Balancing Perspectives on Mathematics Instruction

Focus on Exceptional Children, May 2003 by Jones, Eric D, Southern, W Thomas

Between the two extremes of endogenous and exogenous constructivism is dialectical constructivism. Rather than taking a position that might indicate an extravagant disregard for the bodies of literature on cognitive development or learning, the dialectical constructivist perspective holds that students acquire knowledge through instructional experiences. In some cases the teacher assumes a highly didactic role, and in other cases the teacher provides less obvious support. For example, a teacher provides students with a formula for calculating the area of a surface and then explicitly models the procedure for them to follow. If the students have had prior experience in calculating the area of a surface and the task at hand is to determine the area of a room, the teacher would offer more subtle prompts for the students to recall and apply "the method you already learned for calculating the area of any surface." The teacher determines the extent to which explicit guidance or implicit support is needed to facilitate the desired learning, depending on the student's prior performances in instructional interactions (or the instructional dialogue).

RESEARCH IN MATHEMATICS EDUCATION

Given the history of research in mathematics education, we should not be surprised at how little the quality of mathematics instruction has improved despite continuing calls for the reform of math education. Kilpatrick's (1992) description of the historical development of research in mathematics education considers several issues that have had basic influences on the development of mathematics education and, likewise, on the development of debates about how mathematics should be taught.

Although mathematics has had a long history as a core subject in schools and universities, rigorous research about how to teach has been a relatively recent effort. In the beginning of the 19th century, university-level programs in teacher education began to emerge. By the end of the 19th century mathematics education began to develop as a field of study in several countries, including the United States, several European countries, and Japan. The emphasis of those programs was on the practical and rapid preparation of secondary-level mathematics teachers. Research did not flourish in those programs.

Increasingly, university programs began to assume that mathematics teachers should be rigorously prepared in the content and culture of specialized study in mathematics. Research in mathematics education increased as the status of teacher preparation and mathematics education became more elevated.

The effect that research can have on practice depends substantially on the questions researchers have been asking. Kilpatrick (1992) points out that two disciplines have had pronounced effects on the development of mathematics education: mathematics and psychology. The interests of mathematicians and psychologists have differed substantially. While interest in mathematics education has a long history among mathematicians, it also has been episodic. Mathematicians' flurries of interest in mathematics education have tended to be in response to personal curiosities, as well as concerns over various issues. According to Kilpatrick (1992, "[I]nadequate preparations in lower schools, falling enrollments in advanced courses, the potential erosion of mathematics as a school subject, and threats to national status have from time to time prompted mathematicians to look into what schools are doing and how it might be improved" (p. 5).