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Drijvers, P (2001) The concept of parameter in a computer algebra environment, proc. PME 25, vol.2, 377-284 Dubinsky, E. (1991). Reflective Abstraction in Advanced Mathematical Thinking. In Freudenthal, Hans (1983). The Didactical phenomenology of Mathematcial Structures.

Dodrecht, Holland; Boston: Reidel, Giesen, C. van de (2002). The visualitsation of parameter. In M. Borovcnik &H.

Kautschitsch (Eds.), Technology in mathematics teaching. Proceedings of ICTMT5 pp. 97Vienna: Oebv&hpt Verlagsgesellschaft.

Gravemeijer, K.P.E. (1994). Developing realistic mathematics education. Utrecht: The Netherlands CD B Press Gravemeijer, K.P.E., Cobb, P. Bowers, J &Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb E. Yackel & K. McClain (Eds), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 225-273) Hillsdale, NJ: Lawrence Erlbaum Associates.

Hamdan, M (2006) Equivalent structures on sets: equivalence classes, partitions and fiber structures of functions. Educ. Stud. Math. 62, No. 2, 127-147 (2006).

15TH Annual Conference on Research in Undergraduate Mathematics Education 449 Hamdan M. (2009). Transcribing an Animation: The case of the Riemann Sums proceedings of the September 2009 10th International Conference “Models in Developing Mathematics Education” Presmeg, N. C. (2006) Research on visualization in learning and teaching mathematics.

In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205-235). Rotterdam: Sense Publishers.

Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22, 1 - 36 Shoenfeld A and Arcavi A (1988). On the meaning of variable, Mathematics Teacher, v81 n6 p420-27 Trigueros, M and Martinez P, Rafael (2007). Visualization and Abstraction: geometric representation of functions of two variables, Proceedings of the 29 th annual meeting of PME Zazkis, R., Dubinsky, E., & Dautermann, J. (1996): Coordinating visual and analytic strategies: a study of students’ understanding. Journal for Research in Mathematics Educations, 27(4), 435-437 450 15TH Annual Conference on Research in Undergraduate Mathematics Education !

Interculturally Rich Mathematics Pedagogical Content Knowledge for Teacher Leaders Preliminary Report Shandy Hauk, U. Northern Colorado & WestEd Michelle C. Chamberlin, U. of Wyoming Billy Jackson, St. Xavier Univeristy Nissa Yestness, Kristin King, and Robert Raish, U. Northern Colorado Abstract. We report on our work to build a theory about teacher leader development of interculturally aware mathematics pedagogical content knowledge (PCK). The effort is based on existing and continuing work on developing pre- and in-service teacher classroom PCK and intercultural competence. This preliminary report seeks feedback from RUME-goers on two discussion questions: Discussion Item 1: How do we identify and capture evidence of what might be called “teacher leader pedagogical content knowledge” in interculturally aware ways?

Discussion Item 2: What question formats might be productive for eliciting information from teacher leaders about their awareness of/attention to the intercultural aspects of mathematics instruction?... of mathematics itself?...of teacher leadership? This includes questions for written instruments as well as interview prompts and possible survey items.

Relation of the Work to the Research Literature Teacher leaders are experienced teachers who take on responsibilities and risks to improve students’ educational opportunities while working collaboratively with fellow teachers, administrators, and others (Yow, 2007). Many teacher leaders are mentors to colleagues (e.g., as math coaches or facilitators of teacher professional development, Borko, 2004), conduits of communication with administrators, and collaborators on educational policy, research, and product development – from curriculum to school budget and school law (Dozier, 2007; YorkBarr & Duke, 2004). Many who identify themselves as teacher leaders report entering leadership positions without any formal training (Dozier, 2007; Lieberman & Miller, 2007; York-Barr & Duke, 2004). And, few have preparation in the teaching and learning of adults. Much of the work of a teacher leader involves negotiating meaning across professional and personal cultural differences. While the significance of diversity as a factor in the education of American children has been widely discussed for many years, the nature of “diversity” continues to evolve in U.S.

schools (Aud, Fox, & KewalRamani, 2010). Several frameworks currently exist for professional contexts that involve understanding, interacting, and communicating with people from various “cultures” (see Figure 1 for working definition). In particular, healthcare and international relations groups have generated tools for personal and professional growth based on the theory of intercultural development and communication (Bennett, 1993, 2004; Hammer, 2009). “Culture” can include professional and classroom environments as well as personal or home experience. In this sense, several cultures – sets of values and ways of communicating about them – are developing for teacher leadership in the United States. A university partnership, the Mathematics Teacher Leadership Center (MathTLC), is exploring this area of collegiate mathematics education, and the potential for university-based methods in teacher leadership development.!

Members of the program include teachers whose current or near-future job roles are leadership positions, university mathematics and mathematics education professors as instructors in the program, and graduate student and faculty mathematics education researchers. Our goal includes building a theory about teacher leader development of interculturally aware mathematics pedagogical content knowledge (PCK) that is based on existing and continuing work on classroom PCK (Hill, Ball, & Schilling, 2008; Jackson, Rice, & Noblet, 2011) and intercultural competence development among teachers (DeJaeghere & Cao, 2009). !

15TH Annual Conference on Research in Undergraduate Mathematics Education 451!

Research Questions What is teacher leader PCK (TL-PCK)? How can attention to intercultural competence play a role in the development and refinement of responsive TL-PCK? In what ways do self-awareness and awareness of others as cultural beings support mathematics teacher leadership development?

