# Thinking aloud together: A teacher’s semiotic mediation of a whole-class conversation about percents

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## Abstract

How does classroom interaction support students’ apprenticeship into the ways of speaking, writing, and diagramming that constitute the practice of mathematics? We address this problem through an interpretative analysis of a whole-group conversation about alternative ways of solving a problem involving percent discounts that occurred in a sixth grade classroom. This research study draws upon Dewey’s theory of inquiry, Vygotsky’s cultural–historical psychology, Freudenthal’s realistic mathematics education, and Halliday’s systemic functional linguistics (SFL). From Freudenthal, we borrow the notions of mathematizing and guided reinvention—the former notion offers a view of mathematics as an activity of structuring subject matter and the latter one provides insights into the processes whereby mathematizing is learned and taught in the classroom. We glean from Dewey his view of reflective thinking as inquiry and the role that conversations may serve therein. We rely upon Vygotsky’s notions of a verbal thinking plane and a social phase of learning in order to reconsider the function of whole-class interaction in apprenticing students into mathematizing. Finally, SFL provides us with tools for explaining the choices of grammar and vocabulary students and teachers make as they realize meanings in whole-group conversations. Treating the selected whole-class conversation as a text, we focus our analysis on how this text came to mean what it did. Our central questions are as follows: What meanings were realized in the whole-class conversation by teacher and students and how were these meanings realized? How did the teacher’s lexico-grammatical choices guide the students’ choices? In addressing these questions, we advance an interpretation of the conversation as paradigmatic of students and teacher thinking aloud together about percents.

## Keywords

Reflective thinking Mathematizing Semiotic mediation Whole-class interaction Functional grammar Percents Bar model## Notes

### Acknowledgment

We wish to thank Norma Presmeg as well as the three anonymous reviewers for their very helpful editorial comments and suggestions. We acknowledge Dor Abrahamson for his critical reading of several iterations of this paper. Thanks also to Mary Schleppegrell and Cecilia Colombi for contributing their functional grammar expertise. We also acknowledge Ana María Bressan and Fernanda Gallego, from the Grupo Patagónico de Didáctica de la Matemática (Argentina) as well as Silva Koethe for their invaluable help in coding and analyzing the text. Finally, we are especially grateful to Yeuk-Sze Leong, a teacher in whose classroom we first witnessed the kind of thinking aloud together conversations discussed in this paper. The research described herein has been partially supported by Brooklyn College (City University of New York) through a Leonard and Claire Tow Travel Fellowship.

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