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Math Education

Mathematics As the Most Misunderstood Subject 680

Posted by timothy
from the philosophical-engagement dept.
Lilith's Heart-shape writes "Dr. Robert H. Lewis, professor of mathematics at Fordham University of New York, offers in this essay a defense of mathematics as a liberal arts discipline, and not merely part of a STEM (science, technology, engineering, mathematics) curriculum. In the process, he discusses what's wrong with the manner in which mathematics is currently taught in K-12 schooling."
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Mathematics As the Most Misunderstood Subject

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  • he's right (Score:2, Insightful)

    Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.

    • Re: (Score:3, Interesting)

      by Anonymous Coward

      Yes, the problem teaching Math(s) and programming (applied Math(s)) is that it's just about intelligence - which you can't teach. The smarter you are, the better you'll be able to figure it out. The problem with teaching is all the generalists who think because they have a "Degree in Education" they are able to teach any topic. Traditionally dry disciplines need to be taught by specialists with passion and enthusiasm for their topic, not by generalists who happen to have a gap in their timetable.

      • Re:he's right (Score:5, Interesting)

        by FuckingNickName (1362625) on Wednesday December 22, 2010 @06:25AM (#34639358) Journal

        The brain can be trained and the processes of problem-solving can be generalised - see Polya's How to Solve It. But it doesn't help much to just read the book: you've got to practice, and practice, and practice some more. You must make mistakes and learn from them. You must be prepared to accept multiple inputs rather than merely those which reinforce your strengths and/or prejudices. You must sometimes, as the old 9/11 troll used to say, get some perspective - don't count the angels on a pinhead while Rome burns, even while the most secure of academic positions involves the former and there's such an alluring spirit of mental masturbation in many disciplines and departments.

        Meanwhile a good teacher has spent enough decades on some area that he knows both where to provide you hints on specific complex problems and which direction to guide you in when you're contemplating your whole professional life. But, again, don't just choose the teacher who happens to share your academic and ethical prejudices.

        • by gilleain (1310105)

          ...the processes of problem-solving can be generalised - see Polya's How to Solve It. ..

          "How to Solve It" also talks about more general problem-solving than just mathematical problems - crossword puzzles, for example. Prof. Lewis's article talks about the universal question "Why did they teach me the quadratic formula when I will never use it?" and this is really the answer; doing mathematics (should) teach people how to solve any problem logically. Well, any problem that can be solved logically, of course.

          Meanwhile a good teacher ...knows where to provide you hints

          Heh. Although a bit dry, one fun part of the book is where Pólya talks about givin

          • by rjstanford (69735)

            "How to Solve It" also talks about more general problem-solving than just mathematical problems - crossword puzzles, for example. Prof. Lewis's article talks about the universal question "Why did they teach me the quadratic formula when I will never use it?" and this is really the answer; doing mathematics (should) teach people how to solve any problem logically. Well, any problem that can be solved logically, of course.

            Then why not teach logic and problem solving, possibly using mathematics as the language (but not necessarily)? When we tell ourselves that we're teaching maths, that's all people tend to teach (and learn, for the most part). I agree that teaching logic and deduction is valuable, more valuable than a lot of mathematics to many people (since with skills you can get the maths, but not necessarily vice versa)... but its rarely seen called out on a school curriculum. And that's a shame.

      • heh, that is one of the most unrealistic comments i've seen in a while... specialist teachers with a passion and enthusiasm for maths? i've only ever met one, and he sucked balls as a teacher. maybe we should start recruiting fairies or goblins, maybe they'll get the job done.
        • by digitig (1056110)
          I had teachers with a passion and enthusiasm for maths all the way through school. Condolences for your experience.
          • by tehcyder (746570)

            I had teachers with a passion and enthusiasm for maths all the way through school. Condolences for your experience.

            I had various teachers with passion and enthusiasm for chemistry, geography, German, English literature, physics, Latin, history, PE and maths, and none of them were PhDs or anything, just good teachers.

            • by Stooshie (993666)
              A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.
              • Re:he's right (Score:4, Insightful)

                by vlm (69642) on Wednesday December 22, 2010 @08:35AM (#34639950)

                A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.

                There is also little if any correlation in being able to research, and being able to teach. Culturally, "everyone knows" the purpose of a phd is to become a professor and teach university students while collecting a $100K+ salary. The upper 50% to 10% cream of the crop actually get hired to do that. So, pretty much by definition, as a general cross section of the population, they are in the bottom of the barrel of teaching ability. So I'd be expecting, unless they're education phds, they're almost by definition probably not going to be good teachers.

                • Really? At the University I attended it was crystal clear that Professors were hired primarily to do research and that teaching was a secondary consideration.

