Tuesday, March 12, 2013

The Practical... and the Philosophical

1) Guillermo Bautista recently posted a list of 13 online sites giving calculus tutorials... might be useful for some (I'm not necessarily endorsing them, but just passing along, and also adding the link to the right-hand column "math instruction" list):


2) A different blogger has interestingly spotlighted an older post by Terry Tao on 'rigor in mathematics' and intuitions:


May be a bit too philosophical for some, but anything from Dr. Tao of course is worth consideration.

Tao divides mathematical education into three stages: the "pre-rigorous," "rigorous," and "post-rigorous" stages. I like the approach he takes:
"The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field."

Read the entire post here:


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