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A Mathematician’s Lament (2002) [pdf]

170 points by gshrikant 12y ago

krazydad 12y ago
Love the bit about the misconception that Mathematics is mainly about utility. I remember reading something about G.H. Hardy (which I can no longer find) in which he said he would get a little bit disappointed if he found that one of his results ended up finding a practical use.
- gshrikant 12y ago
  He actually argues quite the opposite in [1] (see footnote, pg 33) - a good read in itself. The quote you're probably referring to is this: "a science is said to be _useful_ if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life" (emphasis mine). And Hardy's response (excerpt): > It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications. > I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. [1] http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%2...
- tujv 12y ago
  He said that the most beautiful mathematics were that which had no practical applications. Not sure how he'd feel that now that his own specialty of Number Theory has turned out to be very practical.
- Retra 12y ago
  That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
- Retra 12y ago
  That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
- Retra 12y ago
  That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
- Retra 12y ago
  That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
- Retra 12y ago
  That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
- Retra 12y ago
  That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
  - saraid216 12y ago
    I'd be more concerned that such a paper would be a lie. When did utility become more important than truth?
    - Retra 12y ago
      Truth is important because it has utility. It enables consistency and reproducibility in a way that nothing else does.
      - saraid216 12y ago
        Really? What consistency and reproducibility does "An individual is a worthwhile human being" have?
4ad 12y ago
A very similar piece by V.I. Arnold[1]: http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html [1]: http://en.wikipedia.org/wiki/Vladimir_Arnold http://en.wikipedia.org/wiki/Vladimir_Arnold
sarciszewski 12y ago
My favorite part is "The Standard School Mathematics Curriculum".
prestonbriggs 12y ago
Lockhart has a book, "Measurement", that elaborates in the same vein. Quite nice.
- mushishi 12y ago
  I personally found it inspiring and beautifully written.
ColinDabritz 12y ago
Such a wonderful essay. It's very applicable to the teaching of computer science/software engineering as well. So much of the problem is the misunderstanding people have about the field. It's a creative, constructive discipline, and so much of the instruction is consumption, mimicry, and repetition. Solving well defined problems is relatively easy. Our real problem is that real problems are not well defined.
- theoh 12y ago
  I think I agree but isn't your last line a bit like saying "hammering in a nail is easy. The problem is that these screws aren't nails." Chuck Close: "I think while appropriation has produced some interesting work … for me, the most interesting thing is to back yourself into your own corner where no one else’s answers will fit. You will somehow have to come up with your own personal solutions to this problem that you have set for yourself because no one else’s answers are applicable." ... "See, I think our whole society is much too problem-solving oriented. It is far more interesting to [participate in] ‘problem creation’ … You know, ask yourself an interesting enough question and your attempt to find a tailor-made solution to that question will push you to a place where, pretty soon, you’ll find yourself all by your lonesome — which I think is a more interesting place to be." http://www.brainpickings.org/2012/12/27/chuck-close-on-creativity/ http://www.brainpickings.org/2012/12/27/chuck-close-on-creat...
mrcactu5 12y ago
This is how people think of Math class: "Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."
- Thrymr 12y ago
  And this is a pretty good description of how it's taught: "Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school"
- nextos 12y ago
  That's very sad and very true. For some reason, I stumbled upon Hubbard & Hubbard Vector Calculus a few days after first reading this essay and it stroke me as the opposite to this wrong math teaching.
MaxScheiber 12y ago
https://news.ycombinator.com/item?id=6187014 https://news.ycombinator.com/item?id=6187014 With that being said, I've always loved this essay. As of recently, I've viewed it as relevant to the recent argument that programming should be a requirement in American public schools, either as a tool in math and science classes or a free-standing course. This kind of mathematical reform might actually be a prerequisite for programming and computer science, given that it would develop mathematical maturity much more effectively than the current system does.
- jnbiche 12y ago
  Yes, but Lockhart is arguing against any practical applications of math in the early years. I'm almost certain he would oppose any attempt to connect math to programming, for the reasons outlined in his essay (roughly, "math should be about imagination and playing, not applications"). In fact, I disagree with this essay, for reasons outlined in [1]. 1. https://news.ycombinator.com/item?id=8847132 https://news.ycombinator.com/item?id=8847132
  - birdsareweird 12y ago
    Who do you think mathematical programming has to be practical?
    - saraid216 12y ago
      You're taking away the wrong thing from the grandparent post. If you teach math as a prerequisite for something else, you're naturally going to be undervaluing it. Math should be taught for the sake of learning math, not because it's the gateway to something else. It is, yes, but that's not the point.
