15 ms·

A Mathematician’s Lament (2002) [pdf]

152 points by kjhughes 13y ago

softbuilder 13y ago
"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.” So painfully spot-on. My mathematical education was horrible. Meanwhile I had been writing code since I was a little kid. It wasn't until I was an adult that I realized how much math I had been learning while programming. And worse, that I had been completely miseducated about what math actually is.
- gxs 13y ago
  Interestingly, the converse is also true. I was always into computers and tech but never really programming. I was a math major in college, however, and after graduating I started programming. The transition was almost seamless, I picked up programming really quickly, it was surprising to me how much the "ways of thinking" are alike.
  - saraid216 13y ago
    One of the key mistakes of educational strategy is dividing up knowledge into a bunch of subjects like "Language", "Math", and so on. You can find linkages in just about everything; this siloing really only serves to make people think they're particularly good at one thing and not another, when really it's a preference for one perspective over another.
    - nsphere 13y ago
      I completely agree! By other's perspectives my hardest semester was when I took PDEs, Geometry Space-time (Minkowski), Algorithms, P-chem (Quantum Mechanics), and Analysis. However there was so much over lap with all the materials and topics. A few weeks after midterms all the course work just fit together like a 100 piece puzzle.
      - banachtarski 13y ago
        Taking as many physics courses as I have, I was getting worried for a while that there was nothing more to life than solving a boundary value problem in some domain or another.
    - marcusf 13y ago
      So the pragmatists question is then naturally, how do you teach it? How do you shard the curriculum if not by subject? It seems a bit overbearing to ask each teacher to have the proper renaissance man's education of knowing a bit of everything.
      - calibraxis 13y ago
        Classes are already discrete, and can thus be taught by different teachers. As I understand, classifying classes into "disciplines" is for the convenience of administrative systems, like university deans. This classification is not a law of the universe, nor of the human mind. One can imagine the negatives of fitting learning into hierarchical administrative models.
      - saraid216 13y ago
        To me, a teacher's proper role isn't having knowledge and dumping it into someone's brain. It's knowing where to find that knowledge. My three Rs are "reading, research, and reflection": a teacher's job is to (1) provide useful material to consume, via lecture or homework or whatnot, (2) point towards larger resources for further exploration, and (3) guide thought processes to make useful conclusions. A teacher's job is not to teach. It's to provide a space in which a student can learn. A focus is useful for this, but the focus doesn't need to be an abstract subject. It's a MacGuffin; it can be anything.
  - dmlorenzetti 13y ago
    Can't vouch for this book, but coincidentally just read about it today-- "The Essential Knuth" by Knuth, Daylight, and DeGrave. Donald E. Knuth lived two separate lives in the late 1950s. During daylight he ran down the visible and respectable lane of mathematics. During nighttime, he trod the unpaved road of computer programming and compiler writing. Both roads intersected! -- as Knuth discovered while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961. "Chomsky theories fascinated me, because they were mathematical yet they could also be understood with my programmer's intuition. It was very curious because otherwise, as a mathematician, I was doing integrals or maybe was learning about Fermat's number theory, but I wasn't manipulating symbols the way I did when I was writing a compiler. With Chomsky, wow, I was actually doing mathematics and computer science simultaneously."
    - hcarvalhoalves 13y ago
      > (...) while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961 Which makes you wonder, what kind of person reads this stuff on the honeymoon.
      - stephencanon 13y ago
        Someone who chose the right person to marry.
- arkj 13y ago
  In India this too never happens. The article is a 1000 times valid here.
adultSwim 13y ago
I thought I had been taking math my whole life until I finally took a math class.
- jackmaney 13y ago
  Back when I was in academia, one of my favorite student evaluations from an abstract algebra class had the following comment (paraphrased from memory): "This course was like climbing Mt. Everest. It's difficult, and sometimes slow going, but at the end, it's breathtaking how much you've learned and accomplished."
- jfarmer 13y ago
  Hah, yes! I hated math in high school but wound up graduating with a BS in mathematics from the University of Chicago. It wasn't until I took my first math course at Chicago using Michael Spivak's Calculus that I thought, "Wait, if this is math, what was I studying all through high school?"
bridgeyman 13y ago
I read this a couple weeks ago and decided to try and find a math book that incorporates some history in it. I found Journey through Genius: The Great Theorems of Mathematics with some great reviews. I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading the book with someone, email me at bridger@understudyapp.com. I could really use some beta testers for the app! If you have already read it, you could still mentor someone in it to review the material again.
- nsphere 13y ago
  Check out this book as well. http://amzn.com/039306204X http://amzn.com/039306204X
  - bridgeyman 13y ago
    Thanks! It is next on my list now.
- aet 13y ago
  Sounds like an interesting idea. Learning is much easier if you can find someone equally interested and at your same level.
- Nate75Sanders 13y ago
  Journey through Genius is an absolutely fantastic book. I took History of Math from a professor in undergrad who had a conjecture named after him and this was his choice of textbook (along with a few other materials) for the course.
- Someone 13y ago
  Another suggestion: http://archive.org/details/TheWorldOfMathematicsVolume1 http://archive.org/details/TheWorldOfMathematicsVolume1 (and volumes 2, 3, and 4). Lots of interesting stories, many of applied mathematics. And, to my surprise, available for free from archive.org.
- 13y ago
nsphere 13y ago
