The secant had it coming

Yesterday DJ Strouse, a student in MIT’s quantum computing summer school, pointed me to A Mathematician’s Lament by Paul Lockhart, the most blistering indictment of K-12 “math” education I’ve ever encountered.

Lockhart says pretty much everything I’ve wanted to say about this subject since the age of twelve, and does so with the thunderous rage of an Old Testament prophet.  If you like math, and more so if you think you don’t like math, I implore you to read his essay with every atom of my being.

Which is not to say I don’t have a few quibbles:

1. I think Lockhart gives too much credit to the school system when he portrays the bureaucratization, hollowing-out, and general doofusication of knowledge as unique to math.  In my experience, science, literature, and other fields are often butchered with quite as much gusto.  Not until grad school, for example, had I sufficiently recovered from eleventh-grade English to give Shakespeare another try (or from Phys Ed do push-ups).

2. Lockhart doesn’t discuss the many ways motivated students can and do end up learning what math is, despite the best efforts of the school system to prevent it.  These side-channels include the web, the books of Martin Gardner, recreational programming, and math competitions and camps.  Obviously it’s no defense of an execrable system to point out how some people learn in spite of it—but these omissions make the overall picture too depressing even for me (which is really saying something).

3. In describing math purely as a soul-uplifting pursuit of beautiful patterns, Lockhart leaves open the question of why, in that case, it’s been in bed with science and technology throughout its history—not merely for the education bureaucrats but for Archimedes, Newton, and Gauss.  (Of course, like most relationships, this one is not without its sniping feuds.)  Personally I have no problem with teachers who want to recognize and celebrate that aspect of math, provided the students respond to it.  “So you say you want theorems that are not only beautiful, but also inspired by physics or economics or cryptography?  Line up then, because here comes a heaping helping of them…”

4. Lockhart doesn’t address an interesting problem that’s arisen in my own teaching over the last few years.  Namely, what happens when you try to teach as he advocates—with history and philosophy and challenging puzzles and arguments about the definitions and improvisation and digressions—but the students want more structure and drill and routine?  Should you deny it to them?  (For myself, I concluded that brains come in different types, and that it would be presumptuous to assume a teaching style that wouldn’t work for me can’t possibly work for anyone else.  Still, before beginning a traditional rote drill session, it’s probably a good idea for all parties involved to agree on a safe-word.)

In the end, Lockhart’s lament is subversive, angry, and radical … but if you know anything about math and anything about K-12 “education” (at least in the United States), I defy you to read it and find a single sentence that isn’t permeated, suffused, soaked, and encrusted with truth.

154 Responses to “The secant had it coming”

  1. sep332 Says:

    The puzzles, arguments, etc. provide motivation and context for the learning, but repetition (“rote” learning) really does work. Sometimes you need to get away from distractions, and practice your integrals for a while.

  2. roland Says:

    I have not experienced math class in the US. But I think Lockhardt is a bit to negative here. For the average person even knowing how to use a calculator and how to look up a formula needs practice. I think the somewhat stupid drill in math class helps to improve logical thinking abilities. And those might come in handy when your bank accountant tries to sell mortgage backed securities to you. I agree that math is an art. But teachers in high schools are not the greatest artists themselves. One shouldn’t expect too much from them.

  3. Larry D'Anna Says:

    Broken link. 🙁

  4. John Sidles Says:

    Scott, this fine blog post targets the same educational points that our QSE Group’s Friday seminar “Litotica: Efficient Quantum Simulation on Manifolds” has been targeting.

    When it comes to mathematical education (in engineering), our QSE Group’s working consensus is the recently released “Final Report of the National Mathematics Advisory Panel, 2008” mostly got it right.

    What we like about the NMAP report is that it is mainly evidence-based. It’s true that the NMAP report focuses on elementary school students learning algebra, but the same lessons (in our opinion) apply to senior undergraduates and first-year graduate students who are learning the practical engineering applications of differential and algebraic geometry.

    A BibTeX summary of the NMAP report is appended (our QSE Group’s reading of it, anyway).

    The (unsolved) underlying problem, obviously, is that NMAP-style education is costly in terms of resources, and at the personal demands enormous commitment from students and teachers alike.

    This is everyone’s lament.

    The only thing that the NMAP report missed (in our opinion) is that increasingly many modern algorithms and software tools are proprietary and/or secret. This is creating a slow-but-inexorable calamity for engineering education.

    Invitation: the Litotica Seminar is held at 1:30 pm, Fridays, ME Building, Room 119 … anyone who is in Seattle, please feel free to stop by.


    Author = {National Mathematics Advisory Panel },
    Date-Added = {2009-06-18 08:50:38 -0700},
    Institution = {U.S. Department of Education},
    Title = {Foundations for Success: the Final Report of the National Mathematics Advisory Panel, 2008},
    Year = {2008},
    Annote = {NMAP Summary (the Litotica Seminar consensus)

    “It is self-evident that teachers cannot teach what they do not know […] Direct assessments of teachers’ actual mathematical knowledge provide the strongest indication of a relation between teachers’ content knowledge and their students’ achievement.”

    U.S. mathematics textbooks are extremely long—often 700–1,000 pages. Excessive length makes books more expensive and can contribute to a lack of coherence. […] Publishers should make every effort to produce much shorter and more focused mathematics textbooks (see also , version 1.6 and later)

    Explicit instruction […] has shown consistently positive effects on performance. […] By the term explicit instruction, the Panel means that teachers provide clear models for solving a problem type using an array of examples, that students receive extensive practice in use of newly learned strategies and skills, that students are provided with opportunities to think aloud (i.e., talk through the decisions they make and the steps they take), and that students are provided with extensive feedback. [Explicit instruction also includes] ensuring that students possess the foundational skills and conceptual knowledge necessary for understanding the mathematics they are learning at their grade level.

    Teachers’ regular use of formative assessment improves their students’ learning, especially if teachers have additional guidance on using the assessment to design and to individualize instruction. Although research to date has only involved one type of formative assessment (that based on items sampled from the major curriculum objectives for the year, based on state standards), the results are sufficiently promising that the Panel recommends regular use of formative assessment for students in the elementary grades.},}

  5. anonymous Says:

  6. Greg Pfeil Says:

    I’m one of those who didn’t survive math education. Teachers always pushed me to do the next level, but it was never anything beyond boring and repetitive. I’ve since learned that math is pretty great, but really have to play catch-up to talk with a lot of my friends.

    I think that your point #4 will be addressed if math education improves for younger students. Drills are familiar to them, so it gives them a bit of comfort. But if they’re raised without that system, then they’re much less likely to desire it at the university level.

    Also, did you slashdot There must be a big theoretical CS reader base …

  7. Stephen Says:

    I think Lockhardt underestimates the degree to which different people are motivated to learn mathematics for different reasons. He claims that his motivation for learning mathematics is the exploration of patterns for their own sake and assumes that others could and should be motivated the same way. Personally, I liked mathematics as a kid mainly because it increased my power to solve classical mechanics problems. I had been exposed at a very young age to the idea of the classical billiard ball universe, in which matter is a bunch of points in Euclidean 3-space following trajectories according to Newton’s second law and various (mostly inverse square) force laws. I deeply believed that if you could exactly measure the current positions and velocities of all the particles and you had a big enough computer you could calculate with absolute precision everything that would ever happen and everything that ever had happened. This beautiful and profound and somehow comforting yet terrifying vision was scorched into my mind and motivated me to learn whatever tools could help me grapple with it. The contrived word problems presented in school seemed to be a clumsy metaphor for the usefulness of mathematics in physics, and the pure mathematicians who delighted in the uselessness of their constructions seemed to be under the influence of some exotic paraphilia. Things appear different to me now. But if teachers had tried to interest me in pure mathematics at age seven I’m not sure whether it would have worked.

  8. Augustus Says:

    The inevitable response, which a couple of people have already used, is that drill and rote learning is an essential part of math. This may be true, but – like many things for which there is no evidence – it probably isn’t. It is more likely that rote learning is an awful thing you can do to get a smattering of unrelated facts to stick for a few days when you have killed the last shred of interest. Try enthusiasm and see what feats of memory and comprehension can be achieved. Don’t assume that the current appalling situation is in any way normal or desirable.

  9. Ken Says:

    Math can be soul uplifting, but I don’t think that necessarily happens right away. The same can be said for learning to play the piano, a person starts with basic disciplines and as she learns more, can come to a point where there is enough core expertise to begin self expression.
    And this is one of the big problems.

    On the one hand a student has to learn the discipline of the actual mechanics of math, the way specific operations are performed to achieve an end. Sometime rote repetition of solving the problem helps understanding to come, in other words, sometimes it is plain hard work before understanding comes.

    In my experience, math has not been cool, especially in the early stages. As my skills progressed enough that I started to catch on to the whole mindset of the discipline, tedium began to give way to appreciation, then appreciation gave way to awe.

  10. EERac Says:

    Can any one point to some concrete proposals for an alternative math curriculum. I’m genuinely curious.

    I have a friend who used to teach at Saint Ann’s in NYC (where Paul Lockhart teaches). I know he spent several days with his students exploring whether they could draw maps that couldn’t be colored in with 4 colors, talking about how these maps could be represented as graphs, and finally trying to get at whether there was a “systematic way” to color the maps.

    This kind of exploratory approach certainly sounds appealing, but it’s hard for me to imagine 12 years of it. I’m sure it’s possible, but have textbooks been written to this effect? Exploration of math would also be a great way to teach people some basic (or even not so basic) programming. For example, what about a unit exploring the structure of the web.

    Finally, if we are going to emphasize more “practical” problem solving, then shouldn’t we at least be focusing on the math people actually use. A huge chunk of college students end up taking some sort of statistics course (and promptly forget most of what they learn). Plus probability and statistics come up in the news all the time. In contrast, almost no one uses any sort of trigonometry or calculus.

  11. Job Says:

    I’ve had both types of teachers, those who try to instill an appreciation of the subject by exposing the art hidden within, and those who mostly just follow the book to the letter.

    One part of the problem is that the books which the “Class II” teachers follow are too exercise oriented, so that’s what the class becomes.

    Maybe what’s missing is a “Math Appreciation”, or “Math History” course.

  12. Qiaochu Yuan Says:

    Schools in Washington and California are currently still suffering through the after-effects of the “New Math” craze. The curriculum in my current district is based around getting students to “discover” concepts for themselves, which is a fine idea if it’s implemented properly – but this exploration happens before students have been taught the basics. Long division has apparently been removed from the curriculum. The result is that I know an otherwise perfectly intelligent girl currently at Stanford who just doesn’t know how to perform arithmetic with fractions.

    Apparently what happened, at least in Washington, is that the math WASL was written incredibly poorly and the curriculum shifted to attempt to “teach to the WASL” – and failed. The result is that an entire generation in this state has been crippled mathematically.

  13. am Says:

    I believe school should teach people to communicate effectively. Initially this implies imprecise communication like English and Spanish, reading and writing.

    But later as we try to describe things more fully we may employ the language of math. For example, lets take a table and see how we can draw it, we can measure it and precisely define its attributes. In fact we can do so so precisely that we can end up telling how heavy it will be, how much room it will take, and how much it will cost.

    As school advances and the need to describe things increases we can use math to describe chemical and biological proceses, physical processes, and sociological processes.

    In fact, as people advance in their education they tend to need math to precisely describe what happened, their theories of what will happen, etc..

    In this manner, students can enjoy the benefits of math in all fields as they advance in its study.

    I know so many people that have studied Differential Equations or even basic Algebra and have no idea how that could ever be useful to them.

  14. Rick Says:

    And of course, there is cheating. Education in general has degenerated into such oppressive, mind-numbing labor that students of all kinds, from the dullards to the geniuses, resort to any means available to fulfill requirements. Their attitude has become, especially in high school, “Yeah cool great whatever. What do I need to do to get into college?”

  15. Ed D. Says:

    My wife who has been involved in K-12 math improvement for more than a decade immediately asked: “Why is it like that?” Frequently we see similar laments from college educators about K-12 education but rarely do the laments recognize that colleges train K-12 teachers. University presidents make the same charges, and yet are not interested in improving their education colleges since that might ruin a cash cow. Many elementary math teachers hated math in school, perhaps not the best preparation for teaching a subject.
    I haven’t yet been able to get the lament to load. Probably the MAA server is overloaded.

  16. EduExpert Says:

    “This may be true, but – like many things for which there is no evidence – it probably isn’t.”

    Just because you aren’t aware of the evidence doesn’t mean there is no evidence. Fifty years of studies and empirical data about K-12 education show that the “direct instruction” method (lecture, drill, etc.) consistently imparts knowledge more effectively than the “discovery learning” method (exploration, questions, participation, etc.). (This isn’t just true of math, but all fields of education.)

    People don’t like direct instruction, of course; it’s less pleasant for both student and teacher. So they reject the decades of empirical data in favor of simply putting their hands over their ears and screaming that cotton candy is more healthy than broccoli. Reality, however, steadily refuses to conform to their desires.

  17. am Says:

    I was always mystified by Chemistry. Lots of kids dream of playing w/test tubes and mixing dangerous chemicals and stuff like that.

    But when they get to school it turns out that they have to memorize the periodic tables, be able to find out what the side products of chemical reactions are, etc..

    Instead of putting people into a lab where they do creative and educational play.. they bury them in side subjects that eventually are useful.

  18. Deane Says:

    Disclaimer: I have *not* yet read the lament, because I can’t seem to dowload it.

    I think any rational and objective analysis of math education would lead to a subversive, angry, and radical lament.

    But I also think that we university math professors need to start with an examination of ourselves and the way we teach mathematics. It’s too easy to blame everybody below us, but I claim we are the ones who need to lead the way and set the proper tone for mathematics education. And I think our own efforts are abysmal. Many secondary school teachers at least have the excuse of not being that comfortable with mathematics. We are supposed to be the masters of our subject, but we do such a lousy job of explaining it to anyone else.

    I don’t have time here to go into detail, but in short I believe we have a muddled schizophrenic approach to teaching mathematics, where we oscillate between teaching it like a humanities course and teaching it like a mechanical rote course.

    I believe we need to think of mathematics as a craft like, say, carpentry, or as a skill, like, say, playing basketball. You want to motivate students by training them in mathematical skills *and* by showing them how to use them in a useful manner. I claim we do this abysmally.

    I am particularly critical of the way we teach freshman calculus. The course focuses on skills that are almost totally useless to students in the long run and completely ignores honestly useful skills. The question I ask everyone is: Six months after a student gets a “B” or “C” in freshman calculus, what useful skill do you trust the student to be able to do correctly?

    If you google “Deane Yang calculus”, you will find some previous rants I’ve written about this.

  19. A parent Says:

    Mostly agree with Qiaochu Yuan above. In NYC schools, they have a new reform every few years. But the basics of algebra (symbol manipulation) don’t seem to be ever taught.
    There was an interesting study (published in Science about a year ago) which said that when teachers introduced ‘real-world’ examples, the students learned less than when they were taught without those examples. See
    What do people think?

  20. A. C. Says:

    “3. In describing math purely as a soul-uplifting pursuit of beautiful patterns,…”

    All too often we get this in school, when for the great majority of students, math is a tool. One math teacher made the analogy that it’s boring like learning to pound nails straight, but you can’t build a house in a reasonable time if you can’t pound nails in straight and quickly. Power nailers don’t work everywhere; a carpenter is using his/her hammer every day. A soul-uplifting pursuit isn’t nearly as worthwhile as a tool that helps you achieve your larger goals, and math as a “mere” set of tools can do that.

  21. Delta Says:

    “This kind of exploratory approach certainly sounds appealing, but it’s hard for me to imagine 12 years of it. I’m sure it’s possible, but have textbooks been written to this effect?”

    I teach at a NYC community college that uses Jacobs’ “Mathematics: A Human Endeavor” for liberal-arts students, written very much in this style, first published in 1970.

    I like it a lot, and a very small subset of my students say it’s their best-ever math class. But in general the class has the most enormous behavioral problems of any class that I deal with, and I routinely have to call security to the room several times per semester. Most students are downright combative when asked to construct their own inductive problem analysis (and being assessed on it).

  22. Leonid Grinberg Says:

    Fascinating and depressing essay, especially for someone who is forced to live the very world he is describing. But there is another important issue here: what he is proposing is almost by nature untestable. I mean, I remember middle school art class. I was horrid at it… but I still got an A because I at least made what seemed like an attempt to follow directions. How are you going to test real, creative math?

    Also, with regard to point #2 — I think the idea is that those students who are motivated got their interest for mathematics from somewhere else before the school managed to destroy it. For instance, long before I entered kindergarten, my grandfather, in much the same way as the author prescribes, had me prove that integers go on forever. In retrospect, it was an honest-to-god inductive proof if I ever saw one, and by the time I got to school, I enjoyed math enough not to be entirely turned off by the mess that the school made of it. Few are lucky enough to be in that position, though

  23. anonmoose Says:

    Some rote practice is probably called for- but which is more likely to inspire real learning, handing out books full of blanks and ordering students to complete them (which is most all our math experience) or giving them insights to motivate on why math really is important, and happening to assign a bit of work along the way? You can probably achieve higher levels of retention with the latter.

  24. Jan Says:

    While I agree with Mr. Lockharts statement that current maths lessons are pretty much useless – I nearly choked on sentences like:

    “Suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?”

    “Here is a simple and elegant question, and it requires no effort to be made appealing”

    Come again?

    I guess Mr. Lockhart hasn’t met many children lately. Or at all. To be blunt, I don’t think most kids (or people of any age) would get his ‘math as an abstract intagible intellectual art’ thing. Intellectual puzzles only appeal to so many kids (usually the nerds who are good at maths anyways). Most neither get it nor do they care.

    I think Mr. Lockhart, who is a smart person, suffers from the delusion that everybode else is just as smart, and has the same interests. Not so.

    Most kids would take his ‘it’s not really useful for anything’ argument and beat him over the head with it. Oh, and I can already hear the wailing ‘pleeeeze tell us some formula or something concrete, no, we don’t care, we just want to know what to learn for the next test’.

  25. EERac Says:

    @ A parent

    Lockhart’s essay, which is definitely worth a full read, doesn’t support teaching math through real world example. What he’s really promoting is having students think about (and struggle with) simple, clean, abstract problems, and then guiding the discussion. As he acknowledges, however, this is difficult (if not impossible) for current math teachers, since they themselves may have never experienced real math.

    Even though Lockhart’s proposal seems bold, he also makes the point that we have very little to lose. Most students retain very little math, and most adults use almost nothing that they “learned” in highschool math.

    As for Qiaochu Yuan’s example of an otherwise intelligent girl being unable to perform arithmetic with fractions, it’s hard to imagine a sensible exploration of mathematical concepts where that doesn’t come up. Regardless, if you’re an intelligent person, and you don’t know how to manipulate fractions, you can still learn that skill at any time. There’s only like 3 or 4 rules you need to know. The real question is what are we currently getting out of our 12 years of math education?

  26. Gary Says:

    I think Mr. Lockhardt misses the point that much of the “exercises” are useful in helping some students “see” how things work out. Similar to practicing sports or painting, until you’ve tried things out and seen the results, you may not realize special twists or variations that can be done.

    Without the practice of looking for factors in numbers, it’s hard to see factor patterns in a string of numbers.

    I agree that the methods of teaching are wrong, but he misses the point that just like sports, music, art and many other skills, math takes practice at the basic skills to be able to apply them in a high level artistic form.

    As a teacher used to tell us… Math is not a spectator sport.

  27. Markk Says:

    Having taught summer “advantage” classes when I was a student, albeit a long time ago, I would say this sounds like a smart guy not getting that a lot of students aren’t as smart as him. The kids I taught needed to learn about balancing a checkbook, needed to learn how to tell if the big can really was a better deal than the small can. They needed how to figure out what a loan really would cost them. They needed to know what that meant when someone said that their risk was 100% increased on a 0.0001% risk vs a 1% increase on a 20% risk. I would rather junk ALL the beauty of math stuff and teach only practical examples and basic probability and stats. Reading this article I have just as much empirical evidence behind my wild a** ideas as this guy.

  28. Colin Says:

    When I was in sixth grade, taking algebra, we were drilling slope-intercept problems. I started thinking a bit, and then raised my hand. “So, suppose you have y = x2, which is y = xx + 0. Does that mean that at any point along the curve, the slope is x?”

    Answer: “No.”

    In hindsight, this was the worst moment in my education. I have never forgiven the teacher in question.

