The Mathematical-Didactic Training of Kindergarten Teachers: the Case of how to Teach Counting

The aim of this paper is to design a training program to be implemented amongst second-year Teacher Training students at the Complutense University of Madrid. The purpose of this training program is to respond to the question: "How can kindergarten pupils be taught to count?" In order to tackle this question, a more basic question of an epistemological nature has to be considered and closely examined, i.e. "What constitutes counting in Kindergarten teaching?" The methodology used is based on the axiom that all training, and in particular kindergarten teachers' pedagogical mathematical training, should be based on the dialectic between the raising of topics in professional work and the construction of responses or response elements to these topics. This methodology, with an essentially qualitative character, has been based on the construction of an environment in order to describe the training component's central questions to which Kindergarten teacher training must respond. The most important results of this investigation are the following: 1) the designed and tested training program allows topics related to doing mathematics to be integrated with those that refer to "teaching and learning mathematics". This result is a very important one since it demonstrates that mathematics teacher training programmes are able to integrate these two, previously separate dimensions; and 2) the methodology used for designing the training process has revealed two main issues. On the one hand, the central role of Didactics of Mathematics in teacher training programmes and, on the other, the theoretical and practical advantages of connecting the proposals and results obtained in two distinct, although very close, pedagogical theories:The Theory of Didactic Situations and the Anthropological Theory of the Didactic.