Math should be uncanny.

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William Brafford

William Brafford grew up in North Carolina, home of the world's best barbecue, indie rock, and regional soft drinks. He just barely sustains a personal blog and "tweets" every now and then under the name @williamrandolph.

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  1. Avatar Jaybird
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    says:

    Building blocks upon building blocks.

    I think that there are, generally, two kinds of math.

    The first kind of math is the kind that absolutely everybody needs… and it’s fairly rudimentary. The ability to add, subtract, multiply, and divide small numbers in ones head, the ability to add simple fractions (1/2 plus 1/3), and quick and dirty estimation skills. You know… balance a checkbook, bake a cake. Life skills.

    The second kind of math is the kind that is also a kind of metaphysical discussion. These are axioms. We use these laws and these axioms to lead to proofs… and upon simple elegant proofs, we can build complex elegant proofs. And, from there, we can move to algebra and trig and from there we can start explaining the universe through calculus and, from there, oh my goodness we are just getting started.

    I would say that I’d be surprised if more than 10% of the population would get any actual benefit from trigonometry. 5%, even… which is not to say that we ought not teach it. Indeed, the 5% of the population that does benefit from it goes on to provide many, many more dividends from having it taught to them than it costs to teach the 95% who won’t benefit particularly.

    There seems, however, to be an assumption that we are doing people a disservice if we do not do everything we can to assume that they are part of the 5%… and, in practice, this seems to lead to such things as, let me quote you here, “Euclidean stuff gets so watered down that it’s practically useless”.

    I don’t know what ought to be done, mind. My old school system had Basic, Regents, and Honors… which, I suppose, was good enough. They discussed math as metaphysics in Honors, they discussed drill and kill in Basic, and Regents was both hot and cold and ended up being lukewarm (Rev 3:16). The kids who saw the Divine in math tended up to Honors… and the kids who did not were scuttled off to the other two depending on their gifts. Maybe stuff like that will have to be good enough, if we want to make sure that everybody gets a crack at trig before they graduate.Report

    • Avatar William Brafford in reply to Jaybird
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      says:

      Something like your distinction works for English class too: basic reading and writing skills vs. literature appreciation and rhetoric.

      My point about the uselessness of Euclidean stuff was that the way geometry is taught in high school doesn’t make sense from either perspective. Really: I have seen tests where the “proof” sections are fill-in-the-blank. (Exercise for the reader: read book I of Euclid, then compare it to the the section on triangles in the textbooks that North Carolina public schools use.)

      I don’t want to say that every student should be taking trigonometry or calculus. But I do think that a lot of students in the main portion of the math-skills bell curve* have a math phobia that they don’t need to have, and that maybe math wouldn’t be so painful for so many if we thought about the aesthetic value of math alongside the practical/scientific value. Basically, don’t treat the second kind of math as if it’s just like the first kind of math.

      *I really have no idea if math skills show up as normally distributed.Report

  2. Avatar Alan Scott
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    says:

    One epiphany I had yesterday after reading the previous post:

    I think there’s too much focus on “how would I use this in real life?”, because the actual applications of higher math in real life are useful but boring. I think there might be more success focusing on “how would I use this if I were Johnny Danger, Action Hero?”Report

  3. From William Bradford:

    “…and still do some recreational math.”

    As a history/anthropology major that statement made my head hurt.

    With that said, as a parent of two kids under 16 I am appalled by their math training. There is no emphasis on basic rote memorization of addition, subtraction, multiplication and division tables like we had when I was growing up.They are taught to still use their fingers well up to middle school. It’s insane.

    As an adult who was never good at math when I was younger, my salvation has been Excel. Using it for work and real-world applications now makes me understand a lot of things better than when i was doing them on paper. My math skill has grown considerably since i began trying to organize numbers for my company 5 years ago. Application is pretty helpful to those who aren’t naturally talented with numbers.Report

  4. Avatar Josh
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    says:

    Your headline is perfect. There’s something arcane and enticingly weird about the possibilities of math, at least when you realize there’s more to it than textbook problems—Cantor’s infinities, the Pythagorean cult, the invention of zero as a concept. I don’t know how exactly a teacher might harness that, but it seems like someone could. (I mean, someone besides the Wu-Tang Clan, bless their hearts.)Report

  5. Avatar David in NY
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    says:

    Anyone interested in math or math pedagogy ought to read Paul Lockhart’s book, A Mathmatician’s Lament. http://www.amazon.com/Mathematicians-Lament-School-Fascinating-Imaginative/dp/1934137170 The “By God They Don’t Subject My Kids to Years of Learning the Times Tables Any More” group will, of course, hate it. But Lockhart’s point, that Mathematics is Art and needs to be taught as such, is worth paying attention to.Report

  6. Avatar M.Z.
    Ignored
    says:

    I never got a good appreciation of math until after my first round of college. By way of biography, I had completed multivariate college calculus (3rd semester generally) by my high school graduation. I haven’t read too many math books since then, but one of Asimov’s book rekindled my flames. John Derbyshire’s book “Prime Obsession” was also excellent, although I had to return it to the library before I could finish it.

