1.2 Mathematical Knowledge for Teaching
In his presidential address to the American Educational Research Association, Lee Shulman (-Shulman (1986)) popularized the concept of knowledge for teaching that included a specialized content knowledge where “the teacher need not only understand that something is so; the teacher must further understand why it is so, on what grounds its warrant can be asserted, and under what circumstances our belief in its justification” (p. 9). Researchers in mathematics education further expanded on this idea in the development of domains of Mathematical Knowledge for Teaching. This textbook focuses on the Subject Matter Knowledge side of the Mathematical Knowledge for Teaching.
![Domains of Mathematical Knowledge for Teaching [@Ball2008]](tikz/ballegg1.png)
Figure 1.1: Domains of Mathematical Knowledge for Teaching (Ball et al., 2008)
1.2.1 Common Content Knowledge
The foundation for much of the other aspects of mathematical knowledge for teaching is common content knowledge, defined as “the mathematical knowledge and skill used in settings other than teaching” (Ball et al., 2008, p. 399). Common content knowledge provides the individual the ability to solve mathematical problems from the related curriculum and to apply the mathematical knowledge to other fields of knowledge outside of mathematics. Common content knowledge also includes understanding how some of the different mathematical subjects build upon one another, but does not require those studying the material to be able to explain broader patterns. Thus, common content knowledge represents the type of knowledge that we expect secondary and introductory post-secondary students to display.
For instance, common content knowledge related to rational algebraic expressions and functions includes understanding the relationships between rational expressions and rational numbers, how long-division of polynomials relates to long-division of integers, and the properties of the quotient and remainder of a rational expression and its effect on the graph of the related function. While this common content knowledge includes a deeper level of knowledge and a richer web of conceptual understanding by the teacher than is common for most K-12 students, the knowledge still falls within the expectations for the most advanced of these K-12 students. Thus, it is paramount that secondary mathematics teachers develop deep fluency in common content knowledge and the interconnectedness of the many different mathematical concepts in the curriculum. Such a deep and interconnected knowledge base of the teacher provides a requisite foundation in helping K-12 students learn mathematics. However, it is only a first step.
1.2.2 Specialized Content Knowledge
Specialized content knowledge incorporates mathematical knowledge that goes beyond knowledge expected of students, with a focus on the mathematical knowledge that improves the ability of the teacher to assist students learning mathematics (Ball et al., 2008, p. 400). Thus, specialized content knowledge supplements the common content knowledge that all K-12 students of mathematics need. For example, while common content knowledge for rational algebraic expressions and rational functions include the relationships to the rational numbers, specialized content knowledge could include knowledge of rings and integral domains, thereby improving the teacher’s ability to help students to make the connections between various pieces of related content knowledge.
A teacher would also use this specialized content knowledge related to rational expressions when explaining the extent to which two rational expressions are equivalent when common factors of the numerator and denominator cancel. For instance the teacher could explain the ways in which the expressions \[\frac{(x-1)(x+2)^2}{(x+1)(x+2)} \quad \mbox{and} \quad \frac{(x-1)(x+2)}{(x+1)}\] are equivalent and distinct.
1.2.3 Horizon Content Knowledge
Excellent teachers use more than just a knowledge of the content in the current curriculum when teaching students. They also draw on knowledge of what their students have previously learned in mathematics, what mathematics content will be covered in the next few years, and how the current mathematical topic relates to applications outside of mathematics. That is, excellent mathematics teachers see their instruction as part of a continuum, of which their work is only a small part. We call this domain of knowledge horizon content knowledge.
For example, a teacher with horizon content knowledge might use students’ familiarity with rational numbers to help high school students develop knowledge of rational algebraic expressions and functions. The teacher could also use her knowledge from differential equations to know that the rational functions play a pivotal role in the Laplace transform, helping to determine the amount of time and detail appropriate for teaching about the quotient and remainder theorem for polynomials and how to rewrite rational expressions using partial fractions. The teacher could also use knowledge of the physics curriculum and Boyle’s law to help students make connections between rational expressions and functions and other fields of study.
Horizon content knowledge can also allow teachers to help students understand content in other subjects as well. For instance, many of the sciences are becoming more reliant on organizing and understanding data and graphs. Understanding level curves can help students to better use topographical maps. Understanding geometric properties provides artists with tools to create projections in paintings. And venn diagrams and basic set theory is used in many other subjects to organize information and structures.
While each of these pieces of knowledge are not essential for mathematics teachers, the more knowledge one has, the better that teacher can help students learn and apply the critical components of the content.
1.2.4 Exercises
Consider the general quadratic expression \(ax^2 + bx+c\), where \(a\), \(b\), and \(c\) are real numbers such that \(a\neq0\).
Write a list of everything you know about the general quadratic expression.
Write a list of everything you can do to the general quadratic expression.
How are quadratic expressions different than quadratic equations?
How does the factorization of a quadratic expression relate to prime numbers?
How much do you know about the ways in which quadratic expressions are used outside of mathematics?
Review the mathematics standards for quadratic expressions for your state. How many of the things you have listed for parts (a) and (b) match up with those standards?
For each of your answers to a. through e., does the content you have listed or described align most with common content knowledge, specialized content knowledge, or horizon content knowledge?
Explain the ways in which the expressions \[\frac{(x-1)(x+2)^2}{(x+1)(x+2)} \quad \mbox{and} \quad \frac{(x-1)(x+2)}{(x+1)}\] are equivalent and distinct.
Consider a triangle. A particular high school geometry textbook defines a triangle as “a polygon with three sides.” A second textbook defines a triangle as “the figure formed by connecting three non-co-linear points with straight segments.” A last textbook defines triangles as “A three-sided figure.”
Are all three definitions accurate, or do some allow for shapes that might not be triangles as you understand them to be included within the category of triangle?
In what ways might each definition be considered sloppy? That is, are there any parts of the definition that might not be well-defined? In what ways might each definition make using triangles in future lessons more difficult?
What information do you think each textbook has presented prior to giving its definition for a triangle?
How might horizon content knowledge help an instructor preparing a lesson on triangles decide whether a definition is appropriate or not for her students?
In order to better understand the importance of definitions we will consider the word of “square”.
Define “square” with as few of words as possible, without using outside resources.
What additional mathematical words would be needed to be defined to understand your definition?
How would your definition of “square” fit within other types of quadrilaterals?