MTEs’ Subject Matter Knowledge MTEs’ Pedagogical Content Knowledge Common Content Knowledge (MKT?) (MTE-CCK) Knowledge at the mathematical horizon (for PTs) Specialized Content Knowledge (MTE-SCK) Knowledge of Content and PTs (MTE-KCS) Knowledge of Content and Teaching PTs (MTE-KCT) Knowledge of Curriculum (recommendations for preparing PTs?) Learn Mentoring Educators • Meaning of educating teachers? • How others learn teacher educating? • Promoting others’ teacher educating? Awareness-in-counsel Learn Educating Teachers • Meaning of math teaching? • How others learn teaching? • Promoting others’ teaching? Awareness-in-discipline Learn Math Teaching • Meaning of math knowledge? • How others develop math? • How to promote others’ math? Awareness-in-action Learn Math • Reason • Communicate • Connect ideas • Compute PCK for a MTE-R College Student Learning Epistemological Development Reflection ??Appropriate Instructional Strategies?? Teacher Learning MT Learning/Change Beliefs Content Knowledge ??Integrating Content & Pedagogy?? CK for a MTE-R • Curricular materials for teaching about mathematics teaching • Experts in the department, college & field (CXK) • Accreditation, state, university & department standards and requirements (CXK – V&L CXK) • MT preparation & M.Ed Programs (CXK - V&L CXK) Mathematical Teacher Educator Knowledge for Teaching University Courses SMCK-MT PCK-MT CK (with CXK)-MT Connections Representations Technology Assessment Problem-Solving Reasoning Algebra Probability Number Concepts Other Proportional Reasoning Geometry Statistics Measurement Professional traditions Teacher-educator- knowledge Mathematics Education sessions Practical wisdom Teacher- knowledge Professional traditions Classroom events Practical wisdom Learner- knowledge Questions • Is a fractal a good metaphor for how these bodies of knowledge are nested within each other? • What are the affordances and constraints of using fractalization as a way of conceptualizing MKTT? o Where does MKT get categorized within the domains of MKTT? o Is MKT a proper subset of MKTT? o How can we start to define the domains of MKTT? • How does MTEs’ knowledge of research fit into a framework of MKTT? We aim to develop a theoretical foundation for the mathematical knowledge for teaching teachers (MKTT) by analyzing and synthesizing the existing literature on MTE knowledge through the lens of elementary mathematics content course development. We use “fractalization” as a metaphor for the many ways in which researchers are theorizing about MKTT, and apply this metaphor to explain many of the theoretical frameworks shown on this poster. The use of this metaphor, however, reveals some unexplored consequences of this framing. We apply this fractal metaphor to the Mathematical Knowledge for Teaching (MKT) (Ball, Thames, & Phelps, 2008) framework to see what happens when we try to define the analogous domains of MKT in terms of MKTT. Rachael M. Welder Western Washington University Priya V. Prasad University of Texas at San Antonio Alison Castro Superfine University of Illinois, Chicago Dana Olanoff Widener University References Ball, D. L. (2012). Afterword: Using and designing resources for practice. In G. Gueudet, B. Pepin & L. Trouche (Eds.), From text to 'lived' resources: Mathematics curriculum materials and teacher development. Dordrecht, Netherland: Springer. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. Chavout, J. (2009). Grounding practice in scholarship, grounding scholarship in practice: Knowledge of a mathematics teacher educator-researcher. Teaching and Teacher Education, 25, 357-370. Cohen, D., Raudenbush, S., & Ball, D. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 1–24. Jaworski, B. (1992). Mathematics teaching: What is it? For the Learning of Mathematics, 12(1), 8-14. Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267. Tzur, R. (2001). Becoming a mathematics teacher-educator: Conceptualizing the terrain through self-reflective analysis. Journal of Mathematics Teacher Education, 4, 259-283. Perks, P. & Prestage, S. (2008). Tools for learning about teaching and learning. In B. Jaworski & T. Wood (Eds.), The international handbook of mathematics teacher education, vol. 4: The mathematics teacher educator as a developing professional (pp. 31–56). Rotterdam: Sense. Zaslavsky, O., & Leikin, R. (2004). Professional development of mathematics teacher educators: Growth through practice. Journal of Mathematics Teacher Education, 7, 5-32. Mathematical Knowledge for Improving the Content Preparation of Elementary Teachers Developing a Framework for Challenging Content for Mathematics Teachers: The Teaching Triad of Mathematics Teachers Challenging Content for Students: Mathematics Management of Students’ Learning Sensitivity to Students Management of Mathematics Teachers’ Learning Sensitivity to Mathematics Teachers Cohen, Raudenbush, & Ball (2003) offered a model of the instructional dynamics that affect student learning, highlighting the interactions between teachers and learners, their interactions with content, and the context in which the learning is taking place. Ball (2012) proposed expanding this model to consider the instructional dynamics surrounding the learning of teachers. Tzur (2001) proposed a four-tier model of teacher educator development (left, in the ellipses), following a progressive hierarchy where each level of foci encompasses all prior levels. Tzur’s framework aligns well with Mason's (1998) hierarchical levels of awareness in his vision for teacher educator development (right, in the tabs). Chauvot (2009) expanded Shulman’s (1986) framework for teacher knowledge to develop a knowledge map for MTEs. Her model follows the fractal metaphor as it is places all domains of knowledge for teaching within the subject matter content knowledge of MTE’s (MTE-SMCK). Chavout’s map is unique in that it considers how the context in which MTEs work, based on Grossman’s (1990) knowledge of context (CXK), affects what they need to know. The model above only illustrates MTE knowledge in the context of teaching university courses. Subject Matter Knowledge Pedagogical Content Knowledge Common Content Knowledge (CCK) Knowledge at the mathematical horizon Specialized Content Knowledge (SCK) Knowledge of Content and Students (KCS) Knowledge of Content and Teaching (KCT) Knowledge of Curriculum Mathematical Knowledge for Teaching (Ball, Thames, & Phelps, 2008) Mathematical Knowledge for Teaching Teachers Perks and Prestage (2008) used a fractal metaphor, as they included the “Teacher Knowledge Tetrahedron” (left) as the “Learner Knowledge” portion of their “Teacher-Educator Knowledge Tetrahedron.” Zaslavky and Leikin fractalized Jaworski’s model of the practice of teaching mathematics, known as “The Teaching Triad of Mathematics Teachers,” to draw attention to the elements involved in creating opportunities for teachers to learn about the practice of teaching math. In developing their “Teaching Triad of Mathematics Teacher Educators," Jaworski's entire teaching triad for math teachers becomes the content to be learned by math teachers as students of teacher education. Teacher educator teachers teachers teacher students students content SMCK for a MTE-R