A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart (PDF)

Description

“One of the best critiques of current K-12 mathematics education I have ever seen, written by a first-class research mathematician who elected to devote his teaching career to K-12 education.” ―Keith Devlin, NPR’s “Math Guy”A brilliant research mathematician reveals math to be a creative art form on par with painting, poetry, and sculpture, and rejects the standard anxiety-producing teaching methods used in most schools today. Witty and accessible, Paul Lockhart’s controversial approach will provoke spirited debate among educators and parents alike, altering the way we think about math forever.Paul Lockhart is the author of Arithmetic, Measurement, and A Mathematician’s Lament. He has taught mathematics at Brown University, University of California, Santa Cruz, and to K-12 level students at St. Ann’s School in Brooklyn, New York.

User’s Reviews

Editorial Reviews: Review “One of the best critiques of current K-12 mathematics education I have ever seen, written by a first-class research mathematician who elected to devote his teaching career to K-12 education.” ―Keith Devlin, NPR’s “Math Guy”“Gorgeous. . . . Lockhart is passionate, contagiously so.” ―Los Angeles Times“Searing and pointed. . . . An easy, thoughtful, and entertaining read. . . . [Lockhart’s] passion makes the critique compelling.” ―Notices of the American Mathematical Society“Provides a fresh way of thinking about math, and education in general, that should inspire practical applications in the classroom and at home.” ―Publishers Weekly“A Mathematician’s Lament is a fascinating argument that anyone interested in mathematics education should read. I promise that they will enjoy the experience, whether they agree with all that Lockhart writes or not.” ―Bryan Bunch, author of The Kingdom of Infinite Number: A Field Guide“This brief and elegant celebration of mathematics is a charming rant against the way you and I learned the subject. Is painting just coloring in numbered regions? Is the sunset just a list of wavelengths and a compass setting? No more, Lockhart argues, than mathematics is just definitions and formulas. To put back play and joy in our mathematics classrooms, he shows, all we need do is restore the real mathematics.” ―Robert P. Crease, author of The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg“Lockhart has written an important, and eloquent, lamentation and exultation: he laments about the state of math education today, but exults in the hope that teachers might be inspired to invite students to experience mathematics as the exciting ‘poetry of ideas’ that it truly is.” ―Barry Mazur, Gerhard Gade University Professor, Harvard University and author of Imagining Numbers (particularly the square root of minus fifteen) About the Author Paul Lockhart dropped out of college after one semester to devote himself exclusively to math. Based on his own research he was admitted to Columbia University where he received his PhD. The author of Arithmetic, Measurement, and A Mathematician’s Lament, he has taught mathematics at Brown University, University of California, Santa Cruz, and to K-12 level students at St. Ann’s School in Brooklyn, New York.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐*I read this for my math class and this is what I wrote in response:Here I was thinking this book was going to be some boring math book. Boy, was I wrong! I very much enjoyed this delightful book. I was more enraptured as each page just rolled by in a blur. Occasionally, two people talked back and forth with one asking questions and the other answering. This was surprisingly helpful and I found myself asking these exact same questions before they came up in the conversation.The theme of this book is that students-people-should be able to play with math on their own without too much instruction. Use your imagination and make something new to put out in the world. Schools have burned the freedom out of kids. They do not allow a certain amount of playfulness into the classroom of math, or any subject for that matter. Minds are made numb in this current school system.The book kicked off with a musician’s nightmare, narrating it as if it was really happening in the waking world. In the dream, everyone was required to do music, no matter where they came from and if they wanted to do it or not. This is a nightmare of a musician because it begs the question: If everyone is forced to do music, what’s so special about it? Across town, an artist has a similar nightmare: students don’t learn how to paint on a canvas until very late in their careers. Educators had the thought that they needed to learn everything about art before they painted so they could better understand it.Mathematicians have the same nightmare, but it’s an actual waking nightmare. Every single school has the teachers introducing math the exact same way because it’s the way they’ve all learned. The curriculum hasn’t changed for a few centuries and there doesn’t seem to be an end to this way of things. Math is a part of the Arts. It’s beautiful, full of possibility and new life. For