      - birdsareweird 12y ago
        You know the problem with that sentiment? Most people do not find abstract patterns and rote symbolic manipulation to be interesting. They simply don't, and if you teach math class mistakenly assuming they do, 90% tunes out. If you can show them how those abstract patterns reflect things in nature, and how those symbolic manipulations represent ideas and algorithms and systems, and can explain complicated things, that's something else. e.g. You don't need computer graphics to teach linear algebra, but how many people who 'know' matrix algebra know that the columns of a matrix are the basis vectors for the principal axes, and that matrix multiplication is the (affine) transform tool from photoshop? I would also suggest that if you want to get kids interested in trigonometry and you fail to mention that every single thing in a 3D video game is made out of triangles, you are a bad teacher and you should feel bad.
ashark 12y ago
HN is big on curated lists lately. Is there one for resources to aid in teaching mathematics the way Lockhart would prefer? I've seen this posted so many places so many times that surely there's a market for materials and support for it. Where are they?
- nextos 12y ago
  For middle school I'm sure Lockhart would love Gelfand's books: http://gcpm.rutgers.edu/books.html http://gcpm.rutgers.edu/books.html At undergrad level, I love Axler but I haven't found a similar calculus book.
  - jinfiesto 12y ago
    Spivak? It's an excellent calculus book. Apostol or Courant are good as well.
- murbard2 12y ago
  "How to solve it" is a good book in this direction, by Polya.
- murbard2 12y ago
  "How to solve it", by Polya.
- murbard2 12y ago
  "How to solve it", by Polya.
- murbard2 12y ago
  Polya's "How to solve it" is pretty good in that respect. I asked this question on math.stackexchange.com a while ago but haven't received many answers yet http://math.stackexchange.com/questions/1040232/can-you-recommend-a-book-to-learn-to-teach-math-to-a-child http://math.stackexchange.com/questions/1040232/can-you-reco...
- GregBuchholz 12y ago
  With a quick search, here's the closest thing I could find: http://mathoverflow.net/questions/5074/are-there-elementary-school-curricula-that-capture-the-joy-of-mathematics http://mathoverflow.net/questions/5074/are-there-elementary-... ...but I too would like to see what you are looking for, at an elementary school age level.
drcomputer 12y ago
I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information. That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me. With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing buggy programs is easy.
drcomputer 12y ago
I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information. That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me. With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.
drcomputer 12y ago
I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information. That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me. With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.
WhitneyLand 12y ago
I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff. However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory. Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.
WhitneyLand 12y ago
I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff. However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory. Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.
deleted 12y ago
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samwilkinson 12y ago
As a Physicist, I feel obliged to mention that black holes were first hypothesised by Physicists (contrary to the essay), albeit through Mathematical enquiry.
- tokai 12y ago
  I would assume that he means that mathematicians worked with singularities, before we knew, or hypothesised, that singularities arises in the real world.
- ninguem2 12y ago
  Pierre Simon de Laplace was the first person to hypothesize black holes. I think he was a mathematician, although such distinctions were not so clear in the eighteenth century.
WhitneyLand 12y ago
I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff. However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory. Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation. Geometry, if I recall correctly, was one of the exceptions in early math where you are allowed to veer off the path a bit. Is everything else algorithmic until college?
jnbiche 12y ago
I actually disagree pretty strongly with this essay. I think Lockhart is advocating for an educational approach that would cater strongly (and exclusively) to a particular learning style, specifically, those learners who thrive when going from abstract to specific (deductive learning). For those of us who do best with inductive learning, the type of education he proposes would bring even greater misery to our grade school education. It was not until I took statistics and probability (for science and engineer majors) in college that I truly began to enjoy math once again. I was able to start with concrete ideas and applications, and then work my way back to the theory behind them. I'm now reading "Concrete Mathematics" and really enjoying it. Knuth's ideas on math education are pretty diametrically opposed to those of Lockhart, as far as I can tell, and they give rise to something very close to my ideal learning environment for math. Why not allow children to follow which ever of the two math paths that is best suited to them, instead of forcing concrete thinkers into an abstract world, and abstract thinkers into a concrete world? Edit: I should probably add that I agree with Lockhart that there's a problem in the way math is taught, but I disagree with him on the solution.
