Over the weekend I started reading a book called Math Wars about the controversy that followed the release of the National Council of Teachers of Mathematics curriculum standards in 1989. A lot of states modeled their standards after them, and a lot of textbooks were written with them in mind.
The controversy was that many people felt the standards were "fuzzy" because they focused on problem solving and allowed the use of calculators. They deemphasized memorization and algorithmic skill.
The author of the book points out that math curriculum has been swinging back and forth between two extremes for a long time. For example, in the 1960's mathematicians themselves had a lot of influence on the curriculum and they developed what was known as "New Math", which focused on things critical to pure mathematics like set theory. Then there was a period based on problem solving, then a shift to "traditional" math focused on arithmetic and algebra.
The one quibble I have with the presentation is that it seems there is not just one axis of variation for the various kinds of curricula. One on axis we have the spectrum between abstract and applied. On the other we have focus on memorization vs depth of understanding. Below is an image of how I would visualize the four quadrants that result:
Each of these quadrants represents an important part of mathematics, but different curricula and different teachers tend to favor one area over another. The 1989 standards focused on the top left quadrant, New Math leaned to the top right, and traditional math often emphasizes the bottom right.
The author of Math Wars likes to emphasize that mathematicians (who developed New Math) and math educators (Phd types who wrote the NCTM standards) don't always see eye to eye. But I think his analysis of this disagreement is a bit muddled because he only has one dimension of variation.
Today I asked my students which kinds of math they felt most comfortable with. Interestingly, a lot of them said they didn't like memorizing arithmetic, although that is clearly important as a foundation. A number of them preferred heuristics. A handful enjoyed the pure math. A lot of them didn't vote at all. I would never base my teaching purely on student preferences, but it was interesting to hear their perspective.
In the end I think that a robust math education needs to include all four quadrants, but how to spend class time is still a big question mark. The right balance probably depends a lot on the age and capabilities of the students.