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.
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