                  Oddly enough their teaching skills were distributed about the same as my high school teachers who were hired to teach and only to teach. That is to say a few were excellent teachers, some were good, the bulk were acceptable and a few were flat out terrible. What we learned from bad teachers was that a bad teacher or professor can't stop you from lear
        • Re:he's right (Score:5, Interesting)

          by CProgrammer98 (240351) on Wednesday December 22, 2010 @07:50AM (#34639738) Homepage

          You should have had Mr Burton, my maths O level teacher. He was brilliant. He was totally passionate about his subject and he was also a fantastic teacher. he encouraged us to think about maths rather than to just blindly follow formulae. I still vividly remember the lesson where he taught us differential calculus from first principles.

          He encouraged us to study outside of lesson time and his door was always open during lunch, or after school. almost every one in his class passed their maths O level with at least a B, over half had A's

          It's no exageration to say I owe my career as a developer to him and his enthusiastic teaching.

          • by jc42 (318812)

            You should have had Mr Burton, my maths O level teacher. He was brilliant. He was totally passionate about his subject and he was also a fantastic teacher. he encouraged us to think about maths rather than to just blindly follow formulae. ...

            You were lucky to have such a teacher. But there are other ways that can work, too.

            Back when I was a high-school sophomore, I decided that math was interesting, so I read that year's math text in the first month, then grabbed copies of the more advanced texts over the following months. By late winter, I'd run out of math texts that the high school had, and asked the teacher for more. The reply was the conventional "You're not ready for those yet", which was clearly BS, but was supported by the other teac

    • Re:he's right (Score:5, Interesting)

      by ShakaUVM (157947) on Wednesday December 22, 2010 @06:29AM (#34639388) Homepage Journal

      >>Mathematics is the foundation for philosophy

      Eh, kinda. Advanced logic is the foundation for a lot of modern philosophy, but Wittgenstein and the rest of the 20th century analytics were just responding to the tremendous success of physics at figuring shit out, and wanted to smear some of that patina on themselves. Well, logic has always been a part of philosophy (think Socrates and his syllogisms) but reading the Tractatus is like reading a modern computer science proof.

      Which isn't surprising, either, given that computer science is essentially applied philosophy in a lot of ways. (cf Bertrand Russell, etc.) If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".

      It does kind of bug me though, that a person who graduates with a degree in mathematics (which is a fairly difficult, hard-nosed subject) gets a wishy-washy BA degree, whereas a hippie with a degree in "environmental engineering" gets a BS, but ultimately I think there's a lot of problems with our current conception with categorizing things into "science" and "not-science". Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science. There's also a huge debate now over Anthropology, after the American Anthropology Association dropped "science" from its official bits.

      • Re: (Score:3, Insightful)

        Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science.

        Well, economics is, especially in its present state, largely influenced by individuals, who can be a lot harder to predict than wind currents. You may identify trends, constants and correlations, but mostly in hindsight. Accurate predictions are as scarce as in cartomancy and useful controlled experiments are hard to imagine. While Climatology shares some of those characteristics, I think we have a much higher chance of predicting a storm than the stock market. Unless tons of people start walking around wit

      • Re:he's right (Score:5, Interesting)

        by grouchomarxist (127479) on Wednesday December 22, 2010 @07:41AM (#34639686)

        If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".

        I understand where you're coming from, but for many philosophers, what they're doing is not just trying create a practical solution to a problem, but describe reality. Your object model might solve the problems from your point of view, but it includes many built in assumptions about the thing modeled.

        In a related way Wittgenstein later came to criticize the Tractatus. Part of the criticism is that if you assume the universe can be fully described with formal logic (logical atomism), then you are already subscribed to a certain type of metaphysics.

        • Re:he's right (Score:5, Insightful)

          by 140Mandak262Jamuna (970587) on Wednesday December 22, 2010 @09:02AM (#34640100) Journal
          In mathematics the reputation of Wittgenstein or Tractatus would not matter at all. The argument, "A great mind, everyone agreed that the mind was great, said this, so we should give this saying more credence" does not hold water.

          In mathematics it is the truthiness of the statement creates "credit" and then we search back in history to find who said it first and then we give the credit to him/her and that is how reputation/respect is created. It flows back in time. Credibility accrues from the statement to the speaker.

          In philosophy a bunch of people agree that some one was/is a great philosopher and so they give more value to a statement from such person. The credibility flows from the speaker to the statement.

          • Re:he's right (Score:4, Insightful)

            by NoSig (1919688) on Wednesday December 22, 2010 @10:06AM (#34640560)
            That happens a lot in mathematics too - it has to, or mathematicians would have to spend all their time refuting amateur "proofs" of famous open problems.
            • by lgw (121541)

              Actually, there was this weird thing going on in Math as a field for much of the 20th century: reinventing Euler. Euler was so very far ahead of the field that odds were that anything you discovered for the next couple of centuries had likely already been discovered by him - thus the saying that theorems are named for the first person after Euler that discovered them.

              But math didn't devolve into a "study of Euler", instead the field plowed ahead happy to rediscover ideas from first principles instead of ju

      • Re:he's right (Score:5, Insightful)

        by tehcyder (746570) on Wednesday December 22, 2010 @08:08AM (#34639828) Journal

        I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers"

        Well, go on then, if it's that fucking simple and obvious. Put those silly old philosophers in their place, what do they know?

