>I don't think I mentioned what usually happened, at least >in the post above (although I do recall writing about it, >so perhaps this was in another post), but my experience >was that almost always the students who could carry out >these coping strategies were already good in math and >usually discovered several on their own while the other >students (with a few very inspirational counterexamples) >found these coping strategies made things more difficult >for them by being yet another thing they were supposed >to learn how to do.
Right on! (As we say in the 'hood.) The reductio ad absurdum is the preposterous little book, "Innumeracy" by John Allen Paulos. Purporting to help the innumerate, "innumeracy" is filled with the kind of cleverness beloved of the mathematically inclined, just the sort of thing that infuriates and frustrates everybody else (about, oh, 98% of the world population), unless it just washes off them like water off a duck's back.
Just this argument is also the refutation to the use of technology in education. The top 5% of students can use Mathematica or Matlab to great advantage. To the rest of them, these programs are just one more barrier to mathematical achievement.
In this forum, we generally have in mind K-12 education when we discuss such issues. An interesting special case is Calculus and Mathematic at the U of Illinois. I seem to recall that, some years ago in this forum, Jerry Uhl gave Wayne Bishop a tongue-lashing for daring to criticize C&M. What Uhl did not supply was results.
Wayne would know much more about this than I, but as I recall, C&M was part of the now defunct Calculus Reform Movement. That movement began circa 1985 with the recognition that far too many college students were falling out of the calculus curriculum. Working purely from memory, I seem to recall that roughly half of calc I students do not progress to calc II; roughly half of calc II students do not progress to calc III; and I do not recall how many students throw in the towel after calc III. I suspect it is a considerable number.
The purpose of the Calculus Reform Movement was to reduce the attrition rate. Uhl sings the praises of C&M in every conceivable way except one: does C&M reduce the attrition rate? In other words, I do not doubt at all that C&M is a better program for better students, but what about the rest of them, the original intended beneficiaries?
If I am mistaken on this point, I would be very pleased indeed to be corrected. But if I am right, however, then C&M is just one more example of technology in education, about which the very best we can say is that it does no harm. Although, for the less able, less motivated, less prepared students, I doubt we can say that about C&M or any other such effort to teach with technology.