Explanation, Justification and Argumentation in Mathematics Classrooms.

Abstract

Current interest in mathematics learning that focuses on understanding, mathematical reasoning and meaning-making underscores the need to develop ways of analyzing classrooms that foster these types of learning. In this paper, I show that the constructs of social and sociomathematical norms, which grew out of taking a symbolic interactionist perspective, and Toulmin’s scheme of argumentation, as elaborated for mathematics education by Krummheuer, provide us with a means to analyze aspects of explanation, justification and argumentation in mathematics classrooms, including means through which they may be fostered. In addition, I use the example of current research in a university-level differential equations class to show how these notions can inform instruction in higher-level mathematics. In his plenary address to PME in 1994, John Mason said, I have long concluded that it is very hard to say anything new that has not been said more eloquently elsewhere.... I see working on education not in terms of an edifice of knowledge, adding new theorems to old, but rather as a journey of self discovery and development in which what others have learned has to be reexperienced by each traveller, re-learned re-integrated and re-expressed in each generation. (p. 177) He went on to say that “All you can do, if you really want to be truthful is to tell a story” (p. 177). My purpose in this paper is to tell a story that intends to capture something of what I have experienced and learned about explanation, justification and argumentation from a variety of mathematics classrooms that I have studied. At the same time, I intend to explain why I am telling you this story—what significance this story might have for someone else. In this regard, I am following the approach that Streefland (1993) took when he said that by analyzing his experiences, in his case in instructional design, his goal was to take what was an after-image for him and make it possible for that to become a pre-image for someone else working in the same arena. Similarly, my intention is to specify various aspects of explanation, justification and argumentation that have emerged from my analyses of classrooms so it becomes possible for others to use these aspects to inform their future efforts in mathematics education research and practice.

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