Learning Mathematical Practices to Connect Abstract Algebra to High School Algebra

Abstract

Making connections between advanced mathematical content, such as abstract algebra, and the mathematics of the school curriculum is a critical component of the mathematical education of future secondary teachers. In this chapter, I argue that engagement in mathematical practices (e.g., constructing arguments, attending to precision) can serve as a link for preservice teachers from their study of abstract algebra to the content they will teach as high school teachers. Using a multiple case study approach, I describe how four preservice teachers had opportunities to learn to engage in mathematical practices in their abstract algebra course. Participants were taking an abstract algebra course specifically designed for future teachers. Data sources include video records from the abstract algebra course and problem-solving interviews with each participant before and after the course. Each participant showed improvement in their mathematical practice engagement and reflected on how valuable a focus on mathematical practices would be in their teaching. These findings demonstrate the key role that mathematical practices play in the preparation of future teachers. There are valuable implications for the design of content courses for teachers and for the ongoing research into connections between advanced mathematics and the school curriculum.

Author Note: An earlier version of this paper was presented at the North American Chapter of the International Group for the Psychology of Mathematics Education Annual Meeting, in November 2015, in East Lansing, MI. The research for this paper was conducted as part of a dissertation study at Stanford University, and was supported in part by a Stanford Graduate School of Education Dissertation Support Grant.

Appendices

Appendix 1: Possible Algebra Pre-task Solution

• Prove the following statement:

• If the graphs of linear functions f(x) = ax + b and g(x) = cx + d intersect at a point P on the x-axis, the graph of their sum function (f + g)(x) must also go through P.

We are given that f(x) = ax + b and g(x) = cx + d. These two functions intersect at a point P on the x-axis. Define P as (p, 0).

The fact that the two functions intersect at point P implies that both contain (p, 0). In other words, f(p) = 0 and g(p) = 0.

To determine whether the sum function also passes through point P, we must consider (f + g)(x) and see if the value of the sum function at p is 0.

$$ \left(f+g\right)(p)=f(p)+g(p)\vspace*{-1pc} $$

$$\hspace*{1.7pc} =0+0 \vspace*{-1pc}$$

$$ \hspace*{.2pc}=0 $$

Since the sum function is defined as the sum of the value of the functions f and g at every point in the domain, we can calculate the value of the sum function at p as shown above. The fact that f and g contain (p, 0) guarantees that their sum also contains (p, 0). ■

Appendix 2: Possible Algebra Post-task Solution

Take a point (p, q) on the Cartesian plane. Reverse the coordinates to obtain a second point (q, p). Prove that on the line between these two points, the x-intercept and the y-intercept are the sum of the coordinates.

Consider the points (p, q) and (q, p) for p ≠ q. Then the equation of the line between the two points is:

$$\begin{array}{rrl} y-q&=&\frac{p-q}{q-p}\left(x-p\right)\\[6pt] y-q&=&\frac{p-q}{-\left(p-q\right)}\left(x-p\right)\\[6pt] y-q&=&-1\left(x-p\right)\\[6pt] y-q&=&-x+p\\[6pt] y&=&-x+p+q \end{array}$$

The y-intercept occurs when x = 0. Plugging in 0 for x gives:

$$ y=-0+p+q\vspace*{-1pc} $$

$$ =p+q $$

And so the y-intercept is the sum of the coordinates p and q.

Similarly, the x-intercept occurs when y = 0. Plugging in 0 for y gives:

$$ 0=-x+p+q\vspace*{-1pc} $$

$$ x=p+q $$

And so the x-intercept is the sum of the coordinates p and q.

If p = q, then (p, q) and (q, p) are the same point, and an infinite number of lines could pass through it. Thus the proposition stated in the problem holds only for p ≠ q. ■

Appendix 3: Tim’s Scratch Work and Formal Proof, Algebra Post-task

Scratch work

Formal work