The "Math Wars" have been framed as a debate between “traditionalists” and “reformers”. I don't take a side in this debate, but rather I think the debate is unproductive.
The reformers actually represent the educational establishment, and their position has been the official position of the National Council of Teachers of Mathematics (NCTM) for over 20 years. They believe in learning by discovery, cooperative work in small groups, and an emphasis on communicating one's thinking. Their philosophy of education is constructivism, and although the word constructivism does not appear in the Principles and Standards for School Mathematics, it is clearly a constuctivist document. The traditionalists, who include many parents and a large number of university mathematicians and scientists, have developed as a reaction to what they see as the excesses of the reformers and a perceived decline in the abilities of college students. A more informal movement, their position seems to be well stated by the Mathematically Correct movement. Traditionalists stress the importance of individual competence, ability to instantly recall number facts, and the ability to perform important algorithms. Although I have not seen them espouse a theory of education, from their prescriptions they implicitly adopt behaviorism and cognitivism.
There is a hidden but clear political dimension to the math wars. The NCTM Principles and Standards states "All students should have the opportunity and support necessary to learn significant mathematics with depth and understanding. There is no conflict between equity and excellence." I see no evidence that the traditionalists agree with this, and my conversations with traditionalists indicate that most believe that mathematics teachers need to put forth a challenging curriculum, and essentially serve those students that are able to rise to the challenge. They also contend that the reform agenda has put equity far ahead of excellence.
My belief is that traditionalism and reformism are not as diametrically opposed as they seem, and that the future will see a convergence in these movements. I think that most reformers now see that students must spend a substantial amount of time on rote learning. For example, the number of high-school graduates who struggle to make change without electronic assistance is disturbing. Without the instant recall of basic number facts, such as the single-digit multiplication table, students are severely handicapped in trying to solve more complex problems. And I think that most traditionalists see value in the goals of the reform movement. I think we can both do a better job of education the top 20% of students who we need as a technological elite and the bottom 80% who will have to find jobs that are increasingly more quantitative and will also need to be informed citizens in a world that depends more and more on numerical analyses.
Those who care about mathematics education need to move beyond the math wars and work together to improve the quality of the teachers and schools that provide this education.