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Article Excerpt
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 proposed a special domain of teacher knowledge that they termed pedagogical content knowledge. What provoked broad interest was the suggestion that there is content knowledge unique to teaching--a kind of subject-matter-specific professional knowledge. The continuing appeal of the notion of pedagogical content knowledge is that it bridges content knowledge and the practice of teaching. However, after two decades of work, this bridge between knowledge and practice was still inadequately understood and the coherent theoretical framework Shulman (1986, p. 9) called for remained underdeveloped. This article builds on the promise of pedagogical content knowledge, reporting new progress on the nature of content knowledge for teaching.
Although the term pedagogical content knowledge is widely used, its potential has been only thinly developed. Many seem to assume that its nature and content are obvious. Yet what is meant by pedagogical content knowledge is underspecified. The term has lacked definition and empirical foundation, limiting its usefulness.
Throughout the past 20 years, for example, researchers have used pedagogical content knowledge to refer to a wide range of aspects of subject matter knowledge and the teaching of subject matter and, indeed, have used it differently across--and even within--subject areas. Besides differences in the breadth of what the term includes, there have been significant differences in how the term is used to relate content knowledge to the practice of teaching. Frequent, for example, are broad claims about what teachers need to know. Such statements are often more normative than empirical. Only a few studies have tested whether there are, indeed, distinct bodies of identifiable content knowledge that matter for teaching.
Without this empirical testing, the ideas are bound to play a limited role in improving teaching and learning--in revamping the curriculum for teacher content preparation, in informing policies about certification and professional development, and in furthering our understanding of the relationships among teacher knowledge, teaching, and student learning. Without this empirical testing, the ideas remain, as they were 20 years ago, promising hypotheses based on logical and ad hoc arguments about the content believed to be necessary for teachers.
For the last 15 years, the work of the Mathematics Teaching and Learning to Teach Project and of the Learning Mathematics for Teaching Project has focused both on the teaching of mathematics and on the mathematics used in teaching. Although the context of our work has been mathematics, we have sought to contribute to a broader discussion by researchers in different school subjects. To consider the knowledge that teaching entails, we began by investigating what teaching itself demands. Instead of reasoning from the school curriculum to a list of topics teachers must know, we developed an empirical approach to understanding the content knowledge needed for teaching. The first project focused on the work teachers do in teaching mathematics. The authors and their colleagues used studies of teaching practice to analyze the mathematical demands of teaching and, based on these analyses, developed a set of testable hypotheses about the nature of mathematical knowledge for teaching. In a related line of work, the second project developed survey measures of content knowledge for teaching mathematics. The measures provided a way to investigate the nature, the role, and the importance of different types of mathematical knowledge for teaching.
In particular, these studies have led us to hypothesize some refinements to the popular concept of pedagogical content knowledge and to the broader concept of content knowledge for teaching. In this article, we focus on the work of teaching in order to frame our conceptualization of the mathematical knowledge and skill needed by teachers. We identify and define two empirically detectable subdomains of pedagogical content knowledge. In addition, and to our surprise, we have begun to uncover and articulate a less recognized domain of content knowledge for teaching that is not contained in pedagogical content knowledge, but yet--we hypothesize--is essential to effective teaching. We refer to this as specialized content knowledge. These possible refinements to the map of teacher content knowledge are the subject of this article.
Because the work of Shulman and his colleagues is foundational, we begin by reviewing the problem they framed, the progress they made, and the questions that remained unanswered. We use this discussion to clarify the problems of definition, empirical basis, and practical utility that our work addresses. We then turn to mathematics in particular, describe work on the problem of identifying mathematical knowledge for teaching, and report on refinements to the categories of mathematical knowledge for teaching. The article concludes with an appraisal of next steps in developing a useful theory of content knowledge for teaching.
Content Knowledge and Its Role in Defining Teaching as a Profession
A central contribution of Shulman and his colleagues was to reframe the study of teacher knowledge in ways that attend to the role of content in teaching. This was a radical departure from research of the day, which focused almost exclusively on general aspects of teaching. Subject matter was little more than context. Although earlier studies were conducted in classrooms where mathematics, reading, or other subjects were taught, attention to the subject itself and to the role it played in teaching or teacher thinking was less prominent. In fact, so little attention was devoted to examining content and its role in instruction that Shulman dubbed this the "missing paradigm" in research on teaching and teacher knowledge (1986).
A second contribution of Shulman and his colleagues was to represent content understanding as a special kind of technical knowledge key to the profession of teaching. In the late 1980s, they conducted case studies of beginning high school teachers as part of their research in the Knowledge Growth in Teaching project. Participants were recent graduates with strong subject matter preparation in mathematics, science, English literature, and history. By examining these novices in the process of learning to teach, the group sought to investigate how strong subject matter preparation translated into the knowledge needed for teaching that subject. Deliberately working across subjects provided a comparative basis for examining more general characteristics of the knowledge that the teachers used in their practice.