        I'm thinking of writing a book called "Why do so many students of Computer Science think they have solved all the riddles of the universe because they know how to write a sorting algorithm?"

    • Re:he's right (Score:5, Insightful)

      by ultranova (717540) on Wednesday December 22, 2010 @06:38AM (#34639434)

      Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.

      Well, we would likely all be malnourished, due to lack of fertilizers, at least those of us who hadn't died at childbirth or soon after. There wouldn't be an Internet to talk on, but that would be okay, since we wouldn't have time to use one due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day. In short, our lives would be miserable, but due to lack of medicine, they would at least be short.

      Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

      Philosophy means you accept the human condition. Technorcacy means you try to do something about it. Hope for a better world in the future lies on the latter, not the former.

      • Re: (Score:3, Insightful)

        So over the past two millennia we have cut the working day by 1/3rd and doubled the average lifespan at birth (if you ignore infant mortality, our lifespan hasn't increased that impressively).

        Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets. We've made up for the opportunity to live a life of leisure surrounded by virtually infinite resources by blasting our population beyond 6 billion.

        Technocracy is for the lazy man w

        • Re:he's right (Score:5, Insightful)

          by ifiwereasculptor (1870574) on Wednesday December 22, 2010 @07:39AM (#34639672)

          Why are people even debating philosophy vs technocracy? Why should someone have to choose one over the other? How do people get dragged into such nonsense? Here a new subject for you: tomatoes vs rainbows. Go.

          • by tehcyder (746570)

            Why are people even debating philosophy vs technocracy? Why should someone have to choose one over the other? How do people get dragged into such nonsense? Here a new subject for you: tomatoes vs rainbows. Go.

            The slashdot hivemind divides the world into a series of either/or choices, e.g. emacs/vi, or pro/anti-Windows

        • by Viol8 (599362) on Wednesday December 22, 2010 @07:58AM (#34639782)

          "Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets."

          I suggest you take off your rose coloured glasses and go read some history, in particular just how "free" your average serf was in feudal times and even later. Don't like what your overload or king does? Tough. Complain and you'll probably at best end up homeless or at worst end up swinging from a tree.

          People in the west have NEVER been as free as they are now.

          So get yourself a fucking clue!

          • Oh, the "we're free because of the First Amendment" fallacy.

            Here's an anecdote from my history book: half of my family comes from fascist Spain, my grandfather a bootmaker and my father asked to do the "backbreaking labo(u)r in the fields" as a boy that everyone likes soundbiting. It seems that lacking the right to whine and be ignored didn't affect either their boldness or sense of freedom nearly as much as today's centralised and surveilled management of corporation and culture.

            • by geekoid (135745)

              Your an idiot. It's really that simple. You've buried yourself in a false premise and refuse to see out relying and anecdotes so far out of context to be useless.

              You offer no arguments, only logical fallacies and irrelevant statements.

              The irony is that you are putting forward philosophy but can't make a logical argument.

          • by mrsquid0 (1335303)

            Swinging from a tree was not the worst thing that could end up happening to a serf who tried to be an "independent man". Many kings and lords were much more sadistic than that when it came to punishing serfs who disagreed with them. The idea that people in the past were generally free-er or more independent than they are today (at least in most democracies) is laughable.

            • What do you think happens today to the average blue collar worker who suddenly decides he's not willing to play society's game? Which of today's government and the Lord of the Dark Ages you are so keen on generalising to everywhere-before-1900 has more resources to catch that man? If a man today, in the middle of the US, considers gathering a group of men to start an uprising to fix the ills of local, regional or national government, do you think he is more or less likely to succeed than a man five hundred

          • by ObsessiveMathsFreak (773371) <obsessivemathsfreakNO@SPAMeircom.net> on Wednesday December 22, 2010 @08:59AM (#34640092) Homepage Journal

            People in the west have NEVER been as free as they are now.

            I don't know. I think we were all a lot freer and happier in the 1990's.

            No Cold War, no War on Terror, no internet filters, no monitoring of habits, no Google Maps/Mail/Panopticon, less sex offender scares, less evolution/abortion debates, less religion, less jihad, didn't hear about "markets" half as much, less news pundits, less foreign wars/quagmires, no Super-China, no airport scans, more newspapers, and Star Trek: The Next Generation was still showing on most terrestrial channels. Sure it wasn't perfect, but it was better than it is now--not that the general public actually gives a shit.

            • by sjames (1099)

              What's interesting is that NOTHING physical has changed and yet there's been needless suffering everywhere. We didn't run out of anything and the crops didn't fail. We have about as much resources now as we ever did, yet because some idiots think a few numbers on a balance sheet are more important than physical reality or human suffering we now have a bad economy.

            • by shermo (1284310)

              Did you know that the first Matrix was designed to be a perfect human world? Where none suffered, where everyone would be happy. It was a disaster. No one would accept the program. Entire crops were lost. Some believed we lacked the programming language to describe your perfect world. But I believe that, as a species, human beings define their reality through suffering and misery. The perfect world was a dream that your primitive cerebrum kept trying to wake up from. Which is why the Matrix was redesigned t

          • "People in the west have NEVER been as free as they are now."

