Middle grades in-service teachers pedagogical content knowledge of student internal representation of equivalent fractions and algebraic expressions



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This study examined teacher pedagogical content knowledge changes through a Middle School Mathematics Program professional development workshop, development of noticing use of student representations, and teacher changes in hypothetical learning trajectories due to noticed aspects of student representation corresponding to the hypothetical learning trajectory model. Using constant comparatives and repertory grid analysis, data was collected in two phases. Phase one, the teacher pre-test, occurred at the beginning of the summer of the 2003 professional development workshop. Phase two, the teacher post-test, occurred at the end of the workshop. Twenty-four teachers supplied data on pre- and post-tests during phases one and two. Eleven teachers were from Texas and 13 from Delaware. Six Texas and eight Delaware teachers worked with the algebraic expression concepts. Five Texas and five Delaware teachers worked with the equivalent fraction concepts. Four mathematics education researchers from Texas, three from Delaware, and two from the American Association for the Advancement of Science participated in facilitating the professional development. The results show that teacher pedagogical content knowledge changes with the help of a professional development partnership. The differences in knowledge can be measured with a hierarchal cluster analysis of the repertory grid by analyzing relationships between constructs and elements. Teacher hypothetical learning trajectories change depending on student representations of what they do and do not know about concepts. The study encourages teachers to use knowledge of students? representation about a concept to determine what to teach next and how the concept should be taught. Teachers should use different types of representations including formal, imagistic, and action representations in teaching mathematical ideas. This will promote student development in all process standards including reasoning and proof, communication, problem solving, and connection. The findings suggest that teacher pedagogical content knowledge can be redefined during professional development partnerships. Furthermore, teachers? knowledge of representation is varied and emphasis on the imagistic representation should be explored further. Finally, professional development models that facilitate how to extract what a student does and does not know based on representation, can be the basis for defining hypothetical learning trajectories.