Argumentation in a Geometry Class: Aligned with the Teacher’s
Conception of Proof1
AnnaMarie Conner
The University of Georgia, USA
aconner@uga.edu
Abstract: While a straightforward relationship between argumentation and proof is not universally accepted, it is
reasonable that argumentation, as a tool for student learning, would be instrumental in the teaching of proof in high
school geometry classes. This study examines the argumentation in one student teacher’s high school geometry
classes and suggests a possible relationship between the observed argumentation and the student teacher’s
conception of proof. Aspects of the student teacher’s conception of proof could be seen in how she supported
argumentation in her classes.
Theme: Research, focused on explanation, argumentation, and proof in geometry education
The importance of the teaching and learning of proof and justification is undisputed
within the mathematics education community, and the relevance of argumentation to student
learning of mathematics has been demonstrated at various levels of school mathematics
(Krummheuer, 2000; Yackel, Cobb, & Wood, 1999; Zack & Graves, 2001). Recent research has
investigated the relationship between argumentation and proof, demonstrating the efficacy of
Toulmin’s (1958/2003) scheme for the analysis of this relationship (Pedemonte, 2007) and the
importance of attending to qualifiers and rebuttals in addition to claims, data, warrants, and
backings in more advanced analyses (Inglis, Mejia-Ramos, & Simpson, 2007). This paper
provides a description of argumentation within a geometry classroom and presents evidence of a
possible relationship between a student teacher’s conceptions of proof and her facilitation of
collective classroom argumentation in a high school geometry class.
Framework
The teaching and learning of proof is of major concern to secondary and tertiary
mathematics education, both because of the centrality of proof to the practice of mathematicians
and because of the status given to proof and justification in curriculum standards (e.g., National
Council of Teachers of Mathematics, 2000). Research on the teaching and learning of proof has
examined students’ proof schemes (Harel & Sowder, 1998; Housman & Porter, 2003), students’
difficulties with proof (Bell, 1976; Chazan, 1993; Moore, 1994; Weber, 2001), students’ ability
to validate texts as proofs (Selden & Selden, 2003; Stylianides, Stylianides, & Philippou, 2004),
the relevance of technology to the learning of proof in geometry (Laborde, 2000; Leung &
Lopez-Real, 2002; Mariotti, 2000), and students’ perceptions of proof (Almeida, 1995; Recio &
Godino, 2001). Building on these researchers’ work in the various aspects of proof and proving, I
define a person’s conception of proof as the person’s ability to prove and analyze arguments,
perception of the role and need for proof in mathematics, and affective perception of proof.
1
This paper is based on the author’s doctoral dissertation, completed at The Pennsylvania State
University under the direction of Rose Mary Zbiek, supported in part by the National Science
Foundation under Grant No. ESI0083429 to the University of Maryland with a major subcontract
to The Pennsylvania State University and by a research initiation grant from The Pennsylvania
State University College of Education Alumni Society. Any opinions, findings, and conclusions
or recommendations expressed in this document are those of the author and do not necessarily
reflect the views of the National Science Foundation or the College of Education Alumni Society.
Toulmin’s (1958/2003) model of argumentation involves claims, data, warrants,
backings, qualifiers, and rebuttals, related as shown in Figure 1. According to Toulmin, a claim
is the statement whose truth is being established, data is evidence presented in support of the
claim, a warrant is a bridge between the data and claim, giving reasons that the particular data
presented is relevant to the claim, and backing, which is usually implicit, is support for the
warrant’s validity in the particular field in which it is used. Qualifiers and rebuttals are less often
used in analysis of argumentation in mathematics education, but were shown to be useful in the
analysis of arguments presented by more advanced students of mathematics (Inglis, MejiaRamos, & Simpson, 2007). A qualifier is indicative of the strength of the warrant (usually a word
such as “probably”), and a rebuttal is a description of circumstances under which the warrant
would not be valid (Toulmin, 1958/2003).
Figure 1: Diagram of Toulmin’s Model, adapted from Toulmin (1958/2003, p. 104)
Krummheuer (1995) adapted Toulmin’s (1958/2003) model of argumentation to
mathematics education research by defining as acceptable warrants and backings those that are
accepted by the particular mathematics community that is engaging in the argumentation rather
than what is accepted by the field of mathematics in general. Like Krummheuer, I define
collective argumentation as “a social phenomenon, when cooperating individuals tried to adjust
their intentions and interpretations by verbally presenting the rationale of their actions”
(Krummheuer, 1995, p. 229). Collective argumentation differs from traditional Aristotelian
argumentation because it is concerned with a group reaching consensus rather than an individual
convincing a group2.