"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." As a Mathematician I've always identified as a creative person, yet struggle to convince "artsy" persons that what I do is creative. As soon as I utter the word math they convulse and shutter as if they were afflicted with a PTSD flashback. I try explaining what the idea involves and draw a diagram or three, but all they do is just nod and say "yeah, yup, okay.." Are the ideas of math that inaccessible to the general population when compared to a work of Art?
- jackmaney 13y ago
  "Are the ideas of math that inaccessible to the general population when compared to a work of Art?" Yes, if only because many (if not most) don't want access. Been there, done that, got the medications to prove it.
  - math_junkie_20 13y ago
    "got the medications to prove it." Please, elaborate?
- sampo 13y ago
  "Are the ideas of math that inaccessible to the general population when compared to a work of Art?" I'd guess at least some fine arts (composing classical or modernist orchestral music, modern abstract painting and sculpting) would also be pretty inaccessible to general population.
  - codyb 13y ago
    Yes, but when a classical composer converses with you and says something along the lines of "There actually is, with all the creativity, a sort of formula I follow to produce my best pieces. You see I do this first, revise it three times, etc, etc." and no one will sit and shutter and have the PTSD style flashback the original comment talked about because they do not know and they do not have preconceived, strongly held, notions in mind about the subject. Or at least I think that'd be true.
mathattack 13y ago
I hated math until I got to calculus. I never knew why, but I might as well quite this mathematician... They took an exciting topic that interested me as a youth and killed it with rote repetition. The question I struggle with is... How do you really get by without math as a mandatory topic? How can you teach science? How can you teach someone to balance a checkbook? The current system is awful, but do we push the subject to electives as a result?
- saraid216 13y ago
  This is one of my current problems in curriculum design. I agree fully with Lockhart, and I strongly feel that requiring math is a mistake. But at the same time, you do need significant math skills in order to appreciate science with any meaningful depth, and I feel that science is a requirement. The proper next step, it seems to me, is to figure out what bits of science we really want to teach and then figure out how to explain math through that. > How can you teach someone to balance a checkbook? Outside of America, no one uses checkbooks anymore. There's no reason to use exact change for tipping, either: just estimate a percentage and then round up. Or splitting the bill: the servers virtually always have a calculator at their disposal.
- ef4 13y ago
  > The question I struggle with is... How do you really get by without math as a mandatory topic? That's the wrong question, because "mandatory learning" is a contradiction in terms. It very clearly doesn't work. Beyond the most primitive pavlovian conditioning, you can't force people to learn if they don't want to. Intrinsic motivation is the dominating factor in how well a person can master an intellectually demanding task. The right question is, how do you inspire people to want to learn?
  - mathattack 13y ago
    True enough. And that's a tough question too. I waited from 4th grade to calculus to get inspired again. And then I think it was the material more than the teaching method.
BgSpnnrs 13y ago
"Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind." That's almost Douglas Adams levels of dry wit. This is a fascinating article, as someone who's never really contemplated the playfulness of maths....I mean, for sure the wonder of maths or the power of maths - I've just never really put the pieces together to link that back to my grounding in maths, beyond the practical, functional stuff I incorporate into my daily life, without considering it's mathiness. Very interesting.
RogerL 13y ago
The article resonated with me on some level, because it does take a long time to learn how to actually do math. If you are at the point of just doing algebraic manipulations on equations to try to figure something out, you've lost the battle (as opposed to using algebraic manipulations to encode your thoughts, and work out the details). On the other hand, I think everybody really did see the beauty in geometry. Yes, the initial manipulation to prove something about symmetric angles is a bit silly, but you aren't being taught symmetric angles here, you are being taught how to do a geometric proof. Which is not easy to learn, so you start with something super simple and obvious. I didn't observe anyone in my classes (well this was back in 1982, so memory is a challenge here) confused about that point. And, as soon as we learned it, the requirement for formalism was dropped. It was the same in algebra. In the first weeks you weren't allowed to go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 = 1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using written out. Annoying yes, once you grasp it, but once you proved you grasped it that was the end of that, and we never had to do it again. But, I had good teachers that always tried to explain the 'why' of what we were doing, and did not make us engage in pointless formalisms. But you need to understand terminology like ABC in geometry; when you get to tough problems you'll be using it. So, learn it with the easy problems.
- gizmo686 13y ago
  My best math teachers never made us engage formalisations. What they would do is manipulate us into having an arguement, or construct an apparent contradiction. We then discussed the problem until we all agree (often without the Teacher talking). The inevitable result of this is that everyone learned formalisation: because it makes for really convinving arguements.
dnautics 13y ago
As an erstwhile math major (I couldn't hack the honors basic algebra class - the difference between a euclidean domain and a principal ideal domain got too confusing; but I rocked proofs in analysis) I have to say that the author is confusing Mathematics (which is an art) and Arithmetic (which is a skill). Part of what make the opening farce absurd is that musical skills and painting are not terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death; whereas understanding sums and compounding processes ARE. While SALVIATI is completely correct in his analysis of the situation, he offers no solution, and I identify with SIMPLICIO more. Perhaps the problem is that in our schools we conflate arithmetic with mathematics. Surely, they are related, but perhaps they need to be delineated and the difference understood.
