Knowing mathematics for teaching: a case study of teacher responses to students' errors and difficulties in teaching equivalent fractions

The goal of this study is to align teachers’ Mathematical Knowledge for Teaching (MKT) with their classroom instruction. To reduce the classroom complexity while keeping the connection between teaching and learning, I focused on Teacher Responses to Student Errors and Difficulties (TRED) in teaching equivalent fractions with an eye on students’ cognitive gains as the assessment of teaching effects. This research used a qualitative paradigm. Classroom videos concerning equivalent fractions from six teachers were observed and triangulated with tests of teacher knowledge and personal interviews. The data collection and analysis went through a naturalistic inquiry process. The results indicated that great differences about TRED existed in different classrooms around six themes: two learning difficulties regarding critical prior knowledge; two common errors related to the learning goal, and two emergent topics concerning basic mathematical ideas. Each of these themes affected students’ cognitive gains. Teachers’ knowledge as reflected by teacher interviews, however, was not necessarily consistent with their classroom instruction. Among these six teachers, other than one teacher whose knowledge obviously lagged behind, the other five teachers demonstrated similar good understanding of equivalent fractions. With respect to the basic mathematical ideas, their knowledge and sensitivity showed differences. The teachers who understood equivalent fractions and also the basic mathematical ideas were able to teach for understanding. Based on these six teachers’ practitioner knowledge, a Mathematical Knowledge Package for Teaching (MKPT) concerning equivalent fractions was provided as a professional knowledge base. In addition, this study argued that only when teachers had knowledge bases with strong connections to mathematical foundations could they flexibly activate and transfer their knowledge (CCK and PCK) to their use of knowledge (SCK) in the teaching contexts. Therefore, further attention is called for in collaboratively cultivating teachers’ mathematical sensitivity