- jacobolus 12y ago
  Endless drilling, memorization, and focus on pushing numbers through “formulas” is not the same as “inductive learning”. It’s entirely possible to do lots of pattern matching and bottom-up “inductive” problem solving in a way consistent with Lockhart’s recommendations. Here’s a great book chapter wherein a mathematician teaches some 6-year-olds in what might be called an “inductive” way: http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf (so much better than a standard first grade mathematics curriculum) Edit: to clarify, I basically think you’re reading something into Lockhart’s essay that isn’t there.
  - deleted 12y ago
    [deleted]
  - jnbiche 12y ago
    Agreed, which is why I included the note (preceding your comment) adding that I agree with Lockhart that there's a problem in the way math is taught. Edit: Can you expand on your idea? I did just re-read the essay (for the 3rd time) and Lockhart seems pretty absolute in his belief that math should be taught as a playful, creative abstraction. And he seems to disdain practical applications of math. So how would you teach math in an inductive manner that maintains the level of abstraction and "playfulness" he's advocating for? By the way, love the book chapter! That's exactly how I'm trying to teach my kids. Not always easy, but they respond well to it. And it's entirely possible I'm reading something into the essay that isn't there. But I'm not yet convinced.
aaronem 12y ago
Vital context: The Underground History of American Education, free online: http://mhkeehn.tripod.com/ughoae.pdf http://mhkeehn.tripod.com/ughoae.pdf
- GregBuchholz 12y ago
  Wow. It certainly has an excellent hook in the very first sentence: "Our problem in understanding forced schooling stems from an inconvenient fact: that the wrong it does from a human perspective is right from a systems perspective."
  - aaronem 12y ago
    It gets better from there. Keep reading.
jackmaney 12y ago
A beautifully written and tragic essay. (Note: What follows is US-centric.) After nine years of teaching mathematics courses (one semester as an undergraduate, 4.5 years as a graduate student, and 4 years as an assistant professor) and navigating university politics, I'm convinced that this is, at its heart, a cultural issue. There's a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).
- chrisdone 12y ago
  America, Britain, France, Italy, etc. education systems built upon "getting results" seem to end up ruining the subject they're trying to teach.
  - sp332 12y ago
    How do you get results then? Ignore the results?
    - wyager 12y ago
      Encourage freedom of thought instead of preparing students for some standardized test. It's crazy how much the AP/IB and SAT tests dictate the direction and content of high school curricula. It gets much better at the college level, because teachers don't have to prepare students for anything in particular.
    - j2kun 12y ago
      Invest in the infrastructure.
    - setr 12y ago
      By developing the argument. The correct answer isn't nearly as important as forming a strong argument, at least in the process of learning, as a well written argument will naturally lead you towards the correct answer. Its the reason why "showing your work" is taken so strongly. The flaw being that you're not showing your work so much as you're showing your ability to follow a defined series of actions. The grade school test doesn't ask you to think of a procedure to resolve some stated question. It asks if you already have it memorized, as the formula was handed to you in a prior lecture. Ideally, the test should ask the student to form an argument, thus showing mastery over the knowledge that precedes its creation. This would be judged, and of the students ability to consistently dot the i's and cross his t's, expected but given a minor role (as it is in "real life", mistakes are inevitable and its why we check our work twice, and then give it to others to check again.)
zodiac 12y ago
FWIW one of our professors here at uwaterloo taught a first year abstract algebra / number theory class in a very Lockhart-esque way (Math 145; he even quoted Lockhart on one of the assignments). I learned a lot of math and enjoyed myself, but the main problem I observed was figuring out how to fairly grade students, and the fact that the homework took a lot more time than a class taught normally.
blahblahblah3 12y ago
I think for most people, math is best learned in the context of some application that they care about (the last four words are very important). Few people appreciate the beauty of the abstract game itself. For example, most people who play poker online quickly learn about expected value, probability, and variance.
- eruditely 12y ago
  Precisely. Poker is one of the most advanced games that legitimately teaches you what you need to know! The language used by poker players should be adapted wholesale.
- saraid216 12y ago
  For such people, is there a good reason to learn math?
mathattack 12y ago
I have a lot of sympathy for his point of view. I loved Math growing up. High school drove the interest out of me, and I didn't get it back until senior Calculus, when I started doing well again. Then I learned to appreciate CS theory, economic theory, etc. Trying to figure out how to break the cycle for my kids: Stats for practical work, and math for curiosity.
eximius 12y ago
Always a good read.
drvortex 12y ago
If the musician woke up in the first sentence, how is he still dreaming in the next one?

