There is a very powerful article on numerical sense in the July Discover magazine. It summarizes a ten year research project ron the human brain by Stanlislas Dehaene in France. Dehaene was originally led into this project by working with a brain damaged fellow who lost all ability to calculate but retained the ability to estimate.
From the article: After getting his masters [in mathematics] Dehaene gave up doing mathematics in favor of trying to understand how the brain does it. One conclusion he has reached after a decade of research is that the brain has a very hard time. Dehaene writes:"Rigorous calculations do not come easily to Homo Sapiens. Like so many other animals, it is born with a fuzzy and approximate concept of number.... While our culture invented logic and arithmetic,our brain remained unchanged and restive to even the simplest of alogorithms."...."the brain isn't capable of doing long division without concentrating on the mechanics. It takes extreme concentration and when you see the brain activity, it's enormous. It's really hard work. And meanwhile it's not concentrating on the meaning of what it's doing and when it makes a mistake, it's an monstrous one. I think it's better that children do their calculations with calculators. At least then they have the result right away. They don't spend a minute thinking about how they got the result, and they can confront the size of the result with the number they statrted with and develop their intuition that way."
Dehaene's book "Number Sense" will be published in the US this fall.
---------------------------------------------------------------------- Jerry Uhl firstname.lastname@example.org Professor of Mathematics 1409 West Green Street University of Illinois Urbana,Illinois 61801 Calculus&Mathematica Development Team http://www-cm.math.uiuc.edu http://www-cm.math.uiuc.edu/dep
[If] logic is the hygiene of the mathematician, it is not his source of food. ---Andre Weil
Only professional mathematicians learn anything from proofs. Other people learn from explanations. ---R. P. Boas
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I repeat: We simply don't have enough evidence about the role of computational skill in algebra, especially at the calculus entry-level, to decide what its place in (to paraphrase Mark Van Doren) the natural history of a mathematical mind might be.
I can't help but note that Klein has responded to just one of the issues I raised...