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A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form
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.
140 pages, Paperback
First published January 1, 2009
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5939 people want to read
About the author
Paul Lockhart
5 books205 followersPaul 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, including Brown University and UC Santa Cruz. Since 2000 he has dedicated himself to "subversively" teaching grade-school math at St. Ann's School in Brooklyn, New York.
Librarian Note: There is more than one author in the Goodreads database with this name.
Librarian Note: There is more than one author in the Goodreads database with this name.
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June 20, 2016
As a mathematics teacher and long-time student of mathematics, I was overjoyed when I came across this book. Finally, I thought, an ode to the profound beauty and elegance of this most precise and direct human languages. And, hopefully, an expose on the state of mathematics education, and a plea to change course, maybe even some practical suggestions on how we may begin to do this.
Lockhart and I started in lock-step. YES. The current state of mathematics education is a TRAVESTY - we are most emphatically not teaching students to fully appreciate its abstractive powers, philosophical implications, and inherent structure. Instead, we are taking the artifacts of doing mathematics and positing that these are what math is about.
An example: numbers.
My absolute favorite. As a math teacher, I get this a lot on my first day of a new school year, "Ms. S., what 36*129?" I am, of course, expected to do this quickly without a calculator. Since, unfortunately, most students have been taught to think that math is about numbers, or shapes, or equations, or graphs, or in general, mathematical objects.
Mathematician Paul Halmos (1916-2006) wrote an excellent essay on this misconception. An excerpt:
I’ve never been a human-computer type of whiz-kid, and I can’t compute ‘in my head’, without the aid of pen and paper. I don’t, in fact, even like numbers particularly, and thankfully never had to deal with them much in school (as a math major). I’m not even a quantitative type of person. And I certainly do not believe in the applicability of mathematics to all human endeavors (well, I believe it can be applied in every circumstance, but not with positive effects; take, for example, the current testing regime in education as a prime example of the devastation that can be wreaked by our belief in numbers to solve all problems).
But, I still love math. And I choose to prove my teacher worthiness by beginning the year-long conversation with my students about how math is not about its objects, but is rather a language. A language unlike any other we speak, one predicated on conciseness, precision, and directness. A language that may also be applied widely, but with caution and attention to ethical and social implications.
---
But let me get back on task. So I absolutely agree with Lockhart: mathematics, as language, is an art. It is indeed a tragedy that in too many schools today, math education consists of futile exercises in computation, memorization of formulas, of solving contrived word problems, and, more recently, manifests as an endless quest to eliminate wrong answer choices on standardized tests. YES of course I agree: math should not be taught in procedural fashion, formulas should not be blindly memorized, problems should not be contrived to be about “real life” situations.
I’m also in agreement that we need to invest in programs that will train all of our math teachers in formal mathematics. At the moment, most math teachers in the US have transitioned from the workforce (engineering mostly, some physics and other sciences) or have degrees in math education. It is my and Lockhart’s contention that a deep understanding of the subject is required in order to be able to relate the essence of math.
[In an ironic twist, badly applied quantitative measures of unquantifiable phenomena (such as the experience of student learning) suggests that math degrees don’t make a difference in terms of student “success” (See this Edweek Article)].
While I agree with Lockhart’s assessment of the inadequacy of the current state of math education, I strongly dissent to his suggestions for how we should move towards reform. A Mathematician’s Lament lacks any kind of historical understanding, and does not at all consult pedagogical and curriculum literature.
For example, Lockhart writes that “word problems” should not be contrived to be about real life (I agree with this point), but then he continues that mathematics is beautiful precisely because it is irrelevant to ‘real life’..
I cannot comprehend how another mathematician could possibly believe the beauty of mathematics comes from the "irrelevance" of its abstractions: 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!!
I teach functions (precalculus, AP calculus) and the main theme is 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 represent 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.
My (and I believe, many) students would be aghast to learn that someone 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 read 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…* (See very long note on Dualism in Lockhart & Real Life Applications)
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, it is absolutely not supported by any peer-reviewed research 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, and I tailor my classes for my particular students that year.
*Very Long Note on Dualism in Lockhart & Real World Applications
Math is a language, and as such, art, that captures the most abstract essence of our world. Some even go as far to say that math is a structure of our very universe (see Penrose and Tegmark); I’m trudging through their work at the moment and am not convinced, but this may change.
Rather, what I’ve always believed is that math is embodied in our cognitive schemas and perception, and that this is precisely what makes it so wonderful: humanity's inherent capacity for thinking about the real world in this abstract way (see Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being by George Lakoff & Rafael Núñez). The point is, these ideas are born in experience: and, in turn, our experiential perceptions are shaped by these ideas, creating the cyclical process of learning and expanding our horizons.
From this perspective, in which experience/perception are perpetually interconnected to our cognitive schemas in a cycle of expansion, to say as Lockheart does, that math, or that anything, for that matter, is purely "of the mind" is basically Descartes all over again, “only the thought exists”. And we all know how well that turned out.
Now, I am not proposing the other side of the dualistic coin: teaching “math for engineering” type courses in which the emphasis is on the application. What I think is essential is to teach math in the context of its history, its applicability, its ethics, its abstractive prowess, its meaning.
Nothing is "born in our mind" alone; nothing exists in our "mind" alone; and for anything to make sense, the very idea of something having a sense, comes from our experiential perception.
Example #1: take the case of Zero (see Charles Seife's Zero: The Biography of a Dangerous Idea). We tend to think of "0" as a number, perhaps like any other, but this is far from accurate. Zero has a complex history within that of counting, and it was not at first even considered a number, but a place holder for "nothing." While anthropologists have discovered potential counting artifacts as old as 30,000 years, zero is only a few thousand of years old, if that. It took many tens of thousands of years after the adoption of numbers to "invent" the concept of "zero"-most likely because zero/the idea of cataloging “nothing” was not part of the daily experience of tracking items, livestock, or people. In fact, mathematicians to this day continue to refer to the set of integers "1" and above as “natural numbers”, and do not include 0 in this set.
Example #2: the concepts of positive and negative, the number line as a construct. The number line parallels our perceptive ability to set dualistic reference points in/with our bodies, such as east-west, up-down, right-left, and so on; this reference-setting tendency is further related to our bipedal structure. Of course, we also think in terms of “continuums”, mostly one-dimensional (linear). I often wonder what our mathematics would be like if humans had the anatomy of octopuses!
I read this book some time ago (2008?). Posted the first review in 2010, which has been significantly edited from its original version in this April 30, 2016 update
Lockhart and I started in lock-step. YES. The current state of mathematics education is a TRAVESTY - we are most emphatically not teaching students to fully appreciate its abstractive powers, philosophical implications, and inherent structure. Instead, we are taking the artifacts of doing mathematics and positing that these are what math is about.
An example: numbers.
My absolute favorite. As a math teacher, I get this a lot on my first day of a new school year, "Ms. S., what 36*129?" I am, of course, expected to do this quickly without a calculator. Since, unfortunately, most students have been taught to think that math is about numbers, or shapes, or equations, or graphs, or in general, mathematical objects.
Mathematician Paul Halmos (1916-2006) wrote an excellent essay on this misconception. An excerpt:
"to begin with, mathematicians have very little to do with numbers. You can no more expect a mathematician to be able to add a column of figures rapidly and correctly than you can expect a painter to draw a straight line or a surgeon to carve a turkey-popular legend attributes such skills to these professions, but popular legend is wrong. There is, to be sure, a part of mathematics called number theory, but even that doesn't deal with numbers..." PDF File here (YES! What number theory is about, by the way, is the concept of counting; numbers are but an artifact…)
I’ve never been a human-computer type of whiz-kid, and I can’t compute ‘in my head’, without the aid of pen and paper. I don’t, in fact, even like numbers particularly, and thankfully never had to deal with them much in school (as a math major). I’m not even a quantitative type of person. And I certainly do not believe in the applicability of mathematics to all human endeavors (well, I believe it can be applied in every circumstance, but not with positive effects; take, for example, the current testing regime in education as a prime example of the devastation that can be wreaked by our belief in numbers to solve all problems).
But, I still love math. And I choose to prove my teacher worthiness by beginning the year-long conversation with my students about how math is not about its objects, but is rather a language. A language unlike any other we speak, one predicated on conciseness, precision, and directness. A language that may also be applied widely, but with caution and attention to ethical and social implications.
---
But let me get back on task. So I absolutely agree with Lockhart: mathematics, as language, is an art. It is indeed a tragedy that in too many schools today, math education consists of futile exercises in computation, memorization of formulas, of solving contrived word problems, and, more recently, manifests as an endless quest to eliminate wrong answer choices on standardized tests. YES of course I agree: math should not be taught in procedural fashion, formulas should not be blindly memorized, problems should not be contrived to be about “real life” situations.
I’m also in agreement that we need to invest in programs that will train all of our math teachers in formal mathematics. At the moment, most math teachers in the US have transitioned from the workforce (engineering mostly, some physics and other sciences) or have degrees in math education. It is my and Lockhart’s contention that a deep understanding of the subject is required in order to be able to relate the essence of math.
[In an ironic twist, badly applied quantitative measures of unquantifiable phenomena (such as the experience of student learning) suggests that math degrees don’t make a difference in terms of student “success” (See this Edweek Article)].
While I agree with Lockhart’s assessment of the inadequacy of the current state of math education, I strongly dissent to his suggestions for how we should move towards reform. A Mathematician’s Lament lacks any kind of historical understanding, and does not at all consult pedagogical and curriculum literature.
For example, Lockhart writes that “word problems” should not be contrived to be about real life (I agree with this point), but then he continues that mathematics is beautiful precisely because it is irrelevant to ‘real life’..
I cannot comprehend how another mathematician could possibly believe the beauty of mathematics comes from the "irrelevance" of its abstractions: 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!!
I teach functions (precalculus, AP calculus) and the main theme is 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 represent 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.
My (and I believe, many) students would be aghast to learn that someone 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 read 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…* (See very long note on Dualism in Lockhart & Real Life Applications)
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, it is absolutely not supported by any peer-reviewed research 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, and I tailor my classes for my particular students that year.
*Very Long Note on Dualism in Lockhart & Real World Applications
Math is a language, and as such, art, that captures the most abstract essence of our world. Some even go as far to say that math is a structure of our very universe (see Penrose and Tegmark); I’m trudging through their work at the moment and am not convinced, but this may change.
