A Case Study Of High School Mathematics Teachers' Pedagogical Content Knowledge

In the past decades, international research on mathematics teaching and teachers has gone through a few major paradigm shifts, from a focus on "how to characterize a good teacher" to "what teachers actually do during mathematics instruction" and from studying "why and how teachers make instructional decisions" to exploring "what teachers know about mathematics teaching". Meanwhile, newly released professional standards and reform-based curricular convey new expectations for teachers'content knowledge for teaching. However, researchers in the past focused mainly on what teachers should know. Few researchers paid attention to the kind of content knowledge for teaching actually used in daily practices.The main objective of this study was to investigate the differences in the pedagogical content knowledge (PCK) between novice and experienced mathematics teachers, and to assess students' perceptions of their teachers' PCK. This mixed-method research was conducted in two stages. In the qualitative stage of this study, six high school mathematics teachers were observed during an instructional unit on mathematical functions. The data sources include field notes, interview transcripts, and classroom observations. The goal was to gain deep insights about the differences in the PCK between novice and experienced teachers. In the quantitative stage of this study,334 high school students of these six teachers were surveyed to assess their perceptions of their own teachers' PCK. The results of the survey showed that novice and experienced teachers' PCK differed significantly in the following four categories:knowledge of mathematics, knowledge of students' understanding, knowledge of feedback, and knowledge of instructional strategies. The survey results were consistent with those obtained from the qualitative case studies of the six teachers.In particular, three general themes for improving mathematics instruction were identified:(1) teachers should develop a deep conceptual understanding of content knowledge and be able to deliver the content clearly to their students. For example, function is one of the core concepts in middle school mathematics and plays a crucial role in the transition from elementary to advanced mathematics. High school teachers should be required to demonstrate solid understandings of the definitions, origins, and structures of mathematical functions. (2) Teachers should develop knowledge of their students. Teachers should be required to identify those mathematical concepts that students struggle the most with and those required of relatively less effort. Being able to spot potential students' misconceptions and challenging mathematical concepts is a key aspect of teachers' understanding of students' development of mathematical knowledge. (3) Teachers should be able to design effective instruction in a target area, specifically, how to match lesson plans, instructional strategies, and assessments with the specific learning objectives of each lesson. They are also expected to develop deeper understanding of each student's strengths and weaknesses, and know when, with what and how to use a specific strategy as they accumulate more teaching experiences.In conclusion, this study has important implications for research in mathematics education and as well as mathematics teacher education. In particular, in order to improve pre-service teachers' PCK, teacher educators may start with improving their content knowledge in mathematics. In addition, we should pay special attention to case studies of individual teacher's PCK. Mathematics teachers apply PCK in specific content on a daily basis. Case studies provide opportunities for teachers to reflect on their PCK and share their PCK with the larger community of mathematics education. This study also shows that assessing students' perceptions of their teachers' PCK provides another valuable perspective for mathematics teachers to reflect on and improve their instruction.