  29. John D'India Says:

    Concerning your 4th point, I had a similar experience as a TA for Calculus. For the bulk of my time, I’d go over the routine drill of solving calculus problems, but for 10-30% of each class, I’d talk more abstractly about connections to other areas and about Calculus as a language for discussing many scientific and engineering problems and a little bit about its history and development. Then one day I just happened to be walking behind a group of my students on the way to class, and they apparently didn’t realize I was right behind them because they were talking about me. Their lament was that I spent too much time discussing big ideas and not enough actually helping them solve homework problems. Their goal was to pass the class, not learn Calculus, and from their perspective, I wasn’t helping them with their goal as much as I could have been.

    I think this brings up a key point. It’s not terribly effective to teach someone something they don’t really want to learn. For Calculus, this difficulty is a little easier to address since some of the students probably don’t need to learn it in the first place, and for those that do a slight reordering of courses and material can probably have a large impact. For example, biology students would probably benefit from taking Calculus at the end of their 4 years rather than the beginning.

    When possible, the learning flow should be “illustrate problem” => “provide solution” as opposed to the default “provide solutions” or the slightly better “provide solutions” => “show connections to problems”.

    High school level mathematics is more difficult in this respect, but I don’t think it’s impossible. Lockheart himself provides some interesting teaching techniques in his lament.

  30. Tyler Says:

    Funny thing, Colin, is that if you’ve taken calculus then you know that the teacher answered your question correctly, since taking the derivative of the function clearly shows that the slope of the tangent line at any given x is in fact 2x, not x.

  31. Daniel Says:

    Tyler: Colin is perfectly aware of that, I’m sure. His problem is that the teacher didn’t take this perfect opportunity to enlighten and motivate.

    (Sorry if I fed a troll. Tyler, sorry if you are not a troll.)

  32. Deane Says:

    I will belabor my point about calculus.

    Please tell me, whether you teach calculus or just took it as a student, what you think the point of first semester freshman calculus. What is the key idea or skill that is being taught?

    I don’t believe I had a clear, coherent answer to this question until after I had taught freshman calculus for many years and then spent some time working as a “quant” on Wall Street. There, I met a person who probably never took calculus, was working essentially as a clerk, was monitoring risk reports for a portfolio of financial securities (interest rate swaps), and (to my shock) had a better concrete understanding of what the meaning of a derivative (the mathematical kind, not the financial) was than every student I had ever taught up to that point. It was only then that I understood how to explain differential calculus to innocent people without snowing them with arcane jargon, formulas, or “concrete” examples from physics that only confused them even more.

    This eventually led me to my own radical and subversive period of teaching calculus. I’ve never been the same since. I very much enjoy visiting friends at other universities, including top-ranked ones, and telling them matter-of-factly that their freshman calculus program is a complete fraud.

    So until we professional mathematicians figure out how to teach “higher” mathematics better ourselves, what right do we have to expect secondary school math teachers to teach better?

  33. Jack Says:

    As a physicist, I see the issues Lockhart raises as endemic to mathematics overall, not just K-12 education. Math papers usually read like the geometry proof example in the article: Beautiful ideas obfuscated and complicated by formalism. Physicists by contrast are much more comfortable discussing the intuitions that led them to particular results, for example Einstein’s “riding on a beam of light” or Faraday’s visual picture of magnetic field lines of force. Why do mathematicians seem uniquely compelled to cover their tracks and present each end result as a monolithic, impenetrable edifice? And why would Lockhart or anyone else be surprised if educators ape this tendency toward formal symbol manipulation? If mathematics is an art, it seems to me mathematicians themselves aren’t wholly comfortable with that idea.

  34. Anthony Staines Says:

    I’m an epidemiologist, basically a numerate doctor, and I used to teach freshman university level general statistics courses. I stopped doing that because I noticed that no-one ever learned anything.

    I’m still the only person I know personally who gained anything from such courses. As far as I can see these are the exact analog of your freshman calculus courses, and they survive for the same reason, inertia and incompetence on the part of university faculty.

    Students learn statistics (and I suspect, calculus) when they need to use it, and not before. They can learn to pass exams in statistics quite readily, but that has nothing to do with knowing anything about data analysis.
    Anthony Staines

  35. Colin Says:

    Tyler, I learned a few years later that the answer is 2x, but that isn’t the point. I just think that “no”, nothing further, was the worst possible answer to my question. “Yes” would actually have been a better answer. I was within delta of having discovered differential calculus, and that incompetent moron kicked my feet out from under me. She could have said a lot of things:

    “No, but we can talk about why after class. It’s interesting.”
    “Not quite – it’s 2x. The reasons are a bit beyond the scope of this class.”
    “Pretty close! It’s a bit more complicated in this case, but you’re asking the first question of calculus.”

    All of these would have been encouragements. Instead she said “no,” making her the worst alleged teacher I’ve ever had. It’s been fifteen years, and I’m still angry about the way she took away my confidence in my mathematical abilities.

  36. Deane Says:


    I really do think you are being unfair to the teacher in question. She answered the way she did, because that was all she knew. She was teaching the best she could, given what she herself was taught. My guess is that she knew what you said was wrong but did not know how to refute your reasoning. She did not know where the logic went wrong. She did not want to encourage discussion or debate, because she didn’t know how to argue your point.

    You know when you’ve taught math properly to a student, because the student feels more powerful and confident than before (just like a successful student of carpentry or basketball). That feeling of power is what inspired me and, I imagine, many others to pursue mathematics or science as a vocation.

    My belief is that the vast majority of math professors and math teachers don’t see how to “empower” their students in this way, and indeed it is a very hard thing to do. So we settle for either a “cultural understanding” of mathematics or a demonstration of some rote skills.

    To be fair, many of my gym teachers taught me basketball and my shop teachers taught me carpentry no better than that, either.

  37. david bandel Says:

    my one objection with the lament:

    i read the lament and found that it perfectly reflected the way i viewed math in the school system; an art portrayed as a cold heartless bureaucracy.

    even as a child i viewed mathematics as more of an art or a philosophy than a science. more a way of thinking about things than a confining set of formulas and carefully derived results.

    i only take objection to his specific examples of mathematical proof. arguably, he practices these misportrayals deliberately, but it doesn’t change the fact that he gives rigorous proofs up as examples of the deplorable state of things and intuitive elegant solutions to problems as the way things really should be.

    elegant solutions aren’t always available. sometimes we just aren’t smart enough to find them. but in their presence, a rigorous proof of some interesting theorem is no less beautiful.

    math is artistic, expressive, creative, and beautiful. but it’s also massive, stretching over hundreds of subsets and benefits from some strict underpinnings.

    maybe these underpinnings shouldn’t be taught in school. his article is almost certainly fundamentally right. if anything, the underlying logical connectivity and standardization of principles and methods should be saved for advanced students who have already found their calling to the world of math. nevertheless, they are in themselves beautiful.

    some methods are more artistic than others but usually it doesn’t mean that the lesser fails to be art. even by his standards.

  38. Deane Says:

    By the way, I didn’t mean to exclude graduate math programs from my attacks. We graduate far too many unqualified math Ph.D.’s, even from top schools, who have memorized amazing amounts of mathematical theorems and proofs but have little or no basic analytical skills.

    I don’t believe mathematicians have ever really understood the extremely damaging consequences of producing such Ph.D.’s and sending them out into both the academic and real worlds.

  39. Ash Says:

    After reading just the opening pages of the lament and these comments I think a good way to approach this problem of teaching and learning mathematics would be to break our concept of math into 2 pieces. Arithmetic and mathematics.
    Arithmetic is what I call the portion of math needed by most people to handle math in modern society. Call arithmetic the tools of math. I suspect that rote memorization may be the fastest way to learn the basic math operations that appear on most calculators. Between drills and memorization most people should be able to learn these within 3-5 years. Some might even learn them faster and should be rewarded by allowing them to move on early.
    In this case, rote memorization and drills have a point of preparing people for skills that use those exact functions. As A.C. mentions above with his carpenter and hammer example, many people have specific needs and practice will get them to the point that they can perform a task with minimal effort. Practice makes perfect with using a tool. Unfortunately, our current system continues to use memorization and drills to add more and more functions over a majority of our mandatory educational life. A large number of these formulas are only useful to introduce the next set, in the order that they were discovered. I’m guessing that someone felt that this was close enough to the process the original masters went through to be the best order to drill each topic and approach the discovery process. I say hogwash to this once you get beyond the buttons on a calculator.
    Mathematics should be taught to everyone that finishes arithmetic. At this point drill and memorization of formulae should be reduced or eliminated in exchange for the exploration approach. I suspect, in this case people will fall into 2 categories, those that are interested, and those that are not. The trick will be to identify the ones that are capable but frustrated, and those that truly have no interest or ability. Give them a few years before pulling people out. Those with an interest should stick with this through the rest of their mandatory education, continuing in college as desired. The rest can settle with learning the tools that the mathematicians have discovered.
    Advanced arithmetic concepts and refreshers should be given to everyone regardless of mathematic interest as they advance, to keep skills that are memorized, and to build on skills that kids learn in other courses in later years. Teach them how to read and perform formulas that are used commonly in various careers. Treat them as tools for performing a task. Don’t worry about the theories and concepts behind the formulas. They will either be covered in the mathematics class, or in other subjects. Those that need math as a tool just need the tool. They may find an interest someday about why a hammer is the way it is, but most people just care that if they need to pound a nail, a hammer is the appropriate tool and how to use one.
    The people that find an interest in mathematics itself will be the ones that question if there is a simpler method to create a large force at a small location than a flat surface on the end of a lever. Or if there is a single formula that will explain why every particle in the universe is at the location that it happens to be at.
    I once heard that every major mathematic discovery was made by people in their youth. I don’t remember the age listed but it seemed to be their early 20’s. If this is true, then the quicker we begin them on the exploration path of mathematics, then the more people will get to that outer edge of known mathematics to make these discoveries while young enough to make them.

  40. Jeremy H Says:

    Part of the challenge with teaching math, as with most other subjects, is that it isn’t just one thing. It isn’t just a tool box, nor is it just an art, nor is it just an intellectual quest for the truth, nor is it just a way of thinking. It is all of these things, and to different people the relative importance can very greatly.

    Further complicating matters, different students need to be taught in very different ways. High-end mathy students require different math instruction from high-end non-mathy students, who require different instruction from low-end students.

    I think Lockhardt’s suggestions would be wonderful to see implemented in Jr/Sr high school courses. In the younger grades, a mix would probably be ideal. Some drills, some high level concepts, some exploration. As with most areas of life, the extremes tend to be bad.

    And when we switch the discussion to college, it becomes a whole new problem again. 🙂

  41. Bram Cohen Says:

    History and foreign languages are also subjects completely obliterated by how they’re taught it schools. One would be hard pressed to find a less effective way of ‘teaching’ them.

    What’s really astounding is how noone of any qualifications is involved in setting school curriculum. No mathematicians are involved in the mathematics, no historians in history, no bioligists in biology. All the professionals view what’s taught as a complete travesty, and yet somehow just because you have a PhD in a subject and work in a university you aren’t considered qualified to teach that subject. You have to get an entirely different set of credentials for that.

    Contrary to what some other commenters have said, noone is advocating for removing arithmetic from the school curriculum. In fact, basic arithmetic is one of the few recognizably mathy things in there. The problem is that its rote aspects are overstressed, and the meaning and derivation of it all is ignored. Most math reform programs actually make everything much worse, because it’s all designed by nonmathematicians who just add a bunch of quizzes about terminology which they made up. A really egregious example of that is TERC, which just plain doesn’t teach math.

  42. Danny Says:

    Lockhart’s Lament has been a staple in explaining Sudbury schools for years. Y’all quite a bit late on this. The deep problem he hints at is our manic obsession with Teaching and our utter mistrust of freewill exploration, play and learning. 48 years of Sudbury schools has shown in independently-verified, detailed, quantitative analysis that children are sailboats we’ve been trying to row.

  43. Scott Says:

    Everyone: Sorry for the problems downloading the lament! The link seems fine; the only explanation I can think of is indeed that the MAA server was unable to handle the “Shtetl-Optimized bump.”

  44. Scott Says:

    Most math reform programs actually make everything much worse, because it’s all designed by nonmathematicians who just add a bunch of quizzes about terminology which they made up.

    I wish to second, third, and fourth this observation of Bram’s. Let no one mistake me for a supporter of “New Math” (which is actually old enough for Feynman to have ridiculed it in the 60s). What Lockhardt and I advocate teaching is not “New Math”, not “Back-to-Basics Math”, but something much more radical than either: math.

  45. PS Says:

    A solution that seems to jump out at me is to separate the teaching of arithmetic from the teaching of mathematics. It doesn’t follow with Lockhardt’s suggestion that everything should follow from understanding, but I hardly think teaching 1st grade students Peano’s axioms will help them understand anything. Separating these two subjects would allow for memorization and drills in one, and exploration and understanding in the other.

  46. Len Ornstein Says:

    Lockhart’s argument that the art and history of math are essential in peaking motivation, is right on!

    But there’s another point which he fails to even mention:

    Math (and formal logic) are the best introductions to precise deductive reasoning. And, despite his protests to the contrary, such a ‘useful’ end is perhaps as important as the ‘art’ it can motivate. When geometry is ‘properly’ introduced as an axiomatic system, it can blow a student’s mind.

    The reason patterns and beauty are motivational is probably because we have built-in, subconscious ‘appreciation’ of a batch of axioms, evolutionarily laid down in our nervous systems as a consequence of the recent evolution of language skills. We use them to build both fanciful models as well as ‘scientific’ models of ‘reality’.

    One of the most important things a good math teacher accomplishes is to help all kinds of students to gradually appreciate that mathematics helps them towards understanding what precise deductive reasoning is – on their road to rationality.

  47. summer schooler Says:


    MIT doesnt have a summer school …. are u referring to the program for high school students ?

  48. Scott Carnahan Says:

    I’m afraid I found the piece rather irritatingly overdramatic, and it really seems like he’s preaching to the choir (of mathematicians who already agree with his views). Although I agree with many of the author’s points, he doesn’t offer any persuasive data that indicate that math education is in the dire situation he describes, and he doesn’t offer any data that suggest that his ideal educational structure (which was not proposed with much in the way of detail) would be an improvement.

    The whole article was suffused with a romanticization of mathematics as a pure creative process, together with comparisons with art classes. My own experiences with art classes in US public schools suggest that a purely exploratory approach is not necessarily the best for everyone. My classes were full of unstructured creative time, and I never received much instruction in art history, or in solid skills like light & shadow, composition, perspective, and color theory. I ended up being quite frustrated with my inability to use art effectively as an expressive medium, and aside from the odd doodle, I don’t really practice any more. When it comes to creating beautiful things, in art as in math, I think technical skills are extremely useful.

    I defy you to read and find a single sentence that isn’t permeated, suffused, soaked, and encrusted with truth.

    Really? I don’t think you’re trying very hard. I strongly disagree with a rather large block of text surrounding the words “you can’t teach teaching.” Does the author really think classroom management is a trivial problem, whose solution comes from the ill-defined notion of “being real”? Do you agree that a planned lesson is automatically false? In my own limited experience, without some sort of plan for a class, my lessons end up rather incoherent and flighty, even if I have an abundance of openness, honesty, willingness to share excitement, and a love of learning.

    My overall impression is that my own experiences in public school grant me some sympathy for his frustration, but his cries for revolution seem both ill-justified and rather insulting to those who work within the system for positive change.

  49. Pat Cahalan Says:

    > I concluded that brains come in different types

    Stop doing research, and just teach. It may be a crime to the research community, but it will be an endless boon to people who actually want to learn.

    Most people who educate at the university level have *no idea* that this is true.

  50. Qiaochu Yuan Says:

    The whole article was suffused with a romanticization of mathematics as a pure creative process, together with comparisons with art classes.

    I’m going to call this artistic license; that is, I’m willing to believe that Lockhart stressed this point not because he thinks it’s the only way to think about mathematics, but because it is so rarely known among non-mathematicians that this aspect of mathematics even exists. Isn’t getting that point across worth a little exaggeration?

  51. Jason Says:

    As one who took a math major in college, three realities finally came to stand out in stark relief after a couple years including time spent tutoring for the school and then TA’ing a calculus course:

    1) I saw most students had one of two problems with mathematics: they couldn’t apply what they knew or they couldn’t get the mechanics to work out. The former is an issue of experience and guidance while the latter is from a lack of drilling and killing the basics. Even proofs based graduate-level texts find a way to drill and kill the most advanced concepts because playing with them is the only way to really learn them.

    2) Almost all of my math-ed friends felt being forced to take a math major made no sense because they would never teach a majority of the material. Most of the engineers didn’t care about the proofs because they were out to build widgets not build new math. Actuaries cared only enough to pass their tests. Almost no one took the approach that mathematics can be taken from its pure forms and then applied to real world problems, instead they took the position that mathematics is a toolkit to be built up as necessary to overcome whatever happens to stand in the immediate front of them.

    3) Almost no one without a formal, collegiate or better level understanding of mathematics has any conception of how much work it takes to build anything in mathematics which can be used in the real world. The only analogy that seemed to work is that mathematicians are like the tool and die makers of the world: everyone needs what we produce, almost no one is interested in what we do, and even fewer understand just how hard it is.

  52. Danny Yee Says:

    I did a degree in pure mathematics and have always loved maths, but I’m firmly convinced that most of what we were taught in school was a complete waste of time.

    We spent huge amounts of time in advanced high school maths learning how to do stuff like integration by parts and similar memorisation of symbol manipulation rules – none of which I have used since, despite doing a physics/maths/compsci degree.

  53. Michael Luvaul Says:

    It would seem to me, as a Junior Math/Physics major, that the best and brightest students in mathematics tend to teach themselves the subject anyway. Not all of us follow the rote memorization and algorithm approaches found in texts either. I for one tend to find as many texts on a subject as I can and take from them what I find useful.

    I fully agree that most of the “tools” of mathematics aren’t really learned until one needs them. A good example in my own experience is that of linear algebra, which sadly enough, I didn’t understand fully how powerful it was until I used it in the context of quantum mechanics.

    His lament could in fact be about nearly any subject taught in K-12 anyway. The fact of the matter is that public schools in this country are not created to teach students. They are created to keep the children out of the streets during the day while the parents are out earning wages. And more importantly, they are there to serve as a method to indoctrinate children into a system that requires them to blindly follow instruction, rarely question authority and become a cog in the machine.

  54. TruePath Says:

    As a working mathematician I had a great deal of sympathy for many things Lockhardt had to say. In particular he couldn’t be more right about the total uselessness of most of the math curriculum to most students. Go ask a working professional (doctor, lawyer, etc..) to solve a system of linear equations in 2 unknowns and it’s immediately apparent they got no direct practical benefit from their math classes.

    I quibble with his ragging on epsilon-delta and other precise definitions. I finally realized math was elegant and exciting precisely because I was so disgusted with (ugly) intuitive arguments about smoothness I went and found a book that taught me the elegant formal definitions that made calculus all fit together. Not that I would recommend this for everyone.


    However, where he really totally blows it is when he assumes that math can be a fun exploratory intellectual adventure for everyone. Yes, virtually everyone has the innate intelligence to do this but no matter what you do math is going to make some people feel dumb and frustrated. There are right and wrong answers in math and not everyone can be above average.

    Sure, everyone might be lackadaisical in HS art class but that’s because few (no?) people’s future depends on their ability to do well in the class. On the other hand the best and the brightest signal their ability by performing well in math. Sure, these students succeed because they are curious and interested but all the other students will struggle to look like the mathematically advanced kids and those who fail will feel bad about themselves for it.

    People don’t like doing things that make them feel stupid or frustrated and learning real math requires genuine curiosity and thought. You just can’t force people who resent the subject to think.

    Perhaps we should simply accept that math is going to be like literature or art. A small percent will have the desire and interest to pursue it in highschool and we should just try to avoid turning off the rest enough they might return in their own time.

  55. Rajesh Says:

    It seems to me that Lockhardt is trying to teach us that its not possible to teach teaching which sounds like a contradiction.

  56. TruePath Says:

    A couple more quick points.

    I think the big elephant in the room here is our background prejudice that somehow you just aren’t ‘educated’ if you don’t know how to handle fractions or never learned the quadratic formula. For 95% of HS or even college graduates this ‘training’ serves the same purpose as most HS english courses: social signaling that your not ignorant. Memorizing the Odyssey could serve the same social purpose and it wouldn’t prejudice everyone against math. It doesn’t matter a jot if your english professor can add fractions or your lawyer remembers the power rule for differentiation.

    Once you accept this point (despite how much it scares those of us employed as math profs) you can get down to serious attempts to make things better instead of worse, e.g., treating math more like geology or anthropology, a subject to lure people into rather than to beat them with.

    Also you can then step back and ask what the best way to teach the rigorous quantitative style thinking that we supposedly want math to convey. My conclusion is that programming would be a much more effective initial introduction since at the outset math can be all frustration and little reward (you just don’t see the proof) but programming offers a much more MUD (or MMORPG if you prefer) like distribution of rewards from the novice.

