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Keywords: authority, mathematical authority, mathematical group settings, model of authority Introduction and Research Questions The role of interviewing in qualitative data collection requires researchers to consider the strategies researchers employ to obtain interview data (Patton, 2002). Authority dynamics between interviewers and participants have been identified as one factor influencing the authenticity of interview data; Langer-Osuna and Engle (2010) and Brubaker (2009) emphasize the need to attend to authority patterns in these settings. Authority can be socially or content based; social authority is defined as charismatic authority derived from social norms, while content authority is derived from the community of practice, instructors, or textbooks (Amit & Freid, 2005). While both types of authority have strong impacts on academic discussions, social authority has been observed to overwhelm content authority, potentially leading groups in directions not based on sound reasoning (Langer-Osuna & Engle, 2010). We use a grounded theory approach to answer the research question: what role does authority play in the guided reinvention process, and what model can be developed to assist researchers in understanding authority dynamics in mathematical group settings?

Esmonde and Langer-Osuna’s model for authority in science discourse has four components: socially negotiated influence, degree of perceived authority, access to the conversational floor, and access to the interactional space (Engle, Langer-Osuna, & McKinney de Royston, in press). While the work by these and other authors have focused on K-12 settings, here we address mathematics educators’ need to understand how authority, both mathematical and social, develops in guided reinvention with undergraduate mathematics students. Relevant work in undergraduate mathematics settings is limited (Langer & Engle, in press), but the work of Szydlik (2000) and Frid, (1994) indicates an aspect of mathematical authority called source of conviction, which dictates how mathematical statements are justified internally (sense-making) or externally (from outside authoritative sources). While SoC can be used as a measure of 400 15TH Annual Conference on Research in Undergraduate Mathematics Education mathematical authority, and Freid and Amit’s (2006) framework can serve as a basis for social authority, a comprehensive model of authority in undergraduate mathematics group settings has not been formulated or used to analyze group dynamics in these settings.

Theoretical Perspective and Methods Examining roles of participation and authority in group settings are two of the basic constructs of situated cognition (Brown, Collins, & Duguid, 1989; Lave & Wagner, 1991;

Salomon & Perkins, 1997). In this perspective, researchers focus on how constructs such as roles and authority contribute to the progressive discourse within the group, which is what the theory defines as learning. We employed this lens to investigate participation and authority and their effects on group dynamics and learning (Bereiter, 1994; Jordan & Henderson, 1995; Sfard’s 1998).

Given the limited literature on authority in guided reinventions, we used grounded theory (Patton, 2002) to develop a model of authority dynamics. We initially open coded the first two days of the guided reinvention of limit concepts (Martin, Oehrtman, Roh, Swinyard, & HartWeber, 2011; Oehrtman, Swinyard, Martin, Roh, & Hart-Weber, 2011; Swinyard, 2011) using a constant comparative method, and then developed our initial categories (Corbin & Strauss, 2008). After we conducted a literature search on social and mathematical authority, we adapted the model and standards of evidence proposed by Engle, Langer-Osuna, and McKinney de Royston (2008) to fit the group size and content discussed by our participants (Figure 1), and then coded the first five days of the guided reinvention using this new framework. The goal of our model was to categorize the types of interaction between the participants and interviewers to model authority dynamics.

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Figure 4 Socially negotiated authority Using an idea as the unit of analysis, we totaled the counts for each day made by the participants. From the table we saw a relatively stable pattern in the number of evaluations made by each participant and the number of times the participant was treated with authority. Belinda tended to have higher evaluations of the group each day of the interview.

Megan Belinda Megan Belinda Megan Math Belinda

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Figure 5 Sources of conviction Again using an idea as the unit of analysis, we totaled the counts of the source of conviction made by each participant. Belinda exhibited more statements coded as internal authority and mathematical authority, while Megan had more external source of authority statements.

Conclusions Understanding the relation between authority and group dynamics is important for mathematical settings such as interviews, focus groups, and teaching experiments like the guided reinvention. We observed diminishing social authority while mathematical authority patterns remain fairly constant in our teaching experiment employing the guided reinvention heuristic.

