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-RCollege Student Learning

Epistemological DevelopmentReflection

??Appropriate Instructional Strategies??

Teacher LearningMT 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-MTPCK-MT

CK (with CXK)-MT

ConnectionsRepresentations

TechnologyAssessment

Problem-SolvingReasoning

Algebra

Probability

Number Concepts

Other

Proportional Reasoning

Geometry

Statistics

Measurement

Professional traditions

Teacher-educator-knowledge

Mathematics Education sessions

Practical wisdom

Teacher-knowledge

Professionaltraditions

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. WelderWestern Washington University

Priya V. PrasadUniversity of Texas at San Antonio

Alison Castro SuperfineUniversity of Illinois, Chicago

Dana OlanoffWidener University

ReferencesBall, 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 TeachersDeveloping 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

studentscontent

SMCK for a MTE-R