In a teaching experiment, I examined a theoretical model of mathematics teaching and learning in practice. In this paper I focus on how the model can guide the teacher's thinking about students' understandings and the generation of activities that foster intended transformations in those understandings. As a research-teacher I taught, twice a week…

I, a teacher-researcher, present my fifth graders with an interesting but incorrect student-constructed proposal first seen in 1996. The students used the patterns they had detected while solving the chessboard task to formulate counterarguments. The five types of counterarguments which emerged offer insights into the children's understanding of…