Rather, what I’ve always believed is that math is embodied in our cognitive schemas and perception, and that this is precisely what makes it so wonderful: humanity's inherent capacity for thinking about the real world in this abstract way (see Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being by George Lakoff & Rafael Núñez). The point is, these ideas are born in experience: and, in turn, our experiential perceptions are shaped by these ideas, creating the cyclical process of learning and expanding our horizons.
From this perspective, in which experience/perception are perpetually interconnected to our cognitive schemas in a cycle of expansion, to say as Lockheart does, that math, or that anything, for that matter, is purely "of the mind" is basically Descartes all over again, “only the thought exists”. And we all know how well that turned out.
Now, I am not proposing the other side of the dualistic coin: teaching “math for engineering” type courses in which the emphasis is on the application. What I think is essential is to teach math in the context of its history, its applicability, its ethics, its abstractive prowess, its meaning.
Nothing is "born in our mind" alone; nothing exists in our "mind" alone; and for anything to make sense, the very idea of something having a sense, comes from our experiential perception.
Example #1: take the case of Zero (see Charles Seife's Zero: The Biography of a Dangerous Idea). We tend to think of "0" as a number, perhaps like any other, but this is far from accurate. Zero has a complex history within that of counting, and it was not at first even considered a number, but a place holder for "nothing." While anthropologists have discovered potential counting artifacts as old as 30,000 years, zero is only a few thousand of years old, if that. It took many tens of thousands of years after the adoption of numbers to "invent" the concept of "zero"-most likely because zero/the idea of cataloging “nothing” was not part of the daily experience of tracking items, livestock, or people. In fact, mathematicians to this day continue to refer to the set of integers "1" and above as “natural numbers”, and do not include 0 in this set.
Example #2: the concepts of positive and negative, the number line as a construct. The number line parallels our perceptive ability to set dualistic reference points in/with our bodies, such as east-west, up-down, right-left, and so on; this reference-setting tendency is further related to our bipedal structure. Of course, we also think in terms of “continuums”, mostly one-dimensional (linear). I often wonder what our mathematics would be like if humans had the anatomy of octopuses!
I read this book some time ago (2008?). Posted the first review in 2010, which has been significantly edited from its original version in this April 30, 2016 update
October 20, 2010
A Mathematician's Lament is more of a long essay than a book--one man's problems with mathematics education without a viable solution. Now, I consider myself, while no mathematician, a mathematics...enthusiast, if you will. I read the occasional recreational mathematics book, I am one of the three people on earth who subscribes to the journal of recreational mathematics, I am constantly sneaking new variations on Tangrams and other puzzles into the house. And I am definitely not a fan of modern American elementary education; I consider my public school education to have prepared me adequately enough for the world, but it did seem largely like a waste of time. Oh, those purple dittos with the blocks that had to be colored in based on the results of math problems -- they were my nemesis, and not because I couldn't do arithmetic.
So partly, I think his argument - that mathematics education gets tripped up in unnecessary formalism and syntax before conceptually interesting problems are tackled, and that the whole thing is defended with the ridiculous "you might need this someday" pragmatism that children will instantly tune out - is a sound one. It would be great if every elementary school teacher were the kind of engaged leader capable of putting his or her students to work on an interesting geometry or abstract algebra problem and wandering around not to give answers, but to provide the occasional hint. Unfortunately, I just don't see how this is going to provide anyone with a well-rounded mathematics education. Lockhart argues that what they get now -- not learning anything because they're so bored -- is worse.
It's tempting to believe all of this. And yet, I couldn't get a nagging thought out of my mind: I managed to get through all that arithmetic, algebra, geometry, and calculus with an ability to apply most of it. I didn't go through school memorizing formulas and I never felt forced to do so either; to this day I couldn't tell you most trig identities without starting at the Pythagorean theorem and deriving them. Is the state of things really that bad?
Even accepting that it is, Lockhart's book is breezy and quick....what might have been called a pamphlet, perhaps, in earlier days. And like satirical pamphlets, it does an excellent job lampooning the state of things but offers very little in the way of realistic alternatives. I am all ears to hear new ways of teaching children math; I imagine classrooms could be greatly improved by incorporating topics from recreational math, calculus, and abstract algebra at an earlier age, and have never understood the obsession with arithmetic that dominates the first years of mathematics education. But Lockhart doesn't provide much in the way of real solutions--just the clue that he really likes circumscribed triangle problems. Spend all of mathematics class playing chess and go? That might be fun, but I'm unconvinced that the primary purpose of school is to entertain.
So partly, I think his argument - that mathematics education gets tripped up in unnecessary formalism and syntax before conceptually interesting problems are tackled, and that the whole thing is defended with the ridiculous "you might need this someday" pragmatism that children will instantly tune out - is a sound one. It would be great if every elementary school teacher were the kind of engaged leader capable of putting his or her students to work on an interesting geometry or abstract algebra problem and wandering around not to give answers, but to provide the occasional hint. Unfortunately, I just don't see how this is going to provide anyone with a well-rounded mathematics education. Lockhart argues that what they get now -- not learning anything because they're so bored -- is worse.
It's tempting to believe all of this. And yet, I couldn't get a nagging thought out of my mind: I managed to get through all that arithmetic, algebra, geometry, and calculus with an ability to apply most of it. I didn't go through school memorizing formulas and I never felt forced to do so either; to this day I couldn't tell you most trig identities without starting at the Pythagorean theorem and deriving them. Is the state of things really that bad?
Even accepting that it is, Lockhart's book is breezy and quick....what might have been called a pamphlet, perhaps, in earlier days. And like satirical pamphlets, it does an excellent job lampooning the state of things but offers very little in the way of realistic alternatives. I am all ears to hear new ways of teaching children math; I imagine classrooms could be greatly improved by incorporating topics from recreational math, calculus, and abstract algebra at an earlier age, and have never understood the obsession with arithmetic that dominates the first years of mathematics education. But Lockhart doesn't provide much in the way of real solutions--just the clue that he really likes circumscribed triangle problems. Spend all of mathematics class playing chess and go? That might be fun, but I'm unconvinced that the primary purpose of school is to entertain.
December 29, 2011
This was TERRIBLE. The first chapter began so well and had me so psyched about the book. Lockhart made an analogy between mathematics and music, where a musician wakes from a terrible dream in which public schools teach only the mechanics of music, but students are not allowed to compose or listen to music until college level. I thought it was a brilliant analogy. But it was downhill from there. His solution to the problem of math not being "fun" seemed to be to no longer teach the mechanics of math, but to just emphasize over and over that math is "art". I drug myself a quarter of the way through the book before giving up. I love math, but I have no clue what the hell he meant by "art". And I'm not sure how anyone who doesn't know the mechanics of math could ever make it to the fun part, which is solving puzzles. A good analogy for his method would be a French teacher who doesn't require that her students know any actual words in French, but instead tells her students over and over what a beautiful language it is. Can't be boring them with the "mechanics" ya know.
March 10, 2017
Ugh, I get it, math is beautiful and no one understands you...
While this essay was eye-opening about the way that the current education system treats math instruction, I overall found the author idealistic and demeaning. Lockhart actually believes that if teachers are just "honest" with their students that the kids will learn. Regardless of an understanding of educational psychology and pedagogy (that's all inauthentic anyway!), teachers just need to be open! Please! He also adds that there should be no required math curriculum, but that students should just play around with numbers and patterns and learn what they learn. (?!?!!)
It was an interesting read, but ultimately unhelpful because Lockhart gives no suggestions for those of us actually trying to teach math under the current standards.
While this essay was eye-opening about the way that the current education system treats math instruction, I overall found the author idealistic and demeaning. Lockhart actually believes that if teachers are just "honest" with their students that the kids will learn. Regardless of an understanding of educational psychology and pedagogy (that's all inauthentic anyway!), teachers just need to be open! Please! He also adds that there should be no required math curriculum, but that students should just play around with numbers and patterns and learn what they learn. (?!?!!)
It was an interesting read, but ultimately unhelpful because Lockhart gives no suggestions for those of us actually trying to teach math under the current standards.
November 4, 2011
I know - 4 stars? Really? The content and the ideas and the presentation are 5-star material. He's a bit crude sometimes, and there's a particularly hedonistic phrase used near the end of the book (part 2, not the free essay material) that I felt was just unneeded. And since I recommended this to all my dear homeschooling friends, some of whom have tender sensibilities, I knocked a star off. Disclaimer done.
Now, for the high praises!! YES, math is supposed to be FUN. It's about noticing, thinking, discovering, beauty, "coincidences," patterns, sense, reasoning and FUN. I typed all sorts of quotes into the goodreads database (just check the right sidebar on this book's page, and you'll see a link to more quotes from this book).
I was blessed, I suppose, to not only inherit nerdy genes, but critical-thinking genes; to have been exposed to puzzle-fun growing up; and to have had a great public school experience over-all; AND to have had a go-at-your-own-pace math curriculum in 5th grade, which was the highlight of every day. I didn't particularly enjoy timed multiplication tests in 3rd grade (I choke under pressure), and Linear Algebra kicked my fanny in college. But everything in between was fun! (Ok, I lied: Mrs. Oyamot used to mark off point for misspelled words on our Algebra tests, which I thought was a bit cranky.)
The point being, Lockhart's Lament is preaching to the choir with me. I've always loved math. And I've always loved it because of the fun of exploring and discovering, and its beauty. I *got* that out of math class, even though it wasn't taught that way.
Also, I'm on the educational fringe already. I homeschool and don't use curriculum; I'm so far away from being concerned with K-12 curriculum it isn't funny.
