A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: toward a model of effective practice

Abstract This study contributes to an emerging body of research on how to develop prospective teachers’ pedagogical content knowledge (PCK) in teacher education. The main focus is on the knowledge transformation process and on the cognitive strategy used to shift prospective teachers’ instructional explanations within the domain of integer subtraction from an instrumental to a relational understanding of mathematics. The strategy rests on the observation that transformation is not a unidirectional process from subject-matter knowledge to pedagogical content knowledge as the literature suggests. The adequacy of prospective teachers’ instructional explanations, and the depth of their subject-matter understanding, is analyzed using a modification of Perkins and Simmons’ levels of subject-matter understanding framework. The study concludes with observations about the interface between student achievement and prospective teachers’ subject-matter and pedagogical content knowledge.

[1]  L. Hedges,et al.  Book Review: Meta Analysis: A Review and A New View , 1986 .

[2]  Edith M. Guyton,et al.  Handbook of Research on Teacher Education: A Project of the Association of Teacher Educators , 1996 .

[3]  J. M. Scholz.  Professional Development for Mid-level Mathematics. , 1995 .

[4]  F. B. Brawer.  The Liberal Arts , 1999 .

[5]  D. Tirosh,et al.  Evoking Cognitive Conflict to Explore Preservice Teachers' Thinking about Division. , 1990 .

[6]  G. Mcdiarmid,et al.  Challenging Prospective Teachers' Beliefs During Early Field Experience: A Quixotic Undertaking? , 1990 .

[7]  The Education of American Teachers , 1963 .

[8]  Christine L. Ebert,et al.  An Assessment of Prospective Secondary Teachers' Pedagogical Content Knowledge about Functions and Graphs. , 1993 .

[9]  J. Kennedy.  Classroom Explanatory Discourse: A Look at How Teachers Explain Things to Their Students. , 1996 .

[10]  Susan S. Stodolsky,et al.  Telling math: Origins of math aversion and anxiety. , 1985 .

[11]  David Niemi,et al.  Assessing Conceptual Understanding in Mathematics: Representations, Problem Solutions, Justifications, and Explanations , 1996 .

[12]  Robert J. Stevens,et al.  Advances in research on teaching , 1981 .

[13]  Mary M. Kennedy,et al.  Knowledge and Teaching , 2002 .

[14]  Hunter Mcewan,et al.  The Pedagogic Nature of Subject Matter Knowledge , 1991 .

[15]  J. R. Edgington,et al.  What Constitutes a Scientific Explanation , 1997 .

[16]  A. Thompson.  Teachers' beliefs and conceptions: A synthesis of the research. , 1992 .

[17]  David F. Treagust,et al.  Developing preservice teachers' pedagogical reasoning ability , 1995 .

[18]  M. Wittrock.  Handbook of research on teaching , 1986 .

[19]  R. Marks.  Pedagogical Content Knowledge: From a Mathematical Case to a Modified Conception , 1990 .

[20]  Donald A. Schön,et al.  Theory in Practice: Increasing Professional Effectiveness , 1974 .

[21]  Martha Stone Wiske,et al.  Teaching for understanding : linking research with practice , 1998 .

[22]  R. Rorty.  Objectivity, Relativism, and Truth: Philosophical Papers , 1990 .

[23]  D. Bob Gowin,et al.  Science, Curriculum, and Liberal Education: Selected Essays , 1978 .

[24]  Martha Stone Wiske,et al.  Teaching for Understanding. Linking Research with Practice. The Jossey-Bass Education Series. , 1998 .

[25]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[26]  David N. Perkins,et al.  Patterns of Misunderstanding: An Integrative Model for Science, Math, and Programming , 1988 .

[27]  L. Shulman.  Those Who Understand: Knowledge Growth in Teaching , 1986 .

[28]  Virginia A. Atwood,et al.  Preservice Elementary Teachers' Conceptions of the Causes of Seasons. , 1996 .

[29]  Frank B. Murray,et al.  The Teacher Educator's Handbook: Building a Knowledge Base for the Preparation of Teachers , 1995 .

[30]  G. Roland,et al.  Council of Teachers of Mathematics , 1999 .

[31]  D. Ball.  Prospective Elementary and Secondary Teachers' Understanding of Division. , 1990 .

[32]  L. S. Schulman.  Paradigms and research programs in the study of teaching , 1986 .

[33]  R. Fuller.  Elementary Teachers' Pedagogical Content Knowledge of Mathematics. , 1996 .

[34]  E. Boardman.  Teacher Education for the 21st Century , 1992 .

[35]  A. Su,et al.  The National Council of Teachers of Mathematics , 1932, The Mathematical Gazette.

[36]  N. Brickhouse,et al.  Teachers' Beliefs About the Nature of Science and Their Relationship to Classroom Practice , 1990 .

[37]  D. Ball.  The Subject Matter Preparation of Prospective Mathematics Teachers: Challenging the Myths. , 1988 .

[38]  Shirley Magnusson.  Teaching Complex Subject Matter in Science: Insights from an Analysis of Pedagogical Content Knowledge. , 1994 .

[39]  L. Shulman.  Knowledge and Teaching: Foundations of the New Reform , 1987 .

[40]  Andrew Carpenter,et al.  The Courage to Teach: Exploring the Inner Landscape of a Teacher's Life , 2007 .

[41]  G. Mcdiarmid.  The liberal arts: Will more result in better subject matter understanding? , 1990 .

[42]  D. Ball.  Unlearning to Teach Mathematics. , 1988 .

[43]  Kathryn F. Cochran,et al.  Pedagogical Content Knowing: An Integrative Model for Teacher Preparation , 1993 .

[44]  P. Grossman.  The Making of a Teacher: Teacher Knowledge and Teacher Education , 1990 .

[45]  Martin A. Simon.  PROSPECTIVE ELEMENTARY TEACHERS' KNOWLEDGE OF DIVISION , 1993 .

[46]  Daniel P. Liston,et al.  Teaching Student Teachers to Reflect , 1987 .

[47]  Deborah Loewenberg Ball,et al.  Breaking with Experience in Learning To Teach Mathematics: The Role of a Preservice Methods Course; Issue Paper 89-10. , 1989 .

[48]  Sam Wineburg,et al.  Peering at History through Different Lenses: The Role of Disciplinary Perspectives in Teaching History , 1988, Teachers College Record: The Voice of Scholarship in Education.

[49]  D. Ball.  Research on Teaching Mathematics: Making Subject Matter Knowledge Part of the Equation. , 1988 .