Mediating between concrete and abstract

One of the problematic issues in mathematics education is the question of how to teach students abstract mathematical knowledge. In the mainstream information processing approach, one usually presents concrete models to help students acquire this abstract knowledge. However, ‘concrete’ in the sense of tangible does not necessary mean ‘concrete’ in the sense of making sense. This observation is in line with research findings (which will be presented later) that the use of manipulad ves1 does not really help students to attain mathematical insight. Moreover, even if a certain mastery is attained, application appears to be problematic. The manipulatives-approach fails probably, because-although the models as such may be concrete-the mathematics embedded in the models is not concrete for the students. Or to put it another way, the manipulatives-approach passes over the situated, informal knowledge of the students. Alternative approaches depart from the idea that situated, informal knowledge and strategies should be the starting point for developing abstract mathematical knowledge.