Prospective teachers' reasoning and responseto a student's non-traditional strategy whendividing fractions

摘要

Recognizing meaning in students' mathematical ideas is challenging, espe-cially when such ideas are different from standard mathematics. This study examined,through a teaching-scenario task, the reasoning and responses of prospective elementaryand secondary teachers to a student's non-traditional strategy for dividing fractions. Sixcategories of reasoning were constructed, making a distinction between deep and surfacelayers. The connections between the participants' reasoning, their teaching response, andtheir beliefs about mathematics teaching were investigated. We found that there were notonly differences but also similarities between the prospective elementary and secondaryteachers' reasoning and responses. We also found that those who unpacked the mathe-matical underpinning of the student's non-traditional strategy tended to use what we call"teacher-focused" responses, whereas those doing less analysis work tended to construct"student-focused" responses. These results and their implications are discussed in relationto the influential factors the participants themselves identified to explain their approach tothe given teaching-scenario task.