Measurementby Paul Lockhart

CAMBRIDGE, MASSACHUSETTS: BELKNAP PRESS OF HARVARD UNIVER-

SITY PRESS, 2012, 416 PP., US $29.95, ISBN 978-0-674-05755-5

REVIEWED BY KAREN SAXE

PPaul Lockhart is by now rather well known for AMathematician’s Lament, published in 2009, severalyears after it first circulated on the internet. The

Lament (which is reviewed in this issue) received muchapplause, and also much criticism from mathematiciansand educators. Measurement is bound to have the samefate: some mathematicians will love it, and some will hateit. Why do I limit this statement to mathematicians as thetarget audience? Because that is who I think will read thisbook. It would be a great book for all high-school teachersto have read, but it is a dangerous choice if it is one of onlyvery few on their ‘‘summer math reading book list.’’Unfortunately, I don’t know many K-12 teachers who havethe luxury of spending hours reading books like this in thename of professional development. As you will read belowin What’s to Love?, a teacher can obtain extraordinarilygood material and pedagogical ideas about how to teachgeometry and trigonometry from this book. As you willread in What’s to Hate?, we can no longer hold on to theidea that the type of math in this book be given centralstage in our school curriculum. Further, we cannot affordto embrace — as a society hoping to be populated andgoverned by internationally competitive problem-solvers —the philosophy about the nature of mathematics andmathematics education promoted in the book.

The book is presented in two long parts, each broken intomany small sections. Part One is titled Size and Shape, andPart Two is titled Time and Space. I urge you to read the bookfront to back, as you would a novel, because stories unfoldgradually and earlier sections are referred to later. The firstpart contains material on static measurements of two- andthree-dimensional shapes — cylinders, triangles, conic sec-tions, and so on. The second part introduces motion andfocuses on the ideas of differential calculus. For me, theexperience of reading the book was akin to watching greatlittle interconnected shows about time-honored math. Theart of the book is hand-drawn and the prose conversational.The overall presentation is whimsical. I think my smart liberal-arts students might think it attractively and endearingly retro.

What’s to Love?The book is fun to read. It is elegant. It contains wonderfulmathematics.

Paul Lockhart is a high-school math teacher, and hiswritings provide sound evidence that both his knowledge ofmath and his skill as a teacher must be at a very high level,putting him in the elite group of truly superb teachers in thecountry. His students are clearly fortunate. As I read thebook, I found myself longing to be in front of a class, ‘‘giving’’

his material as lectures, and working through various activ-ities with students. It is thus an inspiring book. I predict that itwill beparticularly useful to andenjoyedby teachers of planegeometry and trigonometry. Classical results from thesefields are given especially lovely treatment.

Lockhart’s explanation of how Archimedes determinedthe volume of a sphere is one of my favorite parts of the book(Section 14, page 86+). About two thousand years beforeCavalieri was born, Archimedes used the method nowreferred to as Cavalieri’s Principle to determine the sphere’svolume by realizing that it could be sliced up and rearrangedto form a cylinder with a double cone removed. Since thevolumes of cylinders and cones were already known,Archimedes had thus determined the sphere’s volume.

Another favorite is the treatment of ellipses found inSections 23 and 24 (page 139+). The ellipse is realized as theshape with its well-known focal point property, and also as adilated circle. Lockhart treats us to Dandelin’s 1822 argumentjoining the two views, by demonstrating that dilated circleshave the focal property.

A third beautiful section, and one I will discuss in moredetail, provides the build-up to understanding Heron’s for-mula relating the area of a triangle to the lengths of its sides(Section 18, page 111+). Toward the beginning of this dis-cussion, we are considering an arbitrary triangle with sidelengths a, b, and c. Lockhart writes (page 112):

Before we get started, I want to say a few things aboutwhat we should expect. Our problem is to measure thearea of a triangle given its sides. This question is com-pletely symmetrical, in the sense that it treats the threesides equally; there are no ‘‘special’’ sides. … If we were toswitch all the a’s and b’s, for example, the formula shouldremain unchanged.

Another thing to notice is that because of the way that areais affected by scaling, our formula will have to be homog-enous of degree 2, meaning that if we replace the symbolsa, b, and c by the scaled versions ra, rb, and rc, the effectmust be to multiply the whole expression by r2.

In my view, this passage demonstrates what I would call‘‘intentional and thoughtful problem-solving;’’ every singlemath student would be a better math student if s/he followedthis model and thought more often about ‘‘what we shouldexpect.’’ As teachers, we must be more diligent aboutproceeding this way in front of our own classes; although wemight privately think this way, I doubt there are many of uswho actively tell our students to think this way.

