One level of disciplinary content knowledge is teachers’ own understanding of the content they are expected to teach at a particular grade level. The nature and scope of content teachers are expected to teach throughout the K–12 curriculum has changed substantially over the last 15 years, especially evident in the “standards movement” at the national and state levels. Developers of instructional materials have responded to the changing content in national and state standards by adding new topics or moving topics from one grade level to another. The upshot is that teachers may be unfamiliar with content ideas they are required to teach at their own grade level. The argument for teachers to develop student-level knowledge of disciplinary content emphasizes the perhaps obvious importance of teachers understanding the content they are expected to teach, at least at the same depth students are expected to attain at that grade.
The teacher content knowledge literature frequently acknowledges the necessity of teachers having grade-level content knowledge (e.g., Ferrini-Mundy et al., 2005; Carlsen, 1999), but this level of knowledge has not typically been the direct focus of scholarly work because its necessity is generally taken for granted. In practice, given the limited time and resources that can be devoted to deepening teachers’ content knowledge, many efforts may not be able to go much beyond addressing grade-level content. Additionally, key decision-makers within a professional development effort may differ in their views about teacher content knowledge, sometimes even without realizing it. As a result, they may negotiate to focus primary attention on the grade-level content knowledge that fits across their views. Even if it represents a kind of compromise, attention to student-level disciplinary content knowledge is evident in many situations and deserves separate treatment when considering the content knowledge teachers need for effective teaching.
- An illustration of deepening knowledge of student-level mathematics disciplinary content
- An illustration of deepening knowledge of student-level science disciplinary content
For the facets of teacher content knowledge bibliography, click here. [PDF 17K]
Knowledge of advanced disciplinary content
A common view is that teachers’ mathematics and science content knowledge must extend well beyond the content that a teachers’ students are expected to know. In this view, the content knowledge of key importance is understanding the fundamental ideas of the mathematics and science disciplines (Askey, 1999; Cuoco, 2001; Tracy & Walsh, 2004; Wu, 1997). This view on teacher content knowledge is closely associated with the position that K–12 education should establish the foundation for students to develop both knowledge and appreciation of the major concepts and unifying ideas in mathematics and science. The Conference Board of the Mathematical Sciences (2001) recommends that teacher preparation develop in teachers, “a thorough mastery of the mathematics in several grades beyond that which they expect to teach.” Similarly, in its position statement on science teacher preparation, the National Science Teachers Association (2005) strongly recommends that programs enable prospective science teachers to, “Develop robust science knowledge and skills beyond the depth and breadth needed for teaching a curriculum based on the National Science Education Standards at the grade levels they are preparing to teach.”
A missing element in the preparation of many teachers is the opportunity to study the mathematics and science content of the K–12 curriculum beyond their own experience in K–12 schools. Discussing the preparation of secondary mathematics teachers, Wu (1999, p. 8-9) calls for courses “which do not stray far from the high school curriculum” in order to “revisit all the standard topics in high school from an advanced standpoint.” This advanced standpoint includes attending to the historical background of major ideas, inter-connections among them, and complete and rigorous proofs in mathematics or lines of evidence in science. The essential argument is that teacher learning of content that is more advanced than what students are expected to learn should include a clear focus on thoroughly understanding the content of K–12 mathematics and science, rather than only looking forward to the study of more advanced topics. Similarly, Usiskin (2001) recommends that teacher education should attend to advanced topics that will help teachers look back at the K–12 content they have studied, and new topics they will teach, with increased understanding, rather than addressing only topics that set a foundation for graduate-level study. Cuoco (2001, p. 170) recommends a comparable idea, but approaches it from the opposite direction, suggesting that teachers should have experiences “‘mining’ the topics they teach for substantial mathematics” in order to develop the knowledge and disposition to understand “the advanced mathematics that ties them together.”
