Once, when I was teaching a tenth grade class and got the “How is this useful?” question, I gave the quick list of some places where the topic came up, but then said something to the effect of “Really, you can’t understand exactly how this will be applied right now. Remember when you were in 3rd grade, and they explained how fractions are useful because you can make change and count parts of pizzas? That was true, but you now use fractions all the time without thinking about it. Every time we graph a function, we assume it is continuous, and that makes no sense without fractional values.” They actually took this pretty well and seemed to find it a satisfying answer.

If math can be taught as something that’s interesting, rather than as something that’s useful, it changes the way students look at it.

There is a big push in UK schools at the moment for a ‘functional mathematics‘ curriculum as an alternative for weaker students to standard GCSEs, which seems predicated on the belief that weak students will only be interested in maths that is useful in everyday life (see the link for an incredible amount of government nonsense-speak on this topic).

I think you’ve identified several ways in which this is a very bad idea.

An equally bad trend is the need to assert that ‘Maths is Cool’. As Rob Eastaway pointed out at in a keynote talk at a conference I attended recently, saying that something ‘is cool’ or ‘is fun’ does not automatically make it so. If we substitute ‘train spotting’ for ‘maths’ in the assertions, we can see how many students approach what we say!

Huzzah.

The one thing I think might be worth commenting on is this: “Anyone who’s ever taken proof-based math in college knows that mathematicians don’t spend their days doing the stuff that’s taught in high school.” Yes, two column geometry proofs are the best example.

However you also say “This kind of open-ended, puzzle-like problem is a lot more fun to do, and a much better way to show students what math is really like, than the computational topics that take up the current standard curriculum.” Which I agree with, to a point. I’m a new teacher, so maybe I’m totally off base, but there are a lot of students who need the structure of those computational topics. The drilling of the basics. Very specific types of learners find this open-ended environment fruitful. I think that some sort of thought-out mixture of the two is probably the way to go.

You might be interested in reading this (which I sort of disagree with, but it is on the theme of your post) if you haven’t yet: http://www.maa.org/devlin/LockhartsLament.pdf