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Conclusion Hazzan and Zazkis (2005) note “that these interpretations of abstraction are neither mutually exclusive nor exhaustive” (p. 103). This observation is definitely applicable to our data. For example, referring to a familiar game of picking a number among natural numbers can be described in terms of interpretation (1) as well as interpretation (3). Similarly, relying on calculus/limit interpretations of infinity corresponds to (1) as well as (2). Hazzan developed the framework of reducing abstraction and showed its applicability to interpret undergraduate students’ thinking when they struggle with difficult-for-them, at least initially, mathematical concepts. What is partially surprising, that in the case described here, participants with rather strong mathematical background, who demonstrated their ability to approach the task on mathematical/theoretical level, also regressed to reducing abstraction and adding contextual considerations that were at times inconsistent with their formal mathematical solution. However, this finding is in accord with Chernoff (2010) study, that showed prospective elementary school teachers’ tendency toward contextualized interpretation.

References Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In J. Kaput, A.H. Schoenfeld, & E. Dubinsky (Eds.), Research in collegiate mathematics education II (pp. 1-32). Providence: American Mathematical Society.

Cantor, G. (1915). Contributions to the founding of the theory of transfinite numbers. Trans. P.

Jourdain, repr. (1955). New York: Dover Publications.

Chavoshi Jolfaee, S. & Zazkis, R. (2011). Probability zero events: Improbable or impossible?

Proceedings of 35th International Conference for Psychology of Mathematics Education.

Vol. 2, pp. 217-224. Ankara, Turkey: PME.

Chernoff, E. (2011). Investigating relative likelihood comparisons of multinomial contextual

sequences. Proceedings of CERME 7. Available:

http://www.cerme7.univ.rzeszow.pl/index.php?id=wg5 Dubinsky, E., Weller, K., McDonald, M.A., & Brown. A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics 58, 335-359.

Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts.

Educational Studies in Mathematics 40(1), 71–90.

Hazzan, O., & Zazkis, R. (2005). Reducing abstraction: The case of school mathematics.

Educational Studies in Mathematics, 58(1), 101-119.

Mamolo, A. & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in Mathematics Education, 10(2), 167-182.

Noss, R. & Hoyles, C. (1996) Windows on Mathematical Meanings: Learning Cultures and Computers. Dordrecht: Kluwer.

Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits.

Educational Studies in Mathematics, 18, 371–397.

334 15TH Annual Conference on Research in Undergraduate Mathematics Education




Wendy Aaron, Vilma Mesa and Patricio Herbst University of Michigan wendyaar@umich.edu Facilitation in professional development and research contexts is a delicate craft. In the proposed paper we describe the facilitation of study group sessions among community college trigonometry instructors. The study groups were designed to collect data about instructors’ practical rationality (Herbst, 2006; Herbst & Chazan, 2003). W hile these sessions took place to fulfill a particular goal and involved a particular population we believe that our facilitation methods can benefit any group in which the facilitator is responsible for managing public reflection. To this end, we describe questioning strategies for facilitation of the sessions that support productive conversations— that is, conversation that support both professional development and research goals. The notion of a productive conversation is developed in the paper.

Keywords: trigonometry; community college; professional development

Introduction Facilitation in professional development and research contexts is a delicate craft (Borko, 2004;

Elliott, Kazemi, Lesseig, Mumme, Carroll, Kelley-Petersen, 2009; Koellner, Schneider, Roberts, Jacobs, & Borko, 2008; Suzuka, Sleep, Ball, Bass, Lewis & Thames, 2009). In mathematics education these two contexts are often combined which implies that the facilitator often has two competing goals. The first goal is to ensure that the participants feel comfortable enough to share their ideas, that each participant is heard and respected, and that participants’ individual comments join to form a cohesive conversation. The second goal involves uncovering and addressing some knowledge, skill, or disposition that is the target of professional development or eliciting some knowledge or information that is the target of the research (Nachlieli & Herbst, 2010). It is crucial that the first goal is met so that participants will make their reflections public so that both researchers and other participants can learn from them. The second goal ensures that these reflections are of a quality that is valuable to both the participants and the researchers.

In this paper we describe the facilitation of study group sessions among community college trigonometry instructors. The study groups were designed to collect data about instructors’ practical rationality (Herbst, 2006; Herbst & Chazan, 2003), in particular, or the practical knowledge that instructors use to guide their instructional decisions. While these sessions took place to fulfill a particular goal with a specific population we believe that our facilitation methods can benefit any group in which the facilitator is responsible for managing public reflection. To this end, we describe questioning strategies for facilitation of the sessions that support productive conversations—that is, conversation that support both professional development and research goals. The notion of a productive conversation will be further developed in the paper. We recognize that there are many features of a session that support productive conversations besides questioning; we address here only those related to questions proposed by the facilitator.

