Research indicates that students at both high school and university level have difficulty, not only in producing proofs, but also in recognizing what a proof is (Chazan, 1993; Moore, 1994). Research has also informed the field of the difficulty in the understanding of the nature of proof among prospective elementary school teachers (Simon, 1996). Understanding the nature of proof, in addition to its theoretical interest, seems essential for thinking about how to teach students about proof, both at the university level and throughout the K-12 level, as is recommended by the latest NCTM standards (NCTM, 2000). While it is widely agreed that students have difficulty with the nature of proof, there is little agreement on what the nature of proof is? In fact, the nature of proof has been a matter of debate among mathematicians, philosophers, and historians for hundreds of years. For the purposes of the study reported here, the act of proving is defined as the process of constructing a logical and deductive argument that establishes or refutes the truth of a mathematical statement based on a set of previously established set of axioms and mathematically true statements. The finished product of this act of proving is what I will refer to as a proof and this proof should adhere to the standards of the community and should stand up to the scrutiny of the same community.

There have been many studies that examine the possible differences that exist between the understanding of proving between experienced provers and novice provers (e.g. Healy & Hoyles, 2000; Mejia-Ramon & Tall, 2005; Raman 2002). It is well-established that students at all levels have various problems regarding the learning, and construction of proof, and there is precedence within the research community for looking towards the practices regarding proof of more experienced users of mathematics (e.g. Alcock & Inglis, 2008; Inglis, Majia-Ramos, & Simpson, 2007; Raman, 2002).

Research Questions

1. What characterizes the construction of proofs (in terms of Toulmin’s model of argumentation (Toulmin, 1958)) produced by doctoral students in mathematics when confronted with a novel mathematical statement in the area of real-analysis?
2. What is the nature of the warrant-types (Inglis, Mejia-Ramos, & Simpson, 2007) used by doctoral students in mathematics that aid in their construction of proofs of novel mathematical statements in the area of real-analysis?

Methods, Data Sources, and Analysis

The study reported on here is a cross-case analysis of the structure of the process of proving of various experienced doctoral students in mathematics. The subjects are all doctoral students that have already passed their candidacy exams. This requirement is to ensure that an external (to the researcher) community of mathematicians has deemed the subjects as experienced provers of mathematics. The subjects were each given a series of statements to prove from real-analysis. The researcher would routinely ask the subjects clarifying questions that would allow him more access to the reasoning behind the proofs being produced. After the first interview, each completed task from each interview is analyzed to produce a Toulmin Argument Model for that task; i.e. the Data, Claims, Warrants, Qualifiers, and Rebuttals (Toulmin, 1958) are identified for each part of the proofs produced. The formation of these models is almost identical to the method used in the analysis by Conner (2007). The major difference lies in omission of the classroom social norm component in Conner’s work (2007) since classroom social norms do not have a role in this study.

The subjects are then called in for a second interview where the information obtained from the models is provided to them. This interview is primarily for the sake of “member-checking” in order to make sure that the information obtained is accurate. The Toulmin models from each case are examined to look for any emerging patterns in the nature of and the nature of the use of the Data, Claims, Warrants, Qualifiers and Rebuttals. Then a cross-case comparison is conducted to identify any patterns across the nature of and the nature of use of the Data, Claims, Warrants, Qualifiers and Rebuttals.

Preliminary Findings and Implications

Since this is an ongoing study, I can only present some preliminary findings from the analysis performed so far. The analysis of the data will be complete prior to the Research Pre-session.

The most prominent emergent pattern from the initial analysis is the use of the different types of warrants present in the proofs generated by the subjects. Inglis, Mejia-Ramos, and Simpson (2007) show that experienced doctoral students in the field of number theory rely on many different warrant-types. The use of these warrant-types, namely the inductive warrant-type, the structural-intuitive warrant-type, and the deductive warrant-type (Inglis, Mejia-Ramos, & Simpson, 2007) are also evidenced in the proving of experienced doctoral students in the field of real-analysis. The subjects for the most part seem aware of the fact that they do have to return to the non-deductive warrant-types and modify them to be deductive in order to complete the deductive proof. However, the use of the non-deductive warrant-types seem to aid in both the furthering of the proof and to reduce (and in some cases eliminate) uncertainty.

One of the main implications from the results of this study is regarding the design of instruction of proof for undergraduates and even secondary students. If these experienced provers consistently rely on the different warrant-types and in warrants in general to further their understanding required to prove a statement, then perhaps more of the instruction should be focused on making the warrants need more transparent. Also, the consistency observed in the use of different warrant-types in real analysis and in number theory (as in Inglis, Mejia-Ramos, & Simpson, 2007) provides more evidence that the warrants are key in the formation of proofs, regardless of the topic area in mathematics, and as such more attention and emphasis should be placed in the teaching of warrant use in the process of proving.

References

Alcock, L., & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69, 111-129.

Chazan, D. (1993). High school geometry students' justifications for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359-387.

Conner, A. M. (2007). Student teachers' conceptions of proof and facilitation of argumentation in secondary mathematics classrooms. Unpublished doctoral dissertation. The Pennsylvania State University.

Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.

Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66, 3-21.

Mejia-Ramos, J. P., & Tall, D. (2005). Personal and public aspects of formal proof: A theory and a single-case study. In D. Hewitt & A. Noyes (Eds.), Proceedings of the sixth British congress of mathematics education (pp. 97-104): University of Warwick.

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 of school mathematics. Reston, VA: NCTM.

Raman, M. (2002). Proof and justification in collegiate calculus. Unpublished doctoral dissertation. The University of California at Berkeley.

Simon, M. A. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15(3-31). 

Toulmin, S. (1958). The uses of argument. UK: Cambridge University Press.