The Lumber Room

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Is it important to “understand” first?

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The Oxford University Press has been publishing a book series known as “Very Short Introductions”. These slim volumes are an excellent idea, and cover over 200 topics already. The volume Mathematics: A Very Short Introduction is written by Timothy Gowers.

Gowers is one of the leading mathematicians today, and a winner of the Fields Medal (in 1998). In addition to his research work, he has also done an amazing amount of service to mathematics in other ways. He edited the 1000-page Princeton Companion to Mathematics, getting the best experts to write, and writing many articles himself. He also started the Polymath project and the Tricki, the “tricks wiki”. You can watch his talk on The Importance of Mathematics (with slides) (transcript), and read his illuminating mathematical discusssions, and his blog. His great article The Two Cultures of Mathematics is on the “theory builders and problem solvers” theme, and is a paper every mathematician should read.

Needless to say, “Mathematics: A Very Short Introduction” is a very good read. Unlike many books aimed at non-mathematicians, Gowers is quite clear that he does “presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set. I have also avoided topics such as chaos theory and Godel’s theorem, which have a hold on the public imagination out of proportion to their impact on current mathematical research”. What follows is a great book that particularly excels at describing what it is that mathematicians do. Some parts of the book, being Gowers’s personal views on the philosophy of mathematics, might not work very well when directed at laypersons, not because they require advanced knowledge, but assume a culture of mathematics. Doron Zeilberger thinks that this book “should be recommended reading to everyone and required reading to mathematicians”.

Its last chapter, “Some frequently asked questions”, carries Gowers’s thoughts on some interesting questions. With whole-hearted apologies for inserting my own misleading “summaries” of the answers in brackets, they are the following: “1.Is it true that mathematicians are past it by the time they are 30?” (no), “2. Why are there so few women mathematicians?” (puzzling and regrettable), “3. Do mathematics and music go together?” (not really), “4. Why do so many people positively dislike mathematics?” (more on this below), “5. Do mathematicians use computers in their work?” (not yet), “6. How is research in mathematics possible?” (if you have read this book you won’t ask), “7. Are famous mathematical problems ever solved by amateurs?” (not really), “8. Why do mathematicians refer to some theorems and proofs as beautiful?” (already discussed. Also, “One difference is that [...] a mathematician is more anonymous than an artist. [...] it is, in the end, the mathematics itself that delights us”.) As I said, you should read the book itself, not my summaries.

The interesting one is (4).

4. Why do so many people positively dislike mathematics?

One does not often hear people saying that they have never liked biology, or English literature. To be sure, not everybody is excited by these subjects, but those who are not tend to understand perfectly well that others are. By contrast, mathematics, and subjects with a high mathematical content such as physics, seem to provoke not just indifference but actual antipathy. What is it that causes many people to give mathematical subjects up as soon as they possibly can and remember them with dread for the rest of their lives?

Probably it is not so much mathematics itself that people find unappealing as the experience of mathematics lessons, and this is easier to understand. Because mathematics continually builds on itself, it is important to keep up when learning it. For example, if you are not reasonably adept at multiplying two-digit numbers together,then you probably won’t have a good intuitive feel for the distributive law (discussed in Chapter 2). Without this, you are unlikely to be comfortable with multiplying out the brackets in an expression such as (x+2)(x+3), and then you will not be able to understand quadratic equations properly. And if you do not understand quadratic equations, then you will not understand why the golden ratio is \frac{1+\sqrt{5}}{2}.

There are many chains of this kind, but there is more to keeping up with mathematics than just maintaining technical fluency. Every so often, a new idea is introduced which is very important and markedly more sophisticated than those that have come before, and each one provides an opportunity to fall behind. An obvious example is the use of letters to stand for numbers, which many find confusing but which is fundamental to all mathematics above a certain level. Other examples are negative numbers, complex numbers, trigonometry, raising to powers, logarithms, and the beginnings of calculus. Those who are not ready to make the necessary conceptual leap when they meet one of these ideas will feel insecure about all the mathematics that builds on it. Gradually they will get used to only half understanding what their mathematics teachers say, and after a few more missed leaps they will find that even half is an overestimate. Meanwhile, they will see others in their class who are keeping up with no difficulty at all. It is no wonder that mathematics lessons become, for many people, something of an ordeal.

This seems to be exactly the right reason. No one would enjoy being put through drudgery that they were not competent at, and without the beauty at the end of the pursuit being apparent. (I hated my drawing classes in school, too.) See also Lockhart’s Lament, another article that everyone — even, or especially, non-mathematicians — should read.