15TH Annual Conference on Research in Undergraduate Mathematics Education 335 Information about the sessions The sessions analyzed are part of a larger research study that seeks to investigate the nature of mathematics instruction at community colleges (Mesa, 2010, Accepted; Mesa, Celis, & Lande, 2011; Mesa, Suh, Blake, & Whittemore, 2011). Twenty instructors (ten full time and ten part time) were recruited from 12 different community colleges in Michigan and Ohio. Participants meet once per month for five months. Each session is three hours long and includes a mathematical activity (e.g., defining angles, constructing a protractor) and analysis of representations of instruction (e.g., a video of an online tutoring session, video of students’ responses to an interview prompt, an animation of a classroom episode). We seek to fulfill social, mathematical, pedagogical, and research goals with each session. The research goals revolve around seeking information about the rationales that instructors have for doing or not doing certain things as they teach trigonometry.

Methods We analyzed the facilitator’s questions and the participants’ responses to those questions in the sessions described above. Facilitation questions are coded after the challenges that they address, as well as the research or learning goal they advance. Participants’ responses are coded after their usefulness in answering research questions or evidence of participant reflection (Hatton & Smith, 1995). The aim of these methods is to empirically ground the development of a framework for productive facilitation advanced in the paper.

Challenges We have identified at least five challenges to productive conversations that the facilitation needs to overcome to produce productive conversations. The five challenges are: participants are disinclined to discuss mathematical ideas; participants tend to talk about instruction in general terms; participants avoid talking about the mundane features of instruction; participants are disinclined to provide justification for actions that are not supported by reform documents; and participants talk about individual instructors instead of instruction. Below we briefly describe the last three of these challenges.

A void talking about the mundane Participants are not inclined to share mundane details; instead they are inclined to talk about instructional events that are out of the ordinary. However, we are interested in learning about the work that participants do everyday in their classrooms so the challenge to the facilitation is to get participants to share the features of their instructional practice that are unremarkable. We believe that the day-to-day actions and decisions of instruction are the most productive site for making lasting and sustainable changes to participants’ instructional practice.

Hesitant to provide reasons for actions that would be frowned upon Reform documents contain strong support for student-centered approaches to instruction;

however, we have seen that instruction in community college mathematics classrooms is often content-centered. Because content-centered actions are frowned upon, participants are reluctant to admit that they perform them and reluctant to discuss reasons that support them. The challenge to the facilitation is to uncover and document the reasons that instructors have for performing these actions since we believe that instructors have valid reasons (real constraints) for using these forms of instruction.

Talk about features of instructors instead of instruction In the sessions participants may produce long monologues about their own instruction, learning experiences, or in cases where there is a representation of instruction, the participants may talk about the individual instructor in the representation. In our work we are interested in studying the 336 15TH Annual Conference on Research in Undergraduate Mathematics Education work of teaching, so we are interested in hearing about individual instructors insomuch as the experience of the instructor informs us about the work of teaching. It is essential for participant learning and for the research that comments be connected to the work of teaching in general, not just about individual idiosyncrasies of teaching.

These challenges to the facilitation of leading productive conversations could potentially interfere with the learning of the participants and with our research goals. While these examples came from our particular situation we believe that these challenges, or very similar ones, could affect other researchers and teacher educators who endeavor to facilitate sessions in which participants are asked to publicly reflect on their own teaching and decision-making. In the full paper we will provide a more comprehensive list of challenges and examples of how they manifest in the sessions.

Facilitation questions Here we describe questions that the facilitator used as tools to overcome the challenges described. There are other design considerations that can contribute to productive conversations but we limit this discussion to the use of facilitation questions. In Table 1 we list the challenges to facilitation and the types of questions that address the challenge along with examples.

The set of questions listed in the first column of Table 1 address the challenge of avoiding talking about the mundane. These questions overcome the challenge by asking participants to consider specific moments in a representation of instruction. For example, in our sessions we found that our participants found it difficult to explain why they used examples to illustrate mathematical procedures or techniques. We could ask teachers to share other ways in which they might illustrate a mathematical procedure to highlight the benefits of working through examples.

The set of questions listed in the middle column of Table 1 address the challenge of hesitation to provide reason for actions that could be frowned upon. The proposed questions overcome the challenge by making the tacit assumption that there are conditions under which these actions are appropriate and asking participants to provide these conditions. Other questions ask participants about other sources of constraints on their instruction, such as students or administrators, and ask how these stakeholders encourage these actions. For example, in our sessions we found that participants initially claimed that they would never ignore a student question, however this is an action that we have seen happen repeatedly in community college trigonometry classrooms. We could ask teachers when it might be appropriate to ignore a student question to find the reasons that they engage in this action.

The set of questions in the third column of Table 1 address the challenge of talking about features of instructors instead of instruction. The proposed questions overcome the challenge by inviting other participants to share their experiences, expanding the conversation beyond one instructor. Another set of questions asks participants to consider the generality or specificity of the instruction being discussed. This strategy also takes the focus off of a single instructor and moves it to the setting in which the instructional move takes place. For example, one participant in our sessions was inclined to talk at length about her own learning of trigonometry. We could ask other participants if they have students who had similar experiences to find out more about the usual experience for community college trigonometry students.

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