  57. mason Says:

    I have a solution.
    For all the people who hate math and just want to do it so they can get it out of the way, they can continue to do that. For all the people who want to be engineers, or scientists that need to learn formulas and calculations, they can also continue to learn calculus and other maths in the traditional way.

    But I reckon they should offer an elective class that people can take, that teaches maths in the way that Lockhart describes. It would be for the people who want to be creative with numbers and try out new things. It would be for the people who want to try things out for themselves and push the boundaries of what they know.

    What Lockhart is saying would not actually appeal to everyone, because not everyone finds math enjoyable in the same way that not everyone wants to be an artist or a musician. So I say that it should be offered to the people who so badly need it, even if it is a minority.

  58. John Sidles Says:

    Lots of people on this thread are (1) decrying the foolishness of math education “experts”, and (2) lamenting the teaching of mathematics by those who are not “mathematicians”.

    But two inconvenient facts are: (1) Math education experts mostly agree with the points made on this thread. So who is it, exactly, that everyone is disagreeing with? And (2), there is solid evidence in the literature that questions like “What is it that makes a person a mathematician?” and “What is it that makes a good teacher?” cannot be presently be answered by any known certification requirement or testing procedure.

    It is dismaying that academic discussions on mathematical education so often resemble the way right-wing blogs discuss climate change. There’s a superabundance of passion, ideology, and anecdotes — but not much reference to a literature that is full of inconvenient truths.

    A great deal of the literature on mathematical education (IMHO) is summed up in two aphorisms:

    “The spirit will not be aroused in the student by one who is himself not filled with it.” (Charles Schreiffer)


    “The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous.” (Edward Gibbon)

    Obviously these two aphorisms directly contradict one another — that’s because they express a Great Truth of math education (in the sense of Niels Bohr).

  59. Jerry Llevada Says:

    I have taught math from middle school to graduate school, I have taught the good, the bad and some really ugly ones, and I have done it for 32 years. I have also taught physics, economics, and engineering. And to make matters more interesting I also worked at MIT’s EPL labs for 7 years. I know the MIT of the 60s and met and was inspired by most people I met there (they also have a tad collection of duds). But after I left Cambridge, I was only able to find and teach 2 MIT types in all those 32 years. I taught them in HS and they finished their HS math education by taking advanced courses (and I don’t mean AP courses) at the local U; one is now a patent attorney, the other a math prof. My take is this: We don’t have the resources (name your resource) to even approach the periphery of what Lockhart envisions. It will never happen. You guys live in another world. However, I can tell you this: things are better than ever before, and it is people like you and places like this that make it better.

  60. PJ Says:

    I’m a single mom in Oklahoma, somewhere between blue and white collar, ‘some’ college; I work in IT. I’ve always liked math and am homeschooling my daughter who is 12, so currently thinking hard about what to do to improve her instead of ruin her with it.

    She and I looked at an algebra book about a year ago. After, “What’s it for?” went unanswered (I had no good answer) she said — this is hilarious — “uh… could I learn Japanese instead?” You see to her, it looked equally complicated, but she could at least see some *reason* for learning another language and alphabet; wading through the stuff in the math book just seemed like an insane waste of time for no good reason. (I said, “For now.” ) I couldn’t even tell her a good reason! Because you should? Why? Not knowing calculus has never harmed me, to my knowledge, and I’ve used mathematics in various jobs for 25 years but have never to my knowledge used a single algebraic formula. The symbolism in algebra as I learned it has basically zero “mapped meaning” to the psychology. I bet there are equations I could work out which would answer fabulous real-world questions I’ve had but I had no idea that my life issue related to that symbol issue so just never connected them.

    I asked a math professor what the dominant point of algebra+ levels of math were. My translation of his answer kind of boils down to, “You need to learn X so you can learn Y, which you need to learn so you can learn Z, which ….” — a sequence which did not, unfortunately, ever end at some goal-location where I wanted or expected to be. Is it possible that ‘schooling’ has simply bred and multiplied _itself_ to the point that the major point of schooling is to get more schooling? Kind of like making the major point of government to breed more government and more things to govern? Given all the industries related to government, publishing, education, and more are interrelated–if not outright inbred by now–doesn’t that kind of reduce a big % of lifetime to “schooling for the sake of schooling” to being akin to “The Matrix?”

    I once took a statistics course taught by Dr. Jessica Utts online. I loved that but to me, it was like a “logic” course using math tools, which made it interesting and applicable to everything, like a systematic way of thinking about things critically. It wasn’t very much about memorizing formulas and abstracts; you used software tools for the slide-rule-nerd-stuff. Which makes it radically different than another stats class I once looked into. I guess there are kind of two paths in that field depending on philosophy. Anyway, aside from that stats course I have found nothing in post-mathematics “maths” that I comprehend the reason/need for.

    Time, money, memory and attention are ‘resources’. I want to allocate them toward what is useful. Spending 20x the time required (due to little interest and no meaning-mapping so rote-repetition required) to not-or-barely-memorize something with no actual future point to it seems… er, pointless. I want her to be ‘educated’ well, but pounding math repeatedly into someone until repetition-trauma retains the memory by brute force does not seem like the intelligent way to go about anything. (Not that I haven’t done some of that already and will probably do more.) I’ve seen how, when she is interested, she can learn at breathtaking speeds, absorb and remember and synthesize tons of information, and branch it into 10 other subjects, and enjoy it all. I want to find a way to make math interesting so she will be inclined to DO it and really learn it.

    I like Algebra. But it seems about as related to my life as “walnuts” are related to “justice.” I feel sure this must be a lack of understanding on my part. I just cannot seem to find any book, textbook, or person, who can help me connect those dots.

    I was really enjoying the essay and thinking that any minute now, this guy would actually give out that insight, when the author said, “Suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?” and called this simple and elegant and appealing. Uh oh! This is not a question which would inspire me or my kid to want to spend time on it to be honest. If that genuinely interested him as a child then I’d say he has a specific kind of thinking pattern that is not all that common. Find me or my kid a REASON to need an answer, apply it to anything from ‘the real world’ (even to include the online world) and it would be different. His question has the benefit of not being “obscure” like the bureaucratic-symbolism-and-terminology he exampled as an alternative, I will give it that. Still, I suspect it would lead most kids to think, “… who cares?” which is not all that much better.

    In school I hated history and science, which is ironic since as an adult they are subjects I am fascinated with. Too bad I didn’t catch the fever earlier. I have to be a basement mad scientist self-educating on the internet about everything from neural anatomy to statistics now, when I’m old (early 40s) and my kid’s old enough that I “almost” have a life again between her and work, but I’m sure I could have done edu and science more officially when young if I’d only been interested back then. I know that probably my school experience unfairly biased me against everything, and perhaps if I learn about math in a different way (like I did science and history) it will all become clear to me. So maybe the math experts here can help me out with the answer:

    What the hell is this stuff FOR? Algebra, Geometry, Calculus, Trigonometry?

    Does any normal human being outside highly specialized technical jobs use this stuff?

    If I don’t learn more of it (post-Algebra), or my kid doesn’t learn it (pre-Algebra onward), what important thing are we going to be missing, or unable to do?

    Seriously — thanks for any input. The article was great but it actually leads to questions like this.


  61. ScentOfViolets Says:

    Jumping on way late, but I’d like to address this:

    The inevitable response, which a couple of people have already used, is that drill and rote learning is an essential part of math. This may be true, but – like many things for which there is no evidence – it probably isn’t. It is more likely that rote learning is an awful thing you can do to get a smattering of unrelated facts to stick for a few days when you have killed the last shred of interest.

    I’m sorry, but there is loads of evidence that drill is a necessary component to competency – perhaps the component. But mathematics isn’t the only place where this is true. Far from it. Hundreds of years of musicians have resigned themselves to uncounted deadly dull hours of practicing scales. Every year, millions of would-be professional athletes practice the same jump shot over and over. To think that mathematics is somehow exempt from this is nothing short of magical thinking (prompted by the good intentions of the educational establishment, sadly enough.) What does seem to fairly unique to mathematics is that people question the efficacy of drill. Musicians and athletes my not like it, but at least they acknowledge the necessity. Which to me suggests that the profession is relatively powerless, not relatively clueless. Teaching on someone else’s dime is a tricky proposition.

  62. Jim L. Says:

    “I think the big elephant in the room here is our background prejudice that somehow you just aren’t ‘educated’ if you don’t know how to handle fractions or never learned the quadratic formula.”

    That’s right, not learning this makes you uneducated. The notion of what makes someone educated is a social convention. So what if it is “useless” to my lawyer?

    Education serves other purposes. English classes teach critical thinking, as well as writing skills. Math classes teach rigorous thinking, proofs, as well as discipline. A big part of schooling is to separate people out according to their abilities across a variety of different areas. What is wrong with using math as a subject to beat people with? Are you saying that we should only use history classes for signaling?

    In terms of signaling, it is actually an advantage to have a subject that does not have immediate rewards. The ability to pursue a problem despite frustration and lack of immediate reward is an important life skill.

  63. ScentOfViolets Says:

    I like Algebra. But it seems about as related to my life as “walnuts” are related to “justice.” I feel sure this must be a lack of understanding on my part. I just cannot seem to find any book, textbook, or person, who can help me connect those dots.

    This would be a good point were it not everyone’s lament. I’ve had engineers complain that they could have done their jobs with just the first two years of college, lawyers who use only one tenth of the law they ever learned, doctors who never, ever had to use their memorized knowledge of the names of various bones and muscles.

    So, yes, the long odds are that you’re only ever going to use one tenth to one quarter of everything you’ve ever learned. The trick is, which quarter 🙂 If you could reliably answer that question, a lot of folks would be awfully grateful. Plus, you’d be propelled into national prominence.

  64. Jiri Lebl Says:

    I think the lament is a little too radical. I see no problem with the classical lecturing approach. It should be amended with discovery, but unless the students are well motivated (and most of them will not be) I don’t think it works. My problem with K-12 education is that students entering college cannot do basic algebra, do not understand basic concepts. I don’t care if they can do long division, that’s entirely pointless. There are also pointless rules such as FOIL, which I do not understand why they are taught and time is being wasted on them.

    The problem is that there is too much B*S* in this (US) school system. Grade inflation is one culprit. We all pretend like the students know some topic because they have taken a class on it. In my experience, a student who gets a B in a college class probably doesn’t know the basics of what the class was about, but can probably spew a few formulas without knowing what they mean. In my opinion, B should mean you have a reasonable grasp of the subject, but … I’m sure K-12 system is similar. You can’t tell a student that they are doing badly and what they need to improve on so that you do not “hurt their feelings.” Students entering college are unprepared for failing at something. Students entering grad school are even for more of a shock when even the undergraduate hand-holding stops. There is nothing wrong with being bad at something. But there is something wrong with pretending we are all great at everything.

    But I agree with the lament that huge areas of what’s being taught can be scrapped. Spending a year on the inner workings of long division and drilling to do it correctly is pointless torture. There are more interesting and useful things to learn on which you can build further. Long division doesn’t really lead to anything. Understanding algebra, functions, etc… will be genuinely useful in understanding in all technical classes students take in college.

  65. Johan Says:

    Well, I think we all wish that everyone could see the beauty of mathematics and the introduction to the essay was pretty funny. I can not comment on American education specifically but I have some general thoughts.

    We should definitely encourage students to experience math as an art, with lots of problem solving.

    But importantly I believe a lot of people would find that incredibly boring or suck at it. (For various reasons that we probably do not know and the teacher probably cannot affect.)

    For that part of mathematics that everyone needs to know, basic arithmetic, perhaps some statistics, I definitely believe it should be taught in drill form.

    Perhaps we should simply make less math mandatory.

    I think the comparison with art is interesting. My art teaching at school didn’t emphasize drawing skills or other technical skills. Instead we drew pictures. I absolutely hated it. Perhaps for that very reason, since I was forced to do something I really had no idea how to and was not given the chance to learn.

    Later I have learned some perspective drawing and that was in comparison a lot more fun since I could see that I did something right. I wish that my art class at school had emphasized technical drawing skills a lot more.

    Finally I think that the field of education science should become a lot more like medicine with randomized experiments when one evaluates teaching methods. There is far to much ideology and not enough science in the education debate today.

  66. EERac Says:

    I think a lot of people defending math education, and example-based teaching, and the value of drills, aren’t stepping back enough to appreciate just how little students get out of the current curricula.

    As a point of contrast, consider biology, chemistry and physics. Most people take one year of each in high school, and as a result they really do acquire a host of basic facts/concepts from those fields. A typical college graduate is far from a chemist, but they likely understand that the world is made of atoms of different elements, that reactions can occur, that there are different types of matter, etc. In short, they have some idea what a chemist studies.

    In contrast, students take FOUR full years of high school math, and unless they use it in college, *poof*, it’s completely gone. Most college grads have no idea what a mathematician does, and have long lost their to solve the “math problems” they saw in high school.

    We have so little to lose by sacrificing the current highschool curriculum in favor of discussing actual mathematical ideas (i.e. posing questions, trying to come up with answers). Once you get beyond basic algebra, there’s this illusion that you’re working toward this high school-level math skill set, but there’s no need for these skills.

    The potentially valuable aspect of studying high school math isn’t learning to solve certain random types of problems. It’s learning to think clearly about deceptively-simple concepts. In doing this you learn to attack problems, generalize techniques, define concepts, make convincing arguments, and find counterexamples. If you need to know the quadratic formula for some weird reason, then all you need to do is look it up. The skills listed above aren’t just useful to scientists, but to lawyers, writers, doctors, etc…

  67. Michael Maxwell Says:

    PJ asks:
    > What the hell is this stuff FOR? Algebra, Geometry,
    > Calculus, Trigonometry?

    I have the same question about algebra. Geometry–which does not, afaik, rely on algebra (as opposed to analytic geometry, which I guess does) was an eye-opener for me in HS. (I strongly sympathized with Simplicio on pg 23.) It made me realize that everything from politics to religion starts with a few axioms, and all other conclusions are (supposed to be) logical deductions from there. It even changed the way I approached chemistry (all the way through organic chemistry; it did not work, to my chagrin, for biochem, but that’s a different story).

    I suspect that not everyone has that epiphany with geometry, but I believe everyone could have it if the analogies were drawn out, perhaps by combining a course in formal logic with plane geometry.

    As for trig and calculus, I guess they have lots of implications in physical sciences and engineering; trig even has implications for computational linguistics (my field these days).

    I started by saying that I have the same question about algebra: what good is it. Doubtless someone will say you have to understand algebra before you can understand trig or calculus. I’m sure that *some* knowledge of algebra is necessary for those subjects, but I’m not convinced that very much is. But then, it’s been an awfully long time since I studied trig or calculus.

    I’ll close by saying that analytic geometry was interesting, in that (if I remember correctly) you could get to much the same conclusions as you could using plane geometry, but by a very different means. I have heard that mathematicians look for those kinds of connections between apparently disparate fields of math, so that is interesting–to me. Other people’s mileage may vary.

  68. William Newman Says:

    PJ wrote “She and I looked at an algebra book about a year ago. After, ‘What’s it for?’ went unanswered…” and “What the hell is this stuff FOR? Algebra, Geometry, Calculus, Trigonometry?”

    It does seem hard to find many examples where algebra is extremely useful directly. And I don’t think this is just failure of imagination: historically, it seems that useful results obtained using algebra were relatively uncommon until calculus was developed. So I think algebra is primarily useful as a prerequisite for a trinity of skills which are more directly useful: calculus, basic Fourier analysis, and basic linear algebra. (And secondarily, algebra might not be an absolute prerequisite for computer programming, but it sure does seem to be helpful for it.)

    That trinity of more-directly-useful skills is, in turn, is the hard prerequisite for finally the actual, directly-useful skills (engineering, statistics, etc.) for solving a trinity of important problems. First, problems where you need to understand randomness, often involving drawing inferences from noisy data. Second, physical engineering problems. Third, optimization problems.

    Note that even when an instance of such a problem can’t be solved formally (typically because the inputs aren’t known precisely), abstractions at the level of “white/pink/red/whatever noise” and “zero derivative in the neighborhood of the extremum” and “standard deviation” and “second degree diffeq [involving momentum or inductance or whatever]” can be helpful for one’s informal understanding of the problem. IMHO, this matters: it can be surprisingly strange looking back at the way people, even very smart people, thought about problems before Newton.

    Note also that even in a profession dominated by messy problems which resist mathematical formalization (perhaps being a lawyer, or managing an uncomputerized retail outlet) it can be surprisingly handy to understand more formal problems. E.g., the world is full of other people’s digested summaries claiming to provide useful information (such as marketing and psych research). It’s easier to separate valid information from junk science if you have some understanding of statistics and noise.

    If you’re looking for something to cut, I’d nominate (almost all of traditional) trigonometry. Teach the definition of sine and cosine, do a few problems involving looking up values, then move on. Leave any exploration of trig identities until after students know enough calculus that they are naturally comfortable with e-to-the-i-x; a lot of things are easier in that representation. Cutting most of traditional geometry might also be OK with me; people who need geometry itself might learn it on the fly, and people who need to get comfortable with axioms and proofs could learn it in some other course. But algebra-as-prereq-for-calculus is surprisingly broadly applicable and very powerful. (Unless taught sufficiently poorly and/or stubbornly ignored, in which case of course it’s useless.)

  69. Jon Says:

    (First time poster on Scott’s blog)

    I sympathize with the Lament, but there were too many points at which I found myself deviating from it, and saying “That’s not how I learned it”, “It wasn’t *that* bad”, or “How else would you do it?”. His vision is inspiring but it cannot, nor should it, eliminate rote learning and redundant exercises entirely. To his credit, he wasn’t arguing for the elimination of the other paradigms, just their most useless aspects.

    I am a computer scientist, not a mathematician, so there was a lot mentioned in the Lament that I haven’t even thought of since high school. Formal logic and discrete math is directly relevant to my field, but I’ve all but forgotten much of the calculus, trigonometry, and geometry I learned at the high school level. Therefore, I’m not in a position to know what aspects of the traditional curriculum are unique artifacts of K-12 education and outside the scope of “real math”.

    But looking back, I think the best thing we can do to improve the quality of math education, besides ensuring that we accommodate diverse skill levels, is to make sure the teachers know their subject. As a good student, nothing frustrated me more, or made me more painfully aware of the artificial environment of a classroom with a meaningless curriculum, than when a teacher said something I *knew* to be wrong.

    The example that stands out in my mind after all these years was one time in seventh grade when we were taught that you need three points to uniquely determine a two-dimensional plane in three-dimensional space. The teacher justified this by making some kind of cute argument about containing a “box” with three points or else somehow the plane would leak (I’m probably butchering an already ridiculous explanation). On the test, I simply said that the plane could “pivot” along a point or line if fewer than three points were used, and was marked wrong for not adhering to her nonsensical statement.

    There was also an issue in tenth grade with a teacher who swore that subtracting from |x| within the abs function (e.g., |x-2|) would lower the graph on the y axis rather than shift it to the right. We had to break out the graphing calculators when I dissented on that one – I got to be a smug self-righteous bastard about it afterwards.

    In my defense, I was pissed that I was stuck with this teacher in an honors classroom, and she had done other things to annoy me. Searching my memory: I was marked wrong for simplifying (by combining like terms) a product of two polynomials that I was to expand, since when factored out they yielded the simplified version of the original product instead of the exact terms verbatim. I also failed an entire test for leaving out redundant parentheses, even where it would be nonsensical to interpret the expression any other way (like reordering unary operators such as square-root and squared). But most egregious was when opportunities to explain the mathematical rationale behind certain relationships was simply missed, so that I had to fill in these gaps myself and feel damn lucky to have seen it.

    All that was a rant about two particular teachers, however, and not the system as a whole. Hence why I believe that the system as a whole does not need to be scrapped the way Lockhardt describes, but just reinvigorated with more and better trained teachers.

  70. Lucas Rockwell Says:

    I will have to read the paper in more detail, but I have an extremely simple reason why math education in the US is so hard: We use the English system of measurement. Most students in grade school associate math with this horrific measurement system, and once that happens, there is almost no chance to engage students in what math is really all about.

    You want people in the US to embrace math? Then start on a campaign to scrap our horrific measurement system, a system to which logic and simple math can not be applied. In the metric system all you need to know is it is base 10. With the English system you must *know* that there are 12 inches in a foot, and that there are 3 feet in a yard. How many feet in a mile? There is NO WAY to logically derive this information, one must KNOW it! How may teaspoons in a tablespoon? Can math help us with that? Nope, one must know the answer.