As noted earlier, Gowers has some things to say about the philosophy of mathematics. As is evident from his talk “Does mathematics need a philosophy?” (also typeset as essay 10 of 18 Unconventional Essays on the Nature of Mathematics), he has rejected the Platonic philosophy (≈ mathematical truths exist, and we’re discovering them) in favour of a formalist one (≈ it’s all just manipulating expressions and symbols, just stuff we do). The argument is interesting and convincing, but I find myself unwilling to change my attitude. Yuri Manin says in a recent interview that “I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism)”, so it’s perhaps just as well.

Anyway, the anti-Platonist / formalist idea of Gowers is evident throughout the book, and of course it has its great side: “a mathematical object is what it does” is his slogan, and most of us can agree that “one should learn to think abstractly, because by doing so many philosophical difficulties disappear” , etc. The only controversial suggestion, perhaps, follows the excerpt quoted above (of “Why do so many people positively dislike mathematics?”):

Is this a necessary state of affairs? Are some people just doomed to dislike mathematics at school? Or might it be possible to teach the subject differently in such a way that far fewer people are excluded from it? I am convinced that any child who is given one-to-one tuition in mathematics from an early age by a good and enthusiastic teacher will grow up liking it. This, of course, does not immediately suggest a feasible educational policy, but it does at least indicate that there might be room for improvement in how mathematics is taught.

One recommendation follows from the ideas I have emphasized in this book. Above, I implicitly drew a contrast between being technically fluent and understanding difficult concepts, but it seems that almost everybody who is good at one is good at the other. And indeed, if understanding a mathematical object is largely a question of learning the rules it obeys rather than grasping its essence, then this is exactly what one would expect — the distinction between technical fluency and mathematical understanding is less clear-cut than one might imagine.

How should this observation influence classroom practice? I do not advocate any revolutionary change — mathematics has suffered from too many of them already — but a small change in emphasis could pay dividends. For example, suppose that a pupil makes the common mistake of thinking that xa+b = xa + xb. A teacher who has emphasized the intrinsic meaning of expressions such as xa will point out that xa+b means a+b xs all multiplied together, which is clearly the same as a of them multiplied together multiplied by b of them multiplied together. Unfortunately, many children find this argument too complicated to take in, and anyhow it ceases to be valid if a and b are not positive integers.

Such children might benefit from a more abstract approach. As I pointed out in Chapter 2, everything one needs to know about powers can be deduced from a few very simple rules, of which the most important is xa+b = xa xb. If this rule has been emphasized, then not only is the above mistake less likely in the first place, but it is also easier to correct: those who make the mistake can simply be told that they have forgotten to apply the right rule. Of course, it is important to be familiar with basic facts such as that x3 means x times x times x, but these can be presented as consequences of the rules rather than as justifications for them.

I do not wish to suggest that one should try to explain to children what the abstract approach is, but merely that teachers should be aware of its implications. The main one is that it is quite possible to learn to use mathematical concepts correctly without being able to say exactly what they mean. This might sound a bad idea, but the use is often easier to teach, and a deeper understanding of the meaning, if there is any meaning over and above the use, often follows of its own accord.

Of course, there is an instinctive reason to immediately reject such a proposal — as the MAA review by Fernando Q. Gouvêa observes, ‘I suspect, however, that there is far too much “that’s the rule” teaching, and far too little explaining of reasons in elementary mathematics teaching. Such a focus on rules can easily lead to students having to remember a huge list of unrelated rules. I fear Gowers’ suggestion here may in fact be counterproductive.’ Nevertheless, the idea that technical fluency may precede and lead to mathematical understanding is worth pondering.

(Unfortunately, even though true, it may not actually help with teaching: in practice, drilling-in “mere” technical fluency can be as unsuccessful as imparting understanding.)

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Written by S

Tue, 2009-12-01 at 03:38:31 +05:30

11 Responses

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  1. Happened to come across this post, and I personally would put a lot of blame on the teachers. I find that many people start disliking/developing a complex against mathematics from a young age and this continues till high school. This, I feel is due to the the teachers themselves not being so passionate about mathematics (atleast in India, and not all of them) to teach it in a way that kids can grasp it. Ofcourse another argument is that it is not too easy to visualize mathematics as opposed to physics or other areas.

    Interesting blog, by the way.

    Magnus

    Fri, 2009-12-04 at 13:21:13 +05:30

    • True, having passionate teachers would be wonderful, but I’m not sure how justified it is to blame the teachers. Many of them are indeed doing the best they can, and if that isn’t much, it’s not their fault. If anything, it is the fault of those of us who are passionate about maths for not doing something about mathematics education. Note also that rarely are geography teachers passionate about geography, for example — and yet the situation seems worse for maths.

      [I assume you've read Lockhart's Lament, and if you're interested you can read the discussion at Scott Aaronson's blog.]