After some work, a formula for the square of the area ofthe triangle is found:

1

4c2a2 � 1

4c2 c2 þ a2 � b2

2c

� �2

:

At this stage, one is done — one has achieved the goal ofdetermining the triangle’s area given its sides. Lockhartnow takes an opportunity to model further good mathe-matical thinking; he writes:

This is not good. Although we’ve succeeded in measuringthe area of the triangle, the algebraic form of this mea-surement is aesthetically unacceptable. First of all, it is notsymmetrical; second, it’s hideous. I simply refuse to

70 THE MATHEMATICAL INTELLIGENCER � 2014 Springer Science+Business Media New York

DOI 10.1007/s00283-013-9436-y

believe that something as natural as the area of a triangleshould depend on the sides in such an absurd way. It mustbe possible to rewrite this ridiculous expression in a moreattractive form.This section on Heron’s formula is simply wonderful. It

demonstrates the author’s talent as a teacher, highlightingboth his deep understanding of the mathematics and thestrength of his writing. He has a gift for getting the readerengaged, and of emphasizing good habits of mind.

Throughout, I love that Lockhart is so very intentional intalking about what I refer to as ‘‘intellectual risk-taking.’’ Hebeginsonpage5by telling students toexploremathematics—to ‘‘poke it with a stick and see what happens.’’ He remindsthem to do so repeatedly throughout the book. Problemsplay a big role in the book, and many excellent problems areoffered. For example, after discussing areas of circles and rect-angles inscribed in other circles and rectangles, he draws twoof his favorite such (page 62); no questions accompany thispicture but instead the reader has been conditioned to ask (andanswer) his or her own probing questions about the figures.

For slightly more advanced students, there is an excerpton the bottom of page 12 that is worth noting:

… improve your proofs. Just because you have an expla-nation doesn’t mean it’s the best explanation. Can youeliminate any unnecessary clutter or complexity? Can youfind an entirely different approach that gives you deeperinsight?

This sends the signal that there can be different correctproofs, and that correct proofs can perhaps be furthertightened and hence improved. Also, it begs the question:What makes a good proof? Is a good proof one thatilluminates the result? Is a good proof one that is aselementary as possible in the mathematics it uses?

This book is an excellent precursor to a traditional cal-culus sequence. It truly is — if students in my first-yearcalculus class had spent the summer between high-schooland college reading this book, theywould bewell positionedindeed to succeed in calculus. This said, the scope of thebook is limited and does not touch on many parts of themodern mathematics curriculum (this observation is factual,not critical). As a college professor of mathematics, I observethat most students enter college with the idea that calculus isthe mathematical pinnacle. I wish they were disabused of thisidea, and instead came to view calculus as a tremendousachievement of the human intellect but also as a part of abroader and richer field of mathematics. Measurement is old-fashioned, beautifully so, but old-fashioned, and will donothing to dispel the ‘‘calculus as be-all and end-all’’ myth.This brings me to the next section of this review.

What’s to Hate?In his Lament Lockhart asks

… do you really think kids even want something that isrelevant to their daily lives? You think something practicallike compound interest is going to get them excited?People enjoy fantasy, and that is just what mathematicscan provide — a relief from daily life, an anodyne to thepractical workaday world.I find it startling that a math teacher can have this view!

Isn’t the perennial question we receive from students

precisely ‘‘Why is this relevant?’’? He should give credit tohistory and the fact that so much of mathematics has beendeveloped precisely because it could be used to model someaspect of the real world and thus be used to make (veryuseful!) predictions, or to explain observed phenomena. (Tobe fair, he does discuss math as being discovered/inventedout of need on occasion, as for example, when he discussesNapier’s work with logarithms.)

In the Lament he challenges us by asking if we really thinksomething practical like compound interest is going to getstudents excited. I agree with what I think he is getting at —that very (most?) often the ‘‘applications’’ taught in school aredone so in a boring and contrived manner. However,applications needn’t be boring or contrived! In my experi-ence students do care — very much — about election polls,sports rankings, stock and housing markets, public healthdata, the Google PageRank algorithm, coding and internetsecurity, and medical imaging technology. It is much lesscommon for students to show enthusiasm for Heron’s for-mula, or how Archimedes determined the volume of asphere! I don’t want to argue that the applications are betterthan the pure mathematics; I do want to point out thatapplications are important, and can also be beautiful.

