Bootstrapping mental constructions: a learning system about negative numbers

Abstract

In this paper, we propose a new model for learning mathematical concepts which cannot develop informally. This is a “middle-out“ approach in which the objects of the system can be mapped both to formal knowledge and to real-life problems. We hypothesize that the key process which governs the acquisition of knowledge within such learning systems is reasoning with mental models, which we hold to be similar to the acquisition of basic mathematical constructs in non-school settings. The paper reports the development of a system for learning negative numbers and a study in which two pairs of students were exposed to that system.