In this dissertation, I report my professional inquietudes wich were intrinsically connected this discomfort I noted it is producing in my students, from my teaching pratice. Practing, that in many aspects, necessitated reformulations and adjustaments, in accordance to new times. It was a teaching practice based in the purposes of the modern science, that it intended to describe the relity by deterministic, exact, hierarchized and shared laws. Trying to change my practice, I (re)constructed, in the interaction with my pairs, a proposal of a Mathematics Education laboratory. The activities of this laboratory should provide a fertile environment of verbal interactions, arousing an activa participation of the students in the consturction of the knowledge. With the conviction that I needed to increase the reflection about my teaching job, I (re)constructed a new personal view of science. I supported myself in ideas of some philosophers of science, that come near of a transdisciplinary view and what has been called postmodern science. Whith this new conception, I tried to link the Mathematics Education with the principles of the Historical-Cultural Psychology, in order to appropriate myself of theoretician-methodological instruments to investigate my teaching practice. Thus, I put into practice a microgenetic analysis of my interaction with two students in a typical activity of Mathematics Education laboratory: the apprenticeship of the optimized logical-arithmetic solution of a game with rules, the Nim. I filmed the thirty-six games that I played with studentes until they discovered the opitimized strategy of game and after that, I transcribed them, integrally. The analysis pointed out three moments of the optimized strategy construction: the formulation and (re)formulation of the hypothesis; the selection, apllication and test of hypothesis and the transference of the constructed ratiocination to the formulation of new hypothesis. I observed genetics transitions in all moments and changes in the interactive standards, wich sometimes reflected a communicative boarding prodominantly
dialogical (first and third moments) and sometimes they reflected authority (second moment). I perceived how the students, interacting with me, developde the capacity to regulate their own actions to solve up the problem that challenged them.
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