    Sorry, I hope I am not ranting, but this has been a burr in my side for years. Yet it seems that no one ever talks about this aspect when talking about why people in the US are so adverse to math. I guess I should just write my own article about it…

  71. Greg Kuperberg Says:

    In the end, Lockhardt’s lament is subversive, angry, and radical … but if you know anything about math and anything about K-12 “education” (at least in the United States), I defy you to read and find a single sentence that isn’t permeated, suffused, soaked, and encrusted with truth.

    The topic is certainly of interest to me and I think that I know at least something about it. Our two children are enrolled in public school right now and we certainly want them to get the right education in mathematics. They also study music, so I have seen the other side of one of Lockhardt’s analogies.

    Lockhardt’s essay leaves me cold, largely for the same reasons as Scott Carnahan gives. My reading is that all of his sentences are soaked with half-truths. Yes, I know that mathematics is an art, but there is more to it than that.

    Making mathematics creative was the mantra of an attempted mathematics education reform in California in the mid 1990s that was mostly, but not entirely, stopped in its tracks by the Stanford mathematics department. In fact I had several conversations with teachers who rather liked the reform math textbooks and the still-born state math education standards. They rejected the textbooks that we wanted for our children as “drill and kill”.

    The mistake in the reform textbooks is that mathematics is not just an art, it’s an art with established methods. The textbooks were reluctant to explain the division algorithm, the Euclidean algorithm, etc. The students were supposed to creatively discover their own methods. They had a lot less time to learn anyone else’s methods. The end result was to slow down the material substantially. Another common result was that all students were winners no matter how little they learned, except for those students who do not draw enough pictures or did not collaborate enough with other students.

    Yes, mathematics is an art. Violin is also an art, and so is tennis. But you won’t learn them just by having someone hand you a violin or a tennis racket and asking you to be creative. Art with no methods is as bad as artless methods.

    As I review Lockhardt’s essay on this point, it has an insidious element of self-disavowal. The students are all supposed to create and discover. But what if they discover very little, because they had too little practice and got too little explanation? The essay has a ready answer for that; it must be because the curriculum was merely “cutesy”. The essay also rejects anything called reform, even as it demands reform.

    One principle that I think is very important, and which the California reform math movement resisted, is to draw a line between how to teach and what to teach. I do not think of myself as a practiced expert in how to teach. Even if I did, I wouldn’t want to micromanage teachers; I would want them to have some of the same freedom that Lockhardt wants students to have. But I do have things to say about what to teach. Even if I didn’t, I would be suspicious of a teaching establishment that wants to win by moving the goalpost.

    It seems to me that Lockhardt’s essay also lumps together what to teach with how to teach. Again, I assume that Lockhardt himself is a good math teacher. (And again, I personally am not universally popular as one.) But his essay is ready to be exploited by teachers who carry their own goalposts and reject all others. In fact he says as much, when he declares that mathematics should not be mandatory and also rails against standardized testing.

    It is true that some efforts such as No Child Left Behind have taken standardized testing too far. I don’t like the trend of using yardsticks as sticks to beat public schools. It’s also wrong to view standardized tests as the end purpose of education. But some standardized tests — for instance, the SAT, the GRE, and AP tests — are useful tools, and I don’t think that they should be abolished.

    Finally, like Scott Carnahan, I do not like the arrogant tone of the essay. Suppose that you have practiced hard for an hour to learn a new violin piece. Would you like to be told, “My dear fellow, what you think of as ‘music’ isn’t music at all. Music is more than playing a sequence of notes like a robot. Music is an art!” That is so cliched that it is closer to an insult than real advice. I would be ready to brain the kibbitzer with the violin.

  72. Michael Maxwell Says:

    Greg Kuperberg’s comment on violins makes me feel compelled to shoot off my mouth again… Playing a musical instrument and composing music are two very different things. I suspect that what most of us do with math is much more akin to playing a violin than it is to composing music, whereas what the original essay is about is composing.

    (Disclaimer: I got a D in music in eighth grade, which probably means I don’t know what I’m talking about.)

  73. Scott Says:

    Greg: I happen to be a big fan of standardized testing—I think that if you have to test people, it’s certainly fairer than unstandardized, whatever-the-teacher-wants-it-to-be testing.

    I’m also vehemently against wishy-washy curricula where the teacher lets students “discover everything on their own,” with almost no feedback on whether they’re right or wrong. However, I didn’t take Lockhardt to be advocating that. To be fair, he could have stressed more the teacher’s role in correcting and evaluating the student’s ideas, and gently guiding the student toward an understanding that’s mathematically correct (even if it’s not exactly what the teacher had in mind originally).

  74. Joe Shipman Says:

    Many people will eventually find useful some of the math they learn in school, as well as some of the history, some of the biology, etc. But that is not the point of education.

    The point of education is to make you smarter — making you know more is just a byproduct. Good English courses (or “Language Arts as they now call them) will make you a better reader and writer, even if you never see “The Scarlet Letter” again in your life. Good math courses will make you able to think logically and clearly and systematically. The true failure of math (and science) education is that students get through high school without any ability to distinguish correct from fallacious reasoning, or to identify the fundamental relationships and dependencies in a complex situation, leaving them less competent at managing their lives and highly vulnerable to slick advertising and political demagoguery.

  75. Arun Says:

    I hold a Masters in Engineering and have always excelled in Maths exams – but I have never *understood* Mathematics even though (trust you me!) I have tried.

    When learning calculus, I remember puzzling over just how the ratio of two vanishingly small numbers can make sense when we know that division by zero is an invalid operation.

    This only continued into college. What prompted Fourier to come up of all things with his weird series? Why are the cross product, curls and gradients defined the way they’re defined? Why do we have all these theorems after theorems in abstract algebra and real analysis? Don’t even get me started on control theory, Laplace transforms, and later linear systems.

    That said, Lockhardt has wasted 25 pages of virtual paper and quite a bit of his and our time. He laments self-righteously, but offers no solution.

  76. Deane Says:

    I don’t like calling mathematics an “art”. I like calling it a “craft” “Art” implies a need for creativity and imagination. “Craft” implies a skill that almost anyone can learn to do competently, and that a few people can raise to an art.

    There is no question that anyone learning mathematics *must* learn certain basic rote skills, starting with arithmetic and the rules of deductive logic (which, sadly, is almost never taught explicitly). This is no different from saying that if you want to play basketball, you have to know how to dribble and pass.

    But no one should mistake those rote skills as being math itself. The craft of mathematics is the deployment of these basic skills to analyze and maybe even answer certain types of questions that arise in many different situations. Just as playing basketball means not just dribbling and passing a ball but the use of these skills to try to win a game.

    It is this craft that we mathematicians teach very poorly. We do it too much by lecturing and having students watch us do examples. Think about how well you can learn basketball by that means. Why does anyone think you can learn math that way?

    You learn to play basketball by mixing constant drills of basic skills with repeated efforts to play the game. The coach rarely needs to actually play himself; he just provides guidance as the “students” try to play the game themselves.

    Math can be taught quite effectively exactly the same way. We did it at Polytechnic for a while, and it was the most fun I have ever had teaching mathematics.

    And it’s not as if my views are unique. All of this is well known to at least some math education experts and even some mathematicians. But most mathematicians just can’t be bothered and stick stubbornly to the lecture/recitation format. It’s not a lot better at the secondary school level.

  77. Greg Kuperberg Says:

    I happen to be a big fan of standardized testing

    I’m glad that you agree with me, but you invited us to evaluate Lockhart’s essay, not your opinions. The essay is consistently against standardized testing. Could you perhaps be reading what you want to read in this essay at certain points?

    I’m also vehemently against wishy-washy curricula where the teacher lets students “discover everything on their own,” with almost no feedback on whether they’re right or wrong. However, I didn’t take Lockhart to be advocating that.

    Here you have a point, because Lockhart says, “So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.” Here then is another element of self-disavowal in this essay: Lockhardt doesn’t really want to take anything away from the mathematics curriculum, he mainly wants more.

    In fact, the California reform math movement similarly claimed a middle position. It wasn’t that they were against formulas or even a certain amount of drill; they merely wanted to balance it with creativity and student explorations. But in practice, there is no way to have more of one thing without less of another. The reform textbooks spend a lot of time on creativity and not enough time on standard formulas. They undermine balance in the name of restoring it.

    Granted, Lockhart was not part of the California math reform movement; he’s now in New York. But that brings me to the most discouraging disavowal of all in his essay: “Of course what I’m suggesting is impossible for a number of reasons. Even putting aside the fact that statewide curricula and standardized tests virtually eliminate teacher autonomy, I doubt that most teachers even want to have such an intense relationship with their students. It requires too much vulnerability and too much responsibility— in short, it’s too much work!” So he’s not really associated with any failed (or successful) experiment. He sarcastically dismisses the goal of providing useful advice; the minions of the system won’t stand for his great ideas anyway. (And notice the recurrent jab at standardized tests.)

    Maybe the thinking is that an essay like this is a useful manifesto for rallying the troops. Again, I am a parent of children in the system, but this essay doesn’t make me feel like marching. Sure, I recognize and partly share many of the concerns that he raises. But somehow the essay smacks of pseudo-wisdom.

    In fact I can be more specific about that. Both of our kids had very good math teachers this past year (in algebra and calculus, respectively). But these teachers did not succeed by following Lockhart’s advice. They did not ditch the standards, and they did not lead an Arnold-Ross-style revolution in the classroom. They also didn’t spend any time bashing the system. Instead, they succeeded because they know the math fairly well, and they respect both the students and the standards. That was a foundation that allowed both of these teachers some room for challenging, creative math problems.

  78. Bram Cohen Says:

    Greg, there’s plenty of very concrete stuff in Lockhart’s essay about the current mathematics curriculum, mostly having to do with meaningless crud which is in there – obsolete or irrelevant concepts, useless terminology, and pointless odes to formality. If all that crud were dropped, it would be far more pleasant for everyone involved. Calculus was taught for centuries before the formal definition of limit was proposed, why must we force the subtle details of that definition upon every nonmathematician who takes a calculus class?

    I agree with the previous commenter who pointed out that whatever benefits there are to teaching ‘formal’ mathematics are conveyed far better by having people take a programming class. One of the reasons I have hardly any published papers is that I simply don’t understand what the standards for ‘formal’ proofs are – it clearly isn’t computer checkability, and it’s some higher bar than merely being convincing, but everybody acts like its something sacrosanct rather than a mere social convention. Programming teaches rigorous thought much more judiciously, and is a useful skill in and of itself.

  79. Jonathan Vos Post Says:

    I mostly agree with Paul Lockhardt. As evidence for his essay:


    “Mathematics—using abstract symbols to describe, order, explain, and predict—has become essential to human existence.”
    [Framework, p.4]

    I simply disagree with this definition. It is shocking to see such foundational confusion in the opening sentence.

    This comes from

    In general, these is much to praise and much to provoke annoyance in Mathematics Framework 2005. Mathematics Framework for California Public Schools, Kindergarten Through Grade Twelve (2005). Complete Mathematics Framework …
    ISBN 0-8011-1474-8

    Developed by the
    Curriculum Development and Supplemental
    Materials Commission

    Adopted by the
    California State Board of Education

    Published by the
    California Department of Education

  80. Jair Says:

    I confess, I did not read the previous 76 comments, so if this point has been mentioned before than I apologize.

    It seems to me many of the author’s points apply equally to college-level mathematics as to K-12 education. Especially the following:

    “The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense
    when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.”

    If only I could convince more math book authors to realize this. It seems that mathematical ideas go through a long, arduous process of development until they are sophisticated and deep enough to fill a four hundred page book, at which point an author takes these ideas and writes them down in precisely the opposite order they were conceived in. The introductory chapter is a list of mysterious and apparently entirely unmotivated definitions while succeeding chapters coalesce into the ideas that mathematicians had from the start. The goal is to present the ideas as succinctly and logically as possible, but the result is a lot of students who get kicked right in the intellectual nuts at the very beginning of a course, and some never recover. I can’t think of a worse way to begin a book on topology, for example, than defining a topology as a set of sets closed under arbitrary union and finite intersection. How much more removed from the original geometric ideas can you get?

    The problem, I think, is that the authors aim to prove rather than to communicate. Everything is written in strict logical order, viewing the reader as a kind of high-level automated proof-checker. It’s easy enough to check these kind of proofs, but it doesn’t give a lot of insight into the author’s thinking. It’s a one-way function. Why do authors insist on beginning a proof with something like, “Let ε = min({δ, δ^2 /3})…” when surely this was the LAST thing the author scratched on the blackboard when he finished the proof a minute earlier?

    Math books should be written, if not in the order the math was actually created, then in the order in which the ideal mathematician might conceivably discover it. I think the reason the advanced books are more successful than K-12 books is because their audience has more patience and discipline to wait for the good bits where everything suddenly makes sense.

    Of course, many math books are better than this, and many are a goldmine of elegant ideas underneath a hard surface of unmotivated definitions and seemingly meaningless theorems. But they sure make for some harrowing reading.

  81. ~M Says:

    I got the point of Lockhardt’s article and I enjoyed the concept of it. However pointing out the wrong ways we teach and inspire by writing such a mind numbing repetitive article is quite ironic. His math might be beautiful but the article seems to typify all the things he says we’re doing wrong as the good points get lost in a literary misuse of teaching and inspiration. In that sense the article belies some hypocrisy. This could have been written succinctly in probably 5 pages were it not for the repetitious flogging of a dead horse. Seriously, how many “clever” analogies and reworded quips do you need to rehash the same couple points? ~M

  82. Paul Beame Says:

    Scott: I think you’ve confused “New” math and “Reform” math, both of which contrast with “Traditional” or “Drill-oriented” math. There is a big difference in the history here. Despite many complaints in the comments here about the lack of involvement of practicing mathematicians in the development of math curricula, the “New” math actually was developed with a bunch of involvement of university-level mathematicians. Its failure in many ways led to the “reform” movement but it also gave mathematicians a bad name w.r.t. development of school curricula.

    I experienced the change-over to the “new” math in the late 1960’s. Some aspects were great: Even starting in second grade they had “fill-in-the-box” questions that made you think about what operations meant and that made the transition to algebra much easier. (2 plus box gives you 5. Fill in the box to make it true.) It was not just a matter of executing the memorized algorithms, you had to know why you were doing some operations (immortalized in the Tom Lehrer song). There unfortunately were some horrible aspects: The whole enterprise bore the same relation to the elementary school curriculum that Bourbaki did for the rest of math. In 5th grade we were supposed to memorize the axioms of set theory and rings, distinguish between “whole” and “natural” numbers (“natural numbers” started at 1 whereas “whole” numbers started at 0) and between cardinal and ordinal numbers. (BTW: I notiice that my kids’ high school texts are very traditional but retain some of the axiomatic aspects that I associated with the new math.)

    Speaking of high-school math, I find the standard US system of compartmentalizing math topics to be very strange. I was taught in a Canadian high-school where there was a build-up of material from the various topics in roughly 6-week chunks and the level slowly grew in sophistication. My kids did a very traditional Algebra I, Algebra II, Geometry, Precal (Trig etc.) in sequence before Calculus. The notion that you can simply forget one of these topics because you won’t see any of it next year seems very strange. I was particularly appalled by Algebra II: There was a lot of material about finding various points on quadratics or rational functions, and about trig functions, logarithms and exponentials about which students had little maturity and no motivation for beyond a couple of word problems at the end of an entire chapter. They simply had not seen enough other math to make any of it meaningful to them (and it wasn’t always clear that their math teachers had a better idea about that meaning).

    The Lockhart essay is interesting and has some good aspects but completely misses the point in some other ways: Part of doing mathematics is problem-solving – not just discovering and proving patterns. (Without problem solving we might as well require only as much mathematics of students as we do art.) Traditional approaches are focused on problem solving but where they often fail is that they encourage a kind of blind pattern-matching of problems to a very small number of template solution methods without any deeper understanding (or any true math). (I recall a classic study in which students were given word problems that happened to mention extraneous and useless numbers and most students tried to do operations on those useless numbers just because they were there in places where they might be useful for other problems.)

    I had some excellent math teachers in high school. I am sure that they could have made the “reform math” work well, but I suspect that in the hands of weak teachers it would fare much worse even than the traditional drill. Despite my excellent math teachers what really made math appealing for me were the math contests run by the University of Waterloo and by the MAA. Solving these problems did feel like doing math does today. One of the main things that I found attractive is that these contests involve “out-of-the-box” problems that deliberately do not fit the standard kinds of problem templates.

  83. Johan Says:

    “Calculus was taught for centuries before the formal definition of limit was proposed, why must we force the subtle details of that definition upon every nonmathematician who takes a calculus class?”

    I know Lockhardt says something similar but it does not appear to make any sense given his basic idea.

    If we want to teach engineers and physicists to be able to apply math without understanding it then we can teach calculus without limits.

    But for mathematicians, surely epsilon and delta are crucially important. In fact, calculus without formal limits can be hardly be called real math.

  84. Scott Says:

    Could you perhaps be reading what you want to read in this essay at certain points?

    Greg: You’re right, I probably am. Like a mainstream feminist reading Andrea Dworkin, or a mainstream liberal reading Chomsky, etc., I might not agree with everything Lockhardt says, but the urgency and ferocity of his writing resonates with me, as it apparently didn’t with you. That’s why I said his sentences were “permeated, suffused, soaked, and encrusted with truth,” rather than saying they were true. 🙂

  85. John Sidles Says:

    From PJ’s post: [My math professor explained] You need to learn X so you can learn Y, which you need to learn so you can learn Z, which ….” — a sequence which did not, unfortunately, ever end at some goal-location where I wanted or expected to be. …

    What the hell is this stuff FOR? Algebra, Geometry, Calculus, Trigonometry? Does any normal human being outside highly specialized technical jobs use this stuff?

    Without expecting or requiring a consensus (which would be as infeasible and undesirable in mathematics as it is in any democratic community) doesn’t PJ’s observation deserve a more in-depth response than it has gotten?

    To start, people in “highly specialized technical jobs” definitely are “normal human beings.” In this respect mathematics is not essentially different from science, law, engineering, medicine, history, writing, painting, or music … the highest-level practitioners of all of these arts are a mixed lot that includes plenty of richly human personalities.

    The false perception that mathematics is learned by ascending a rigid ladder of logical deduction is an artifact of the (bad) ways that mathematics is taught, rather than a reflection of the (good) ways that mathematics is learned and practioners.

    As for what these professions are for, at the highest level their purpose is generally radical—to change the world. This is the aspect that is omitted at the grade-school and high-school level … it’s too controversial for the public schools.

    Our QSE Group maintains a shelf of literature that focuses on the world-changing aspect of math, science, engineering, and literature. The best concordance we have found to this literature is Jonathan Israel’s two-volume (so far) history Radical Enlightenment and Enlightenment Contested … these accounts are an effective antidote to the pablum that passes for intellectual history in the schools.

    Conclusion: fixing historical education would go a long way toward fixing mathematical education.

  86. Jerry Llevada Says:

    I have heard most of the discussion before. Sometimes it is a different take, but the similarities are striking. PJ is a classical example of somebody at the window watching from the outside. Please PJ, visit us at, or write to me I want to bring you and your daughter inside, I want you to learn as a participant.

    Because I had to go through MechEng with a slide rule and punch cards to a computer I was not allowed to see, I can perhaps shed some light into this “what is this algebra good for.”

    First, most math is designed for a need; secondly, math for the sake of math is almost a hobby. I will assume here that the former is what we are talking about. Let’s keep both separate!

    Now, understand this: The world has only two types of movements. We either go straight (linearly) or wiggling (non-linearly). And we do it three-dimensionally.

    Algebra I is how we start learning about how to get a handle on the variability that surround us, but to keep it simple, we do it in two dimensions only. Approx. 60% of it is dedicated to teach linearity, the other 40% gets us started on non-linearity. Then you ask: What is geometry doing between the two algebras? Geometry is there because geometry is easy (compared to the analysis needed for non-linearity in algebra II) for the young mind. Many students get hooked on math at this level, but only if the instructor is above par. Geometry gets you started on the shapes you will later analyze in algebra II and above. You see, the graphical nature of geometry is its best attribute, but it’s also its downfall. Some students that go to great achievements in physics do poorly in HS geometry.

    In algebra II, all those non-linear shapes we learned in geometry are now set to equations and we continue to see how non-linear functions with square roots, exponents, and absolute values can be manipulated. Pre-cal is more of the same, where applications to what is learned in algebra II is expanded (trig., analytic geo., logs…).

    And this brings us to calculus (because this is about HS math and eventually I have to stop, this the last topic).
    Calculus has been improperly named, every mathematician knows that, but other names would be too long. Calculus has two main functions: Finding the slope of points on any line, and finding the area of shapes. That is all you do.

