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A Mathematician's Lament: An essay on mathematics education

  1. Nov 14, 2009 #1


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    I've found this fantastic essay (with a very apt title) by Paul Lockhart on the (sad, sad) state of mathematics education in North America.

    Here is the link: http://www.maa.org/devlin/LockhartsLament.pdf [Broken]

    It's definitely worth reading for all educators out there.

    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Nov 14, 2009 #2
    I've only read the first 7 pages so far, but plan to save this and read the rest later. It's an amazing essay. It's just how I have come to think of math, but I have never been able to express it nearly as clearly as Paul Lockhart. I wouldn't say it's just worth a read for educators, but pretty much everyone. This should be mandatory reading in school to make sure students at least get a chance to glimpse what they may be missing.
  4. Nov 14, 2009 #3


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    Very interesting. Here is what Lockhart describes as a "completely honest course catalog for K-12 mathematics."

    The Standard School Mathematics Curriculum
    LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not
    something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

    MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.
    Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.

    ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

    GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of
    students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

    ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

    TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.

    PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked
    attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.

    CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.
  5. Nov 15, 2009 #4

    Due to particular circumstances (being an ignorant dipgarbage until my mid-twenties), The only course here that I've taken was Geometry in 9th grade (standardized school testing placed me in the "upper level" of math maturity after that, which put me in the AP math section, which were no longer mandatory...so by flaw of the system, I never took another math course).

    Looking over this...many things make sense.
    Realizing in my mid-twenties that math wasn't just putting numbers into a calculator, I self studied up through Calc I.
    So, I started my "official" math education in Calc II at the University.

    What is said in this quote is very interesting to me. Seeing the strange looks when I derive a trig identity that I can't remember, trying to explain why we don't actually have to calculate something because we can just picture the graph and see that the derivative has to be zero *here*, or arguing (to refusing minds) that you can see why the derivative of 1/x is ln(x) by just calculating out the change in 1/x to see that you get the value of ln(x) all fit in perfectly if what is said here is true.

    There have been a number of times where my lack of formal education has "bit" me to some extent, but reading this makes me feel a whole lot better about myself.

    Sorry to turn the thread into a "look at me!" post, but I had to post after reading Mark44's post.
    I hope I can help my children see that mathematics doesn't have to be "that way."
  6. Nov 15, 2009 #5


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    He misses out the bit where an attempt will be made to teach 'modern maths' by having kids draw Venn diagrams. They will then stop in order to get back to 'proper maths'. This process will be repeated later when they will be expected to remember which of a bunch of weird symbols means which kind of set - but will basically be the Venn diagrams again with more complex typos
  7. Nov 15, 2009 #6
    It's a lament on mathematics education in the North America. Schools in East Asia have *VERY* different mathematical education systems.

    One thing that I find weird is that when it's been noted that students in the US generally do worse on math tests than students in East Asia, you'd think that the logical thing to do is to translate East Asian textbooks, find some East Asian math teachers to give talks, and basically change the system to work like the system in East Asia...... But no.....

    What seems to end up happening is that people look at the low test scores of US math education, and conclude that the thing to do is to teach the system that doesn't work, even harder....
  8. Nov 15, 2009 #7
    The vast majority of us have no use for but a small fraction of the education provided. How academia is so ignorant of this is beyond me.
    Last edited: Nov 15, 2009
  9. Nov 15, 2009 #8


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    It's an old complaint, in 1900 British scientists were complaining that British education in science,particularly chemistry and engineering was so far behind Germany's that British industry would never be able to compete.

    A century later it's clear that British chemical and car industries have nothing to fear from Germany's. America is in the same boat - it was able to invent the atomic bomb and go to the moon without relying on foreign education.
  10. Nov 15, 2009 #9
    And yet Americans are consistently in the top 3 by the number of gold medals on IMOs. So are Russians, whose education is very similar to what's described here.

    But we must be doing something wrong, according to Lockhart. We must drop all the useless trigonometry and quadratic equations and make mathematics an elective course. We must definitely stop teaching elementary school students to count. We can make it a high-school course (after all, numbers were regarded only a few centuries ago as too difficult for the average adult) and offer a one-year course of using the calculator.

    This reminds me of another essay:

  11. Nov 15, 2009 #10
    What is IMO?
  12. Nov 15, 2009 #11
    What does that have anything to do with any of the topics under discussion here? There are some very good things about the US educational system, but primary/secondary math education isn't one of them.

