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Article Excerpt
Across the curriculum teachers are being asked to delve into and make use of students' thinking. Mathematics is no exception. Mathematics education researchers have gathered consistent evidence of the benefits of attending to students' thinking (Franke, Kazemi, & Battey, 2007; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Sfard & Kieran, 2001; Silver & Stein, 1996). During the past 20 years, researchers investigating Cognitively Guided Instruction have worked with teachers, sharing research about the development of students' mathematical thinking and studying teachers' use of that information. These researchers have found that teachers readily begin asking students open-ended questions after the students have solved a problem (e.g., "How did you solve that problem?") and can elicit an initial student explanation. Teachers find it more difficult, however, to follow up on student explanations and pursue students' thinking in ways that support students as they try to detail their strategies or connect with other students' strategies (Franke, Fennema, Carpenter, Ansell, & Behrend, 1998). Little research-based evidence exists to help teachers make the transition from asking the initial question to pursuing student thinking. We know little about the details of teacher practice, specifically the kinds of questions a teacher asks when supporting students in making their thinking explicit.
Background
Student talk is considered a major component of classroom discourse and a vehicle for increasing student learning:
Students must talk, with one another as well as in response to the teacher. ... When students make public conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community. (National Council of Teachers of Mathematics, 1991, p. 34)
Student talk can lead to increased student mathematical knowledge and understanding in two interrelated ways. First, listening to students talk makes it possible for the teacher (and other students) to monitor students' mathematical thinking. Teachers can use information gleaned from student talk to inform their instructional decision-making practices, including problems to pose and follow-up questions to ask (Franke, Fennema, & Carpenter, 1997). Similarly, when students converse with each other, their talk makes it possible for students to gauge each other's strategies and comprehension, providing opportunities for students to help each other build more complete mathematical understanding. Second, the act of talking can itself help students develop improved understanding. Describing, explaining, and justifying one's thinking all help students internalize principles, construct specific inference rules for solving problems, become aware of misunderstandings and lack of understanding (Chi, 2000), reorganize and clarify material in their own minds, fill in gaps in understanding, internalize and acquire new strategies and knowledge, and develop new perspectives and understanding (Bargh & Schul, 1980; King, 1992; Rogoff, 1991).
Not just any kind of student talk is expected to be productive for supporting or challenging students' thinking. Providing explanations is positively related to achievement outcomes, even when prior achievement is controlled for, whereas giving only answers is not related or is negatively related to achievement outcomes (Webb & Palincsar, 1996). Beyond providing answers, students must describe how they solve problems and why they propose certain strategies and approaches. Moreover, when describing their thinking, students must be precise and explicit in their talk, especially providing enough detail and making referents clear so that the teacher and fellow classmates can understand their ideas (Nathan & Knuth, 2003; Sfard & Kieran, 2001).
Teacher Moves to Support Student Explanations
Despite the demonstrated importance of students' explaining their thinking, "teacher-centered instruction continues to dominate elementary and secondary classrooms" (Cuban, 1993). In most classrooms students infrequently ask questions (Graesser & Person, 1994), and teacher talk typically dominates classroom discourse (Cazden, 2001). Moreover, the vast majority of teacher queries consist of short-answer, low-level questions that require students to recall facts, rules, and procedures (Graesser & Person, 1994), rather than high-level questions that require students to draw inferences and synthesize ideas (Hiebert & Wearne, 1993; Webb, Nemer, & Ing, 2006). International comparisons mirror these findings. The lack of opportunities in U.S. classrooms for students to discuss connections among mathematical ideas and to reason about mathematical concepts constituted one of the most prominent findings of the Third International Mathematics and Science Study (Hiebert et al., 2003; Stigler & Hiebert, 1999). Additionally, these descriptions of the level of student participation echo those made two or more decades ago (e.g., Cazden, 1986; Doyle, 1985; Gall, 1984; Mehan, 1985).
Yet we know from a growing body of work that teachers' questions scaffold students' engagement with the task, shape the nature of the classroom environment, and create opportunities for learning high-level mathematics (Boaler & Brodie, 2004; Kazemi & Stipek, 2001; Smith, 2000; Stein, Remillard, & Smith, 2007). We also know that teachers' questions can serve as a way to move students through the task in a specified way, ensuring they get the correct answer (Wood, Cobb, & Yackel, 1991). Finding the balance in the types of questions and when to ask them can make a large difference in how students continue to participate.
A number of researchers have begun to make explicit the moves a teacher may make to support students in making their mathematical thinking explicit, such as asking students to share their ideas publicly and using those ideas as the basis of conversation. For example, Wood (1998) examined the role of the teacher when supporting students to make explanations and found that teachers used different approaches: taking on some of the mathematical work and moving students in a direction teachers thought most critical ("funneling") versus encouraging students to do most of the mathematical work by focusing attention on particular aspects of students' explanations without guiding students in a specific, predetermined direction ("focusing").
Much remains to be learned about how teacher questioning in mathematics classrooms can help students participate in ways that allow them to make explicit their mathematical thinking and lead them to formulate complete and correct strategies. In this study, we look closely at the questions teachers ask as they engage with their students in mathematical conversation and the ways in which students participate in relation to teacher questioning.
Method
Building on the large-scale study of professional development by Jacobs and colleagues (2007), we selected teachers who had been engaged in the algebraic reasoning professional development for more than a year. In this article, we focus on three classrooms and the ways the teachers asked questions to help students make public and extend their mathematical thinking. We chose these teachers for observation and analysis because they came from similar schools, taught similar concepts and skills, used similar classroom structures (a combination of collaborative group and whole-class discussion of problem-solving strategies), but showed substantial differences in student achievement on posttests of algebraic thinking. We videotaped and audiotaped conversations in these classrooms in ways that allowed us to document what students said to the teacher and to each other so that we could closely analyze the relationship between teacher practice and student participation.
Participants
The three elementary school classrooms (two second grade, one third grade) analyzed here come from a large urban school district in Southern California. Prior to the algebraic thinking professional development and the large-scale study, the district administrators and teachers recognized the value of engaging in algebraic reasoning in elementary school, and long-term plans for overall school improvement were under way. The district, in its 2nd year of new leadership when the study began, had a history of poor performance and a long-standing sense from those outside the district that it would never do well. According to the state's ranking system and standardized test scores, it was one of the lowest performing school districts in California. As in many urban school districts, hiring and retaining qualified teachers was a struggle. Although the district was making progress, at the beginning of the study, only 57% of the teachers in the district held credentials and 30% of the teachers were in their 1st or 2nd year of teaching. The community served by this district had shifted from being predominantly African American to being predominantly Latino, and at the time of our work, the schools served students of whom 99% were minority, 52% were classified as English language learners, and 93% received free or reduced-cost lunch.
Professional Development Program
Participating teachers engaged in professional development to explore the development of students' algebraic reasoning and, in particular, how that reasoning could support students' understanding of arithmetic. The professional development content, drawn from Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School (Carpenter, Franke, & Levi, 2003), highlighted "relational thinking," including (a) understanding the equal sign as an indicator of a relation, (b) using number relations to simplify calculations, and (c) generating, representing, and justifying conjectures about fundamental properties of...
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