# Ball, Thames, & Phelps (2008)

Content Knowledge for Teaching: What Makes It Special?

## Abstract

This article reports the authors' efforts to develop a practice-based theory of content knowledge for teaching built on Shulman's (1986) notion of pedagogical content knowledge. As the concept of pedagogical content knowledge caught on, it was in need of theoretical development, analytic clarification, and empirical testing. The purpose of the study was to investigate the nature of professionally oriented subject matter knowledge in mathematics by studying actual mathematics teaching and identifying mathematical knowledge for teaching based on analyses of the mathematical problems that arise in teaching. In conjunction, measures of mathematical knowledge for teaching were developed. These lines of research indicate at least two empirically discernible subdomains within pedagogical content knowledge (knowledge of content and students and knowledge of content and teaching) and an important subdomain of "pure" content knowledge unique to the work of teaching, specialized content knowledge , which is distinct from the common content knowledge needed by teachers and nonteachers alike. The article concludes with a discussion of the next steps needed to develop a useful theory of content knowledge for teaching.

## Summary of Content Knowledge for Teaching: What Makes It Special?

Ball, Thames, and Phelps begin by looking at the 20+ years since Shulman (1986) introduced his theories of pedagogical content knowledge (PCK). Despite the PCK's widespread use, Ball and colleagues claim it "has lacked definition and empirical foundation, limiting its usefulness" (p. 389). 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 foundations, 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, Thames, and Phelps, 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. The following descriptions use concepts from algebra.

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, algebra includes the knowledge that $\displaystyle{ (x + y)^2 = x^2 + 2xy + y^2 }$. Most everyone who has studied basic high school algebra has learned this, and this knowledge would not be unique to math teachers.

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, an algebra teacher needs to have a specialized understanding of how $\displaystyle{ (x+y)^2 }$ can be expanded using FOIL or modeled geometrically 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 a teacher needs to know both to better understand student strategies and sources of error. Namely, the error that $\displaystyle{ (x + y)^2 = x^2 + y^2 }$.

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 the example of $\displaystyle{ (x + y)^2 = x^2 + 2xy + y^2 }$, algebra teachers need to understand how previous topics like order of operations, exponents, and the distributive property relate to this problem. Looking forward, algebra teachers would 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 $\displaystyle{ (x + y)^2 = x^2 + y^2 }$, 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 gives algebra teachers insight about why students mistakenly think $\displaystyle{ (x + y)^2 = x^2 + y^2 }$, KCT is the knowledge that allows teachers to decide what to do about it. This may mean choosing a geometric representation for instruction over using FOIL, which lacks the geometric representation and does little to address students' difficulty in recognizing that $\displaystyle{ (x + y)^2 = (x + y)(x + y) }$.

Knowledge of Content and Curriculum

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 the textbook series 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, Ball, Thames, and Phelps 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.

## Corrolary

APA
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
BibTeX
@article{Ball2008,
author = {Ball, Deborah Loewenberg and Thames, Mark Hoover and Phelps, Geoffrey},
doi = {10.1177/0022487108324554},
journal = {Journal of Teacher Education},
keywords = {mathematics, teacher knowledge, pedagogical content knowledge},
number = {5},
pages = {389--407},
title = {{Content knowledge for teaching: What makes it special?}},
url = {http://jte.sagepub.com/content/59/5/389},
volume = {59},
year = {2008}
}