            Eh, that's pretty iffy.

            It would be more accurate to say that people in the West have never been better off in terms of material wealth, true. We've never had as high a level of technology or cheap access to gadgets or advanced medicine.

            But free? I guess it depends on your definition of freedom. We're certainly more free than the Russian serf of the 1700's or the Spaniard under the Caliphate of the middle ages or the Greek and Serbian living under

      • Re:he's right (Score:5, Insightful)

        by CRCulver (715279) <crculver@christopherculver.com> on Wednesday December 22, 2010 @07:21AM (#34639592) Homepage

        due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day.

        While agriculture requires backbreaking labour, hunter-gatherer societies only worked a couple of days a week. Not that I advocate a return to it, but backbreaking labour all the livelong day was not universal in ancient society.

        As far as I can tell, most of them chose their field because it doesn't punish sloppy work.

        Philosophical journals have the same rigorous standards for papers as journals for the various sciences. Your view of philosophy is about as valid as a grizzled mountain man who mutters about hard science being all book-learnin' and mumbo-jumbo.

        Philosophy means you accept the human condition. Technorcacy means you try to do something about it.

        Even that is a statement of philosophy. Furthermore, you seem unaware that many calls for improving human lives came from works of philosophy: More's Utopia, Kirkegaard's questions of metaethics, even what is often called the beginning of the Western tradition, when Socrates hung out in the agora and asked passersby "What if what you comfortably believe is wrong?"

        • While agriculture requires backbreaking labour, hunter-gatherer societies only worked a couple of days a week.

          Only thought to be the case by Europeans who didn't think that hunting was "real work".

      • Re: (Score:3, Insightful)

        by Kashgarinn (1036758)

        Philosophy means you accept the human condition.

        No.. Philosophy means questioning the human condition. it's confronting the status quo and asking "why?"

        So exactly the opposite in every way of what you think it is.

        You're also wrong in your assumption that philosophy and technocracy are mutually exclusive, in fact if they aren't mutually inclusive, then as a technocrat you're trying to find solutions when you don't even know what the problem is.

        Philosophy is a very powerful way of thinking, and in no way whatsoever does it represent conformity or acceptan

      • Re:he's right (Score:4, Interesting)

        by horigath (649078) on Wednesday December 22, 2010 @07:43AM (#34639694) Homepage

        Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

        There's plenty of philosophy-types who think that Searle is an idiot, too, for the Chinese Room and other things. Guy loves to position himself as a defender of rationality and realism because it lets him belittle poststructuralists with oversimplifications and straw men while acting like a hero of a scientific worldview that he clearly doesn't know that much about.

        In some ways his antagonistic materialsm is quite similar to your dismissal of philosophy in general, actually.

      • Re:he's right (Score:5, Insightful)

        by digitig (1056110) on Wednesday December 22, 2010 @08:06AM (#34639816)

        Missing these kinds of little details is why I have very little respect of philosophers.

        They don't "miss" those details, they're not in scope.

        As far as I can tell, most of them chose their field because it doesn't punish sloppy work.

        Philosophy does punish sloppy work. relentlessly. Philosophical work is subject to more scrutiny and criticism than any discipline I know of, and that includes pure maths.

        And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

        Laughing something out doesn't work in philosophy. Unlike whatever discipline you work in, it seems, in philosophy you have to show the reasons why something is wrong. And if you think the issue of emergent properties hasn't been considered in excruciating detail in connection with Searle's Chinese Room thought experiment then you clearly have no idea what philosophy is doing.

        Philosophy means you accept the human condition.

        Say what? Some philosophy is abstract, but so is some maths. Lots of philosophy (philosophy of science, political philosophy, ethics) is concerned with changing the human condition. Maybe you criticise philosophy because it didn't discover antibiotics (although it did lay a lot of the foundations), but do you criticise biology because it didn't invent democracy? Both changed the human condition, in ways appropriate to their respective disciplines.

      • by hey! (33014)

        Well leaving aside the dubious notion that studying applied subjects is really pursuing "technocracy", I think we're engaging in a bit of false dichotomy here. You don't have to choose as an individual or as a society to pursue liberal arts or applied arts; to study philosophy or to study engineering.

        The medieval liberal arts curriculum had two levels. The Trivium consisted of grammar, logic and rhetoric. These are the basic tools of expression, thinking and persuasion. A student versed in the Trivium c

  • by pieisgood (841871) on Wednesday December 22, 2010 @06:29AM (#34639380) Journal

    I can attest that "true" math is very removed from computation. The computational classes are all regarded as the "easy" classes. This is in contrast to the "hard" classes, real analysis and abstract algebra. Being thrown into real analysis after just one quarter of study in proofs is extremely rough going. If proofs were introduced as puzzles or just introduced earlier in education the whole of America would be better off for it.

