Symbole, Referenzkontexte und die Konstruktion mathematischer Bedeutung – am Beispiel der negativen Zahlen im Unterricht
When introducing negative numbers mathematics teaching is confronted with the following developmental problem: The students interpret natural numbers within the frame of concrete, empirical conditions of counting, adding and taking away etc. Negative numbers are no longer immediately applicable to real situations nor is it possible to deduce them logically starting from concrete contexts. According to the methodical principle to begin with the concrete and to go up to the abstract, mathematics teaching handles this contradiction between the concrete, visible perception and the formal-relational structure of the concept of negative numbers in a 'linear' way: The aim is to deduce this new concept in a methodically natural manner from the already known epistemological frame of the natural numbers. This didactical intention contradicts the epistemological insight, that the new concept cannot be reduced to empirical facts. The new symbol system of negative numbers, which is constructed by an 'operational extension' needs a generalised re-interpretation: These symbols no longer refer to empirical properties of real objects, they reflect relational structures within and between objects, and these relations have to be socially negotiated and to be controlled by the mathematical rule-structure.
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