A Four-Stage Model of Mathematical Learning

Research in education and applied psychology has produced a number of insights into how students think and learn, but all too often the resulting impact on actual classroom instruction is uneven and unpredictable (Sabelli & Dede, in press; Schoenfeld, 1999). In response, many in higher education are translating research in education into models of learning specific to their own disciplines (Buriak, McNurlen, & Harper, 1995; Felder, Woods, Stice, & Rugarcia, 2000; Jensen & Wood, 2000). These models in turn are used to reform teaching methods, to transform existing courses, and even to suggest new courses. Research in mathematics education has been no less productive (Schoenfeld, 2000). This article is in the spirit of those mentioned above, in that I combine personal observations and my interpretation of educational research into a model of mathematical learning. The result of this approach can be used to address issues such as the effective role of a teacher and appropriate uses of technology. That is, the model can be viewed as a tool that teachers can use to guide the development of curricular and instructional reform. Before presenting this model, however, let me offer this qualifier. In my opinion, good teaching begins with a genuine concern for students and an enthusiasm for the subject. Any benefits derived from this model are in addition to that concern and enthusiasm, for I believe that nothing can ever or should ever replace the invaluable and mutually beneficial teacher-student relationship.