I define a proof as logically correct deductive argument built up from given conditions,
definitions, and theorems within an axiom system. This definition of proof suggests that a proof
is a particular kind of argument, and thus presupposes a relationship between argumentation and
proof. The nature of this relationship is not, however, straightforward. Balacheff (1999) suggests
that argumentation may present an epistemological obstacle to a person’s learning to prove,
while others (e.g., Pedemonte, 2007) propose that argumentation may lead to proof, although the
structure of the proof may be different from the structure of the related argumentation. The
relationship that I propose is linked to the collective nature of argumentation in classrooms and
to beliefs about the role of the teacher in facilitating learning in classrooms. In brief, part of the
teacher’s role in the classroom is to act as a representative of the mathematics community in
negotiating sociomathematical norms, including normative standards of argumentation (Yackel
& Cobb, 1996). In that role, the teacher would ideally facilitate argumentation within the
classroom that is consistent with acceptable mathematical justification that would eventually lead
2
Within this paper, the word argumentation will be used to imply collective argumentation, since
the context is a class of cooperating individuals (high school geometry students and their
teacher).
to proof (Conner, 2007). Thus, the teacher’s conception of proof has the potential to be very
important in the kinds of argumentation that he or she might facilitate.
Methods
3
Lynn was a mathematics major who was also seeking certification in teaching 6-12
mathematics at a large university on the east coast of the United States. She completed her
culminating student teaching experience in the final semester of her university experience.
Lynn’s mentor teacher, Mr. B, allowed Lynn to function independently during her student
teaching experience. She was responsible for all instruction, discipline, and assessment during
the time she was observed. Mr. B was available to assist if she had questions, but he did not
attempt to determine or constrain her teaching methods or techniques, and he allowed her to set
learning goals for students within the constraints of the curriculum.
Lynn’s conception of proof was inferred from her responses in two semi-structured
interviews during her culminating student teaching experience. Her conception of proof included
an emphasis on reasons as personally important (she claimed, “I’m a reason person” (Lynn,
Interview 2, line 153)). She described the purpose of proof in mathematics as to provide reasons
“why it works” (Lynn, Interview 2, line 154). Similar to some of the secondary teachers who
participated in Knuth’s (2002) study, Lynn focused on proof as a way to know why things
worked (she gave the example of knowing where the Pythagorean Theorem came from) rather
than, as Knuth phrased it, “an argument’s illumination of the underlying mathematical concepts
which determine why a statement is true” (Knuth, 2002, p. 80). Her conception of proof was also
context-dependent, with both her affective reaction to proof and her ability to prove and analyze
proofs differing based on whether she was dealing with geometry, topology, or number theory.
This is consistent with Housman and Porter’s (2003) findings that an individual may exhibit
multiple proof schemes.
A method of analysis based on an adaptation of Krummheuer’s (1995) interpretation of
Toulmin’s (1958/2003) model of argumentation to collective argumentation in mathematics
classrooms was used to examine Lynn’s facilitation of the argumentation in her classes. Classes
were observed and audio-recorded. These recordings were transcribed and augmented with
figures and additional information from field notes. Episodes of argumentation were identified,
many consisting of a main argument with several supporting sub-arguments. Within arguments,
claims, data, and warrants were identified and attributed to Lynn, her students, or both (meaning
both Lynn and her students contributed to the part of the argument). Each argument and episode
was diagrammed using a scheme adapted from Toulmin’s diagram in Figure 1, and the diagrams
were examined for patterns within contributions. The diagrams included notations for who
contributed each component of each argument, which components were left implicit, and how
arguments were connected within an episode of argumentation (see Figures 3 and 4 for
examples).
Results
While much of the discourse within Lynn’s classes conformed to an IRE4 model of
instruction, complex models of argumentation were constructed from this discourse. The
discourse in the transcript excerpt below is typical of discourse in Lynn’s classes. In this
example, the Lynn and her students are exploring the relationship between two angles, angle
AFB and angle BFD, in quadrilateral ABDC (see Figure 2). Within this single episode of
3
All names are pseudonyms.
Initiation (by teacher) – Response (by student) – Evaluation (by teacher), see Forman & Ansell
(2002).
4
argumentation there are multiple layers of argumentation, including several sub-arguments
leading up to a main argument.
T: I’d like you to look at measure of angle AFB and measure of angle BFD, so AFB, this angle right here, I
want you to compare that to measure of angle BFD. Now, looking at this figure, do we have two triangles
that we could say have two congruent sides?
Ss nod [indicating agreement].
T: Yes we do?