- saraid216 13y ago
  > whereas understanding sums and compounding processes ARE [terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death]. That's not actually true.
  - dnautics 13y ago
    in an ideal world, I would agree with you. We do however live in the real world.
    - saraid216 13y ago
      Please give one example applicable to a reasonable majority of human beings on the planet at this current time where "understanding sums and compounding processes" are a matter of life and death.
      - jameshart 13y ago
        It seems probable that the subprime mortgage crisis was a contributing factor in a number of deaths, via suicide, stress-induced illness, or, with the help of alcohol, violent or vehicular incidents.
ctdonath 13y ago
In contrast, my 5- and 3-year-olds are enjoying DragonBox on the iPad, not knowing they're learning Algebra.
rtfeldman 13y ago
This is intensely thought-provoking and beautifully well-written. It's not directly about hacking, but it's the type of treasure that hackers love to stumble upon. It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.
- gizmo686 13y ago
  >It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else. At risk of getting to meta, this piece seems to pop up everywhere. I don't mean that in a bad way, but this is at least the fifth time over the course of many years that I have seen this piece pop up on completely unrelated sites.
  - ColinWright 13y ago
    Not to mention it's been here on HN at least 8 times, probably 10 or more. I've stopped bothering to direct people to previous conversations - they just say the same things over and over again anyway.
stiff 13y ago
Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. "Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes http://en.wikipedia.org/wiki/Zeno%27s_paradoxes The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it: http://archive.org/stream/elementsofdiffer00woolrich#page/n17/mode/2up http://archive.org/stream/elementsofdiffer00woolrich#page/n1... This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox. If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first! One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.
calebm 13y ago
I had to overcome my school-age brainwashing in order to enjoy math. One book that helped me was "Who is Fourier? A Mathematical Adventure" (http://www.amazon.com/ho-Fourier-Mathematical-Adventure-Edition/dp/0964350432 http://www.amazon.com/ho-Fourier-Mathematical-Adventure-Edit...). I highly recommend it for those looking to find enjoyment in math.
hcarvalhoalves 13y ago
Reflects my experience. During school I had interest and good grades in all classes, including physics and geometry (those are separate from math in Brazil), the only exception being math (which covers algebra, polynomials, etc.). How a kid can get an A in physics and a D in math is beyond my comprehension, but I did. I only started appreciating math much later in life.
gizmo686 13y ago
During high school, at the end of the year everyone in my Calculus class had to make a math presentation (to go along with a paper). During lunch one day we stood with our posters to explain our project the the students that choice/were encouraged to attend. I explained my topic (re-ordering conditionally convergent series) to a group of students. They seemed to understand what I was saying, showed awe at some of the key insights/tricks, and even interupted me to excitadly finish the arguement. Then, when I was done, they pointed to the equations and asked me to explain them. My response was that those equations are exactly what I had just said, and I repeated the arguement while showing them where it is in the equations.
ArbitraryLimits 13y ago
Richard P. Feynman “Physics is like sex: sure, it may give some practical results, but that's not why we do it.”
lmartel 13y ago
I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?
- abecedarius 13y ago
  http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674057554 http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755... (by guess who) looks like the best starting point since this resonated with you. (I've dipped into it but not seriously tackled it yet.) It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.) More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.
- westoncb 13y ago
  Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not. I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.
whiddershins 13y ago
This is how I feel about doing most coding tutorials versus the MIT intro to CS through python, or project euler. When coding is presented as: - here is a problem - how would you solve this problem? - here are some hints to get you started it is incredibly fun for me. when it is presented as: - follow along - look what you did! it can be a bit dry.
- abecedarius 13y ago
  Hi whiddershins, here's a little present for you then: https://github.com/darius/regexercise https://github.com/darius/regexercise (It's not quite finished and I'd love to get feedback.)
ctl 13y ago
I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum? I'm really asking -- my friend is about to start as a high-school math teacher. I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Lockhart sense -- real puzzles, not exercises.) But how often are "techniques" actually taught in high school math, especially algebra and precalculus? A lot of high school math consists of digesting new definitions, or the generalization of old definitions. A fair amount of it consists of learning theorems that go unproven, or that are proven (by the teacher) too quickly for students to understand where they come from -- and in general it isn't satisfying to solve a puzzle with a theorem that one doesn't actually understand. On top of that... students have to spend time with problems before they become genuinely interested in their solutions, so progress would be slower with this method. It's not clear that you could teach a whole year's curriculum in one year like this. (And if you fail to do that you'll eventually get fired.) Any insight? I believe that it's possible to teach math, even standard high school curriculum, in such a way that students are at all times intrinsically interested in what's presented. But it would be awfully hard to do at scale, at the standard pace, as a high school teacher would have to. How might a teacher start in that direction?
