Pedagogical Content Knowledge: Knowledge of Student Thinking
Teachers clearly need to understand mathematics/science content in order to teach it. However, the fact that not everyone who has deep content knowledge is an effective teacher makes it equally clear that content knowledge alone is not sufficient. The “additional” understandings that teachers need are typically referred to as “pedagogical content knowledge,” a term initially suggested by Shulman (1986).
A key area of pedagogical content knowledge is knowledge of students’ thinking in mathematics and science. Three general and related aspects of knowledge of students’ thinking have been identified as important to the work of teaching science and mathematics. First, teachers’ knowledge might include understanding which ideas are prerequisites or foundations for more sophisticated understandings (e.g., AAAS, 2001). Second, teachers might know the ways that students typically think about ideas, ideally from research that has identified informal or intuitive ways that students commonly approach problems involving specific content ideas. For some ideas, students’ informal or intuitive thinking may be very close to correct understandings; for other ideas, students’ prior experiences may result in initial conceptions that are counter to established disciplinary understandings. Third, teachers might understand the cognitive development of ideas (e.g., Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Catley, Lehrer, & Reiser, 2004; Lehrer & Schauble, 2000). Knowledge of cognitive development of ideas offers teachers frameworks for guiding students’ growing understandings of specific concepts.
- An illustration of deepening knowledge of student thinking in mathematics
- An illustration of deepening knowledge of student thinking in science
For the facets of teacher content knowledge bibliography, click here. [PDF 17K]
Pedagogical Content Knowledge: Knowledge of Implications for Instruction
As research has identified both the importance of student prior conceptions, and the ways students are likely to think about particular concepts, professional development has increasingly focused on helping teachers understand student thinking. But just as content knowledge is necessary but not sufficient for effective teaching, understanding of student thinking addresses only part of the challenge. Teachers need also to understand how particular instructional experiences can build on students’ thinking to provide opportunities for them to learn specific mathematics or science ideas.
Understanding representations of mathematics or science concepts and how they might be used in instruction is an example of this type of knowledge (Ferrini-Mundy et al., 2005; Magnusson, Krajcik, & Borko, 1999; Usiskin, 2001). Teachers’ knowledge of representations includes understanding both the conceptual integrity of each representation and its comprehensibility to an audience of learners (Kennedy, 1997). Knowledge of instructional activities is another example in this area. Content-specific knowledge of activities attends to what aspects of the content being addressed are highlighted in a given activity, and which might be obscured (Magnusson, Krajcik, & Borko, 1999). Such knowledge supports understanding of what aspects of a targeted concept can and cannot be addressed well with a particular instructional activity.
- An illustration of deepening knowledge of implications for mathematics instruction
- An illustration of deepening knowledge of implications for science instruction
For the facets of teacher content knowledge bibliography, click here. [PDF 17K]
Pedagogical Content Knowledge: Knowledge of Curriculum
State and national content standards describe the goals of mathematics and science education, sometimes for individual grades or courses, and other times for grade bands such as 6-8. Although targeting the same sets of standards, instructional materials developers make different decisions about the relative emphasis to devote to particular topics, the sequence with which to address topics, and how to engage students with the mathematics/science ideas. Instructional materials selection committees, or sometimes individual teachers, then decide which materials are likely to help their students achieve the designated goals.
Ideally, developers of student instructional materials make use of what is known about student thinking, instructional experiences, and applications for teaching specific mathematics/science concepts. Deepening teachers’ pedagogical content knowledge when they are learning to use specific instructional materials involves helping them understand how the materials organize the mathematics or science content for classroom teaching. This knowledge includes understanding how content ideas are sequenced (which ideas are introduced earlier that are used as the foundation for learning other ideas later), how connections are made (which ideas are tied together and in what ways), and how the various activities and their sequencing in the instructional materials are intended to contribute to mathematics/science learning goals. Especially if the instructional materials are not explicit about the “storyline” of the various activities, teachers need opportunities to consider how and why the materials are designed the way they are.
Instructional materials generally rely on applications and context to structure learning experiences for students. Ferrini-Mundy and colleagues (2005) describe an area of teacher content knowledge labeled “applications and context,” suggesting that teacher knowledge includes understanding situations and circumstances in which particular content ideas arise. These situations could be purely within the discipline (e.g., in science, while studying catalysis in a chemistry class, looking at how enzymes function as catalysts in cells; or in mathematics, examining the ratios and proportional relationships that arise in similar geometric figures) or could come from applications outside of the discipline (e.g., in science, examining the construction of different kinds of bridges in the study of balanced forces; or in mathematics, creating mathematical models from data to optimize pricing for a cookie sale). This type of knowledge is considered a part of pedagogical content knowledge because teachers’ knowledge should take into account how instructional activities that rely on applications will help students learn important content ideas and generalize them from the specific context in which they were learned, not just how they can demonstrate that content ideas are useful in the “real world” (NCTM, 2000). As is the case with representations, models, and analogies, teachers need to be able to evaluate applications for their affordances and limitations.
- An illustration of deepening knowledge of mathematics curriculum.
- An illustration of deepening knowledge of science curriculum.
For the facets of teacher content knowledge bibliography, click here. [PDF 17K]