Wednesday, May 2, 2018

Geometry Snacks

What a sweet book!

These are two amazingly creative and fun thinkers about math. I feel a bit like that theme in The Shack, where everybody is one of God's favorites, but these are two of my favorite people on math twitter.

I thought of Ed as @solvemymaths for years, with great resources, dead clever problems and, of course, the math Mr. Men.  Check his blog for all this and insightful writing on teaching and curriculum. He's inspired some GeoGebra work from me, too.

Vincent, @panlepan, I think, came to my attention through his math art tweets, but maybe it was in a Simon Gregg discussion, or possibly a tessellation deep dive... He's introduced me to many interesting bits of math history, art, and people on Twitter. He's always willing to think aloud and do problem solving in public.

So I was very positively predisposed to like a book from the two of them together, and I was not disappointed.

For more of a preview of the book, check Ed's Twitter feed, or an Alex Bellos introduction. But probably you should just go ahead and eat it, er, order it. Snacks this good will make you hungry for more.

These puzzles and problems are so good. It is composed of five sections, with the somewhat surprising order of What Fraction Is Shaded?, What's the Angle?, Prove It!, What's the Area? and Sangoku. Each section is followed by solutions. Not just answers, but real solutions that guide the reader through the thinking of one way to solve each problem. And these problems tend to admit multiple solutions each, so knowing way to solve a problem does not make it worth much less for thinking about.

The problems are all visually presented, in beautiful black red and grey, almost crossing the border into pop art. Often they are stark in their simplicity. "This can't be enough information to solve it!" But then one relationship occurs to you, then "if that's true, this must be the case, and what if I..." In other words, the visual problem posing invites connections and problem solving. In my classes, one of my favorite definitions of math is "Math is the study of what else do we know?" and this book exemplifies it. I am not sure about this, maybe it's just me, but there is almost a sense of humor in these problems. Maybe it's whimsy? Maybe just the authors' sense of delight in the mathematics coming through.

While I can conceive of another book of problems that are this accessible and engaging, what would still set this book apart is the organization. The sequencing of the problems is intuitive, almost curriculum-like, but in a good way. The principle that helps solve one problem is often applied in a new context in the next problem, or needs to be extended in an upcoming puzzle. The fraction problems familiarize you with the shapes in the angle problems. The sequenced reasoning about angles is a lead into the idea of other sequenced reasons in the proofs. The proof reasoning prepares the reader for computation with measurements in the area section. The Sangaku problems are not the classical how do you construct the image precisely, but problems posed about measurements and relationships in those harmonious arrangements.

A nice indicator of how accessible these problems are is that a similar problem of Ed's blew up into an internet sensation, the Pink Triangle. What fraction is shaded?
People have proposed roughly a jillion different ways to think about this, which explore traditional geometric techniques (add extra lines, transformations, similarity), estimates, discovering properties of partitioning fractions, and so on. Such a simple prompt, and I would argue even noticing the conditions is doing mathematics. This book encourages Polya's first phase: understand the problem, so often neglected.

So obviously I'm recommending this book. For yourself, for your role as a teacher or parent, or as an appreciator of the mathematical aesthetic. You will snack to satisfaction and return for more.

It was that or end with Bon Appetit, which I'm sure has been used in a healthy fraction of reviews for this book already.

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