    Someone point out that if you went to 1900 and tried to find the best universities in the world, *none* of them would be American. American universities became great, by looking at the excellent universities that Germany had, and copying everything then could.

    And the facts are wildly wrong. Quick name three German car brands. VW, Mercedes, Audi. Quick, name three British car brands? Ummmm... Ahhh.. Rolls-Royce (owned by BMW). The German chemical industry is twice the size of UK. The German auto industry is three times that of UK. And remember that this is *after losing two world wars*.

    Whoa!!!! That's an incredibly silly statement.

    Fermi? Einstein? Von-Braun? Von Neumann? Bethe? Szilard? Von Karman? Edward Teller? Felix Bloch? J. Robert Oppeheimer was US-born but studied in Germany. Let's go down the names of the people that were involved in the Manhattan and Apollo projects, and see how many of them were foreign educated. Also go to any 1950's monster movie, and note that the stereotypical crazy scientist had a German accent. If you go into any research lab in the US, you'll find that most of the people there are Indian or Chinese.

    The US has benefited hugely from foreign education, since it has an open door to scientists and engineers that were educated with school systems that are a lot better than the US for teaching math, but that doesn't mean that the US wouldn't be better off it it improved the school system.

    Relying on foreign immigrants to do US science and engineering is good, but in the long run, it's not sustainable. As the economies of China and India improve, people are going to be much less willing to come to the US for education, and people in the US will be looking at going back. Right now, China is looking at the US back and forth, up and down to look at ways that it can improve it's educational system. If the US also doesn't do the same thing, then all of the improvements will be one way.
  13. Nov 15, 2009 #12
    The American system is pretty good at helping individuals who are already motivated or naturally sees through the stark presentation. The problem are the people who don't really find math as presented interesting.

    Also is the IMO really a good measure of the kind of ability an average student should have? I remember when I was at some training sessions for IMO (not the American team) there were a few people who spent so much time focusing on learning useless inequality techniques devised explicitly to work on IMO problems, but never apply to anything of real interest. This time could be so much better spent learning more general problem-solving techniques or more advanced theory, but this wouldn't give as good a score.

    This isn't what he's arguing. The most important part is that it shouldn't be forced on the student as a rule set in stone, but rather as the beautiful end-product of an interesting process. Consider solving quadratic equations. If I ask a student how to solve a quadratic equation they will reply:
    (-b +- sqrt(b^2-4ac})/2a
    But this is a spoonfed formula. Why not teach it to the students by giving them some examples to try, get them to come up with the formula themselves. As for trigonometry it's also taught wrongly. Mnemonics should play no part in the teaching of math. They should understand trigonometry, not remember the trigonometric identities. The problem isn't that they know this stuff, but that they know the formulas, but doesn't understand them.

    I don't think you've really gotten the point. The point isn't to suspend all teaching or dumb it down, but to make it appear more natural. Why not let the young kids devise an optimal strategy for playing tic-tac-toe? That would teach them something about how to split a problem into cases, how to experiment, how to THINK. Sure they should learn addition and multiplication, but try to postpone it a bit and see if you can get them to come up with algorithms themselves (maybe just for simple cases such as 2-digit + 2-digit number). Get them involved in the thought process. And for god's sake don't waste several years of their time trying to teach them to follow algorithms efficiently. I don't think there is an easy answer as to how math should be taught, but I think Lockhart makes a good point that something should be done and he points at the areas where something really is needed. I remember my math teacher in 7-grade who often set an hour aside for the purpose of letting us work on a problem. I remember working out the formula for an arithmetic progression myself, establishing some facts about angles in general polygons using induction (though we didn't name it induction), etc. This is the kind of thing more math teachers should do. Still today I distinctly remember the experience of working out the formula for an arithmetic progression; the whole class found it a fairly enjoyable experience even though few succeeded, but most had guesses, or ideas, or computations.

    An abbreviation for the International Mathematical Olympiad; an international contest for high school students.
  14. Nov 16, 2009 #13
    I really don't think that it's his point.

