Letting go [electronic resource] : cultivating agency and authority through number talks in the secondary mathematics classroom
Keywords: Education.

Abstract Mathematics teaching in the United States has remained substantively unchanged in most schools for almost a century (Hiebert, 1999). Focused on demonstrating procedures to obtain right answers, however, this "school mathematics" tradition (Richards, 1991) has the drawback of providing few opportunities for students to learn to solve unfamiliar problems. This deficiency has caused increasing concern, not only because of U.S. students' relatively poor performance on international assessments (OECD, 2014), but also because success in today's world requires more than knowing how to use routine procedures to solve predictable problems (Darling-Hammond and Adamson, 2010). The Common Core State Standards for Mathematics (CCSS-M) (2010) therefore stress the need for students to understand mathematics well enough to apply their knowledge to unfamiliar situations and problems. Eight additional Standards for Mathematical Practice explicitly identify how students should regularly engage with mathematics; for example, students should explain their reasoning, justify their conclusions, make and analyze conjectures, and "attend to the meaning of quantities, not just how to compute them" (CCSS-MP, p. 6). Implementation of these standards implies centralizing students' ideas in instruction, and therefore requires significant changes in the practices of most mathematics teachers, (Cobb & Jackson, 2011). Lessons organized around students' ideas are fundamentally different from lessons in traditional mathematics classrooms (Brown, Stein, & Forman, 1996). This is especially apparent during whole class discussions, which have been shown to be challenging to enact (Boerst, Sleep, Ball & Bass, 2012; Kazemi, Franke, & Lampert, 2009). Teachers who are accustomed to explaining procedures, for example, need to learn to invite students' ideas for solving problems; they also need to learn to probe students' thinking by asking authentic questions rather than the more typical questions for which they have answers already in mind (Nystrand, Wu, Gamoran, Zeiser, and Long, 2003). Students, in turn, need to generate their own ideas rather than repeat what they think their teachers want to hear, and learn to listen to and learn from each other. These interactions are so different from traditional classroom practices that both teachers and students need considerable support in learning these new roles. This participant-observation study investigates the challenges faced by two preservice high school teachers as they learned to organize discussions around student thinking in an instructional activity called Number Talks. Number Talks are short daily lessons in which students solve computation problems mentally and share their strategies in whole class discussions. Through this replicable routine, teachers can learn to elicit students' ideas and probe for understanding, while students can learn to author and share their mathematical ideas. Number Talks are therefore an ideal canvas for the study of teacher learning. During university coursework prior to the study, teachers learned to enact Number Talks through carefully structured pedagogies of practice (Grossman et al., 2009). The study itself comprised ten Number Talks for each teacher over five weeks, with the teachers observing each of the other's Number Talks interspersed with Guided Collaboration Meetings (GCMs) before and after each lesson. The GCMs, which were loosely structured (Levine & Marcus, 2010) to be responsive to issues that arose, offered opportunities for teachers to reflect, analyze, and plan for the next lesson. As one of the instructors for the university coursework and the coach for the GCMs, my role therefore was primarily participant-as-observer. Video records comprise the majority of data in this study. Analysis can best be described as the progressive refinement of hypotheses (Engle, Conant, & Greeno, 2007). Analyses of the videos of the GCMs found that teachers were considerably challenged by the dominance of rote procedures in students' thinking, and that the subsequent search to nurture sense making was an important theme of the meetings. Analysis of these meetings also demonstrated that guided collaboration helped the teachers learn to notice (Sherin, Jacobs, & Philipp, 2011) important moments in students' explanations -- such as a student willingly sharing their confusion -- and consider together how to respond. The Number Talks themselves were analyzed through several lenses. One analysis, which considered intellectual authority, showed that students in both classes became increasingly able to draw on their own understandings rather than relying on rote procedures in their mathematical explanations, but that they reverted to rote procedures when the topic changed. A second view of the lessons examined how the teachers interpreted and incorporated questioning and responding strategies they had learned. These findings showed that both teachers consistently asked probing sequences of specific questions that helped students give correct and complete conceptual explanations (Franke et al., 2009). The analysis also suggested, however, that when teachers continued to ask leading or "funneling" (Wood, 1998) questions (in addition to probing questions), student agency and intellectual authority may have suffered. These findings also considered praise and evaluation, which indicated that a teacher's ability to relinquish a strong evaluative stance may be associated with growth in her students' public risktaking behaviors. Finally, this study shows that even in five short weeks, teachers can learn to work with student thinking in ways the promote students' sense making and agency. Understanding the particular challenges and potential trouble spots teachers face as they begin using unfamiliar instructional practices in the classroom can help teacher leaders and coaches forestall the discouragement that may lead teachers to give up and revert to familiar practices. These findings indicate that teaching practices cannot be considered independent of students' readiness and willingness to respond in the reflexive environment of the classroom. As much as teachers would like to be able to "establish" particular environments, they can only do so within a continual process of negotiation and renegotiation with their students.