Lockhart's Way...

"My idea with this book is that we will design patterns. We'll make patterns of shape and motion, and then we will try to understand our patterns and measure them. And we will see beautiful things!
But I won't lie to you: this is going to be very hard work. Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and I can't help it. Once you've been to the jungle, you can never really leave. It haunts your waking dreams." 

-- from the Introduction to "Measurement" by Paul Lockhart

How could anyone read the above words and not want to proceed to read the book that follows!?

I'm only 3/4 of the way through Paul Lockhart's new volume but will go ahead with an overview blurb on it, because it's obvious to me that I like this idiosyncratic offering and want to recommend it! The book is composed of just two main parts: Part 1 on "Size and Shape," and Part 2 on "Time and Space." Geometers especially will love Part 1 (a sort of geometry primer), though Part 2 (which I'm only partially into) is likely the richer, more fascinating, and slower, more arduous read (essentially an introduction to calculus).

Lockhart's writing style is conversational; refreshingly so, compared to the usual prescriptive tone of much math-writing. He stresses intuition over, or at least equal to, logic. One feels at times as if he is in the same room, ever-standing over your shoulder, talking to you, or, perhaps, like a kindly grandfather, holding your hand as he takes you on a stroll pointing out things along the way that he finds exciting. The book's tone very slightly reminds me of David Berlinski's "The Advent of the Algorithm," another writer with a unique and passionate style.

One of the things I particularly like about the book is that Lockhart is bluntly honest with his audience right from the start (as indicated in the quote above). So many popular books these days imply that they will make math fun and easy for you:  "Learn Calculus In Your Sleep" or "Quantum Mechanics in 3 Easy Steps" (ok, so I made those titles up, but you get the idea). But this offering doesn't pretend to be a "Math For Dummies" book. For most of us, math (at some level) is hard, and Lockhart acknowledges that; some folks who are very bright in other areas, have real mental blockages for mathematical thinking. Once again from Lockhart's intro:

"…expect it to be slow going. I have no desire to baby you or to protect you from the truth, and I'm not going to apologize for how hard it is. Let it take hours or even days for a new idea to sink in -- it may have originally taken centuries!
"I'm going to assume that you love beautiful things and are curious to learn about them. The only things you will need on this journey are common sense and simple human curiosity." 

Thanks grandpa ;-)
So I'm sure there are math-challenged or -phobic individuals out there who will simply find Lockhart's effort just as dry and indecipherable as any other math volume. But for those with an inclination toward the subject matter this volume will likely be a gem.

Unfortunately though, the title doesn't convey that gem-like quality. I suspect "Measurement" was thought to be an elegantly simple title, but I fear it will sound boring and even misleading to many prospective readers, for whom the word conjures up tedious, rote procedures. This book is full of 'elucidation' or 'illumination' (through math), and even play. It is no casual or beach read of course, but nor is it a cold textbook or instruction manual either; and certainly it's a valuable book for teachers to have on hand (so many wonderful examples/ideas herein). The title is not inappropriate, but it may not be the attention-getter the book deserves.

One of my favorite quotes from the volume comes when Lockhart is explaining the calculation of the area of a circle by utilizing an infinitely-sided inscribed polygon:

"In other words, an infinite sequence of lies with a pattern can tell you the truth. It is arguable that this is the single greatest idea the human race has ever had." [bold added]

Gotta love that enthusiasm, and it beams forth from the book repeatedly.

I do have one major quibble with the volume though:

Lockhart regularly tosses out various thought questions or problems for the reader to figure out on their own to more fully fill out the ideas being discussed. Some are easier than others, but they are good and instructive, and it is ashame (even annoying and unsatisfying) that he nowhere offers the answers to these lobbed exercises, so readers can check themselves or see the intended answer if need be. Given Lockhart's joy at elucidating logical steps for the reader I can't imagine why he drops the ball on these interspersed problems, and leaves the reader potentially hanging somewhat -- a simple appendix or addendum at the end could've covered many of them.

Toward the conclusion Lockhart notes, interestingly, that there has been precious little in the book about "reality;" rather, he is discussing mathematics in the abstract, as part of an imaginary world, or a world inside our own heads, while 'reality' is little more than the 'possibly illusory sensory input' from the world around us.
He also writes at one point, "…we love patterns. Mathematics is a meeting place for language, patterns, curiosity, and joy. And it has given me a lifetime of free entertainment."
To which I say, 'Thanks for sharing, Paul!!'

Despite the simple title, this is not a simple or altogether easy book… but it is an easy one to give a thumbs-up to for NON-math-phobes! 

Lockhart's own YouTube promo for the book here: