More Invisibility

At the end of my most recent post I quoted the following idea from a paper on the cognitive analysis of mathematics comprehension:

(1) In order to do any mathematical activity, semiotic representations must necessarily be used even if there is the choice of the kind of semiotic representation. (2) But the mathematical objects must never be confused with the semiotic representations that are used.

The crucial problem of mathematics comprehension for learners, at each stage of the curriculum, arises from the cognitive conflict between these two opposite requirements: how can they distinguish the represented object from the semiotic representation used if they cannot get access to the mathematical object apart from the semiotic representations? And that manifests itself in the fact that the ability to change from one representation system to another is very often the critical threshold for progress in learning and for problem solving.

Multiplication, for example, can have a number of different semiotic representations, including both formal and everyday language referring to multiplication or multiplication processes:


But none of these is multiplication. Mastering the above representations is only minimally helpful when students start multiplying fractions and decimals, so new representations must be employed. And mastering these is only minimally helpful when students work with polynomials, matrices, etc.

This is the starting point for real-world, multiple-approaches proponents. But is it enough to justify their methods?

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