduced to proofs in the context of the second and third levels. Namely, proofs are first taught in the context of Euclidean geometry, where facts are proven formally (the third level) and where the truths of these facts are seldom in question (the second level). In these teaching experiments, on the other hand, students begin with the first level (e.g., the Sunshadow problems) and proceed toward the second and third levels. It is also interesting to note a historical observation: Euclid Elements--the first Western documentation of a deductive system of rigorous proofs--may be interpreted as an enterprise residing at the second level in this authors' scheme of proof levels. For, to a large extent, the Elements deal with facts whose certainty is seldom in doubt. With regard to this scheme of proof levels, the authors have emphasized an important lesson they had derived from their research; namely, that "the uncertainty status of the truth of a statement is crucial for the initial construction of the meaning of theorems and calls for careful didactical choice of problem-solving situation, where the production of a conjecture is required". Their teaching experiment involving geometrical perspective provides an excellent example of a field of experience in which measurement (i.e., empirical reasoning) ceases to constitute for the students a convincing tool for the validation of observation. I strongly agree with the authors that the uncertainty
status of the truth of a statement is essential to the
initial construction of the meaning of theorems. My own
research observations concur with this position. The
implementation of this curricular recommendation is not
easy. It is particularly difficult in contexts such as "real
analysis" and "abstract algebra" courses. For example,
students are usually satisfied with their intuitive
explanation of why This is an example of what I call "need for formalization" (see Harel, in press). It is less robust and effective than the need for computation, because it requires an adequate level of mathematical maturity, which beginning students usually lack. I am referring to students' lack of appreciation for the need to justify assertions with strong visual or kinesthetic interpretations, such as, the Intermediate Value Theorem (IVT) and the Extreme Value Theorem (EVT). While it is not hard to show students the application power of these theorems, it is not always easy to convince them of the need to prove them. I applied an experimental treatment to this problem, where I assigned an additional purpose to the proofs of these theorems. Namely, not just showing that these theorems' assertions are true--that was unnecessary to the students--but also to test the formalization of intuitive notions, such as those of "limit", "continuity", and "there are no gaps on the real line". So, if we (the class) can, for example, prove the IVT, which we all regard true, by using "only" the e - d definitions of limit, the formal definition of continuity, and the completeness axiom, then we can be certain that we were successful in formalizing the corresponding intuitive notions. Another valuable observation made by the authors is the notion of "cognitive unity". They have indicated that their analysis of the work done by geometers highlights the continuity between the process of producing a conjecture and the process of the construction of its proof or refutation. Their teaching experiments were based on the hypothesis of the existence of cognitive unity in their students' mathematical behavior. As they have indicated, this hypothesis was found to have important didactic implications, "since it calls into question the traditional school approach to theorems" Although I did not investigate the phenomenon of cognitive unity explicitly, I found it very useful to describe my own observations. The authors' definition of this notion, however, requires clearer criteria for analyzing and interpreting the interrelationship between the processes of conjecturing and proving in students' thinking. Also, some of the authors' conclusions regarding cognitive unity demand further research, both quantitative and qualitative. As an example, I mention their conclusion that "the fact that poor argumentation during production of the statement always corresponded to lack of arguments during construction of the proof seems to confirm the close connection that exists between production of the conjecture and construction of the proof. What is meant by "poor argumentation"? Does "this close connection" indicate causality or mere corrolation? What is the nature of the process that mediates between "good" argumentation during the production of the statement and the construction of its proof?. |
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which typically is something to the effect: "
,
because, by their own words, "the larger n gets the
closer is 1/n to -1". This exchange usually results--as my
experience suggests--in a sort of a conflict with the
students, whereby they see a need to modify their conception
of limit.