      Thanks, BTW. :-)

      Shreevatsa

      Fri, 2009-12-04 at 18:45:14 +05:30

  2. Having known you for the last 4.5 years I see no change in your ways and means. You seem to consistently continue with your art of writing things which just report on what every other person in the world had to say without ever for a second letting the reader know what your opinion is!

    You seem to very efficienntly do the work of a new reporter in this case too without ever putting foth a set of proposals or efforts about what can be done to improve the state of mathematics education.

    You usual methods of poking fun at every side while continuing to conveniently sit on the fence in between. ;-)

    Not a very progressive approach if I may say so.

    And about the specific topic I would like to enquire whether there is any research which disproves a possible fact that mathematics is actually a very specialized subject and there is some say intrinsic physiological or psychological or genetic difference between people who like and who don’t like mathematics?

    The possibility arises from the fact that in day to life I do feel that there is a distinctive difference between how people good in mathematics go about life and how the others do. Ability in mathematics seems to reflect at large in general about how the person goes about in various others aspects.

    It is possible that there is a fundamental attitudinal difference between people good and bad at mathematics. This possibility needs to be ruled out by serious research before embarking on an argument of the kind that mathematics teaching is bad in general.

    Further you seem to give quite a free ride to the mathematics teachers whereas I would content that most mathematics teachers in school in India are simply incompetent and are ignorant of any undergraduate level mathematics and hence can’t give the students any larger view point which in principle for example you are capable of. (though a largely unused ability in your case!)

    Anirbit

    Tue, 2009-12-08 at 05:01:02 +05:30

    • I’m not (right now) seriously trying to change mathematics education, just report on an intriguing idea. So I don’t feel obliged to give out ill-formed opinions on every matter. :p

      Speculation on intrinsic differences etc. must always be done carefully — and so far little is known, with the answer tending to No (see this, this, this.) My own belief coincides with Prof. Gowers’s quote: “I am convinced that any child who is given one-to-one tuition in mathematics from an early age by a good and enthusiastic teacher will grow up liking it.”

      Finally, as I said already, incompetence is not a crime. There are economic factors which make the demand for mathematics teachers higher than the supply of good ones. So instead of blaming the teachers we have, we must either decrease the amount of mathematics that is taught, induce better teachers to teach, or make the teachers better. All of which would be more fruitful than assigning blame.

      Shreevatsa

      Tue, 2009-12-08 at 09:32:35 +05:30

  3. You need not have an opinion but it is still important that you try to explain to the reader what your analysis is of a topic. It need not lead to a very firm result. But still if the reader gets to know that you have an analysis of your own that lends a greater credibility to your blog than seeing you just report what others have to say!

    An ill-formed opinion is still valuable since that signals to the reader that you are thinking instead of a situation where you have no opinion!

    I expect blogs to be different from news paper reports.

    Thanks for telling me of the references which try to explain that there is no intrinsic capability difference. I will try to read up the links.

    A question of intrinsic difference is a crucial one that needs to be established by thorough research and not by just what one believes even if it is a belief of Tim Gowers.

    Its not a question of blame game. Pinning down the cause of the trouble is very necesasary to ensure that one’s reform efforts are not dispersed but are concentrated at the core source of the problem. The standard of the mathematics teaching at the school level is appalling and thats a question one has to solve.

    Any ideas how to solve that? (instead of just reducing the demand for mathematics which might be disastrous)

    Any ideas how to actually get good mathematics people to come and teach in the schools?

    We have a serious knowledge gap problem at the school level where most teachers simply don’t know enough mathematics to teach!

    Any ideas about solving that problem?

    Anirbit

    Tue, 2009-12-08 at 12:57:14 +05:30

    • General note: this “blog” may be on WordPress.com, but I do not care to follow any conception of what a blog should be like, to establish “credibility” with hypothetical/mythical readers or indeed care about any readers other than myself, to “signal” that I’m thinking, and least of all for your expectations. :P

      Slightly general note: Please click on “reply” instead of starting a new thread.

      Specific note: No, I don’t have any ideas that would be worth discussing here.

      Shreevatsa

      Thu, 2009-12-10 at 21:27:33 +05:30

  4. I wrote up a long comment, but it was far too incoherent for even me to read. Just two things:

    1. I can totally relate to “being put through drudgery that they were not competent at”. Perfect!!

    2. Thank you for the Yuri Manin quote – one sore spot feels better, at least until the next bout of cognitive dissonance comes by :-)

    KVM

    Mon, 2009-12-28 at 17:53:32 +05:30

    • I just re-read Lockhart’s Lament, and I distinctly remember reading it earlier – maybe you had posted a link many months ago?

