| | Devlin's Angle |
Things change dramatically around the sophomore university level, when almost everything a student learns has significant applications.
I am not arguing that utility is the only or even the primary reason for teaching math. But the question of utility is a valid one that deserves an answer, and there really isn't a good one. For many school pupils, and often their parents, the lack of a good answer is enough to persuade them to give up on math and focus their efforts elsewhere.
Just over a decade ago, I worked with PBS on a major six-part television series Life by the Numbers that set out to provide classroom teachers with materials to show their students the major role mathematics plays in today's world. [The series is still available on DVD at http://www.montereymedia.com, though many of the examples are now looking decidedly dated.] But most of the mathematics referred to in the series is post K-12 level, so at best the series simply added some valuable flesh to the "The math you learn now leads on to stuff that is important" argument.
Another possibility to try to motivate K-12 students (actually, in my experience from visiting schools and talking with their teachers, it is the older pupils who are the ones more likely to require motivation, say grades 8 or 9 upward) is for professional mathematicians to visit schools. I know I am not the only mathematician who does this. There is nothing like presenting pupils with a living, breathing, professional mathematician who can provide a first-hand example of what mathematicians do in and for society.
I recently spent two weeks in Australia, as the Mathematician in Residence at St. Peters College in Adelaide. This was only the second time in my life that a high school had invited me to spend some time as a visitor, and the first time overseas - over a very large sea in fact! In both cases, the high school in question was private, and had secured private endowment funding to support such an activity. For two weeks, I spent each day in the school, giving classes. Many classes were one-offs, and I spent the time answering that "What do mathematicians do?" question. For some 11 and 12 grade classes, we met several times and I gave presentations and mini-lessons, answered questions, engaged in problem sessions, and generally got to know the students, and they me. You would have to ask the students what they got from my visit, but from my perspective (and that of the former head of mathematics at the school, David Martin, who organized my visit), they gained a lot. To appreciate a human activity such as mathematics, there is, after all, nothing that can match having a real-life practitioner on call for a couple of weeks.
Thought of on its own, such a program seems expensive. But viewed as a component of the entire mathematics education program at a school, the incremental cost of a "mathematician in residence" is small, though in the anti-educational and anti-science wasteland that is George Bush's America it may be a hard sell in the U.S. just now. But definitely worth a try when the educational climate improves, I think. If it fails, the funds can always be diverted elsewhere.
Sadly, none of the letters published stated what I think is the main reason why we teach mathematics: to develop mathematical thinking. Since our ancestors invented numbers about 10,000 years ago, we humans have developed a way of thinking about the world we live in (and more recently the worlds we create) using a mode of thought we call "mathematics". It is a curious - and not well understood - blend of discrete and continuous quantitive measurement, abstract objects, abstract relations, abstract structures, rule-governed reasoning, and various other stuff. (I tried to flesh out the components in my 2000 book The Math Gene.)
I used to tell students that mathematical thinking is just "formalized common sense," but then I realized this is not true. Sure, it seems an apt description for some of the more basic parts of mathematics, taught in the lower school grades. But it is way off target when it comes to much advanced mathematics, which is a highly specialized form of thinking - a "language game" in the sense of Wittgenstein, some would say - that in many cases is actually counter to "common sense reasoning."
The amazing thing is that this strange way of thinking has proved to be extremely useful, in highly practical ways! Without mathematical thinking, we would still be living in caves or mud huts, breathing in the smoke from our fires. You need mathematical thinking to design and construct buildings, do science, develop technologies, and do all the other things Homo sapiens modernus takes for granted.
In fact, so important is this way of thinking to our lives, so powerful is it, and so strange - just think about it for a moment, it really is a most "unworldly" and unnatural, somewhat stilted way of thinking - that it surely must be taught to every living person. (Not to mastery; for most people, general awareness and a general competency is surely enough. There is no shortage of the dedicated experts, indeed the world has an oversupply.)
At least, that's the way I see it.