Measurement: A Review

Measurement by Paul Lockhart

My rating: 3 of 5 stars

I believe that Measurement is meant to be more or less a math text book for students around the middle-school or high-school grades. I’m not in the intended audience, and I’m not a math teacher. I’m just a professional mathematician, so take my opinions with a grain of salt.

I first encountered Paul Lockhart (as many did) through A Mathematician’s Lament, an essay highly critical of the status quo in math education. Lockhart’s essay was controversial, but many people loved it. I was one that did.

Maybe this excerpt will give an idea of Lockhart’s approach:

So no, I can’t tell you how to do it, and I’m not going to hold your hand or give you a bunch of hints or solutions in the back of the book. If you paint a picture from your heart, there is no ‘answer painting’ on the back of the canvas. If you are working on a problem and you are stuck or in pain, then welcome to the club. We mathematicians don’t know how to solve our problems either.

With Measurement, Lockhart demonstrates how much more difficult it is to do something right than to point out the flaws in how others are doing it. That being said, Measurement is a brave and important shot at doing math education right. It is far from being a failure. I really liked Part One: Size and Shape, which deals mostly with geometry. Part Two: Time and Space, which introduces calculus, seemed much weaker.

Lockhart included lots of great hand-drawn diagrams. (I’m assuming they are hand-drawn. Maybe they were just made to look that way.) He also has included lots of exercises. Here is an example of his diagrams and exercises in one:

I didn’t actually do a lot of the exercises. They just looked too darn hard.

Like I said, Part One I enjoyed. I had a lot of fun with it. Two things that really sparked my interest were projective space and parabolas. Yes I’ve seen both of these things many times before, but Lockhart gave me some brand new perspectives on them. In fact, I became inspired (jointly by Lockhart and my good friend Jason Lee) to compile a list of mathematical properties of parabolas. You may hear more from me on that in the future…

Part Two was interesting, but it seemed much more contrived. When calculus was first introduced to me (in high school) I was blown away by it. Maybe calculus has just become too trite to me. Any careful introduction of it seems tedious. I’m afraid I’ve become a poor judge of how engaging an introduction to calculus will be to fresh minds.

In Part Two I did like Lockhart’s discussion of space and dimension:

What about four-dimensional space? Is there such a thing? If we’re asking whether four-dimensional space is real we might as well ask about three-dimensional space: Is there such a thing? I suppose it appears that there is. We’re walking around (apparently), and things certainly look and feel as though they are part of a three-dimensional universe, but when you come right down to it, three-dimensional space is really an abstract mathematical object–inspired by our perception of reality, to be sure, but imaginary nonetheless. So I don’t think we should put four-dimensional space in any special mystical category. Spaces come in all sorts of dimensions, and none are any more real than any other. There are no one-dimensional or two-dimensional spaces in real life, and the only thing that gives the number 3 any special status is that our senses appear to offer us that particular illusion.

I would have eaten that kind of stuff up when I was in middle school or high school. Well, OK. I still will eat it up.

Here is something that I think Lockhart gets wrong. He says:

A mathematical structure is what it is, and anything we discover about it is the truth. In particular, if we choose to model an imaginary curve or motion by a set of equations, we are not making any guesses or losing any information through oversimplification: our objects are already (for aesthetic reasons) as simple as they can be. There is no possibility of conflating reality and imagination if everything is imaginary in the first place.

I say to that: “yes and no.” In pure mathematics it is true that we are not modeling real-world phenomena with mathematical structures. We are studying the mathematical structures themselves. However, (and it took me a lot of years in pure mathematics to realize this) we can and do model mathematical structures with other mathematical structures. As Lockhart says, the model may be equivalent to the original structure. On the other hand it may not be! It may be that the original structure is just too complicated for us to make any progress, and we need to model it with a simpler structure. Then learning about the simpler structure can tell us things about the original structure, but there is room for error. See my post on the Eternity Puzzle (and the links that it points to) for more on this. In other words, there is (in pure mathematics) the possibility of conflating imagination with imaginary imagination.

Measurement is thought provoking. I hope that it finds its way into some young people’s hands.

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