Supporting Whole-class Collaborative Inquiry in a Secondary Mathematics Classroom

Recent mathematics education reform efforts call for the instantiation of mathematics classroom environments where students have opportunities to reason and construct their understandings as part of a community of learners. Despite some successes, traditional models of instruction still dominate the educational landscape. This limited success can be attributed, in part, to an underdeveloped understanding of the roles teachers must enact to successfully organize and participate in collaborative classroom practices. Towards this end, an in-depth longitudinal case study of a collaborative high school mathematics classroom was undertaken guided by the following two questions: What roles do these collaborative practices require of teacher and students? How does the community’s capacity to engage in collaborative practices develop over time? The analyses produced two conceptual models: one of the teacher’s role, along with specific instructional strategies the teacher used to organize a collaborative learning environment, and the second of the process by which the class’s capacity to participate in collaborative inquiry practices developed over time. Whole-class collaborative inquiry 3 In recent decades, there have been a multitude of reform efforts in mathematics education in the United States. One major focus of these efforts is to effect classroom-level changes. Specifically, efforts have aimed to create more collaborative and student-centered environments, where students have opportunities to reason and construct their understandings as part of a community of learners (National Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000). The value of these reform practices has been demonstrated with respect to student learning particularly students’ conceptual understanding (Boaler, 1997; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997; Silver & Stein, 1996; Wood & Sellers, 1997), and in the production of potentially more equitable learning environments (Boaler, 1997; Cohen & Lotan, 1997; Silver, Smith & Nelson, 1995). Furthermore, students participating in these kinds of practices tend to develop positive identities with respect to mathematics (Boaler & Greeno, 2000; Cobb & Hodge, 2002) and their beliefs about mathematics more accurately reflect the nature of the discipline than those of their traditionally schooled peers (Schoenfeld, 1988, 1992). Despite these compelling successes, traditional models of instruction, particularly at the secondary level, still dominate the educational landscape (Jacobs, Hiebert, Givvin, Hollingsworth, Garnier, & Wearne, 2006; Kawanaka & Stigler, 1999; Stigler & Hiebert, 1999). Teachers find it very challenging to organize and support student participation in these discourse-intensive practices that centralize students’ ways of thinking. There are myriad factors contributing to the staying-power of traditional models and the lack of implementation of more collaborative, inquiry-oriented environments. An assumption of this study is that this limited success, in part, can be attributed to an underdeveloped understanding of the nature of the practices of these communities and the roles required of teachers and students to enact these practices successfully (Boaler, 2003). This article offers two conceptual models of a teacher’s instructional strategies that supported students’ participation in whole-class discussions, a central event in inquiry-oriented environments. Whole-class discussions are used by teachers to promote and extend students’ mathematical understandings and develop their proficiency with a range of mathematical practices such as conjecturing, justifying and reconciling (Ball & Bass, 2000). The conceptual models are based on data from a longitudinal case study of a teacher and class that regularly engaged in collaborative inquiry practices. The first model represents the role of the teacher during collaborative whole-class discussions. It aims to capture the critical work that the teacher did as she supported students’ participation in collaborative inquiry practices. The second model represents the process by which the class’s capacity to enact collaborative practices developed over time. In their prior mathematics courses, the students had Whole-class collaborative inquiry 4 experienced primarily traditional modes of instruction. Thus an important part of the teacher’s work was establishing new norms and expanding the class’s repertoire of learning practices (Boaler, 2002; Cohen & Ball, 2001). Although whole-class discussion is the identified participation structure, this study focuses on collaborative inquiry mathematics practices as a specific instantiation of reform ideals. Rochelle and Teasley (1995, as cited in Dillenbourg, 1999, p. 17) define collaboration as “a coordinated, synchronous activity that is the result of a continued attempt to construct and maintain a shared conceptualization of a