*******
From my notes:
p46 "Honest intellectual relationship" caused me pause - I suppose because it looks so different depending on the setting and the age. And I take honesty for granted.
p48 "We lean things because they interest us now, not because they might be useful later..." There's truth in that, and it's certainly more true for the young and the immature and some personalities. But I hope that if someone is becoming truly educated and not just "taught" they develop a desire to learn useful things. Of course, that's gradual.
p49 [Large-digit addition is too advanced for most 3rd graders.... Wait until they're naturally curious about numbers.] I mostly agree with this; primarily because this is how I handled reading. I didn't push reading on my kids, rather I exposed them to lots of books, I modeled reading and I read TO them. Especially because math is so maligned in today's world, I think exposure to "playing" math is essential. If you wait until someone develops a natural curiosity without exposing them to said subject, it might never happen. Exposure in a playful, fun way is essential.
p50 "Preparing tomorrow's workforce today" is such a sarcastic, loaded remark!!
p52 He contrasts memorizing poetry (the "awful" way of teaching) to writing your own (the only "good" way), and I believe them to be points on a continuum. It's asking a lot for a child to write poetry without any exposure. Exposure, again, is key.
p63 "Cogs in a great soul-crushing machine" rivals the oft-quoted Titanic reference in the beginning.
p64 More staggering cruelty towards The System: "The problem is not that the students can't handle it, it's that none of the teachers can."
p78-9 This pretty much epitomizes his criticism: "The problem with the standard geometry [or any math] curriculum is that the private, personal experience of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a rigid step-by-step pattern of uninspired formal deductions. The textbook presents a set of definitions, theorems, and proofs, the teacher copies then onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the "good" students.
"The result is that the student becomes a passive participant in the creative act.... They are being trained to APE arguments, not to INTEND them. So not only do they have no idea what their teacher is saying, THEY HAVE NO IDEA WHAT THEY THEMSELVES ARE SAYING."
*******
Part 2 of the book models a few instances of questioning and working for the answer. My favorite was the one about odd numbers and square numbers. I never knew that one! :) The second part of the book was a nice complement to the first, but simply reading the essay online certainly communicates the point.
His detractors are fair when they say that he gives little concrete suggestions about how to *go about* teaching math without curriculum, but on the other hand, one of his main points is that you can't teach math if you don't love it. Your enthusiasm and guidance will facilitate math-learning, based on where your students are and your time together, etc. BUT, the world would certainly benefit from more "this is what we did to explore math today" ideas.
Highly recommended if you think you're "not good at math." If you dislike math there's a good chance you don't understand what it is AT ALL.
Happy mathing!
(see also http://www2.ed.gov/about/bdscomm/list...
http://www.noetic-learning.com/gifted...
http://www.artofproblemsolving.com/Vi... especially "doodling in math class")
Now, for the high praises!! YES, math is supposed to be FUN. It's about noticing, thinking, discovering, beauty, "coincidences," patterns, sense, reasoning and FUN. I typed all sorts of quotes into the goodreads database (just check the right sidebar on this book's page, and you'll see a link to more quotes from this book).
I was blessed, I suppose, to not only inherit nerdy genes, but critical-thinking genes; to have been exposed to puzzle-fun growing up; and to have had a great public school experience over-all; AND to have had a go-at-your-own-pace math curriculum in 5th grade, which was the highlight of every day. I didn't particularly enjoy timed multiplication tests in 3rd grade (I choke under pressure), and Linear Algebra kicked my fanny in college. But everything in between was fun! (Ok, I lied: Mrs. Oyamot used to mark off point for misspelled words on our Algebra tests, which I thought was a bit cranky.)
The point being, Lockhart's Lament is preaching to the choir with me. I've always loved math. And I've always loved it because of the fun of exploring and discovering, and its beauty. I *got* that out of math class, even though it wasn't taught that way.
Also, I'm on the educational fringe already. I homeschool and don't use curriculum; I'm so far away from being concerned with K-12 curriculum it isn't funny.
*******
From my notes:
p46 "Honest intellectual relationship" caused me pause - I suppose because it looks so different depending on the setting and the age. And I take honesty for granted.
p48 "We lean things because they interest us now, not because they might be useful later..." There's truth in that, and it's certainly more true for the young and the immature and some personalities. But I hope that if someone is becoming truly educated and not just "taught" they develop a desire to learn useful things. Of course, that's gradual.
p49 [Large-digit addition is too advanced for most 3rd graders.... Wait until they're naturally curious about numbers.] I mostly agree with this; primarily because this is how I handled reading. I didn't push reading on my kids, rather I exposed them to lots of books, I modeled reading and I read TO them. Especially because math is so maligned in today's world, I think exposure to "playing" math is essential. If you wait until someone develops a natural curiosity without exposing them to said subject, it might never happen. Exposure in a playful, fun way is essential.
p50 "Preparing tomorrow's workforce today" is such a sarcastic, loaded remark!!
p52 He contrasts memorizing poetry (the "awful" way of teaching) to writing your own (the only "good" way), and I believe them to be points on a continuum. It's asking a lot for a child to write poetry without any exposure. Exposure, again, is key.
p63 "Cogs in a great soul-crushing machine" rivals the oft-quoted Titanic reference in the beginning.
p64 More staggering cruelty towards The System: "The problem is not that the students can't handle it, it's that none of the teachers can."
p78-9 This pretty much epitomizes his criticism: "The problem with the standard geometry [or any math] curriculum is that the private, personal experience of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a rigid step-by-step pattern of uninspired formal deductions. The textbook presents a set of definitions, theorems, and proofs, the teacher copies then onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the "good" students.
"The result is that the student becomes a passive participant in the creative act.... They are being trained to APE arguments, not to INTEND them. So not only do they have no idea what their teacher is saying, THEY HAVE NO IDEA WHAT THEY THEMSELVES ARE SAYING."
*******
Part 2 of the book models a few instances of questioning and working for the answer. My favorite was the one about odd numbers and square numbers. I never knew that one! :) The second part of the book was a nice complement to the first, but simply reading the essay online certainly communicates the point.
His detractors are fair when they say that he gives little concrete suggestions about how to *go about* teaching math without curriculum, but on the other hand, one of his main points is that you can't teach math if you don't love it. Your enthusiasm and guidance will facilitate math-learning, based on where your students are and your time together, etc. BUT, the world would certainly benefit from more "this is what we did to explore math today" ideas.
Highly recommended if you think you're "not good at math." If you dislike math there's a good chance you don't understand what it is AT ALL.
Happy mathing!
(see also http://www2.ed.gov/about/bdscomm/list...
http://www.noetic-learning.com/gifted...
http://www.artofproblemsolving.com/Vi... especially "doodling in math class")
March 3, 2025
There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.
i have recommended this essay to a lot of people who subsequently saw the length and went, "...oh," and did not read it. and okay, yeah, i don't know a lot of people excited to dive into twenty-five pages of Discussion About Math. but you should want to! because this essay is clever and bitingly funny and really fucking furious, and if you are or were ever a high schooler sitting in math class wondering WHAT the point of it all was, you’re gonna feel pretty fucking vindicated by this one.
yes, some of lockhart’s turns of phrase and even some of his points might be overdramatic, but regardless of whether you agree, this is worth reading. the first time i read this, i was just mad about my algebra homework and hoping for some validation, and guess what! it changed my entire view of math as a field forever. it’s worth reading not just because it’s a scathing takedown of the way math is taught in schools, but also because it’s a beautiful description of what math SHOULD be from someone whose clear love for math as an art shines through in every sentence. it’s also worth reading because sometimes you need someone with authority to go, “hey, yeah, you’re right, math classes ARE boring and chock-full of busywork.” and if none of those reasons are persuasive to you, it’s worth reading for this lethal jump cut:
(find it here!)
i have recommended this essay to a lot of people who subsequently saw the length and went, "...oh," and did not read it. and okay, yeah, i don't know a lot of people excited to dive into twenty-five pages of Discussion About Math. but you should want to! because this essay is clever and bitingly funny and really fucking furious, and if you are or were ever a high schooler sitting in math class wondering WHAT the point of it all was, you’re gonna feel pretty fucking vindicated by this one.
yes, some of lockhart’s turns of phrase and even some of his points might be overdramatic, but regardless of whether you agree, this is worth reading. the first time i read this, i was just mad about my algebra homework and hoping for some validation, and guess what! it changed my entire view of math as a field forever. it’s worth reading not just because it’s a scathing takedown of the way math is taught in schools, but also because it’s a beautiful description of what math SHOULD be from someone whose clear love for math as an art shines through in every sentence. it’s also worth reading because sometimes you need someone with authority to go, “hey, yeah, you’re right, math classes ARE boring and chock-full of busywork.” and if none of those reasons are persuasive to you, it’s worth reading for this lethal jump cut:
(find it here!)
March 9, 2023
If math had been taught like this I would have actually enjoyed it in school.
May 29, 2024
This essay is so brilliant. SO brilliant.
There are some criticisms (at the end) but still, this is written in such a striking, creative and yet truthful/accurate way that I give it 5 stars.
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters— the ones who know their colors and brushes backwards and forwards— they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.”
I recognize this exact set up from SO many of the classes I've been in. Classes in which we memorize and regurgitate, and when we ask what this is all for - we are told it will be useful later in life with no further explanation. I.e. a teacher's way of saying "Please shut up". But I get it - the teachers are also beholden to the same system of fulfilling a curriculum.
This is taken form the musician’s nightmare:
In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.
The textbook presents a set of definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the “good” students.
Spot on!
Even though I was a good student - one who succeeded and one who enjoyed learning, so many mindless drills and exercises in school lowered my interest in subjects like physics, and biology - which later returned when I started reading about these topics on my own.
The one criticism I have of this essay is Lockhart's insistence that what makes math fun and inspiring is that it's not connected to real life. He claims we shouldn't force a connection to everyday life, we should embrace the pure joy of abstraction. I understand the joy he's talking about - I experience it - but I think one has to already be somewhat advanced at an abstract topic to experience this joy.
In the beginning, there is an initial effort to start learning about the topic - the best way to get students to learn is almost always to wake up their already existing curiosity and show them how this new topic applies to what they already know.
I think back to learning to program - just in the past few years. Coming to new concepts like loops, classes, objects - as an abstraction it all means nothing. It was only when I started to write loops to do something useful and tangible for me that I understood what they are and what they do - then the abstract properties of loops became clear - automatically and effortlessly - due to the fact that I could navigate the "real life" application of the loop.
So I think Lockhart is right that our math education is extremely boring, and he's right that we should make it more interesting and intuitive - but his approach to achieving this end goal is questionable.
There are some criticisms (at the end) but still, this is written in such a striking, creative and yet truthful/accurate way that I give it 5 stars.
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters— the ones who know their colors and brushes backwards and forwards— they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.”
I recognize this exact set up from SO many of the classes I've been in. Classes in which we memorize and regurgitate, and when we ask what this is all for - we are told it will be useful later in life with no further explanation. I.e. a teacher's way of saying "Please shut up". But I get it - the teachers are also beholden to the same system of fulfilling a curriculum.