    Everything that is taught in HS math has a purpose, and just because you don’t understand it doesn’t mean it does not have a purpose. It all comes down to who you land as an instructor. The more math you know (other things being equal), the higher your probability for success. If you don’t believe me, see the oodles of dough that former physics majors make in Wall S. before and after the collapse. If you cannot hit a baseball on steroids, stick to math.

    The beautiful thing about math is that you can get off at any stage and still feel good about it. In HS students really can’t tell where they’re heading. We may hate history, dislike religion (I was educated by Jesuits), be bored with science…, so what of it? You have to accept guidance because education is what makes you a better person, and that is precisely the reason to get a universal education. Nothing entrenched, focused, like Japanese. You cannot learn a language one hour a week, there is no theory, no base. In HS we should not be teaching applications, just general knowledge to see where your appetite will take you.
    The four main topics in HS are: Math, Language Arts (to us is English), Science, and Social Studies, and that is a lot. Moon watching and sun worshiping is for adults. If you don’t have an appetite for math in HS, you don’t have to do it, but forget about college. If you need or like algebra as an adult, take it then; it’s never too late.

    Sorry, but today is father’s day and my family wants me to take them out to eat. More later.

  87. Vladimir Levin Says:

    I don’t agree with all the details of the lament. For example, I am not the kind of person who would be drawn to questions about just the relationship between two numbers; I would be much more interested in the fact that math be *applied* to give quantitative answers to many different kinds of interesting questions. However, I agree with Scott that the lament has the ring of truth in the sense that it bemoans an approach to education that currently emphasizes rote learning over real understanding. Teaching any subject should include the following:

    1) A teacher with a solid and authentic grasp of the material, i.e. does not just stay one lesson ahead of the class
    2) A teacher with genuine enthusiasm for his/her subject
    3) Interaction with the class, not just having students passively absorb information
    4) Presenting a variety of interesting problems and applications
    5) Providing a context that demonstrates how different concepts can be connected – within the same subject and even across subjects so that students develop a rich conceptual understanding they can apply to problems they haven’t seen before
    6) Offering some historial background that also puts the material in context

  88. Greg Kuperberg Says:

    Like a mainstream feminist reading Andrea Dworkin, or a mainstream liberal reading Chomsky, etc., I might not agree with everything Lockhardt says, but the urgency and ferocity of his writing resonates with me, as it apparently didn’t with you.

    By some standards I am a mainstream liberal and a mainstream feminist. With all three of these authors, I feel an overdose of ferocity. I also perceive the urgency as partly real and partly counterfeit. Wouldn’t you agree in the case of your other two examples?

    Calculus was taught for centuries before the formal definition of limit was proposed, why must we force the subtle details of that definition upon every non-mathematician who takes a calculus class?

    I don’t know that we necessarily should, nor that we always do. Many universities teach a course called “Short Calculus”, in which the formal definition of a limit is either skipped entirely or reduced to a bare mention.

    But on the other hand, Cauchy devised the epsilon-delta definition of a limit expressly as an explanation for college students. He published his definition in 1821, which was 188 years ago; while at the time, modern calculus was at most 155 years old. This definition may be subtle, but it is not malicious. And even as math education, it goes back almost two centuries.

    Lockhart has a grab bag of other complaints. I completely agree with some of them — I rejected statement-reason tables when I first saw them in geometry and I still do. But some of his other complaints don’t make much sense to me. For instance, yes, mixed and improper fractions are two different notations; the fractions themselves are the same. But this terminology has been useful when we talk about mathematics with our children. I don’t see the problem with defining these terms.

  89. Bram Cohen Says:

    Johan, ‘informal’ calculus keeps the epsilons but drops the deltas. In some sense its less mathy than full formality, but it was all Leibniz knew, and it would be silly to say he wasn’t doing ‘real’ math. Sure you can make the formal stuff available to students who are interested, but that should only be for the ones who show a direct interest in it.

    Aside from the too much meaningless crud and not enough recreational math problem, the other big issue with school mathematics is that the teachers are hopelessly unqualified. I had a high school math teacher teaching honors math at one of the most elite high schools in the country and she somehow took an entire semester to cover row reduction in linear algebra. She had no understanding of determinants other than the recursive algorithm for calculating them, and subjected the students to ungodly hours of homework calculating them the long way (which I never did, and was always on the verge of flunking as a result). She one time asked if you were to have three white and four black marbles in a bag and remove them WITH replacement, what’s the probability of the fourth one being white, and had considerable skepticism when I did the problem in no time flat using the obvious calculation. As of just a few years ago, she was still teaching there. If literally some of the best math students in the entire country are being taught by someone who couldn’t get through an undergraduate mathematics program, what does that say about the rest of the schools?

    The giant elephant in the room is that that problem won’t get better until someone breaks the backs of the teachers’s unions. The problem is that for any politician to say that would be career suicide.

  90. Jerry Llevada Says:

    It’s not an elephant, it is a gorilla. And the gorilla is all of us, our wants and desires. But forget about the managerie and stick to reality. Unions are a necessary evil to counter school boards, period. We receive the quality we request. Bram, if you went to a private school, you probably enjoyed smaller classes, but that wasn’t the answer. You had a lousy teacher, but you didn’t do anything about it. It is your fault. Get involved. Good, certified, math teacher are abundant, but your alma mater doesn’t want to pay for them. So the net result is that you must do it yourself, which is what you are doing, anyway. By the way, most of you are not AVERAGE, you are the exception. And it will stay that way until you die.

  91. John Sidles Says:

    Bram says: The problem [with math education] won’t get better until someone breaks the backs of the teachers’ unions. The problem is that for any politician to say that would be career suicide.

    With respect, Brian, your assertion explifies Mencken’s aphorism “For every problem, there is a solution that is simple, neat, and wrong.”

    My wife served for two years on the Seattle School Board, and during these years our family definitely learned a ton about educational facts-on-the-ground (and no, neither of us have ever been members of a teachers’ union).

    The inconvenient truth is, the challenges of public education—in math or any other area—aren’t easy to meet, simple to meet, or cheap to meet … and too-simple ideology-driven Mencken-esque educational reforms have (historically) been as likely to make things worse as better.

    “The best teacher is not the one who knows most but the one who is most capable of reducing knowledge to that simple compound of the obvious and wonderful. . .” (Mencken)

    My wife asks me to pass on the message that (surprisingly many) elementary teachers are committed agents of the enlightenment whose most radical teachings are lessons in conservation, democracy, individual enterprise, and tolerance.

    Gosh … a peaceful, prosperous planet with a healthy global ecosystem … that would be revolutionary.

  92. Zephir Says:

    Mathematics can describe only deterministic things completely and what we know about quantum mechanics world is, it’s indeterminism is fundamental and experimentally proven to agree with Bell’s theorem. In addition, by Gödel’s theorems even the formal math as such has it’s own limits, as it cannot describe itself completely. By AWT such similarity isn’t accidental, because number concept is based on countable objects, i.e. concept of colliding particles fulfilling Fermi-Dirac statistics. Formal math has nothing very much to say about bosons, which are violating Peano algebra like ripples at water surface: they cannot be counted. In my opinion formal math isn’t equivalent, but subgroup of physical reality, which is subgroup of observable world. Not all things which we can feel we can observe, not all things, which can be observed (a physical reality) can measured, not all things which we can measure can be measured in reproducible way, not all things, which we can measure can be computed, not all things, which can be computed can be falsified. In addition, not all things, which can be computed numerically can be derived in formal way (for example the formal model of N gravitational bodies). Formal math is in fact quite poor subclass of observable reality – which shouldn’t prohibit us in extending of its relevance scope, whenever possible.

  93. Jonathan Vos Post Says:

    Mathematics is both a Meditative Art and is a Performing Art. As performance, it is only appreciated by other artists of the genre (unlike painting or film or music where anyone off the street has an aesthetic reaction). The purpose of Mathematics is ENLIGHTENMENT and insight, either of an application represented (as in the Sciences), or of oneself (as meta-cognition), or of the glory of pure structure and symmetry and pattern of abstract ideas. A curriculum that prevents insight is a dungeon, and a torture chamber for young minds.

    We are all born creative geniuses. We are all born with a novel, or painting, or opera, or skyscraper, or theorem bursting to be born. Bad neighborhoods, bad parenting, bad management, bad schooling crushes that out of a majority of humans. We must hold tight to our stem cell genius, our innate creativity, and create because we are human. Money doesn’t enter into this, except as a side-effect of establishing Professionalism in the mechanism of distribution.

  94. KristofU Says:

    The document opened my eyes and provided me with a completely different perspective on math.
    An aha-erlebnis if I ever had one.

  95. Rachel Says:

    With regard to point 2 — it is also worth noting that while it’s not impossible, it’s culturally more difficult for girls to become aware of what mathematics really is and why they might like it than boys (math as taught in k-12 school is dull and uninteresting, which fits into the cultural narrative that girls think math is dull and uninteresting).

  96. Paul Says:

    “Fifty years of studies and empirical data about K-12 education show that the “direct instruction” method (lecture, drill, etc.) consistently imparts knowledge more effectively than the “discovery learning” method (exploration, questions, participation, etc.). (This isn’t just true of math, but all fields of education.)”

    2 things:
    1. Can you site some of these studies or give a link that does
    2. Rote memorization may impart knowledge more effectively but it will not impart love of the topic, which is essential to real learning.

  97. Bram Cohen Says:

    Jerry, I was the second-best math team student in my year in the entire city of new york, and I complained bitterly about having that teacher. If you think I caused being subjected to that teacher to myself, you should pay a visit to planet reality.

  98. sigma Says:

    For Lockhardt: Sure, there are some pretty patterns in mathematics. It’s important to understand the role of discovery in mathematics.

    But Lockhardt goes too far, beyond reason, out of control. Mostly the pretty patterns in mathematics are not visible to students in K-6. There are serious practical reasons to teach decent abilities with arithmetic and simple applications in K-6.

    In grades 7-12, the problem is the teachers: Nearly none of them know enough to make good use of the learning abilities of the students of those ages.

    There is a fundamental problem: Our society wants children ages 5-17 under some adult supervision and off the streets and to leave the parents free to neglect their children. Instead, children are good at learning at the knees of their parents, 24 x 7.

    Many women are not very busy as wives and mothers or in valuable careers so can be K-12 babysitters, uh, ‘teachers’. So, mostly we just throw away the time, money, and effort of both the students and the teachers waiting for the students to be old enough to need less adult supervision. That gigantic waste is what we call K-12.

    That’s basically all our society knows to do.

    For college, that’s mostly just for teaching teachers or pre-med or pre-law. Again, our society doesn’t know much else what to do with the time, money, and effort.

    But there is some interesting, and possibly useful, material between the covers of some books or, now, as PDF files on the Internet. For math, with TeX, finally now we can type, and many more authors do!

    If a child wants to learn piano, violin, opera singing, music composition, sculpture, computer programming, restaurant cooking, auto repair, landscape architecture, carpentry, math, etc., fine, but the K-12 educational system is from nearly irrelevant down to seriously counterproductive. Nearly no teacher in K-12 in the US knows enough about math to help a student who wants to learn.

    Students: Don’t take any advice on how to learn math from anyone other than a full professor at a top research university. The professor can be in pure or applied math or mathematical physics. ASAP get past any materials intended for K-12 and get to the best college materials.

    Someone should write some materials to bridge quickly between sixth grade arithmetic and a respected college calculus text. That is, just junk all of 7-12 math educational materials. I looked at AP calculus and was horrified: Those materials look like they were written by assistants to the devil trying to kill calculus. looked at some North Carolina materials on probability (where I’ve published peer-reviewed research) and linear programming (I’ve taught in graduate school and an elementary part of my specialization in optimization where I’ve published peer-reviewed research) and just screamed at the wild incompetence. No student should touch that sewage. There’s just nearly no competent content anywhere in K-12.

    College can help students learn math, but in graduate school sometimes the explicit message is that the students are expected to learn the basic material on their own and the courses are just introductions to research. So, net, to learn math, have to do a lot of learning alone or nearly so.

    K-12 is not just a big waste but also highly destructive: The ratio of adults to children is far too low so that mostly the children are ‘learning’ from their peers. The result is full of nastiness, cruelty, ridiculous conflicts, bullying and some just horribly degenerate, degrading, and destructive ‘pop’ culture. Well educated parents should turn off their TVs, carefully supervise any interactions with children from public schools, and home-school their children, even if mostly the children apprentice in the family business. For the basic lessons, some good Internet systems would be terrific: Learn the basics there and for the rest have lessons, say, violin or piano, and, especially, at the knee of the parents.

    US K-12 public education is a cesspool. Learn math there? HA!

  99. Arun Says:

    I think Mathematicians should come up with documents that describe the kind of education *they* would have loved to have had. How would you do it if you had to do it all over again?

    The methodologies should use as much scalable resources as possible. For instance, it’s not easy to get great teachers and plenty of great teachers have better things to do than spend time at a low-paying job.

    Again, Mathematicians should do some introspection as they are doing Mathematics and describe the dynamics of problem solving they observe within them. Don’t just give us fish. Instead, teach us to fish. Just imagine how *richer* our lives would have been had we had intimate knowledge of the thought processes of the likes of Newton, Euler, Gauss and Riemann.

    We need more Jaques Hadamards observing themselves at work in this world.

  100. Bram Cohen Says:

    Arun: Scott answered your question right at the top – Martin Gardner books and math team.

  101. Patrick Cahalan Says:

    @ John Sidles (#58)

    You can certify excellent teachers, and you can certify excellent mathematicians.. you just can’t do it with a standardized test.

    Teaching is a wicked problem. You have to be able to do about seven things at once, and only one of them is “get knowledge out of your own head and into the students’ noggins”. Education experts don’t know how to test for this on a standardized test, but if you get 5 education experts behind a one way mirror with a teacher and a classroom on the other side, they’ll be able to rate that teacher pretty accurately.

    One of the things that complicates teacher evaluation, though, is that scaling is really difficult. There are lots and lots of bad teachers. There are lots of mediocre teachers (ones who might teach one subject or class of subjects to one type of thinker really well, and can’t do much else). There are many decent teachers (ones who can teach a class of subjects to multiple types of thinkers). There are not very many excellent ones (ones who can teach a class of subjects to multiple types of thinkers, and can adjust teaching methodologies according to which types of thinkers are absorbing what bits of knowledge on the fly, during a class, while eliminating disruptions and keeping everyone engaged, even those that aren’t actively learning right that second). Seniority has very little correlation with making the jump more than one classification: training can turn crappy teachers into mediocre ones, or mediocre ones into decent ones, and every once in a while a decent one into an excellent one, but it’s almost impossible for any teacher to make two of those jumps, and the decent and excellent ones don’t start at the top of the pay scale, where they belong.

    I personally can keep about 15 kids engaged and moving forward, if they’re an average distribution. Throw in 10 more… and if they’re not all pretty motivated to begin with, I’m going to start losing the edge cases. Throw in two really disruptive kids at the 15 number, I can keep ’em under control; in a class of 30 I can’t without it really impacting knowledge transfer.

    Excellent teachers are like wizards; they’re the ones people make movies about. They can simultaneously engage the disinterested kids, challenge the smart ones, keep the daydreamers focused, and keep the rowdy kids from cracking the dynamic to bits… and they can do it whether the class size is 20 or 35. The ability to surf the mood of the class and keep it *going forward* is something that’s almost impossible to train.

  102. linzel Says:

    Personally, this is not a math problem, it is a problem with the structure, design and purpose of the educational system. What is the purpose of the system? Originally to train individuals for industrialized jobs. It was not to produce thinkers but doers. Society needs both. [there may be more categories] We need to ask what we are trying to do with students. First: ask the students themselves, then their parents, then what society needs, then what they are capable of actually doing?? Not sure that order is correct except the first one. In a period of incredible technological advances I suggest curriculum CAN be individualized to a large extent. It should be process based and driven by the student. The curriculum changes as the student changes. Their entire path of learning kept within a digital learning log. Stating and explaining, why they wanted to learn something, how they decided to study the material, the sequence and structure they chose, how they demonstrated their learning, how it was built upon by the next layer of learning, how they used what they learned to impact those around them to demonstrate leadership, service, growth.
    It is the educational structure that needs changing, not simply a curriculum or pedagogy.

  103. Jerry Llevada Says:

    Look Bram, I’m a father–and grandfather–with an attitude. If I had had a son as brilliant as you, in private school, that lady wouldn’t had been your teacher. My daughter, a brilliant dancer and great organizer, went through three geometry teachers at the local public high school until she, and I, were satisfied. And although she still hates math at 28, she credits me for always having good teachers. By the way, I had the privilege of teaching the number one math student in the nation in 1998. Great kid with a tough mother. Perhaps you should had gone to public school.

  104. Bram Cohen Says:

    Jerry, I went to a public high school, Stuyvesant high school in New York City, which is the big feeder school and gets all the best students. How this particular teacher wound up teaching the honors math program there I’m not privy to exact details on, I assume she had a lot of seniority.

  105. Y Says:

    @ John Sidles (#58) and Patrick Cahalan (#97)
    It’s been asserted here that it is impossible to quantify the features of a good teacher, but with no citations or descriptions of failed studies to back the claims. This assertion is arguably false.

    In fact, there is a promising quantitative assessment that predicts teacher quality, at least among elementary school teachers. This is the work pioneered by Deborah Ball.

    Ball’s assessment, tested the assessment on a random sample of Michigan elementary school teachers, shows positive correlation between performance by teachers on the assessment and gains by students over time (how much the teacher’s students performance improved on a test that the state of Michigan often uses). It is the first ever assessment that shows positive correlation with student performance.

    The usual studies cited when discussing the difficulty of quantifying/predicting teacher quality are by Begle (1979) and Monk (1994). These looked at teacher background.

    What distinguishes Ball’s work is meticulous analysis of mathematics that teachers do on a daily basis, which is more than just the curriculum! It includes, as Patrick Cahalan phrased, “getting into the noggins” of students, which most people don’t ever need to know or have a chance to learn. The assessment is more than just careful analysis of math, as it stands on about 15 years of observation work and consultation with statisticians to ensure reliability/validity. There are several recent papers that talk about the work (one in Journal for Research in Mathematics Education (2008), and another published by the American Educational Research Association).

  106. Pat Cahalan Says:

    @ Y

    Uh, sort of disingenuous to call somebody out for not citing studies, and then not provide them yourself 🙂 I think you misunderstood me, in any event… I wasn’t saying that you couldn’t assess teachers via a standardized *assessment*, just via a standardized *test*.

    Ball’s paper’s list is pretty interesting (
    I’d like to see her assessment methodology, when I see “meticulous analysis”, that leads me to believe the assessment is more qualitative than quantitative. Not that I have anything against qualitative assessment, but that’s not the easiest thing in the world to standardize. A cursory look at one of the assessment papers ( pretty much reflects what I was saying above: you can measure it, but it’s not easy.

  107. John Sidles Says:

    Patrick Callahan’s posts are (IMHO) excellent, and furthermore, his conclusions from (personal-experience)+(common-sense)+(parental-interest) are broadly supported by the research cited in the above-referenced Final Report of the National Mathematics Advisory Panel.

    As pretty much everyone posting here recognizes, there’s a lot of literature on education, but most of it is not well-grounded in evidence:

    p.81 Standards of Evidence The Panel as a whole reviewed more than 16,000 research studies and related documents. Yet, only a small percentage of available research met the standards of evidence and could support conclusions.

    The studies that did make it through the NMAP quality-control filter affirm that no known method of high-quality mathematics education is simple, easy, or cheap; neither is there any known one-size-fits-all math curriculum or reliable math teacher certification method.

    So we’d all better be prepared to invest for the long term, offer diverse choices to students and parents, and make wise compromises as educators … three things that are mighty tough to do well.

  108. SheGeek Says:

    I spent my undergraduate days as a mathematician (B.S, thank you very much) and left with the intention of becoming a New York City school teacher. I left not because I didn’t want to teach but because I wanted to be a scientist more. During my brief tenure as a potential teacher I had all the fire and flame of the author for revolutionizing this punchline to the joke that is the ill-preparation of our youth, and in many ways I still agree with him. However, much as the poet studies intensely the modes and methods of verse and sometimes rhyme, this structure is done for a reason: to allow the author to be present with his work and escape his own pretense. If we are to follow the math-as-art simile(?) to its conclusion, then we must apply this philosophy here as well. It would be foolish to discard the current methods entirely as they are necessary to allow the mature mathematician to be present with the numbers (or variables, or rings and fields, etc.). We need a more rounded and robust curriculum that complements the current methods with the kinds of explorations the author discusses. We must learn from the inner city ‘process writing’ failure of the last two decades to teach students what they really desired out of an english class: the language with which to communicate in the formal, academic, and business communities.

