Measurement by Paul Lockhart CAMBRIDGE, MASSACHUSETTS: BELKNAP PRESS OF HARVARD UNIVERSITY PRESS, 2012, 416 PP., US $29.95, ISBN 978-0-674-05755-5 REVIEWED BY KAREN SAXE
aul Lockhart is by now rather well known for A Mathematician’s Lament, published in 2009, several years after it first circulated on the internet. The Lament (which is reviewed in this issue) received much applause, and also much criticism from mathematicians and educators. Measurement is bound to have the same fate: some mathematicians will love it, and some will hate it. Why do I limit this statement to mathematicians as the target audience? Because that is who I think will read this book. It would be a great book for all high-school teachers to have read, but it is a dangerous choice if it is one of only very few on their ‘‘summer math reading book list.’’ Unfortunately, I don’t know many K-12 teachers who have the luxury of spending hours reading books like this in the name of professional development. As you will read below in What’s to Love?, a teacher can obtain extraordinarily good material and pedagogical ideas about how to teach geometry and trigonometry from this book. As you will read in What’s to Hate?, we can no longer hold on to the idea that the type of math in this book be given central stage in our school curriculum. Further, we cannot afford to embrace — as a society hoping to be populated and governed by internationally competitive problem-solvers — the philosophy about the nature of mathematics and mathematics education promoted in the book. The book is presented in two long parts, each broken into many small sections. Part One is titled Size and Shape, and Part Two is titled Time and Space. I urge you to read the book front to back, as you would a novel, because stories unfold gradually and earlier sections are referred to later. The first part contains material on static measurements of two- and three-dimensional shapes — cylinders, triangles, conic sections, and so on. The second part introduces motion and focuses on the ideas of differential calculus. For me, the experience of reading the book was akin to watching great little interconnected shows about time-honored math. The art of the book is hand-drawn and the prose conversational. The overall presentation is whimsical. I think my smart liberalarts students might think it attractively and endearingly retro.
What’s to Love? The book is fun to read. It is elegant. It contains wonderful mathematics. Paul Lockhart is a high-school math teacher, and his writings provide sound evidence that both his knowledge of math and his skill as a teacher must be at a very high level, putting him in the elite group of truly superb teachers in the country. His students are clearly fortunate. As I read the book, I found myself longing to be in front of a class, ‘‘giving’’ 70
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his material as lectures, and working through various activities with students. It is thus an inspiring book. I predict that it will be particularly useful to and enjoyed by teachers of plane geometry and trigonometry. Classical results from these fields are given especially lovely treatment. Lockhart’s explanation of how Archimedes determined the volume of a sphere is one of my favorite parts of the book (Section 14, page 86+). About two thousand years before Cavalieri was born, Archimedes used the method now referred to as Cavalieri’s Principle to determine the sphere’s volume by realizing that it could be sliced up and rearranged to form a cylinder with a double cone removed. Since the volumes of cylinders and cones were already known, Archimedes had thus determined the sphere’s volume. Another favorite is the treatment of ellipses found in Sections 23 and 24 (page 139+). The ellipse is realized as the shape with its well-known focal point property, and also as a dilated circle. Lockhart treats us to Dandelin’s 1822 argument joining the two views, by demonstrating that dilated circles have the focal property. A third beautiful section, and one I will discuss in more detail, provides the build-up to understanding Heron’s formula relating the area of a triangle to the lengths of its sides (Section 18, page 111+). Toward the beginning of this discussion, we are considering an arbitrary triangle with side lengths a, b, and c. Lockhart writes (page 112): Before we get started, I want to say a few things about what we should expect. Our problem is to measure the area of a triangle given its sides. This question is completely symmetrical, in the sense that it treats the three sides equally; there are no ‘‘special’’ sides. … If we were to switch all the a’s and b’s, for example, the formula should remain unchanged. Another thing to notice is that because of the way that area is affected by scaling, our formula will have to be homogenous of degree 2, meaning that if we replace the symbols a, b, and c by the scaled versions ra, rb, and rc, the effect must be to multiply the whole expression by r2. In my view, this passage demonstrates what I would call ‘‘intentional and thoughtful problem-solving;’’ every single math student would be a better math student if s/he followed this model and thought more often about ‘‘what we should expect.’’ As teachers, we must be more diligent about proceeding this way in front of our own classes; although we might privately think this way, I doubt there are many of us who actively tell our students to think this way. After some work, a formula for the square of the area of the triangle is found: 2 1 2 2 1 2 c2 þ a2 b2 c a c : 4 4 2c At this stage, one is done — one has achieved the goal of determining the triangle’s area given its sides. Lockhart now takes an opportunity to model further good mathematical thinking; he writes: This is not good. Although we’ve succeeded in measuring the area of the triangle, the algebraic form of this measurement is aesthetically unacceptable. First of all, it is not symmetrical; second, it’s hideous. I simply refuse to
