The Math Wars
For those who may be unaware, we are in the middle of a philosophical war about what math children should be taught and how they should be taught it. During the 1990s in the United States, a reform movement in mathematics and science education gained ascendance based on the constructivist theories of the famous Swiss psychologist, Jean Piaget. A number of views characterize constructivism as applied to mathematics education, including: 1) that children construct their own understanding of mathematical principles, rather than being passive receptacles of knowledge imparted by others; 2) that the goal of instruction is to aid children in developing their own understanding, rather than to teach mathematical facts and procedures or to impart conventional conceptions of mathematics; 3) that children construct an understanding of mathematics by working with concrete, real life referents, not by learning facts and procedures in isolation; and 4) that children learn through curiosity and a desire to understand their world, not through the imposition of external rewards.
In keeping with these views, 1990s constructivist reforms in math involved problems and examples drawn from real-life situations and designed so that children could discover underlying mathematical principles in the process of working out their own solutions. These student-directed activities, sometimes called discovery learning, took precedence over direct instruction by teachers. Professional development for teachers emphasized the teacher's own conceptual understanding of mathematics as well as the teacher's ability to figure out what children are thinking in real time in order to guide children's efforts to make sense of math.
In addition to these pedagogical shifts, constructivist reforms redefined standards for mathematics content and performance. The definition of mathematical proficiency was broadened from the ability to perform the computations necessary to solve math problems, to the ability to understand mathematical concepts, to apply mathematics to novel problems, and to reason mathematically. Broadly speaking, reformers focused on the development of conceptual understanding of mathematics and lessened the emphasis on fluent mastery of mathematical facts and procedures. Reports, recommendations, and standards from such respected sources as the National Council of Teachers of Mathematics and the National Research Council indicated that instruction should address all strands of proficiency simultaneously, and rejected the "old ways" of teaching math, such as mastering computational procedures followed by problem solving.
Critics of the 1990s math reforms charged that in shifting the focus from computation to understanding, from teacher-directed to student-directed learning, and from sequenced instruction along a hierarchy of skills to simultaneous instruction in all strands of mathematical competence, reformers generated a recipe for poor preparation of students for mathematically challenging content. The critics argue that the 1990s math reforms ignore the importance of fundamental building blocks, are not sufficiently rigorous, do not cover aspects of math content that are necessary for proficiency, and over-generalize the role of curiosity and discovery as core principles of mathematics learning.
It would be very satisfying if I could tell you that the math wars have been resolved based on high quality research. Unfortunately, we are far from that. However, there is research that suggests where some of practices and assumptions of both the constructivists and their critics may require more nuanced implementation.