problem” (p. 70). The term collaborative then implies a joint production of ideas, where students offer their thoughts, attend and respond to each other’s ideas, and generate shared meaning or understanding through their joint efforts. Collaboration is distinct from cooperation which only implies sharing. Students can cooperate to accomplish a task, by sharing answers or creating a poster or other product together, but may not engage in collaboration with respect to the joint production of mathematical ideas. Collaboration is also distinct from practices involving extensive participation or talk from students. Too often “surface features” associated with reform practices, such as high levels of interaction among students or groupwork, are taken as evidence of successful realization of reform ideals. Yet, classrooms supporting these surface features may not give sufficient attention to the quality of students’ engagement with mathematical ideas (Nathan & Knuth, 2003; NCTM, 2000). Inquiry is a practice or stance, and indicates a particular way of engaging with and making sense of the world. In this article, I focus on inquiry as it relates to students’ mathematical activities and I take the term to mean both inquiry into mathematics and inquiry with mathematics. Inquiry into mathematics involves delving into mathematical ideas and concepts and trying to understand the structure, power, and limitations of mathematics. Inquiry with mathematics involves using mathematics as a tool to make sense of problem situations and come to some reasonable resolution. This type of work involves problem solving, modeling, and applications to business, physics or other “real world” phenomena. The focus of this research on practices is driven by developments in learning theories, particularly the advent of sociocultural and situative perspectives. These perspectives understand learning as a fundamentally participatory and social process. From a situative perspective, “mathematics is a set of practices of inquiry and sense-making that include communication, questioning, understanding, explaining, and reasoning” and “learning mathematics is marked by increasing participation in an expanding range of such practices” (Greeno & MMAP, 1997, p. 104). Learning results from, and is evidenced by, student participation in both standard disciplinary practices (e.g., justifying, representing algebraically) and an array Whole-class collaborative inquiry 5 of other practices of mathematical communities (e.g., questioning, communicating, informal reasoning). Thus, understanding the social context and practices of a classroom, and how the teacher organizes and supports these practices, is essential to understanding the learning opportunities afforded students. This study builds on prior research on the successful enactment of collaborative inquiry practices in mathematics classrooms. Several studies offer insights into the pedagogical demands of these socially and intellectually intensive learning communities with analyses of teaching practices or particular aspects of pedagogy that support student participation in reform-oriented mathematics classrooms (e.g., Ball, 1993; Chazan, 2000; Fraivillig, Murphy, & Fuson, 1999; Lampert, 1990, 2001; Lobato, Clarke, & Ellis, 2005; Rasmussen & Marongelle, 2006). For example, Fravillig et al. (1999) developed a framework, Advancing Children’s Thinking (ACT), to describe teaching at the elementary level that supported the advancement of students’ mathematical thinking in an Everyday Mathematics classroom. They identified three overlapping but distinct “patterns of practice”: eliciting children’s solution methods; supporting children’s conceptual understanding; and extending children’s mathematical thinking along with corresponding instructional strategies for each pattern. Such a framework offers a conceptualization of the many facets of the teacher’s role in a reform-aligned classroom and the specific actions teachers may employ to support student participation and mathematics learning. Other studies have focused on analyzing the role of the teacher in organizing student participation in practices authentic to communities of mathematicians (e.g., Ball, 1993; Forman, Larreamendy-Joerns, Stein, & Brown, 1998; Lampert, 1990, 2001) such as argumentation, justifying, and revising. Another line of research has analyzed the critical work of teaching that supports students’ participation in a new set of learning practices (e.g., Boaler, 2002; Chazan, 2000; Goos, 2004; Heaton, 2000; Hufferd-Ackles, Fuson, & Sherin, 2004; Lampert, 1990, 2001; Wood, 1999). These classroom cultures and communities represent a radical departure from more conventional classrooms, which requires students to participate in new ways. For example, Goos (2004) studied an expert secondary teacher of an advanced mathematics course. Drawing on a Vygotskian perspective, she identified teacher actions and assumptions that helped to create a culture of inquiry in this classroom and