- jfarmer 13y ago
  I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult, even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy, have to teach to a common set of state-wide standards, and have students of varying levels of interest. Here are my scattered thoughts, though. I'm going to try to not suggest a pie-in-the-sky solution like "new curriculum!" First, I majored in mathematics at the University of Chicago, but I hate, hate, hated mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication? And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work. Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it. So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on. If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them. Second, I think the way math is taught is overly symbolic. What most non-mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed. I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa. Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations. Hey class! Look at this expression: 45+6. What does it equal? A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer." A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like... Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong per se. We could have chosen to live in either world, but we have to choose one consistent set of rules. These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and . So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it. It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't. I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)
detcader 13y ago
We really need to teach people _how_ to teach induction, which is only done right when you put quotes around your Boolean statements; the "implies" symbol gets jumbled up with everything else otherwise, and not using it at all is passing up on a great tool. One can do simple proofs-by-induction without a single English word, completely symbolically, and have it be understood easily, if one uses quotes and correct LaTeX formatting (or good handwriting) Induction doesn't just involve numbers and equality signs, it involves _statements_ with variables inside of them, and non-programmers need to be made well-aware of this (and taught Boolean logic early, PLEASE)
- hcarvalhoalves 13y ago
  There was a time people were taught logic and geometry and math and history all at the same time. Also, philosophy. The reductionist approach of our education is it's major failure.
jobenjo 13y ago
Love this essay. I read it years ago when my brother was working with Paul Lockhart, who deeply influenced him as a math teacher. My brother and his wife have since started an organization called Math For Love (http://www.mathforlove.com http://www.mathforlove.com) focused on changing the way math is taught. They run workshops for teachers and provide great material for students. If you're in Seattle and interesting in pedagogy and math, you should check them out.
Myrmornis 13y ago
"The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.”" +1 Also s/math/science/ +1
dwaltrip 13y ago
This article just blew my mind. I feel little silly, especially since I majored in my math. I have never seen these types of ideas expressed so clearly and powerfully. He is 100% right. We should not be teaching the formulaic rules of basic arithmetic until high school, and only then as a part of life skills class. It should be taught as the art form that it is.
dsego 13y ago
Still haven't finished reading this and all the painful memories came back from middle school. Not math (although it too wasn't a fun experience) but art. For four years we had a teacher that thought music and visual arts by dictating. For 45 minutes we would write down everything word for word. Then after a few weeks she would ask us broad questions and we'd have to recite everything back to her. I still remember (I'm almost 30 now) rote learning from my notebook, walking around the house, repeating those stupid sentences in my head, memorising descriptions of works of art I've never seen, painting techniques I've never witnessed, music I've never heard. Wow, just talking about it makes me angry, makes me want to to choke the bitch's neck. Sadly, that's the state of schools in Croatia, people equate schools with rote learning and memorising. You have to do it to get into a good university and get a good job. Who will create those jobs? No wonder the country is in the shitter.
- dsego 13y ago
  oops, meant taught instead of thought.
dominotw 13y ago
The author of this article teaches a course on coursera that I can't recommend enough, he put so much effort into that course. https://www.coursera.org/course/maththink https://www.coursera.org/course/maththink As you might have guessed its not really about 'math' in traditional sense.
dominotw 13y ago
Attempts to present mathematics as relevant to daily life inevitably appear forced and contrived: “You see kids, if you know algebra then you can figure out how old Maria is if we know thatshe istwo years older than twice her age seven years ago!” (Asif anyone would ever have access to that ridiculous kind of information, and not her age.) I wince every time I see these examples in my niece's textbook. So ridiculous.
kazagistar 13y ago
For music, the formal curriculum does involve memorizing songs and scales and whatnot. The problem is that the best way to start a subject is to fuck around with it (playing random songs you like on your guitar, drawing cool shapes and noticing patterns in them, etc), but our schools are ill equipped for such things, and enabling this style is difficult and incredibly expensive.
kazagistar 13y ago
A classic issue that mathematicians, philosophers, and even computer "scientists" have is the idea that they can somehow reason their way to the truth. Oh, he makes a good argument. But the greeks made great arguments about how the sun goes around the earth. I could say that this or that argument is flawed. But really, the only valid arguments are data. His data is severely lacking, and many modern methods of teaching are much more supported by experimentation. Logic is a tool for finding logically consistent imaginary realities. Science is a tool for finding out about this reality.

