Effects of social metacognition on geometric reasoning and micro-creativity: Groups of students constructing proofs
Pawlikowski, Michael James
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As students have historically performed poorly on geometric proofs, curriculum and instruction reforms often target this topic. While previous research has focused on defining cognitive aspects of proof such as student proof schemes or types of geometric reasoning, they have neglected the ways in which face-to-face student interactions affect learning, reasoning, and knowledge construction in geometry. Indeed, little is known about how students influence one another while working together on geometric proofs. This study helps fill this research gap by synthesizing theoretical frameworks (proof schemes, geometric reasoning and social metacognition) to create a new framework to analyze students' creation of ideas (micro-creativity) and geometric proof processes. Then, this new framework is applied to two groups of students working on proofs. Two small groups of four high school geometry students each engaged in a series of tasks to investigate their proof ability and proving processes. A pre-test proof was given to each individual student to identify a baseline of their proof ability. Then students worked together on a group proof which was coded along the dimensions of social metacognition, geometric reasoning, and micro-creativity for statistical discourse analysis. Following the group-proof, a post-test proof was given to each individual student. The progression of tests helped identify students' conceptual, computation, and conclusion errors at each point in time. The regression analyses showed that both group's social metacognitive factors predicted different types of geometric reasoning and micro-creativity. When a student gave a command, it was often a new idea. When a student individually positioned themselves, it was often a recognition, a conjecture or a new idea. When a student made a statement, it was often a systematization but seldom a recognition. When a student made a suggestion, it was often a recognition, inference, or a correct, new idea. After a groupmate disagreed, a student was more likely to express a conjecture or a correct idea. The post-test proof provided mixed results as student errors varied from the pre-test. One significant factor from the post-test revealed that between groups the weaker students showed overall improvement whereas the stronger students showed an overall decline. The findings of this study have implications for students, teachers, and researchers. When students work together on geometric proofs, they can correct one another's misconceptions, misuse of formulas, and false assumptions. For teachers, using instructional strategies like small group work can be beneficial for teaching geometric proof, but these groups need teacher facilitation to ensure all students acquire and retain good proof habits. For researchers, it provides a new perspective on assessing students' geometric proof and proving processes, as well as evidence for the importance of examining socially constructed proofs.