Abstract
The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what percentage of the population really cares about it.
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Notes
Some people might prefer to use the term “claim” instead of “conjecture”. I deliberately chose “conjecture” because it is a kind of technical notion in the philosophy of science and it is supposed to remind us of Popper’s (1963) “Conjectures and Refutation”.
Note that in many explanations in this paper, I say, “for example.” It demonstrates the crucial role that examples have in explanations. Sometimes, when I use this expression, its role is to construct a class of objects as mentioned above. I am not sure that this is always the case, but I do not want to deal with this kind of sophistications in a mathematics education paper. To avoid the exaggerated use of the expression “for example”, I will also use the expression “for instance.”
Elsewhere (Vinner, 2007), when I spoke about the role of rituals in teaching, I even suggested for them some names. Here, I suggest calling this a ritual “the ritual of pointing at an object and calling it by its name.”
I myself have a hard time identifying objects, which are claimed to be chairs, designed by post-modern industrial designers.
Some of the explanations given in this paper were taken from The Free Dictionary (http://thefreedictionary.com).
There is no point in arguing whether to accept or reject it as a definition of a poem in the context of this paper. However, it is important to distinguish between this kind of definitions and mathematical definitions.
Note that classes or sets exist only in our mind. They do not exist in the real world. In the real world we have only different objects. It is our mind that considers them as a unity, namely, a class or a set. It is clear that this holds also for abstract notions. Their examples (a metaphor, for instance) were, from their very beginning, products of our mind.
Note that some classical dictionaries (the Merriam Webster, for instance) tell us the year in which the notion officially appeared as a “legal element” of the language.
This might be the Kindergarten teachers’ concept image and, sometimes, even their personal concept definition.
Namely, a function is any correspondence between two non-empty sets such that to every element in the first set (the domain), it assigns exactly one element in the second set (the range).
When I introduced such sequences to my students, some of them argued that these were not sequences, because in a sequence the nth element depends on n. In other words, it should be a function of n. This was their concept image, which was probably formed by the examples they had in their previous examples or even by the common notation a n . To convince them that the constant sequence is a legitimate one, I suggested to them the sequence: a n = (−1)2n c.
I am aware of the fact that many people who are involved in mathematics education (especially mathematicians, as well as some mathematics educators) disagree with this recommendation.
Note that I speak about the mind and not about the brain. There are two reasons for that: first, I am not a brain expert; second, even brain experts cannot point at this stage, in spite of their enormous scientific achievements, at brain structures and processes that can explain phenomena such as curiosity or the need to generalize.
This word combination is almost an oxymoron. The use of the word “conjecture” usually implies that its status is temporary and, theoretically, it can be refuted.
Of course, it is relevant also to scientific thought processes, whether these occur in natural sciences, social sciences or medicine, as well as to business administration, accounting and more.
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Vinner, S. The role of examples in the learning of mathematics and in everyday thought processes. ZDM Mathematics Education 43, 247–256 (2011). https://doi.org/10.1007/s11858-010-0304-3
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DOI: https://doi.org/10.1007/s11858-010-0304-3