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flexibeasti just finished reading Lockhart's Lament [PDF], more formally titled "A Mathematician's Lament". It's an excellent essay critiquing the way mathematics is often taught - in the US in particular, although i feel it applies to maths education here in Australia, and probably in other Anglophone countries as well.
Lockhart invites us to imagine worlds in which music or painting is taught to children the same way we teach them maths, demonstrating how these subjects would probably seem as sterile and pointless as mathematics is to many (most?) people in our world. He then goes on to argue that we should not be teaching mathematics as simply an arcane tool requiring a variety of mysterious incantations in order to solve particular problems, but as a form of art that should be pursued for its own sake, regardless of its practical applications (thus echoing the sentiments of the famous essay by G.H. Hardy, "A Mathematician's Apology"). That is, just as we don't teach children about music primarily in terms of music notation, or in terms of being able to write music for ads and movies, we shouldn't talk to children about mathematics unnecessarily formally, or in terms of "real world" examples (e.g. being able to work out how much money we actually end up spending when we buy something on credit).
As someone who takes pleasure in certain forms of mathematics; as someone who feels that my society is poorer for its lack of popular interest in mathematics1; and as someone who has in recent times had the privilege of tutoring a friend's teenage daughter in maths, i found Lockhart's essay very thought-provoking. The teen in question often complains about how she feels that none of what she's having to learn is relevant, and i've usually hastened to provide real-world examples of what the maths she's learning is good for. However, irrelevance per se is not actually the issue, as she has no problem (for example) learning to sing, which might or might not end up being relevant to her career. The problem is that it's irrelevant and it's boring. It's boring because the primary emphasis is on learning what seems to be set after set of rules, rather than trying to get student engaged in the overall idea of mathematics; and attempts to make learning set after set of rules 'fun' is as lame to kids as is parents trying to relate to their teenage children by using the teen lingo du jour.
So Lockhart effectively wants the teaching of mathematics to be 'humanised', so to speak: to involve history, personalities, opinions, debates, imagination and intuition. Young people often have active imaginations; but as they interact with the world, and develop a sense of the internal lives of people other than themselves, they also tend to develop rules and intuitions about how the world, and various human interactions, 'work' - rules and intuitions which they then go on to apply in imaginary scenarios. Isn't it possible that maybe, by encouraging kids to play with various mathematical objects, they can, of their own accord, develop at least some rules and intuitions about mathematics, and thus develop a good foundation on which to build further mathematical knowledge at the time they actually need it?
1. And indeed, the active pride that some people take in not having a clue about mathematics, the history of mathematics or significant mathematicians. To paraphrase something i once read, if a person boasted about not knowing who these "Beethoven" or "Picasso" people were, they'd often be labelled a philistine; but not knowing about Euler, Gauss or Gödel is fine.
Lockhart invites us to imagine worlds in which music or painting is taught to children the same way we teach them maths, demonstrating how these subjects would probably seem as sterile and pointless as mathematics is to many (most?) people in our world. He then goes on to argue that we should not be teaching mathematics as simply an arcane tool requiring a variety of mysterious incantations in order to solve particular problems, but as a form of art that should be pursued for its own sake, regardless of its practical applications (thus echoing the sentiments of the famous essay by G.H. Hardy, "A Mathematician's Apology"). That is, just as we don't teach children about music primarily in terms of music notation, or in terms of being able to write music for ads and movies, we shouldn't talk to children about mathematics unnecessarily formally, or in terms of "real world" examples (e.g. being able to work out how much money we actually end up spending when we buy something on credit).
As someone who takes pleasure in certain forms of mathematics; as someone who feels that my society is poorer for its lack of popular interest in mathematics1; and as someone who has in recent times had the privilege of tutoring a friend's teenage daughter in maths, i found Lockhart's essay very thought-provoking. The teen in question often complains about how she feels that none of what she's having to learn is relevant, and i've usually hastened to provide real-world examples of what the maths she's learning is good for. However, irrelevance per se is not actually the issue, as she has no problem (for example) learning to sing, which might or might not end up being relevant to her career. The problem is that it's irrelevant and it's boring. It's boring because the primary emphasis is on learning what seems to be set after set of rules, rather than trying to get student engaged in the overall idea of mathematics; and attempts to make learning set after set of rules 'fun' is as lame to kids as is parents trying to relate to their teenage children by using the teen lingo du jour.
So Lockhart effectively wants the teaching of mathematics to be 'humanised', so to speak: to involve history, personalities, opinions, debates, imagination and intuition. Young people often have active imaginations; but as they interact with the world, and develop a sense of the internal lives of people other than themselves, they also tend to develop rules and intuitions about how the world, and various human interactions, 'work' - rules and intuitions which they then go on to apply in imaginary scenarios. Isn't it possible that maybe, by encouraging kids to play with various mathematical objects, they can, of their own accord, develop at least some rules and intuitions about mathematics, and thus develop a good foundation on which to build further mathematical knowledge at the time they actually need it?
