Research in Brief - February 2008 - Volume 108 (2)

Views of Content and Pedagogical Knowledges for Teaching Mathematics

Diana Piccolo

The ability to teach mathematics content is influenced by general pedagogy, pedagogical content knowledge and mathematical content knowledge. Shulman (1987) stated several categories of teacher knowledge, including content knowledge, general pedagogical knowledge, and pedagogical content knowledge. He described content knowledge as the knowledge, understanding, skills, and dispositions that students learn. General pedagogy knowledge is described as broad teaching strategies, such as classroom management and organization and pedagogical content knowledge as the “blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction” (p. 8). Mathematics education researchers have expounded on several of Shulman's ideas and focused on specific pedagogies and content knowledge needed for teaching mathematics.

Research is replete with strategies and methods pertaining to pedagogical content knowledge and content knowledge for teaching mathematics (Ball, Hill, & Bass, 2004; Leinhardt & Smith, 1985; Borko & Livingston, 1989; Tirosh, 2000; Wilson, Floden & Ferrini-Mundy, 2002). The National Council of Teachers of Mathematics (NCTM, 2000) outlined in the Teaching Principle, that development and utilization of pedagogical content knowledge required teachers to continually increase their knowledge about mathematical content and pedagogy. Ball and Bass (2000) discuss the importance of interweaving content and pedagogy, despite the gap that exists between how to effectively organize and develop teachers' knowledge in these two areas.

General pedagogy skills used in planning and structuring a mathematics lesson include managing the classroom, organizing activities, lesson planning, motivating students, and assessing mathematics content (Fennema & Franke, 1992; Interstate New Teacher Assessment and Support Consortium, 1992). The Interstate New Teacher Assessment and Support Consortium (INTASC) Standards were developed to reflect what beginning teachers needed to know in teaching. They included both general and content pedagogy skills, such as planning lessons and activities that meet the needs of varied learners and understanding subject matter concepts. Leinhardt and Smith (1985) described this knowledge as “lesson structure knowledge” that is separate from content knowledge. Components of this knowledge included skills needed to plan a lesson and transitions from one part of the lesson to another. If general pedagogy skills are planning and preparation of mathematics teaching, then pedagogical content knowledge is the inclusion of that pedagogy into the actual teaching of mathematical content. Even (1993) stated that pedagogical decisions, such as designing and planning activities, are based partially on content knowledge.

Multiple views have been presented on how mathematical content knowledge should be taught in order to ensure quality instruction in the classroom (Ball, Hill & Bass, 2005; Eisenhart, Borko, Underhill, Brown, Jones, & Agard, 1993; Leindhardt & Smith, 1985; Ma, 1999; NCTM, 2000; Schoenfeld, 2007; Sherin, 2002). Ma (1999) found that mathematical content should be taught as a developmental-coherent whole in which teachers needed a deep, vast, and thorough understanding of that knowledge. Leinhardt and Smith (1985) showed that subject matter knowledge needed in mathematics instruction included “concepts, algorithmic operations, the connections among different algorithmic procedures, the subset of the number system being drawn upon, the understanding of types of student errors, and curriculum presentation” (p. 247). The understanding of this content knowledge was depicted through a combination of semantic nets, planning nets, and flow charts. Another view of mathematics content knowledge is ‘common’ and ‘specialized’ knowledge (Ball, Bass, & Hill, 2004). ‘Common” knowledge is the essential algorithmic and procedural knowledge necessary for computing mathematical solutions. ‘Specialized’ knowledge is the skills, procedures, and competencies needed for teaching mathematics to students. Mathematics teachers need both types of knowledge to teach effectively so they can “unpack” mathematical ideas and procedures for their students (Ball, 2001).

Pedagogical content knowledge (PCK) is a nexus of both content and pedagogy into a form of knowledge that comprises representations of analogies, illustrations, examples, explanations, and demonstrations so that the content is understandable to students. Ball and Bass (2000), stated that pedagogical content knowledge are “representations of particular topics and how students tend to use them . . . it is the close interweaving of subject matter and pedagogy in teaching” (p. 87). The development of teachers' pedagogical content knowledge is influenced by several factors, beginning with content knowledge learned during teacher preparation program and initial teaching experiences (Capraro, Capraro, Parker, Kulm, & Raulerson, 2005). Students need a combination of content knowledge, teaching for understanding, and curriculum to gain a more of a conceptual understanding of mathematics (An, Kulm, & Wu, 2004).

The framework for effective mathematics teaching comprises these three components: general pedagogy, content knowledge for teaching mathematics, and pedagogical content knowledge. Nevertheless, all pedagogical competencies and understandings are based upon having a deep, vast, and thorough understanding of mathematical content. The inter-relatedness of these components and how they influence mathematics teaching and learning makes this a multifaceted topic for contemporary mathematics educators and researchers.

References

An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7, 145-172.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83-104). Westport, CT: Ablex .

Ball, D. L. (2001, February). Developing a mathematically proficient public: What are the problems, what do we know about them and what would it take to solve them? Paper presented for the Aspen Institute congressional conference on Promoting Excellence in the New Economy: The Challenges to National Policy, St. Petersburg, FL.

Ball, D. L., Hill, H., & Bass, H. (2004, July). Knowing and using mathematical knowledge in teaching: Learning what matters. Paper presented at the Southern African Association for Research in Mathematics, Science, and Technology Education, Capetown, South Africa.

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Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding. Journal for Research in Mathematics Education, 24, 8-40.

Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24, 94-116.

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Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22.

Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31, 5-25.

Wilson, S., Floden, R., & Ferrini-Mundy, J. (2002). Teacher preparation research: An insider's view from the outside. Journal of Teacher Education, 53, 190-204.