S1: Yeah, we do because um if F bisects AD, then AF and FD are congruent.
T: Okay, AF and FD are congruent and they’re already marked congruent because I said F was our midpoint.
S1: And then BF is reflexive to itself so it’s the other congruent side.
T: Right, so these two triangles here we can compare. We can compare AF to FD, those are congruent, and then
FD is congruent to itself because of the reflexive property, very good. Now, do we have an angle measure?
T: No, so which inequality do you think we are going to use?
S2: SSS
T: Right, side-side-side because what do we know about these sides? What do we know? What is AB?
S3: AB is greater than BD
T: You said AB is greater than what?
S3: BD
T: BD, what is AB?
S4: 5
T: And BD is?
S5: 3
T: Okay, so if AB is bigger than BD, S6 what do you think we can say about angle AFB and BFD? Which
one’s larger?
S6: AFB
T: AFB is larger. So AFB is greater than BFD.
Figure 2: Quadrilateral ABDC
The episode above, which is diagrammed in Figure 3, consists of a main argument and
several sub-arguments that lead to and support the data for the main claim. Lynn and her students
provide the data and warrant for their main claim before actually stating that claim. The main
claim is that the measure of angle AFB is greater than the measure of angle BFD. Early in the
episode, Lynn states that they are looking for a relationship between the two angles, but does not
indicate what the relationship might be. Together, Lynn and her students establish what is known
about other measures and relationships in the figure in order to be able to make the claim that the
measure of angle AFB is greater than the measure of angle BFD.
Lynn’s main mechanism of support for argumentation was asking questions. Lynn asked
questions that prompted her students to make claims and provide data and warrants for their
claims. As can be seen in the above excerpt, her claim-eliciting questions often supported
students’ provision of claims by suggesting particular geometric objects to which a claim might
apply. She asked for reasons to support students’ claims. These questions sometimes were
interpreted as requests for data; in other arguments students provided warrants in response to
similar questions. Norms regarding argumentation that were negotiated in her classes included
students’ use of theorems and definitions previously discussed in class as warrants in episodes of
argumentation.
Figure 3: Diagram of Argument5 in Geometry Context
The IRE nature of the discourse is observable, as Lynn asks a question, a student or
students answer it, and Lynn reiterates, rephrases, or validates the student’s answer and then asks
another question. This brief episode illustrates that, although the discourse proceeds in what may
appear to be a linear pattern, the argument being constructed may be complex. This particular
argument begins with a claim, prompted by Lynn, that is actually both the first sub-claim of the
episode of argumentation and part of the data for the main claim. A student provides the data for
this claim, treating each part of the data as a separate claim and providing data for one subargument (Lynn provides the warrant) and both data and warrant for the other sub-argument.
This episode is different from episodes of argumentation in which Lynn and her students
discuss algebraic ideas. When discussing algebraic ideas, Lynn and her students generally did
not provide warrants within the argumentation, leaving them implicit rather than providing them
explicitly. This is illustrated by the argumentation diagrammed in Figure 4, where in this and
other circumstances, Lynn did not ask for reasons for the various operations suggested by
students for the solutions of the equations. She asked how to solve the equations, but not why
students’ suggestions were valid, thus leaving the warrant unstated. The sub-argument in Figure
4 occurred during a larger episode of argumentation in which Lynn and her students were finding
missing values of x and y as they were examining relationships within a parallelogram.
5
Components of the argument attributed to students are represented in blue boxes with dashed
edges; components attributed to Lynn are represented in red boxes with dot-dashed edges;
components attributed to both are represented in violet boxes with double-lined edges, and
components that were not specified are represented in black boxes with solid edges.
Figure 4: Sub-argument in Algebraic Context
The observed differences in Lynn’s support of argumentation in algebraic versus
geometric contexts aligns well with the context-dependence of her conception of proof. Lynn
was much more comfortable proving and talking about proof in the context of geometry than in
any other mathematical context. For instance, she described her experience with proof in a
topology class as “unsuccessful” and having “too many definitions and theorems” (Lynn,
Interview 1, lines 195-205), and she was not able to correctly validate a proof in a number theory
context. However, she was confident in her ability to prove theorems for and with her students in
a geometry context, and she was able to competently evaluate whether several geometric
arguments were indeed proofs of a geometric theorem during her second interview. The contextdependence of her conception of proof was evident in her support for argumentation in algebraic
versus geometric contexts. She asked for or provided warrants for most arguments when the
context of the argumentation was geometric, but seldom requested or provided warrants when
the context was algebraic.
