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.

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.

This thesis is submitted to the University of Manchester for the degree of Doctor of Philosophy (PhD). This study developed an experimental small group teaching method in the Realistic Mathematics Education tradition for teaching fractions using models and contexts to year 7 children in Brunei (N=89) whose effectiveness was evaluated using a treatment-control design: the E1 group was given the experimental lessons, the E2 group who was given “normal” lessons taught by the experimenter, and a whole class (E3) group which acted as the control group. The experimental teaching was video recorded and subject to semiotic analysis, aiming to describe the objectifications that realized ‘learning of fractions’ by the groups.The research addresses two research questions:1. How effective was the experimental teaching in helping learners make sense of fractions, with respect to equivalence of fractions and flexibility of unitizing?2. What were the semiotic learning and teaching processes in the experimental group of the RME-like lessons? This study used a mixed method approach with a quasi-experimental design (QED) for the quantitative side, and a semiotic analysis for the qualitative side. Quantitatively, the experimental teachings proved to be relatively effective with an effect size of 0.6 from the pre- to the delayed post-teaching test, compared to the E2 and the control groups.The basic findings pertaining to the semiotic analyses were:a. The mediation of the production of fractions in terms of length, from the production of fractions in terms of the number of parts which led to equivalence of fractions;b. The use of language and gesture help to objectify the equivalence of fractions and the flexibility of unitizing–in some case it involved gesturing to the self;c. The role of the Hour-Foot clock (HFC) as a model in a realistic context; andd. The complexity of the required chains of objectifications reflects the difficulties of the topic.