170 points by gshrikant 12y ago

Love the bit about the misconception that Mathematics is mainly about utility. I remember reading something about G.H. Hardy (which I can no longer find) in which he said he would get a little bit disappointed if he found that one of his results ended up finding a practical use.

gshrikant 12y ago
He actually argues quite the opposite in [1] (see footnote, pg 33) - a good read in itself. The quote you're probably referring to is this: "a science is said to be _useful_ if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life" (emphasis mine). And Hardy's response (excerpt): > It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications. > I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. [1] http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%2...
tujv 12y ago
He said that the most beautiful mathematics were that which had no practical applications. Not sure how he'd feel that now that his own specialty of Number Theory has turned out to be very practical.
Retra 12y ago
That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
Retra 12y ago
That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
Retra 12y ago
That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
Retra 12y ago
That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
Retra 12y ago
That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
Retra 12y ago
That's not a misconception. It would be great if all mathematics had obvious utility, but that's demanding perfection. It doesn't mean mathematics would be worth doing if it had no utility. In fact, we often try to do mathematics in a way that guarantees some amount of utility; that's why we use proof-based methods in leu of wishful thinking. Should I write a mathematics paper concluding "true = false" and argue that mathematics is not about utility? No. Such a thing is utterly useless, and it would be ridiculous to propose that I'm doing mathematics without taking the necessary care to ensure my work is useful.
- saraid216 12y ago
  I'd be more concerned that such a paper would be a lie. When did utility become more important than truth?
  - Retra 12y ago
    Truth is important because it has utility. It enables consistency and reproducibility in a way that nothing else does.
    - saraid216 12y ago
      Really? What consistency and reproducibility does "An individual is a worthwhile human being" have?

4ad 12y ago

A very similar piece by V.I. Arnold[1]: http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html [1]: http://en.wikipedia.org/wiki/Vladimir_Arnold http://en.wikipedia.org/wiki/Vladimir_Arnold

sarciszewski 12y ago

My favorite part is "The Standard School Mathematics Curriculum".

prestonbriggs 12y ago

Lockhart has a book, "Measurement", that elaborates in the same vein. Quite nice.

mushishi 12y ago
I personally found it inspiring and beautifully written.

ColinDabritz 12y ago

Such a wonderful essay. It's very applicable to the teaching of computer science/software engineering as well. So much of the problem is the misunderstanding people have about the field. It's a creative, constructive discipline, and so much of the instruction is consumption, mimicry, and repetition. Solving well defined problems is relatively easy. Our real problem is that real problems are not well defined.

theoh 12y ago
I think I agree but isn't your last line a bit like saying "hammering in a nail is easy. The problem is that these screws aren't nails." Chuck Close: "I think while appropriation has produced some interesting work … for me, the most interesting thing is to back yourself into your own corner where no one else’s answers will fit. You will somehow have to come up with your own personal solutions to this problem that you have set for yourself because no one else’s answers are applicable." ... "See, I think our whole society is much too problem-solving oriented. It is far more interesting to [participate in] ‘problem creation’ … You know, ask yourself an interesting enough question and your attempt to find a tailor-made solution to that question will push you to a place where, pretty soon, you’ll find yourself all by your lonesome — which I think is a more interesting place to be." http://www.brainpickings.org/2012/12/27/chuck-close-on-creativity/ http://www.brainpickings.org/2012/12/27/chuck-close-on-creat...

mrcactu5 12y ago

This is how people think of Math class: "Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."

Thrymr 12y ago
And this is a pretty good description of how it's taught: "Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school"
nextos 12y ago
That's very sad and very true. For some reason, I stumbled upon Hubbard & Hubbard Vector Calculus a few days after first reading this essay and it stroke me as the opposite to this wrong math teaching.

MaxScheiber 12y ago

https://news.ycombinator.com/item?id=6187014 https://news.ycombinator.com/item?id=6187014 With that being said, I've always loved this essay. As of recently, I've viewed it as relevant to the recent argument that programming should be a requirement in American public schools, either as a tool in math and science classes or a free-standing course. This kind of mathematical reform might actually be a prerequisite for programming and computer science, given that it would develop mathematical maturity much more effectively than the current system does.