This is taken form the musician’s nightmare:
In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.
The textbook presents a set of definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the “good” students.
Spot on!
Even though I was a good student - one who succeeded and one who enjoyed learning, so many mindless drills and exercises in school lowered my interest in subjects like physics, and biology - which later returned when I started reading about these topics on my own.
The one criticism I have of this essay is Lockhart's insistence that what makes math fun and inspiring is that it's not connected to real life. He claims we shouldn't force a connection to everyday life, we should embrace the pure joy of abstraction. I understand the joy he's talking about - I experience it - but I think one has to already be somewhat advanced at an abstract topic to experience this joy.
In the beginning, there is an initial effort to start learning about the topic - the best way to get students to learn is almost always to wake up their already existing curiosity and show them how this new topic applies to what they already know.
I think back to learning to program - just in the past few years. Coming to new concepts like loops, classes, objects - as an abstraction it all means nothing. It was only when I started to write loops to do something useful and tangible for me that I understood what they are and what they do - then the abstract properties of loops became clear - automatically and effortlessly - due to the fact that I could navigate the "real life" application of the loop.
So I think Lockhart is right that our math education is extremely boring, and he's right that we should make it more interesting and intuitive - but his approach to achieving this end goal is questionable.
February 8, 2022
I wasn’t smart enough to understand this book entirely but I did appreciate it. I just felt like his critique of math classes and curriculum went on forever but that was because Calculus means nothing to me at this time 😂
The second half was very interesting and actually got me interested in math. Slightly miraculous.
The second half was very interesting and actually got me interested in math. Slightly miraculous.
December 10, 2024
This book inspired me to spend four hours trying to work out a conjecture about how the areas of regular polygons relate to the number pi.
February 7, 2016
This is brilliant! After reading this I finally remember what fascinated me about maths before I was trained to sit maths exams.
The author's stunningly poethic approach to math as a study of world and its transcendent nature that is so eloquently explained in this work can make even the most antimathematically thinking person to fall in love with maths!
A sure must reread.
The author's stunningly poethic approach to math as a study of world and its transcendent nature that is so eloquently explained in this work can make even the most antimathematically thinking person to fall in love with maths!
A sure must reread.
April 25, 2017
This is far less a "lament" and much more of a rant. Like most rants, it starts off with some reasonable points/objections and is amusing to listen to at first ... and then, like most rants, it veers right off the road, through the fence, and ends up upside down in a pond with a nearby cow just staring at the wreckage, slowly chewing its cud.
I have no objection to the claim(s) that the way math is taught today is illogical and stultifying. But the notions that math is an "art" and that is was created by humans for their own amusement are absurd. You know, pottery and weaving can both produce artistic end-products but the notion that they were created just for amusement would also be dismissed as ridiculous...
The author also veers off into some rather unproductive and uncalled-for attacks on teachers themselves. While the institutional approach to math education may be lousy and many math teachers may not be the inspired math-artisans that the author wants them to be, I know too many of them who work too hard to do the best they can to accept his attacks as valid.
Frankly, this was a waste of time and not worth reading - if you can find it in a bookstore or library, read the first chapter and call it a day there.
I have no objection to the claim(s) that the way math is taught today is illogical and stultifying. But the notions that math is an "art" and that is was created by humans for their own amusement are absurd. You know, pottery and weaving can both produce artistic end-products but the notion that they were created just for amusement would also be dismissed as ridiculous...
The author also veers off into some rather unproductive and uncalled-for attacks on teachers themselves. While the institutional approach to math education may be lousy and many math teachers may not be the inspired math-artisans that the author wants them to be, I know too many of them who work too hard to do the best they can to accept his attacks as valid.
Frankly, this was a waste of time and not worth reading - if you can find it in a bookstore or library, read the first chapter and call it a day there.
March 19, 2018
The author of this book really loves math, and believes that what is bring taught as “math” in schools is not math, but rather an exercise in soul-crushing drudgery and torture. He has some interesting points. He says that true mathematics is an art form and has no practical purpose other than to bring joy to those who practice it. It should be taught as an art form just for the sake of enjoying its true beauty, just like we would study great literature, art, or music. As a Charlotte Mason-inspired educator, I am intrigued. Miss Mason believed in a “liberal education for all,” and did not think all learning had to be related to preparing for a particular trade. But I digress.
The first part of the book is a mouthy rant against the current system and the second part is an introduction to “true mathematics,” namely, problem solving. Figuring out how numbers work together, how patterns work, how lines relate to each other in space, etc. What is sorely lacking is any kind of practical description of how a person would lead a child in practicing “real mathematics.” Perhaps he would say I am missing the point.
The first part of the book is a mouthy rant against the current system and the second part is an introduction to “true mathematics,” namely, problem solving. Figuring out how numbers work together, how patterns work, how lines relate to each other in space, etc. What is sorely lacking is any kind of practical description of how a person would lead a child in practicing “real mathematics.” Perhaps he would say I am missing the point.
June 21, 2020
A passionate clarion call for educators, and a genuinely fun read. This will be more palpable to a wider audience than Hardy's "Apology" but it is no less opinionated. Not having any real relation to the US math curriculum, I read this for the opinions and the math, which were both fantastic. The only thing I have against it is that it ended too quickly. The conversational approach towards proofs was most invigorating.
July 5, 2011
Disclaimer 1) This is only a review of the 25 page essay, which can be found here: http://www.maa.org/devlin/LockhartsLa.... Why am I reviewing the essay instead of the book? Well, I don’t have the book, but I did read the essay and thought that posting a review of even part of it would be of worth to some poor, sad, math-challenged-but-don’t-know-why soul.
Disclaimer2) I know next to nothing about mathematics, but am endeavoring to want to learn it. God bless you, Sally B., for sending me the link to this paper before I went out and bought a set of Saxon math!
Okay, I admit it. I’m math-o-phobic. Always have been. What are you going to do? Bad teachers. Boring subject. No natural tendency nor talent. Can’t see the relevancy (what are calculators for, anyway?) I just plain don’t get it. I’m just not a math person, I guess.
Does that sound familiar? Do you suffer too? Well, I think that some relief may lie herein.
But seriously, this essay turned me on my head concerning all things mathematical. I have a 12 yo whom I have been feeling needs to “begin” learning math, he’s getting on in years, it’s time I suppose. I truly was all set to get down to the dirty business of it and buy him the Saxon books this year and then give him the “some things are just plain boring, hard, and hateful, and we have to do them anyway so we can someday get into college” lecture. It could work.
Lucky for my kid. Lockhart saves the day (well, Sally saved the day really…).
You must know that Lockhart’s essay is more a treatise on great, inspirational (and inspired) teaching than mathematics. It also happens to be very well-written and laugh-out-loud hilarious, which really helps, as the subject matter (that millions of souls have been damaged or destroyed by being “taught” the all-time worst of subjects, math) can get a little downheartening. Truly it is so. You don’t believe it? Give this little essay a try.
Lockhart’s premise is this, if I can adequately sum up: Mathematics is the world’s oldest and most beautiful form of art. It is inspiring. It is gorgeous. It is simple. It is fun! Oh, except to everyone who has ever been taught it. I quote: “There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum.” The way we are taught math is as if we were mandatorily taught art by first learning theory and how to properly identify colors, mediums & utensils. Then if we’re lucky we can move on to “pre-paint-by-numbers” and then further on to real “paint-by-numbers.” And this is all without ever seeing a real work of art or hear a story of the life of a famous artist. If they are really good at that, and somehow still retain interest, maybe in college we’ll let them actually paint on a blank canvas and look at some art. (Lockhart’s analogy.)
Lockhart says this is exactly how we teach mathematics. We’ve extracted all the miserably boring parts and shunted off all the beautiful parts, because, well, because it’s only the boring parts that can be adequately tested. HOLY BUCKETS!! You mean all the hell we all went through “learning” “math” was only a devilish device for testing? To see if we can follow directions? That is it?
There were only two things I disagreed with in this essay (at this reading, anyway.) One was Lockhart’s assumption that math is beautiful because it is so totally irrelevant (just flights of fancy and such). I’m not sure I read it right, and I’m not sure that’s even what he meant. Assuming that he meant what I think he said, I disagree in that I think it is very probable that as there are mathematical patterns in every living thing, God knows pretty advanced mathematics and He used them for a purpose. I think it is highly applicable to have fun discovering that relevancy.
Second, I disagree with this statement: “English teachers know that spelling and pronunciation are best learned in a context of reading and writing. History teachers know that names and dates are uninteresting when removed from the unfolding backstory of events.” So, if you get what I disagree with, sue me or wink too, whichever suits you best.
Really, there’s too much to say about this essay. I’d end up regurgitating it all and I really think it would do EVERY. SINGLE. PERSON. ON. THE. PLANET to give it a read. You don’t have to agree, but it’s worth a look. Sorry for the enthusiasm. But hey, it’s the first time EVER I’ve been enthusiastic about something mathematical. Let’s have a party.
I’ll leave you with a little excerpt from one of the little delightful question and answer “dialogues” Lockhart intermingles within the essay.
SIMPLICIO: But don’t we need people to learn those useful consequences of math? Don’t we need accountants and carpenters and such?
SALVIATI: How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is
excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide.
SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?
SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful.
SIMPLICIO: But not everyone is cut out to be an artist. What about the kids who aren’t “math people?” How would they fit into your scheme?
SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be “good at math.” 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.
SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?
SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.
SIMPLICIO: So you would remove mathematics from the school curriculum?
SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.
SIMPLICIO: But how many math teachers know enough about their subjects to teach it that way?
SALVIATI: Very few. And that’s just the tip of the iceberg….
Heck, someday, if I go about it right, I may even end up as math nerdy as Sally. But I’m not sure I can ever get to that level of cool.
--
P.S. Per reading some of the unfavorable reviews here on goodreads. Sure it is true that Lockhart does not offer many solutions to his problems. However, we live in an age where we always WANT other people to do the work and give us the solutions. I think half the fun of this journey (and definitely ALL of the reward) will be in finding our own way. I am so grateful for the freedom & opportunity & drive to educate my kids the way I see fit.
