My last two posts summarized the underpinnings of Shulman's

*pedagogical content knowledge*and Deborah Ball's early work building upon and extending Shulman's theories. Now we jump from Ball's 1988 article to one she co-authored in 2008 with University of Michigan colleagues Mark Thames and Geoffrey Phelps, titled

*Content Knowledge for Teaching: What Makes It Special?*

*This article starts by looking at the 20+ years we've had to further develop Shulman's theories of pedagogical content knowledge (PCK). Despite the theory's widespread use, Ball and colleagues claim it "has lacked definition and empirical foundation, limiting its usefulness" (p. 389). (See also Bud Talbot's 2010 blog post and related efforts.) In fact, the authors found that a third of the more than 1200 articles citing Shulman's PCK*

do so without direct attention to a specific content area, instead making general claims about teacher knowledge, teacher education, or policy. Scholars have used the concept of pedagogical content knowledge as though its theoretical founcations, conceptual distinctions, and empirical testing were already well defined and universally understood. (p. 394)

To build the empirical foundation that PCK needs, Ball and her research team did a careful qualitative analysis of data that documented an entire year of teaching (including video, student work, lesson plans, notes, and reflections) for several third grade teachers. Combined with their own expertise and experience, and other tools for examining mathematical and pedagogical perspectives, the authors set out to bolster PCK from the ground up:

Hence, we decided to focus on the work of teaching. What do teachers need to do in teaching mathematics -- by virtue of being responsible for the teaching and learning of content -- and how does this work demand mathematical reasoning, insight, understanding, and skill? Instead of starting with the curriculum, or with standards for student learning, we study the work that teaching entails. In other words, although we examine particular teachers and students at given moments in time, our focus is on what this actual instruction suggests for a detailed job description. (p. 395)

For Ball et al., this includes everything from lesson planning, grading, communicating with parents, and dealing with administration. With all this information, the authors are able to sharpen Shulman's PCK into more clearly defined (and in some cases, new) "Domains of Mathematical Knowledge for Teaching." Under

*subject matter knowledge*, the authors identify three domains:

- Common content knowledge (CCK)
- Specialized content knowledge (SCK)
- Horizon content knowledge

And under

*pedagogical content knowledge*, the authors identify three more domains:

- Knowledge of content and students (KCS)
- Knowledge of content and teaching (KCT)
- Knowledge of content and curriculum

Ball describes each domain and uses some examples to illustrate, mostly from arithmetic. For my explanation, I'll instead use something from high school algebra and describe how each domain applied to my growth of knowledge over my teaching career.

#### Common Content Knowledge (CCK)

Ball et al. describe CCK as the subject-specific knowledge needed to solve mathematics problems. The reason it's called "common" is because this knowledge is not specific to teaching -- non-teachers are likely to have it and use it. Obviously, this knowledge is critical for a teacher, because it's awfully difficult and inefficient to try to teach what you don't know yourself. As an example of CCK, my knowledge includes the understanding that \((x + y)^2 = x^2 + 2xy + y^2\). I've known this since high school, and I would have known it whether or not I became a math teacher.#### Specialized Content Knowledge (SCK)

SCK is described by Ball et al. as "mathematical knowledge and skill unique to teaching" (p. 400). Not only do teachers need this knowledge to teach effectively, but it's probably not needed for any other purpose. For my example, I need to have a specialized understanding of how \((x+y)^2\) can be expanded using FOIL or modeled geometricaly with a square. It may not be all that important for students to understand both the algebraic and geometric ways of representing this problem, but I need to know both so I can better understand student strategies and sources of error. Namely, the error that \((x + y)^2 = x^2 + y^2\).#### Horizon Content Knowledge

This domain was provisionally included by the authors and described as, "an awareness of how mathematical topics are related over the span of mathematics included in the curriculum" (p. 403). For my example of \((x + y)^2 = x^2 + 2xy + y^2\), I need to understand how previous topics like order of operations, exponents, and the distributive property relate to this problem. Looking forward, I need to understand how this problem relates to factoring polynomials and working with rational expressions.#### Knowledge of Content and Students (KCS)

This is "knowledge that combines knowing about students and knowing about mathematics" (p. 401) and helps teachers predict student thinking. KCS is what allows me to expect students to incorrectly think \((x + y)^2 = x^2 + y^2\), and to tie that to misconceptions about the distributive property and exponents. I'm not sure I had this knowledge for this example when I started teaching, but it didn't take me long to figure out that it was a very common student mistake.#### Knowledge of Content and Teaching (KCT)

Ball et al. say KCT "combines knowing about teaching and knowing about mathematics" (p. 401). While KCS gave me insight about why students mistakingly think \((x + y)^2 = x^2 + y^2\), KCT is the knowledge that allows me to decide what to do about it. For me, this meant choosing a geometric representation for instruction over using FOIL, which lacks the geometric representation and does little to address the problem if students never recognize that \((x + y)^2 = (x + y)(x + y)\).#### Knowledge of Content and Curriculum

For some reason, Ball et al. include this domain in a figure in their paper but never describe it explicitly. They do, however, scatter enough comments about knowledge of content and curriculum to imply that teachers need a knowledge of the available materials they can use to support student learning. For my example, I know that CPM uses a geometric model for multiplying binomials, Algebra Tiles/Models can be used to support that model, virtual tiles are available at the National Library of Virtual Manipulatives (NLVM), and the Freudenthal Institute has an applet that allows students to interact with different combinations of constants and variables when multiplying polynomials.Some of the above can be hard to distinguish, but thankfully Ball and colleagues clarify by saying:

In other words, recognizing a wrong answer is common content knowledge (CCK), whereas sizing up the nature of an error, especially an unfamiliar error, typically requires nimbleness in thinking about numbers, attention to patterns, and flexible thinking about meaning in ways that are distinctive of specialized content knowledge (SCK). In contrast, familiarity with common errors and deciding which of several errors students are most likely to make are examples of knowledge of content and students (KCS). (p. 401)

In their conclusion, the authors hope that this theory can better fill the gap that teachers know is important, but isn't purely about content and isn't purely about teaching. We can hope to better understand how each type of knowledge above impacts student achievement, and optimize our teacher preparation programs to reflect that understanding. Furthermore, that understanding could be used to create new and improved teaching materials and professional development, and better understand what it takes to be an effective teacher. With this in mind, you can gain some insight to what Ball was thinking when she gave this congressional testimony:

References

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. doi:10.1177/0022487108324554