    My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.

    In any case, at least math attracts the curious.

    • My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.

      Trigonometry predates calculus by a long time (see Ptolemy's table of chords [rutgers.edu] which were calculated purely geometrically, since algebra wasn't invented then either). Trigonometric functions are incredibly rich and important, there are so many different ways of looking at them, and

    • by Fnkmaster (89084)

      At Harvard, at least back in the day (circa mid-1990s), the boys were separated from the men in the first semester of math freshman year.

      Those who thought they were hot shit all started in a class called Math 25/55 and were beaten down with point set topology and real analysis. Those of us who had never gone beyond AP Calculus BC, or even multivariable calculus, in high school got our asses handed to us rapidly.

      It was basically all kids from math- and science-focused honor schools who had been exposed to p

  • by LambdaWolf (1561517) on Wednesday December 22, 2010 @06:33AM (#34639402)

    I've seen the following link in many a Slashdot thread before, but it certainly bears repeating here: "A Mathematician's Lament" by Paul Lockhart [maa.org] It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way. I think it would be great if more students saw math as something worth doing for its own sake, like art or athletics, and hey, it lets you do science and engineering too.

    In fact, this summary sounds similar enough to "Lament" that I wouldn't be surprised if this Dr. Lewis was inspired by and/or cited it. But this is Slashdot, so I'll let someone else check that out.

    • by dcollins (135727) on Wednesday December 22, 2010 @07:49AM (#34639734) Homepage

      As a part-time college math teacher, I almost totally disagree with Lockhart's Lament. (Ironically, the K-12 school where he teaches is close to the neighborhood where I live.)

      It's not that it's bad to see that math can be an art and a pattern-finding exploration (some part of the time), but someone has got to teach and be held accountable for the nuts-and-bolts of how to read and write mathematical vocabulary, notation, and justification (algebra and geometry, for starters). Knowing about the scientific method is necessary, but exclusively spending your K-12 time re-inventing the wheel is inefficient at best. It's the same problem as in English nowadays -- I was told last weekend that teachers in junior high schools are forbidden from teaching the rules of grammar. That is, it's exclusively about expressing "big ideas", no matter how poorly-formed or unreadable. The more this produces crippled students, the more we seem to run deeper in the same direction -- if you abandon teaching the basic structure of our shared communication systems, then we thereby just generate more and more unreadable nonsense as time goes on.

      The remedial math I teach (basic algebra; about half my assignment load) is almost entirely about just reading & writing. Even the first unspoken step of simply transcribing symbols (i.e., an expression) from one page to another is almost impossible for about half my students, because no one has ever asked for any level of precision in their reading, writing, or observation skills (whether in English, math, or anything else). To me, basic math is an opportunity to focus on precision in thinking and writing -- applications belong in other classes! No, that's not what a professional mathematician works at on a daily basis, but frankly, not every K-12 class can be an independent research opportunity. At some point you've got to eat your vegetables, and if you run entirely away from that, then it truly is a monumental waste of time.

  • by Chrisq (894406) on Wednesday December 22, 2010 @06:36AM (#34639426)

    I have a cousin who is great at mathematics, and really can see mathematics as an art. Whereas I am happy if I can solve a problem, he will look for an "elegant solution". I had a number of equations that I solved, trying to optimise the buffer size for various input queues. I shown him, and he quickly said that I had the right answer. A day later he came and shown me how he derived an equation that could simply solve all problems of this type. He also generalised it to allow buffer sizes that were complex numbers. The first part was very useful to me, the second absolutely useless - but to him it was all just interesting.

    This is one way that mathematics as an art is unlike any other art. It gives useful results. I have heard time and time again about engineers going to the mathematics department of a University asking how they can solve a "new" problem - to be told that the solution had been discovered a century before. I am sure most of these solutions came from someone just wanting to find an elegant way of expressing something without thought of any use. So if its an art and is useful why do so few people follow it?

    The answer is obvious, because its hard! In many forms of art you can slap anything down and convince someone that it has value and its art. This may not always have been true, before photography accurate representational art was highly valued - but today someone producing a lifelike portrait will not be values as much as someone slapping their name on an unmade bed! Mathematics has to be right, you can't just slap down a few numbers and call it an equation. This is the basic problem that anyone will have in persuading someone to follow maths for its art, there are a lot easier ways to become an artist.

  • Not just maths (Score:5, Interesting)

    by Schiphol (1168667) on Wednesday December 22, 2010 @06:41AM (#34639454)
    I wish science in general was considered part of what a learned person has to know. I mean, if you want to pass for an intellectual you have to read your Dante, your Beckett and you at least need to know who Lautreamont was. But, apparently, you can very well get away with thinking that you can suck gravity out of a room the way you suck air, or with not having even heard about string theory. That divorce makes no sense, and it was impossible in the history of ideas till very recently. And Euler's formula is more beautiful than most poems.
  • by dtmos (447842) on Wednesday December 22, 2010 @06:46AM (#34639462)

    This is by far the best defense of mathematics I've ever read. It's a shame that the poor quality of grade school math education has made it necessary, though. Can one imagine a similar essay on any other subject? Only math is so poorly taught.