---
To Sally: Perhaps I ought to have found more to disagree with…but maybe that will come as I actually learn what math can and should be. I’ll have to reread it a year or so from now. For now, it was like lightning. And I can hardly wait to talk to you more about the whole subject. You are my self-chosen mathematics mentor, should you yourself choose to be appointed. With humble, yet enthused, gratitude; Krislyn
Disclaimer2) I know next to nothing about mathematics, but am endeavoring to want to learn it. God bless you, Sally B., for sending me the link to this paper before I went out and bought a set of Saxon math!
Okay, I admit it. I’m math-o-phobic. Always have been. What are you going to do? Bad teachers. Boring subject. No natural tendency nor talent. Can’t see the relevancy (what are calculators for, anyway?) I just plain don’t get it. I’m just not a math person, I guess.
Does that sound familiar? Do you suffer too? Well, I think that some relief may lie herein.
But seriously, this essay turned me on my head concerning all things mathematical. I have a 12 yo whom I have been feeling needs to “begin” learning math, he’s getting on in years, it’s time I suppose. I truly was all set to get down to the dirty business of it and buy him the Saxon books this year and then give him the “some things are just plain boring, hard, and hateful, and we have to do them anyway so we can someday get into college” lecture. It could work.
Lucky for my kid. Lockhart saves the day (well, Sally saved the day really…).
You must know that Lockhart’s essay is more a treatise on great, inspirational (and inspired) teaching than mathematics. It also happens to be very well-written and laugh-out-loud hilarious, which really helps, as the subject matter (that millions of souls have been damaged or destroyed by being “taught” the all-time worst of subjects, math) can get a little downheartening. Truly it is so. You don’t believe it? Give this little essay a try.
Lockhart’s premise is this, if I can adequately sum up: Mathematics is the world’s oldest and most beautiful form of art. It is inspiring. It is gorgeous. It is simple. It is fun! Oh, except to everyone who has ever been taught it. I quote: “There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum.” The way we are taught math is as if we were mandatorily taught art by first learning theory and how to properly identify colors, mediums & utensils. Then if we’re lucky we can move on to “pre-paint-by-numbers” and then further on to real “paint-by-numbers.” And this is all without ever seeing a real work of art or hear a story of the life of a famous artist. If they are really good at that, and somehow still retain interest, maybe in college we’ll let them actually paint on a blank canvas and look at some art. (Lockhart’s analogy.)
Lockhart says this is exactly how we teach mathematics. We’ve extracted all the miserably boring parts and shunted off all the beautiful parts, because, well, because it’s only the boring parts that can be adequately tested. HOLY BUCKETS!! You mean all the hell we all went through “learning” “math” was only a devilish device for testing? To see if we can follow directions? That is it?
There were only two things I disagreed with in this essay (at this reading, anyway.) One was Lockhart’s assumption that math is beautiful because it is so totally irrelevant (just flights of fancy and such). I’m not sure I read it right, and I’m not sure that’s even what he meant. Assuming that he meant what I think he said, I disagree in that I think it is very probable that as there are mathematical patterns in every living thing, God knows pretty advanced mathematics and He used them for a purpose. I think it is highly applicable to have fun discovering that relevancy.
Second, I disagree with this statement: “English teachers know that spelling and pronunciation are best learned in a context of reading and writing. History teachers know that names and dates are uninteresting when removed from the unfolding backstory of events.” So, if you get what I disagree with, sue me or wink too, whichever suits you best.
Really, there’s too much to say about this essay. I’d end up regurgitating it all and I really think it would do EVERY. SINGLE. PERSON. ON. THE. PLANET to give it a read. You don’t have to agree, but it’s worth a look. Sorry for the enthusiasm. But hey, it’s the first time EVER I’ve been enthusiastic about something mathematical. Let’s have a party.
I’ll leave you with a little excerpt from one of the little delightful question and answer “dialogues” Lockhart intermingles within the essay.
SIMPLICIO: But don’t we need people to learn those useful consequences of math? Don’t we need accountants and carpenters and such?
SALVIATI: How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is
excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide.
SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?
SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful.
SIMPLICIO: But not everyone is cut out to be an artist. What about the kids who aren’t “math people?” How would they fit into your scheme?
SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be “good at math.” 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.
SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?
SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.
SIMPLICIO: So you would remove mathematics from the school curriculum?
SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.
SIMPLICIO: But how many math teachers know enough about their subjects to teach it that way?
SALVIATI: Very few. And that’s just the tip of the iceberg….
Heck, someday, if I go about it right, I may even end up as math nerdy as Sally. But I’m not sure I can ever get to that level of cool.
--
P.S. Per reading some of the unfavorable reviews here on goodreads. Sure it is true that Lockhart does not offer many solutions to his problems. However, we live in an age where we always WANT other people to do the work and give us the solutions. I think half the fun of this journey (and definitely ALL of the reward) will be in finding our own way. I am so grateful for the freedom & opportunity & drive to educate my kids the way I see fit.
---
To Sally: Perhaps I ought to have found more to disagree with…but maybe that will come as I actually learn what math can and should be. I’ll have to reread it a year or so from now. For now, it was like lightning. And I can hardly wait to talk to you more about the whole subject. You are my self-chosen mathematics mentor, should you yourself choose to be appointed. With humble, yet enthused, gratitude; Krislyn
January 29, 2024
This is an unexpectedly wonderful book. As someone who hated math class even though I was supposedly good at it, and then someone who had trouble teaching it to my own children, I thoroughly enjoyed it. To use words that the author used about math, it was beautiful and elegant!
This is definitely a book to read many times. It is a fun book to read.
This is definitely a book to read many times. It is a fun book to read.
September 14, 2024
If I read it for class, I read it for goodreads
May 27, 2009
This book is fantastic. I recommend it to all those people who, upon hearing from me that I do math, have replied, "Oh, I suck at math" or "Oh, I always hated math in school." For years I've encountered a recurring frustration at the fact that, when I tell people that I'm studying mathematics, I tend to discover that they have a completely wrong impression of what it is that I do (or at least try to do), and that it is not easy to correct this impression. I try to tell them: you hate math because you don't know what it is, because the stuff they taught you in school was not math, but almost something else entirely; real math is fun and interesting and makes you think both logically and imaginatively, it is both beautiful and surprising. But I never quite feel like I get the whole truth across.
This book gets the whole truth across. It says everything I've ever wanted to say (as well as many things I hadn't thought of) in regards to all of this, but does a much better job than I ever could. He accurately portrays exactly how it is that people commonly misunderstand what mathematicians really do, the common misperceptions that they tend to have. Then he goes on to critique what is pretty obviously at the root of all this misunderstanding: the state of math education itself. Through a few clever analogies and a bit of simple explanation, the author demonstrates how the average student's assessment of the 'mathematics' that we are taught in school is completely accurate: namely, that it is arbitrary, stupid, and boring. He clearly and succinctly articulates the tragedy of the current state of the public math education system, and how it poisons the general public's understanding of what mathematics is, and never even comes close to giving the general public a real sense of what the subject is even about. He gleefully and accurately (and also quite humorously) tears to shreds the current K-12 curriculum, exposing it's idiocy.
But the book doesn't stop there; after showing us what is wrong with the current state of mathematics education, he goes on to give a brief and wonderful little demonstration, through several very accessible examples, of what mathematics really is, the types of things that mathematicians actually think about, and why they are interesting. I found this to be wonderful-the best part of the book.
The whole book is also superbly written; his choice of words had me laughing out loud at times.
Finally, reading this book actually made me feel somewhat ashamed of my sometimes aloof personal attitude towards my area of study; my tendency to think of my subject as something intractably esoteric and advanced, inaccessible to the general public (because, of course, only the intellectual elite such as myself are capable of comprehending it). At one point he makes note of the fact that many people make it most of the way through most of graduate school believing (because they've always been told) that they are good at math, only to discover, when they attempt to do some real mathematics, that all they were really good at is following directions. Ouch. I hope that isn't me...
So again, I recommend this book to all my friends and family who wonder what it is that I'm trying to do, and I may even be buying it for some of you, come Christmas time!
This book gets the whole truth across. It says everything I've ever wanted to say (as well as many things I hadn't thought of) in regards to all of this, but does a much better job than I ever could. He accurately portrays exactly how it is that people commonly misunderstand what mathematicians really do, the common misperceptions that they tend to have. Then he goes on to critique what is pretty obviously at the root of all this misunderstanding: the state of math education itself. Through a few clever analogies and a bit of simple explanation, the author demonstrates how the average student's assessment of the 'mathematics' that we are taught in school is completely accurate: namely, that it is arbitrary, stupid, and boring. He clearly and succinctly articulates the tragedy of the current state of the public math education system, and how it poisons the general public's understanding of what mathematics is, and never even comes close to giving the general public a real sense of what the subject is even about. He gleefully and accurately (and also quite humorously) tears to shreds the current K-12 curriculum, exposing it's idiocy.
But the book doesn't stop there; after showing us what is wrong with the current state of mathematics education, he goes on to give a brief and wonderful little demonstration, through several very accessible examples, of what mathematics really is, the types of things that mathematicians actually think about, and why they are interesting. I found this to be wonderful-the best part of the book.
The whole book is also superbly written; his choice of words had me laughing out loud at times.
Finally, reading this book actually made me feel somewhat ashamed of my sometimes aloof personal attitude towards my area of study; my tendency to think of my subject as something intractably esoteric and advanced, inaccessible to the general public (because, of course, only the intellectual elite such as myself are capable of comprehending it). At one point he makes note of the fact that many people make it most of the way through most of graduate school believing (because they've always been told) that they are good at math, only to discover, when they attempt to do some real mathematics, that all they were really good at is following directions. Ouch. I hope that isn't me...
So again, I recommend this book to all my friends and family who wonder what it is that I'm trying to do, and I may even be buying it for some of you, come Christmas time!
February 26, 2023
FUCK TOK and this guy gets it
September 14, 2018
I didn't realize when I checked this out from the library that I had already read Part 1 of the book in essay format for a math education class I took, but it is so persuasively written that I was happy to read it again. In this tiny book, Lockhart makes a compelling case for transforming the way math is taught in the United States, demonstrating that mathematics is an incredibly fun, artistic, and intellectually stimulating endeavor that is merely viewed as boring and practical because of the distorted way it is taught in schools.
Lockhart is not very practicality-oriented, which makes sense for a mathematician, and at times this made his writing less convincing. I strongly disagree with his idea that teachers don't need training on how to teach math or that they don't need to plan anything—teaching math the way he wants would be very different from how people normally think of teaching, and thus would certainly require a fair amount of retraining (see Jo Boaler's books for implementation strategies). Furthermore, scaffolding mathematical exploration to maximize learning does require a lot of thought and planning.
