Already KIENEL (1977) distinguishes in his dissertation five categories of mathematical “problems”. Problems of Types I – III can be solved by applying a rule or an algorithm or a procedure. In problems of Type I the rule or algorithm or procedure is mentioned explicitly, in problems of Type II the rule or algorithm or procedure is known to the problem solver, but it is not mentioned explicitly. The rules or algorithms or procedures necessary to solve a problem of Type III first must be constructed by the problem solver via combining known rules or algorithms or procedures. Problems of Type IV are given verbally as “a real world” problem, and the mathematical content first must be analyzed and transformed into a mathematical problem to get then a problem of Type I-III. Type V puts together all those “real world” problems, where neither the knowledge of rules or algorithms or procedures nor a specific knowledge of facts, of data, of relations, of properties, etc. is sufficient to get a solution. Especially "open" problems or "challenges" are problems of Type V. To solve these problems we need a new idea or a “cognitive jump”, "a divergent or creative thinking is necessary" (KIENEL, p. 122). This leads us to a central question.
How can we describe or define "creative thinking"? Many experts from different disciplines give various descriptions, but there is no standardized answer. "Creativity" is a highly complex phenomenon, and for some people it seems to be somehow incompatible with mathematics teaching. The traditional style of working in the mathematics classroom seems not to allow many creative ideas. But again, how can we characterize "creative thinking"? We will not provoke the impression that creativity can be described by a long list of isolated items nor that such a list may help to identify or to develop creative ideas. But, as a first step to further the development of creativity, the flavor of such lists may help teachers and text book writers when they prepare classroom lessons. To develop and further creativity in mathematics education teachers and students need more than a correct and solid mathematical knowledge.
REN ZIZHAO (1999) demands independence ("instead of simply repeating other's old tricks") and relative originality. For KIESSWETTER (1983) flexible thinking is one of the most important abilities. For BISHOP (1981) two complementary modes of thinking are necessary, a logical, one-dimensional, language orientated aspect and a visual, more-dimensional, intuitive view. KRAUSE et al. (in ZIMMERMANN 1999, p. 129ff) even distinguish four basic components for creative thinking, finding analogies, double representations (visual-perceptual / formal-logical), multiple classifications and reducing complexity. When GRAY and TALL (1991) and others refer to "procepts" or "encapsulation of a process" they also describe abilities necessary for being creative. Again, there are various aspects, but there is no standardized description of "creativity".
A mathematics teaching which furthers creative thinking needs specific environments. In our research group in Muenster we try to concentrate on three aspects. First, we must further individual and social components, like motivation, curiosity, self-confidence, flexibility, engagement, humor, imagination, happiness, acceptance of oneself and others, satisfaction, success, ... We need a competitive atmosphere which still allows spontaneous actions and reactions, we need responsibility combined with voluntariness, ... We need tolerance and freedom, for the individuals to express their views, and within the group to further a fruitful communication. We need profound discussions as well as intuitive or spontaneous inputs and reactions.
Second, we need "challenging problems". They must be fascinating, interesting, exciting, thrilling, important, provoking, ... Open ended problems are welcome or challenging problems with surprising contexts and results, ... We must connect the problems with the individual daily life experiences of the students, we must meet their fields of experiences and their interest areas. The students must be able to identify themselves with the problem and its possible solution(s). And third, the children must develop important abilities. They must learn to explore and to structure a problem, to invent own or to modify given techniques, to listen and argue, to define goals, to cooperate in teams, ... We need children who are active, who discover and experience, who enjoy and have fun, who guess and test, who can laugh on own mistakes, ... That means, another step to further creative thinking is to further the development of these abilities. But they are demanding abilities and not simple skills. They rely and depend on a complex system of cognitive processes. Perhaps it may help a bit to analyze these internal processes.
A successful problem solving depends on the cognitive structure the problem solver has. An appropriate “internal representation” or concept image is necessary. The problem solver must have adequate “Vorstellungen” (MEISSNER 2002). These "Vorstellungen"1 are like scripts or frames or micro worlds, and they are personal and individual. They are “Subjective Domains of Experiences”2 (BAUERSFELD 1983).