  • -- The parenthetical comment "(if it was done right!)" in "Ready For The Big Play" should, of course, be, "(if it were done correctly!)"

    -- References in "Cargo Cult Education" to the "south Pacific" should be to the "South Pacific"

    -- Also in "Cargo Cult Education", "But of course nothing came. (except, eventually, some anthropologists!)" should be, "But of course nothing came (except, eventually, some anthropologists!)."

  • Simple that... I honestly can not understand where there can be "beauty" in a mathematical expression that covers the entire blackboard. And more so when the teacher fails miserably to show practical uses for the expression.
    • by gilleain (1310105)

      There was a BBC4 program recently called "Beautiful Equations" where an art critic went round various mathematicians asking about E=MC^2, F=G(m1m2/r^2), S=A/4, and er the Dirac Equation.

      The point about most of these examples they chose - apart from being conveniently in the UK - was that they were short. Also that they are directly related to important ideas about how the Universe works. So mass can be converted to energy, bodies attract each other, black holes can shrink, and antimatter exists. Dirac was p

    • by dtmos (447842)

      I honestly can not understand where there can be "beauty" in a mathematical expression that covers the entire blackboard.

      No one else can, either.

      The beauty is in the simple relations between apparently unrelated things that, while provably true, still seem magical and mysterious. One example:

      You're probably aware that the ratio of the circumference to the diameter of a circle has been given a special name, pi. This is a practical, useful thing that seems purely geometric; you can measure the diameter of a circular hole, multiply by pi, and get the circumference of the hole. Fine.

      Well, it was shown in the 17th Century (!) t

  • by tonywestonuk (261622) on Wednesday December 22, 2010 @07:12AM (#34639568)

    I remember been taught differentiation at school – One lesson, lecturer puts a parabolic curve, x=y*y, on the board, and asks the problem, determine the angle of the line

    Then, he didn’t say anything else.. Just, for the rest of the lesson, responded with ‘Yes’, ‘No’, or ‘Maybe’. So, after a frustrating 20 minute discussion, trying to work out how the hell to do this problem, someone came up with the idea of adding a ‘little bit’ of x, to x..

    We worked out, as a group, the concept differentiation, with only the smallest bit of guidance from the lecturer. This is how things should be taught – allowing people to discover concepts themselves, rather than preaching the correct ways to do things.

    • by dtmos (447842)

      Brilliant -- a live version of "A Pathway into Number Theory [slashdot.org]". That's the kind of teaching for which awards should be given.

  • "A Mathematician's Lament" [maa.org], an article that's been making the rounds among mathematicians since 2002 (but was only published in 2008), expresses some similar views, and is also a good read.

  • Recommended and relevant reading is "A Mathematician’s Apology" by G. H. Hardy.

    Available online at http://web.njit.edu/~akansu/PAPERS/GHHardy-AMathematiciansApology.pdf [njit.edu]

  • The first part comprises the results of previous work by mathematicians; the finished product. That's what underpins most of physics and engineering nowadays.

    The second part is the "live" Mathematics, i.e. the process of actually doing Mathematics in the sense of figuring something out. That's a slow, arduous, iterative and groping process. Starting with an observation that confuses or amazes us, incrementally and tentatively formulating concepts (definitions, constructs of previously known mathematics),

  • Hi,

    in school mathimatics is mostly execution of algorithms provided by your teacher, learning when and how to apply them. This changes a lot with university. At first, mathematics is a language to be learned. You have to be able to express your problems in a normed language. This is the first art. If you read papers, you can distiguish easily between those peoples who truely have mastered that language and those who don't have. Later on, you learn how to prove things. The interesting things you cannot prove

  • by Charliemopps (1157495) on Wednesday December 22, 2010 @08:05AM (#34639810)
    My highschool math teacher was a retired NASA programmer. According to her, teaching Mathamatics was about leaching logic and problem solving. If you forgot all the formulas taught in her class, she said, it wouldn't matter. The real skill learned was how to deal with an entirely new mathematical problem. WHY is area "height x width"? How to build your own sort of equations. Sure enough, decades later I have forgotten every single equation I had been taught there, but when faced with a logic problem I'm still able to work it out.
  • by AB3A (192265) on Wednesday December 22, 2010 @08:46AM (#34640012) Homepage Journal

    In practice, there are two forms of teaching. The first is applied subject matter in school. In this specific case, it is applied mathematics. They give you the calculation tools for describing a relationship and then they expect you to find similar relationships and apply that formula. The goal is to teach the use of a tool. It is no different than teaching one to write a coherent paragraph, communicate in a foreign language, or to be a good citizen in a democracy. Teaching applied mathematics is a necessary element of any school curriculum.

    The second is one of discovery. My journey began as a teen, when I read about fractals in an article from Scientific American. Since then I've gone on and explored prime number theories, methods of calculation, the history of these discoveries, and I've gone looking for the blind alleys that may not have been explored as thoroughly as we might think.

