The interplay of intimations and implementations: Generating new discourse with new symbolic tools

The analysis of the Batteries and AIDS episodes presented in this article is guided by the assumption that thinking can be conceptualized as an activity of communication, and learning can be regarded as modifying and extending one's discursive ways. Within this framework, 1 of the aims of mathematical learning is to become skillful in the discursive use of designated symbolic artifacts supposed to mediate solving certain types of problems. My analysis of the learning episodes is aimed at uncovering the ways in which the discursive uses of such new symbolic tools were interactively constructed by the students. I argue that the construction process is extremely complex because of a certain inherent circularity of this process. The analysis reveals that this difficulty may be overcome by the gradual dialectic adjustment of former discursive habits to new contexts. The adjustment happens gradually, through cycles of intimations about the applicability of the old habits followed by implementations in which this applicability is examined. This intricate interplay of intimations and implementations is found in both the Batteries and the AIDS episodes and is presented in detail in the article. I also show that, with time, students significantly increased their mastery of this particular discursive mechanism.