The Teaching and Learning of Mathematics at the University Level, Crucial Questions for Contemporary Research in Education, Volume 46, Number 11

F or more than thirty years, research in mathematics education has tried to clarify , through work both theoretical and experimental , the processes of learning in mathematics. It has tried also to develop strategies for teaching that take into account the knowledge that it progressively constructed, and then to test it. Research was first carried out on the earliest stages of learning, those of elementary school, with work concerning high school and the university, beyond the required schooling, being relatively marginal. But the great increase in the number of students taking mathematics at these more advanced levels today poses educational problems that again constitute new challenges for research. In this article we will be interested in these problems and will try to make more precise the potential as well as the limitations of work conducted so far. The discussion is from a personal point of view, marked by my own European and French culture; other researchers would doubtless have a considerably different overview. In effect, it is not as easy in education as in mathematics to give an overall picture of the state of development of research in a given area and to identify clearly the results that have been obtained there. This results initially from the fact that this field of research is not at all unified. Diverse approaches coexist, making generalizations difficult, as the recent ICMI study edited by A. Sierpinska and J. Kilpatrick and dedicated to mathematics education as a research domain attests. This diversity is doubtless tied to the relative youth of the field, but it results also from the complexity of the studied phenomena; a single point of view seems insufficient to encompass this complexity. The diversity results also from the fact that the processes of teaching and learning are partially dependent on the social and cultural environments in which they develop; the results of the research thus depend on the social and cultural environments also, and it is not always easy to make precise the field of validity for them. The diversity results finally from the fact that although the benchmarks acquired by research allow one to understand better the difficulties that students have, as well as the dysfunctions of our teaching, they more rarely give us inexpensive means of action to make teaching better in an immediate and sensible way. In spite of these difficulties it is undeniable that research advances, producing …