Teachers' knowledge of student thinking and their instructional practices in algebra
Erbas, Ayhan Kursat
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This study was designed to address and contribute to our emerging knowledge and understanding of teachers’ knowledge of student difficulties and related issues with instructional practices in algebra. The perspectives suggested by the constructs of teaching in context, cognitively guided instruction, and pedagogical content knowledge influenced the theoretical orientations of the present study. Since “knowing” is a variety of separate entities, I found the distinction of knowing as knowing-about (i.e., knowing-that, knowing-why, knowing-how) and knowing-to useful to look at different degrees of knowing. Qualitative case study research design and methodologies were used in generating data collected from two inservice mathematics teachers of first year algebra (one eighth-grade and one ninth-grade teacher) who were selected purposefully. Data collection strategies included conducting audio-recorded semi-structured interviews, making video-recorded classroom observations, and collecting archival documents. Data stories about each case included thick descriptions of each participant’s beliefs, knowledge, and practices concerning student thinking. Findings revealed that even though both teachers presented an awareness and recognition of students’ thinking and difficulties in terms of “knowing-that,” their knowledge in terms of “knowing-why” and “knowing-how” was narrow and even problematic in some cases. Such insufficient knowledge might have limited the teachers’ pedagogical content knowledge of student thinking in terms of “knowing-to” and hampered the teachers when acting in the moment. Issues other than conceptual, cognitive, and epistemological problems characterized both teachers’ knowledge of and beliefs about general sources of students’ difficulties in terms of “knowing-why.” Those issues were: lack of arithmetical and geometrical knowledge, lack of motivation, lack of experience with nontraditional curricula, lack of practice in similar type of problems, carelessness, and inability to understand and apply definitions. However, the teachers were able to give explanations for students’ difficulties and mistakes in specific concepts they were teaching. Both case studies revealed that textbook dependence was central to the teachers’ practices at different stages of instruction such as when planning lessons, assigning homework, or assessing students’ learning. This dependence served as a blocking factor for teachers in trying to get more elaboration and knowledge of student thinking. Commonalities and differences arise for individual reasons for textbook dependence.