Why problem solving?
Problem solving has long been recognized as an exciting and engaging approach to doing mathematics. "A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion."
Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc., 2000.
Schoenfeld calls problems "starting points for serious explorations, rather than tasks to be completed."
Schoenfeld, A.H. "Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics." Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. Ed. D.A. Grouws. New York, NY: Macmillan, 1992. 334-370.
Ball and her colleagues observe that the knowledge that math teachers need consists of more than knowing math well or understanding how children think at particular developmental stages. It comes from knowing how to apply mathematical knowledge to students understanding.
Cohen, D. K., & Ball, D. L. (2001). Making change: Instruction and its improvement. Ball, D., & Bass, H. (2000). Interweaving Content and Pedagogy in Teaching and Learning to Teach: Knowing and Using Mathematics. In J. Boaler (ed.), Multiple Perspectives on Mathematics Teaching and Learning, Westport, CT: Ablex.
Ball, D., & Bass, H. (2004). Knowing Mathematics for Teaching, in R. Strasser, G. Brandell, B. Grevholm and O. Helenius, Educating for the Future Proceedings of an International Symposium on Mathematics Teacher Education, Sweden: The Royal Swedish Academy of Sciences.