*Rediscovering Mathematics: You Do the Math*, by Shai Simonson is a great book. It starts with his piece on

How to Read Mathematics, which I love, and wends its way through lots of mathematical problems, and lots of ideas about how to teach and learn math.

When my review copy arrived yesterday I was disappointed to see the hard cover. I suspected that would mean a higher price than most people would find comfortable. The damage is even worse than I expected - $65 (or $53 for MAA members). I think it's a shame to limit the distribution of such good material with a cover price like that. [I had considered approaching the MAA about publishing my book, but perhaps they don't know how to produce books at affordable prices.] Note added five years later: Now it's available used for just over $30. Much better!

The only other objection I have to this lovely book is that solutions immediately follow the problems posed, so it's hard to resist the temptation to peek. I just now managed to resist temptation, and sat with the problem below (from page 175) until I got it. I don't teach geometry, so my geometric intuitions aren't well-honed. It took me a while, and I reverted to algebraic reasoning for parts of it.

Given the square ABCD, with side length 1 and circular arcs centered at each vertex, find the area of the region at the center - *without* using calculus.

I broke it down into shapes I could handle, and got an answer something like his but not quite. I haven't yet found the discrepancy. (My process was very different from his.)

Simonson includes a number of problems I haven't seen before, which is quite a feat after all the grazing I've done online in the past few years. And the problems are at lots of levels, so there is much to chew on whatever your mathematical sophistication.

I like his perspective on how math should be taught and learned. Here he is on memorization (page 169):

Every mathematical idea has a story. To remember the idea, just recall the story. In mathematics, the stories are proofs and the endings are theorems. The more you turn a proof into a story, the easier it is to remember the ending. Can you tell me what you did last summer? Of course you can. Did you memorize that? Surely not; there is a context and one thought leads to another. Of course it can get a little tedious recalling a story a hundred times just to get to the ending, so sooner or later one just knows the ending. This is the kind of memorizing that a student should do with mathematics.

An example of this comes earlier. On page 131 he writes, "In the 1960's, it was popular in the U.S. for the middle school math curriculum to include a square root algorithm. The logic of the algorithm ... is clever but cumbersome, unintuitive, and inaccessible to students and most teachers." He compares this method to a Babylonian method: To find the square root of x, guess (r1), divide x by your guess (x/r1), for a better estimate average your guess with this quotient (r2=(r1+x/r1)/2), repeat. Simple, elegant, transparent.

I didn't learn an algorithm for finding the square root in my U.S. math classes, but I did learn one during my junior year, while attending high school in Brazil (as an exchange student). I've wished I could remember how it worked, but never could dredge it up. I don't think it was anything this straightforward.

Most students will not forget the Babylonian method because it makes sense. It is a story. It can be remembered because it can be reconstructed. There is context, purpose, and structure. (page 132)

I'll be sharing quite a few nuggets from this book with my students. If you can afford it, I highly recommend getting this book.