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Socio-Cultural Issues
CREATING OPPORTUNITIES FOR ALL:UNPACKING EQUITABLE PRACTICES IN MATHEMATICS CLASSROOMS Melissa Sommerfeld Gresalfi The University of Alabama
Victoria M Hand The University of Alabama
Lynn Liao Hodge The University of Alabama
[email protected]
Research on equity in mathematics and other school subjects has tended to focus on what is meaningful for particular racial and ethnic groups. With increasingly diverse classrooms and concerns about the danger of essentializing, there is a need for the field of mathematics education to identify classroom practices that provide equal access to opportunities to engage in mathematical inquiry and develop positive dispositions towards mathematics for a range of students. This paper reports three studies that examined the structure of activity in mathematics classrooms that were successful at promoting broad-based participation among their students. The analyses focus on four aspects of activity systems including instructional tasks, classroom discourse, and the roles of the teacher and students. Additionally, the analyses illustrate how interactions among these elements created access to opportunities for all students to be successful. Introduction The purpose of this paper is to problematize a current conceptualization of equitable teaching that focuses primarily on what is meaningful for particular racial and ethnic groups, and encourage a focus instead on how classrooms create opportunities for all students to negotiate competent participation and thus positive dispositions towards mathematics learning. The motivation for this paper stems from several pressing concerns. First, we are wary of approaches that simplify a complicated and rich process of locally constructing cultural practices in a particular setting by reducing them either to a list of topics to include in a curriculum or practices that may be easily stereotyped. Second, we believe that accommodating every student’s home experiences is often untenable in America’s increasingly heterogeneous schools, wherein one classroom may include students from many different cultural backgrounds. For these reasons, we believe that researchers who are concerned with issues of equity need to expand their focus to explore how particular classroom practices create opportunities for all students to be successful (Delpit, 1995; Gutierrez, 1999; Moschkovich, 2002). Doing so requires thinking about success as more than achievement scores to include a consideration of both the ideas that students learn and how they come to appreciate and see value in mathematics. Our examination of opportunities to learn in the local space of the mathematics classroom shapes our conceptualization of equitable teaching in terms of students’ access to classroom mathematical practices. We consider the processes by which students gain or lose access to mathematical practices in the design of classroom tasks, in classroom discussion, and in the broader classroom practices and norms that get organized over time Cobb & Nasir, 2002). This perspective of equity stands in contrast to one in which broader socio-political structures and processes around race, culture, and power outside of the classroom are considered with respect to classroom life (Martin, 2000, 2003; Tate, 1994) In the proposed paper, we will present three studies that investigate aspects of the classroom social context in supporting students’ access to ideas and forms of reasoning as well as a sense of _____________________________ Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds) (2006). Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, México: Universidad Pedagógica Nacional.
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affiliation with mathematics. These three studies are united in their approach to the study of equitable teaching both in their perspective of the classroom as a system of activity comprised of different elements (Greeno,& MMAP, 1998; ), as well as the use of research methods that capture the functioning of these elements both in moment-to-moment classroom interaction and over time. Considering Access in the Classroom Activity System Creating access to opportunities to engage deeply with mathematical content is a theme that cuts across these studies. Focusing on the distribution of access in this way highlights the role of the organization of activity systems (Cohen & Lotan, 1995; Engle & Conant, 2002), and the nature of students’ experiences in mathematics in order to account for students’ successes and difficulties (Boaler, 1998; Martin, 2000). This perspective draws attention to the complex interactions between elements of a classroom system in constructing and reifying mathematical practices, and thus what counts as competent participation (XXXX). From the perspective of access, aspects of the classroom system can be seen to support or delimit students’ experiences to be successful in learning mathematics. Considering the construction of equitable access to meaningful engagement ensures that practices of engaging with content become the focus of analysis, rather than assumptions of what may be meaningful, and thus “motivating,” for particular groups of students (XXXX). In the proposed paper, we will draw on data from three studies to delineate aspects of the classroom activity system that contributed to of the creation of opportunities for students to engage competently with mathematical content. In doing so, we focus our discussion on four themes: instructional activities, classroom discourse, the role of the teacher, and the role of the student. These aspects are closely related and can be seen to contribute to how mathematics becomes realized in particular classrooms (Bowers & Nickerson, 2001; Cobb, Yackel, Wood & McNeal, 1992; Lampert, 2001). The sorts of mathematical conversations that take place in a classroom can be viewed as situated within a space of possibilities that are afforded by instructional activities. In turn, classroom discourse can be seen as developing in interaction in which the teacher’s role, students’ roles, and their negotiation of what counts as mathematical competence are critical. Data Sources and Method of Analysis All three studies took place in the United States in urban middle and high school classrooms. The duration of the studies ranged from one semester to one year. All analyses drew on field notes, videotaped recordings of class sessions, and student interviews. The studies share common aspects of systematically analyzing data through multiple stages of coding (Glaser & Strauss, 1967). These common and overlapping phases include first working through field notes in order to identify critical incidents and potential shifts in patterns of classroom discourse, the role of the teacher, and the role of the students. A second phase involves working through videotaped recordings session by session in order to isolate critical incidents and to make conjectures on patterns of participation. A third phase involved examining the analyses of sessions at a metalevel in order to test conjectures about patterns and to isolate sessions that illustrate significant shifts in discourse, roles, and the opportunities that were afforded students to engage with mathematics in substantial ways. Particular attention was paid in each of the studies to situations that seemed to contradict ongoing conjectures and to offer explanations for these situations and how they informed the ongoing analyses.