I also found that Lockhart has the same frustrating dismissiveness of practicality that I often see among humanities professors and students. I love his passion for mathematics as a pure intellectual pursuit, and I admire him for unabashedly supporting that, but I worry that he undermines his effectiveness by his frequent assertions that he does not care if math is in any way useful or practical. I think for parents, teachers, and policymakers to agree to changes in math education, it's important to note that even when mathematics is being studied purely for its own sake, that study has many incredibly useful byproducts.
Despite these qualms, I think this is an insightful, convincing, and (perhaps most importantly) concise essay on the wonders of mathematics and the necessary changes for how it is taught.
Lockhart is not very practicality-oriented, which makes sense for a mathematician, and at times this made his writing less convincing. I strongly disagree with his idea that teachers don't need training on how to teach math or that they don't need to plan anything—teaching math the way he wants would be very different from how people normally think of teaching, and thus would certainly require a fair amount of retraining (see Jo Boaler's books for implementation strategies). Furthermore, scaffolding mathematical exploration to maximize learning does require a lot of thought and planning.
I also found that Lockhart has the same frustrating dismissiveness of practicality that I often see among humanities professors and students. I love his passion for mathematics as a pure intellectual pursuit, and I admire him for unabashedly supporting that, but I worry that he undermines his effectiveness by his frequent assertions that he does not care if math is in any way useful or practical. I think for parents, teachers, and policymakers to agree to changes in math education, it's important to note that even when mathematics is being studied purely for its own sake, that study has many incredibly useful byproducts.
Despite these qualms, I think this is an insightful, convincing, and (perhaps most importantly) concise essay on the wonders of mathematics and the necessary changes for how it is taught.
July 28, 2013
Oh boy, oh boy, oh boy, where do I start!
Perhaps with this: YOU, YES YOU, READ THIS ASAP, I'm strongly convinced you won't regret it, especially if you're involved with maths in one way or the other!
This is by far the most inspiring book on mathematics I've ever stumbled upon and I honestly doubt that I'll stumble again on something so honest, so true, so passionate and human! I'm sure many of you mathematics lovers will experience the same feeling of joy and understanding when you hear what Lockhart has to say about how education destroys mathematics and about why is it such a beautiful form of art.
I could go on and on and on about how great I think this book is, but hey... it's only hundred and something pages long, it's better to check for yourself... go on, find it, open it and start reading, you can always close it if you don't like it...
Perhaps with this: YOU, YES YOU, READ THIS ASAP, I'm strongly convinced you won't regret it, especially if you're involved with maths in one way or the other!
This is by far the most inspiring book on mathematics I've ever stumbled upon and I honestly doubt that I'll stumble again on something so honest, so true, so passionate and human! I'm sure many of you mathematics lovers will experience the same feeling of joy and understanding when you hear what Lockhart has to say about how education destroys mathematics and about why is it such a beautiful form of art.
I could go on and on and on about how great I think this book is, but hey... it's only hundred and something pages long, it's better to check for yourself... go on, find it, open it and start reading, you can always close it if you don't like it...
September 1, 2015
It's a brilliant little rant on the state of maths teaching in schools. In summary, he is arguing that mathematics should be taught for its intellectual pleasures and not because it is useful. To that end, he urges for moving to a more problem-centric approach to maths teaching instead of the current disconnected learning of techniques and notations devoid of any context.
Loved the dialog parts of the treatise which very closely resemble the actual conversations you are likely to have if you agree with author's thesis.
Loved the dialog parts of the treatise which very closely resemble the actual conversations you are likely to have if you agree with author's thesis.
March 13, 2010
Slightly expanded from the essay online (pdf) in that it has a Part II: Exultation where Lockhart wants to "tell you more about what math really is and why I love it so much." (p.92).
Seriously, this is a great essay/book. Worth reading probably once a semester, if not more. And before structuring a class (curriculum). The faux dialog at the end of every section is awesome, and indicates good ways to respond to nay-sayers (are there any?), even if not all of the questions/concerns are fully addressed.
When in doubt, remember that math is an art. Argue for it as you might for music or painting. Of course, to a Platoist, math is discovered, instead of created, like I think of for music or painting. But there's some famous sculptor who said something about freeing a form from the marble, instead of making it, right?
Part I: Lamentation
p.23:
p.24: "If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary."
p.25: "... major theme in mathematics: things are what you want them to be." and later "This is the amazing thing about making imaginary patterns: they talk back!"
p.27: "Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn't that what art is all about?"
p.29: "Mathematics is the art of explanation."
p.32-33: "gross misconception that mathematics is somehow useful to society!" later: "Music can lead armies into battle, but that's not why people write symphonies."
p.37: "The mathematics curriculum... needs to be scrapped." later: "Mathematics is the music of reason." later still: "you gave it sense and you still don't understand what your creation is up to" within "To do mathematics is... to be alive, damn it."
p.39: "We don't need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience."
p.43 (emphasis mine):
p.44: "The trouble is that math, like painting or poetry, is hard creative work."
p.46: "Teaching is not about information. It's about having an honest intellectual relationship with your students."
p.47: "... your lesson will be planned, and therefore false." later: "Teaching means openness and honesty, an ability to share excitement, and a love of learning."
p.48: "We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas."
p.51: "... get curious about a question..."
p.52: part of one of the dialogues:
p.53: "Mathematics is not a language; it's an adventure."
p.54: "We teach to enlighten everyone, not to train only the future professionals. ... think creatively and independently."
p.55-56: "the exact same things are being said and done in the exact same way and in the exact same order" <-- next two which I wrote: "internet!" Make the process more efficient. Of course, this isn't what Lockhart wants, and, thinking about it, likely not what I want.
p.56: "Art is not a race.... seeing mathematics as an organic whole."
p.58: "Of course, it is far easier to test someone's knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning."
p.59-60: "Mathematics is about problems.... Painful and creatively frustrating as it may be..." Later on p.60: "English teachers know that spelling and pronunciation are best learned in a context of reading and writing. History teachers know that names and dates are uninteresting when removed from the unfolding backstory of events. Why does mathematics education remain stuck in the nineteenth century?"
p.62: "Teaching is a messy human relationsihp... if you need a method you're probably not a very good teacher."
p.63: "I'm sure most of them love their students and hate what they are being forced to put them through. They know in their hearts that it is meaningless and degrading. They can sense that they have been made cogs in a great soul-crushing machine, but they lack the perspective needed to understand it, or to fight against it."
p.64: done right, according to Lockhart, "there would obviously be a range of student interest and ability... but at least students would like or dislike mathematics for what it really is..."
p.66: "Doing mathematics should always mean discovering patterns and crafting beautiful and meaningful explanations."
p.72: ranting on geometry, the "two-column proof": "The effect of such a production being made over something so simple is to make people doubt their own intuition." I'd argue that making people doubt their intuition is actually a good thing. Later: "Rigorous formal proof only becomes important when there is a crisis"
p.75: "This is what comes from a misplaced sense of logical rigor: ugliness."
p.76: "Mathematics is about removing obstacles to our intuition, and keeping simple things simple."
p.78: "... private, personal experience of being a struggling artist..."
p.79: "definitions matter... And they are problem generated."
p.80: "I don't want students saying, "the definition, the theorem, the proof," I want them saying, "my definition, my theorem, my proof."" Later: "Efficiency and economy simply do not make good pedagogy."
p.81: "It's hard to completely ruin something so beautiful..." Later:
Plus, in all that time, students will have learned how to think and learn, and so picking something up later will be easier.
p.82: "a good teacher can guide the discussion and the flow of problems so as to allow the students to discover and invent mathematics for themselves.... individuals doing what they think best for their students."
p.87-88: "How ironic that people dismiss mathematics as the anitthesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract."
Part II: Exultation
p.91: "School has never been about thinking creating. School is about training children to perform so that they can be sorted."
p.92: "... mathematics is an art. Math is something you do. And what you are doing is exploring a very special and peculiar place - a place known as "Mathematical Reality." ... elegant, fanciful, wonderful, imaginary, fascinating, curious..." Later: "In this way, being a mathematician is a lot like being a field biologist." a nice analogy that he expands on.
p.94: "the difference between the thing itself and the representation of the thing.... The only thing that matters in mathematics is what things are, and more important, how they act."
p.100: "Mathematical objects... are still nothing more than figments of our imagination.... they are what we ask them to be."
p.101: "we mathematicians do not like being told what we can and cannot do."
p.103: "we play and create and try to get closer to ideal beauty."
p.104: "Being a mathematician is not so much about being clever... it's about being aesthetically sensitive and having refined and exquisite taste."
p.106: "The only thing I am interest in using mathematics for is to have a good time and to help others do the same. And for the life of me I can't imagine a more worthwhile goal. We are all born into this world, and at some point we will die and that will be that. In the meantime, let's enjoy our minds and the wonderful and ridiculous things we can do with them. I don't know about you, but I'm here to have fun."
p.108: "This is the Frankenstein aspect of mathematics - we have the authority to define our creations, to instill in them whatever features or properties we choose, but we have no say in what behaviors may then ensue as a consequence of our choices." Later: "I am drawn in by the possibility of a connection..."
p.109: "Nothing I have ever seen or done comes close to having the transformative power of math." Later: "Mostly I love the abstraction of it all, the sheer simplicity."
p.110: (on checking the first umpteen cases) "We could then say that it's true for all practical purposes, and be done with it. But that's not what mathematics is about.... Math is about reasoning and understanding." Later: "That is the goal of the mathematician: to explain in the simplest, most elegant and logically satisfying way possible."
p.111-112: "This is a unique art form within the world of rational science." Later: "imagine a Two-Headed Monster of mathematical criticism. The first head demands a logically airtight explanation... The second head wants to see simple beauty and elegance, to be charmed and delighted..."
p.113: "a lot of pain and frustration and crumpled-up paper."
p.114-115: "mathematics... is our most quintessentially human art form... We are biomechanical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are."
p.117-118: "what it's like to do mathematics. Playing with patterns, noticing things, making conjectures, searching for examples and counterexamples, being inspired to invent and explore, crafting arguments and analyzing them, and raising new questions."
p.118: "That's really what it means for something to have a pattern - if we can capture it with language."
p.119: "For a brief shining moment we lifted the veil and glimpsed a timeless simple beauty. Is this not something of value?"
p.120-121: "We're talking about a perfectly innocent and delightful activity of the human mind - a dialogue with one's own mentality. Math requires no pathetic industrial or technological excuses." Later:
p.126: "Mathematics is fundamentally an act of communication."
p.128: "problems can be classified... The point of this sort of framework...: to help us understand... helps us make predictions and to know what to look for. Classifications are a guide for our intuition."
p.131-132: "The historical development of mathematics...: first come the problems,... connections are made... structures are then devised... New questions then arise... And then the process continues."
p.133: "as modern mathematicians we are always on the lookout for structure and structure-preserving transformations."
p.135: "math problems... come from playing."
p.137: "How bizarre that something so simple should turn out to be so hard!"
p.138: "Does math ever come to an end? No". Later: "learning and playing are the same thing."
p.139: "if you are a math teacher... throw the stupid curriculum and textbooks out the window!"
p.140: final paragraph:
Seriously, this is a great essay/book. Worth reading probably once a semester, if not more. And before structuring a class (curriculum). The faux dialog at the end of every section is awesome, and indicates good ways to respond to nay-sayers (are there any?), even if not all of the questions/concerns are fully addressed.
