### It all adds up

Jul. 14th, 2009 03:49 pm**matthewdaly**

Alas,

It's hard to have this conversation without an agreement on what mathematics IS, and what mathematical skills we need from the general population. This won't come without a struggle, and the way we do things now reflects our lack of consensus. Math is traditionally the language of both accountants and engineers, who each use their own fields with their own language. And you would work your way up the tower until you hit the limits of either your talent or leisure time and that would determine whether you were qualified to be a laborer, a manager, or an expert in some field like surveying or astronomy or what have you. This has served us for centuries all the way up through the time that Generation X (including me) was in high school, with the prize that people who have mastered calculus could train to study science and engineering in college.

There has been a rebellion against that model of education over the past ten or fifteen years, and quite a bit of it was well-deserved. The main problem (as I see it) was that we have been holding people in high school for the same length of time regardless of their ambition. When high school lasts for eight periods a day for four years whether you're training for a prestigious diploma or a lesser one, one might well ask what the sense is of anyone signing up for the lesser one besides the obvious conclusion that educators can't be bothered to challenge everyone. Plus, of course, mid-level bureaucrats had far too much power to limit the potential of women and minorities through the self-fulfilling argument that they didn't seem like the sorts of folk who could become engineers.

So now everyone is on the pre-college math track, which is probably great except that we didn't actually change the curriculum when we made that decision. The train is still making all of the local stops even though everyone is going to the end of the line, resulting in a fair amount of busywork that makes little sense in the broader context. For example, one spends quite a bit of time learning strategies for factoring polynomials in "Algebra II" that never get applied outside that cocoon because virtually all polynomials in "the real world" are irreducible. If we were to take side trips for the sake of showing off the breadth of mathematics, would that really be a part of anyone's plan?

And, needless to say, we haven't talked about the elephant in the room which is whether the four years of high school math should be obligated to shoot its entire wad in the name of satisfying the prereq for college freshman physics. Some of my best friends are engineers, but there are other things to be too. There are some mathematicians (including me and evidently Lockhart) who would argue that mathematics is bigger than the science of creating abstract models of physical phenomena for the sake of making better and simpler scientific predictions of real-world behavior, and should be broadened to consider the entire range of intellectual strategies to solve problems. Ordinary folk in my experience can get by without the Fundamental Theorem of Calculus but would be well served with some discrete topics like logic and graph theory. But it would take a larger mathematical revolution than I've ever experienced or even read about to knock calculus off the top of the mountain.

**rivka**doesn't crosspost her livejournal stuff to dw and I don't want to get another lj account just to respond to her posts (as thoughtful and marvelous as they are), but I was struck by her article here referencing this thesis (in PDF form) by Paul Lockhart (evidently a private high school teacher with the cache to teach an elective math course) that the high school mathematics curriculum in the United States has no redeeming values at all. Of course, his conclusion is nearly entirely correct, but his dualistic over-reaching and insipid straw-man dialogs are far more amusing than persuasive. His argument glorifying his research-focused methods and lambasting the soul-killing of everyone else seems to say "My students are able to see so far (due to *my* training, natch) in spite of the fact that giants are standing on their shoulders." Meh, it lacks that intuitive ring, which really strikes at the heart of whether Lockhart is the sort of mathematical leader we want to lead us into the next generation.It's hard to have this conversation without an agreement on what mathematics IS, and what mathematical skills we need from the general population. This won't come without a struggle, and the way we do things now reflects our lack of consensus. Math is traditionally the language of both accountants and engineers, who each use their own fields with their own language. And you would work your way up the tower until you hit the limits of either your talent or leisure time and that would determine whether you were qualified to be a laborer, a manager, or an expert in some field like surveying or astronomy or what have you. This has served us for centuries all the way up through the time that Generation X (including me) was in high school, with the prize that people who have mastered calculus could train to study science and engineering in college.

There has been a rebellion against that model of education over the past ten or fifteen years, and quite a bit of it was well-deserved. The main problem (as I see it) was that we have been holding people in high school for the same length of time regardless of their ambition. When high school lasts for eight periods a day for four years whether you're training for a prestigious diploma or a lesser one, one might well ask what the sense is of anyone signing up for the lesser one besides the obvious conclusion that educators can't be bothered to challenge everyone. Plus, of course, mid-level bureaucrats had far too much power to limit the potential of women and minorities through the self-fulfilling argument that they didn't seem like the sorts of folk who could become engineers.

So now everyone is on the pre-college math track, which is probably great except that we didn't actually change the curriculum when we made that decision. The train is still making all of the local stops even though everyone is going to the end of the line, resulting in a fair amount of busywork that makes little sense in the broader context. For example, one spends quite a bit of time learning strategies for factoring polynomials in "Algebra II" that never get applied outside that cocoon because virtually all polynomials in "the real world" are irreducible. If we were to take side trips for the sake of showing off the breadth of mathematics, would that really be a part of anyone's plan?

And, needless to say, we haven't talked about the elephant in the room which is whether the four years of high school math should be obligated to shoot its entire wad in the name of satisfying the prereq for college freshman physics. Some of my best friends are engineers, but there are other things to be too. There are some mathematicians (including me and evidently Lockhart) who would argue that mathematics is bigger than the science of creating abstract models of physical phenomena for the sake of making better and simpler scientific predictions of real-world behavior, and should be broadened to consider the entire range of intellectual strategies to solve problems. Ordinary folk in my experience can get by without the Fundamental Theorem of Calculus but would be well served with some discrete topics like logic and graph theory. But it would take a larger mathematical revolution than I've ever experienced or even read about to knock calculus off the top of the mountain.