jnbiche 12y ago
Yes, but Lockhart is arguing against any practical applications of math in the early years. I'm almost certain he would oppose any attempt to connect math to programming, for the reasons outlined in his essay (roughly, "math should be about imagination and playing, not applications"). In fact, I disagree with this essay, for reasons outlined in [1]. 1. https://news.ycombinator.com/item?id=8847132 https://news.ycombinator.com/item?id=8847132
- birdsareweird 12y ago
  Who do you think mathematical programming has to be practical?
  - saraid216 12y ago
    You're taking away the wrong thing from the grandparent post. If you teach math as a prerequisite for something else, you're naturally going to be undervaluing it. Math should be taught for the sake of learning math, not because it's the gateway to something else. It is, yes, but that's not the point.
    - birdsareweird 12y ago
      You know the problem with that sentiment? Most people do not find abstract patterns and rote symbolic manipulation to be interesting. They simply don't, and if you teach math class mistakenly assuming they do, 90% tunes out. If you can show them how those abstract patterns reflect things in nature, and how those symbolic manipulations represent ideas and algorithms and systems, and can explain complicated things, that's something else. e.g. You don't need computer graphics to teach linear algebra, but how many people who 'know' matrix algebra know that the columns of a matrix are the basis vectors for the principal axes, and that matrix multiplication is the (affine) transform tool from photoshop? I would also suggest that if you want to get kids interested in trigonometry and you fail to mention that every single thing in a 3D video game is made out of triangles, you are a bad teacher and you should feel bad.

ashark 12y ago

HN is big on curated lists lately. Is there one for resources to aid in teaching mathematics the way Lockhart would prefer? I've seen this posted so many places so many times that surely there's a market for materials and support for it. Where are they?

nextos 12y ago
For middle school I'm sure Lockhart would love Gelfand's books: http://gcpm.rutgers.edu/books.html http://gcpm.rutgers.edu/books.html At undergrad level, I love Axler but I haven't found a similar calculus book.
- jinfiesto 12y ago
  Spivak? It's an excellent calculus book. Apostol or Courant are good as well.
murbard2 12y ago
"How to solve it" is a good book in this direction, by Polya.
murbard2 12y ago
"How to solve it", by Polya.
murbard2 12y ago
"How to solve it", by Polya.
murbard2 12y ago
Polya's "How to solve it" is pretty good in that respect. I asked this question on math.stackexchange.com a while ago but haven't received many answers yet http://math.stackexchange.com/questions/1040232/can-you-recommend-a-book-to-learn-to-teach-math-to-a-child http://math.stackexchange.com/questions/1040232/can-you-reco...
GregBuchholz 12y ago
With a quick search, here's the closest thing I could find: http://mathoverflow.net/questions/5074/are-there-elementary-school-curricula-that-capture-the-joy-of-mathematics http://mathoverflow.net/questions/5074/are-there-elementary-... ...but I too would like to see what you are looking for, at an elementary school age level.

drcomputer 12y ago

I loved mathematics throughout school. I don't know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information. That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I've learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I've studied makes it sound like classical music does to me. With math, I don't really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don't know if that's irrational reasoning, but I know that understanding math correctly is hard, and writing buggy programs is easy.

drcomputer 12y ago

WhitneyLand 12y ago

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff. However I don't see why they couldn't start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory. Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.

WhitneyLand 12y ago

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samwilkinson 12y ago

As a Physicist, I feel obliged to mention that black holes were first hypothesised by Physicists (contrary to the essay), albeit through Mathematical enquiry.

tokai 12y ago
I would assume that he means that mathematicians worked with singularities, before we knew, or hypothesised, that singularities arises in the real world.
ninguem2 12y ago
Pierre Simon de Laplace was the first person to hypothesize black holes. I think he was a mathematician, although such distinctions were not so clear in the eighteenth century.

WhitneyLand 12y ago

jnbiche 12y ago

I actually disagree pretty strongly with this essay. I think Lockhart is advocating for an educational approach that would cater strongly (and exclusively) to a particular learning style, specifically, those learners who thrive when going from abstract to specific (deductive learning). For those of us who do best with inductive learning, the type of education he proposes would bring even greater misery to our grade school education. It was not until I took statistics and probability (for science and engineer majors) in college that I truly began to enjoy math once again. I was able to start with concrete ideas and applications, and then work my way back to the theory behind them. I'm now reading "Concrete Mathematics" and really enjoying it. Knuth's ideas on math education are pretty diametrically opposed to those of Lockhart, as far as I can tell, and they give rise to something very close to my ideal learning environment for math. Why not allow children to follow which ever of the two math paths that is best suited to them, instead of forcing concrete thinkers into an abstract world, and abstract thinkers into a concrete world? Edit: I should probably add that I agree with Lockhart that there's a problem in the way math is taught, but I disagree with him on the solution.

