The Tacit-explicit Dimension of the Learning of Mathematics: An Investigation Report**

This paper reports on study that investigated the tacit-explicit dimension of the learning of mathematics. The study was carried out in a secondary school and consisted of an episode analysis related to a class discussion about the difference between plane figures and spatial figures. The data analysis was based on integration between some aspects of Polanyi’s theory on tacit knowledge and Ernest’s model of mathematical knowledge, with reference to its mainly explicit and mainly tacit components. This integration has involved not only the types of knowledge – mainly explicit or mainly tacit – the students used in a psychological way to perform a mathematical task involving conversation, but also and particularly how much the projection of those types of knowledge on the task were manifest tacitly or formalized by the students. Among the results of the research, a strong finding was that the lack of correspondence between the students’ utterances and their original understandings is directly related to the manner in which the tacit co-operates with the explicit in the process of articulation.This paper is a development of three short papers presented at ICME-10, Copenhagen, Denmark, 4–11 July 2004, and PME-28, Bergen, Norway, 14–18 July 2004.