152 points by kjhughes 13y ago

"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.” So painfully spot-on. My mathematical education was horrible. Meanwhile I had been writing code since I was a little kid. It wasn't until I was an adult that I realized how much math I had been learning while programming. And worse, that I had been completely miseducated about what math actually is.

gxs 13y ago
Interestingly, the converse is also true. I was always into computers and tech but never really programming. I was a math major in college, however, and after graduating I started programming. The transition was almost seamless, I picked up programming really quickly, it was surprising to me how much the "ways of thinking" are alike.
- saraid216 13y ago
  One of the key mistakes of educational strategy is dividing up knowledge into a bunch of subjects like "Language", "Math", and so on. You can find linkages in just about everything; this siloing really only serves to make people think they're particularly good at one thing and not another, when really it's a preference for one perspective over another.
  - nsphere 13y ago
    I completely agree! By other's perspectives my hardest semester was when I took PDEs, Geometry Space-time (Minkowski), Algorithms, P-chem (Quantum Mechanics), and Analysis. However there was so much over lap with all the materials and topics. A few weeks after midterms all the course work just fit together like a 100 piece puzzle.
    - banachtarski 13y ago
      Taking as many physics courses as I have, I was getting worried for a while that there was nothing more to life than solving a boundary value problem in some domain or another.
  - marcusf 13y ago
    So the pragmatists question is then naturally, how do you teach it? How do you shard the curriculum if not by subject? It seems a bit overbearing to ask each teacher to have the proper renaissance man's education of knowing a bit of everything.
    - calibraxis 13y ago
      Classes are already discrete, and can thus be taught by different teachers. As I understand, classifying classes into "disciplines" is for the convenience of administrative systems, like university deans. This classification is not a law of the universe, nor of the human mind. One can imagine the negatives of fitting learning into hierarchical administrative models.
    - saraid216 13y ago
      To me, a teacher's proper role isn't having knowledge and dumping it into someone's brain. It's knowing where to find that knowledge. My three Rs are "reading, research, and reflection": a teacher's job is to (1) provide useful material to consume, via lecture or homework or whatnot, (2) point towards larger resources for further exploration, and (3) guide thought processes to make useful conclusions. A teacher's job is not to teach. It's to provide a space in which a student can learn. A focus is useful for this, but the focus doesn't need to be an abstract subject. It's a MacGuffin; it can be anything.
- dmlorenzetti 13y ago
  Can't vouch for this book, but coincidentally just read about it today-- "The Essential Knuth" by Knuth, Daylight, and DeGrave. Donald E. Knuth lived two separate lives in the late 1950s. During daylight he ran down the visible and respectable lane of mathematics. During nighttime, he trod the unpaved road of computer programming and compiler writing. Both roads intersected! -- as Knuth discovered while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961. "Chomsky theories fascinated me, because they were mathematical yet they could also be understood with my programmer's intuition. It was very curious because otherwise, as a mathematician, I was doing integrals or maybe was learning about Fermat's number theory, but I wasn't manipulating symbols the way I did when I was writing a compiler. With Chomsky, wow, I was actually doing mathematics and computer science simultaneously."
  - hcarvalhoalves 13y ago
    > (...) while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961 Which makes you wonder, what kind of person reads this stuff on the honeymoon.
    - stephencanon 13y ago
      Someone who chose the right person to marry.
arkj 13y ago
In India this too never happens. The article is a 1000 times valid here.

adultSwim 13y ago

I thought I had been taking math my whole life until I finally took a math class.

jackmaney 13y ago
Back when I was in academia, one of my favorite student evaluations from an abstract algebra class had the following comment (paraphrased from memory): "This course was like climbing Mt. Everest. It's difficult, and sometimes slow going, but at the end, it's breathtaking how much you've learned and accomplished."
jfarmer 13y ago
Hah, yes! I hated math in high school but wound up graduating with a BS in mathematics from the University of Chicago. It wasn't until I took my first math course at Chicago using Michael Spivak's Calculus that I thought, "Wait, if this is math, what was I studying all through high school?"

bridgeyman 13y ago

I read this a couple weeks ago and decided to try and find a math book that incorporates some history in it. I found Journey through Genius: The Great Theorems of Mathematics with some great reviews. I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading the book with someone, email me at bridger@understudyapp.com. I could really use some beta testers for the app! If you have already read it, you could still mentor someone in it to review the material again.