  109. sigma Says:

    @ Jerry Llevada #99

    I just looked at your ‘Algebra I’ DVD set. You say, “Complete Algebra I course based on NCTM** recommendations”.

    I don’t know who wrote the “NCTM recommendations”, but it looks like they were going through cases of vodka and smoking kilograms of funny stuff in their meetings.

    The list of topics looks just AWFUL: It’s packed with made up, junk-think, busy-work, irrelevant NONSENSE. Some of the nonsense is just contrived; some of it is destructive.

    For high school courses in algebra, geometry, trigonometry, and solid geometry there can be some benefit for many people with some passing knowledge. E.g., it can be good to understand perpendicularity and the Pythagorean theorem and the connection with projection and least distance. But mostly the purpose of these courses has to be just to get students ready for calculus. So, if some material is not very important for calculus, then JUNK IT.

    E.g., you have

    5.1 Systems of Equations: Solve by Graphing (6:23)


    5.5 Graphing Systems of Inequalities (4:42)

    NONSENSE. Made-up nonsense. The idea of ‘solving’ by graphing is nonsense. ‘Graphing’ systems of linear inequalities is almost entirely nonsense. You seriously misinterpret the reason for graphs and apply graphs where they are just absurd.

    The main purpose of graphs is just to build a little intuition about functions. The goal of graphs is certainly not a method of solution of anything; e.g., we no longer use nomographs. Moreover, graphs are just pictures, and there is nearly no serious question for which graphs provide a solid answer. You are making up work to do, and it’s silly, wasteful work.

    For linear inequalities, there is now essentially only one practical reason for that topic, and the reason is not very important in Algebra I or, now, hardly anywhere: The reason is the subject of linear programming, e.g., as in (with TeX markup)

    George L.\ Nemhauser and Laurence A.\ Wolsey, {\it Integer and Combinatorial Optimization,\/} ISBN 0-471-35943-2, John Wiley \& Sons, Inc., New York, 1999.\ \

    The intuition to be obtained from ‘graphing’ a system of linear inequalities is to see intuitively that the set of all solutions to a linear inequality is a ‘closed half-space’ which is convex and the set of all solutions to a system of linear inequalities is an intersection of closed half-spaces and, thus, closed and convex. Also get to illustrate the extreme points and their role. That’s the point of interest and the point trying to illustrate.

    To find a solution to a system of linear inequalities are essentially forced into just linear programming. That is, just finding one solution requires essentially the same effort as solving a linear programming optimization problem. So, there is no royal road here: To find solutions, can’t avoid linear programming. Can’t hope to do anything practical with linear programming with graphs. Instead need at least one good (i.e., effective, not necessarily polynomial!) algorithm, some corresponding software, and a computer.

    E.g., the last linear programming problem I solved had 40,013 constraints and 600,000 variables. Actually it was a 0-1 integer linear programming problem, and I wrote some software that found a feasible solution within 0.025% of optimality in 905 seconds on a 90 MHz PC via 500 steps of Lagrangian relaxation and the OSL. Intuition about convexity was crucial; graphs of linear inequalities played no role at all.

    Linear programming is part of ‘operations research’, and over several decades the US economy has given its opinion: Operations research is not worth the effort, a ‘late parrot’, a dead subject. Sorry ’bout that.

    Thus teaching systems of linear inequalities anywhere, especially in algebra I, is likely a waste of effort.

    Saw your topic

    10.4 Dividing Polynomials (4:51)

    Sure, it can be done. Have any idea why one would want to? Trying to get students ready for rings of polynomials? Galois theory? Poles and zeros in passive electrical circuits? Algebraic coding theory? Chebyshev numerical approximation?

    Moreover, you have too much emphasis on polynomials. This topic is partly one that should have been left on the scrap heap of history. Yes, at one time some people were all aflutter with polynomials, roots of polynomials, and finding closed form expressions for the roots in terms of the coefficients. That work is done and over with. The questions were not very important and mostly neither were the answers.

    Polynomials became a quasi-religion: Once computers were widely available, too many people rushed to ‘fit’ polynomials to data. NOT promising since the stupid thing is guaranteed to rush off to positive or negative infinity far too quickly; the fit is unstable; the usual numerical approach (least squares normal equations) is numerically unstable (i.e., leads to the notorious Hilbert matrix); etc. One should need a special license to be permitted to fit a polynomial with degree higher than 3. In particular, all the material in your course on polynomials of degree 3 should be chucked unless you want to mention poor, couldn’t shoot straight Galois for 90 seconds.

    Your interest in polynomials is from the characteristic polynomial of a square matrix, Taylor series, analytic functions, that the complex numbers are algebraically closed but the reals are not, the differences between the rational numbers, the algebraic numbers, and the real numbers? Not worth it in algebra I. Besides, often material on polynomials is misleading: E.g., mostly we do NOT use Taylor series to construct numerical approximations of functions. We definitely do NOT find the eigenvalues of a square matrix by constructing the characteristic polynomial and finding its roots.

    Noticed your

    Chapter 11: Statistics

    11.1 Measures of Central Tendencies and Spread

    11.2 Introduction to Probability

    11.3 The Counting Principle, Permutations, and Combinations

    and I’m horrified.

    First, 11.2 is definitely NOT a topic under 11.0. Sorry ’bout that. Instead 11.2 is the most important tool in 11.0.

    Second, for 11.1, definitely JUNK it. The idea of ‘central tendency’ is some quasi-religious nonsense left over from about 100 years ago. The idea is, in any measurement, should always get the same result except for ‘experimental error’ and by averaging repeated measurements the errors will cancel and will converge to the mean and ‘central tendency’. This is narrow, simplistic nonsense. Instead, the world is just awash in data where we can want to think of a distribution and where ‘central tendency’ and ‘experimental error’ mean nothing at all. E.g., consider system management data, generated by the terabyte in server farms daily. Or consider economic data. Or, what is the ‘central tendency’ of the age of the next person you see on the street? The idea of ‘central tendency’ has misled a lot of people. Your syllabus is drawing from the dregs of nonsense from 100 years ago. JUNK it.

    Avoid mentioning unless you also want to discuss the law of large numbers.

    Avoid mentioning the Gaussian distribution unless you also want to discuss the central limit theorem.

    For ‘spread’ or ‘dispersion’, just mention variance and standard deviation. If you want an application, then give the main result of the W. Sharpe capital asset pricing model. The students might enjoy hearing that the SAT scores are supposed to have standard deviation 100.

    For 11.3, there is NO easy connection with probability unless you also discuss probabilistic independence, and that is a much more difficult topic than permutations and combinations.

    For statistics, if you want some, do some simple hypothesis tests and maybe some confidence intervals. There are some non-parametric ideas that are easy to teach.

    Competent teaching of probability and statistics is rare in the US, at every level. Mostly the subject is too difficult for high school and/or high school teachers.

    Watched your

    1.6 The Number Line (12:36)

    Definitely JUNK it. It’s contrived NONSENSE. It’s not just a waste of time but HARMFUL. No student should watch that. You are teaching that we should do numerical addition and subtraction by using our fingers to walk left and right on the number line. Contrived, made-up, busy-work, misleading NONSENSE.

    Apparently someone with no significant knowledge of the real numbers was told to tell students about “the number line” and wanted a reason why and a connection with ‘solving real world problems’ so cooked up that finger exercise as the ‘reason’ for the number line. Ignorant. Brain-dead. Destructive. While the students will likely watch porn sometime, we have a shot at keeping them from your number line lecture forever.

    Yes, the real numbers are important, one of the most important topics in all of mathematics. Some of the properties are amazing, nearly beyond belief. E.g., see the long, gorgeous dessert buffet in

    John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \

    One not fully wrong definition of calculus is “some of the elementary properties of the completeness property of the real numbers.”. But at the level of algebra I, it is plenty just to say, “The X axis can be called ‘a number line’.”. DONE. Do NO MORE. Get it OFF THE HEAT. STOP. QUIT. CEASE. DESIST. Saying more is TOO MUCH.

    For algebra I, squeeze it down to about six weeks. A good student should get through it in one week, maybe a weekend. The rest is a BIG WASTE.

    Use the extra time to move on to plane geometry, algebra II, trigonometry, solid geometry, a little analytic geometry, especially the conic sections, and then calculus. Then linear algebra, abstract algebra, advanced calculus, ordinary differential equations, partial differential equations of mathematical physics, applications in physics and engineering, metric spaces, general topology, measure theory, functional analysis, optimization, probability, stochastic processes, statistics and various applications, filtering, control, exterior algebra, Maxwell’s equations, special and general relativity, quantum mechanics, etc. or a better list.

    There is a larger pattern here: Somehow high school math has taken on huge quantities of excess baggage. Somehow there is a ‘system’, apparently dominated by graduates of schools of education who have no decent knowledge of math. In the pursuit of make-work, or some neurotic fear of being criticized for doing too little, these people have dragged in excess baggage, junk-think nonsense. Need to get rid of ALL the high school algebra I up to calculus nonsense and start over. And for AP calculus, JUNK it and just take a respected college calculus text.

    S. Eilenberg said, “Elegance in mathematics is directly proportional to what you can see in it and inversely proportional to the effort it takes to see it.” He was correct, and your algebra I syllabus is the opposite of ‘elegance’.

  110. sigma Says:

    @ Patrick Cahalan #97

    The problems in ‘teaching’ that you describe are correct and illustrate that the system you describe is nearly hopeless.

    Children are TERRIFIC at learning but not in an environment such as you describe.

    In one important sense you are trying to do too much, and in another, too little.

    Too much: You keep trying to ‘teach’, that is, transfer knowledge from your head to that of the students. So, you talk, write on the board, interact, ‘motivate’, ‘engage’, etc. Mostly that doesn’t work. If a ‘good’ teacher can to it, then good for them; still mostly it doesn’t work. You are trying to make learning too much of a spectator sport. Learning, especially math, is mostly not a spectator sport. In particular, cannot do much in teaching someone math by talking and writing at a board. Instead, the student has to take some basic learning materials and do the learning themselves by whatever magic it is in people that lets them read, think, and learn.

    Too little: Most students need some guidance, e.g., help in finding suitable learning materials. Then occasionally a student needs some guidance to say such things as, “You need another explanation of this topic. Try books Y and Z. Rarely does one author give a really good explanation of everything; so, usually need one main source plus two or three others to use when the main source is not so good.” “You are doing too much here; the real goals, needs, and content are simpler than that. Here’s a fast overview of what is going on and what you should get …”. “You seem not to have gotten some of the main ideas in this topic. Here’s some of what you might look at again ….”

    You explain some of the problems with the environment well, e.g., disruptive students. One solution is to let a student who is trying to learn go to the library and just STUDY alone. Tell them when the next test is and how they can come to you for help and then just let them get to work.

    So, (1) Provide an environment that is not destructive and lets students learn. (2) Provide some good guidance. (3) Be available to let students ask questions. (4) Give and grade some appropriate tests, even standardized tests.

    Does it work? It’s about the only way I learned anything about anything. It’s overwhelmingly the foundation of the software part of the US computer industry: People learn from materials on the Internet and books. A huge fraction of most bookstores are just books teaching computer software. LOTS of competing books. The whole system has been a big success for the US, and schools have had next to nothing to do with it. For me, all through school, K-Ph.D., the time in class was nearly always from wasteful down to destructive. We’re talking a LOT of waste here, folks.

    In elementary school, I got contempt and criticism from teachers. So I tried harder. I still got contempt and criticism, even when it seemed to me I did well. So I concluded that the opinions of the teachers were not accurate or fair and that, sometimes, the teacher was really just my enemy. As I look back I can see that I was not fully wrong on these points. In K-8 the main problem was simple: I was a boy and nothing like a girl. All the teachers were women, and about half of the girls were teachers pets.

    Here’s a complete course in high school math teaching by example:

    In plane geometry the teacher was possibly the most offensive person I have ever known. Her favorite statement was a nasty, “Get that mess OFF the board.”. But the book was good enough — that is, it was actually about proving theorems. So, each day in class I had my head down with my eyes closed mostly just resting. Then that evening, alone, I read the section for the day and then attacked the exercises. I started with the easiest exercises and skipped until I found one that took more than 10 seconds to see. Then I found solutions to all the rest. If an exercise took more than a minute, then I’d write out a solution, in the margin of the book or on scrap paper. Then I attacked the more difficult supplementary exercises in the back of the book and did all of those. If an exercise took me more than an hour, then I’d write out a solution on a clean sheet of paper. The teacher’s idea of homework was for students to write out, in the usual laborious format, never seen earlier or later in math, three not very difficult exercises. I only used that format for the most difficult exercises, and in class I never showed any homework. The teacher viewed me with scorn and contempt.

    One of the supplementary exercises took me the weekend, all Friday evening, most of Saturday, most of Sunday, and into Sunday evening. I got it. Early in the book there was an easy exercise with the same figure. So, in class on Monday, after the class did the easy exercise, I mentioned, the first time I spoke in class, that there was an exercise with the same figure in the back of the book. About 30 minutes later, with the teacher at the board shouting, “Think, class. THINK.” with no real progress, I didn’t want to be accused of ruining the whole class period so raised my hand and gave a hint. The teacher was livid, and shouted, “You knew it all the time.”. Of course I knew it all the time. I never said I didn’t. If I didn’t know it, then I’d certainly would not have mentioned it in class. I got every non-trivial exercise in the book, never let myself miss even one. Period. Given the hint, she didn’t bother to complete the solution. Payback time, sweetheart!

    She was a case: She started the course saying that she was “Miss” but still hoping. She had all the feminine prettiness and figure of a city bus. Try the lottery, sweetheart, where you have a MUCH better chance. Also, that class was half full of some of the most gorgeous women on the planet. E.g., Cybil Sheppard went to that school and was NOT one of the prettiest girls. The irony, that offensive city bus standing before drop dead gorgeous Hollywood casting material and hoping to get married.

    On the state test, I did second best in the class. The last problem on the state test was to inscribe a square in a semi-circle. I didn’t get it until after the test. So, I asked to show my solution to the teacher after school. She was angry and confused at my interest and asked why I still cared about the problem but gave in. I started by constructing a square, and she said “You can’t do that”. I continued: Bisect one side of the square, take that point as a center of a circle with radius the distance to a non-adjacent corner of the square. Now have a figure similar to the desired one. For the crucial length, say, half the length of one side of the square, in the original figure, construct a fourth proportional. Done. Still she said, “You can’t do that.” Her solution was to construct a square with one side the diameter of the given circle and then draw a line from the center of the circle to essentially the same square corner I used. So, she did the same but her similar figure was positioned so that she didn’t have to use the usual construction of a fourth proportional. Okay. My solution used a general approach and hers was more specific. By usual mathematical values, my solution was better. The test was multiple choice and asked for the first step in “the” solution; my first step was likely not theirs which means that the test question was badly written.

    In college freshman chemistry, a senior English major struggling with math was taking a strange course in geometry and gave me a problem: “Given triangle ABC, construct D on AB and E on BC so that lengths AD = DE = EB.” So, I started with angles CAB and CBA, went to the side, constructed a figure similar to the desired one, and found the crucial length AD in the original figure by constructing a fourth proportional, again. Next day in chemistry, I showed her my solution, and she said: “How’d you know to do that? You just reinvented ‘simitude’ that we are studying now.” I explained that I had invented the technique when I was in high school but the teacher said, “You can’t do that”.

    The guy who beat me on the state test in geometry also beat me by a little on the Math SAT. We were 1-2 in the school.

    When the teacher with the SAT scores opened my Math SAT envelope, she said, “Uh, there must be some mistake.” That’s right, sweetheart, “some mistake”, and not mine, for 12 long, agonizing, painful, wasteful, destructive years for me.

    An important lesson here is, the way I learned plane geometry was really the best way. For all the ‘teaching’ the teacher tried to do, I just put my head down, closed my eyes, and waited until the nonsense stopped and I was permitted to get back to learning plane geometry. It’s NOT a spectator sport. Instead, just have to study it, especially to do good exercises. This work is best done alone in a quiet room.

    That learning technique is the main way I learned all the math I’ve learned and, of course, similar to how I’ve done all applications and peer-reviewed research I’ve done. Net, there’s just NO WAY to be any good as a professional mathematician, in academic research or in business, or really go very far in math, without doing nearly all the learning just the way I did in plane geometry. In the course I was fully correct: I did the right stuff. The teacher thought I was a terrible student, but I was likely one of the best she ever had.

    Why did I like math so much? I part because when I got correct answers, which I nearly always did, the teachers had one heck of a time saying I was wrong. They had to swallow all their nonsense conclusions from their nonsense irrelevant evidence, whatever that was, and admit I was correct. However painful that admission was for them, it was not nearly painful enough. I liked standardized tests because the grading was objective; my ‘reputation’ as circulated in the ‘teacher’s lounge’ was irrelevant.

    Those 12 years were a disaster: No student should go through it. No taxpayers should pay for it. In K-12 I was a motivated, talented, hard working math student, eager, even desperate, to acquire competence in the fields said to be important — math, physical science, engineering — but as I look back I can see that not one of the teachers I had was any good at math.

    For our society, here are some ways to do better:

    The students, starting at age 10 or so, can do MUCH, MUCH better than they are, EASILY. The bottleneck is the K-12 system where the teachers just do not know anywhere near enough math.

    So, there is a big issue about quality.

    For college math, the situation is fairly good: Math quality, teaching quality, and math variety are all high. There is plenty of connection with applications in physical science and engineering. There does need to be better connection with applications in social science and business and maybe computing.

    In graduate math in US research universities, the quality is very high although in the pure departments the lack of connection with applications is crippling.

    So, we know where the quality is and where it isn’t. For the quality in K-12 after basic arithmetic and its applications, there’s no practical fix — no way to turn all those babysitters into mathematicians — and we have to circumvent.

    So, we need some external materials. Past arithmetic and it’s applications, the goal is just to rush to get to college calculus. For the rest, JUNK it.

    Is Bill Gates listening? Robert Compton? Barack Obama? So, we need some funding to develop several, really MANY, sets of good materials and some associated tests.

    For each of the courses, we cannot permit just a single list of topics and need several competing lists and, for each, several competing sets of materials. E.g., we can’t have just a single ‘algebra I’ course: If someone wants to drag themselves through some algebra I full of excess baggage, so be it. Similarly for someone with good sense who just wants to get on to calculus and knock off algebra I in a week or weekend. Once a student has done well in college calculus, NO ONE will care that they junked the excess baggage in algebra I. There should be a lot of Internet and PDF access. No doubt there will be a lot of fora, blogs, and clips on YouTube. TERRIFIC! For the students’ discovering what’s available, ah, the Internet! The K-12 teachers can maybe administer the standardized tests and, maybe, for some really weak students tutor them a little or arrange some tutoring. Teachers: Do no harm.

    Then in K-12 we can cut out 6-8 years of highly destructive waste and nonsense and get our country going in high school math.

  111. Y Says:

    @ Pat Cahalan (#101)
    Point taken. 🙂 Specific references: Monk, and three papers with Ball as co-author. These papers discuss the methodology in more detail. (I can’t find a copy of Begle online; however, it has ERIC id ED171515.)

    It is indeed difficult to test for good teachers, and I was not clear enough in my previous post. The reason why Ball’s result is so remarkable is that it relies on a multiple choice test of math problems. So while the foundation for the work was in part grounded in qualitative methods (such as reviewing the tapes of an year’s worth of first grade classes) and probably could not have been achieved without careful qualitative analysis, the ultimate goal was to produce a quantitative instrument that was easily scorable and scalable to large numbers of responses. In the third Ball paper linked, pp. 387-388 summarize how test items were conceived, p. 394 gives a table with correlations, and general discussion begins on p. 399. Sample sizes for this paper are: 1190 first grade students, 1173 third grade students, 334 first grade teachers, 365 third grade teachers, 115 schools. Admittedly, this is a small sample; as John Sidles says, we still need to invest for the long term. However, a positive result in an endeavour with no known positive results still gives hope.

    I think we agree with each other; producing this instrument was expensive in terms of labor and time. However, the ultimate product takes only a scantron to grade, which is promising.

  112. Arun Says:

    Which country in the first world has the best high school education in mathematics? How do they do it? Are those methods useable in the US? If not, why not?

  113. Arun Says:


    I wish they told us how many standard deviations from the top the US was.

    I suppose the goal of producing some math geniuses might require different methods from producing a high level of math literacy in the population as a whole. I’m assuming the latter is the goal. So shouldn’t we go to Singapore and Taiwan and Japan and observe? (Observe, adapt, practice, improve what they do?)