Ongoing research is aimed at understanding the causes of such patterns. Being aware that interviewers often have authority over the learners is important, especially in our case, where we saw how a multi-day teaching experiment created an environment where authority dynamics initially established persisted over later days. We suggest this could be because the participants perceived the researchers as authority figures (an aspect of social authority), causing the participants to attempt to foster appeals to mathematical reasoning (an aspect of mathematical authority). Overall, our current model describes the authority relationships in this interview. In our preliminary report, we are interested in obtaining feedback, particularly the following questions: (1) what factors may foster shifts in authority dynamics?, (2) what are typical considerations interviewers take regarding authority?, and (3) what additional information would need to be incorporated into a model of authority dynamics to usefully inform data collection methods?

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Bereiter, C. (1994). Implications of Postmodernism for Science, or, Science as Progressive Discourse. Educational Psychologist, 29, 3-12.

Bianchini, J. A. (1999). From here to equity: The influence of status on students access to and understanding of science. Science Education, 83, 577-601.

Brown, J. S., Collins, A., & Duguid, P. (1989), "Situated Cognition and the Culture of Learning," Education Researcher, 18(1), 32-42.

15TH Annual Conference on Research in Undergraduate Mathematics Education 403 Brubaker, N. (2009). Negotiating Authority in an Undergraduate Teacher Education Course: A Qualitative Investigation. Teacher Education Quarterly, 36(4), 99-118.

Corbin, J., & Strauss, A. (2008). Basics of qualitative research (3rd ed.). Thousand Oaks, California: SAGE Publications, Inc Engle, R. A., Langer-Osuna, J., & McKinney de Royston, M. (2008). Toward a model of differential influence in discussions: Negotiating quality, authority, and access within a heated classroom argument. In B. C. Love, K. McRae & V. M. Sloutsky (Eds.), Proceedings of the 30th Annual Conference of the Cognitive Science Society (pp. 2010-Austin, TX: Cognitive Science Society.

Engle, R. A., Langer-Osuna, J., & McKinney de Royston, M. (in press). to appear in Cognition and Instruction.

Esmonde, I. & Langer-Osuna, J. (2009). Power plays: How students in a small group negotiated mathematical discussions in hybrid spaces. Manuscript under review.

Esmonde, I. & Langer-Osuna, J. (in press). Power in numbers: Student participation in mathematical discussions in heterogeneous spaces. To appear in The Journal for Research in Mathematics Education.

Frid, S. (1994). Three Approaches to Undergraduate Calculus Instruction: Their Nature and Potential Impact on Students’ Language Use and Sources of Conviction. Research in College Mathematics Education I, Conference Board of the Mathematical Society, Issues in Mathematics Education, Volume 4, American Mathematical Society, 69-100.

Fried, M., & Amit, M. (2006). Proof, authority, and agency: intimations from an 8th grade classroom. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 113-120. Prague: PME.

Jordan, B. & Henderson, A. (1995). Interaction analysis: Foundations and practice. Journal of the Learning Sciences, 4(1), 39-103.

Langer-Osuna J., & Engle, R. (2010). “I study features; believe me I should know!”: The mediational role of distributed expertise in the development of student authority. In Glodmna, S. & Pelligrino, J (Eds). International Society of Learning Sciences Proceedings of the 9th International Conference of the Learning Sciences, Vol. 1, Chicago: ISLS.

Lave, J., and Wenger, E. (1991). Situated Learning. Cambridge: Cambridge University Press.

Martin, J., Oehrtman, M., Roh, K., Swinyard, C., & Hart-Weber, C. (2011). Students’ reinvention of formal definitions of series and pointwise convergence. Paper presented at 404 15TH Annual Conference on Research in Undergraduate Mathematics Education the 14th Annual Conference on Research in Undergraduate Mathematics Education, Portland, OR: Portland State University Oehrtman, M., Swinyard, C., Martin, J., Roh, K., & Hart-Weber, C. (2011). From intuition to rigor: Calculus students’ reinvention of the definition of sequence convergence. Paper presented at the 14th Annual Conference on Research in Undergraduate Mathematics Education, Portland, OR: Portland State University Palincsar, A. & Harrenkohl, L.(2002). Designing collaborative learning contexts. Theory into Practice, 41(1), 26-32.

