Research in the Learning of Mathematics: Some Genuinely New Directions

Summary Research in how people learn and use mathematics has been changing in fundamental ways during the past decade. Increasingly it draws its organizing principles, perspectives and methodologies from the new Cognitive Science and Artificial Intelligence disciplines rather than from classical educational psychology, which traditionally has informed mathematics education research. This change, part of a larger evolution associated with the arrival of the Information Age, has led to detailed observations of college students doing increasingly realistic mathematical tasks. The resulting analyses are used to build explicit cognitive models describing the mental structures and processes involved in doing these tasks and related errors. This work provides scientific bases for the following general results. (1) Students at all age levels have little ability to relate their school mathematics to the wider world of experience. (2) All students bring to every learning situation primitive but remarkably robust intuitive conceptions that are the primary means by which they interpret their experience, including any school explanations ofr theories. (3) Novice/Expert differences include more than quantity of information, and differences in the amount of informal, qualitative, imagistic, metaphoric, and heuristic knowledge that experts use (and of which they are mainly unaware. (4) Students must construct knowledge by actively acting on and through ol knowledge structures — communication metaphors about learning need to be replaced by construction metaphors. Information technology is radically altering educational objectives as wellas the means by which they may be achieved. Thus we must expand our understandings of student thinking so new curricula and pedagogy will fit the cognitive realities being uncovered. Fortunately, new software developments promise both improved means to examine the workings of students’ minds as well as the means to improve them.