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

Keith Devlin , Paul Lockhart
5.0 out of 5 stars  See all reviews (3 customer reviews)
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Book Description

14 May 2009 1934137170 978-1934137178
"One of the best critiques of current mathematics education I have ever seen."-Keith Devlin, math columnist on NPR's Morning Edition A brilliant research mathematician who has devoted his career to teaching kids reveals math to be creative and beautiful and rejects standard anxiety-producing teaching methods. Witty and accessible, Paul Lockhart's controversial approach will provoke spirited debate among educators and parents alike and it will alter the way we think about math forever. Paul Lockhart, has taught mathematics at Brown University and UC Santa Cruz. Since 2000, he has dedicated himself to K-12 level students at St. Ann's School in Brooklyn, New York.

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Product details

  • Paperback: 144 pages
  • Publisher: Bellevue Literary Press (14 May 2009)
  • Language: English
  • ISBN-10: 1934137170
  • ISBN-13: 978-1934137178
  • Product Dimensions: 18.3 x 12.4 x 1.3 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 447,006 in Books (See Top 100 in Books)

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About the Author

Paul Lockhart became interested in mathematics when he was 14 (outside the classroom, he points out). He dropped out of college after one semester to devote himself exclusively to math. Based on his own research he was admitted to Columbia, received a PhD, and has taught at major universities. Since 2000 he has dedicated himself to "subversively" teaching grade-school math.

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4 of 4 people found the following review helpful
5.0 out of 5 stars Maths is art - not science 22 Mar 2013
Format:Paperback|Verified Purchase
Every so often, I read a book which I cannot put down. Paul Lockhart's book is one of them. I received it this morning and finished it this afternoon, including some time to work through one or two of his 'maths games.'

As reported in other reviews, Lockhart brings a wealth of experience as a university level maths teacher, who decided to take his talents to benefit K12 level students in school. Lockhart is exactly the kind of teacher everyone should have in their maths class. His approach is simple and intuitively sound; namely, that maths as it is currently taught in most school classrooms is not really maths per se; rather it is a training process that rewards those who are good at learning a multitude of facts in the shape of formulae and algorithms, but who are not necessarily inclined towards or even competent at thinking 'outside the box.' As the Forward to the book by Keith Devlin (a maths professor at Stanford University) points out, many successful high-school mathematics students come unstuck when arriving at university to study mathematics, since the approach and character of the subject is so very different. The analogy is that pre-university maths is similar to learning to paint by numbers and that only when one 'arrives' at university is true maths introduced into the curriculum and the student is allowed to pick up a blank canvas to construct a painting. Many cannot make the transition, largely because they lack the mind-set necessary for this unstructured approach.

Lockhart appeals to us to appreciate that this transition is not something which should simply occur for a minority of students arriving at university.
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7 of 7 people found the following review helpful
5.0 out of 5 stars maths as it should be 18 Dec 2010
Format:Paperback
A wonderful read. It should be noted by the potential purchaser that it is only the foreward that is by Keith Devlin. This book is in fact Paul Lockhart's brilliant 140 page arguement that maths is 'the purest of the arts, as well as the most misunderstood.' And his arguement is very persuasive, writing that the 'maths' we are presented with in school is not the real thing at all, but a frighteningly dummed down version. That what is done to maths in school is the equivalent of painting-by-numbers being presented as art's true essence. Lockhart states 'Mathematics is fundamentally an act of communication, and, as if to prove his point, it is clear that the author has communication down to an art form. As a non-mathematician I was fully able to follow and appreciate the arguements and mathematical problems presented in this book. Perhaps best summed up in Lockhart's own phrase; 'If tears aren't streaming down your face, maybe you should read it again.' Not that that would be a chore. Five stars are not enough.
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1 of 2 people found the following review helpful
5.0 out of 5 stars A must read! 13 Jan 2014
Format:Kindle Edition|Verified Purchase
A book every maths teacher needs to read and use these ideas and other ideas coming from the first ideas to create a love of maths in his/her pupils. A great book to set one thinking way beyond the confines of the set syllabi!
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Amazon.com: 3.8 out of 5 stars  48 reviews
103 of 106 people found the following review helpful
5.0 out of 5 stars An Important Read 11 May 2009
By Robert Potter - Published on Amazon.com
Format:Paperback|Verified Purchase
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.)
65 of 66 people found the following review helpful
5.0 out of 5 stars Brilliant and a little devastating 15 Jun 2009
By R. Wright - Published on Amazon.com
Format:Paperback|Verified Purchase
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.
34 of 35 people found the following review helpful
3.0 out of 5 stars An interesting indictment of our current approach to mathematics education 31 Jan 2010
By Dunyazad - Published on Amazon.com
Format:Paperback
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.
40 of 44 people found the following review helpful
5.0 out of 5 stars Math is fun, after all. 1 May 2009
By John J. Turley - Published on Amazon.com
Format:Paperback|Verified Purchase
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...
35 of 43 people found the following review helpful
2.0 out of 5 stars Wonderful Ideas in Theory, with tragic flaws in argument 11 Sep 2011
By Ioana Stoica - Published on Amazon.com
Format:Paperback|Verified Purchase
As a mathematics teacher and long-time student of mathematics, I was overjoyed to begin reading this book, finally one that attempts to explain the beauty and elegance of mathematics and to expose the way in which we are teaching it, which does not do justice to it at all. I absolutely agree with *most* of Lockhart's assessment on many points, for example, that mathematics is an art, that it should not be taught as procedures and formulas and meaningless word problems that contrive to be about "real life."

