# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

**Wittmann, M.C., Steinberg, R.N., and Redish, E.F. (2005). Activity-Based Tutorials Vol 2:**

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Vygotsky, L.S. (1962). Thought and language. Cambridge, MA: MIT Press.

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Keywords: Graduate teaching assistant education, calculus, theories of learning and teaching.

Abstract: The purpose of this research study was to explore mathematics graduate teaching assistants’ (GTA) beliefs about the nature of mathematics, their pedagogical approaches toward teaching mathematics and how these evolve over a span of a year. The GTAs participated in four open-ended interviews designed around the planning, performing and assessing framework of Speer and Kung (2009). Our preliminary analyses revealed hierarchical stages of GTA knowledge of their students as well as a separation between their ontological and pedagogical stances.

Introduction: It cannot be denied that communication of a subject is in part a reflection of the individual’s view of that subject. Many have documented pre-service K-12 teachers’ beliefs and have gone on to study how these beliefs influenced classroom practice. (For example: Cooney, T., Shealy, B. and Arvold, B., 1998; Day, R., 1996; Thompson, A., 1984; Thompson, A., 1992;

Vacc, N. and Bright, G., 1999.) The harmony between the two is crucial. Austin (2002) discusses a number of issues related to professional development for future faculty members in general and further points out the importance of biography in understanding how beginning graduate students develop. An important aspect of doing so is understanding how GTAs’ beliefs about mathematics might be similar to or different from those of their students.

Methodology and framework: The five participants of this study were graduate students pursuing Ph.D.s in mathematics at a top research university in the Unites States. At the beginning of this study, the participants were in the first or second year of their program and all held degrees in mathematics. Except one participant all other had no prior experience in teaching college mathematics. The participants primarily led discussion sections and followed a standard curriculum while reserving the autonomy in how they structure the discussion sections or write the quizzes they give.

Data were collected in a series of clinical interviews, the first one being recorded during the GTA orientation week prior to the start of fall classes. The participants were interviewed three more times approximately at intervals of one semester. The open-ended interviews were designed following the framework of Kung and Speer (2009) of stages of planning, performing and assessing. Our premise was that initially GTA expectations of their students’ view of mathematics were formed by their own experience and ideas about how mathematics is (ontological stance) and how a teacher of mathematics is (pedagogical stance). These expectations drove their practice (planning, performing, assessing). The practice generated assessments of their students (in-class/exam).

522 15TH Annual Conference on Research in Undergraduate Mathematics Education Research Question: Our goal was to answer the following question. Can the assessment results portraying in part how students view mathematics, reform the expectation which is formed by how the GTAs view mathematics (and thereby the practice that follows), if those are at a contradiction with one another? And what do the GTAs do faced with the reformed expectation?

Do the GTAs separate their pedagogical stance from their ontological stance, in that they present a different view of mathematics to their students while keeping their own view for themselves?

A preliminary analysis of the results of the first three interviews led us to gather further

**clarification in the following categories:**

1. GTA’s ontological stance versus pedagogical stance a. GTAs’ view of mathematics and the view they portray to their students.

i. Theory-based (abstract) versus example-based (concrete) ii. Connected (deep) versus stand-alone (shallow) b. GTAs’ preferred style of teaching (when they were taught) and the style they adopt as they teach, including

2. GTAs’ views of struggles and rewards of teaching as well as remedies they suggested.

Preliminary results: The preliminary analyses of this interview confirmed the following levels in GTAs’ knowledge of their students

We classify the stages B1 and B2 as behaviorist, and the latter two as constructivist.

15TH Annual Conference on Research in Undergraduate Mathematics Education 523 Stages B1, B2: Teacher centered-knowledge of how students react to tasks (behaviorist) Stages C1, C2: Student-centered- knowledge used to create cognitive theories (constructivist) Below we present a glimpse of the GTA thoughts and practices with the example of Clara.

Example of Clara Clara, a GTA who has been teaching for four semesters, is a graduate student pursuing her doctorate in the field of logic. She likes math for its abstract and rigorous nature, “Like if you prove something, you know it’s true and there is no discussion about it”, prefers theory and proofs to examples and applications for herself but not abashed about offering the students quite the opposite, because she knows “they don’t care about it” (the theories and the proofs) and that she doesn’t want to “make them feel confused about it”. She holds a rich and connected view of mathematics for herself admitting that challenging questions enriched her own understanding as a student, whereas provides her students with a user’s manual approach to calculus and straightforward questions so that “they don’t feel the pressure” to understand. She calls her approach as “adapting to her students” based on her knowledge that “they are not like me”.

At first we see her claiming how she is not bothered by this dichotomy of her as a student and her as a teacher, as she offers dismissive sentences such as “They pay me, so I do it” or the excuse that it is “harder to bring these features in Calculus”. However, as we go further the inner conflict becomes apparent by her disappointment that her students are not like her, that those “really amazing cool results” she learned in linear algebra are not the same for her students, when she finds that “most of the students probably don’t see or don’t care about this at all”. Or her admittance that if all her students exhibited the curiosity or cared about the material, she would “definitely” change her questioning patterns to include more challenging questions, “questions they can think about”. It gets further reconfirmed by the end when she chooses her one wish if she could change anything to make it more exciting for her as a teacher as having students who “want to be in the course”, who “want to understand”; perhaps not to the extent of her as a student, perhaps not to the level of a guaranteed understanding of mathematics, but as a step merely necessary in that direction.