nsphere 13y ago
Check out this book as well. http://amzn.com/039306204X http://amzn.com/039306204X
- bridgeyman 13y ago
  Thanks! It is next on my list now.
aet 13y ago
Sounds like an interesting idea. Learning is much easier if you can find someone equally interested and at your same level.
Nate75Sanders 13y ago
Journey through Genius is an absolutely fantastic book. I took History of Math from a professor in undergrad who had a conjecture named after him and this was his choice of textbook (along with a few other materials) for the course.
Someone 13y ago
Another suggestion: http://archive.org/details/TheWorldOfMathematicsVolume1 http://archive.org/details/TheWorldOfMathematicsVolume1 (and volumes 2, 3, and 4). Lots of interesting stories, many of applied mathematics. And, to my surprise, available for free from archive.org.
13y ago

nsphere 13y ago

"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." As a Mathematician I've always identified as a creative person, yet struggle to convince "artsy" persons that what I do is creative. As soon as I utter the word math they convulse and shutter as if they were afflicted with a PTSD flashback. I try explaining what the idea involves and draw a diagram or three, but all they do is just nod and say "yeah, yup, okay.." Are the ideas of math that inaccessible to the general population when compared to a work of Art?

jackmaney 13y ago
"Are the ideas of math that inaccessible to the general population when compared to a work of Art?" Yes, if only because many (if not most) don't want access. Been there, done that, got the medications to prove it.
- math_junkie_20 13y ago
  "got the medications to prove it." Please, elaborate?
sampo 13y ago
"Are the ideas of math that inaccessible to the general population when compared to a work of Art?" I'd guess at least some fine arts (composing classical or modernist orchestral music, modern abstract painting and sculpting) would also be pretty inaccessible to general population.
- codyb 13y ago
  Yes, but when a classical composer converses with you and says something along the lines of "There actually is, with all the creativity, a sort of formula I follow to produce my best pieces. You see I do this first, revise it three times, etc, etc." and no one will sit and shutter and have the PTSD style flashback the original comment talked about because they do not know and they do not have preconceived, strongly held, notions in mind about the subject. Or at least I think that'd be true.

mathattack 13y ago

I hated math until I got to calculus. I never knew why, but I might as well quite this mathematician... They took an exciting topic that interested me as a youth and killed it with rote repetition. The question I struggle with is... How do you really get by without math as a mandatory topic? How can you teach science? How can you teach someone to balance a checkbook? The current system is awful, but do we push the subject to electives as a result?

saraid216 13y ago
This is one of my current problems in curriculum design. I agree fully with Lockhart, and I strongly feel that requiring math is a mistake. But at the same time, you do need significant math skills in order to appreciate science with any meaningful depth, and I feel that science is a requirement. The proper next step, it seems to me, is to figure out what bits of science we really want to teach and then figure out how to explain math through that. > How can you teach someone to balance a checkbook? Outside of America, no one uses checkbooks anymore. There's no reason to use exact change for tipping, either: just estimate a percentage and then round up. Or splitting the bill: the servers virtually always have a calculator at their disposal.
ef4 13y ago
> The question I struggle with is... How do you really get by without math as a mandatory topic? That's the wrong question, because "mandatory learning" is a contradiction in terms. It very clearly doesn't work. Beyond the most primitive pavlovian conditioning, you can't force people to learn if they don't want to. Intrinsic motivation is the dominating factor in how well a person can master an intellectually demanding task. The right question is, how do you inspire people to want to learn?
- mathattack 13y ago
  True enough. And that's a tough question too. I waited from 4th grade to calculus to get inspired again. And then I think it was the material more than the teaching method.

BgSpnnrs 13y ago

"Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind." That's almost Douglas Adams levels of dry wit. This is a fascinating article, as someone who's never really contemplated the playfulness of maths....I mean, for sure the wonder of maths or the power of maths - I've just never really put the pieces together to link that back to my grounding in maths, beyond the practical, functional stuff I incorporate into my daily life, without considering it's mathiness. Very interesting.

RogerL 13y ago

The article resonated with me on some level, because it does take a long time to learn how to actually do math. If you are at the point of just doing algebraic manipulations on equations to try to figure something out, you've lost the battle (as opposed to using algebraic manipulations to encode your thoughts, and work out the details). On the other hand, I think everybody really did see the beauty in geometry. Yes, the initial manipulation to prove something about symmetric angles is a bit silly, but you aren't being taught symmetric angles here, you are being taught how to do a geometric proof. Which is not easy to learn, so you start with something super simple and obvious. I didn't observe anyone in my classes (well this was back in 1982, so memory is a challenge here) confused about that point. And, as soon as we learned it, the requirement for formalism was dropped. It was the same in algebra. In the first weeks you weren't allowed to go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 = 1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using written out. Annoying yes, once you grasp it, but once you proved you grasped it that was the end of that, and we never had to do it again. But, I had good teachers that always tried to explain the 'why' of what we were doing, and did not make us engage in pointless formalisms. But you need to understand terminology like ABC in geometry; when you get to tough problems you'll be using it. So, learn it with the easy problems.

gizmo686 13y ago
My best math teachers never made us engage formalisations. What they would do is manipulate us into having an arguement, or construct an apparent contradiction. We then discussed the problem until we all agree (often without the Teacher talking). The inevitable result of this is that everyone learned formalisation: because it makes for really convinving arguements.

