An Oral Proof in a Geometry Class: How Linguistic Tools Can Help Map the Content of a Proof

Abstract

Recommendations that teachers promote argument and discourse in their mathematics classrooms anticipate researchers' needs for methods for examining and analyzing such talk. One form of discourse is oral arguments, including proofs. We ask: How can we track the development of an oral argument by a teacher and her/his students? We illustrate a method that combines Systemic Functional Linguistics (SFL) and Toulmin's argumentation scheme to examine how speakers logically connect different parts of an argument. We suggest that conjunction analysis can aid a researcher to map the content of a proof that has been constructed in class discussion. Using data from a discussion of a geometry proof, we show that different types of conjunctions enabled the teacher and the students to connect various components of an argument and, also, different arguments. The article illustrates how conjunction analysis can support and deepen what Toulmin's scheme for arguments can reveal about oral discussions.

ACKNOWLEDGMENTS

Research reported in this article was carried out with the support of NSF CAREER grant REC 0133619 to the second author and while the first author was a doctoral student, under the direction of the second author, at the University of Michigan. While contributing to that project, the first author was supported by a Rackham Merit Fellowship from the University of Michigan. Opinions expressed by the authors do not necessarily represent the views of the National Science Foundation or the University of Michigan. A prior version of this article was presented at the 2007 Annual Meeting of the American Educational Research Association in Chicago. We thank Mary J. Schleppegrell for feedback on the linguistic analysis. We appreciate the careful comments by the executive editor, John P. Smith III, and the anonymous reviewers to the content of this article.

Notes

We recognize that there is a difference between Martin and Rose's (2007) discursive definition of conjunction and the narrower, lexicogrammatic definition by Halliday and Hasan (1976). What we are calling conjunctions, after Martin and Rose, includes not only what Halliday and Hasan called conjunction but also what they called continuatives.

According to Eggins and Slade (1997), “The Subject is the pivotal participant in the clause, the person or thing that the proposition is concerned with and without whose presence there could be no argument or negotiation” (p. 75). Also, “The Finite expresses the process part of the clause that makes it possible to argue about the Subject participant” (Eggins & Slade, 1997, p. 77). The Finite corresponds to the verbal group in traditional grammar, whereas a nominal group represents the Subject (Halliday & Matthiessen, 1999, p. 9).

There is a third metafunction of language, the interpersonal metafunction, that has some relevance regarding conjunctions, for example, when considering expectancy and counterexpectancy. However, we did not explore the use of conjunctions to realize the interpersonal metafunction in this analysis because our focus on the construction of meanings and the logical organization of the text. Morgan (1996) discusses how a focus on understanding how the three metafunctions of language are realized in mathematical texts can provide a new way to understand fundamental aspects of the language of mathematics.

As Livingston (1999) argued, the artifacts that humans create to represent proofs (e.g., in writing or other verbal accomplishments) can be considered “descriptions of proof” rather than proofs themselves. According to Livingston, “a proof—as an ongoing activity—is the pairing of, on the one hand, a description of that proof and, on the other, the reasoning of the proof and the organization of the practices of proving that the proof-description describes” (p. 877).

Thanks to an anonymous reviewer for providing this viewpoint and directing us to Leron's (1983) work.

In a situation of calculation, the norm is to set up and solve operations between quantities or algebraic expressions using properties of the figure.

In the parsed transcript, we removed notes regarding overlapping speech, interruptions, pauses, and interpersonal gestures to simplify the reading of the parsed transcript. We also removed fillers and interjections.

TABLE 2 Parsed Transcription of the Rectangle Episode

^a Alana started turn 14 by naming a possible theorem of triangle congruency, when she said “By side.” However, Alana elided the reason that could justify why triangles MCN and PCO would be congruent. We take “by side” as a clause with elided text, in response to the teacher's question. Alana's use of the conjunction by suggests that the expected answer to the teacher's question in turn 13 is a reason by which the two triangles are congruent.

^b In formal mathematics, there is a difference between congruence and equality. However, teachers and students sometimes use the term equal for parts that are congruent. In this episode, we consider that when speakers talk about equal parts of triangles or equal triangles they mean that these are congruent

^c We take Alana's use of a and side as a mistake in speech, when she could have said, “And then MN and PO are sides.”

^d We take Alana's use of two processes is and are as a self-correction. The clause can read as “that {MN and PO} are equal.”

^e Anil repeated because in two turns, 18 and 20. We do not consider because in 18.2 as a clause because it is a false start. We consider 20.1 as a clause because he included more information that he elided in 18.2, when interrupted by Ms. K. This information included a statement with an implicit process: in a rectangle, the opposite sides are equal. In clause 18.1, we consider that Anil referred to the diagram of quadrilateral MNOP. However, in clause 20.1, we take that Anil referred to a general property of rectangles.

^f In 26.1, we cannot reconstruct the elided text. Therefore, 2.6 is a pseudo clause, even though Anil's use of the conjunction because suggests that Anil intended to give a reason to the question posed by the teacher in turn 25.

^g We consider Ms. K's question in 34.11, “What else do I get?,” as a probe for Anil to identify other congruent parts.

^h Because there is not a process, we do not consider turn 41 as a clause. However, Anil gave a reason for Ms. K's question about the diagonals (40.7) in 43.1. We consider turns 41 and 43 as an interrupted clause, with elided text about the diagonals MO and NP. Later in our analysis of conjunctions of this episode, we only count the conjunction because in turn 41 once because it was expanded in turn 43. We do not count the repetition of because by Ms. K in clause 44.1, which we added to make the elided text part of the clause.

There are two subtypes of conjunctions to denote cause (Martin & Rose, 2007, pp. 128–129, 132). One subtype includes expectant cause, which signals a cause-and-effect relation that usually happens. Some examples of expectant cause conjunctions are because, so, and therefore. The other subtype includes concessive cause, which denotes a result that counters usual expectations. Some examples of concessive cause conjunctions are although, even though, but, and however. We group these two subtypes of cause conjunctions here because we are focusing on consequence cause conjunctions in general.

TABLE 4 External Conjunctions in the Rectangle Episode

There were two mentions of but in the episode. In one clause, we coded but as denoting consequence (clause 9.4) and in another clause as denoting comparison (clause 14.3).

The classification of a conjunction as a subtype of consequence conjunctions that denotes purpose indicates that the speaker made a linguistic choice to indicate that a clause (or a pseudo clause) expresses the purpose of action in another clause (or pseudo clause).

^aThe reason that would usually be attributed to this statement in a geometry class would be the reflexive property. However, here we are using the evidence in the text of the rectangle episode to record the reasons that they stated in the oral discussion.

For the sake of clarity, we only included statements and reasons that the teacher approved. For example, even though one can conceive that Alana's focus on a set of triangles that they could not prove congruent (turns 10–14) prompted the teacher's request to focus on another set of triangles, we did not include Alana's statement in the model because the teacher did not use these triangles to prove that the diagonals of a rectangle are congruent.

FIGURE 4 Representation of the argument made in the rectangle episode using a simplified version of Toulmin's (1958) argument layout.

According to Martin and Rose (2007, p. 139), speakers use so to show a conclusion in spoken language. In clause 34.399, we take so as an internal conjunction with the purpose of showing the conclusions of an argument. Here there is a connection to something that had been said before, beyond connecting immediate steps.

The conjunction then could also be coded as an external conjunction to denote time when there is an immediate succession of events. Here, I take then as a marker of condition. In mathematical discourse “if…then” statements are usual (O'Halloran, 2005, p. 124) to show the conditions in a mathematical statement.