The following learning requirements, which are development-adapted on the one hand and development-improving on the other hand, are, on the basis of corresponding targets and contents of the maths lessons in connection with a creativity-friendly learning atmosphere, suitable to produce creativity in maths lessons:
• It is allowed to make a mistake and a piece of rough paper is allowed, too (assessment-free moments during the lessons, time to think – possibility of stress-free learning)
• Every idea is taken seriously (mutual esteem of students and teachers; unprejudiced contacts)
• Different proceedings are not only tolerated and accepted but also adopted – for example in tests.
• The students’ need for orientation is taken into account by clear targets and valuation standards as an orienting guideline for the personal learning and by avoiding small-step procedures which are rather obstructive to independent acting and thinking
Concerning the targets and issues of maths lessons it should be aimed at
• a balanced relation between formal and application-oriented competencies,
• coordinated preparation of the courses to avoid insular knowledge,
• teaching of different mathematical terms, correlations and procedures as well as of methods and techniques which can be used for one special topic (modular subject-oriented curriculum structure versus one-dimensional systematically theory-oriented structure) so that a large fund is available.
Concerning the nature of the learning requirements the following methodical aspects are of importance:
• optional exercises (to be chosen personally from an exercise fund on one topic with different degrees of difficulty and sometimes in different variations)
Prerequisite that these possibilities are made use of (avoidance of over- or underchallenging) is that the students are gradually enabled to come to a realistic self assessment (realization of capabilities and weak points) and to take over responsibility for their personal learning.
• “open” exercises
as learning requirements which can be managed by different procedures and with different results (e.g. find a story on a path-time-diagram; compare several exercises and their solutions under different aspects) and especially:
• invention of own exercises
as learning requirements from which further, additional or more detailed questions can be developed by modifications/variations/supplements of a given exercise (blossom model of the variation of exercises - especially suitable for the inner mathematical formulation of questions) in the form of problems where mathematical questions have to be developed first (funnel model of the variation of exercises – e.g. in application- or modelling exercises)
• problematic exercises in the “zone of the next development phase”, e.g.: “River width” Find as many different methods as possible to determine the width of a river with side marker, protractor and tape measure!
There are more than 10 different solutions to this problem. It can already be solved with the basic knowledge on angles and right-angled isosceles triangles available in classes 5 and 6. In the course of the school years the mathematic instruments to determine a distance are gradually increasing (parallelogram, intercept theorems, trigonometry, vector methods).
The following aspects are essential to make students want to learn problem solving:
• Praise and appreciation for particular achievements, for unusual, new solutions; support of sound competition
• The experience that maths can be generated, the connection between sense and significance becomes transparent
• Choice of subjects in dependence of the students’ age, consideration of their changing leisure interests and surroundings without completely orienting the subjects to them – students must also be confronted with facts they are not (yet) interested in
• The exercise contains some surprise and arouses astonishment, which leads to questions like: How is this possible? Is this always the case? Is this really right?
• The formulation of a question is addressed to the students’ supposed competences:
How would you decide?
Discuss your decision!
Is there another way – or another possibility?
Another condition connected with motivational aspects consists in an adequate quality of the requirements:
If exercises in a concrete learning situation are appropriate to and supporting the development of a student, they can be assumed as a challenge. The art of teaching consists for the most part in finding such adequate and thus motivating demands to take learning action and to assist the initiated process of dealing with it. A solution to this problem are the so-called “open exercises” which have differentiating effects as the students are allowed to choose the level of processing on their own. However, this procedure implies another problem: how can a desired learning success be reached (only!) by open questions? We can state that: a multitude of exercises is not yet a guarantee for successful learning and: exercises which offer more room to construct individually suitable learning exercises improve the learning success. However, a certain quality level must be guaranteed.
If we take a large exercise concept which covers the different demands for learning activities, the teaching substance of the lessons can be described as “working with problems”. This includes the construction, choice and formulation of exercises as well as the kind of assistance administered to the students during their processing of the exercises, cf. Bruder (2000d).
Working with problems includes:
• choosing or constructing, varying, formulating, solving, comparing, evaluating and setting problems by the teacher
• finding, modifying, comparing, setting and solving problems by the students and the assistance in this process by the teacher.
To determinate these multiple activities from the students’ point of view, we are talking about problem processing in contrast to – what has always been expected most – problem solving. Working with problems has the following functions in the lessons:
• problem processing as means (way) to achieve skill and knowledge
• problem processing as diagnostic instrument for the proceeding and results in the learning process
• capabilities in problem processing (problem solving!) as intended objective
Why should students achieve also abilities in the processing of problems in the sense of the above-mentioned activities? Isn’t it sufficient to solve carefully chosen and representative exercises? If students find, modify and compare exercises, these activities go clearly beyond the solving of current exercises without being automatically much harder. The requirements are a bit different but appropriate to meet the initially mentioned claim for the assumption of responsibilities concerning the own learning and the reflection about the personal proceeding.
The varying of exercises by the students can easily be justified as preparation of a test: Whoever is in a position to transform the formulation of an exercise or to join another aspect to the given problem will be less confused by unfamiliar exercises. Unfortunately it can often be observed that students who are fixed to a certain kind of questions are feeling helpless in front of even smallest modifications of the formulation. Such phenomena will occur especially in case of a teachers change. To gain more flexibility students should learn explicitly how to ask sense-making questions. They should also learn something about the center of interest in the different sciences.
The comparing of problems has particular importance. Actually the problems are meant, not the results! Once a series of several problems or tasks was processed and their results compared or presented, these problems and their solutions are often put aside without taking further activities. In reality this is the phase to gain available skill and knowledge! If the students are asked now to find out common aspects and differences in the just solved problems, the exercises will be analysed once again from another point of view. The given questions can be compared as well as the solution ways and possibly also the results, e.g. with respect to their existence and clearness. The purpose is to grasp the sense of the questions: what was the nature of the problem? Which concepts, proceedings, solution methods and strategies were helpful?
Such working with problems requires prospective working in the important reflection phases as well as a sensible guidance of the students by the teachers in frontal teaching talks. Relevant meta-tasks to compare different problems could be introduced in upper classes or in the form of learning reports or learning diaries which have to be kept independently. There are lots of suitable methods and organisation forms – decisive is above all the quality of the learning requirement.
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