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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.

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Notes

  1. Adler, 1999; Bartolini Bussi, 1998; Cobb, Wood, & Yackel, 1993; Forman, Larreamendy-Joerns, Stein, & Brown, 1998; Herbst, 2002; Inagaki, Hatano, & Moritas, 1998; Lampert & Blunk, 1998; Leikin & Dinur, 2007; Lobato, Clarke, & Burns Ellis, 2005; Lubienski, 2000; Moschkovich, 1999; Nathan & Knuth, 2003; O’Connor, 2001; Sherin, 2002; Staples, 2007; Steinbring, 2007; Zack & Graves, 2001; Zolkower & Shreyar, 2007.

  2. Within SFL, the relationship between semantics and grammar is conceived as one of realization: wording “realizes” or encodes, meaning and, in turn, wording is “realized by” sound or writing (Halliday, 1994, p. xx).

  3. An instance of language use is said to be a text when it has texture, that is, when it is “coherent with respect to the context of situation, and therefore consistent in register” and “coherent with respect to itself, and therefore cohesive” (Halliday & Hasan, 1976, p. 23).

  4. SFL has proven a well-suited toolbox for studying spoken, written, and/or diagrammed texts in science and in mathematics classrooms (Atweh, Bleicher, & Cooper, 1998; Lemke, 1990; Morgan, 1998, 2006; O’Halloran, 2003; Veel, 1997, 1999; Wells, 1999, Zolkower & Shreyar, 2007).

  5. The above “speech roles are available even when the speaker is talking to him or herself, assuming roles of both speaker and addressee, in an interior dialogue” (Gerot & Wignell, 1994, p. 23).

  6. ‘Representation’ conveys the sense that language communicates experiences that exist prior to their linguistic communication. This is in tension with Halliday’s view of language as a system for construing rather than communicating meanings.

  7. Most likely, the students’ medium to high socioeconomic status inclined them to view the difference between the two discounted prices as a rather insignificant one. This fact further pushed to the background the specifics of the shopping scenario, making it easier to frame the conversation as a reflection on the meaning of the ideas/tools proposed for calculating the sought percents, thereby treating the problematic reality paradigmatically.

  8. This is a typical error in dealing with percents whereby the % sign is ignored and the percent discount is treated as a number (cf. Parker & Leinhardt, 1995).

  9. Elliptic constructions are characteristic of the predicated grammar of inner speech (Vygotsky, 1986).

  10. Bromme and Steinbring (1994) discuss how a teacher’s manner of addressing the collective student during a whole-group discussion allowed for a “consistent presentation/development of the subject matter”.

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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.

Author information

Authors and Affiliations

  1. Brooklyn College, City University of New York, New York, USA

    Betina Zolkower

  2. Teachers College, Columbia University, New York, USA

    Sam Shreyar

  3. Grupo Patagónico de Didáctica de la Matemática, San Carlos de Bariloche, Río Negro, Argentina

    Silvia Pérez

Authors
  1. Sam Shreyar
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  2. Betina Zolkower
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  3. Silvia Pérez
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Corresponding author

Correspondence to Betina Zolkower.

Additional information

An earlier version of this paper was presented at the 2008 Meeting of the American Education Research Association (New York City).

Appendix

Appendix

Transcription conventions

Layout:

Turns are numbered consecutively

Clauses:

Within turns are also numbered

Speakers:

Speakers are indicated by pseudonyms. Students indicates more than one speaker but not the whole class (which is indicated as Chorus)

Italics:

Nonverbal action (e.g., bodily gesture, facial expression)

Underlining:

Overlapping words spoken by more than one speaker at a time

Caps:

Words spoken with emphasis

?:

Rising intonation in clauses judged to have an interrogative intent

,:

Breathing space

…:

Short pause within a clause or a turn

… …:

Long pause within a clause or a turn

-:

Interrupted clause or turn

Codes for analyzing the interpersonal grammar

Mood or grammatical form

Speech function

Declarative: Dec

Statement: S

Interrogative:

Question: Q

Wh: Int(wh)

Polar or yes/no: Int(p)

Imperative: Imp

Command: C

Offer: O

Check: Ch

Minor (mood-less) clauses: (Min)

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Shreyar, S., Zolkower, B. & Pérez, S. Thinking aloud together: A teacher’s semiotic mediation of a whole-class conversation about percents. Educ Stud Math 73, 21–53 (2010). https://doi.org/10.1007/s10649-009-9203-3

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