    We need to recognize that education is not about discovery. It is about teaching a person the tools of modern society. However, in our zeal to teach the applied aspects of these subjects, we need to realize that we are failing to nourish the creative spirit of discovery. Mathematics is no different than reading, writing, civics, history, geography, or language. Learning to write a coherent text does not make one appreciate literature.

    Our schools are obsessed with application, not discovery. We spend ridiculous time teaching application, application, and more application. Then we sit and wonder why our children lack the will to explore...

  • by gringer (252588) on Wednesday December 22, 2010 @08:48AM (#34640032)

    This article frustrates me. He talks a lot about some particular thing, claims that it relates to maths, but doesn't really say what particular part of maths it relates to, nor does he get into specifics, nor does he spend much (if any) time on how to improve matters.

    Okay, I'll try to explain my confusion with a parable. When I was fifteen, I did a school certificate maths exam. It had a whole bunch of questions, none of which we had ever answered earlier in the year, but somehow the examiner thought I could answer them, and unfortunately I was unable to answer all questions "correctly" according to the examiner.

    What does that have to do with mathematics education over the past 25 years? Unfortunately a great deal. We were required to have exams for mathematics, because every subject had exams. The end result was that some people didn't do well in exams, even failing enough to be unable to continue on in their maths education in the next year. The truth is that exams cannot alone be used to evaluate a person's effectiveness as a mathematician. The only way to get around this is to teach mathematics properly, and make sure each person understands maths at all levels.

  • Math is just...math. (Score:4, Interesting)

    by wickerprints (1094741) on Wednesday December 22, 2010 @10:15AM (#34640646)

    Really. Must we contextualize mathematics, or try to talk about what it is or is not? Do we really need to point to a particular cognitive framework as "the reason" why math is not taught "properly?"

    To use a slightly loathsome phrase, math "is what it is." Instead of talking about how people should relate to it, I suggest a radical approach: just LEARN it. Teach it for what it is.

    I struggled with arithmetic when I was in grade school, not because I didn't understand the rules, but because I kept making mistakes. And my teachers had the wisdom to know that those errors had to be drilled out of me before I could proceed any further. I suffered. I *hated* the tedium. We were asked to multiply two twelve-digit numbers with no assistance from any computing devices or tables; divide four-digit numbers into twenty-digit numbers, until we could do it with 100% accuracy every time. It didn't have to be lightning fast. It just had to be CORRECT.

    And when I mastered that skill, it felt fantastic. We moved on to more advanced topics, and each time the teacher made sure we had firmly laid down the next conceptual brick of this vast mathematical edifice we were building for ourselves. It was hard but rewarding. To those critics who might say such an approach would discourage some students, and that some kids just need to be excited by what they learn, clearly you have never really understood what it means to build that foundation. It's got to be ROCK SOLID. No crap about trying to make math "fun" or "interesting" or "relevant." That sort of stuff comes when it comes; they are merely ornaments on the pillars. There's no point in making the structure pretty before you make it sturdy.

    So then, how do you get students motivated? It's really quite simple. You challenge them and you force them to bust their asses, and when all their hard work pays off, that sense of accomplishment is better than any drug. To know that you did it on your own, and you have complete confidence in your mastery of the concept, is precisely what must drive them forward. You can't entice them with anything else. You can't try to swaddle the math in some cutesy real-world application, because that is going to be fake, and they know it.

    That's the story of how I graduated with my BS in mathematics from one of the most prestigious scientific universities in the world. It was purely the early appreciation for persistence toward understanding mathematics for its own sake. I'm not saying everyone has to keep math "pure." If your goal is to apply it in some other discipline, go for it. But the learning process has to build upon that foundation of math for math's sake.

  • by Jason Levine (196982) on Wednesday December 22, 2010 @10:17AM (#34640660)

    My older son is in the 2nd grade and is gifted (IQ somewhere around 140). Right now, they're learning simple addition. There's only one problem. He already learned this last year. He was doing complex subtraction with my wife (a teacher) over the summer break. But the class is doing simple addition so that's what he's stuck on.

    It gets worse. They're using a so-called "spiral curriculum" this essentially means they learn one way of figuring out that 8+3=11, then learn another way, then a 3rd, 4th and 5th way. My son gets it the first time, yet he has to sit through all of the other ways. He yearns for more advanced math. He asked me about multiplication and division and, when I showed him an example using Legos, he got the concept right away.

    He already knows his times tables up to 5 and wants more. But school is boring to him because they don't push him. He isn't being challenged at all. He tends to act out when he's bored too which makes everything more complicated. If you have a child who is falling behind in school, there are resources to help them catch up. If you have a child who is gifted and wants to pull ahead, your kid needs to sit down, be quiet and learn for the fifth time what 8+3 equals.

    • by Buelldozer (713671) <cliff@gindu l i s . net> on Wednesday December 22, 2010 @12:04PM (#34641874)

      I logged on for the sole purpose of replying to your post as our situations are so similar I couldn't let it pass without comment.

