Purposes and Methods of Research in Mathematics Education, Volume 47, Number 6

T he first quotation above is humorous; the second serious. Both, however, serve to highlight some of the major differences between mathematics and mathematics education—differences that must be understood if one is to understand the nature of methods and results in mathematics education. The Cohen quotation does point to some serious aspects of mathematics. In describing various geometries, for example, we start with undefined terms. Then, following the rules of logic, we prove that if certain things are true, other results must follow. On the one hand, the terms are undefined; i.e., “we never know what we are talking about.” On the other hand, the results are definitive. As Gertrude Stein might have said, a proof is a proof is a proof. Other disciplines work in other ways. Pollak’s statement was not meant as a dismissal of mathematics education, but as a pointer to the fact that the nature of evidence and argument in mathematics education is quite unlike the nature of evidence and argument in mathematics. Indeed, the kinds of questions one can ask (and expect to be able to answer) in educational research are not the kinds of questions that mathematicians might expect. Beyond that, mathematicians and education researchers tend to have different views of the purposes and goals of research in mathematics education. This article begins with an attempt to lay out some of the relevant perspectives and to provide background regarding the nature of inquiry within mathematics education. Among the questions explored are the following: Just what is the enterprise? That is, what are the purposes of research in mathematics education? What do theories and models look like in education as opposed to those in mathematics and the physical sciences? What kinds of questions can educational research answer? Given such questions, what constitute reasonable answers? What kinds of evidence are appropriate to back up educational claims? What kinds of methods can generate such evidence? What standards might one have for judging claims, models, and theories? As will be seen, there are significant differences between mathematics and education with regard to all of these questions.