dnautics 13y ago

As an erstwhile math major (I couldn't hack the honors basic algebra class - the difference between a euclidean domain and a principal ideal domain got too confusing; but I rocked proofs in analysis) I have to say that the author is confusing Mathematics (which is an art) and Arithmetic (which is a skill). Part of what make the opening farce absurd is that musical skills and painting are not terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death; whereas understanding sums and compounding processes ARE. While SALVIATI is completely correct in his analysis of the situation, he offers no solution, and I identify with SIMPLICIO more. Perhaps the problem is that in our schools we conflate arithmetic with mathematics. Surely, they are related, but perhaps they need to be delineated and the difference understood.

saraid216 13y ago
> whereas understanding sums and compounding processes ARE [terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death]. That's not actually true.
- dnautics 13y ago
  in an ideal world, I would agree with you. We do however live in the real world.
  - saraid216 13y ago
    Please give one example applicable to a reasonable majority of human beings on the planet at this current time where "understanding sums and compounding processes" are a matter of life and death.
    - jameshart 13y ago
      It seems probable that the subprime mortgage crisis was a contributing factor in a number of deaths, via suicide, stress-induced illness, or, with the help of alcohol, violent or vehicular incidents.

ctdonath 13y ago

In contrast, my 5- and 3-year-olds are enjoying DragonBox on the iPad, not knowing they're learning Algebra.

rtfeldman 13y ago

This is intensely thought-provoking and beautifully well-written. It's not directly about hacking, but it's the type of treasure that hackers love to stumble upon. It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.

gizmo686 13y ago
>It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else. At risk of getting to meta, this piece seems to pop up everywhere. I don't mean that in a bad way, but this is at least the fifth time over the course of many years that I have seen this piece pop up on completely unrelated sites.
- ColinWright 13y ago
  Not to mention it's been here on HN at least 8 times, probably 10 or more. I've stopped bothering to direct people to previous conversations - they just say the same things over and over again anyway.

stiff 13y ago

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. "Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes http://en.wikipedia.org/wiki/Zeno%27s_paradoxes The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it: http://archive.org/stream/elementsofdiffer00woolrich#page/n17/mode/2up http://archive.org/stream/elementsofdiffer00woolrich#page/n1... This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox. If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first! One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.

calebm 13y ago

I had to overcome my school-age brainwashing in order to enjoy math. One book that helped me was "Who is Fourier? A Mathematical Adventure" (http://www.amazon.com/ho-Fourier-Mathematical-Adventure-Edition/dp/0964350432 http://www.amazon.com/ho-Fourier-Mathematical-Adventure-Edit...). I highly recommend it for those looking to find enjoyment in math.

hcarvalhoalves 13y ago

Reflects my experience. During school I had interest and good grades in all classes, including physics and geometry (those are separate from math in Brazil), the only exception being math (which covers algebra, polynomials, etc.). How a kid can get an A in physics and a D in math is beyond my comprehension, but I did. I only started appreciating math much later in life.

gizmo686 13y ago

During high school, at the end of the year everyone in my Calculus class had to make a math presentation (to go along with a paper). During lunch one day we stood with our posters to explain our project the the students that choice/were encouraged to attend. I explained my topic (re-ordering conditionally convergent series) to a group of students. They seemed to understand what I was saying, showed awe at some of the key insights/tricks, and even interupted me to excitadly finish the arguement. Then, when I was done, they pointed to the equations and asked me to explain them. My response was that those equations are exactly what I had just said, and I repeated the arguement while showing them where it is in the equations.

ArbitraryLimits 13y ago

Richard P. Feynman “Physics is like sex: sure, it may give some practical results, but that's not why we do it.”

lmartel 13y ago

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?

abecedarius 13y ago
http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674057554 http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755... (by guess who) looks like the best starting point since this resonated with you. (I've dipped into it but not seriously tackled it yet.) It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.) More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.
westoncb 13y ago
Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not. I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.

whiddershins 13y ago

This is how I feel about doing most coding tutorials versus the MIT intro to CS through python, or project euler. When coding is presented as: - here is a problem - how would you solve this problem? - here are some hints to get you started it is incredibly fun for me. when it is presented as: - follow along - look what you did! it can be a bit dry.

abecedarius 13y ago
Hi whiddershins, here's a little present for you then: https://github.com/darius/regexercise https://github.com/darius/regexercise (It's not quite finished and I'd love to get feedback.)