Conceptual Framework Our efforts rely on two theories: one theory for intercultural competence development for mathematics teaching and learning in post-secondary settings and one for PCK. The first framework is based on the Developmental Model of Intercultural Sensitivity (Bennett & Bennett, 2004). As a developmental model, it includes lower and upper anchor orientations, intermediate orientations, and descriptions of the transitions among the orientations. Associated with this framework is an explicit attention to aspects of discourse based on effective intercultural conflict resolution (Hammer, 2005). The continuum of orientations runs from a monocultural or ethnocentric “denial” of difference based in the assumption “Everybody is like me” to an intercultural and ethnorelative “adaptation” to difference. The development from denial to the “polarization” orientation comes with the recognition of difference, of light and dark in viewing a situation (e.g., Figure 2a). The polarization orientation is driven by the assimilative assumption “Everybody should be like me/my group” and is an orientation that views cultural differences in terms of “us” and “them.” A developing tendency to deal with difference by minimizing it and focusing on similarities, commonality, and presumed universals (e.g., biological similarities – we all have to eat and sleep; and values – we all know the difference between good and evil) leads to the minimization orientation. A person in minimization will, however, be blind to deeper recognition and appreciation of difference (e.g., Figure 2b, a “colorblind” view). Transition from a minimization orientation to the “acceptance” of difference involves attention to nuance and a growing awareness of oneself as having a culture and belonging to cultures (plural) that differ in both obvious and subtle ways. While aware of difference and the importance of relative context, how to respond and what to respond in the moment of interaction is still elusive. The transition to “adaptation” involves developing frameworks for perception, and behavior shifting skills, that are responsive to a full spectrum of detail in an intercultural interaction (e.g., the detailed and contextualized view in Figure 2c). Adaptation is an orientation wherein one may shift cultural perspective, without loosing or violating one’s authentic self, and adjust communication and behavior in culturally and contextually appropriate ways. There are several ways that knowing one’s orientation, or the normative orientation of a group, can inform teacher leader work.

In thinking about TL-PCK we have relied on the layered model shown in Figure 3, where the yellow region (classroom) is the “C” of “content” in TL-PCK. In our presentation we will talk about how intercultural aspects of TL-PCK and PCK live in the model as we frame the research questions and forms of their answers and engage in RUME Session Discussion Item 1 (next page). We note that we have not yet tackled the other kinds of socio-cultural knowledge needed for teacher leaders to work with administrators, policy makers, and others.!

Goals for RUME 2012. The work on the research questions is shaped by the program goals (see Figure 4). For example, intercultural theory gives a language for thinking and talking about how we come to communication – including communication across orientations – and how we each respond to the variety of orientations in a room (e.g., meet people where they are). The theory also gives a language to develop awareness, as someone who has perspectives about difference and similarity in educational contexts, and for calibrating self-efficacy (e.g., adjust judgments of ability to successfully complete task X to take into account how others involved in task X define “success”).

In particular, at the conference we will focus on:

452 15TH Annual Conference on Research in Undergraduate Mathematics Education!

RUME Session Discussion Item 1: How do we identify and capture evidence of what might be called “teacher leader pedagogical content knowledge” in interculturally aware ways?

RUME Session Discussion Item 2: What question formats might be productive for eliciting information from teacher leaders about their awareness of/attention to the intercultural aspects of mathematics instruction?... of mathematics itself?...of teacher leadership? This includes questions for written instruments as well as interview prompts and possible survey items.

Research Methods The exploration of the culture of teacher leadership being developed by the members of the project and the nature of pedagogical content knowledge for teacher leaders is mixed-methods.

All members completed a 50-item validated and reliable Intercultural Development Inventory (see idiinventory.com) that provided intercultural orientation profiles of stakeholder groups.

These profiles were shared with all groups. To date we also have completed thematic and categorical coding of teacher leader application essays (coding of subsequent reflective essays by teacher leaders and university staff is ongoing), and initial cognitive interviews and piloting of written assessments of teacher leader pedagogical content knowledge. Further interviews with teacher leader experts developing and facilitating the program are being collected and will be analyzed, preliminary results may be shared at RUME2012 (not reported on here).

Preliminary Results To give a sense of the population and a preliminary portrait of their TL-PCK and cultural awareness, analysis of application essays for 14 teacher leaders (the first of four planned cohorts) is summarized in Figures 5 and 6. Essay prompts were about (1) ideal classroom, (2) significant experiences prompting a move to leadership, and (3) personal and professional goals. Many talked about the desire to understand another persons’ perceptions: “I hope the program will help me gain a deeper understanding of how other teachers view their teaching of mathematics” and a to “translate my knowledge and skills as a classroom teacher into pedagogical knowledge about adult teachers learning math and learning to teach math to diverse populations.” Reports on goals included “My hope would be that through my participation in this program I would gain the skills and knowledge to improve my own teaching, better meet the needs of the diverse population of County High School and to influence more classroom teachers to be involved in the school improvement process from the classroom to the national level.” For context, we offer Figure 7, showing the distributions of intercultural orientations of program members along with a reference set of additional stakeholders: secondary mathematics teachers (the “students” of the program’s teacher leaders). As a group, the teachers’ orientation was normatively in polarization while the teacher leaders were largely at the lower end of minimization and university folk were largely in minimization. As part of the research process, we have conducted group profile debriefing sessions with teachers, teacher leaders, and university staff and asked how knowledge of these orientations (for oneself and awareness that they exist for others) might play a part in their professional work. We have also created items used on a written instrument and in interviews with teacher leaders to look at the various aspects of the TL-PCK model shown in Figure 3. Below, we give an example of such an item and will share others at the conference as we explore RUME Session Discussion Item 2.

Part 1. Create a story problem whose solution would require 8th grade students to solve the following for x: 5x – 3 = 12.

Part 2. What challenges might you expect the students to encounter in doing your story problem?

15TH Annual Conference on Research in Undergraduate Mathematics Education 453!

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