Listening as a Vital Element in Synergistic Argumentation during Mathematics Problem Solving
Cassel, Darlinda, Reynolds, Anne, Focus on Learning Problems in Mathematics
From research observations of activities during a second-grade mathematics problem-centered learning classroom, synergistic argumentation emerged as a class norm for discussing the students' mathematics. In this paper we analyze the contrast between two students who were participants in this class. Both students were accustomed to sharing during whole-class discussions in this classroom environment. Both were very capable mathematical thinkers. One student, Brett, depicts the use of argumentation successfully while the other, Miriam, depicts the use of argumentation ineffectively. An important aspect of Brett's argumentation was identified as hermeneutic listening. Brett's engagement enhanced the learning environment whereas Miriam's stance was counter-productive, at least for her.
Our research over the last several years investigated a second-grade mathematics classroom where the teacher enacted a problem-centered learning environment (Wheatley, 1991). During the last three years, we focused in particular on the quality of discourse and argumentation that occurs during the whole class sharing time in the lesson. The purpose of this study has two fold: (a) to describe the mathematics whole class sharing session and (b) to analyze its function for effective learning occurring through conflict and disagreement in the open explanation of solutions and strategies. In the problem-centered learning environment students are encouraged to express their ideas freely, try to make sense of each other's methods, listen, question, and carry on a conversation between and among themselves. Our overall goal was to analyze the whole class interaction patterns and learning opportunities.
In the second year of this investigation, two students in particular provided us with contrasting pictures of discourse and argumentation during their involvement in the whole-class discussions. In this paper we will describe and analyze episodes involving these two students. Examining their contrasting stances of sharing or not sharing of ideas, provides a deeper understanding of the nature of discourse and argumentation and its degrees of success or failure in learning enhancement from the young students' perspective.
Framework for this Study
In recent years there has been increasing numbers of investigations into classroom environments where children talk openly about mathematics and explain their mathematical solutions. In these environments the teacher's role is different; s/he (a) facilitates the development of students' mathematical understands and (b) acts as a facilitator by organizing instruction so that students are interactively engaged in the doing and talking dynamically about mathematics (NCTM, 2000; Wood, 1999). A number of researchers argue that collaboration and whole-class sharing encourages children to learn with understanding through opportunities to explain, justify, and listen to one another's ideas (Cobb, 1998; Cobb & Yackel, 1996; Cobb, Yackel, & Wood, 1992; Kazemi, 1998; Mevarech, 1999; Wheatley, 1991). Further, they argue that explanations are the best means for students and teachers to elaborate meaning and to make connections to other mathematical topics as well as to their own prior knowledge. These conditions help students to construct rich networks of meaning. As students share their explanations they seek meaningful ways to communicate and elaborate their ideas with each other and the teacher, negotiating meanings as opposed to just reciting facts. As students negotiate, they adjust their interactions by presenting rationales for their strategies while attempting to make sense of each other's ideas.
Sfard (2001) contends that this open communication is equivalent to thinking itself. She states that our thinking is a dialogical endeavor as we inform, argue, reflect, and question others (as well as ourselves). Thus, thinking is communication; not necessarily verbal, and not necessarily inner. …