      It is fantastic. I thoroughly agree with what he says. I won’t bother if this is incoherent, so advance apologies. I hate, hate, hate this idea of ‘respecting’ ‘education’ as if it is some kind of alien God who demands our servility. This nauseating, suffocating talk of ‘respecting’ education is the cause of many ills in the Indian education system. It becomes very unbearably meta, with almost no one bothering to talk of the beauty of the subject itself (consider a common candidate, ‘Mathematics’ in all its unqualified unity), and instead choosing to talk of how great and important ‘it’ is, how great it is to study it and how great those who study it are. This deification makes it impossible to talk of a nice problem without seeming like a clueless nerd. This ‘respect’ also spawns ultra-competitiveness and judgment by absolutely anyone. It is always about who ‘knows more’ and who is ‘respected more’ than what the heck it is people are talking about. Once it gets to well-ordering people, it is impossible to talk of how nice some algorithm or formula is. Either you are a fool because you didn’t know it before, or “why are you telling me this?”

      I was particularly gratified to read the part about most of today’s school math being ‘useless’, and so we might as well make it interesting. I conducted a survey of my friends who are working, and asked a few tens of them if they had ever used differentiation in their jobs. Only one person replied he had, and that too to add spice to a presentation. But _accepting_ that math is just for fun is heresy – No, it must be Respected. It is Important.

      His displeasure towards ‘teaching methods’ is also very gratifying. Now it is made into an elaborate exercise – powerpoint slides with the right kind of images, planning the ‘flow’ of the lecture, printing out elaborate lecture notes that are ‘useful’ for the exam, ‘sticking to the plan’ and not deviating with ‘unnecessary’ history, being ‘professional’ about homework – perhaps the next in line would be writing on a tablet PC so that students can download your notes!

      I am disgusted with talk of rigour without necessity. Everything that makes sense should be acceptable unless there is a counterexample! And any ‘correct’ form MUST not be presented as an edifice – there should be SOME history of how it came to that, about _why_ all the frills are necessary and sufficient.

      KVM

      Mon, 2009-12-28 at 18:41:13 +05:30

      • Heh, it seems to have hit a spot. :-)
        [I essentially agree, so advance apologies for disagreeing out of compulsion.]

        I don’t see anything wrong with respecting education (except in the trivial sense that “respect” sometimes/often does not mean what it says, and is just a proxy for keeping at a distance.) Most things worthy of love are also worthy of respect, and it would be a sad world where they were mutually exclusive. That not many would dwell on the intrinsic beauty of things is an independent social problem…
        Of course, how to retain your own aesthetics and doggedly disregard discouragement and other signals from outside remains an open problem. :-)

        Also, I like notes on a tablet PC! It’s slightly harder to search than “proper” notes, but still better than nothing (I like listening in class and not having to take notes), and handwriting has its warmth. :p
        (I also like rigour! Rigorous does not mean boring and dry; a proof is a beautiful thing. (IIRC, Lockhart has examples of his students coming up with interesting rigorous proofs, a very rewarding experience.) It’s ok if not everyone finds it so, but a few children — some of them future mathematicians — do find enjoyable the dance of high-school geometry (which is where they first encounter proofs), even if not the silly two-column format, and removing even that trace of real mathematics from the curriculum would make it too desolate to bear.)

        S

        Wed, 2009-12-30 at 19:26:08 +05:30

        • LOL, you have a knack for phrases that very succinctly express ideas that would otherwise take a long meandering sentence. “disagreeing out of compulsion” – perfect! That’s right on top in my list, too :-)

          Respect and rigour – I see your point. It goes to show that one extreme can be as painful as another, and my judgment of the less-respect,less-rigour extreme is probably made very rosy and cloudy because of being pained too much at the other. Sort of like how a starving man can’t grok a campaign against factory farms :P [*]

          I still think that a forced respect for an idea is an important cause for the social problem you mention, of not dwelling on intrinsic beauty – but I need to think a lot more about this.

          But notes on a tablet PC – the horror, the horror! Handwriting has its warmth and it’s certainly nicer to just listen instead of a juggling act of writing, listening and planning – absolutely true, but I strenuously object to those abominable tablets! A plague upon their hard, low-tracking-resolution touch-screens! :P

          [*] – The farms of non-rigorous writing, where innocent mathematical ideas are tortured and brutally slaughtered :P

          KVM

          Sun, 2010-01-03 at 12:50:26 +05:30

          • I haven’t used a tablet PC myself; I was only thinking of notes that the teacher has written on a tablet (instead of in TeX).

            Speaking of non-rigorous writing, random quotes I stole from I-forgot-where:

            When I give this talk to a physics audience, I remove the quotes from my ‘Theorem’.
            — Brian Greene (invited talk at Joint Math Meetings, Washington, DC, Jan. 19, 2000)

            The first law of Engineering Mathematics: All infinite series converge, and moreover converge to the first term.

            S

            Wed, 2010-02-03 at 10:18:40 +05:30


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