Lynn’s conception of proof included aspects of a personal insistence on reasons, having
reasons for why things worked the way they did. In her support for argumentation, Lynn asked
her students to provide reasons for many of the claims made in her classes, and she provided
those reasons for other claims. Lynn was consistent in asking for reasons and justifications both
verbally in class and on the tests that she developed. Lynn’s desire for reasons for her own
understanding is apparent in her teaching as a desire for students to give or hear reasons
(warrants) within their own mathematical experiences.
Conclusion and Future Directions
Using Toulmin’s (1958/2003) model of argumentation allows us to foreground some of
the complexities inherent in the discourse in this classroom. In the case of this student teacher,
her conception of proof seems to parallel aspects of her support for argumentation. Using
Toulmin’s model to diagram the argumentation allowed for the details of this support to become
evident. In addition, it enables us to analyze how different constituencies within the classroom
contribute to the argumentation that occurs.
Much more remains to be done in the analysis of argumentation in secondary geometry
classes. In the particular case reported here, I have just begun to examine differences in
arguments when the class was constructing a two-column proof versus when they were exploring
a geometric situation or solving a problem. Preliminary analysis indicates a difference in the
order in which components of arguments are presented. Further analysis is necessary to begin to
describe how proof and argumentation may be related in this and other classrooms. In addition, if
a teacher’s conception of proof influences the support for argumentation in his or her classroom,
future research must begin to describe what conceptions of proof cultivate productive
argumentation and how these conceptions can be fostered in teacher preparation.
References
Almeida, D. (1995). Mathematics undergraduates' perceptions of proof. Teaching Mathematics and its Applications,
14, 171-177.
Balacheff, N. (1999). Is argumentation an obstacle? Invitation to a debate... [Electronic Version]. Preuve:
International Newsletter on the Teaching and Learning of Mathematical Proof, May-June 1999. Retrieved
November 1, 2007, from http://wwwleibniz.imag.fr/DIDACTIQUE/preuve/Newsletter/990506Theme/990506ThemeUK.html
Bell, A. W. (1976). A study of pupils' proof-explanation in mathematical situations. Educational Studies in
Mathematics, 7, 23-40.
Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and
mathematical proof. Educational Studies in Mathematics, 24, 359-387.
Conner, A. (2007). Student teachers' conceptions of proof and facilitation of argumentation in secondary
mathematics classrooms. Unpublished doctoral dissertation, The Pennsylvania State University, University
Park.
Forman, E. A., & Ansell, E. (2002). Orchestrating the multiple voices and inscriptions of a mathematics classroom.
The Journal of the Learning Sciences, 11, 251-274.
Harel, G., & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. In CBMS Issues in
Mathematics Education (Vol. 7, pp. 234-283): American Mathematical Society.
Housman, D., & Porter, M. (2003). Proof schemes and learning strategies of above-average mathematics students.
Educational Studies in Mathematics, 53, 139-158.
Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of
qualification. Educational Studies in Mathematics, 66, 3-21.
Knuth, E. J. (2002). Teachers' conceptions of proof in the context of secondary school mathematics. Journal of
Mathematics Teacher Education, 5, 61-88.
Krummheuer, G. (2000). Mathematics learning in narrative classroom cultures: Studies of argumentation in primary
mathematics education. For the Learning of Mathematics, 20, 22-32.
Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity
of proving. Educational Studies in Mathematics, 44, 151-161.
Leung, A., & Lopez-Real, F. (2002). Theorem justification and acquisition in dynamic geometry: A case of proof by
contradiction. International Journal of Computers for Mathematical Learning, 7, 145-165.
Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational
Studies in Mathematics, 44, 25-53.
Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249-266.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston,
VA: NCTM.
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies
in Mathematics, 66, 23-41.
Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational
Studies in Mathematics, 48, 83-99.
Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an
argument proves a theorem. Journal for Research in Mathematics Education, 34, 4-36.
Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2004). Undergraduate students' understanding of the
contraposition equivalence rule in symbolic and verbal contexts. Educational Studies in Mathematics, 66,
133-162.
Toulmin, S. E. (2003). The uses of argument (Updated ed.). New York: Cambridge University Press. (First
published in 1958).
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies
in Mathematics, 48, 101-119.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal
for Research in Mathematics Education, 27, 458-477.
Yackel, E., Cobb, P., & Wood, T. (1999). The interactive constitution of mathematical meaning in one second grade
classroom: An illustrative example. Journal of Mathematical Behavior, 17, 469-488.
Zack, V., & Graves, B. (2001). Making mathematical meaning through dialogue: "Once you think of it, the Z minus
three seems pretty weird". Educational Studies in Mathematics, 46, 229-271.