On the one hand, there's the Tiger-mother-inspired, "achievement"-oriented approach to math instruction, which is based largely on the theory that kids—for the most part—learn unconsciously, and need to practice something in order to acquire a certain skill level. On this theory, math is something not to be too much enjoyed, but to be drilled and internalized. It is, in another words, a set of basic skills involving numbers and computations that, once practiced and learned, will set a solid foundation for future quantitative endeavors.
On the other side of the coin, we have the idealist philosophy on math instruction. An idealist's first question about the drilling approach is always: what's the point? Yes, what we think of as math is hard for children to learn, and in order to learn it they certainly do need to practice it, but why do we think it's so important in the first place? Why should they spend hours learning the finer points of long division, fraction addition, and decimal conversion? They will never need to do those things ever again, even if they become engineers! To the idealist, math is something completely different; it's about patterns and elegance, puzzles and strategies, creativity and exploration, rather than mindless problem solving and following instructions. According to such a philosophy, there's little purpose to teaching children computations, especially if they find them boring and tedious. [For a more complete rendition of the idealist's argument, including answers to various rebuttals, see the essay here).
Perhaps my descriptions transparently reveal my own inner-leanings. As much as I can, I try to make my classes interesting and challenging for the kids (intellectually challenging, that is...examples to come!). This involves, above all, sitting back when they protest that they "don't get" something, and letting them struggle with it, over their fervent protestations. Thus, it can be immensely difficult, both for them and for me because they don't really like having to think things through, or be challenged in math class. They like hearing how to do something, doing it, and being validated by getting the correct answers. Getting them to switch to a different mindset, where the primary importance is in the way they arrive at an answer, or the reasoning they use behind an explanation, is next to impossible. They just keep asking "is this right? Wait, is this right?"
But the idealist approach has other drawbacks. The simplest is: testing. And by that I don't mean that I (or they) will be judged by this or that particular standardized test this year or next. The problem is, the whole system is set up to emphasize, reward and capitalize on a certain type of knowledge, ie, the standard math curriculum. Whether or not a student can learn and master the standard math curriculum is a pretty good predictor of the quality of college he or she will attend, how well he or she might do, and even how well he or she might do in life beyond college! But I don't think people do better in life in because they can do long division, or even that the ability to learn long division is terribly important to learning other important skills in life. But when everybody is judging people, up through college and even graduate school, based on the same skewed criteria, those criteria become excessively important merely by virtue of others' judgments. In other words, the importance of being good at math becomes a self-fulfilling prophecy (or a tinkerbell).
As I see it, in my job I face a continuous tension between emphasizing to my students the skills that I truly believe would help them succeed in an ideal world (perseverance, curiosity, creativity, reasoning), and the skills that will help them succeed in the flawed world that we have. Hopefully I'm finding a decent balance!