    Also, what happens in England between the 4th and 8th grades?

  114. John Armstrong Says:

    Seriously, sigma, settle down. Nobody is worth remembering and writing out pages and pages of vitriol years after the fact. Can we just stipulate that you’re insanely smart and this woman was insanely stupid and have done with it?

  115. Scott Carnahan Says:

    You seem to have some very strong opinions about how mathematics should be taught and what content should go into a curriculum, but I was unable to understand some of your arguments. It sounds like you had an unpleasant experience in school, but perhaps if you had approached your education with a less adversarial stance, your teachers would have been a little more accommodating.

    Contrary to your assertions, my own experience in elementary and middle school suggests that instruction on the number line and graphical solutions to inequalities can be very helpful. Most of us seem to have some brain hardware that is well-suited for building geometric intuition for problem solving, and I don’t see how it is so positively harmful to teach students to use it. I’m not sure how your reference to a graduate measure theory textbook is relevant to your discussion of a high-school algebra course.

    I also don’t understand your antipathy toward polynomials. They may be ill-suited to approximation schemes, but they are useful pedagogically and theoretically – they are the simplest functions to manipulate algebraically, and they behave well under differentiation. It seems rather shortsighted to restrict our education to those subjects that are in current favor among numerical analysts.

    Finally, I feel that it’s rather unkind to tear someone’s hard work apart without writing a solid analysis of why it is specifically harmful to students (rather than boldly asserting that it is so), and offering a concrete alternative. I’d be interested to see if you have created any education-related media that you would like to share with the world.

  116. John Sidles Says:

    Sometimes its more interesting to think positively.

    … Let’s imagine a peaceful, prosperous planet with ten billion people living on it. 🙂

    … Educational institutions have improved to the point that one person in a thousand becomes a professional mathematician. 🙂

    … Applied or pure, it doesn’t matter … `cuz peaceful prosperous planets need *lots* of *both* kinds of mathematicians. 🙂

    … And each of these mathematicians publishes one peer-reviewed articles per year. 🙂

    … Which is a global flux of 30,000 new articles per day.

    Which is why every optimistic person foresees a wonderful not-too-distant future for mathematics.

    Of course, things could stay pretty much the same as they are now … but wouldn’t this require that the overall course of near-future earth history be radically dystopian?

  117. sigma Says:

    > You seem to have some very
    > strong opinions about how
    > mathematics should be
    > taught and what content
    > should go into a
    > curriculum, but I was
    > unable to understand some
    > of your arguments.

    For content, my view past arithmetic is just to rush to college calculus. Calculus need NOT be a very difficult subject, and getting ready for it is NOT nearly as difficult as the high schools keep assuming.

    > It sounds like you had an
    > unpleasant experience in
    > school, but perhaps if you
    > had approached your
    > education with a less
    > adversarial stance, your
    > teachers would have been a
    > little more accommodating.

    K-12 for me was like being tortured in prison. I wasn’t at all “adversarial”, only defensive. Except for a few times, I just sat in class and waited for the time to pass so that I could get on with learning or whatever else. If I didn’t say anything, then I couldn’t be accused of saying anything wrong and wouldn’t get criticized.

    > Contrary to your
    > assertions, my own
    > experience in elementary
    > and middle school suggests
    > that instruction on the
    > number line and graphical
    > solutions to inequalities
    > can be very helpful.

    Again, “the number line” is just the X axis. Before calculus, don’t say more.

    For your “inequalities” and the number line, sure, it’s okay to illustrate: So, can illustrate the set of all x so that x – 3 = 5, the distance from x to 3 is >= 5, that is, the boundary of the disk and outside it. So, here get two closed half lines. So, the solution is not convex. But with inequalities, mostly the algebraic techniques are more important than the geometric ones.

    The inequalities I discussed were linear inequalities, e.g., 3x -2y + z >= 4, and systems of them. The set of all solutions is a convex polytope, and the point of drawing graphs is just for building intuition about such polytopes. The graphs are not a “solution” technique.

    Those linear inequalities are central to one of my favorite subjects, optimization, but are not worth teaching on the way to calculus.

    > Most of us seem to have
    > some brain hardware that is
    > well-suited for building
    > geometric intuition for
    > problem solving, and I
    > don’t see how it is so
    > positively harmful to teach
    > students to use it.

    Geometric intuition is terrific. My objection was in teaching linear inequalities at all on the way to calculus.

    > I’m not sure how your
    > reference to a graduate
    > measure theory textbook is
    > relevant to your discussion
    > of a high-school algebra
    > course.

    Oxtoby covers some of the more astounding properties of the real numbers and just supports my claim that the real numbers are astounding. I was in part saying that ‘the number line’ is close to something very important. But I went on to say that on the way to calculus, don’t try to cover the more advanced properties of the reals.

    > I also don’t understand
    > your antipathy toward
    > polynomials. They may be
    > ill-suited to approximation
    > schemes, but they are
    > useful pedagogically and
    > theoretically – they are
    > the simplest functions to
    > manipulate algebraically,
    > and they behave well under
    > differentiation.

    Polynomials can be manipulated? Sure, but to what end? Manipulate functions when need to and not just for the exercise of manipulating. High school algebra is big on identifying and practicing various cases of ‘algebraic manipulation’ but with usually vague objectives. E.g., supposed to ‘factor’. Nonsense: Factor when there is a reason to and, then, of the possibly several ways to factor, for the reason in mind. For some suggestion of a most aesthetic way to write expressions, f’get about it.

    Polynomials are infinitely differentiable, but supposed to learn this in calculus.

    My objection is that, on the way to calculus, polynomials are overemphasized and actually are not very important. Likely the greatest importance is the connection with the complex numbers being algebraically closed, but the list of topics for that algebra I course seemed to want to emphasize just playing with polynomials without mentioning algebraic closure. I’m not big on teaching algebraic closure on the way to calculus, but if want to push polynomials then that is one reason to do so.

    > It seems rather
    > shortsighted to restrict
    > our education to those
    > subjects that are in
    > current favor among
    > numerical analysts.

    It would be. Just need a good reason to do much with polynomials on the way to calculus, and there isn’t such a reason.

    > Finally, I feel that it’s
    > rather unkind to tear
    > someone’s hard work apart
    > without writing a solid
    > analysis of why it is
    > specifically harmful to
    > students (rather than
    > boldly asserting that it is
    > so), and offering a
    > concrete alternative.

    For an alternative, I illustrated with how I learned geometry and, really, later essentially all the math I learned. I’m saying that, for students to learn math on the way to calculus, have them sit in a quiet room with a good book. In particular, don’t try to ‘teach’ the material. I was quite clear and relatively “concrete” on my “alternative”. The “alternative” is not vague or merely theoretical; instead, I lived it. Uh, we might notice, research professors continue to learn material but rarely attend courses. So, how do they do that? After some one hour seminar or a review paper that introduces the subject, they just sit and study, alone, as I described.

    I outlined some of what can be harmful: (1) Having scorn and contempt for students for no good reason. (2) Pushing students to do make-work (e.g., the way my geometry teacher wanted me to do homework) instead of solid work learning (the way I did the homework). (3) Teaching nonsense such as claiming that should do addition and subtraction by walking fingers right and left on the number line.

    For a more “solid analysis”, why was my way in geometry better? (A) I did MANY more problems than just the three the teacher assigned. (B) I learned to work problems almost entirely in my head instead of starting out with the usual geometry format for a proof. Generally high school teaches that to solve a problem, start writing: WRONG. That only works for trivial problems and otherwise is a waste of effort. Instead, for significant problems, start by THINKING. Get some crucial IDEAS that will be the key to the solution. End up doing nearly all the important work before writing much of anything. The high school teaching implies that the work is ‘algebraic manipulations’, pushing symbols around until get the desired result — not good. I had to learn this lesson on my own when doing research. Nearly all the important work was done while I worked with intuitive concepts with my eyes closed. When I started much writing, I had the ideas, knew about what to write, and was nearly done. Never heard that from a teacher. (C) When tackle some really hard problems, say, one for an entire weekend, then learn how to organize the attack between the ears. Else can spend a week instead of a weekend. (D) What was important was the way I wrote out the solutions, WITHOUT the usual format. The teacher implicitly agreed: When I gave her the hint, she saw that the important work was done and didn’t write more either, certainly didn’t spend five minutes in mechanical drawing to get a pretty figure.

    > I’d be interested to see if
    > you have created any
    > education-related media
    > that you would like to
    > share with the world.

    When I was teaching applied math in college and graduate school, I didn’t have a good text and I developed some course notes so that the students would mostly not have to take notes in class. But I ended that teaching just before TeX became common. So, I never had TeX to let me type in some decent quality teaching materials, a fact I hated. I was eager to write out the subject so that the students could just sit in the library, read, work lots of good exercises, check their answers in the back, and, thus, learn the subject. Then the classes would be only to give the students overviews and intuitive views to help them read the material. Alas, I was blocked by a word processing bottleneck. But, I was a professor in a research university, and there writing teaching materials is not considered good work! Then I left teaching for business.

    Now with a recent PC, Windows XP SP3, TeX, a 600 DPI printer, and software to write PDF, for writing math it’s a totally different world.

    So, no, unfortunately I have no educational materials to offer the world.

    But for good writing in math, I would recommend: Calculus by Protter and Morrey (I taught from it as a graduate student — it’s beautifully clear) or Thomas. But there are several more. I learned from Johnson and Kiokemeister and liked it. I used another book with an excellent chapter on conic sections but loaned it, never got it back, and don’t remember the author! There are a lot of good college calculus books.

    My favorite writer of mathematics is Halmos. His ‘Finite Dimensional Vector Spaces’ was a major event in my life! It was wonderful. The polar decomposition was my favorite result — I was astounded.

    FDVS by Halmos is a great place to get started as a mathematician. Yes, at one time it was one of three books, along with ‘baby’ Rudin and Spivak’s ‘Calculus on Manifolds’, for Harvard’s famous Math 55.

    I like Royden — tough not to — but also like Rudin, at least his first two books on analysis, but can see why many people do not.

    Simmons ‘Introduction to Topology and Modern Analysis’ is elegant.

    As a college senior, I read through Kelley ‘General Topology’ and gave a lecture a week. It was a lot of work but will make one a better mathematician! In most ways the writing is very polished, but it’s tough for a student to see just what the goals are!

    For 19th century style vector analysis, STILL important in physics and engineering — the first edition (NOT the second) of Apostol’s ‘Advanced Calculus’.

    Ordinary differential equations — Coddington.

    In probability, Breiman is my favorite. He’s a wonderful author. Neveu is more elegant but more challenging.

    I have stacks of books on statistics but don’t know of a good book on that subject.

  118. sigma Says:

    Armstrong #114

    My goal is to help K-12 math from the end of arithmetic to the start of calculus.

    The relevance of my experience? It was very wasteful. The teachers said I was a poor student when I was an excellent student.

    Conclusion: The teachers didn’t know enough math to know good work from poor.

    I have to believe that this situation is still nearly universal in US high school math teaching.

    Then, what to do about this sad situation?

    There are some people quite concerned: Bill Gates, Robert Compton, etc.

    I explained that the algebra I course referenced here had a lot of excess baggage.

    I contrasted what Carnahan was encountering and doing with how I learned that math: F’get about the teacher, take a good book, study in a quiet room, and LEARN. Moreover, this technique continues to work, really is a key technique in math.

    My solution would let all the more promising math students make MUCH better progress.

    E.g., for Carnahan’s problem of some students disrupting the class and the more general problem discussed here of evaluating (by social science techniques complete measures with reliability and validity, etc.) teacher quality, get to set nearly all that aside. E.g., when I was in a quiet room, there were no disrupting students around.

  119. jerry llevada Says:

    There seems to be a lot that you don’t know. You could have saved a lot of typing by giving real, coherent reasons why you detest the way algebra one is taught. Perhaps you come from another planet I have not mingled with terrestrials enough. I’ll attempt, and I will probably fail, to answer some your diatribe.
    In your first three paragraphs I gather you only want us to teach a small % of the HS student body, the rest we sent home. I gather by your narrow approach to education that we should only teach some elite that you have fashioned. Because you sound to be a very intelligent person, at least as it relates to some level of math, I fail to see why you can’t look beyond a very narrow definition of what a human is capable of doing and what educational systems all over the world are trying to achieve against awful odds.
    And then you try to explain yourself by name-dropping a very long list of math subject that probably only 0.0000000001% of the college-level students will ever see—let me remind you the subject here is HS algebra one. I don’t know with certainty how many student I have taught and I don’t remember their names; however, I remember all of their faces. Let’s say I have taught in 32 years over 5,000 students; of those 5000+ I have only met 1 that can stand up to you in nonsense.
    I have only one comment to add: Please, find yourself a corner somewhere and go and develop a General Theory about everything, perhaps you can develop one. And, please again, don’t come out until you develop one.

  120. dreamer Says:

    “I defy you to read it and find a single sentence that isn’t permeated, suffused, soaked, and encrusted with truth…”

    Okay, here’s a sentence that does not ring true to me: “The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.”

    Math is totally different from music or painting. Consider: I walk into a store, and purchase an item that cost a dollar and a quarter. I hand the checker a 5 dollar bill. The checker gives me my change of $2.75. I walk out of the store, content with my purchase.

    If I don’t know math, and by that I mean the K-6 variety (fractions, addition and subtraction, etc) I won’t realize that the checker made a mistake that cost me money. This has nothing to do with art. This is basic commerce, the way our society runs.

    Yes, I agree that very high level math (post-graduate) does have aspects that can be considered art. Memorizing the circle of fifths is not required in our society, but memorizing multiplication tables is. Two very different things.

  121. innumerate Says:

    jerry llevada Comment #117 said:”… a very long list of math subject that probably only 0.0000000001% of the college-level students will ever see”

    Please explain, why did you bother typing a 1 after all those zeroes?

  122. Charles Jones Says:

    I would like to know what educational system xx and yy have been monitoring when they suggest implementing what is essentially a version of “new math” or, on a broader scale, the “self esteem movement”. Everything I’ve been reading for years indicates that the method proposed has been the norm for the last 20 years or so. Interestingly, the drop in American math scores corresponds very well with the introduction of these methods. In other words, this method has demonstrably failed while the more rote methods of the prior 50+ years demonstrably succeeded. More evidence of this is found when comparing techniques and results of European and Asian schools.

    The analogy to music and art provided by xx doesn’t make sense. Doing multiplication tables is doing math the same way that practicing scales or learning the color wheel is doing music or doing art. Finding a professional musician that didn’t practice basics for years or doesn’t practice the same song over and over again would be very rare.

    There seems to be a lack of understanding of the “standing on the shoulders of giants” situation. Do you really propose that first graders discover what one is? Or that they develop their own notation for arithmetic? How long would that take? If you’re aware of the Moore Method (which is a graduate level version of the technique proposed) you’ll know that a successful semester long course is lucky to develop a proof for a single known theorem. Although creative teaching shouldn’t be completely ignored, the approach takes time that just isn’t available.

    Please. Stop championing the educational philosophy that got us into the current mess.

  123. Deane Says:

    “…There seems to be a lack of understanding of the “standing on the shoulders of giants” situation. Do you really propose that first graders discover what one is? Or that they develop their own notation for arithmetic? How long would that take?…”

    I think this illustrates a lack of understanding of what many thoughtful and experienced math educators think is the best way to teach. None of us are proposing anything remotely resembling what you claim we are.

    No one is saying that *everything* has to be discovered by the student. But at every stage of learning math, even at the very beginning, it is *very* useful to challenge the students to figure out what can be done with the basic skills they have learned so far and allow a student (or, better, a team of two or three students) to struggle with solving a non-trivial problem using the tools they have already mastered. This for me has the following productive outcomes:

    a) Students learn how to solve problems in the simplest way possible, rather than the most sophisticated.

    b) When you go on to teach students about fancier tools that solve the same problems in a more efficient manner, students really appreciate the power of the new concept being taught and therefore are more receptive to learning it.

    But you *are* right that this better approach requires more time. It also requires properly trained teachers or coaches (to use the basketball analogy). But it *has* been implemented successfully in many different settings, so it *is* possible.

    I’m just going to keep saying this: We need to teach math more like the way good basketball coaches teach basketball

  124. Jerry Llevada Says:

    innumerate (Comment 119), if this is your real name (I changed mine because my HS friends rhymed it with a very important part of female genitalia), the number is equivalent to the concentration of chlorine in my pool.

    To more important things:
    I finally read all of Lockhardt’s article and I am in total disagreement in labeling math Art. Math, the way he likes to define it, is a mental, entertaining effort. Once you put in on paper it becomes something else, but not art. A graph perhaps, archaic symbols. Art is expression in the most broad form. Not all painting is art, neither is all music, or dancing, or theater, or film, or poetry… Personally, I use sex to define art. If it cannot express sex, it is not art. Most porn is not art, but art can be porn. Writing can be art, so can some very specific engineering. I remember watching Steven Coons develop CAD from art. He was quite an accomplished artist. I wonder if some of his stuff survived.
    If we use Lockhardt’s definition, then philosophy is also art, and so is chess when is played at a very high level. Anyway, I admire Lockhardt immensely, and his argument is not all lost. He brings to the table an excellent discourse, but it is all wishful thinking. Sex cannot be represented through math, therefore, math is not art. Of course, some of my students, in class, were caught trying to prove me wrong.

  125. David Komp Says:

    Charles Jones,

    In other words, this method has demonstrably failed while the more rote methods of the prior 50+ years demonstrably succeeded. More evidence of this is found when comparing techniques and results of European and Asian schools.

    You seem to be implying that schools in high-achieving European and Asian schools owe their success to rote methods. But one of the countries that consistently ranks in the top is Japan, whose elementary schools practice lesson study, which is far from “rote”. (You can see examples of videos at ).

    I also would like to know where you are getting the idea that the education system 50 years ago is “demonstrably” better than the education system now. While it’s true that the current system is disappointing, and no one wants to see a resurgence of the movements that you mention, international tests like the TIMSS and PISA only began collecting data after these movements began. If you are talking about the SAT scores, does show a drop in the 1980’s but also shows improvement since then (even accounting for recentering), so I don’t see the dramatic difference between 50 years ago and now, especially when considering the increasing number of students that take the SAT every year.

  126. Chris Monson Says:

    You misspelled “Lockhart” throughout.

  127. John Sidles Says:

    Lockhart asserts:

    At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement; that works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste. A piece of mathematics is like a poem…

    Surely Lockhart is expressing a Great Truth … his words are “permeated, suffused, soaked, and encrusted with Great Truthiness” … and therefore, the opposite of what he says is a Great Truth too.

    We can ask, what is this opposite Great Truth?

    Obviously, a great portion of humanity’s literature has been created for reasons other than amusement. Some of these reasons have been noble, many more were not.

    It is a Great Truth that if humanity’s literature had been written for just one reason, or had been taught in just one way, or had been embraced just for noble reasons, humanity would be greatly diminished thereby.

    Surely, the same is true for humanity’s mathematics. Right?

  128. Scott Says:

    You misspelled “Lockhart” throughout

    Chris: Thanks for catching! Fixed.

  129. Greg Kuperberg Says:

    A comment for today:

    Some of the standard recurrent accusations against the educational system are that it has been damaged by bad teaching methods; that it has been damaged by unions; or otherwise that the teachers are unknowledgeable or anti-intellectual. Many people who put forward these theories also see private schools or home schooling as a rescue from the failed public system.

    I agree that some of the teaching mantra are bad, or at best a distraction. I agree that most grade school math teachers don’t know all that much math, and that it would help if more of them liked math. And I agree that union power can be a distraction. The purpose of the union is to bargain for better labor terms for teachers, and not mainly to press for a better education for students.

    But despite all that, I think that the main problem is that even as learning math has become economically more important, schools have to compete ever more with the private sector to find mathematically literate teachers. Which means that if any of us were a school principal in a district with zero union power, and with complete freedom to set the curriculum, we still wouldn’t be able to solve the main problem. Public grade schools cannot afford six-figure salaries, and they also cannot afford university-level teaching loads.

    In fact there are schools with no union influence and nearly unrestricted curriculum freedom. Many charter schools enjoy all of that. And yet, according to credible studies, charter schools educate students less well on average after adjusting for demographics and start-up subsidies. The same is true of religious schools, and even secular private schools are only about tied. Without adjusting for demographics, secular private schools look great, as everyone knows. On average, home schooling is almost certainly the worst system; one deception is that home schooling with voluntary testing looks better than public school with involuntary testing.

    My impression is that for all of their failures, public schools cope reasonably well with unfavorable labor economics. I suspect that they make more efficient use of what they have than they did 30 years ago. Institutional experience and economy of scale both work.