example, we should be able to take a rectangle and imagine another shape inside of it. Help it to become something…more. Look at something and see another thing.Math is that subject that seems to be the most confusing to a lot of people. They don’t understand it as it should be understood. We do the math problems, but do we really understand it? I don’t think so. Teachers give the students a bunch of math problems and tell them to do it, but they’re not learning a scrap of what is out there. Math is someone doodling all over the back of their assignment. Doodles turn into math and that turns into a thing that can be molded into something amazing. It’s simply easy and schools tend to make it more complicated; math is a simple thing.Page 26 talks about asking questions. A drawing of a rectangle with a triangle drawn inside of it must be measured. How can we tell how big this triangle is and how much space it takes up? Well, let us draw a line down the middle of it. Here’s the question: What made me draw that line? I saw a gamble and took it. Making mistake after mistake is how those Great mathematicians, artists, and musicians become famously smart. The line that was drawn-that was a little detail that made all the difference. Take an image-a math problem-and draw something to add those details that can change your whole outlook. Appreciate what is in front of you, see something beneath the surface. Teachers make the kids memorize an absurd amount of content without really understanding what any of it means because they’re so busy memorizing it.I love this book and would recommend it to any one person. It’s for everyone, but especially educators. We, as educators, are the manipulators of the arts in universal form. Page 30 has us telling ourselves to drill our own way. Don’t sit back and take what your teachers throw at you. Page 31 tells us about society and math. Teachers are so busy trying so hard to incorporate real life scenarios into their lessons. What does math have to do with anything I’m going to be doing? Sometimes, it seems like they’re trying too hard because the math they teach in schools doesn’t have much to do with real life situations; memorize formulas and do the assignments. Math is just fun and logical, not needing any explanation. Who remembers anything they learned in math class? Not very many.Page 36: There’s no better way to make passion disappear than to make certain classes a requirement. Find a way to teach the kids so they can understand. Take their interests and use those to shape the student. Be that teacher who can listen with an open mind. Start small and go from there. There is an order to learning that is imperative to follow. Although, kids can do anything they set their minds to without the faintest idea of the specifics. Try not teaching them and tell them to come up with their own problems to fix; build your own mind.Page 73 gives us an example of a circle with a triangle in the middle. Circles are curious things. If a triangle is in the middle of a circle, it’s always a right angle no matter which way it rotates. How do we know this? We use imagination and see the triangle/circle rotating, pictures are drawn. We take something bland and turn it into something to be considered, a thing that is beautiful.This book was useful to me because it opened my mind to new possibilities: methods are not useful in the slightest-they’re restricting. We are comfortable learning the way we do in schools because that’s how we’ve done it our whole lives. It’s the same with teaching. We learn from teachers who have been teaching for years, but they are so stuck in old ways that you are liable to fall into the same hole. We get so used to math problems being too hard for us that when we see an easy problem, we look at it from every possible angle. The same goes for any other subject. My vision has broadened into a hard, tangible monster. I was soft and now my mold is a little more stable. I now know that I can do so much more with math than I once thought. Math is like English. It’s poetry, art, and so much more. My favorite parts in this book are those of wisdom, which was the whole book. I will keep this book as a reference in the future and I will recommend it to educators. It’s the type of book that I can’t imagine not reading and gaining something different from it every time I pick it up.

⭐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

⭐).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).

⭐I think the main contribution that this book provides is a basic narrative context of mathematical thought, which allows you to ask further mathematical questions. Unfortunately, this kind conversation about math is completely ignored in most educational systems around the world. Any criticism of modern education will need to be much larger and more expansive than this, but it does provide a good series of arguments that highlight the most important aspects of how our education can fail.Because of that, I think about this book often. So much so that I’m writing a review of it 7 years after I first bought it. It’s a quick read and I highly recommend it.

