Margaret Mohr, Universi_4 of Kentucky Mathematics Knowledge for Teaching Research indicates that U. S. teachers' mathematical knowledge continues to be weak, and there is an inherent difference between the mathematical knowledge needed to be an effective teacher and that needed by a mathemati- cian. Shulman (1987) defined pedagogical content knowl- edge as the ability of the teacher to transform the content knowledge into forms which are "pedagogically powerful and yet adaptive to the variations in ability and background presented by the students" (p. 15). This pedagogical con- tent knowledge links content, students, and pedagogy, revealing a special kind of teacher knowledge (RAND Mathematics Study Panel, 2003). Fennerna and Franke (1992) used Shulman's model as a base in discussing the five components of their model of teachers' knowledge: the knowledge of the content of mathematics, knowledge of pedagogy, knowledge of stu- dents' cognitions, context specific knowledge, and teach- ers' beliefs. The content of mathematics includes teachers' knowledge of the concepts, procedures, and problem-solv- ing processes within the domain in which they teach. Pedagogical knowledge is teachers' knowledge of teaching procedures. Learners' cognitions include knowledge of how students think and learn. Hill, Schilling, and Ball (2004) argued that specific measures of pedagogical content knowledge and mathe- matical content knowledge were not yet in place in math- ematics education. They set out, at the elementary level, to map what elementary teachers knew regarding pedagogi- cal content knowledge and found that teachers' mathemat- ics knowledge for teaching the elementary grades was partly domain specific rather than relating to their teaching or mathematical ability. Hill, Rowan, and Ball (2005) for- mally defined mathematics knowledge for teaching: By "mathematical knowledge for teaching," we mean the mathematical knowledge used to carry out the work of teaching mathematics. Examples of this "work of teaching" include explaining terms and con- cepts to students, interpreting students' statements and solutions, judging and correcting textbook treatments of particular topics, using representations accurately in the classroom, and effects of teachers' mathematical knowledge on student achievement providing students with examples of mathematics concepts, algorithms, or proofs. (Hill, Rowan, & Ball, 2005, p. 373) Hill, Rowan, & Ball have recently provided a model that includes several components of MKT (Ball, 2006). Common Content Knowledge (CCK) refers to the mathe- School Science and Mathematics matical knowledge share by most educated adults, such as knowledge of the curriculum. Specialized Content Knowledge (SCK) refers the mathematical knowledge of teachers that goes beyond the knowledge of the curricu- lum. An example of this specialized content knowledge is providing explanations. Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT) are complementary domains concerning the knowl- edge about mathematics, the knowledge about teaching, and the knowledge about students. Hill, Rowan, and Ball's (2005) study of mathematical knowledge for teaching in the elementary grades present- ed remarkable and groundbreaking research for the math- ematics education community. In addition to finding that teachers' mathematical knowledge for teaching predicted mathematics achievement during the first and third grades, their results suggested that measures of teacher knowledge should be at least content specific and better yet, specific to the teaching of grade level. References Association for Supervision and Curriculum Development [ASCD]. (2003). Research-based character- istics of high-quality teacher preparation. Research Brief 1(4), 1-3. Ball, D. L. (2006, March). Who knows math well enough to teach third grade-and how can we decide? Presentation to the Wolverine Caucus, Lansing, MI. Fennema, F., & Franke, M. L. (1992). Teachers' knowedge and its impact. In D. A. Grouws (Ed.), Handbook of mathematics teaching and learning (pp. 147- 164). New York: Macmillan Publishing Company. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on stdent achievement. American Educational Research Journal, 42, 371-406. Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers' mathematics knowledge for teaching. Elementary School Journal, 105, 11-30. RAND Mathematics Study Panel [RAND MSP]. (2003). Mathematical proficiency for all students: Towards a strategic development program in mathematics educa - tion. Santa Monica, CA: RAND Corporation MR-1643.0- OERI. Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 5 7(l), 1-22. 219 RESEARCH IN E 0