    That does not mean that there is little room for reform, only that it doesn’t help anything to slam the system as incompetent and malign. If I could ask for one reform, it would be to concentrate on content and try not to micromanage teaching methods. The credentialing system demands ten times over that the teachers know how to teach — and does not always demand it properly — but only asks fairly faintly that they know what to teach. Likewise the worst part of many of the reforms is that they conflate method and content.

  130. John Sidles Says:

    Greg, that was an outstanding post. Thank you.

    Another point that deserves to be made relates to the earlier praise of Sudbury-style (democratic) schools.

    Sudbury schools have two tremendous advantages: (1) the students and their parents want to be there, and (2) students whose personalities are deemed “poor fits” to Sudbury-type communities can be (and are) denied admission and/or expelled.

    Public schools have neither advantage. The people who teach in public schools are true heroes (IMHO).

  131. Chris W. Says:


    Regarding the analogy to music, you and many commenters on this post might have found much of interest in listening to a recent interview with Branford Marsalis and the new drummer in his quartet, Justin Faulkner, who is 18 years old. I don’t think it was recorded, unfortunately.

    In a number of ways, what Marsalis had to say about formal music education echoed what Lockhart is saying about mathematics education. He openly admitted that he hated to practice, and his raw technical command of his instrument was not up to that of many of his contemporaries until fairly late in his career, but he held his own with some first-rate players because he had learned to critically listen—both to recordings and his colleagues in live performance. He repeatedly emphasized the importance of listening thoughtfully to lots of music of many genres.

    Of course, he knew he wanted to be a musician. He heard the real thing growing up, and had ample basis for making an informed choice. Many people exposed to more or less the same environment made different choices. (More often than not they had to, because they didn’t have the talent to be professional musicians.) Perhaps the most sensible way to view Lockhart’s manifesto (inspired rant?) is simply as a plea to allow young people some exposure to what real mathematics is. If most of them respond with indifference, or with anxiety that they won’t master certain practical skills that are sure they need, then so be it, but don’t pass off contrived exercises that exist only in pre-college math courses as the real thing.

    By the way, a sad fact has been reiterated by Lockhart’s essay and many of the responses to it: What most children are exposed to in school and arguably in most other areas of life is the result of a lot of fairly ugly practical (and often bureaucratic) compromises by parents, teachers, and other authorities. With surprising frequency the native intelligence, imagination, and energy of children gleans far more from the dross than one might hope for. I doubt there is any way to escape from this reality on a large scale.

    (A final remark: Branford Marsalis made no attempt to hide how tough a career in jazz generally is, and how essential a strong personality can be in dealing with other musicians as well as the other demands of the profession. This made for an exceptionally bracing and candid interview.)

  132. John Sidles Says:

    Michael Nielsen has posted a essay on Hacker News titled How to read mathematics and physics. This essay can be read (among other ways) as an meditation on math education.

    Michael’s essay is 1/10 the length and (IMHO) has 10X the value of most essays on education. A Google search for the starting phrase “I was a theoretical physicist for 13 years” … will find it (and Michael’s blog has a link too).

  133. Bram Cohen Says:

    Greg, among demand of the teachers unions is that, first and foremost, there be no merit-based assignments whatsoever. Everything goes on a strict formula – whoever’s been teaching longer gets priority, end of story. Before the situation can be improved, it’s necessary that at least a hint of meritocracy be an option.

  134. John Sidles Says:

    Bram, with respect, please cite sources. Try searching the recent news for “Merit Pay Gains Among Teachers” …

    Unions aside, most teachers are just as (rationally) skeptical of merit pay as they are of educational reforms in general.

    The reason is simple: they have seen too many ideology-driven educational reforms fail miserably.

  135. Scott Says:

    By some standards I am a mainstream liberal and a mainstream feminist. With all three of these authors, I feel an overdose of ferocity. I also perceive the urgency as partly real and partly counterfeit. Wouldn’t you agree in the case of your other two examples?

    Greg: Interesting question! Here’s what I see as the central difference between Dworkin and Chomsky on the one hand and Lockhart on the other. Most people have probably known since the 60s that there exist radicals who are against heterosexuality and Western civilization (meaning not their abuses, but the things themselves). They don’t agree with the radicals, but at least they’re aware of them.

    By contrast, most people seem completely unaware that there exists a Lockhartian “radical critique” of K-12 math education. They honestly think that what was drilled into them in school is math. They might still resent it, they might see it as torture that was ultimately worth the price, they might have even liked it, but in any case, it would probably astound them to know that many mathematicians consider what they were exposed to not math at all—indeed, a cruel parody of math. For this reason, I think essays like Lockhart’s serve a genuine consciousness-raising function, whereas with Dworkin and Chomsky there’s just the ferocity (eloquent and interesting though it sometimes is).

  136. Pat Cahalan Says:

    @ Y

    Thanks for the email; I’ll try to respond tomorrow. Getting ready to move buildings, so things are somewhat chaotic.

    > However, the ultimate product takes only a scantron
    > to grade, which is promising.

    That *is* interesting. I’ll have to read her stuff. I’m curious if this is a correlation instead of causation issue, however. Implementing this in the present moment may very well give a good metric as to the good vs. the bad teachers, but I’m curious what would happen after teachers start studying to pass the test… does it make them better teachers?

    I expect you’ll see at best that “level jump”, from bad to mediocre, or mediocre to average, etc.

    Note, this isn’t a problem limited to the subject of mathematics. Teaching, in general, is really hard, regardless of the subject. Personally, I don’t expect that my childrens’ teachers will all be excellent (even were I willing to spend a college tuition for private grade school); it’s just a really hard standard to meet.

    If the entire body of teachers my children intersect with in their educational experience do a decent job, I’ll be happy with that. I’m not really expecting my children to get a love of learning from their teachers in any event, that’s sort of my job IMO. Finding a half-dozen real mentors in the bunch would be great.

  137. Josef A Banks Says:

    Ha ha ha … the lament is beautiful and hilarious.

    But not more hilarious than the comments posted on this blog! Those who disagree with it only provide in them more evidence for the correctness of the lament’s arguments.

    Thank you Lockhardt for doing this.

  138. Qiaochu Yuan Says:

    For this reason, I think essays like Lockhart’s serve a genuine consciousness-raising function, whereas with Dworkin and Chomsky there’s just the ferocity (eloquent and interesting though it sometimes is).

    Unfortunately, really the only people I know who’ve read the essay are mathematicians. I guess there is some substance to the accusations of preaching to the choir. The MAA isn’t exactly the best way to reach non-mathematicians.

    And just for reference, God Plays Dice also discusses the lament.

  139. Lazlo Toth Says:

  140. John Sidles Says:

    Many are claiming that Lockhart’s essay is “radical”. But is this really true? (1) The Lockhart essay contains no new ideas (similar laments have appeared for centuries). (2) The Lockhart essay (by common consensus) contains no feasible prescriptions for change.

    It has happened that well-written essays have laid philosophical foundations for radical, successful, global transformation in education. Good examples are William Osler’s essays on medical education (“Aequanimitas” and “A Way of Life” are famous examples).

    Just as important, Osler successfully blended the ideas of his essays with radically transformational—and successful—methods for organizing and financing medical schools. That is why Osler’s ideas and methods have been universally adopted.

    Mathematical education has not yet found its Osler.

  141. David Komp Says:

    For this reason, I think essays like Lockhart’s serve a genuine consciousness-raising function, whereas with Dworkin and Chomsky there’s just the ferocity (eloquent and interesting though it sometimes is).

    I’m as disappointed and at times mad as any other mathematician/scientist/person who appreciates math that many K-12 students don’t get to experience competence in basic math or problem solving. But Lockhart’s essay is flawed as a consciousness-raiser. It claims to blame the “culture”, but the two manifestations of culture — teachers and curricula — that it spends the most time maligning are created by people who generally agree with what Lockhart has to say. I find Lockhart’s Lament harmful and divisive because it encourages a naive view of whatever systemic problems we have. A natural extension of his Lament is pitting mathematicians against educators and teachers — the opposite of what is necessary for progress. What we need, among other actions, is for more mathematicians to cooperate with educators and teachers and try to understand each other’s work.

  142. Jonathan Vos Post Says:

    I have had, to put it mildly, mixed results in trying on and off since 1973 to teach Math in secondary, post-secondary, senior citizen, and corporate institutions, while publishing Math, Computer Science, Mathematical Physics, Mathematical Biology, and Mathematical Economics papers in my own time.

    I agree with Scott that Lockhart is correct that nearly all of my students had previously been taught was “not math at all—indeed, a cruel parody of math.”

    The secondary schools have the insane curricula, and are mostly taught by people already brainwashed by the failed system. Community colleges where I’ve taught make their profit on students taking remedial Math classes because High School socially promoted them. These colleges have smatterings of Math at or above Calculus, but deeply Applied according to departmental objectives (i.e. Math for Computer Graphics, Math for Architecture, Statistics for Social Sciences) and teachers (instructors, adjuncts) who question the approach actually get fired (I’ve seen this several times).

    The Corporate courses I’ve developed and taught are necessarily Applied (Statistics for Software Engineers, modern methods in Computational Aerodynamics, simulation of Moonbase economics).

    Universities? Your mileage may vary. Our best is unsurpassed worldwide. Our worst? I never experienced our worst at this level.

    Don’t get me started on Colleges of Education. I’ve just emerged from 2 absurd, expensive, frustrating years of having ideology drummed into my head by professors who almost boast that they “don’t GET Math” but feel free to attack my approaches to teaching Math. Colleges of Ed have been dumbed down, crammed with “Ed Speak” jargon, and faddish nonsense, as teachers must be created from the students who emerged from the failed secondary and college population.

    Since Secondary schools are prerequisites for post-secondary, senior citizen, and corporate institutions, and either put people on the right track for pure or applied Math, or lobotomize and castrate them for ideological reasons, Lockhart’s critique is of existential importance for the United States of America, both for reasons both internal (cultural) and external (post-industrial global competitiveness).

    I don’t know all the answers. But Lockhart asks the right questions.

  143. Greg Kuperberg Says:

    What we need, among other actions, is for more mathematicians to cooperate with educators and teachers and try to understand each other’s work.

    I said yesterday that if I could change one thing, it would be not to conflate methods with content, and to concentrate on content in state standards. Actually, I overlooked another aspect that might be even more important to change, although it is related to what I did say.

    Namely, that the education establishment has filled space with low-quality teacher certification requirements. The establishment gives the public a choice between more training and less training, when the real problem is the type of training that it requires. These required classes lecture prospective teachers in basic common sense, things like how to write a grade sheet. And they are laden with a degree of psychobabble. Content takes a back seat.

    So yes, you can cooperate with education schools and school districts on teacher training, but only on their terms. They’re happy to have you train teachers, some of it using paid time away from the classroom. But the certification system creates the cadre of teachers who you can train, and that part is not negotiable. On top of that, most schools have one teacher for all subjects through grade six, and even in later grades teachers are sometimes reassigned. Putting it all together, teacher training by mathematicians is drops into a huge bucket that the education establishment has already filled.

    Private schools and some charter schools are exempted from rigid teacher certification. I said before that on average, and after adjusting for demographics, these schools are not better than public schools. But that just means that they may or may not put their extra breathing room to good use. It has to be a win to avoid the weird outcome of teachers with master’s degrees who are hazy on math that they teach to sixth graders.

    See this related article in the New York Times.

  144. David Komp Says:

    Putting it all together, teacher training by mathematicians is drops into a huge bucket that the education establishment has already filled.

    You’ve mentioned three entities, the education establishment (this presumably includes state and federal policymakers), the school districts, and schools of education. I’m not sure which of these entities had the most influence in creating NCLB, the appalling consequences that the NYT article exposes.

    If I understand correctly, you make three arguments: (1) teacher training affords precious little time for interaction with mathematicians, because of the overload of coursework and duties imposed by schools of education and school districts; (2) the certification system allows low-quality candidates to teach; (3) the certification system can be composed of low-quality training (both method and content?).

    (2) and (3) seem anecdotally true. Teachers I know have come to the conclusion that their training was a waste of time, not because teaching methods are easy or natural or purely common sense, but because the challenges they faced weren’t addressed by their training. On the other hand, other teachers I know have said that the only reason they survived their years teaching was because of an excellent methods course. This is a difficult issue to solve, because there is a bootstrapping problem: it is often hard to teach something that there is no apparent motivation for learning; i.e., you don’t know what problems you want to learn to solve until you encounter them.

    With respect to (1), we should distinguish elementary from secondary preparation. Mathematicians may not have much interaction with future elementary school teachers (because time needs to be split with all the other subjects they are asked to teach as well as methods courses) — this is something that policy makers and school districts may be able to change, but schools of education probably cannot. However, teaching at the secondary level often requires a mathematics minor. This is, granted, not as much as a mathematics major, but it seems like more than a drop in the bucket.

  145. Greg Kuperberg Says:

    I’m not sure which of these entities had the most influence in creating NCLB, the appalling consequences that the NYT article exposes.

    No Child Left Behind is and is not the point. It is true that NCLB is a cynical masterpiece of treating the patient with the disease. But the diseases were there before NCLB and they will be there after NCLB.

    The New York Times article described two highly competent but uncertified teachers. The only reason that those teachers were allowed into public schools at all was because they were charter schools. In an ordinary school district, they never would have been able to start, with or without NCLB, for lack of certification.

    (1) teacher training affords precious little time for interaction with mathematicians, because of the overload of coursework and duties imposed by schools of education and school districts

    No that’s not the argument. A good course in any subject is both a filter and a pump. But post-certification teacher training is only a pump and not a filter at all. The public school system tells education-minded mathematicians, “we picked them, you train them”. This is a structurally inefficient form of training.

    It’s true that pre-certification training, as with a mathematics minor or an education-track major, is better. But for the reasons listed below, required coursework is not the best model anyway.

    (2) the certification system allows low-quality candidates to teach; (3) the certification system can be composed of low-quality training (both method and content?).

    To some extent, this is splitting an indivisible hair. As the teachers in that New York Times article said, the certification courses are patronizing. Yes it is low-quality training. But it’s not that the training “allows” low-quality candidates to teach. Rather, required training of this type discourages good teachers.

    It’s all well and good that some teachers like this or that method course. But as you say, many teachers consider many of the certification courses a waste of time. It’s unfortunate that the courses are required for everyone.

    The reform that I think would help is one that is already in place for the state bar, at least in California. In order to practice law in California, you’re not strictly required to take courses in anything. You have to pass the bar exam, and you have to either attend law school or work as an apprentice for a judge or in a law firm. What the system says is that if you can prove yourself with an exam and with apprenticeship, you could be a good lawyer even if you don’t have the time or money for law school.

    Likewise to teach public school, tests and evaluated apprenticeship ought to be good enough. If you can prove your qualifications in the classroom and with content exams, you shouldn’t have to enroll in education school. I’m sure that this would portrayed as undercutting standards. It would actually allow stricter standards for content mastery. It would even allow stricter in-class evaluation of teaching methods.

    Here is another New York Times article on this point.

  146. David Komp Says:

    Likewise to teach public school, tests and evaluated apprenticeship ought to be good enough. If you can prove your qualifications in the classroom and with content exams, you shouldn’t have to enroll in education school.

    Okay, I see your point. Going back to one of your earlier posts, I suppose if I had two wishes, it would be that the school system wouldn’t collapse from overworked staff if we only hired the people who passed these tests, and that there on-the-job mentorship in methods and content was guaranteed to all teachers.

    (Potential point of interest: England has a strict in-class evaluation of teaching methods that must be passed by the second or third year of teaching if the teacher wishes to keep his/her job. Otherwise, they must find another school and go through the evaluation again in a year or two’s time, I believe. I don’t know about the requirements for contest mastery.)

  147. Greg Kuperberg Says:

    Okay, I see your point. Going back to one of your earlier posts, I suppose if I had two wishes, it would be that the school system wouldn’t collapse from overworked staff if we only hired the people who passed these tests, and that there on-the-job mentorship in methods and content was guaranteed to all teachers.

    Actually, California already has an internship program along the lines of what should exist. The impetus of this credentialing track is to ease teacher shortages. The problem is that instead of simply evaluating whether the interns do a good job in the classroom, the interns still have to take all of the education courses concurrently.

    California even sometimes issues so-called emergency teaching credentials to waive the education school requirement. The impetus, again, is teacher shortages. But instead of trading coursework for better evaluation, the system strictly labels these teachers as unprepared, beginning with the word “emergency” in their teaching permits.

  148. Bram Cohen Says:

    John Sidles: As absurd as it is to provide references for the obvious, there are good writeups here and here.

    Color me unimpressed by the recent news on merit-based pay. For teachers to be willing to grudgingly accept a raise despite it requiring some amount of merit based distribution is hardly a sign of things being good. That there could be context for such a statement to be made is a sign of how sick things have gotten. For there to be even a suggestion of merit based retention would go fundamentally against current union dogma.

  149. John Sidles Says:

    Bram, it’s never absurd to provide references for the obvious … but your first reference used the word “union” only once … your second reference used the word “union” only in the context of anecdotal evidence … and neither reference was peer-reviewed.

    What we teach our medical residents applies here too: “The plural of anecdote is not data.”

  150. Bram Cohen Says:

    John, the unions’s opinions on these subjects are not a secret, it says it right in the article you referred to. The NEA is against merit pay under any circumstances. The AFT supports it if it’s extra money. Both are against merit for retention discussions under any circumstances.

    Also, I’ll provide ‘peer reviewed’ references when you provide me a ‘peer reviewed’ reference that Barack Obama is the president of the united states. None of the statistics in those studies are in dispute, nor are the details of what any teacher contracts look like. For you to request this ‘peer review’ thing is just a rhetorical gambit. Teaching in US public schools is by design flat-out unmeritocratic, and this is not even a matter of dispute.

  151. John Sidles Says:

    I’ll provide ‘peer reviewed’ references when you provide me a ‘peer reviewed’ reference that Barack Obama is the president of the united states.

    OK … here’s mine! 🙂


    Seriously … we require that our medical residents base their case presentations solidly on evidence-based medicine. Its surprising how many medical beliefs and practices that “are not even a matter of dispute” have little or no basis in evidence.

    If math education is somehow different … then gosh … I for one would sure like to see the evidence! 🙂

  152. John Sidles Says:

    No thread lasts forever, but this one is so great that perhaps it can be rekindled by mention of a topic that, so far, has not been discussed: mathematical education in fiction.

    If we read with a focus on education, Ursula LeGuin’s 1974 novel The Dispossessed is a classic of the genre and (IMHO) is still worth reading today.

    But the times have passed-by LeGuin’s worldview … as Feynman foresaw in that same decade:

    We are very lucky to live in an age in which we are still making discoveries. It is like the discovery of America—you only discover it once. The age in which we live is the one in which we are discovering the fundamental laws of nature, and that day will never come again. It is very exciting, it is marvelous, but this excitement will have to go.

    Neil Stephensen’s The Diamond Age or, A Young Lady’s Illustrated Primer offers a post-Feynman/LeGuin worldview whose characters embrace informatic and nanotechnological engineering.

    Diamond Age can be read as Stephensen’s twenty-first century answer to Feynman’s 1965 challenge:

    Of course, in the future there will be other interests. There will be the interest in the connection of one level of phenomena with another—phenomena in biology and so on […] but there will not still be the same things that we are doing now.”

    It seems to me that we’re about due for another great novel that embraces educational themes in math, science, and engineering.

    Perhaps this novel has already been written? Can someone suggest some candidates?

  153. Jonathan Vos Post Says:

    John Sidles sends us on a fascinating tangent. The case could be made that Timescape, the 1980 novel by science fiction writer Gregory Benford (and uncredited co-author Hilary Foister), which won the 1980 Nebula Award, was about Math Education, if we include Mathematical Physics and Postdocs. Similarly, EVERY novel by Greg Egan is, obliquely, about Math education (as protagonists try to figure out the nonstandard axioms of their cosmos). Films and plays do the job quite well, including “Proof” (daughter of Mathematician is home-schooled and in university) and the somewhat fictionalized “A Beautiful Mind” (teacher and student who becomes wife are both learning Math). John Forbes Nash, Jr., wife Alice, and son John all told me that they loved that film. There’s too much qualifying short fiction to list. Nor is this the place for me to talk of my unpublished novels on the subject, such as “Axiomatic Magic.”

  154. John SIdles Says:

    JVP, the stories you list all use math/science breakthroughs to carry the plot.

    But how likely is it (really) that our planet will work its way through its troubles via math/science breakthroughs?

    Math/science breakthroughs may come … but we cannot rely upon this as our sole “Plan A”.

    We have better have, also, a “Plan B” that relies on making good use of the math and science that we’ve got.

    IMHO, this will suffice.