⭐You may have heard of Kant’s Dove. The dove is flying while thinking to itself how much faster and easier flight would be without air to impede it. But, as we know, in a vacuum the dove would plummet like a brick.Kant’s dove has been flying through education for many decades. Within the teaching of English, there was a thought that, if students were freed from the encumbrances of spelling and grammar, their creativity would naturally flourish. And we can only speculate about the creative heights that Shakespeare might have reached if he hadn’t been subjected to the petty restriction of a mere 14 lines for a sonnet!Within the teaching of mathematics, in the mid-80s, outdated ‘didactic’ teaching methods were now under attack: to be displaced by open-ended mathematical investigations. The mantra was that ‘Maths must be fun.’ To the best of my knowledge, this approach was adopted without any evidence that it would actually result in more creative mathematicians: rather, it was an experiment which was conducted on a large cohort of children and, as far as I can see, was largely forgotten about when it failed to deliver. For me, this book was reminiscent of this failed experiment.Where I would agree with the author is that mathematics is an art, and a highly creative one at that. However, I don’t think I came across a single reference in this book to the differing abilities of individuals. For example, there are some people who will struggle to leave school with even a basic level of numeracy. The author seems to think that many will be put off by the irrelevance of algebra and trigonometry to their daily lives, but doesn’t seem to realise that some won’t even be exposed to these ideas because they will still be struggling with basic numeracy when they leave school.Others, like myself, realise at university that we simply do not have the ability to become creative mathematicians. This is not because we have been taught badly or because we aren’t inspired by the subject: it is simply a function of intelligence. The author says that many gain self-esteem from the ‘mindless manipulation of symbols’. There is another way of looking at this though: for those of us for whom our abilities will take us no further, there is considerable satisfaction in this limited activity. It also puts us in a position where we can admire the brilliance of other mathematicians, whilst fully realising that we are not going to reach those lofty heights ourselves. This is something which many pupils enjoy, at varying levels, and the author risks removing this, and for what?What the author wants is for mathematics to be taught by creative mathematicians. As he says on p.43, ‘Of course what I’m suggesting is impossible for a number of reasons.’ He then goes on to insult teachers for being too irresponsible and too lazy to implement his (admittedly impossible) schemes. Back in the real world, the subject is often taught by non-specialists: sometimes very competently. There is also an argument for saying that creative mathematicians should be doing mathematical research rather than teaching, and that the chance of finding a creative mathematician who is also a skilful teacher is going to be slim.If his experiment were to be adopted, maybe the most appropriate setting would be at university level? My memory of Pure Mathematics at university was: Definition / Theorem / Proof / Definition / Theorem / Proof and so on. The only variety being the occasional use of the word ‘lemma’ or ‘corollary’. If what is happening in secondary schools is a perversion of the subject, is there nothing to be said here?I embarked on this book with high hopes. Having read it, there is very little with which I have sympathy. In broad outline it is an experiment which I witnessed in the mid-80s, and one which was quietly swept under the carpet when it failed to deliver. I hope we won’t get fooled again.

