(28C) Mathematics & Pedagogy

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An excerpt from The Limits of Rationality, For the Learning of Mathematics, Vol. 8, No. 1 (Feb., 1988), pp. 14-17, 24

7. The achievements of scientific rationality may seem so substantial that we choose to forget its tendency to tip over into irrationality. The process is more apparent in the human sciences where the danger of pushing rationality too far and forcing it to tip over is only too obvious. Or should be.
  Pedagogy provides an illuminating example …
…
  All pedagogical devices cast their imprint on the matter they are designed to teach. And in case one would be so naïve as to suppose that this difficulty might be avoided by suppressing pedagogical devices altogether, let us remember that when we teach anything to someone who does not yet know it, we cannot proceed without offering the person at least an implicit model of what is to be learned.
  The need for pedagogy comes from another source too. There is an inevitable tension between engaging with mathematics in order to use it and engaging with it in order to teach it. The teacher and the mathematician do not have the same professional insights into mathematics; what is illuminating for one is not necessarily so for the other. The Hindu-Arabic notation, when it reached Europe, played hell with the teaching of arithmetic, causing teachers to substitute "ciphering" for the counting and manipulation of beads and other objects. Giving the number system a solid foundation in set theory was a liberation for mathematics and an aberration in the classroom. The HP 28C is a remarkable mathematical aid, but it is not the calculator that educators would like to have been able to design to sort out some of the difficulties for the learner of college mathematics. Indeed, what is best for mathematics and the mathematician is not always best for teachers and would-be mathematicians. …

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