I agree, most of our math teachers do not have this kind of appreciation for mathematics, which is tragic because it means our kids will grow up scared and intimidated by math ("math anxiety") instead of awed at its power of abstract interpretation. I agree our approach needs to be completely overhauled.

My 2 star rating is due to the fact that Lockhart's analysis is strongly lacking in a historical understanding as well as pedagogical/curriculum knowledge. For example, he says that word problems should not be contrived to be about real life (I agree with this point), and that math is beautiful precisely BECAUSE it is irrelevant to real life.

As a mathematician I cannot possibly comprehend how another mathematician could possibly believe the beauty of mathematics comes from its "irrelevance" of abstraction: in fact, the reason math is SO powerful is that these abstract representations have all been historically "discovered" or "invented" (depending on what you believe math is: inherent in the world, or a human game of abstraction)--particularly in order to try to model and explain phenomena observed in "the real world."

Lockhart says math was created by humans "for their own amusement" (p. 31), but ignores that in fact all branches of mathematics in the past were created in response to actual world problems, and not only that, but now, some of the most fascinating mathematics is being created again in response to solving some of the most complex problems we have imagined, such as the mathematics behind string theory. I don't know how Lockhart could possibly consider that humans invented counting, ways to measure their plots of land and keep track of money, or ways to measure the orbits of planets (thus leading us to the current "space age") as "purely amusement"--perhaps, if LIFE is amusement in general, but really, all of these inventions had a very REAL, concrete, specific historical cultural purpose and are not "just made up" for fun!!!

In fact, math is EMBODIED in our cognitive schemas and perception, and THIS IS PRECISELY what makes it so WONDERFUL: its RELEVANCE to EVERYTHING in real life and humanity's inherent capacity for thinking about the real world in this abstract way! Math is not "just" "fantasy" (as on p. 39) (see especially Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being).

I teach functions (precalculus, AP calculus) and the main theme and point of math for example at this level is to teach kids how basically, in life, we track patterns of change in anything and everything--public health data, unemployment, polling, the stock market, baseball stats, etc. Functions are just the most abstract way to model these changing patterns over time (or some other variable) and thus give us the powerful tool of projecting into the future/past and otherwise analyzing trends. Yes, functions are abstract, but they are NOT "just fantasy play," irrelevant to the real world, or made up simply for the fun of it, in fact, quite the opposite of all of these.

Further, my (and I believe, many) students would be aghast to learn that a math teacher is suggesting an overhaul of math education based on the idea that"kids don't really want something that is relevant to their daily lives." This is the most absurd statement I have ever heard, so I am guessing Lockhart knows nothing about adolescent/child development, interest, and pedagogical literature. Learning in general is based on making connections to prior knowledge, and I have never heard any question asked more often in math class when I didn't explain the relevance in advance than "Why do I need to know this? How is this relevant to my life?" This is probably the MOST pressing question for adolescents in general..

Other examples of pedagogical tragedies in this book include Lockhart's admonitions that "you can't teach teaching," that "schools of education are a complete crock" and that teachers shouldn't lesson plan because this is somehow "not real" or authentic (p. 46-47). While I agree schools of education are not preparing our teachers well and what we need is much more systemic training in content knowledge (for example, math teachers should all have to double major in math/pedagogy or education), IT IS absolutely not true or supported by any research (except perhaps by the current corporate brand of the reform movement) that teaching is something you "have" that you don't need to "learn" and, further, that you shouldn't plan because this is inauthentic.

A plan should of course never prevent a teacher from moving in new directions as suggested by the course of the class, but coming in without a plan is certainly not considered sound practice in any theory of learning and from any angle, and in general is not a sound principle of life (i.e., just doing everything by the seat of your pants and counting on your "genius" to lead you through whatever you should have planned usually doesn't work, unless you are in a feel-good movie). Only in Lockhart's fantasy "lala land" of irrelevancy is planning a vice and not a virtue. Plus, there's so much more to "planning" than thinking about the flow of the lesson, how you will help students make connections, etc. I assess and plan hand in hand for example, so I will grade the last night's HW and that day's Exit Slip and plan the next days's and week's lessons all the while incorporating items my students did not fully understand the first time, and also while addressing specifically the mistakes they made (and each class/year of students tends to have different problems and make different mistakes so it is important to constantly plan and reflect as a teacher on what is best for your particular students NOW).
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