⭐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. Rather, real maths should be the starting point of a child’s introduction to the subject, so that the beauty and creativity that is at the heart of mathematics can be truly appreciated and crafted by the student. Unfortunately, the existing educational system tends to wrongly assume that maths is really a branch of science and as such should be taught to prepare students to be competent in the use and manipulation of calculus to support their studies in the sciences. In order to reach this level at school, the student is therefore ‘trained’ from day one in basic maths, to be followed in sequence by more maths, algebra 1, geometry, algebra 2, pre-calculus and finally calculus. Of course, many students leave school or drop the subject at aged 16 and don’t really even get exposure to calculus. Unfortunately, well before aged 16, many more students have simply been ‘turned off’ by maths, since its method of ‘training’ tends to reward the students who invest in the recipe of learning a mathematical operation, then practising the skill to a level where it is ingrained into the psyche. A good example being the algebraic formula for determining the factors of a quadratic equation where x=(-b+/-SQRT(b^2-4*a*c))/2*a. Whilst useful for solving the specific kind of problem, its relevance to the vast majority of students is such that once they have sat and passed or possibly failed their 16 year old maths exam, then like all the other formulae learnt for ‘the exam’ it will be willingly forgotten and never used again for the rest of their lives.However, it would be wrong to paint Lockhart as being some free thinking spirit who denies the importance of learning certain facts, even formulae. His point is however, that frequently the student at school is introduced to such topics and concepts like the above formula as simply the next thing to learn and be mastered on the curriculum. Most of the time, the most important question of why does this formula work, or what is the history and reason behind its development is never mentioned. Lockhart’s argument is that without expecting a student to be familiar with everything that has been developed in mathematical thinking during the past 3,000 years, it would at least make sense to introduce students to the various areas of the subject by way of exploration; by way of playing games and looking at maths as something to enjoy and experience without artificial exercises. One example of an artificial exercise that Lockhart uses which I enjoyed was his illustration of how in algebra one might be asked to solve the ‘real life’ problem of the age of your friend Maria, who is “. . . 2 years older than twice her age seven years ago.” Lockhart’s heartfelt retort to such attempts to make the subject interesting is that these kinds of unrealistic and ridiculous examples are not what algebra is all about. Instead, simply ask the question, “Suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?” Lockhart states that “Algebra is not about daily life, it’s about numbers and symmetry – and this is a valid pursuit in and of itself.”Furthermore, Lockhart is not out to attack school maths teachers. He fully recognises that most are ‘trapped in the system,’ but he appeals to maths teachers to rethink what they are trying to achieve in their classes.Perhaps most importantly, Lockhart’s observations go right to the heart of one of the problems of modern education in general, namely, that its objective is primarily to train people for the workforce. Setting aside any Orwellian undertones to such criticism, I wish that government ministers and policy makers would take note of Lockhart’s messages. Maths is an art that should inspire and encourage thinking outside the box from the youngest ages. It should not be taught as a series of facts simply to be learnt for performing computations in a series of exams. Such an approach suffocates the intellectual development that real maths can so easily nurture.One may not agree with everything said in this book, since it is first and foremost a lament, but also a call to arms as such it is naturally subjective in nature. However, like all good ideas, anyone reading it could not possibly fail to be stimulated into thinking about these important ideas.Well worth the read.

⭐A penetrating comment on the teaching of mathematics – if taken seriously by the US and UK Education Departments, the teaching of mathematics would become unrecognizable – and a whole lot better. It’s well-written and entertaining, but behind the entertainment is a serious message that the maths taught at school bears no resemblance to the real thing, which most young people don’t meet until University. He argues passionately that maths should be fun, i.e. cut out the dreary utilitarian attitude to the subject, and also a field of open-ended enquiry. His thesis is that real mathematics (enquiry, conjecture and proof) is nonexistent in what passes for maths at school.I’ve had my eye on this book for some time – up till now I’ve relied on a shorter version available online. The book version completes the work and is everything I wanted.

⭐I hated maths at school. Deadliest of subjects. All I can remember is the class having to stand on chairs and recite our 10x table. Torture was the double-period once a week. My teacher…can’t tell you the name. Basically l switched off. Came across this book by chance. Wasn’t sure, so dowloaded a sample and was hooked. This book goes to the heart of the problem which is at the heart of my problem with maths, in otherwords the disastrous way in which the subject is taught. The idea of mathematics as an art form…what a joy.

⭐I ended up disappointed with the ending where he said that intellectual games (that’s me been polite!) were more important than relating maths principles to the real world. Hence the reasons I gave it three stars. He should read John Neumann’s essay on maths in the real world.

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