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

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 wholegroup 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 Educ Stud Math (2010) 73:21-53 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.

Developing a contextualization of students’ mathematical problem solving

“…However, it is not clear what "to abstract a given situation" (NGAC, CCSSO, 2010, p. 6) would mean in practice. Systemic Functional Linguistics may argue that abstraction has a discursive use for semiotic mediation by clustering similar situations and offering a general representation (Shreyar, Zolkower, & Perez, 2010), while cognitive theorists may prefer to look at abstraction as a reification of interiorization (Piaget, 1970) offering a purely mathematical representation void of any situational elements. Hence, there is some ambiguity in this language of the SMP.…”

Mathematical meaning-making in whole-class conversation

“…Paramount to apprenticing students into mathematizing is developing in them a paradigmatic approach toward reality so that, when facing with a problem that begs to be structured by mathematical means (Freudenthal, 1991), they would be inclined to consider whether or not it shares some commonalities with other, previously solved problems so that, if that were the case, they could borrow the tools used for the already resolved conundrum to tackle the one currently facing (Shreyar, Zolkower, and Pérez, 2009). "I wonder if anything from that you can borrow that we can use to solve this problem" is, thus, paradigmatic of a teacher's verbal gesture aimed at initiating students into this way of thinking, this habit of mind.…”

Equity in Discourse for Mathematics Education

“…Shreyar et al ( 2010 ) offer a similar analysis with additional linguistic features in focus, in order to show how a sixth grade teacher in a Spanish-English bilingual classroom in Argentina supports an understanding that multiple approaches are possible as she engages students in a whole-class discussion of a problem about percentages. As Zolkower and Shreyar point out, teacher commands are not always suited to such verbal mediation of mathematics and, at fi rst glance, "the authoritative manner in which the teacher mediated the whole-class discussion may have seemed unconducive to engaging her students in genuine mathematics exchanges" (p. 200).…”