(snipped; 3 items about proofs and explanations; following are 2 additional
items)
(4) Won't all mathematics teachers recognize the phenomenon of having
someone grasp how to use an algorithm by way of a few examples? Or at
least how to use it except for exceptional cases of some kind? When an
algorithm fails, e.g. when it leads to results which are contradicted by
experiences or experiments of some kind, do we give up an algorithm which
works well (i.e., well enough) in a myriad of cases? Or are we maybe
inspired to see what's wrong in the exceptional cases, and perhaps to amend
the algorithm in some way? And: Do some of our minds generalize such
procedures without our ever consciously formulating, in some natural or
unnatural language, a general statement of the procedure, capable of being
proved by way of statements in some language (natural or formal), perhaps
with suitable restrictions likewise stated?
(5) I expect that minds have to be prepared or receptive to mathematical
explanations in some way. Can't you say "Behold!" to some people when
showing them a tile pattern which illustrates (say) the Pythagorean
theorem, and then see their eyes light up (maybe because you're a
mathematics teacher, and they're expect some trick like this, or maybe
because they've seen the Pythagorean theorem in some other form before, or
maybe because they're like the slave in Plato's dialogue, the *Meno*) as
they see the desired (non-metrical!) relationship between triangles and
squares --- while others think you're referring to some dirt on the tiles?
What does this signify about proofs and explanations?
Gordon Fisher gfisher@shentel.net