    It is very useful for children's minds to study math, to memorize formulas, to solve problems. Yes, math is monotonous for someone who grew up on Spongebob. That does not mean that we should replace it with song and dance. Vegetables are good for you, but, if you're used to snack on cheetos and chocolate bars, you won't appreciate vegetables. That does not mean that we should give up trying to feed children vegetables and replace them with vegetable-shaped sweets.
  15. Nov 16, 2009 #14
    Ummmm....... I don't see this

    And China is #1


    In 2009, the US ranked #6, and three of the people on the US team had Chinese names......

    One of the purposes of the mathematical education is to produce large numbers of technically trained factory workers, and it does a very good job at that. One thing that even non-math majors from East Asia note is how trivially easy college math is.

    No, because that's not how math is taught in East Asia.

    US has a wonderful system of higher education, but just like the US financial system is being propped up by Chinese money, the US system of higher education is being propped up by Chinese and Indian brains. That's not going to last for more than another ten years.
  16. Nov 16, 2009 #15
    The trouble is that you end up with large numbers of people without the ability to solve problems. One thing that is really interested in East Asian math education is that there is hardly any memorization of formula.

    I think this is the "no pain, no gain" principle. Math must be painful, because if it isn't painful then obvious it's not good for you. The trouble is that this principle is total nonsense. It's possible for a system of education to be *both* painful and useless. One reason that I don't think very highly of the US primary/secondary math system is that I've taught Algebra 101 at the University of Phoenix, and I've found myself to be half therapist trying to undo the damage of badly taught math courses and taught some basic skills that everyone in Taiwan would have learned by the 7th grade.

    If math is monotonous then it's being badly taught.

    Another useless random analogy. What's any of this got to do with math education?
  17. Nov 16, 2009 #16
    #3 in '02, '03 and '08, #2 in '01, '04 and '05. The three people with Chinese names are probably third-generation immigrants from Taiwan whose grandparents immigrated in the 60's.

    Some people are fundamentally incapable of solving problems. It's not the matter of the right or wrong approach. They will get some benefit from rigorous math treatment, but they won't become good technically trained factory workers, even in Taiwan. Dumbing down the subject per Lockhart's suggestions won't help them much, but it will lower the bar even further for the ones capable of abstract thinking.

    I heard the same thing about school students coming to the United States from Russia. American school mathematics has already been dumbed-down to a remarkable level (in many cases, using exactly the methods Lockhart describes, to make it more "intuitive") and that is definitely something that needs to be reversed.

  18. Nov 16, 2009 #17


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    The logger bit isn't a joke anymore - see;

    The problem with counting medal winners is that there is a small difference between medal winners and average.
    African-americans win most athletics gold meals - this does not translate into African-Americans having the best health in the population.
    Similarly, as described in another thread on health care, the USA has more surgeons and MRI machines than any other country - it does not mean that there are no problems in US health care.

    ps. two-fish, read the sig!
    Last edited by a moderator: Apr 24, 2017
  19. Nov 16, 2009 #18

    Ben Niehoff

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    Last edited by a moderator: Apr 24, 2017
  20. Nov 16, 2009 #19


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    That's simply insulting. Do you have ANY evidence at all to back up your claim? You're so sure that their grandparents came from Taiwan, not from Hong Kong or the mainland? You're so sure they're not one of the many children born in China (or Hong Kong, or Taiwan) who come to America and find the math insultingly easy? When was the last time you attended school? I'm attending school RIGHT NOW and I can tell you that the best math students are recent Chinese immigrants, not third-generation immigrants.
  21. Nov 16, 2009 #20


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    Where did you get this information? How many American high school students know how to express tan (x/2) in terms of sinx and cosx? Be honest: do you know? How many know the half-angle formulas ? All Chinese students who intend to study the sciences (as opposed to the humanities) are required to know these off the top of their heads. I don't know about the situation in Japan, but I assume it's similar.

    Math involves a negligible amount of memorization, considering the variety of dates, events, names, and implications that a history or philosophy student has to remember. I really don't think students are having trouble because they can't memorize a few formulas; they're having trouble because they can't understand them.

    The difference between China's math education and that of the West is that there's very little plug-and-chug in Chinese math classes: all the problems are difficult and take a lot of thinking to solve. There's also a huge, almost unbearable amount of homework because the competition for admission to university is intense. When I was in grade 6, I did about an hour of math homework every evening, plus half an hour of math at lunchtime. And how many hours does the typical American sixth-grader do?
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