1. And indeed, the active pride that some people take in not having a clue about mathematics, the history of mathematics or significant mathematicians. To paraphrase something i once read, if a person boasted about not knowing who these "Beethoven" or "Picasso" people were, they'd often be labelled a philistine; but not knowing about Euler, Gauss or Gödel is fine.
no subject
Date: 2008-03-08 21:49 (UTC)But in a tangential vein, the thing that bugs me these days is "math envy". So many fields are poisoned with this deep-seated fear that anyone who knows more math than they is somehow smarter, and so they seek to encode their findings in the most abstruse jumble of greek letters and newfangled notation in order to convince people that they know more math than the other guy. People in these fields try to keep math as arcane and unreachable as possible because it keeps the other guy out and shows how smart they must be to be so close to mathematicians (of course, the mathematicians don't seem to notice this envy, and the humanists don't care how much math you know).
I can't help but wonder if this buffer zone between mathematicians and humanists isn't made so wretched precisely because of the way math is taught. That is, when things like "calculus" and "monads" and "Abelian groups" and "lattices" are taught as bizarre and challenging ideas, it makes sense that anyone wanting to imitate mathematicians would just start to make shit up. Without any history or intuition behind the different ideas, without seeing how they all interconnect, of course it would take a genius to keep it all straight.
no subject
Date: 2008-03-09 12:35 (UTC)i lol'd when i read this. :-)
On the other hand, it seems that sometimes just making shit up can turn out to be useful: "Hey, what if we could take the square root of a negative number? What if we assume that Euclid's fifth postulate is false?" "Yeah, good on you bozo, why don't you just assume that we can walk on the sun whilst you're at it? . . . . Oh, those are interesting results . . . ." ;-)
*nod* Excellent point.
no subject
Date: 2008-03-14 01:09 (UTC)The trick is we only keep the bits which are useful/fun/interesting, and so most of the longest lived bits of math are really straightforward if you can get past the jargon. E.g. if I hear another person describing functions as "one-to-one", "one-on-one", or "onto" (instead of "in-/sur-/bijective") I'm going to murder the guy who thought those names up. One thing that I think would help is if we taught some set theory and graph theory instead of all this focus on (non-abstract) algebra. Sets and graphs are pretty easy to understand the basics of and they give an arsenal of alternative perspectives for looking at problems.
Then again, I suppose a bunch of it only comes as "familiar" after you've been toying around with math long enough. Like rings and semi-rings: if you haven't seen enough of the examples, I could see how it'd be hard/odd to explain; but if you've seen the examples its a duh generalization. Abstract algebras, similarly.
no subject
Date: 2008-03-14 06:41 (UTC)Only if one is not a Platonist. ;-)
Perhaps - as long as it doesn't get over-rigorous, à la the geometric proof cited by Lockhart. (i personally think it's better to start out with approximately-correct explanations that are straightforward and easily retainable, and which are refined as education proceeds, than to be so rigorous that people are completely bored, turned off the subject, and retain nothing.)
no subject
Date: 2008-03-15 20:35 (UTC)But then, I think most of K-12 education should be more about introducing people to the high-level concepts of any field rather than focusing on the details of one particular sub area. History, for example, is made awful by focusing on memorizing dates of particular events rather than creating a narrative for what was happening and why people responded in the ways they did.
no subject
Date: 2008-03-09 00:50 (UTC)Mrs Easton was my math teacher in 3rd and 4th grades, who had the perception to see that I wasn't bad at/hated math, I was bored with arithmetic, and cultivated my mathematical ability by showing me Fun Bits, and encouraging me to engage with the subject and discover rules/intuitions for myself. In the process (I have no idea how intentionally or accidentally), she immunized me against the "girls are bad at math and science" trope - I have no idea whether I was told this a lot; if I was, I didn't hear it because it so obviously had nothing to do with me - and that in turn shaped all sorts of things related to empowerment and feminism and such. Hence the devotional, whenever it comes up from one angle or another, which is a lot because it intersects with so many things.
Here, it also serves the purpose of showing enthusiastic agreement, in a more interesting way than just saying, "I agree!"
Sunflower
no subject
Date: 2008-03-09 12:46 (UTC)As for the "girls are bad at math and science" trope, i don't know if you've already read about this (http://www.eurekalert.org/pub_releases/2006-10/uobc-wmp101306.php), this (http://www.eurekalert.org/pub_releases/2007-01/afps-isa012407.php), this (http://www.eurekalert.org/pub_releases/2007-05/uoc-sma052107.php), and this (http://www.eurekalert.org/pub_releases/2008-02/sfri-fsa020108.php), or seen this recent comic on that subject (http://xkcd.com/385/)?
no subject
Date: 2008-03-10 02:35 (UTC)Sunflower
no subject
Date: 2008-03-11 01:04 (UTC)no subject
Date: 2008-03-09 13:33 (UTC)(no, I don't think it's irrelevant).
If I had to take those formal maths classes again these days I'd literally flake out.