**EMP **Introduction Documentation

## Mathematical Knowledge for Teaching

Reports and policy recommendations from the Mathematical Association of America, the American Mathematical Society, NCTM, and others (Conference Board of the Mathematical Sciences, 2012; Committee of Science and Mathematics Teacher Preparation, 2001; US DOE, 2008) have acknowledged the role of subject matter knowledge and pedagogical content knowledge in effective mathematics teaching and have targeted both domains as important elements in the preparation of elementary teachers. The *Mathematical Education of Teachers II *report reiterates the importance of content knowledge, “At every grade level—elementary, middle, and high school—there is important mathematics that is both intellectually demanding to learn and widely used. …Teachers need to have more than a student’s understanding of the mathematics in these grades. … For example, an elementary teacher needs to know how the associative, commutative, and distributive properties are used together with place value in algorithms for addition and multiplication of whole numbers, and the significance of these algorithms for decimal arithmetic in later grades.” (CBMS 2012, p. 1-2). Others such as Liping Ma (1999, 2010) describe the need for elementary teacher candidates to develop “profound understanding of fundamental mathematics.”

The mathematical knowledge needed for teaching is more than what is needed by the average adult; it is even different from that needed by mathematicians (Ball, 2003). Yet, subject matter knowledge for teaching is not a watered down version of formal mathematics (Ball & Bass, 2003). In fact, knowledge for teaching includes details that are unnecessary for everyday functioning. For example, people with common knowledge of mathematics can divide accurately and solve division word problems. Teachers with specialized content knowledge possess common knowledge of division as well as the knowledge to explain why division procedures work, how division can be interpreted using equal sharing, measurement and missing factor interpretations, how remainders can be interpreted, and under what circumstances quotients are smaller, equal to, or larger than the dividend. They can generate different types of application problems that are solved using the operation.

Analyses of teachers’ practices reveal that the mathematical demands of teaching are substantial even in daily tasks such as assigning student work, listening to student talk, grading or commenting on student work (Ball, 1999; Ball & Bass, 2000; Ball & Bass, 2003). Not only do teachers need to know how to carry out common procedures, but they also must be able to identify incorrect answers and faulty methods and analyze errors efficiently and fluently, just like mathematicians do in the course of their work. Teachers must be able to make sense of students’ non-standard procedures, even when they have never encountered them before. They have to validate the mathematical soundness of students’ solutions and discern whether a particular approach would work in general. Interpreting student error and evaluating alternative algorithms is not all that teachers do, however. Teaching also involves explaining computational algorithms. Teachers need to be able to provide justification for the steps in algorithms and procedures, meanings for terms, and explanations for concepts. The choice of mathematical representations and examples to illustrate ideas and the sequencing of the examples are also a component of the task of teaching algorithms.

The variety of these many complex tasks has led researchers to posit that teaching requires a specialized form of mathematical knowledge that is intertwined with knowledge of pedagogy, students, curriculum, or other non-content domains. This specialized knowledge is different than other knowledge of mathematics in that it is relevant only in the context of teaching (Hill et al., 2007). Ball and colleagues (2008) elaborate Shulman’s (1986) content knowledge and pedagogical content knowledge into six distinct domains as shown in Figure 1.

This framework divides the fundamentals of mathematical knowledge for teaching into two main categories and six sub-categories. The two main categories are subject matter knowledge and pedagogical content knowledge. The six sub-categories are common content knowledge (CCK), specialized content knowledge (SCK), horizon content knowledge, knowledge of content and teaching (KCT), knowledge of content and students (KCS), and knowledge of content and curriculum (Ball et. al., 2008).

This project will assist pre-service teachers in developing both their subject matter knowledge and their pedagogical content knowledge in ways that can be translated into the work of a teacher; they will discuss and explain mathematical ideas, procedures, terms, and methods and justify and explain their answers using models, equations, metaphors and evidence.

Ball and colleagues define common content knowledge as the knowledge of mathematical concepts and procedures that all educated adults are expected to know. For example, knowing how to determine if a number is divisible by 2, 3 or 5 is considered common content knowledge because it is knowledge that many adults outside of the teaching profession possess – these procedures are not inherently specific to teaching. Specialized content knowledge, on the other hand, is mathematical knowledge that is specific to teaching. This type of knowledge “involves an uncanny kind of unpacking of mathematics that is not needed – or even desirable – in settings other than teaching” (Ball et al., 2008, p. 400). Effectively choosing, making, and using appropriate representations, evaluating the validity of students’ mathematical responses, appreciating the differences between various strategies, and knowing how to explain why something is true require teachers to have specialized knowledge of mathematics. For example, while most adults can determine if a number is divisible by 2 or 5, only teachers need to be aware of why these divisibility rules work and why and how these two rules differ from the rule for divisibility by 3. Teachers need to be able to explain the relationship between our base ten numeration system and division and how these concepts are connected to divisibility rules. Such knowledge is specialized because it represents the work that is particular to the daily work of teachers.

The EMP units focus primarily on developing pre-service teachers’ common content knowledge and specialized content knowledge. While the lessons include problems that attempt to foster pre-service teachers’ pedagogical content knowledge (e.g., problems addressing common student misconceptions and computational errors), such knowledge is a secondary focus of the EMP units. This should not be interpreted to mean that subject-matter knowledge is more important than pedagogical content knowledge. We believe that both types of knowledge are important for teachers. The choice to focus on one type of knowledge over another was due to multiple factors including the design of our teacher preparation program and the courses in which these materials would be used. Following their work with the EMP materials, pre-service teachers enroll in a Methods of Teaching Mathematics course that works to develop their pedagogical content knowledge.

#### References

Ball, D.L. (1999). Crossing boundaries to examine the mathematics entailed in elementary teaching. *Contemporary Mathematics, 243*, 15-36.

Ball, D.L. (Chair, Rand Mathematics Study Panel). (2003). *Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education*. Santa Monica, CA: Rand.

Ball, D.L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmt (Eds.), *Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group*, 3-14. Edmonton, AB: CMESG/GCEDM.

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, Thames, M.H., & Phelps, G. (2008). *Content knowledge for teaching: What makes it special?* Journal of Teacher Education, 59, 389-407.

Committee on Science and Mathematics Teacher Preparation (CSMTP). 2001. *Educating Teachers of Science, Mathematics and Technology: New Practices for the New Millennium.* Washington, DC: National Academy Press.

Conference Board of the Mathematical Sciences. (2012). *The Mathematical Education of Teachers II*, CBMS Issues in Mathematics Education, Volume 17. Providence: American Mathematical Society.

Hill, H.C., Sleep, L., Lewis, J.M., & Ball, D.L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts? In F. Lester (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 111-156). Charlotte, NC: Information Age Publishing.

Ma, L. (2010). *Knowing and teaching elementary mathematics (anniversary edition)*. New York: Routledge.

National Mathematics Advisory Panel. (2008). *Foundations for success: The final report of the National Mathematics Advisory Panel*. Washington, DC: U.S. Department of Education.

Shulman, L.S. (1986). Those who understand. Knowledge growth in teaching. *Educational Researcher, 15*(2), 4-14.