Conjecturing and Argumentation in High-School Geometry Students
Kenneth R. Koedinger Carnegie Mellon University
There is a tendency to think of individuals who can discover new ideas or develop convincing arguments as having special "talent" or superior "intelligence." This view of conjecturing and argumentation abilities as fixed traits suggests that instruction directed toward such abilities is pointless for all but the most gifted of students. In direct contrast to that view, this chapter argues that successful conjecturing and argumentation performances are the consequence of particular skills and knowledge. In an appropriately structured learning environment, such skills can be acquired by anyone.
One reason for doubt regarding the instructability of conjecturing skills is our limited understanding of what these skills are. Developing a model of these skills is a key step toward creating effective learning environments for conjecturing. This model can then provide design guidance in creating elements of a learning environment: conjecturing tasks that appropriately challenge students and forms of assistance (including manipulatives, facilitative talk, and computer software) that support student learning. The learning approach being advocated here has important similarities with the Vygotskian notion of assisted performance ( Vygotsky, 1978) and more recent variations like cognitive apprenticeship ( Collins, Brown, & Newman, 1989).
This chapter presents a cognitive analysis of student conjecturing that includes a task analysis and an initial model of conjecturing, observations of student performance in a dynamic assessment, and modifications to the proposed model as guided by the results of this assessment. This cognitive analysis is then used to suggest forms of student assistance, including computer software tools, activities that draw out conjecturing skills, and facilitative "talking points" (hints or prompts), to help students through the most difficult terrain on their ways to conjecturing skill.