Tacit models, treasured intuitions and the discrete—continuous interplay

Abstract

We explore conditions for productive synthesis between formal reasoning and intuitive representations through analysis of college students’ understanding of the limit concept in the definition of the derivative. In particular, we compare and contrast cognitive processes that accompany different manifestations of persistence of intuitions and tacit models that coexist with students’ logical reasoning. The students are highly trained in mathematics. We encounter expressions of the persistence and impact of intuitions and tacit pictorial models as described by Fischbein. But we also observe some new characterization of persistence of tacit models in which the tacit pictorial model continues to interfere in the student’s reasoning process, coexists with a logical reasoning but does not prevent the student from reaching a feeling of logical consistency. The empirical analysis and the theoretical discussion offered in the present paper permit us to highlight this very special integration of the formal and the intuitive components of the reasoning process.