Customer Reviews

A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

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Once in a while we read books that we just know are especially important, and that we know we will be thinking and talking about long after reading them. This book is one of them for me.

I am a returning adult student, and I am about to finish my training to become a math teacher. Having gone through my education program, my enthusiasm was just about completely drained, and I've been having trouble remembering why I ever wanted to become a math teacher in the first place. Why would anyone?

Paul Lockhart knows, and his book has reawakened my desire to help students discover the joy of mathematics. His argument is concise, and he makes it forcefully. His book is a joy to read, mainly because his understanding of the subject and his passion for it are clear in every page. He reinforces ideas I already had about how school sucks the life out of math (and all subjects), but he also challenges some of my opinions. I think this will happen with most people who read it.

Once he finishes making his argument about math education in about the first two-thirds of this short book, he devotes the remaining section to describing what he finds wonderful about mathematics itself. This section should make just about anyone want to become either a mathematician or a math teacher.

I want people to read the book for the specifics of his arguments, but I want to discuss one important point that he makes. Many people in math education claim that in order to make math more understandable and interesting to students, we need to show how practical it is and how it is used in everyday life. I've always felt like this idea was wrong, or at least limited in its usefulness in that regard. Well, Lockhart demolishes the idea, essentially claiming that practical uses are simply by-products of math, and that the real excitement and beauty of mathematics is in the abstract, imaginary, and creative world of mathematical ideas that have no specific connection to the everyday. By-products and applications can make math seem boring and secondary to the uses it serves. I agree with him--and much more now after having read his argument.

I honestly think just about everyone should read this book. Of course math teachers should, as should anybody involved in math education in any way. But I think people outside of math education should read it too. The specific mathematical ideas discussed in the book do not require a strong mathematical background, and I can't think of a better book that so concisely conveys the nature of the subject and the way it is viewed and misunderstood in society. I'm still not sure I agree with Lockhart's every point, but I love this book. (And I might agree with his every point after more thought and experience in the classroom.)

When I began to read Lockhart's Lament, I was skeptical -- particularly with his view of mathematics as more of an art than a science. I am an applied mathematician, and I most enjoy teaching applied mathematics, but after serious and humble reflection, I came to fundamentally agree with Lockhart. Mathematics was developed as an expression of human creativity, and teaching it as such is really the only viable option for most students to be able to appreciate it and therefore fully apply it (if they ever need or want to).

As a relatively new mathematics teacher, I appreciate Lockhart's observations of the mathematics curriculum. I taught (college) trigonometry just before reading his Lament for the first time, and I was blown-away (and a little devastated) by the accuracy of his scathing description of that course:

"Two weeks of content are stretched to semester length by masturbatory definitional runarounds... students must learn to use the secant function, 'sec x,' as an abbreviation for the reciprocal of the cosine function, '1 / cos x' (a definition with as much intellectual weight as the decision to use '&' in place of 'and.') That this particular shorthand, a holdover from fifteenth century nautical tables, is still with us... is mere historical accident... Thus we clutter our math classes with pointless nomenclature for its own sake."

This book is an absolute necessity for anyone who wants to make sure their students actually enjoy mathematics. But be warned, if you view teaching mathematics as just a job, this book probably isn't for you.

Paul Lockhart is a mathematics teacher. He obviously loves the subject that he teaches. He is also a writer, and a good one at that. In his delightful little book, Paul explains why mathematics is so interesting, and what is wrong with the current curriculum in our schools.

What many of us have accepted as a "fact" our whole lives is patently false: Mathematics is neither boring nor difficult. Rather, it is creative and fun. Anyone can do it, if only shown how. And they would enjoy it, as much as if they were painting or playing music.

Paul laments that mathematics is not taught like painting or music. It has instead evolved into something unnecessarily dreadful, with rigid terminology and pointless memorization. Is it any wonder why so many otherwise intelligent people think they are bad at mathematics?

I would recommend this book to anyone who "knows" that mathematics is difficult or boring. It is neither! And while we're at it, let's throw out all those boring textbooks...

This is an interesting indictment of our current system of mathematics education, a subject that I almost always enjoy reading about (I very nearly became a mathematics teacher myself not long ago). Lockhart makes his point clearly, eloquently, and succintly--this is a very quick read at only 140 pages of fairly large type.

