Knowing and Teaching Elementary Mathematics Teachers' Understanding of Fundamental Mathematics in China and the United States
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...Several issues about our proposed categories are worth addressing—their relationship to pedagogical content knowledge, the special nature of specialized content knowledge, our use of teaching as a basis for defining the domains, and problems with the categories that need to be addressed. From our definitions and examples it should be evident that this work may be understood as elaborating on, not replacing, the construct of pedagogical content knowledge. For instance, the last two domains—knowledge of content and students and knowledge of content and teaching—coincide with the two central dimensions of pedagogical content knowledge identified by Shulman (1986): “the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons” and “the ways of representing and formulating the subject that make it comprehensible to others” (p....
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...Most people would agree that an understanding of content matters for teaching. Yet, what constitutes understanding of the content is only loosely defined. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. Lee Shulman (1986) and his colleagues...
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...As a concept, pedagogical content knowledge, with its focus on representations and conceptions/misconceptions, broadened ideas about how knowledge might matter to teaching, suggesting that it is not only knowledge of content, on the one hand, and knowledge of pedagogy, on the other hand, but also a kind of amalgam of knowledge of content and pedagogy that is central to the knowledge needed for teaching. In Shulman’s (1987) words, “Pedagogical content knowledge is the category most likely to distinguish the understanding of the content specialist from the pedagogue” (p....
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...There was immediate and widespread interest in the ideas presented by Shulman and his colleagues. In the two decades since these ideas were first presented, Shulman’s presidential address (1986) and the related Harvard Education Review article (1987) have been cited in more than 1,200 refereed journal articles....
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...Several issues about our proposed categories are worth addressing—their relationship to pedagogical content knowledge, the special nature of specialized content knowledge, our use of teaching as a basis for defining the domains, and problems with the categories that need to be addressed. From our definitions and examples it should be evident that this work may be understood as elaborating on, not replacing, the construct of pedagogical content knowledge. For instance, the last two domains—knowledge of content and students and knowledge of content and teaching—coincide with the two central dimensions of pedagogical content knowledge identified by Shulman (1986): “the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons” and “the ways of representing and formulating the subject that make it comprehensible to others” (p. 9). However, we also see our work as developing in more detail the fundamentals of subject matter knowledge for teaching by establishing a practice-based conceptualization of it, by elaborating subdomains, and by measuring and validating knowledge of those domains. We have been most struck by the relatively uncharted arena of mathematical knowledge necessary for teaching the subject that is not intertwined with knowledge of pedagogy, students, curriculum, or other noncontent domains. What distinguishes this sort of mathematical knowledge from other knowledge of mathematics is that it is subject matter knowledge needed by teachers for specific tasks of teaching, such as those in Figure 3, but still clearly subject matter knowledge. These tasks of teaching depend on mathematical knowledge, and, significantly, they have aspects that do not depend on knowledge of students or of teaching. These tasks require knowing how knowledge is generated and structured in the discipline and how such considerations matter in teaching, such as extending procedures and concepts of whole-number computation to the context of rational numbers in ways that preserve properties and meaning. These tasks also require a host of other mathematical knowledge and skills—knowledge and skills not typically taught to teachers in the course of their formal mathematics preparation. Where, for example, do teachers develop explicit and fluent use of mathematical notation? Where do they learn to inspect definitions and to establish the equivalence of alternative definitions for a given concept? Where do they learn definitions for fractions and compare their utility? Where do they learn what constitutes a good mathematical explanation? Do they learn why 1 is not considered prime or how and why the long division algorithm works? Teachers must know these sorts of things and engage in these mathematical practices themselves when teaching. Explicit knowledge and skills in these areas are vital for teaching. To represent our current hypotheses, we propose a diagram as a refinement to Shulman’s categories. Figure 5 shows the correspondence between our current map of the domain of content knowledge for teaching and two of Shulman’s (1986) initial categories: subject matter knowledge and pedagogical content knowledge....
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