      I realized in 1st grade that my son was the same as yours. His IQ doesn't test quite as high, somewhere around 130, but he has an intuitive grasp of certain things that's almost breathtaking. I remember when he, at 5, described to me the mechanics behind a lunar eclipse! It wasn't even a topic of conversation, just out the blue. Apparently he had been mulling it over and had worked it out. Anyway, back to the subject.

      Let me say you rock as a dad, not only for noticing the problem but working with your son. My son has also been subjected to the "spiral curriculum" and it's alternately made me want to rage or laugh. Far too much time is spent teaching different ways to accomplish the same tasks and there is no way to speed it up for those who are bored. I solved this problem by advancing the curriculum at home. When my son got bored with addition and subtraction I made the numbers bigger, when that became trivial I made them harder by including decimals, then harder again by using fractions. When he became bored with multiplication and division I started teaching him Algebra. When his class moved on to kiddie Geometry and he grew bored with it I started him on Geometry I. You get the idea. It was in Geometry this year where the teacher caught on to me.

      His teacher and I had a major blowout when one of his Geometry papers was returned with a score of zero. My son was freaked and so was I. What did I do wrong? I went back and forth through that paper for two hours looking for what had happened and couldn't find it. I called in the wife who has a Degree in Math and she couldn't find anything. I called in the Grandpa with dual Masters (Chemistry and Physics) and 45 years experience as a High School teacher and he didn't find anything. I went to the school the next day and had his teacher explain why and you know what the answer was? He forgot the damn degree symbols. Yes, that's right a 10 year old doing math work years ahead of his level received a zero with a page full of correct answers and a companion page showing all of the work because he FORGOT THE DAMN DEGREE SYMBOLS.

      Further she told me that she didn't like me teaching him this stuff because my way was different than hers which made it difficult for her to grade his papers and he confused other children when he tried to help them! I didn't know whether to cry or murder her. The depth of willful stupidity on display at that moment still staggers me. In the end I politely told her that I wasn't going to stop doing it because his education was more important than her classroom. I left shaking my head and wondering how our education system got this screwed up.

      So you keep rocking on Dad, you keep pushing his curriculum and teaching him. What the idiots at the school won't do for him is your privilege, and responsibility, to provide. When he grows bored you up the ante and make it more challenging by showing him the "Big Boy" way and giving him something new to explore. Someday when he outgrows your ability you can sit back and proudly tell him "Son, I don't have anything left to teach you." and then watch him start learning it for himself.

      • Harry Chapin (Score:4, Informative)

        by geek2k5 (882748) on Wednesday December 22, 2010 @03:04PM (#34643984)

        The mindset of the teacher reminds me of the Harry Chapin song "Flowers Are Red."

        Teachers that are that narrow minded should be transferred to places where they can't do any damage to students. Perhaps a prison environment would be best for them. They could at least try to help some of the people they screwed over.

  • by Animats (122034) on Wednesday December 22, 2010 @03:22PM (#34644164) Homepage

    Some in their 50s or so may remember "New Math", which was an attempt to teach elementary math with more emphasis on the underlying theory. It's now widely considered to have been a disaster. The author of the original article seems to date from that era.

    One of the approaches to fundamental mathematics is to start with axiomatic set theory and build up from there. (That's not the only approach; one can also start with the Peano axioms and build up to set theory via lists, as is done in constructive Boyer-Moore theory.) This is minimalist and elegant (which is why mathematicians like it) but it requires considerable theoretical development before you get to addition. Teaching kids arithmetic that way was a disaster.

    Euclid's approach to axiomatic geometry is like that, too. There's a lot of abstract logical structure that has to be built up before you can do anything. That's how math was taught up to 1900 or so, and 7th grade geometry is still often taught that way.

    That's the "liberal arts" approach to mathematics. It's an intellectual exercise forced onto little kids. Even if you use advanced mathematics in your work, it's very rare to need either axiomatic set theory or axiomatic plane geometry.

    A completely different approach can be found in some math courses given during WWII courses to soldiers who needed to do technical work. These were utterly practical. Trigonometry was taught with direct applications to surveying and static structural analysis. After that trig course, you could calculate the size of the beams required for a truss bridge. The calculus course covered subjects like the ballistics of big guns. (I especially liked the "tables method" of integration, which taught you how to use those tables of integrals in the back of the book.)

    There's a mindset in math teaching that math is about "puzzles". It's not. (Mathematics in England at the university level went off into that dead end for a century, with rated "wranglers" and "senior wranglers", until Hardy kicked them out of it.) But the school version of mathematics overstresses puzzles, because they're easy to assign and grade. That's a bigger problem than the "liberal" aspect.

    For a non-puzzle curriculum, see PSSC Physics, which was taught in the 1960s. Lots of little experiments which required some calculation and data analysis.

[Crash programs] fail because they are based on the theory that, with nine women pregnant, you can get a baby a month. -- Wernher von Braun

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