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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)

    by FuckingNickName (1362625) on Wednesday December 22, 2010 @05:07AM (#34639294) Journal

    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.

  • by Joce640k (829181) on Wednesday December 22, 2010 @05:29AM (#34639390) Homepage

    Basic math is easy enough for nobody to have an excuse for not knowing it.

  • by AliasMarlowe (1042386) on Wednesday December 22, 2010 @05:31AM (#34639394) Journal

    (A+B)(A-B)=A(A-B) , divide like terms

    Divide by zero error! After this point, every conclusion is invalid since the results are undefined.
    Depressingly, some people (adults as well as kids) would not spot that.

  • by Palmsie (1550787) on Wednesday December 22, 2010 @05:35AM (#34639418)

    This is exactly the kind of thinking that has got us into the mess we're into now.

    Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner. No one is born knowing any of those things, therefore they all must be learned. The entire point of the OP is to say that the way we go about teaching math is wrong and that people need to reconceptualize how they teach the information because it doesn't make sense to the learner. In the end, its all difficult to some degree. It's when you have that "A-Ha!" moment, it clicks, and you get it. But if you have some terrible algebra teacher who doesn't understand advanced math or someone who doesn't care that you learn, only that you can complete problems 1-50 in a mechanic fashion, then of course it's going to seem difficult (or more difficult than it should be).

  • by Chrisq (894406) on Wednesday December 22, 2010 @05:36AM (#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.

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

    by ultranova (717540) on Wednesday December 22, 2010 @05:38AM (#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.

  • by txoof (553270) on Wednesday December 22, 2010 @05:43AM (#34639460) Homepage

    The way math is taught in schools is atrocious. Most math texts that I've used with 5th and 6th graders emphasize learning processes and methods for solving a set of problems. The texts do not hold all of the blame, however. The texts are written to follow state and national standards. The standards are written in such a way to emphasize process and not necessarily apprehension of greater concepts. For example:

    5th Grade Level Expectation 1. Differentiate between the term factor and multiple, and prime and composite (N-1-M)

    While these vocabulary items are important and these skills are definitely useful, learning this skill in isolation (which most texts teach) is pretty useless as students do not connect these skills to a greater picture.

    A revision of mathematics standards and teaching methods will go a long way to improving the quality of mathematics education. A holistic approach that includes some wrote learning of basic skills and lots of real application problems. Real application problems are not word problems. How many "real" word problems have you had to solve in the last ten years?

    Some texts such as Every Day Math from the University of Chicago does a much better job at integrating all sorts of skills and teaching in a much more holistic method. It includes some excellent modeling exercises, games that rely on a real understanding of mathematical principals for mastery and interesting lessons. But even the best text can't help a kid if they don't have a good teacher that really understands mathematics. Watching an uniformed teacher try to explain what a prime number is, or a different method for division (such as repeated subtraction) is painful. They simply can't do it. Unfortunately, in my experience most of the teaching candidates that were in my classes thought that math was "hard" and "didn't really matter." They scraped by with the lowest possible scores in the required math classes and one even told me she "wasn't going to bother teaching math." While this is pure anecdotal evidence, the declining math scores in the US show that we really do suck and producing math teachers.

    The problem stems from bad math teachers badly teaching math which of course leads to more poorly instructed math teachers. Placing a real emphasis on reading and mathematics, with highly qualified and well-supported specialists is the only way we're going to solve this problem. Unless we have some real political will akin to that found during the space race, we're not going to solve this problem any time soon. We'll just keep cranking out kids that think that math is done by computers and a few nerds that wave their magic math wand over problems to find solutions.

  • Re:he's right (Score:3, Insightful)

    by FuckingNickName (1362625) on Wednesday December 22, 2010 @06:05AM (#34639538) Journal

    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 who wishes to be controlled and for the fascist who wishes to control others. The technocrat only has to think about one thing. But philosophy regards technology as one of many tools, not as a master. The philosopher-ruler (for philosophy is a basis for living, not an alternative) must not let prejudice cause him to dismiss the possibility that he can do better and for more.

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

    by CRCulver (715279) <crculver@christopherculver.com> on Wednesday December 22, 2010 @06:21AM (#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?"

  • Re:he's right (Score:3, Insightful)

    by ifiwereasculptor (1870574) on Wednesday December 22, 2010 @06:33AM (#34639644)

    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 with nuclear powered, oversized fans. If you catch my drift.

  • Re:he's right (Score:3, Insightful)

    by Kashgarinn (1036758) on Wednesday December 22, 2010 @06:38AM (#34639670)

    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 acceptance, it represents freedom of thought to think critically.

    In fact philosophy really should be tought in schools, it's the basis of how we view the world today, and if the future is bright, it will be philosophy to thank and the people who dared to question the status quo.

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

    by ifiwereasculptor (1870574) on Wednesday December 22, 2010 @06:39AM (#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.

  • Re:he's right (Score:3, Insightful)

    by FuckingNickName (1362625) on Wednesday December 22, 2010 @06:46AM (#34639710) Journal

    Philosophy is the path by which every man continually asks questions of his condition and can thereby strive to improve it. It is something practiced while living, not instead of living (as "pursuit of happiness" is the ongoing enjoyment of happiness, not the singular and final goal of happiness). You may as well argue that man should not breathe because people who breathe are wasting their time only breathing when they should be doing other things.

    Philosophy does not give a single solution to the world's ills and it does not force you to do anything or to make others subordinate to your will. I'm not sure what you're afraid of, but it's not philosophy.

  • by Anonymous Coward on Wednesday December 22, 2010 @06:50AM (#34639740)

    There is a whole world of fascinating computational mathematics out there, young learner. Try reading Trefethen's Numerical Linear Algebra.

  • by Viol8 (599362) on Wednesday December 22, 2010 @06:58AM (#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!

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

    by digitig (1056110) on Wednesday December 22, 2010 @07:06AM (#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.

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

    by tehcyder (746570) on Wednesday December 22, 2010 @07:08AM (#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:4, Insightful)

    by vlm (69642) on Wednesday December 22, 2010 @07:35AM (#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.

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

    by tehcyder (746570) on Wednesday December 22, 2010 @07:44AM (#34639996) Journal

    Philosophy is only a pretty word for wild speculation/daydreaming/brainstorming

    You are exactly right, if your definition of "philosophy" is "wild speculation/daydreaming/brainstorming".

    Unfortunately for you, words have generally agreed upon meanings, not just whatever brainshite you happen to vomit forth.

  • by ObsessiveMathsFreak (773371) <obsessivemathsfreak@nosPAm.eircom.net> on Wednesday December 22, 2010 @07:59AM (#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.

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

    by 140Mandak262Jamuna (970587) on Wednesday December 22, 2010 @08:02AM (#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 @09:06AM (#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.
  • Re:he's right (Score:4, Insightful)

    by AmiMoJo (196126) <[ten.3dlrow] [ta] [ojom]> on Wednesday December 22, 2010 @10:40AM (#34641570) Homepage

    Just because the lack of a final answer to a problem is dissatisfying does not mean that there must be one. Some problems simply cannot be resolved absolutely.

    The example Sartre used is a good one. How would you make a decision in that instance which ends the debate?

    The world does not conform to your desire for resolution.

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

    by afidel (530433) on Wednesday December 22, 2010 @11:44AM (#34642398)
    They do, it's called a university and the school of hard knocks. Now we could refine that and go with the European system of dual (or triple) tracks for secondary education but that would be admitting that some snowflakes are less special than others.

In seeking the unattainable, simplicity only gets in the way. -- Epigrams in Programming, ACM SIGPLAN Sept. 1982

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