jacobolus 12y ago
Endless drilling, memorization, and focus on pushing numbers through “formulas” is not the same as “inductive learning”. It’s entirely possible to do lots of pattern matching and bottom-up “inductive” problem solving in a way consistent with Lockhart’s recommendations. Here’s a great book chapter wherein a mathematician teaches some 6-year-olds in what might be called an “inductive” way: http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf (so much better than a standard first grade mathematics curriculum) Edit: to clarify, I basically think you’re reading something into Lockhart’s essay that isn’t there.
- deleted 12y ago
  [deleted]
- jnbiche 12y ago
  Agreed, which is why I included the note (preceding your comment) adding that I agree with Lockhart that there's a problem in the way math is taught. Edit: Can you expand on your idea? I did just re-read the essay (for the 3rd time) and Lockhart seems pretty absolute in his belief that math should be taught as a playful, creative abstraction. And he seems to disdain practical applications of math. So how would you teach math in an inductive manner that maintains the level of abstraction and "playfulness" he's advocating for? By the way, love the book chapter! That's exactly how I'm trying to teach my kids. Not always easy, but they respond well to it. And it's entirely possible I'm reading something into the essay that isn't there. But I'm not yet convinced.

aaronem 12y ago

Vital context: The Underground History of American Education, free online: http://mhkeehn.tripod.com/ughoae.pdf http://mhkeehn.tripod.com/ughoae.pdf

GregBuchholz 12y ago
Wow. It certainly has an excellent hook in the very first sentence: "Our problem in understanding forced schooling stems from an inconvenient fact: that the wrong it does from a human perspective is right from a systems perspective."
- aaronem 12y ago
  It gets better from there. Keep reading.

jackmaney 12y ago

A beautifully written and tragic essay. (Note: What follows is US-centric.) After nine years of teaching mathematics courses (one semester as an undergraduate, 4.5 years as a graduate student, and 4 years as an assistant professor) and navigating university politics, I'm convinced that this is, at its heart, a cultural issue. There's a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).

chrisdone 12y ago
America, Britain, France, Italy, etc. education systems built upon "getting results" seem to end up ruining the subject they're trying to teach.
- sp332 12y ago
  How do you get results then? Ignore the results?
  - wyager 12y ago
    Encourage freedom of thought instead of preparing students for some standardized test. It's crazy how much the AP/IB and SAT tests dictate the direction and content of high school curricula. It gets much better at the college level, because teachers don't have to prepare students for anything in particular.
  - j2kun 12y ago
    Invest in the infrastructure.
  - setr 12y ago
    By developing the argument. The correct answer isn't nearly as important as forming a strong argument, at least in the process of learning, as a well written argument will naturally lead you towards the correct answer. Its the reason why "showing your work" is taken so strongly. The flaw being that you're not showing your work so much as you're showing your ability to follow a defined series of actions. The grade school test doesn't ask you to think of a procedure to resolve some stated question. It asks if you already have it memorized, as the formula was handed to you in a prior lecture. Ideally, the test should ask the student to form an argument, thus showing mastery over the knowledge that precedes its creation. This would be judged, and of the students ability to consistently dot the i's and cross his t's, expected but given a minor role (as it is in "real life", mistakes are inevitable and its why we check our work twice, and then give it to others to check again.)

zodiac 12y ago

FWIW one of our professors here at uwaterloo taught a first year abstract algebra / number theory class in a very Lockhart-esque way (Math 145; he even quoted Lockhart on one of the assignments). I learned a lot of math and enjoyed myself, but the main problem I observed was figuring out how to fairly grade students, and the fact that the homework took a lot more time than a class taught normally.

blahblahblah3 12y ago

I think for most people, math is best learned in the context of some application that they care about (the last four words are very important). Few people appreciate the beauty of the abstract game itself. For example, most people who play poker online quickly learn about expected value, probability, and variance.

eruditely 12y ago
Precisely. Poker is one of the most advanced games that legitimately teaches you what you need to know! The language used by poker players should be adapted wholesale.
saraid216 12y ago
For such people, is there a good reason to learn math?

mathattack 12y ago

I have a lot of sympathy for his point of view. I loved Math growing up. High school drove the interest out of me, and I didn't get it back until senior Calculus, when I started doing well again. Then I learned to appreciate CS theory, economic theory, etc. Trying to figure out how to break the cycle for my kids: Stats for practical work, and math for curiosity.

eximius 12y ago

Always a good read.

drvortex 12y ago

If the musician woke up in the first sentence, how is he still dreaming in the next one?