Listening as a Vital Element in Synergistic Argumentation during Mathematics Problem Solving

Cassel, Darlinda, Reynolds, Anne, Focus on Learning Problems in Mathematics

Abstract

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.

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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. Body movements, situational clues, and histories of interactions all influence communication and its effectiveness.

Thus, in contrast with classrooms that follow what Richards (1991) labeled the school mathematics tradition--where students passively follow the teacher's direction and practice teacher prescribed procedures--problem-centered classrooms place emphasis on collaboration and negotiations among students to promote learning. A greater responsibility is placed on the student to enter into the classroom conversation in order to learn mathematics. The term argumentation is used throughout this research literature with several positive and negative connotations associated with it. We propose the use of the term "synergistic" argumentation (Cassel, 2002) as a better descriptor of what takes place within a problem-centered classroom. Synergistic argumentation is a circular back-and-forth process where students listen actively, explain their ideas and solutions, defend them in the face of questions, and question other students' ideas; all to make sense of the mathematics.

Because of the importance placed on such discourse our goal for the three-year study was to analyze the whole-class interaction patterns and learning opportunities of the students. In examining the synergistic argumentation as it developed in the classroom under investigation our focus question was, "Are there certain characteristics of discourse/argumentation that lead to successful learning?" The two students whose actions we analyze in this paper provided an opportunity to probe in greater depth some assumptions implicit in this learning setting.

The Research Setting

This research was conducted in a second-grade classroom in a low to middle income suburban public school that was also a site for the district's emotionally handicapped program. The students were from various backgrounds and had varying abilities and disabilities. For a number of years the teacher had enacted a problem-centered learning environment (Wheatley, 1991). This teacher expected children to think and reason about the mathematics in ways that promoted making sense of mathematical ideas. Tasks given provided opportunities for students to test ideas, develop reasoning, and gain confidence in their understandings of mathematics. The teacher encouraged students to participate while making sense of the mathematics. She would ask questions based on the situation and the students' actions. Examples of these questions are, "Did anyone do it differently?" "Can you explain how you got that?" "What do you (meaning other students) think of that way of doing the task?" Sometimes the teacher would ask the students if they understood how another child solved the task. In this way the teacher was asking the students to not only make sense of their own ways of thinking about mathematics but also to listen to and make sense of the other students' explanation and ways of thinking. Thus, teacher and students were negotiating the socio-mathematical norms for their classroom (Yackel & Cob, 1996). These norms define what kind of talk is valued (Cobb, 1998; Kazemi, 1998). They are not rules that students must follow, but they deal with the process of making …

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Publication information: Article title: Listening as a Vital Element in Synergistic Argumentation during Mathematics Problem Solving. Contributors: Cassel, Darlinda - Author, Reynolds, Anne - Author. Journal title: Focus on Learning Problems in Mathematics. Volume: 29. Issue: 4 Publication date: Fall 2007. Page number: 30+. © 2008 Center for Teaching - Learning of Mathematics. COPYRIGHT 2007 Gale Group.

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