Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult questions, I propose to base research on a metaphor of thinking-as-communicating. This conceptualization entails viewing learning mathematics as an initiation to a certain well defined discourse. Mathematical discourse is made special by two main factors: first, by its exceptional reliance on symbolic artifacts as its communication-mediating tools, and second, by the particular meta-rules that regulate this type of communication. The meta-rules are the observer’s construct and they usually remain tacit for the participants of the discourse. In this paper I argue that by eliciting these special elements of mathematical communication, one has a better chance of accounting for at least some of the still puzzling phenomena. To show how it works, I revisit the episodes presented at the beginning of the paper, reformulate the ensuing questions in the language of thinking-ascommunication, and re-address the old quandaries with the help of special analytic tools that help in combining analysis of mathematical content of classroom interaction with attention to meta-level concerns of the participants. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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