The pedagogical content knowledge of two middl-school mathematics teachers
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The purpose of this study was to investigate two middle‐school mathematics teachers’ pedagogical content knowledge in terms of how it was manifested in their classroom instruction and ultimately to identify its components. The study also explored the teachers’ beliefs about mathematics, about teaching, and about learning mathematics and how those beliefs, as a backdrop, were related to their pedagogical content knowledge. Two eighth‐grade mathematics teachers participated in the study. Data were collected in the form of classroom observations, individual interviews, questionnaires, and documents. Data were analyzed using a strategy of case study and grounded theory methods. Mr. Smith’s pedagogical content knowledge included: (a) knowledge of mathematics, consisting of purposes of teaching mathematics, connections among topics, concepts to teach, various ways of solving problems, and textbook knowledge; (b) knowledge of students’ understanding, which involves particular students’ understanding, students’ errors and common misconceptions, and students’ difficulties and confusions; and (c) knowledge of pedagogy as revealed in learning activities, attempts to motivate students, and realistic applications. Ms. King’s pedagogical content knowledge included: (a) knowledge of mathematics that involved purposes of teaching mathematics, understanding topics, and curricular knowledge or topic organization; (b) knowledge of students’ understanding, consisting of students’ learning styles, learning difficulties, and common errors and misconceptions; and (c) knowledge of pedagogy, which includes ways of representation, lesson planning and organization, and teaching strategies that involved designing learning activities, incorporating student presentations, using various teaching styles and realistic applications, and using textbooks and journals. The two teachers had slightly different structures of pedagogical content knowledge. Mr. Smith was more dependent on his knowledge of mathematics and knowledge of students’ understanding, and Ms. King on her knowledge of pedagogy and, to a lesser extent, knowledge of mathematics. Consequently, no single model fits the pedagogical content knowledge of both teachers, perhaps because they used their knowledge differently or focused it differently in teaching mathematics.