Apostolos grips my heart: a swift kick to remind me what I know in my soul – that the subject I love lights me up as much as a transforming piece of music or a long hike in the mountains. Mathematics speaks to – and improves upon – my humanity.

And my brain will follow: I can see how doing what I consider selfish (spending the summer exploring the math that interests me, without regard to how it will directly impact the very particular details of the courses I am teaching next year) is actually an important thing. I’d like to study math in the context of the “complex, adventurous, brave, struggling human beings” who developed it. I’d like to figure out how to invite students to see, understand, and join me on that quest.

@Jason: “It certainly happened with square roots.” I would argue that it is *extremely important* for students to learn to compute square roots. I’m not sadistic enough to suggest they should do it beyond the tenths place, and a handful of practice problems would be enough to satisfy me.

Knowing how to show that the square root of 2 is about 1.4 (i.e. because 1.4^2 is 1.96) is, to me, a very accessible skill (14^2 = 196!), and a very important one to master. How else will students come to understand square roots *conceptually?* Wait a minute — they don’t! And that’s why it’s so hard to teach them abstract, algebraic rules about square roots.

The student who can compute with square roots will have a very easy time learning that the cube root of x^12 is x^4 — it is simply that thing whose cube is x^12.

Having argued for the necessity of computing with square roots, I admit I can make no such argument for a great many other math skills.

We may well ask of any item of information that is taught … whether it is worth knowing? I can only think of two good criteria and one middling one for deciding such an issue: whether the knowledge gives a sense of delight and whether it bestows the gift of intellectual travel beyond the information given, in the sense of containing within it the basis of generalization. The middling criterion is whether the knowledge is useful. It turns out, on the whole, … that useful knowledge looks after itself. So I would urge that we as school men let it do so and concentrate on the first two criteria. Delight and travel, then.

Narrative certainly provides both delight and travel (including the basis for generalization through use of allegory and metaphor). And for those few students for whom algebra skills actually will be useful later, perhaps we are now in an era where that knowledge can take care of itself (although experience with current college students suggests to me that a desire to be, say, a physical chemist or an actuary along with pointers to information about basic skills that were forgotten or never learned is not usually sufficient for the knowledge to “take care of itself”.)

@Jerzy — great cartoon!

@Bowen — Logarithms are a wonderful example of where historical narrative is essential for making the topic bearable! Perhaps prosthaphaeresis could do the same for trig identities!

However, I disagree with you about the need for rational expressions, at least for students going on to do calculus. As you indicate, they are the prime example for understanding the difference between an expression and the function it defines (more precisely, the phenomenon of equal expressions defining different functions). This particular misunderstanding seems to me the precise reason why students do not understand why all limits aren’t computed simply by “plugging in the value”, which itself leads to them not having a basic understanding of the definition of derivative (even though they might very well understand the intuitive description of a limit of slopes of secant lines.) Now you can argue whether there is a practical need for students for understand the definition of derivative, but having proceeded this far it is easy to introduce history and narrative by talking about differentials, the long process of developing the definition of limit, early successes in computing derivatives, etc.

Rational expressions, in my opinion, really needs to be shown the door. In the curriculum it mostly exists for rewriting expressions, notably for calculus integrals like the integral of 1/(1-x^2). That stuff is getting deprecated in two ways: the computation can be performed by technology, and there is no longer a need for an explicit solution. At the high school level, there’s not much really happening with it other than talking about function domain, and giving messier and messier algebra work (and factoring). Bah.

There’s a lot of stuff like this — why do we care about converting the form of an ellipse? — but rational expressions is one of easiest topics to zap cleanly out of the way.