Brief comments on school (K-12) mathematics

Professional mathematicians may be the least appropriate people to comment on pre-college education -- as two of many reasons, because they have been successful in the existing system, and because they implicitly regard it merely as a preliminary to college mathematics. So I have no desire to make some crazy proposal of my own, just to relate some points I like to keep in mind when reading other people's proposals.

Surely the fundamental question is

why do we devote such a huge amount of time to mathematics in K-12 education?

One can imagine four possible goals.

Some math is useful in the everyday life of a typical adult.
Further math is necessary for certain professions.
The point of working out in the gym is not to become proficient at using particular machines, but to attain a more generalized state of physical fitness; analogously the main point of learning math is not to be able to solve quadratic equations but to attain a more generalized state of "intellectual fitness".
Conveying the intrinsic beauty and elegance of math.

The 4th goal -- mathematics as poetry -- hardly addresses why one should spend so much time on mathematics rather than, say, poetry. As for the 3rd goal, proponents of Latin and chess can and do make the same argument -- it is surely hard to pin down the "intellectual transferability" of any subject. So it seems to me that in thinking about K-12 math education one should focus on goals 1 and 2.

Obviously no-one disagrees with goal 1, and the existing curriculum partly addresses it. But if you made a list of 100 representative things related to math that you would like every adult to know ("the proportion of doctors who are men is different from the proportion of men who are doctors") then I strongly suspect that the typical adult actually understands rather few. There is a huge imbalance between the number of classroom hours devoted to math and the amount of "everyday useful" comprehension that sinks in.

In public discussion it is goal 2 that is most emphasized, both at a societal level (skills lacking in the labor force) and at the individual level (parents recognizing that quantitative skills help their children attain a well-paid career). But translating this recognition into actual K-12 teaching is a slow and indirect process. On one side is a kind of academic "trickle down". From what is actually needed in a profession, to what is taught in a College Major program related to the profession, to what is taught in lower division college courses, to what is on SAT tests, to what is taught in high school, to elementary school. On the other side are separate attempts to change all or some of the K-12 curriculum, which may or may not be motivated by some notion of specific employment skills.

As another point, the current system strikes me as based on a certain traditional perception of the internal logical structure of math -- how one topic builds upon another -- rather than explicitly addressing any explicit reason for learning topics.

My only conclusion is that one should separate the two main issues: the high-level issue of what broad topics should be taught to what percentage of citizens, and the classroom issue of how to teach a topic. The latter should be strictly evidence-based; the former is inevitably somewhat a matter of opinion. Alas most mathematicians engage discusssions of teaching at the level of "this is my opinion of how to teach Algebra 1/Calculus 1" which completely misses the point.

On a positive note, How to Fix Our Math Education by Sol Garfunkel and David Mumford strikes me as one of the most sensible proposals suggesting a brief outline of what to teach. "Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering".