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Melissa Mills Oklahoma State University Abstract This study uses ethnographic methods to investigate the teaching practices of mathematics faculty members when presenting proofs in class. Four case studies of faculty members at a large research institution who are teaching in different mathematics content areas are used to describe the ways in which examples are used in proof presentations in upper-division proof-based undergraduate mathematics courses.

Keywords: proof presentations, examples, teaching practices, ethnographic methods Introduction There have been few studies addressing the teaching practices of university teachers, although there have been calls for such studies (Harel & Sowder, 2007; Harel & Fuller, 2009; Speer, Smith, & Horvath, 2010). In particular, there has been very little research addressing the teaching practices of faculty members in upper-division proof based courses (Weber, 2004). This study will contribute to our knowledge of teaching practices of mathematics faculty members as they teach courses in which students are expected to construct original proofs. Interview data and video data from four different faculty members teaching abstract algebra, analysis, number theory, and geometry will be analyzed to determine the ways these instructors use examples to motivate and support their presentations of proofs in class.

Research Question In what ways are examples used to motivate and support proof presentations in an upper-division proof-based mathematics course? What is the pedagogical motivation of the instructor for the use of particular examples in proof presentations? How does the instructors’ usage of examples contribute to their overall presentation style?

Literature Review At the collegiate level, there are few studies focusing on teaching practice, i.e.

“what teachers do in and out of the classroom on a daily basis” (Speer, et al, 2010). A foundational understanding of teaching practice contributes to our understanding of the phenomenon of teaching and learning. In particular, there is value in focusing in on small, meaningful aspects of practice that mathematicians already use in the classroom (Speer, 2008). Many studies have emphasized the importance of using examples in teaching, particularly when the examples are generated by the students themselves (Watson & Mason, 2005; Watson & Shipman, 2008). Examples serve to make connections between what students already know and the new material that is being presented. When the content involves mathematical proofs, the use of examples may be even more important. Exploration of examples is often part of the process for constructing an original proof (Alcock & Inglis, 2008), and the ability to generate a 512 15TH Annual Conference on Research in Undergraduate Mathematics Education specific example of a proof strategy is an important facet of proof comprehension (MejiaRamos, Weber, Fuller, Samkoff, Search, & Rhodes, 2010).

Attending to proof presentations in class is one of the primary ways in which students construct their understanding of what constitutes a proof (Weber, 2004). There is evidence that instructors spend large portions of their class time (between one third and two thirds) presenting proofs (Mills, 2011). Several recent studies have used faculty interviews to investigate the pedagogical views of faculty members concerning proof presentations in class (Weber, 2010; Yopp, 2011; Alcock, 2009; Harel & Sowder, 2009;

Hemmi, 2011). Some of these studies discussed a relationship between proof presentations and the use of examples. Several instructors mentioned that they often accompany a proof with an example (Weber, 2010). Alcock (2009) identified ‘instantiation of definitions and claims’ as one of the four proof-related skills that instructors are trying to teach. Observations of a particular professor throughout the course of a semester revealed that he modeled the mathematical behavior of ‘example exploration and generalization’ when presenting lectures in class (Fukawa-Connelly, 2010). Fukawa-Connelly, Newton, & Shrey (2011) focused on the use of examples in a proof based course by describing in detail how a faculty member used examples to instantiate the definition of a mathematical group in an abstract algebra class.

The contribution of the present study is that it combines faculty interviews with observation data to investigate what faculty members think about the pedagogy of proof presentations as well as to catalog their actual behaviors in the classroom. This will allow us to investigate their teaching practices, which is more in line with the type of studies that Speer, et. al. (2010) called for.

Methodology Faculty members at a large comprehensive research university who were teaching proof-based upper division mathematics courses during between August 2010 and August 2011 were asked to participate in the study. Three instructors agreed to participate in a one hour interview and agreed to allow their lectures to be video-taped approximately every two weeks throughout the semester. All of the faculty members taught in a lecture style, with the instructor was primarily teaching from the board, and the students were listening, taking notes, and sometimes answering questions and participating in class discussions.

Interviews were transcribed and analyzed using the constant comparative method (Glaser & Strauss, 1967) to determine the pedagogical views of the participant concerning proof (Weber, 2010). The analysis of the video data occurred in several phases. First, I viewed the videos and took notes about what was happening in each time interval. Then all of the instances of proof presentation in the observation data were transcribed. For this study, I have pulled out all of the instances in the data when examples are used to support the proof of a claim in different ways. A careful search of

the literature provided initial categories. These categories are:

1. Start-Up Examples – Motivate basic intuitions and claims (Michner, 1978)

2. Generalization of a Pattern – Examples are used to help the students generalize the statement of a claim from a small number of numerical computations (Bills & Rowland, 1999; Harel, 2001; Inglis, Mejia-Ramos, and Simpson, 2007) 15TH Annual Conference on Research in Undergraduate Mathematics Education 513 3. “Generic” Example – When an example is used to go through the steps of a proof, and then the general method of the proof can be extracted from the example, or viceversa (Rowland, 2001; Weber, 2010) Similar to Michner’s (1978) “Model” Examples.

3a. Pictorial “Generic” Examples – When a diagram or picture is used to organize the proof of a statement, and the proof is based on the diagram (Weber, 2004)

4. Instantiation – 4a. Instantiation of Claims – An example to help students understand the statement of a claim, or the necessity of given conditions in a claim (Michner, 1978; Alcock, 2009) 4b. Instantiation of Definitions – An example of a mathematical object that satisfies a definition (Alcock, 2009) Pictorial examples are often used for this purpose.

4c. Instantiation of Notation – When an example is used to introduce a new notation Each example will be categorized and the usage of examples as well as comments about example usage in the interviews will contribute to the construction of the characteristic style of the instructor. Since the instructors are teaching different content areas, it makes sense to consider these as separate case studies. Similarities and differences among the faculty members will be highlighted, however, the goal of this study is to describe and catalog, not to evaluate the methods used.

Preliminary Results Though they were not explicitly asked about how they use examples in class, three of the faculty members in this study mentioned the use of examples in the interviews. They gave several reasons for using examples, and described different ways in which they use examples. One professor said that he gives simple examples to warm the students up for the statement of the theorem. Another participant said that if it is a proof of a pattern, he would emphasize computation to try to get the students to figure out what the pattern is. In other words, he would use the examples to get the students to conjecture the statement of the theorem. He also said that he uses examples to help the students know how the proof of the theorem should go. The use of pictorial examples as a guide to organize the proof was also mentioned. One participant said that he takes the statement of the theorem and produces examples from the statement, which is similar to what Alcock (2009) calls ‘instantiation of claims.’ Initial analysis showed that examples were used by the instructors in 27% to 67% of the proofs they presented in the observation data (Mills, 2011).

There are many other dimensions that contribute to a professor’s proof presentation style. In my dissertation work, I will investigate other aspects of the instructors’ characteristic style, and follow-up interviews will explore the pedagogical reasoning behind each instructor’s moves when presenting proofs in class.


1. How can these ways to use examples help us understand undergraduate teaching of mathematical proof?

2. How can this be linked to student learning? What’s the next step?

514 15TH Annual Conference on Research in Undergraduate Mathematics Education References Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69, 111-129.

Alcock, L. (2009). Mathematicians’ perspectives on the teaching and learning of proof. In Hitt, F., Holton, D., & Thompson, P. (Eds.), Research in Collegiate Mathematics Education VII (63-91). Providence, RI: American Mathematical Society.

Rowland, T. and Bills, L. (1999). Examples, generalization and proof. In Brown, L. (ed.) Making Meanings in Mathematics: Refereed Proceedings of the British Society for Research into Learning Mathematics. York, QED Publications, 103-116.

Fukawa-Connelly, T. (2010). Modeling mathematical behaviors; Making sense of traditional teachers of advanced mathematics courses pedagogical moves.

Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education. Raleigh, NC.

Fukawa-Connelly, T., Newton, C., & Shrey, M. (2011). Analyzing the teaching of advanced mathematics courses via the enacted example space. Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education.

Portland, OR.

Glaser, B. & Strauss, A. (1967) The Discovery of grounded theory: Strategies for qualitative research. New Brunswick, NJ: Aldine Transaction.

Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zaskis (Eds.). Learning and Teaching Number Theory, Journal of Mathematical Behavior. New Jersey, Ablex Publishing Corporation, 185-212.

Harel, G. & Fuller, E. (2009). Current Contributions toward Comprehensive Perspectives on the Learning and Teaching of Proof. Teaching and Learning Proof Across the Grades: A K‐16 Perspective. Reston, VA: National Council of Teachers of Mathematics.

Harel, G., & Sowder, L (2007). Toward a comprehensive perspective on proof, In F.

Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, Reston, VA: National Council of Teachers of Mathematics.

Harel, G. & Sowder, L. (2009). College instructors views of students vis-a-vis proof. In M. Blanton, D. Stylianou, & E. Knuth (Eds.) Teaching and Learning Proof Across the Grades: A K-16 Perspective. Routledge/Taylor & Francis.

Hemmi, K. (2010). Three styles characterizing mathematicians’ pedagogical perspectives on proof. Educational Studies in Mathematics, 75, 271-291.

Inglis, M. & Mejia-Ramos, J. (2009). Argumentative and proving activities in mathematics education research. Proceedings of the International Commission of Mathematical Instruction. Taipei, Taiwan.

Mejia-Ramos, J., Weber, K., Fuller, E., Samkoff, A., Search, R., & Rhoads, K. (2010).

Modeling the comprehension of proof in undergraduate mathematics.

Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education. Raleigh, NC.

Michener, E. R. (1978). Understanding understanding mathematics. Cognitive Science, 2, 361–383.

15TH Annual Conference on Research in Undergraduate Mathematics Education 515 Mills, M. (2011). Mathematicians’ pedagogical thoughts and practices in proof presentation. Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education. Portland, OR.

Rowland, T. (2001). Generic proofs: setting a good example. Mathematics Teaching 177, 40-43.

Speer, N. (2008). Connecting beliefs and practices: A fine-grained analysis of a college mathematics teacher’s collection of beliefs and their relationship to his instructional practices. Cognition and Instruction, 26, 218-267.

Speer, N., Smith, J., Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. The Journal of Mathematical Behavior, 29, 99-114.

Watson, A. & Mason, J. (2005). Mathematics as a Constructive Activity: Learners Generating Examples. Mahwah: Lawrence Erlbaum Associates.

Watson, A. & Shipman, S. (2008). Using learner generated examples to introduce new concepts. Educational Studies in Mathematics, 69, 97-109.

Weber, K. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course.

Journal of Mathematical Behavior, 23, 115-133.

Weber, K. (2010). The pedagogical practices of mathematicians with respect to proof.

Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education. Raleigh, NC.

Yopp, D. (2011). How some research mathematicians and statisticians use proof in undergraduate mathematics. Journal of Mathematical Behavior, 30(2), 115-130.

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