24 SES 02 A, Issues in Mathematics Teacher Education (Part 1)
Paper Session: to be continued in 24 SES 03
It is widely accepted that pupil achievement is dependent to a large extent on the quality of teaching (Stronge, Ward, & Grant, 2011). Besides, mathematical content knowledge of teachers is regarded as an important factor in the teaching and learning of mathematics (Williams, 2008). Nevertheless, researchers identified that teachers have limitations in their mathematical content knowledge (e.g., Ball, 1990; Ma, 1999). Therefore, mathematics educators around the world attempted to develop measures or generate theories for deepening teachers’ mathematical content knowledge (e.g., Ball, Hill, & Bass, 2005; Rowland, Huckstep, & Thwaites, 2005). Ball et al. (2005) developed items to test both common and specialized content knowledge of teachers. According to Rowland, Turner, Thwaites, and Huckstep (2009), Ball et al.’s questionnaire might give some clues about teachers’ pedagogical content knowledge but might not reflect how teachers act in practice. Rowland et al. (2009) added that in order to assess teachers in their actual practice, we need to observe those teachers while they are teaching. By adopting this idea, in this paper we attempted to reflect on a middle school mathematics teachers’ teaching of integer addition and subtraction by using the Knowledge Quartet framework generated by Rowland, et al. (2005). This framework includes four broad categories: foundation, transformation, connection and contingency (Rowland, Turner, 2007; Turner, 2012). In particular, we used the transformation category of the Knowledge Quartet. This category is about “teacher demonstrations; use of instructional materials; choice of representation; choice of examples” (Turner, Rowland, 2011, p.200).
We selected the topic of integer addition and subtraction as the unit of analysis of the current study for several reasons. First, integers and integer operations are conceptually difficult for students (Gregg and Gregg, 2007; Janvier, 1983; Vlassis, 2004). Second, the topic of integer addition and subtraction strongly displays the teaching practice under scrutiny (Mitchell, Charalambos, & Hill, 2014). Third, since there is no single representation that can be used to teach integers and its operations (Stephan & Akyüz, 2012), teachers usually need to utilize several representations in order for students to make sense of the underlying ideas of integers.
We asked ourselves the following questions while reflecting on teacher’s teaching of integer addition and subtraction. Does the teacher use appropriate equipment; select appropriate forms of representation; choose appropriate examples; clearly explain ideas or concepts possibly by using an analogy; and demonstrate clearly and accurately how to carry out procedures? By focusing on the transformation category we tried to delve into teacher’s ability to transform his knowledge of integers to students in an accessible and appropriate way. Thus, the research question of this study was formulated as follows: How do middle school mathematics teachers transform their knowledge of integer addition and subtraction to their students?
Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. Elementary School Journal, 90(4), 449-466. Ball, D. L., Hill, H. C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), p. 14-17, 20-22, 43-46. Gregg, J., & Gregg, D. U. (2007). A context for integer computation. Mathematics Teaching in the Middle School, 13(1), 46-50. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. London: Lawrence Erlbaum. Mitchell, R., Charalambous, C. Y, & Hill, H. C. (2014). Examining the task and knowledge demands needed to teach with representations. Journal of Mathematics Teacher Education, 17, 37-60. Rowland, T., & Turner, F. (2007). Developing and using the knowledge quartet: A framework for the observation of mathematics teaching. The Mathematics Educator, 10(1), 107-123. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-281. Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing primary mathematics teaching. London: SAGE. Stephan M. & Akyüz D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464. Stronge, J. T., Ward, T., & Grant, L. (2011). What makes good teachers good? A cross-case analysis of the connection between teacher effectiveness and pupil achievement. Journal of Teacher Education, 62, 339-347. Turner, F. (2012). Using the knowledge quartet to develop mathematics content knowledge: The role of reflection on professional development. Research in Mathematics Education, 14(3), 253-271. Turner, F., & Rowland, T. (2011). The knowledge quartet as an organizing framework for developing and deepening teachers' mathematics knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.195-212). New York: Springer. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469-484. Wiersma, W. (2000). Research Methods in Education: An introduction (7th ed.). Allyn & Bacon. Williams, P. (2008). Independent review of mathematics teaching in early years settings and primary schools. London: HMSO. Yin, R. K. (2003). Case study research: Design and Methods (3rd ed.). Thousand Oaks, California: Sage Publications.
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