    In the end I don’t think you can have a proper appreciation for things unless you have some experience. There seems to be a false dichotomy that if you don’t enjoy something that you can’t learn it and learn to appreciate it, as if enjoyment were a prerequisite for learning. It is very difficult to enjoy most any activity until you get the basics down.Report

  7. Avatar Steve Sailer
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    says:

    How should we teach math to the half of all children who are below average in math ability?Report

    • Avatar Freddie in reply to Steve Sailer
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      says:

      I’d say recognizing that there is such a thing as child who will never be above-average in math proficiency would be a very useful start!Report

      • Avatar William Brafford in reply to Freddie
        Ignored
        says:

        Freddie’s right — I’ve known people for whom algebra was just totally unnatural, to say nothing of calculus. Like I said in a different comment, I am not really sure where the average is.

        Steve, I don’t have a good answer for your question. Sorry!Report

    • Avatar M.Z. in reply to Steve Sailer
      Ignored
      says:

      That would be a gross misrepresentation of data. Certainly we can say half the children will have an easier (or more difficult) time acquiring any sort of knowledge. That is merely the logical possibilities by definition. However, we can not use that to state that half will have trouble acquiring requisite knowledge. We can state with relative certainty that 97% of children in the United States will learn the alphabet, despite the logical postulates that half will have an easier time doing so than the other half. Likewise we can say 80% of people will not acquire the knowledge to perform Leibniz transformations, despite the logical postulates that half will have more difficulty doing so than the other half.Report

    • Avatar Mike Schilling in reply to Steve Sailer
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      says:

      Suppose “ability to do math” were both one-dimensional and possible to measure accurately, say using the same scale as IQ where average is 100. There’s going to be a limit to how precisely it can be measured (note that I’m using ‘accurately’ and ‘precisely’ with their technical meanings here). Let’s define “101” as the lowest score that we can be certain reflects above-average prowess, and “99” as the highest score that we can be certain reflects below-average prowess. A significant number of students are going to score exactly 100, leaving fewer than half to be 99 or lower, that is, below average. And even that’s making unrealistic assumptions about the precision of testing. In the real world, we’d define average as something more like “between 95 and 105”, leaving even fewer students in the ranges below.

      And that’s assuming that mathematical ability can be represented accurately with a single number. In real life, things are more complex (and here I don’t mean “part real and part imaginary”.) Some students have great facility with numbers but find the abstractness of algebra difficult. Some do fine with algebra but lack the ability to visualize geometric objects (and vice versa) Others are quite successful until calculus defeats them, while still others can grasp enough calculus to use it as a tool, but will never understand it at the deeper levels that mathematicians require. (Limits are hard; the naive use of infinitesimals that Bishop Berkeley objected to is much simpler.) There are going to be students who are below average in each of these, but since the sets aren’t co-extensive, their intersection is again going to be less than half.

      At any rate, much as “half are below average” sounds tough-minded and realistic, it doesn’t stand up to any real scrutiny. I think there’s a lesson there.Report

      • Avatar Freddie in reply to Mike Schilling
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        says:

        Your argument is totally shorn from any context; here in America, we aren’t anywhere near accepting that some large number of students simply are not going to achieve at the level of mathematics that we are expecting all students to. Indeed, both our rhetoric and our legal policy requires 100% competence. 100% compliance is not a mature goal, and it is the mark of a society that is far more interested in talking tough than in actually making practical gains on vexing problems. But that’s America– image and symbolism over substance, always.Report

      • Avatar zic in reply to Mike Schilling
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        says:

        There are very specific methods for testing for abilities, math and its subset skills included.

        That there are kids who will never master algebra is obvious; that there are kids who will never master Chaucer, equally obvious.

        What is really sad as that we focus on math/language arts and leave out a host of other skills; particularly the mechanical, musical, visual, and kinetic.

        As the mother of two kids with extreme learning differences — top 2% in math, bottom six in some parts of language arts — I can speak volumes to the woes of teaching to the average instead of teaching to the child.

        But I have noticed one important thing: many kids ‘outgrow’ their learning difficulties in their early 20’s, my assumption is that the change is related to the development of the frontal cortex. For my two kids, that’s when their lives as ‘well-rounded’ students began.

        Patience is a virtue, particularly with education and the habits of life-long learning.Report

  8. Avatar A.R.Yngve
    Ignored
    says:

    Speaking of uncanny math, it doesn’t get much uncannier than this:
    http://en.wikipedia.org/wiki/Chaitin%27s_constantReport

  9. Avatar Simon K
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    says:

    I learned a lot of things in school that I did not fully appreciate at the time, and even when I was told why they were important, I didn’t really get it. Later in life, I’ve been surprisingly moved by discovering why what I was taught actually mattered. No 16-year-old is ever going to empathize with King Lear, or understand the practical importance of solving systems of linear equations, or really appreciate the subtle tactics of ski-racing. But there’s no better time to teach these things – if you wait until people need the knowledge to teach it to them, they’ll be too busy and their habits of mind will probably be too set. So we really have to teach kids to write essays on fileal ingratitude, solve silly little systems of 3 equations, and run gates (well, that ones not so important unless you grow up in Truckee), knowing that they won’t really understand the full significance of these things until they’re much older. This is why school is boring, and I don’t think there’s much we can do about it.Report

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