Central to all in mathematics education is the activity of teaching-or, if you prefer, the activity of helping others learn to think about mathematical situations. This is the heart of school mathematics and, for most people, this is the part of mathematics education that has the greatest importance in their lives.
It is not, however, the easiest to study or to understand. What teachers do is subtle and profound, so studying it must also have these same two attributes. The report in this section-by Philipp, Flores, J. Sowder, and Schappelle-is clear evidence of the growing sophistication of mathematics education. Philipp et al. look carefully at matters that, a few years earlier, might have passed unnoticed, in a prior age when "teaching mathematics" was commonly thought of as little more than showing students how to carry out a few specific algorithms, such as "adding fractions" or "solving linear equations." Because of the way they study teaching, Philipp et al. show us professional people who work on complex tasks that reach far deeper into how students think, and who must consequently delve far more deeply into reflections on what they themselves are trying to accomplish.