believe that something as natural as the area of a triangle should depend on the sides in such an absurd way. It must be possible to rewrite this ridiculous expression in a more attractive form. This section on Heron’s formula is simply wonderful. It demonstrates the author’s talent as a teacher, highlighting both his deep understanding of the mathematics and the strength of his writing. He has a gift for getting the reader engaged, and of emphasizing good habits of mind. Throughout, I love that Lockhart is so very intentional in talking about what I refer to as ‘‘intellectual risk-taking.’’ He begins on page 5 by telling students to explore mathematics — to ‘‘poke it with a stick and see what happens.’’ He reminds them to do so repeatedly throughout the book. Problems play a big role in the book, and many excellent problems are offered. For example, after discussing areas of circles and rectangles inscribed in other circles and rectangles, he draws two of his favorite such (page 62); no questions accompany this picture but instead the reader has been conditioned to ask (and answer) his or her own probing questions about the figures. For slightly more advanced students, there is an excerpt on the bottom of page 12 that is worth noting: … improve your proofs. Just because you have an explanation doesn’t mean it’s the best explanation. Can you eliminate any unnecessary clutter or complexity? Can you find an entirely different approach that gives you deeper insight? This sends the signal that there can be different correct proofs, and that correct proofs can perhaps be further tightened and hence improved. Also, it begs the question: What makes a good proof? Is a good proof one that illuminates the result? Is a good proof one that is as elementary as possible in the mathematics it uses? This book is an excellent precursor to a traditional calculus sequence. It truly is — if students in my first-year calculus class had spent the summer between high-school and college reading this book, they would be well positioned indeed to succeed in calculus. This said, the scope of the book is limited and does not touch on many parts of the modern mathematics curriculum (this observation is factual, not critical). As a college professor of mathematics, I observe that most students enter college with the idea that calculus is the mathematical pinnacle. I wish they were disabused of this idea, and instead came to view calculus as a tremendous achievement of the human intellect but also as a part of a broader and richer field of mathematics. Measurement is oldfashioned, beautifully so, but old-fashioned, and will do nothing 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 is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide — a relief from daily life, an anodyne to the practical 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 to history and the fact that so much of mathematics has been developed precisely because it could be used to model some aspect of the real world and thus be used to make (very useful!) predictions, or to explain observed phenomena. (To be fair, he does discuss math as being discovered/invented out of need on occasion, as for example, when he discusses Napier’s work with logarithms.) In the Lament he challenges us by asking if we really think something practical like compound interest is going to get students excited. I agree with what I think he is getting at — that very (most?) often the ‘‘applications’’ taught in school are done so in a boring and contrived manner. However, applications needn’t be boring or contrived! In my experience students do care — very much — about election polls, sports rankings, stock and housing markets, public health data, the Google PageRank algorithm, coding and internet security, and medical imaging technology. It is much less common for students to show enthusiasm for Heron’s formula, or how Archimedes determined the volume of a sphere! I don’t want to argue that the applications are better than the pure mathematics; I do want to point out that applications are important, and can also be beautiful. What mathematician wouldn’t agree that math is beautiful, that our work is an art, and that the process of doing math is creative? These thoughts do not, however, lead me to conclude that math is also not useful, that math doesn’t have a rich history of important applications, and that teaching applications destroys students’ sense of enjoyment of mathematics. Mathematics is (perhaps even