Clara clearly demonstrates different ontological and pedagogical stances. But what stage is she at when it comes to her knowledge of her students?

We will present further results along with more details about the framework for the behaviorist and the constructivist stages at the meeting.

**Questions for audience:**

1. How do you classify Clara?

2. Can this phenomenon be viewed as dichotomy or is it a duality?

3. Did you experience this as a GTA?

**Bibliography:**

524 15TH Annual Conference on Research in Undergraduate Mathematics Education

1. Austin, A. (2002). Preparing the next generation of faculty: Graduate school as socialization to the academic career. Journal of Higher Education, 73(1), 94-122.

2. Cooney, T. J., Shealy, B. E., and Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306-33.

3. Gutmann, T. (2000). Mathematics and Socio-cultural Behavior: A Case Study of the Enculturation of a New Mathematician. Unpublished doctoral dissertation, University of New Hampshire, Durham, NH.

4. Gutmann, T. (2004, October 21-24). The Burden of Mathematics. Paper presented at the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Toronto, ON.

5. Kung D. & Speer N. (2009) Mathematics teaching assistants learning to teach: Recasting early teaching experiences as rich learning opportunities, Journal of Graduate and Professional Student Development,12, 1-23.

6. Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127.

Thompson, A. (1992). Teachers’ beliefs and conceptions: a synthesis of the research. In D. A.

7. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 127-146).

New York: NCTM/Macmillan.

8. Vacc, N. N. and Bright, G. W. (1999). Elementary preservice teachers’ changing beliefs and instructional use of children's mathematical thinking. Journal for Research in Mathematics Education, 30(1), 89-110.

15TH Annual Conference on Research in Undergraduate Mathematics Education 525 What’s the big idea?: Mathematicians’ and undergraduates’ proof summaries

Abstract In this study, seven mathematicians and seven undergraduates were asked to read and summarize mathematical proofs that they read to investigate which ideas they consider to be important in a proof. Mathematicians’ ideas were generally a) important equations, theorems or facts used in the proof, b) general methods used in the proof, c) diagrams or graphs, or d) overarching goals of the proof. Additionally, mathematicians and students sometimes included details or computations in their summaries that were unfamiliar, subtle, or not routine for them.

**Key words: Proof, key ideas, proof reading, proof summaries**

Introduction A large part of lectures in advanced undergraduate mathematics courses consists of professors presenting proofs of mathematical theorems to their students (e.g., Weber, 2001). Many mathematics educators contend that the purpose of presenting these proofs goes beyond convincing students that theorems are true; these proofs should also communicate to students some form of explanation or insight (deVilliers, 1990; Hanna, 1990; Knuth, 2002). However, as Raman (2003) and Weber (2010) noted, there is not a shared standard as to what this explanation and insight means.

The purpose of this paper is to investigate what mathematicians and undergraduates think are the important and useful ideas in the proofs that they read. We explore this broad issue by addressing these specific research questions: How do mathematicians and undergraduates create personal summaries of the proofs that they read? What aspects of these proofs do they choose to include in their summaries?

The decision to focus on summaries has practical importance. When reading any text, including mathematical proof, readers rarely remembers every word or sentence of that text. Rather, most readers remember main ideas from the text and use these to reconstruct the details, if possible and necessary. Knowing how mathematicians and undergraduates summarize text is there useful for two reasons. First, exploring how undergraduates summarize a proof provides insight into what insights undergraduates gain from the proofs that they read. Second, knowing how mathematicians summarize proof illuminates what aspects of proofs that they find important, and suggests aspects of proofs that could be emphasized to students.

Theoretical perspective Mathematical proofs are written so that each assertion that is not an acceptable premise (e.g., a hypothesis or a previously established fact) is a logical consequence of previous assertions in the proof. One way that a proof can be understood is at a line-byline level, where the reader of the proof identifies the mathematical reasons for how new 526 15TH Annual Conference on Research in Undergraduate Mathematics Education assertions follow from previous ones (e.g., Weber & Alcock, 2005). However, many mathematicians and mathematics educators argue that a proof can be understood in terms of its global ideas, and that this understanding can be as valuable, if not more valuable, than understanding the proof at a line-by-line level (e.g., deVilliers, 1990; Hanna, 1990;

Leron, 1983; Thurston, 1994; Mejia-Ramos et al, 2012). In this paper, we aim to investigate the different global mathematical ideas that mathematicians and students find important enough to include in their summaries.

Mathematics educators have posited different types of main ideas that may be

**present in a proof, which we summarize below:**

Explanation by characteristic property: Building on the philosophical work of Steiner (1978), Hanna (1990) argued that proofs can explain why a theorem is true by revealing a crucial property that a mathematical object has that causes the theorem to be true.

Mental models: Thurston (1994) and Weber (2010) suggested that mathematicians are sometimes less concerned about the step-by-step logic in a proof but more concerned with inferring the mental models that the author used to explain why the theorem was true and support the construction of the proof.

Key ideas: Raman (2003) contended that a proof can be understood in terms of its “key ideas”, where a key idea is a mapping from a “private” informal way of understanding why the theorem is true to a “public” formal rigorous proof.

High-level ideas: Leron (1983) argued that a proof can be summarized by its high-level ideas that describe the proof in terms of a few major steps while not including the logical details needed to support the proof.

Central equation or theorem: Weber (2006) claimed that a proof can be understood in terms of the central theorem or mathematical principal being applied.

We use these constructs as a preliminary means for analyzing our data.

Methods Participants.