A closely related purpose was to draw from these categories of teacher knowledge to inform the development of a National Board system for the certification of teachers that would "focus upon the teacher's ability to reason about teaching and to teach specific topics, and to base his or her actions on premises that can bear the scrutiny of the professional community" (Shulman, 1987, p. 20). Attention to certification was deliberately geared toward informing debates about what constitutes professional expertise and what such expertise implies for teacher preparation and for policy decisions. In particular, Shulman was concerned with prevailing conceptions of teacher competency, which focused on generic teaching behaviors. He argued that "the currently incomplete and trivial definitions of teaching held by the policy community comprise a far greater danger to good education than does a more serious attempt to formulate the knowledge base" (Shulman, 1987, p. 20). Implicit in such comments is the argument that high-quality instruction requires a sophisticated, professional knowledge that goes beyond simple rules such as how long to wait for students to respond.
Figure 1 Shulman's Major Categories of Teacher Knowledge * General pedagogical knowledge, with special reference to those broad principles and strategies of classroom management and organization that appear to transcend subject matter * Knowledge of learners and their characteristics * Knowledge of educational contexts, ranging from workings of the group or classroom, the governance and financing of school districts, to the character of communities and cultures * Knowledge of educational ends, purposes, and values, and their philosophical and historical grounds * Content knowledge * Curriculum knowledge, with particular grasp of the materials and programs that serve as "tools of the trade" for teachers * Pedagogical content knowledge, that special amalgam of content and pedagogy that is uniquely the province of teachers, their own special form of professional understanding (Shulman, 1987, p. 8)
To characterize professional knowledge for teaching, Shulman and his colleagues developed typologies. Although the specific boundaries and names of categories varied across publications, one of the more complete articulations is reproduced in Figure 1.
These categories were intended to highlight the important role of content knowledge and to situate content-based knowledge in the larger landscape of professional knowledge for teaching. The first four categories address general dimensions of teacher knowledge that were the mainstay of teacher education programs at the time. They were not the main focus of Shulman's work. Instead, they functioned as placeholders in a broader conception of teacher knowledge that emphasized content knowledge. At the same time, however, Shulman made clear that these general categories were crucial and that an emphasis placed on content dimensions of teacher knowledge was not intended to minimize the importance of pedagogical understanding and skill: Shulman (1986) argued that "mere content knowledge is likely to be as useless pedagogically as content-free skill" (p. 8).
The remaining three categories define content-specific dimensions and together comprise what Shulman referred to as the missing paradigm in research on teaching--"a blind spot with respect to content that characterizes most research on teaching, and as a consequence, most of our state-level programs of teacher evaluation and teacher certification" (1986, pp. 7-8). The first, content knowledge, includes knowledge of the subject and its organizing structures (see also Grossman, Wilson, & Shulman, 1989; Wilson, Shulman, & Richert, 1987). Drawing on Schwab (1961/1978), Shulman (1986) argued that knowing a subject for teaching requires more than knowing its facts and concepts. Teachers must also understand the organizing principles and structures and the rules for establishing what is legitimate to do and say in a field. The teacher need not only understand that something is so; the teacher must further understand why it is so, on what grounds its warrant can be asserted, and under what circumstances our belief in its justification can be weakened or denied. Moreover, we expect the teacher to understand why a particular topic is particularly central to a discipline whereas another may be somewhat peripheral. (p. 9)
The second category, curricular knowledge, is "represented by the full range of programs designed for the teaching of particular subjects and topics at a given level, the variety of instructional materials available in relation to those programs, and the set of characteristics that serve as both the indications and contraindications for the use of particular curriculum or program materials in particular circumstances" (p. 10). In addition, Shulman pointed to two other dimensions of curricular knowledge that are important for teaching, aspects that he labeled lateral curriculum knowledge and vertical curriculum knowledge. Lateral knowledge relates knowledge of the curriculum being taught to the curriculum
that students are learning in other classes (in other subject areas). Vertical knowledge includes "familiarity with the topics and issues that have been and will be taught in the same subject area during the preceding and later years in school, and the materials that embody them" (Shulman, 1986, p. 10).
The last, and arguably most influential, of the three content-related categories was the new concept of pedagogical content knowledge. Shulman (1986) defined pedagogical content knowledge as comprising:
The most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations--in a word, the most useful ways of representing and formulating the subject that make it comprehensible to others.... Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: 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. (p. 9)
The claim for pedagogical content knowledge was founded on observations that effective teachers in the Knowledge Growth in Teaching study represented key ideas using metaphors, diagrams, and explanations that were at once attuned to students' learning and to the integrity of the subject matter (see also Carlsen, 1988; Grossman, 1990; Marks, 1990; Wilson, 1988; Wilson et al., 1987; Wineburg, 1990). Some representations are especially powerful; others, although technically correct, do not open the ideas effectively to learners.
A second important idea is that representations of the subject are informed by content-specific knowledge of student conceptions. A focus on conceptions, and in many cases a particular interest in student misconceptions, acknowledges that accounting for how students understand a content domain is a key feature of the work of teaching that content. Grossman (1990) pointed out that these ideas
are inherent in Dewey's admonition that teachers must learn to "psychologize" their subject matter for teaching, to rethink disciplinary topics to make them more accessible to students.... Teachers must draw upon both their knowledge of subject matter to select...
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