ctl 13y ago

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum? I'm really asking -- my friend is about to start as a high-school math teacher. I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Lockhart sense -- real puzzles, not exercises.) But how often are "techniques" actually taught in high school math, especially algebra and precalculus? A lot of high school math consists of digesting new definitions, or the generalization of old definitions. A fair amount of it consists of learning theorems that go unproven, or that are proven (by the teacher) too quickly for students to understand where they come from -- and in general it isn't satisfying to solve a puzzle with a theorem that one doesn't actually understand. On top of that... students have to spend time with problems before they become genuinely interested in their solutions, so progress would be slower with this method. It's not clear that you could teach a whole year's curriculum in one year like this. (And if you fail to do that you'll eventually get fired.) Any insight? I believe that it's possible to teach math, even standard high school curriculum, in such a way that students are at all times intrinsically interested in what's presented. But it would be awfully hard to do at scale, at the standard pace, as a high school teacher would have to. How might a teacher start in that direction?

jfarmer 13y ago
I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult, even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy, have to teach to a common set of state-wide standards, and have students of varying levels of interest. Here are my scattered thoughts, though. I'm going to try to not suggest a pie-in-the-sky solution like "new curriculum!" First, I majored in mathematics at the University of Chicago, but I hate, hate, hated mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication? And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work. Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it. So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on. If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them. Second, I think the way math is taught is overly symbolic. What most non-mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed. I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa. Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations. Hey class! Look at this expression: 45+6. What does it equal? A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer." A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like... Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong per se. We could have chosen to live in either world, but we have to choose one consistent set of rules. These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and . So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it. It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't. I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)

detcader 13y ago

We really need to teach people _how_ to teach induction, which is only done right when you put quotes around your Boolean statements; the "implies" symbol gets jumbled up with everything else otherwise, and not using it at all is passing up on a great tool. One can do simple proofs-by-induction without a single English word, completely symbolically, and have it be understood easily, if one uses quotes and correct LaTeX formatting (or good handwriting) Induction doesn't just involve numbers and equality signs, it involves _statements_ with variables inside of them, and non-programmers need to be made well-aware of this (and taught Boolean logic early, PLEASE)

hcarvalhoalves 13y ago
There was a time people were taught logic and geometry and math and history all at the same time. Also, philosophy. The reductionist approach of our education is it's major failure.

jobenjo 13y ago

Love this essay. I read it years ago when my brother was working with Paul Lockhart, who deeply influenced him as a math teacher. My brother and his wife have since started an organization called Math For Love (http://www.mathforlove.com http://www.mathforlove.com) focused on changing the way math is taught. They run workshops for teachers and provide great material for students. If you're in Seattle and interesting in pedagogy and math, you should check them out.

Myrmornis 13y ago

"The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.”" +1 Also s/math/science/ +1

dwaltrip 13y ago

This article just blew my mind. I feel little silly, especially since I majored in my math. I have never seen these types of ideas expressed so clearly and powerfully. He is 100% right. We should not be teaching the formulaic rules of basic arithmetic until high school, and only then as a part of life skills class. It should be taught as the art form that it is.

dsego 13y ago

Still haven't finished reading this and all the painful memories came back from middle school. Not math (although it too wasn't a fun experience) but art. For four years we had a teacher that thought music and visual arts by dictating. For 45 minutes we would write down everything word for word. Then after a few weeks she would ask us broad questions and we'd have to recite everything back to her. I still remember (I'm almost 30 now) rote learning from my notebook, walking around the house, repeating those stupid sentences in my head, memorising descriptions of works of art I've never seen, painting techniques I've never witnessed, music I've never heard. Wow, just talking about it makes me angry, makes me want to to choke the bitch's neck. Sadly, that's the state of schools in Croatia, people equate schools with rote learning and memorising. You have to do it to get into a good university and get a good job. Who will create those jobs? No wonder the country is in the shitter.

dsego 13y ago
oops, meant taught instead of thought.

dominotw 13y ago

The author of this article teaches a course on coursera that I can't recommend enough, he put so much effort into that course. https://www.coursera.org/course/maththink https://www.coursera.org/course/maththink As you might have guessed its not really about 'math' in traditional sense.

dominotw 13y ago

Attempts to present mathematics as relevant to daily life inevitably appear forced and contrived: “You see kids, if you know algebra then you can figure out how old Maria is if we know thatshe istwo years older than twice her age seven years ago!” (Asif anyone would ever have access to that ridiculous kind of information, and not her age.) I wince every time I see these examples in my niece's textbook. So ridiculous.

kazagistar 13y ago

For music, the formal curriculum does involve memorizing songs and scales and whatnot. The problem is that the best way to start a subject is to fuck around with it (playing random songs you like on your guitar, drawing cool shapes and noticing patterns in them, etc), but our schools are ill equipped for such things, and enabling this style is difficult and incredibly expensive.

kazagistar 13y ago

A classic issue that mathematicians, philosophers, and even computer "scientists" have is the idea that they can somehow reason their way to the truth. Oh, he makes a good argument. But the greeks made great arguments about how the sun goes around the earth. I could say that this or that argument is flawed. But really, the only valid arguments are data. His data is severely lacking, and many modern methods of teaching are much more supported by experimentation. Logic is a tool for finding logically consistent imaginary realities. Science is a tool for finding out about this reality.