Abstract
This article reports on an analysis of episodes of invalid or controversial arguments that occurred while two different groups of students worked on similar fraction tasks and examine the role that these types of arguments played in the development of students' reasoning. One group consisted of suburban, middle-class, fourth graders who worked on these tasks during the regular school day. The other group was comprised of sixth-graders from an urban community working on similar tasks as part of an informal learning after-school project. The findings of this study indicate that allowing students to share and discuss incorrect arguments promoted rich mathematical discourse and argumentation. The invalid arguments triggered the use of varied reasoning by other students and cleared up previous [mis]understandings.
Highlights
Researchers generally agree that reasoning and proof are the basis of mathematical understanding, and that learning to reason and justify is critical for growth in mathematical knowledge (Polya, 1981; Hanna, 2000; Maher, 2009). Thompson (1996) defines mathematical reasoning as “purposeful inference, deduction, induction, and association in the areas of quantity and structure” (p.267). Yackel and Hanna (2003) extend this definition to recognize the social aspects of reasoning and describe it as a communal activity in which learners participate as they interact with one another to solve mathematical problems (p.228)
We identified sessions and tasks during which students used multiple forms of reasoning, or during which students built arguments following several iterations and in response to the comments of other students. These occurrences highlighted instances of varied and complex argumentation, respectively. Analysis of both data sets highlighted that these situations were more often than not triggered by a fallacious or controversial arguments that led to varied argumentation and student engagement
The results of this study indicate that affording students opportunities to share and discuss incorrect arguments promoted mathematical discourse and argumentation (Stein, Engle, Smith, & Hughes, 2008)
Summary
Researchers generally agree that reasoning and proof are the basis of mathematical understanding, and that learning to reason and justify is critical for growth in mathematical knowledge (Polya, 1981; Hanna, 2000; Maher, 2009). Thompson (1996) defines mathematical reasoning as “purposeful inference, deduction, induction, and association in the areas of quantity and structure” (p.267). Yackel and Hanna (2003) extend this definition to recognize the social aspects of reasoning and describe it as a communal activity in which learners participate as they interact with one another to solve (resolve) mathematical problems (p.228). Researchers generally agree that reasoning and proof are the basis of mathematical understanding, and that learning to reason and justify is critical for growth in mathematical knowledge (Polya, 1981; Hanna, 2000; Maher, 2009). Yackel and Hanna (2003) extend this definition to recognize the social aspects of reasoning and describe it as a communal activity in which learners participate as they interact with one another to solve (resolve) mathematical problems (p.228). Reasoning is a process that enables the revisiting of ideas and can trigger the re-construction of previous knowledge in order to build new arguments (Davis, 1992; Maher, 2009). Reasoning and justification of arguments are essential to learning new mathematics, but are necessary to relate mathematical knowledge to other circumstances. Reasoning and justification are the foundation of problem solving
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