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Measuring Secondary School Mathematics Teachers’ Knowledge of Mathematics for Teaching: Issues of Conceptualization and Design
Joan Ferrini-Mundy, Sharon Senk, Raven McCrory – USA ICMI Study Group 15 Águas de Lindóia , Brazil – May, 2005 |
Researchers working on the Knowing Mathematics for Teaching Algebra project at Michigan State University hypothesize that Algebra Knowledge for Teaching consists of three constructs called Knowledge of School Algebra, Advanced Mathematical Knowledge and Teaching Knowledge. We focus here on the highlighted dimension of the matrix, Algebra Knowledge for Teaching.
Knowledge of School Algebra:
This category includes knowledge of mathematics in the intended algebra curriculum for middle and high school. This is knowledge we expect students to learn in secondary school algebra in the United States. The algebra standard described in Principles and standards for school mathematics (NCTM, 2000) is an overview of the big ideas in school algebra. More specific grade-level algebra expectations on which we build our conception of Knowledge of School Algebra is provided in standards issued by various states and in textbooks and other instructional materials. Sample Items.
Advanced Mathematical Knowledge:
This category includes other mathematical knowledge, in particular college level mathematics, which gives a teacher perspective on the trajectory and growth of mathematical ideas beyond school algebra. The Mathematical Education of Teachers (CBMS, 2001) gives examples of advanced knowledge, which is both broader and deeper than what is taught in school algebra. Some general areas of mathematics that are presumed to provide breadth and depth of understanding of secondary school algebra are calculus, linear algebra, number theory, abstract algebra, real and complex analysis, and mathematical modeling. According to Usiskin, Peressini, Marchisotto and Stanley (2003) knowing alternate definitions, extensions and generalizations of familiar theorems, and a wide variety of applications of mathematics are also characteristics of an advanced perspective of mathematics.
Teaching Knowledge:
This category includes mathematical knowledge specific to teaching
algebra. It includes such things as what makes a particular concept
difficult to learn and what misconceptions lead to specific
mathematical errors. It also includes mathematics needed to identify
mathematical goals within and across lessons, to understand students’
thinking, to select what to emphasize with curricular trajectories in
mind, and to enact other tasks of teaching. The knowledge referred to
here may fall into the category of pedagogical content knowledge
(Shulman, 1986, 1987) or it may be pure mathematical content applied in
teaching. It may or may not be taught in advanced mathematics courses.
References
Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Providence, RI: American Mathematical Society.
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4 – 14.
Shulman, L. S. (1987). "Knowledge and Teaching: Foundations of the New Reform." Harvard Educational Review 57(1), 1 - 22.
Usiskin, Z., Peressini, A., Marchisotto, E. & Stanley, D. (2003). Mathematics for high school teachers: An advanced perspective. Upper Saddle River, NJ: Prentice Hall.
Sample Items