The goal of mathematics education is to develop mathematical "Vorstellungen" which are extensive and effective, which are rich and flexible. We distinguish two kinds of "Vorstellungen", which we call "spontaneous Vorstellungen" and "reflective Vorstellungen". Thus we refer to a polarity in thinking which already was discussed before by many other authors. “Reflective Vorstellungen” may be regarded as an internal mental copy of a net of knowledge, abilities, and skills, a net of facts, relations, properties, etc. where we have a conscious access to. Reflective Vorstellungen mainly are the result of a teaching. The development of "reflective Vorstellungen" certainly is in the center of mathematics education. Here a formal, logical, deterministic, and analytical thinking is the goal. To reflect and to make conscious are the important activities. We more or less do not realize or even ignore or suppress intuitive or spontaneous ideas. A traditional mathematics education does not emphasize unconsciously produced feelings or reactions. In mathematics education there is no space for informal pre-reflections, for an only “general” or “global” or “overall” view, or for uncontrolled spontaneous activities. Guess and test or trial and error are not considered to be a valuable mathematical behavior. But all these components are necessary to develop "spontaneous Vorstellungen". And these spontaneous Vorstellungen mainly develop unconsciously or intuitively.
Both types of “Vorstellungen” together form individual “Subjective Domains of Experiences” (SDE). For a well developed and powerful SDE both is essential, a sound and mainly intuitive “common-sense” and a conscious knowledge of rules and facts. Both aspects belong together like the two sides of a coin. And whenever necessary the individual must be able, often unconsciously, to switch from the one side to the other. But the way of looking at things is different. Spontaneous Vorstellungen and reflective Vorstellungen often interfere, positively or negatively. The view on facts, relations, or properties “suddenly” changes. There also is a competition between different SDEs to become dominant when a new problem is presented. Intuitively, the then chosen SDE often remains dominant even when conflicts arise. The individual rather prefers to ignore the conflict than to modify the SDE or to adopt another SDE. And in mathematics education it is quite natural that a “analytical-logical” behavior remains dominant and that conflicting common-sense experiences or spontaneous ideas get ignored4. The chosen SDE even then remains dominant when the reflective Vorstellungen obviously are not sufficient to solve the problem. Then the related rules and procedures just get “reduced” or “simplified” or get replaced by easier “mechanisms”.
Summarizing, we distinguish two types of internalizing “external representations” and experiences. We get (mainly conscious) reflective Vorstellungen and (often intuitive) “spontaneous” Vorstellungen. Related to the momentarily situation both types together create or modify an individual “Subjective Domain of Experiences” in which this situation is imbedded then. But what does creativity mean in this context?
Studying the diverse descriptions how to define “creativity” we can detect some common aspects. We need more than reflective Vorstellungen (not only “simply repeating other's old tricks"). Flexible thinking is demanded (KIESSWETTER), especially in two complementary modes (BISHOP, KRAUSE). And now we should add, we need flexible thinking with regard to the reflective Vorstellungen and with regard to the spontaneous Vorstellungen and we need a flexible thinking to combine or interweave or expand these two types of Vorstellungen. Flexible thinking includes independence. We must not rely on a few dominant SDEs, we need the chance to experience, to construct, and to reflect many divergent SDEs. We then can detect and discuss analogies and differences and multiple classifications. Reflecting these activities we gain more insight and reduce the complexity. A social communication is the vehicle to widen consciously the horizon.
To further creative thinking in mathematics education we need more than powerful reflective Vorstellungen. Intuitive and spontaneous components are necessary. Each SDE is a mixture which allows different ways of looking at things. A balance between an reflective arguing and a common-sense thinking must become the goal in the class room: Let us start with a typical situation and try to collect and to discuss then all the individual SDEs coming up. Let us try not to separate artificially our daily life knowledge and experiences from the development of the “scientific concepts” in mathematics education. Above we had listed aspects which we think are necessary for a creative mathematics teaching:
1. We must further individual and social abilities and
2. we need challenging problems.
And now we can add:
3. We need more spontaneous ideas and more (unconscious and intuitive) common sense knowledge.
Creative thinking then may develop as a powerful ability to interact between reflective and spontaneous Vorstellungen.
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