When in doubt, remember that math is an art. Argue for it as you might for music or painting. Of course, to a Platoist, math is discovered, instead of created, like I think of for music or painting. But there's some famous sculptor who said something about freeing a form from the marble, instead of making it, right?
Part I: Lamentation
p.23:
... there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind-blowing as cosmology or physics..., and allows more freedom of expression than poetry, art, or music.... Mathematics is the purest of the arts, as well as the most misunderstood.
p.24: "If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary."
p.25: "... major theme in mathematics: things are what you want them to be." and later "This is the amazing thing about making imaginary patterns: they talk back!"
p.27: "Somehow, I was able to create a profound simple beauty out of nothing, and change myself in the process. Isn't that what art is all about?"
p.29: "Mathematics is the art of explanation."
p.32-33: "gross misconception that mathematics is somehow useful to society!" later: "Music can lead armies into battle, but that's not why people write symphonies."
p.37: "The mathematics curriculum... needs to be scrapped." later: "Mathematics is the music of reason." later still: "you gave it sense and you still don't understand what your creation is up to" within "To do mathematics is... to be alive, damn it."
p.39: "We don't need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience."
p.43 (emphasis mine):
So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and levels of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.
p.44: "The trouble is that math, like painting or poetry, is hard creative work."
p.46: "Teaching is not about information. It's about having an honest intellectual relationship with your students."
p.47: "... your lesson will be planned, and therefore false." later: "Teaching means openness and honesty, an ability to share excitement, and a love of learning."
p.48: "We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas."
p.51: "... get curious about a question..."
p.52: part of one of the dialogues:
Simplicio:Yes, but before you can write your own poems you need to learn the alphabet. The process has to begin somewhere. You have to walk before you can run.
Salviati: No, you have to have something you want to run toward. Children can write poems and stories as they learn to read and write. A piece of writing by a six-year-old is a wonderful thing, and the spelling and punctuation errors don't make it less so. Even very young children can invent songs, and they haven't a clue what key it is in or what type of meter they are using.
p.53: "Mathematics is not a language; it's an adventure."
p.54: "We teach to enlighten everyone, not to train only the future professionals. ... think creatively and independently."
p.55-56: "the exact same things are being said and done in the exact same way and in the exact same order" <-- next two which I wrote: "internet!" Make the process more efficient. Of course, this isn't what Lockhart wants, and, thinking about it, likely not what I want.
p.56: "Art is not a race.... seeing mathematics as an organic whole."
p.58: "Of course, it is far easier to test someone's knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning."
p.59-60: "Mathematics is about problems.... Painful and creatively frustrating as it may be..." Later on p.60: "English teachers know that spelling and pronunciation are best learned in a context of reading and writing. History teachers know that names and dates are uninteresting when removed from the unfolding backstory of events. Why does mathematics education remain stuck in the nineteenth century?"
p.62: "Teaching is a messy human relationsihp... if you need a method you're probably not a very good teacher."
p.63: "I'm sure most of them love their students and hate what they are being forced to put them through. They know in their hearts that it is meaningless and degrading. They can sense that they have been made cogs in a great soul-crushing machine, but they lack the perspective needed to understand it, or to fight against it."
p.64: done right, according to Lockhart, "there would obviously be a range of student interest and ability... but at least students would like or dislike mathematics for what it really is..."
p.66: "Doing mathematics should always mean discovering patterns and crafting beautiful and meaningful explanations."
p.72: ranting on geometry, the "two-column proof": "The effect of such a production being made over something so simple is to make people doubt their own intuition." I'd argue that making people doubt their intuition is actually a good thing. Later: "Rigorous formal proof only becomes important when there is a crisis"
p.75: "This is what comes from a misplaced sense of logical rigor: ugliness."
p.76: "Mathematics is about removing obstacles to our intuition, and keeping simple things simple."
p.78: "... private, personal experience of being a struggling artist..."
p.79: "definitions matter... And they are problem generated."
p.80: "I don't want students saying, "the definition, the theorem, the proof," I want them saying, "my definition, my theorem, my proof."" Later: "Efficiency and economy simply do not make good pedagogy."
p.81: "It's hard to completely ruin something so beautiful..." Later:
Simplicio: So we're supposed to just set off on some free-form mathematical excursion, and the students will learn whatever they happen to learn?
Salviati: Precisely. Problems will lead to other problems, technique will be developed as it becomes necessary, and new topics will arise naturally. And if some issue never happens to come up in thirteen years of schooling, how interesting or important could it be?
Plus, in all that time, students will have learned how to think and learn, and so picking something up later will be easier.
p.82: "a good teacher can guide the discussion and the flow of problems so as to allow the students to discover and invent mathematics for themselves.... individuals doing what they think best for their students."
p.87-88: "How ironic that people dismiss mathematics as the anitthesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract."
Part II: Exultation
p.91: "School has never been about thinking creating. School is about training children to perform so that they can be sorted."
p.92: "... mathematics is an art. Math is something you do. And what you are doing is exploring a very special and peculiar place - a place known as "Mathematical Reality." ... elegant, fanciful, wonderful, imaginary, fascinating, curious..." Later: "In this way, being a mathematician is a lot like being a field biologist." a nice analogy that he expands on.
p.94: "the difference between the thing itself and the representation of the thing.... The only thing that matters in mathematics is what things are, and more important, how they act."
p.100: "Mathematical objects... are still nothing more than figments of our imagination.... they are what we ask them to be."
p.101: "we mathematicians do not like being told what we can and cannot do."
p.103: "we play and create and try to get closer to ideal beauty."
p.104: "Being a mathematician is not so much about being clever... it's about being aesthetically sensitive and having refined and exquisite taste."
p.106: "The only thing I am interest in using mathematics for is to have a good time and to help others do the same. And for the life of me I can't imagine a more worthwhile goal. We are all born into this world, and at some point we will die and that will be that. In the meantime, let's enjoy our minds and the wonderful and ridiculous things we can do with them. I don't know about you, but I'm here to have fun."
p.108: "This is the Frankenstein aspect of mathematics - we have the authority to define our creations, to instill in them whatever features or properties we choose, but we have no say in what behaviors may then ensue as a consequence of our choices." Later: "I am drawn in by the possibility of a connection..."
p.109: "Nothing I have ever seen or done comes close to having the transformative power of math." Later: "Mostly I love the abstraction of it all, the sheer simplicity."
p.110: (on checking the first umpteen cases) "We could then say that it's true for all practical purposes, and be done with it. But that's not what mathematics is about.... Math is about reasoning and understanding." Later: "That is the goal of the mathematician: to explain in the simplest, most elegant and logically satisfying way possible."
p.111-112: "This is a unique art form within the world of rational science." Later: "imagine a Two-Headed Monster of mathematical criticism. The first head demands a logically airtight explanation... The second head wants to see simple beauty and elegance, to be charmed and delighted..."
p.113: "a lot of pain and frustration and crumpled-up paper."
p.114-115: "mathematics... is our most quintessentially human art form... We are biomechanical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are."
p.117-118: "what it's like to do mathematics. Playing with patterns, noticing things, making conjectures, searching for examples and counterexamples, being inspired to invent and explore, crafting arguments and analyzing them, and raising new questions."
p.118: "That's really what it means for something to have a pattern - if we can capture it with language."
p.119: "For a brief shining moment we lifted the veil and glimpsed a timeless simple beauty. Is this not something of value?"
p.120-121: "We're talking about a perfectly innocent and delightful activity of the human mind - a dialogue with one's own mentality. Math requires no pathetic industrial or technological excuses." Later:
To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits.
p.126: "Mathematics is fundamentally an act of communication."
p.128: "problems can be classified... The point of this sort of framework...: to help us understand... helps us make predictions and to know what to look for. Classifications are a guide for our intuition."
p.131-132: "The historical development of mathematics...: first come the problems,... connections are made... structures are then devised... New questions then arise... And then the process continues."
p.133: "as modern mathematicians we are always on the lookout for structure and structure-preserving transformations."
p.135: "math problems... come from playing."
p.137: "How bizarre that something so simple should turn out to be so hard!"
p.138: "Does math ever come to an end? No". Later: "learning and playing are the same thing."
p.139: "if you are a math teacher... throw the stupid curriculum and textbooks out the window!"
p.140: final paragraph:
And if you are neither a students nor teacher, but simply a person living in this world and searching as we all are for love and meaning, I hope I have managed to give you a glimpse of something beautiful and pure, a harmless and joyful activity that has brought untold delight to many people for thousands of years."
December 8, 2021
I picked this book up after I saw a friend add it to their goodreads list. What a pleasant surprise! Anyone who has any relationship to math should read this short opinion work.
I've included the following for my own personal remembering of what I enjoyed about this book.
...