What mathematician wouldn’t agree that math is beauti-ful, that our work is an art, and that the process of doingmathis creative? These thoughts do not, however, lead me toconclude that math is also not useful, that math doesn’t havea rich history of important applications, and that teachingapplications destroys students’ sense of enjoyment ofmathematics. Mathematics is (perhaps even uniquely)beautiful, because it is at once an art and also gives criticaltools for solving many of humanity’s most pressing problems(I’m thinking of disease eradication and materials develop-ment, to name just two). We should teach our students thebeauty of Heron’s formula and also quantitative skills(modeling, approximation techniques, statistics, etc.) forbeginning to understand and address societal problems.Lockhart has led us to the front lines of the pure math versusappliedmathwars. There is no reasonwhyour students needchoose sides and learn one to the exclusion of the other.There is no reason why we, the designers of math curricula,need choose sides and teach one at the exclusion of theother. It seems to me unarguable that the pure math thatLockhart promotes is less useful for almost all of our studentsthan is learning a little bit about modeling, for example.Although Measurement is a beautifully written book con-taining some real gems, its tone pushes some well-wornbuttons and assertions such as

‘‘Peopledon’t domathematics because it’s useful’’ (page49)will certainly offend and turn off many potential readers.My experience is that a whole lot of people do mathematicsprecisely because it is useful!

Lockhart opens the book with a section titled Reality andImagination. His observation therein, that ‘‘[A]ny measure-ment made in this universe is necessarily a roughapproximation,’’ struck me as promising, and I was hopefulthat he might expand on this at some later point in the book.He did not take opportunities in this regard. For example, hecould have talked about the ellipse as a model for planetarymotion. This model is not perfect; modeling requires finding

� 2014 Springer Science+Business Media New York, Volume 36, Number 2, 2014 71

a good balance between a model rich enough to yielddesired information, with a model simple enough so thatusing it is computationally possible and also efficient. I wasbothered that he didn’t even make passing mention ofnumerical solutions in his treatment of differential equations(Section 21 would have been a place to do this). Finally — onpage 295 — Lockhart mentions the role of modeling forengineers, architects, and scientists. Inmymind, it is too little,too late. He is of course aware of the tensions surroundingpure and applied math, and is straight about telling us (page397) that he

didn’t want to talk about the applications of mathematicsto the science (which are fairly obvious anyway) because[he] feel[s] that the value of mathematics lies not in itsutility but in the pleasure it gives.

Fair enough. But, the parenthetic remark made me scratch myhead—are they really all that obvious? I regularly see students(aswell as nonmathematician colleagues) surprised and impressedwith math’s wide variety of substantial applications.

Final ThoughtsI understand Lockhart’s goals with the book as described, forexample, in Section 30. And I think he understands perfectlywell whom this book will please, whom it will offend, andprecisely why. Lockhart’s philosophy about math education isshown in his Lament, in which he is scathing in his indictmentof the ‘‘packaging’’ of school mathematics. Measurementprovides an antidote for the way math is taught in schools, butnot really for what is taught. Lockhart is particularly critical ofthe way geometry is taught, which perhaps explains why thefocus of this book is, in large part, geometry.

He observes in his Lament that ‘‘[People] are apparentlyunder the gross misconception thatmathematics is somehowuseful to society! … Mathematics is viewed by the culture assome sort of tool for science and technology.’’ Presumably,with this observation comes the understanding that there issome genuine reason for this cultural view, that somemathematicians do in fact spend their time developing toolsmotivated by real-life problems in science and technology.

We should at the least give credit to mathematicians whowork on applications for being able to communicate theirwork effectively to the public. Going further, we might evenallow that they enjoy this work and find it gratifying! Thepoint of Measurement is to demonstrate the creative ele-ments of doing mathematics, and that mathematics can bethoroughly — and solely — enjoyed for its own sake.Lockhart succeeds terrifically in giving engaging versions oflots of classical school mathematics.

Unfortunately, I think that this book — read alone — willwork to further position Lockhart on thepure side in the puremath versus applied math wars. I’d rather our students didnot learn about this harmful dichotomy. I’d rather Measure-ment’s readers’ attention be focused on his excellentmethods for getting students to approach problemsthoughtfully, take intellectual risks, and develop intellectualpersistence.

Reading Measurement is a richer experience after readingthe Lament, and I encourage you to read them in tandem.There are many important messages found in the latter,including a good closing message for this review:

Weare losing somanypotentially giftedmathematicians—creative, intelligent people who rightly reject whatappears to be a meaningless and sterile subject. They aresimply too smart to waste their time on such piffle.

We must take this observation seriously, consider it a call-to-arms, and start treating our students’ intellects and thecurriculum we offer with much more respect.

REFERENCE

Lockhart, Paul. A Mathematician’s Lament. Bellevue Literary Press,

2009.

Department of Mathematics, Statistics, and Computer Science

Macalester College

St. Paul, MN 55105

USA

e-mail: saxe@macalester.edu

72 THE MATHEMATICAL INTELLIGENCER