Patton, M. Q. (2002). Qualitative research & evaluation methods (3rd ed.). Thousand Oaks, California: SAGE Publications, Inc.

Salomon, G. & Perkins, D. (1997). Individual and Social Aspects of Learning. In Review of Research in Education, 23, 1-24 Sfard, A. (1998). On Two Metaphors for Learning and the Dangers of Choosing Just One.

Educational Researcher, 27, 4-13.

Swinyard, C. (2011). Reinventing the Formal Definition of Limit: The Case of Amy & Mike.

Journal of Mathematical Behavior, 30(2), 93-114.

Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31(3), 258-276.

15TH Annual Conference on Research in Undergraduate Mathematics Education 405 Learnin tra g jectories and formative assessment in first semester calculus: A case study

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While formative assessment, assignments given for feedback rather than grades, raise student achievement, the literature lacks an explanation for how these assessments affect student learning. The purpose of this case study of an introductory calculus class using the approximation framework was to investigate how adding formative assessments to an introductory class using the approximation framework changed the learning trajectory for the class. The preliminary analysis of the formative assessments suggested that the assessments appeared to scaffold metacognition, self-reflection, and transfer of the approximation framework between units.

Keywords: approximation framework, formative assessment, learning trajectory, transfer 406 15TH Annual Conference on Research in Undergraduate Mathematics Education Introduction and Research Questions Formative assessments, low stakes assignments given to assess students’ current level of understanding, increase student achievement (Black & Wiliam, 2009; Clark, 2011), but little is known about how implementing formative assessments facilitates this achievement gain. The purpose of this case study is to study the impact of formative assessment on students’ learning trajectories in a calculus course with Oehrtman’s (2008) approximation framework; the main research question that guided my investigation was: How does formative assessment impact students’ Zone of Proximal Development of the Approximation Framework (Oehrtman, 2008) between contexts in introductory calculus?

Understanding how formative assessment affects how undergraduates learn and transfer the approximation framework helps to advance the theory of formative assessment, which has been primarily developed on European students in primary and secondary school (Black & Wiliam, 1998, 2001, 2003, 2006, 2009). Furthermore, a better understanding of how formative assessments scaffolds student achievement allows us to improve our calculus pedagogy.

Black & Wiliam’s (2009) framework of formative assessments suggests that there are five major benefits of formative assessment: (1) to communicate clearly what the learning goals are, (2) allowing instruction to be based on students’ current level of understanding, (3) providing learners with feedback that scaffolds learning, (4) giving peers a common experience to talk to each other about, and (5) raising students ownership of learning. Researchers have found that transferring concepts from the initial contexts in which the concept is learned is difficult for students (Barnett, 2002; Lobato & Siebert, 2002), but since formative assessment can increase student self-monitoring (Clark, 2010), which can facilitate transfer (Ning & Sun, 2011), we hypothesized formative assessments could also help facilitate transfer, which could also impact students’ learning trajectories.

Theoretical Perspective and Methods Examining the role of peripheral participation in group settings, such as a formative assessment, is a basic constructs of situation cognition (Brown, Collins, & Duguid, 1989; Lave & Wagner, 1991; Salomon & Perkins, 1997). A situated cognition perspective allows researchers to focus on how these constructs contribute to the progressive discourse within the group, which is what the theory defines as learning. Using the established frameworks of situated cognition, we chose to use this lens to investigate participation and its effect on group dynamics and learning (Bereiter, 1994; Jordan & Henderson, 1995; Sfard’s 1998). These frameworks guided my standards of evidence and were effective tools for investigating our research question about how authority dynamics can be modeled for mathematical group settings.

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