Wu (1999) further recommends that teachers develop a viewpoint of understanding advanced topics which build on topics studied in K–12 mathematics. Ferrini-Mundy and colleagues (2005, p. 28) describe the need for teachers to understand content trajectories in mathematics, which they define as “understanding both the origins and extensions of core concepts and procedures – knowing the basis for ideas in the domain, and understanding how those ideas grow and become more abstract or elaborated.” Although written in the context of mathematics, these needs appear equally applicable in science.
- An illustration of deepening knowledge of advanced mathematics disciplinary content
- An illustration of deepening knowledge of advanced science disciplinary content
For the facets of teacher content knowledge bibliography, click here. [PDF 17K]
Knowledge of profound disciplinary content
The term “profound knowledge” comes from Liping Ma’s (1999) work in which she defined “profound understanding of fundamental mathematics” as “understanding of the mathematical terrain that is deep, broad, and thorough.” Those who stress the importance of profound knowledge are primarily concerned with content issues that arise in teaching practice, arguing that the instructional decision-making teachers do in planning, carrying out, and reflecting on lessons depends on their ability to use mathematical or scientific knowledge. When teachers choose tasks to assign, ask questions of students, interpret students’ responses, and assess students’ understanding, they employ content knowledge differently than academic mathematicians and scientists, or those working in applied fields, use their content knowledge. Ma (1999) described how teachers organize “knowledge packages” of closely related ideas that they use to think about instruction and student learning. These knowledge packages consist of “decompressed” knowledge of content, breaking down topics into very specific, connected, key understandings that can guide interpretations of student thinking and instructional decisions.
Understanding an idea with depth includes connecting it to more conceptually-powerful ideas which form the foundation not only for the idea at hand, but for many other ideas as well; these foundational ideas form the substantive structure of the discipline itself. Ball (1989) suggested that deep understanding provides a basis for (1) establishing the correctness of ideas, (2) giving meaning to ideas, and (3) connecting ideas. Similarly, when discussing the preparation of physics teachers, McDermott, Heron, and Shaffer (2005) wrote:
Teachers should be given the time and guidance necessary to develop concepts in depth and to construct a coherent conceptual framework. They need to be able to formulate and apply operational definitions so that they can recognize precisely and unambiguously how concepts differ from one another and how they are related. (p. 20)
The Conference Board of the Mathematical Sciences (2001) recommended a focus on deep understanding for the preparation of teachers, particularly the “need to understand the fundamental principles that underlie school mathematics,” further recommending that “[a]ttention to the broad and flexible applicability of basic ideas is preferable to superficial coverage of many topics.” Kennedy (1997) identified the knowledge to distinguish these central ideas in the discipline from minutiae as a central aspect of conceptual knowledge of a discipline. In science, the physics education community has led the field in designing courses for prospective teachers consistent with these recommendations, including Physics by Inquiry (McDermott, 1996) and Physics for Elementary Teachers (Goldberg, 2006). In contrast to traditional survey courses, these physics courses focus on a small number of topics in order to develop deep understanding of the fundamental ideas within those topics.
Understanding an idea with breadth entails both connecting it to ideas at a similar conceptual level, and identifying examples that illustrate the idea (Ma, 1999). Kennedy (1997) and McDermott, Heron, and Shaffer (2005) similarly describe conceptual knowledge of an idea as including both an elaborated understanding of the idea itself, and a sense of its relationships to other ideas. Understanding relationships among ideas entails knowledge such as how the same idea can underlie different procedures or can be represented in different ways for different purposes.
Understanding ideas thoroughly is to weave them together in ways that facilitate navigation from one idea to another, bringing coherence to what otherwise might appear to be an unrelated set of ideas (Ma, 1999). The Conference Board of the Mathematical Sciences (2001) recommends that teachers “learn mathematics in a coherent fashion that emphasizes the interconnections among theory, procedures, and applications.” Such coherent knowledge provides pathways to and from key foundational ideas, and across topics.
- An illustration of deepening knowledge of profound mathematics disciplinary content
- An illustration of deepening knowledge of profound science disciplinary content
For the facets of teacher content knowledge bibliography, click here. [PDF 17K]