数学创造力外文翻译资料

Mathematical Creativity

Creativity plays a vital role in the full cycle of advanced mathematical thinking. It contributes in the first stages of development of a mathematical theory when possible conjectures are framed as a result of the individualrsquo;s experience of the mathematical context; it is also plays a part in the formulation of the final edifice of mathematics as a deductive system with clearly defined axioms and formally constructed proofs. It is an essential factor in research mathematics when new ideas are formulated in a manner previously unknown to the mathematical community. Yet it is external to the theory of mathematics. It is a human activity, a meta-process, which acts upon and generates new mathematics. As such it is often viewed as a mysterious phenomenon. Most mathematicians seem to be not interested in the analysis of their own thinking procedures and do not describe how they work or conceive their theories. Only a few (such as Poincareacute;, Hadamard) explicitly attempt to describe ideas related to mathematical creativity. The best known reference (at least to mathematicians) is probably Hadamard (1945), which has been followed recently by Muir (1988).

The present chapter will not attempt to give an exhaustive description of the nature of mathematical creativity and how it works. From a somewhat closer look at the aspects of different kinds of mathematical activity as an heuristic procedure to register examples of mathematical creativity, we derive some striking characteristics of the phenomenon and frame a tentative definition. The reader is invited to activate his/her own imagination and attentiveness and to rectify deficiencies in the text with their own personal observations.

1. The Stages of Development of Mathematical Creativity

Mathematical creativity does not, presumably cannot, occur in a vacuum. It needs a context in which the individual is prepared by previous experiences for the significant step forward in a new direction. Such preparation occurs through previous activities which form an appropriate environment for creative development. We hypothesize that the context for creativity is set by a preparatory stage in which mathematical procedures become interiorized through action before they can be the objects of mathematical thought. Preliminary to this may be an initial stage where the procedures might be used without even a full appreciation as to their theoretical status.

Stage 0: A preliminary technical stage

We hypothesize that genuine mathematical activity may be preceded by a preliminary stage consisting of some kind of technical or practical application of mathematical rules and procedures, without the user having any awareness of their theoretical foundation. We refer here to the art of the craftsman who applies a set of mathematical procedures as a toolkit providing him with the necessary tools to manufacture his product. The justification for these procedures is that it has been checked empirically that they work, in the sense that a correctly applied rule always yields the desired result. An example of such a practical procedure is the rule used in Ancient Mesopotamia and Egypt to stake out a right angle: they used a rope with marks dividing it into three parts of length 3, 4 and 5. Forming the contour of a triangle, they obtained a right angle between the sides of length 3 and 4. This preparatory stage has become part of modem theories of mathematics learning, for instance, the “tool- object” dialectic of Douady (1986) which proposes that an idea should be introduced first as a tool as part of a problem-solving activity, to become part of the individualrsquo;s experiential cognitive structure before being reflected on as an object in its own right.

Stage I : Algorithmic activity

At this stage procedures are used to carry out mathematical operations, to calculate, manipulate, solve. Algorithmic activity is essentially concerned with performing mathematical techniques. Examples of such techniques are: application of an algorithm, working out formulae, factorizing a polynomial, calculating an integral, computational activities involving computer programs such as in numerical methods for solving differential equations. A characteristic of such activities pertaining to this first stage is that they need to be quite explicit. All intermediate steps have to be considered, at least implicitly; if not, a serious error may occur and totally invalidate the result. In the case of a computer algorithm, no steps, not even trivial ones, may be forgotten. There is no regeneration of missing steps in an algorithm.

Such activities are an acceptable part of advanced mathematics because they may be seen as part of an overall theory, created in accordance with the principles of the higher activities to be described in stage 2. We hypothesize that algorithmic activity is an essential part of the learning of mathematics because such processes must be interiorized and become routinized before they can be reflected upon as manipulable mental objects in a higher order theory. (See chapter 7 for a discussion of reflective abstraction, in which a process is encapsulated as a concept). As with the tool-object dialectic, it is essential that the tool become familiar in action before it becomes the focus of reflective activity.

Stage 2: The creative (conceptual, constructive) activity

It is at this second stage that true mathematical creativity is likely to occur and act as a motive power in the development of a mathematical theory. A non-algorithmic decision is taken in a manner which seems to signify a bifurcation of the underlying concept structure. Mathematical creativity is the ability to perform such steps. The decisions that have to be taken may be of a widely divergent nature and always involve a choice, such as a choice of a certain concept to b

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