Foreword: "In my view, this book, like the original essay it came from, should be obligatory reading for anyone going into mathematics education, for every parent of a school-aged child, and for any school or government official with responsibilities toward mathematics teaching. You may not agree with everything Paul says. You may think his approach to teaching is not one that every teacher could successfully adopt. But you should read what he says and reflect on his words. "A Mathematician's Lament" is already a recognized landmark in the world of mathematics education that cannot and should not be ignored. I am not going to tell you how I think you should respond. As Paul himself would agree, that is for every individual reader to do. But I will tell you this. I would have loved to have had Paul Lockhart as my school mathematics teacher. Keith Devlin, Stanford University"
p 49 "Simplico: Then what should we do with young children in math class? Salviati: Play games! Teach them chess and Go, Hex and backgammon, Sprouts and nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don't worry about notation and technique; help them to become active and creative mathematical thinkers."
p 50-51 "Salviati: All right, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them and you remember what matters to you. We have millions of adults wandering around with "negative b plus or minus the square root of b squared minus 4ac all over 2a" in their heads, and absolutely no idea whatsoever what it means. And the reason is that they were never given the chance to discover or invent such things for themselves. They never had an engaging problem to think about, to be frustrated by, and to create in them the desire for technique or method. They were ever told the history of mankind's relationship with numbers - no ancient Babylonian problem tablets, no Rhind Papyrus, no Liber Abaci, no Ars Magna. More important, no chance for them to even get curious about a question; it was answered before they could ask it."
p 51-52 "Let's be clear about this. I'm complaining about the complete absence of art and invention, history, and philosophy, context and perspective from the mathematics curriculum. That doesn't mean that notation, technique, and the development of a knowledge base have no place. Of course they do. We should have both. If I object to a pendulum being too far to one side, it doesn't mean I want it to be all the way on the other side. But the fact is, people learn better when the product comes out of the process. A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own."
p 114 "To me, this kind of mathematical experience goes to the heart of what it means to be human. And I'll go even further and say that mathematics, this art of abstract pattern-making - even more than storytelling, painting, or music - is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are."
p 123 "Again, the right thing for me to do as your math teacher would be nothing. That's a thing most teachers (and adults generally) seem to have a hard time doing. Were you my student (and assuming this problem interested you) I would simply say, "Have fun. Keep me posted." And your relationship to the problem would develop in whatever way it would."
p 138 "YOU: Does this process every stop? Does math ever come to an end? ME: No, because solving problems always leads to new problems..."
I've included the following for my own personal remembering of what I enjoyed about this book.
...
Foreword: "In my view, this book, like the original essay it came from, should be obligatory reading for anyone going into mathematics education, for every parent of a school-aged child, and for any school or government official with responsibilities toward mathematics teaching. You may not agree with everything Paul says. You may think his approach to teaching is not one that every teacher could successfully adopt. But you should read what he says and reflect on his words. "A Mathematician's Lament" is already a recognized landmark in the world of mathematics education that cannot and should not be ignored. I am not going to tell you how I think you should respond. As Paul himself would agree, that is for every individual reader to do. But I will tell you this. I would have loved to have had Paul Lockhart as my school mathematics teacher. Keith Devlin, Stanford University"
p 49 "Simplico: Then what should we do with young children in math class? Salviati: Play games! Teach them chess and Go, Hex and backgammon, Sprouts and nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don't worry about notation and technique; help them to become active and creative mathematical thinkers."
p 50-51 "Salviati: All right, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them and you remember what matters to you. We have millions of adults wandering around with "negative b plus or minus the square root of b squared minus 4ac all over 2a" in their heads, and absolutely no idea whatsoever what it means. And the reason is that they were never given the chance to discover or invent such things for themselves. They never had an engaging problem to think about, to be frustrated by, and to create in them the desire for technique or method. They were ever told the history of mankind's relationship with numbers - no ancient Babylonian problem tablets, no Rhind Papyrus, no Liber Abaci, no Ars Magna. More important, no chance for them to even get curious about a question; it was answered before they could ask it."
p 51-52 "Let's be clear about this. I'm complaining about the complete absence of art and invention, history, and philosophy, context and perspective from the mathematics curriculum. That doesn't mean that notation, technique, and the development of a knowledge base have no place. Of course they do. We should have both. If I object to a pendulum being too far to one side, it doesn't mean I want it to be all the way on the other side. But the fact is, people learn better when the product comes out of the process. A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own."
p 114 "To me, this kind of mathematical experience goes to the heart of what it means to be human. And I'll go even further and say that mathematics, this art of abstract pattern-making - even more than storytelling, painting, or music - is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are."
p 123 "Again, the right thing for me to do as your math teacher would be nothing. That's a thing most teachers (and adults generally) seem to have a hard time doing. Were you my student (and assuming this problem interested you) I would simply say, "Have fun. Keep me posted." And your relationship to the problem would develop in whatever way it would."
p 138 "YOU: Does this process every stop? Does math ever come to an end? ME: No, because solving problems always leads to new problems..."
August 29, 2023
A blazing critique of humanity's societal and cultural misunderstanding of mathematics that I believe should be mandatory reading for every student who enters high school. With a free-flowing and provocative tone that matches his educational philosophy, Lockhart shreds apart the K-12 education curriculum and the gut-wrenching notion that mathematics is the antonym of creative thinking, expression, and imaginative play. Mathematics is an art, rich with experimentation and exploration, and deserves to be taught as one, for it possesses more self-expression, abstract thinking, and possibilities than a painter could wish out from their canvas.
While he presents more appropriate teaching philosophies and lesson structures from his own classes (on a third-grade and university level), ideally, Lockhart should have proposed a solution as cohesive as his problem analysis. But...this is a lament. And for a mournful piece riddled with outbursts, he does a good job of keeping us hopeful. A free PDF can be found online easily. Please, just read the opening analogies, and it will lift mathematics (or, at least, your idea of mathematics) out of the class memories' stress and grime and lay into a comfortable velvet chair: one that you are happy to see it enjoy.
"Real mathematics doesn't come in a can- there is no such thing as an Algebra 1 idea. Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of the subject, and a teacher's own path through the standard curriculum reinforces this myth and prevents him or her from seeing mathematics as an organic whole. As a result, we have a math curriculum with no historical perspective or thematic coherence, a fragmented collection of assorted topics and techniques, united only by the ease in which they can be reduced to step-by-step procedures."
While he presents more appropriate teaching philosophies and lesson structures from his own classes (on a third-grade and university level), ideally, Lockhart should have proposed a solution as cohesive as his problem analysis. But...this is a lament. And for a mournful piece riddled with outbursts, he does a good job of keeping us hopeful. A free PDF can be found online easily. Please, just read the opening analogies, and it will lift mathematics (or, at least, your idea of mathematics) out of the class memories' stress and grime and lay into a comfortable velvet chair: one that you are happy to see it enjoy.
"Real mathematics doesn't come in a can- there is no such thing as an Algebra 1 idea. Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of the subject, and a teacher's own path through the standard curriculum reinforces this myth and prevents him or her from seeing mathematics as an organic whole. As a result, we have a math curriculum with no historical perspective or thematic coherence, a fragmented collection of assorted topics and techniques, united only by the ease in which they can be reduced to step-by-step procedures."
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May 24, 2025La lectura m’ha entretingut i renovat les ganes de jugar amb les matemàtiques. Tant els problemes que planteja, com recordant-ne d’altres que en el passat he anat gaudint.
Pel que fa a la crítica que fa dels ensenyaments de mates, en comparteixo una bona part. D’entrada, però, crec que me n’enduc una bona part d’autocrítica perquè certament fa temps que penso que em sé molt bé les regles tal i les aplico com me les han ensenyat (i em sembla fàcil atendre-hi) més que no pas jugo creativament amb els problemes. Sobretot, coincideixo en la inutilitat de fer memoritzar problemes amb notacions que es carreguen la intuïció, i que s’acaben convertint en llistats de passos a memoritzar. Una frustració meva durant la carrera era precisament haver d’anar amb pressa i aprendre a sobreviure més amb aquests mecanismes que no realment poder dedicar estones i treure conclusions dels problemes gaudint-los. Aquí escombro cap a casa, una de les meves frustracions amb les mates i la física, és pròpiament que estudiar-les també deixi fora tota reflexió sobre perquè són el llenguatge que descriu la natura. També sobre el seu significat, si descriuen la realitat o només en són models, i si hi ha diferències entre diferents branques, que es relacionen de manera molt diferent amb les equacions.
Res, m’agrada haver-lo llegit acabant el curs per reflexionar des de quin lloc m’he relacionat amb el contingut que hem ensenyat i per proposar-me fer-m’hi més de cara el curs vinent.
Pel que fa a la crítica que fa dels ensenyaments de mates, en comparteixo una bona part. D’entrada, però, crec que me n’enduc una bona part d’autocrítica perquè certament fa temps que penso que em sé molt bé les regles tal i les aplico com me les han ensenyat (i em sembla fàcil atendre-hi) més que no pas jugo creativament amb els problemes. Sobretot, coincideixo en la inutilitat de fer memoritzar problemes amb notacions que es carreguen la intuïció, i que s’acaben convertint en llistats de passos a memoritzar. Una frustració meva durant la carrera era precisament haver d’anar amb pressa i aprendre a sobreviure més amb aquests mecanismes que no realment poder dedicar estones i treure conclusions dels problemes gaudint-los. Aquí escombro cap a casa, una de les meves frustracions amb les mates i la física, és pròpiament que estudiar-les també deixi fora tota reflexió sobre perquè són el llenguatge que descriu la natura. També sobre el seu significat, si descriuen la realitat o només en són models, i si hi ha diferències entre diferents branques, que es relacionen de manera molt diferent amb les equacions.
Res, m’agrada haver-lo llegit acabant el curs per reflexionar des de quin lloc m’he relacionat amb el contingut que hem ensenyat i per proposar-me fer-m’hi més de cara el curs vinent.
October 9, 2018
if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they're right...
[Excerpt free here]
August 6, 2024
Powered through this short book in one sitting last night and: wow. If the only thing Paul Lockhart did was convince me math is beautiful, I'm glad I read this. But also he argues math, essentially, is art. (!?) Why have I never heard this before? It would have revolutionized my math education! This is very new for me. I feel like I'm at the bottom of a mountain trying to describe the peak, but: I'm actually interested in the mountain now... and that is pretty significant. Excited to learn more.
June 21, 2025
Soms beetje zweverig, maar wel zalig geschreven. Een ode aan wiskunde en creativiteit. Elke persoon die wiskunde leerkracht wil zijn zou dit moeten lezen al kan het voor andere mensen ook geen kwaad imo
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September 1, 2023Todosdeberianleeresto
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