Middle School Mathematics Teachers' Reasoning about Students' Nonstandard Strategies: Division of Fractions

Abstract

Although students' nonstandard strategies have great importance in understanding students' thinking and creating effective mathematics classrooms, much remains unexplored in the literature. This study investigated 22 middle school teachers' reasoning about a student's nonstandard strategy for the division of fractions. The data were collected through semi-structured interviews and a task consisting of a student's nonstandard strategy within a classroom excerpt which simulates how mathematical work emerges in the context of teaching. Six categories of layers were formed based on their reasoning about the validity, generalizability, and efficiency of the nonstandard strategy. These layers were categorized as a surface, intermediate, and deep level of reasoning. It was found that while half of the teachers had a surface level of reasoning, only one-third of teachers are at a deep layer of reasoning. On the other hand, teachers' reasoning approaches of how and when the nonstandard strategy works for all problems were determined as equating the answer, equating the process, being multiples of each other, and equating the denominators. The results and implications are discussed, and recommendations are presented in accordance with the findings of the study.