I agree with much of what he says, though I do think that his claims sometimes go a bit too far: he doesn't seem to see much point in learning to add in an age of calculators, for example.

The main problem I had with this book is that, as far as I'm concerned, Lockhart doesn't offer up a viable alternative to the status quo. It's always easy to criticize, but it's a lot harder to come up with a better way of doing things. Lockhart does offer some ideas about how the ideal mathematics education should function: a mathematics teacher should be a practicing mathematician himself, and should be so engaged in the subject that he has no need for lesson plans or curricula, but can rely solely on his passion for mathematics. Teacher training should be abolished, since someone either is a good teacher or isn't, and nothing can change that. While this sounds nice in theory, it just doesn't seem feasible. I'm not convinced that all these perfect mathematics teachers will suddenly appear, and if they don't, we're left with nothing (which I suppose Lockhart would say is better than the current state of affairs). To me, this doesn't seem like a solution. After reading about how terrible the current system is, I'd like to have seen some real suggestions for how it could be reformed.

Still, this is a worthwhile read, and one that should generate a lot of interesting discussion; despite the fact that I wasn't entirely satisfied with it, I plan to encourage my family to read it so that I can see what they think. And there were parts of it that I loved, particularly the initial description of a musician's nightmare that provided a powerful insight into how ridiculous mathematics education can be. So, even with its shortcomings, this is a book that I would recommend.

I recently came across Lockhart's lament. He has very beautifully articulated much of what is problematic with math education today. Indeed, many teachers have lost sight of process, as we focus solely on product. But the problems go deeper, and the solutions are more complex.

As the Math department head at a high school, I constantly grapple with integrating his ideas in my classes. How can I spend time exploring the beauty and power of zero when they have SAT's junior year? How can I examine the relationship between the binomial distribution, Pascal's triangle, and the Normal curve, when the AP is looming in May?
But the key here, is that I grapple with these questions, and *I teach the honors classes*!!!
When I discuss "teaching the poetry of math" with teachers of weaker students, I might get a look of "Are you kidding me?".

Lockhart correctly surmised that the whole system is broken, and the scope of the changes necessary goes well beyond what one math teacher can do in his or her classroom. But it isn't just because of SATs and AP's and college calculus courses that they need to be prepared for. It is because math is 3-4 YEAR REQUIREMENT in high school. This, more than anything else, is the key impediment to change.

Let's face it - I am not an artist. Still, I can see value in having a mandatory year of art in high school. It would expose me to a rich field, make me a more well rounded person, etc. But a student who doesn't appreciate or enjoy math is forced to take 12 years of it! Imagine majoring in a subject that doesn't speak to you in the least.

This is what forces many teachers to teach by rote - we are teaching students who should be spending their time more productively, taking courses that they enjoy more and can excel at. We are NOT all artists, and for a student with no aptitude or appreciation for math, teachers are relegated to a formulaic, step by step approach that students can do, even if they wish they could teach it otherwise.

Thus, with so many of Lockhart's examples, I can imagine many students asking "Who cares?" and "How is this useful?" not because they have been indoctrinated as such, but because that is *who they are*. One day, many of them will have great careers as doctors, accountants, plumbers, bus drivers, and more, and they will have wasted untold minutes taking useless math classes, no matter how it is taught.

A solution to the problems Lockhart brings up involves us recognizing that math is art for some, practical for some, and downright odious to others in any way, shape and form. We should be open to appreciating the many ways that math can be beautiful, or ugly, to students.

First off, I am not a parent, I have no children, so the topic of this book (the teaching of math in American schools, grades K-12) doesn't directly concern me. The author is not speaking about anything that I am called upon to deal with personally, either as a parent or teacher (I am neither); so as I read the book, I was not reading with any hope that the author would -- clearly, forcefully, and with the authority of print and the approbation of popular mathematicians -- express any frustration I have with the current ways of teaching math in grades K-12. I read it only because I was curious what he had to say. I am not the intended audience, in other words.

The subtitle "How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form", along with the approving back-cover comments by Keith Devlin, Robert Crease, Bryan Bunch, and Barry Mazur, caught my attention. Devlin's forward calls it "a recognized landmark in the world of mathematics education that cannot and should not be ignored."