uniquely) beautiful, because it is at once an art and also gives critical tools for solving many of humanity’s most pressing problems (I’m thinking of disease eradication and materials development, to name just two). We should teach our students the beauty of Heron’s formula and also quantitative skills (modeling, approximation techniques, statistics, etc.) for beginning to understand and address societal problems. Lockhart has led us to the front lines of the pure math versus applied math wars. There is no reason why our students need choose 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 the other. It seems to me unarguable that the pure math that Lockhart promotes is less useful for almost all of our students than is learning a little bit about modeling, for example. Although Measurement is a beautifully written book containing some real gems, its tone pushes some well-worn buttons and assertions such as ‘‘People don’t do mathematics because it’s useful’’ (page 49) will certainly offend and turn off many potential readers. My experience is that a whole lot of people do mathematics precisely because it is useful! Lockhart opens the book with a section titled Reality and Imagination. His observation therein, that ‘‘[A]ny measurement made in this universe is necessarily a rough approximation,’’ struck me as promising, and I was hopeful that he might expand on this at some later point in the book. He did not take opportunities in this regard. For example, he could have talked about the ellipse as a model for planetary motion. This model is not perfect; modeling requires finding 2014 Springer Science+Business Media New York, Volume 36, Number 2, 2014
a good balance between a model rich enough to yield desired information, with a model simple enough so that using it is computationally possible and also efficient. I was bothered that he didn’t even make passing mention of numerical solutions in his treatment of differential equations (Section 21 would have been a place to do this). Finally — on page 295 — Lockhart mentions the role of modeling for engineers, architects, and scientists. In my mind, it is too little, too late. He is of course aware of the tensions surrounding pure and applied math, and is straight about telling us (page 397) that he didn’t want to talk about the applications of mathematics to the science (which are fairly obvious anyway) because [he] feel[s] that the value of mathematics lies not in its utility but in the pleasure it gives. Fair enough. But, the parenthetic remark made me scratch my head — are they really all that obvious? I regularly see students (as well as nonmathematician colleagues) surprised and impressed with math’s wide variety of substantial applications.
Final Thoughts I understand Lockhart’s goals with the book as described, for example, in Section 30. And I think he understands perfectly well whom this book will please, whom it will offend, and precisely why. Lockhart’s philosophy about math education is shown in his Lament, in which he is scathing in his indictment of the ‘‘packaging’’ of school mathematics. Measurement provides an antidote for the way math is taught in schools, but not really for what is taught. Lockhart is particularly critical of the way geometry is taught, which perhaps explains why the focus of this book is, in large part, geometry. He observes in his Lament that ‘‘[People] are apparently under the gross misconception that mathematics is somehow useful to society! … Mathematics is viewed by the culture as some sort of tool for science and technology.’’ Presumably, with this observation comes the understanding that there is some genuine reason for this cultural view, that some mathematicians do in fact spend their time developing tools motivated by real-life problems in science and technology.
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We should at the least give credit to mathematicians who work on applications for being able to communicate their work effectively to the public. Going further, we might even allow that they enjoy this work and find it gratifying! The point of Measurement is to demonstrate the creative elements of doing mathematics, and that mathematics can be thoroughly — and solely — enjoyed for its own sake. Lockhart succeeds terrifically in giving engaging versions of lots of classical school mathematics. Unfortunately, I think that this book — read alone — will work to further position Lockhart on the pure side in the pure math versus applied math wars. I’d rather our students did not learn about this harmful dichotomy. I’d rather Measurement’s readers’ attention be focused on his excellent methods for getting students to approach problems thoughtfully, take intellectual risks, and develop intellectual persistence. Reading Measurement is a richer experience after reading the 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: We are losing so many potentially gifted mathematicians — creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their time on such piffle. We must take this observation seriously, consider it a callto-arms, and start treating our students’ intellects and the curriculum we offer with much more respect.
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: [email protected]