Socio-Cultural Issues
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Results A number of significant findings emerged from each study, and the purpose of the proposed paper is to draw attention to connections as well as significant differences across studies in order to inform more equitable classroom practices. We discuss key findings in terms of the four themes identified above. Our analyses revealed that instructional activities were critical in supporting substantial conversations about mathematics in that they opened up a space of possible mathematical topics that could become the focus of discussions. The structure of activities served both to support students’ initial engagement with the task and their later use of mathematics for solving problems. Classroom discourse and in particular what counted as a mathematical explanation afforded students access to multiple task interpretations from which they could become interested in informing the problem situation and in teasing out important math ideas as they compared analyses. As the analyses show, classroom discourse also presented students with opportunities to make their thinking and mathematics public, thereby creating opportunities for revision and contributing to a shared purpose and a sense of community in supporting everyone’s learning. Both the roles of the teacher and students proved to be critical as well in opening up spaces of opportunities. For instance, all three teachers adeptly included students’ comments as part of discussions while advancing the mathematical agenda. At the same time, all three teachers drew on different strategies and practices to navigate this tension. The roles that became constituted for students differed across studies, but one key commonality was that students were active contributors to the ideas that came to matter in each class, and in this process students were able to experience voice while engaging deeply with mathematics. Significance The proposed paper is positioned to make significant theoretical and pragmatic contributions. Theoretically, the paper will contribute to an understanding of how the dynamics of equity and processes involving access and opportunities are created, developed and sustained in classrooms. Pragmatically, the proposed paper will contribute to an understanding of more equitable instructional practices and resources that support students’ access to mathematical ideas and to a view of mathematics as a worthwhile and important activity. References Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29, 41-62. Bowers, J. S., & Nickerson, S. (2001). Identifying cyclic patterns of interaction to study individual and collective learning. Mathematical Thinking and Learning, 3, 1-28. Cobb, P. and N. Nasir (2002). "Diversity, Equity, and Mathematical Learning." Mathematical Thinking and Learning 4(2&3). Cobb, P., Wood, T., Yackel, E., & McNeal, G. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research Journal, 29, 573-602. Cohen, E.G., & Lotan, R.A. (1995). Producing equal-status interaction in the heterogeneous classroom. American Educational Research Journal, 31,1, 99-120. Delpit, L. (1995). The silenced dialogue: Power and pedagogy in educating other people’s children. In L. Delpit Other People’s Children: Cultural Conflict in the Classroom. New York, NY: W.W. Norton and Company.
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Engle, R. A., & Conant, F. (2002). Guiding principles for fostering productive disciplinary engagement: Explaining an emergent argument in a Community of Learners classroom. Cognition and Instruction, 20,4, 399-483. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. New York: Aldine. Greeno, J. G. & MMAP (1998). The situativity of knowing, learning, and research. American Psychologist 53(1), 5-26. Gutierrez, K. D., P. Baquedano-Lopez, et al. (1999). "Rethinking Diversity: Hybridity and Hybrid Language Practices in the Third Space." Mind, Culture, and Activity 6(4): 286-303. Lampert, M. (2001). Teaching Problems and the Problems of Teaching. New Haven, Yale University Press. Martin, D. (2000). Mathematics Success and Failure Among African-American Youth. Mahwah, New Jersey, Lawrence Erlbaum Associates. Martin, D. (2003). "Hidden assumptions and unaddressed questions in mathematics for all rhetoric." The Mathematics Educator 13(2): 7-21. Moschkovich, J. N. (2002). "A Situated and Sociocultural Perspective on Bilingual Mathematics Learners." Mathematical Thinking and Learning 4(2&3): 189-212. Tate, W. F. I. V. (1994). "Race, retrenchment, and the reform of school mathematics." Phi Delta Kappan 75: 447-485.
