Anthyphairesis, the “originary” core of the concept of fraction
Résumé
In spite of efforts over half a century, the results of teaching and learning fractions are not satisfactory. In response to this trouble, we have proposed a radical rethinking of the didactics of fractions, that begins with the third grade of primary school. In this presentation, we propose some historical reflections that underline the “originary” meaning of the concept of fraction. Our starting point is to retrace the anthyphairesis, in order to feel, at least partially, the “originary sensibility” that characterized the Pythagorean search. The walking step by step the concrete actions of this procedure of comparison of two homogeneous quantities, results in proposing that a process of mathematisation is the core of didactics of fractions. This process begins recording the act of comparison by a pair of natural numbers, and is realized in the Euclidean division. Classroom activities ensure that children perceive the Euclidean division as the icon of their active process of learning. The Euclidean division becomes the core of many circular paths along which the teaching process is developed.
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