Two Student Teachers' Varying Support for Collective Argumentation

Collective argumentation in a mathematics class involves multiple people working together to make claims and support those mathematical claims. Krummheuer (1995, 2007), Yackel (1995), and others demonstrated the usefulness of collective argumentation for student learning of important mathematical ideas and practices. Engaging students in collective argumentation allows their reasoning to be highlighted, as illustrated by Forman, Larreamendy-Joerns, Stein, and Brown (1998). Yackel highlighted the importance of the teacher in this endeavor, calling for more research examining the teacher's role. Krummheuer (1995) introduced Toulmin's (1958/2003) model of argument structure to the mathematics education community. We modified this structure to reflect the teacher's support for argumentation (as shown in Figure 1). Toulmin posited that an argument involves claims, data, warrants, backings, qualifiers, and rebuttals, some of which remain implicit unless specifically requested.

Figure 1: Typical episode of argumentation in Bridgett's[1] class with accompanying transcript

An episode of argumentation involves the argument itself, any sub-arguments related to it, and the support the teacher provides for the argument. Support includes questions asked that prompt parts of the argument, reactions to parts of the argument (often affirmations), and other supportive actions, such as writing pertinent information on the board. Our analysis of teachers' support for argumentation additionally draws upon research on facilitating mathematical discussions (e.g., Hufferd-Ackles et al., 2004; Stein, Engle, Smith, & Hughes, 2008; Williams & Baxter, 1996). We undertook this research to answer the question: How do student teachers support collective argumentation in secondary mathematics classes?

Methods

As part of a larger project, we observed and video recorded two student teachers, each teaching one geometry unit: Bridgett, 7 class days, in an accelerated, integrated ninth grade class; Kylee, 9 class days, in an eleventh grade geometry class. Kylee and Bridgett participated together in a sequence of mathematics education courses in an undergraduate university program that emphasized implementing good tasks and facilitating mathematical discourse.

We transcribed the video recording of each class. The transcripts and recordings were used as primary data sources, supplemented by field notes and written class materials. We identified and diagrammed episodes of argumentation from the transcripts. We used a modification of Toulmin's (1958/2003) diagrams to create and vet 277 diagrams (100 from Bridgett and 177 from Kylee). Each diagram included the parts of an argument and the teacher's support for argument components (see Figure 1). We categorized the teacher's support first into questions, direct contributions, and other support, and then inductively classified each of these supportive moves into more specific categories. This presentation will focus on the patterns observed in the questions and direct contributions.

Results

Bridgett and Kylee structured their classes differently: Bridgett often asked her students to work in small groups, and Kylee's class consisted of primarily whole class discussions. We focused our analysis on the whole class discussions because we are interested in the role of the teacher in collective argumentation. Because of the different classroom structures, we have more arguments from Kylee's class (173) than Bridgett's class (51). We began by examining how often Bridgett and Kylee directly contributed argument components and which components each contributed. We found differences in the relative number of components contributed by each: Kylee contributed almost half of the argument components; Bridgett contributed approximately 1/8 of the components. Both student teachers often contributed the initial data for arguments. That is, they contributed the problem statement or initial information needed to begin a problem. Bridgett rarely contributed warrants or claims that were also used as data, whereas Kylee provided more warrants than other parts of arguments. When the number of warrants contributed by students without the teacher's support is examined, Bridgett's students contributed 63 warrants in 51 arguments, more than one per argument, illustrating the normative practice of making reasoning explicit in her class. In Kylee's class, students contributed only 60 unprompted warrants in 173 arguments; Kylee usually prompted her students to provide reasoning when they did so. The students' contributions were qualitatively different in the two classes; we believe the kinds of questions the teachers asked give insight into these differences.

Asking questions was a normative activity in both classes. Elaboration questions, questions asking for an explanation, interpretation, or justification, made up the largest category of Bridgett's questions; but this percentage was closely followed by questions requesting an immediate answer, action, or idea. In general, Bridgett's questions were relatively evenly divided between categories. In contrast, in Kylee's class, the largest percentage of questions requested an immediate answer, such as identifying a segment or angle on the board, recalling a known fact, giving a term, or providing an already calculated result. These were questions that could be answered quickly without elaboration. Kylee also asked questions requiring an action, elaboration, or idea, but her questions were clustered more around immediate answers; even the ones requesting an action were often requests for a simple calculation such as five times nine.

Implications

            The different kinds of questions asked in the classes and the amount of direct contributions by each student teacher illustrate differences in the discourse and the norms related to justification in each class. Bridgett provided some analytic scaffolding (as used by Williams & Baxter, 1996) in her class, but she also expected students to provide scaffolding for each other, and there were clear norms for explicating reasoning in her class, as evidenced by the proportion of warrants students provided in her class. Kylee also provided analytic scaffolding in her class, but she, by the action of providing warrants in her class, implicitly suggested that while explicating reasoning is important, she would provide or prompt most of that reasoning. Justification was important in both classes, as evidenced by the number of warrants provided in each class. However, the teachers' moves suggest differences in their views of who should provide the justification within a mathematical argument.

Presentation Structure

We will provide an overview of the major results of the study in the first 15-minute presentation and discuss examples, categories, and coding during the roundtable discussions.

References

Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). "You're going to want to find out which and prove it": Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527-548.

Hufferd-Ackles, K., Fuson, K., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81-116.

Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures (pp. 229-269). Hillsdale, NJ: Lawrence Erlbaum Associates.

Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroomTwo episodes and related theoretical abductions. The Journal of Mathematical Behavior, 26(1), 60-82. doi: 10.1016/j.jmathb.2007.02.001

Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10, 313-340.

Toulmin, S. E. (2003). The uses of argument (updated ed.). New York: Cambridge University Press. (First published in 1958.)

Williams, S. R., & Baxter, J. A. (1996). Dilemmas of discourse-oriented teaching in one middle school mathematics classroom. The Elementary School Journal, 97(1), 21-38.

Yackel, E. (2002). What we can learn from analyzing the teacher's role in collective argumentation. Journal of Mathematical Behavior, 21, 423-440.