Is this really as good as it gets? Forgive me for saying this, but the book lacks rigor. The two main sections of the book are entitled "Lamentation" and "Exultation", not, please notice, "Problems" and "Solutions". Therein lies the book's weakness. A no doubt uncharitable, but I think more accurate, description would rename the sections "Rant" and "A Few Examples of Cool Math that Everyone Can Understand".

Parents who high-five the author are not seeing how weak the book actually is in its not offering any real method of getting kids from learning how to count from one to ten and beyond (and behind), to understanding the procedures and applications of simple arithmetic, to the more complex meaning and use of symbolic algebra, to the inherent logic of dimensional figures in geometry and trigonometry, to the beginnings of higher math and whatever else math teachers teach in K-12 these days.

The author likens math to music and painting, when in fact he should liken it to learning another language that cannot be picked up the way a natural language can. If mathematics is an "art", that is because there can be an aesthetic response to it and to the form and structure of mathematical proofs (e.g. "elegant"), but that does not thereby make it fully analogous to music and painting.

Any child can whistle and hum and sing, and any child can finger-paint and draw and make designs and figures and use crayons and paint to color the scene, but it takes, among other things, the understanding and grasp of an algorithm to divide numbers, and more than that to get the initial gist of algebra. It takes, that is, intellect, and the comprehension of rules, meanings, and methods.

You don't need to know how to read or write music to create it or to "know what it means" when you hear it; but without knowing how to read or write mathematics, you're certainly not going to be coming up with much of it on your own, and you're not going to get very far in actually understanding what most of it means.

The section on Exultation uses geometry and figured drawings to show the beauty of math. That's cheating, as far as the author's purported wish goes, because what he wants is for the entire math program in grades K-12 to be junked and rewritten from scratch, yet in showing only the beauty of what I'm going to call "pictographic math" he misrepresents the range of what needs to be taught if you're actually going to say you're teaching math and not merely the pictographic areas of math, or worse yet, quasi-math.

If there are indeed serious faults in the current K-12 math curriculum, and if this book is a "recognized landmark" in highlighting those faults, then I hope its landmark status is only in marking the first stage of the path to repairing those faults. Much needs to be said beyond this book if anything cogent and viable is going to be said at all.

This book is life changing. No joke. I read Paul Lockhart's essay, before he published this book, when i had him as a math teacher in 11th grade. He was amazing as a teacher and this book is a true expression of his teaching. While the first half is a tirade on not only math but much of our schooling system itself. The second half of the book is a math lesson as it SHOULD be taught. It is brilliant, elegant, and completely moving. The sadist moment for me was thinking that i only had Paul for 1 year of my schooling. How much more could i have done with math had it been taught this way?

This book is mind blowing. It has an authenticity that I think is very rare. It should be read by everyone.

My daughter started high school (honors) geometry two weeks ago. She knows how excited I usually get when invited to help with math homework. My heart sank when I saw those ridiculous proof charts, and I couldn't remember the difference between a postulate and a theorem. I thought that my involvement with her math homework had come to an end.

Then I read Lockhart's book. It made me realize that I provide the most help by looking for patterns, and drawing pictures, and asking questions - what happens if we try to do this? It's not just knowing the language.

Anyone who wants to help their children be good at and enjoy math should read this book. It will remind them about the joy of problem-solving and encourage them to pass on that joy on to their children by thinking of math as Lockhart suggests - as an art.

While I enjoyed reading the book, learned about mathematics beauty and agree with many of the arguments, I was still waiting to see recommendations on how to implement some of the ideas into practice. I am a mother of a 9 yr old boy, who is not particularly doing well in math and who does not enjoy it. I have been very successful in instilling love for reading and to teach him full fluency in two languages, including writing and reading (well beyond just speaking at home). In math though, I have difficulty getting there. I want him to enjoy and play with mathematical concepts and I was looking forward for some illumination in the book but there were very few hints on what to do. I guess it is a quest that this book started...

As a high school student that read this on the recommendation of my philosophy teacher, I must say it is a necessary read for anyone who is in or works at a high school. It is a great read about mathematics and education in general, and was interesting enough for me to write two college application essays about it. For anyone in academia or loves learning this book is for you.