Archive for the ‘pedagogical’ Category

Excitement!

December 29, 2009

Sorry about the lack of new post; it’s coming. It turned out to be a more interesting problem than I at first thought; look for it around New Years’.

I’m working through some of the holes in my graph knowledge with my shiny new copy of Bollobas’ Modern Graph Theory. Chapter 1, Exercise 19 is a problem I’ve done before, but the way it’s presented makes me want to do it all over again:

Characterize the degree sequences of forests!

Exercise 17 is about the degree sequences of trees, and 18 extends it to forests with a fixed number of components — so this isn’t totally out of the blue. Still, it makes me wonder why more textbooks don’t end problems with exclamation marks.

Quasi-bleg: Why are there bump functions?

September 4, 2009

When I was learning analysis (beyond, say, first-year calculus), one of the facts that most surprised me was the fact that there are functions that were smooth (i.e., infinitely differentiable) and yet compactly supported. Of course, I didn’t think about it with that phrasing; there’s a pretty simple geometric interpretation of smoothness for most functions one encounters in calculus (actually, one rarely sees differentiable functions that aren’t smooth!) Specifically, if a function isn’t smooth at x, then there’s some sort of a “kink” at that point, or at least “around” that point.

Is this justified? Well, not totally, but let’s give a couple of examples to at least show why it’s a good first approximation. (more…)

Little insights

July 2, 2009

Qiaochu has a post up on “going beyond your comfort zone” in mathematics. Towards the end of it, he lists many of the most commonly given good reasons for learning new areas of mathematics — and all of them are very good reasons. But there’s one that I want to talk about now which I don’t think I’ve seen mentioned before; it’s not so much a pedagogical principle as an autodidactical one.

Mathematical research is largely driven by insights — seeing that trick X can be applied to problem Y, for instance, or formulating conjecture C based on an interesting similarity between areas P and Q. Of course, top-quality work (especially theory-building work) is often focused around “big” insights, like the Langlands conjectures, or Serre and Grothendieck’s radical restructuring of algebraic geometry, or almost everything Riemann ever wrote. But as anyone who’s ever done serious mathematical research knows, those “big” insights are virtually always the culmination of a series of “small” insights that grew progressively larger, backed up by calculation or experimenting with special cases.

When you learn new mathematics, you get to have these “little insights” in spades — because the foundational level of any theory is built on them, and it’s much more fun to try to anticipate them in advance than to sit back and learn the material passively. Examples of these “little insights” might include:

  • If you know that a vector space is in a sense “an abelian group over a field,” and you’ve learned the definition of a ring, you might start thinking about the properties of “a vector space over a ring.” Congratulations — you’ve rediscovered module theory.
  • The realization that group presentations really just specify quotient groups of the free group — and the subsequent realization that every group is a quotient group of the free group on some number of generators.
  • The realization that group representations and their characters can tell us a lot about the structure of the group, including, often, whether it is simple or solvable.
  • The realization that powers don’t matter when we’re considering the solution set of a system of polynomials. (This leads directly to the Nullstellensatz.)
  • Noticing that every group action is continuous under the discrete topology.
  • Various and sundry other “hey, I’ve seen this before!” moments, for instance: seeing that taking the radical of an ideal is in many ways formally similar to taking the topological closure of a set; seeing that tensor products and/or direct sums are “essentially the same thing”; and many of the lower-level tricks in combinatorics that pop up time and again (e.g., counting mod 2).

I believe that every math course above perhaps freshman calculus should be taught in such a way as to maximize the number of these “aha!” moments where these little insights are gained, and something that’s taught in the next chapter or the chapter after that is anticipated, even if very informally. The insight muscle is among the mathematician’s most powerful tool, and we should exercise it early and often.

The derivative of x^2

June 21, 2009

A commenter on Scott Aaronson’s blog, in a discussion of K-12 mathematics education in the U.S., brought up the following (I would imagine fairly common) Algebra I fallacy:

The slope of the line y = mx + b is equal to m. Consider the equation y = x*x + 0. Around x = c, this reduces to y = c*x, and so the slope of the parabola y = x^2 at the point (x, x^2) is x.

Of course, anyone who has taken freshman calculus knows that the slope of the parabola at that point is 2x, not x, and therefore the above reasoning is flawed in some way. But from the perspective of a beginning Algebra I student, there’s nothing at all obviously wrong with it!

So how would one go about deriving (no pun intended) the correct slope of a parabola using only basic Algebra I knowledge? You might imagine that you could just calculate the “rise over run,” and that’s a good start — ((x+h)^2 – x^2)/h = 2x + h. But (without the crucial “take the limit as h approaches 0”) this is ill-defined!

It’s clear that if you could prove the product rule for derivatives, then the correct expression for the slope would follow. Still, though, it seems impossible to do this with such a basic level of knowledge, without resorting to the concept of a limit or an infinitesimal.

The most promising path would likely be to show that y = x^2 actually looks locally like y = 2cx – c^2, rather than y = cx. Actually, it isn’t that difficult to show that y = cx is wrong; if you draw “tangent lines” to the parabola for large enough values of c, it’s clear that those lines don’t come anywhere near the origin. Still, 2cx – c^2 seems incredibly arbitrary without foreknowledge of the calculus.

So, here’s the challenge: How do you convince a reasonably bright Algebra I student, without getting into calculus, that the slope of the parabola is not x but is, in fact, 2x?