Subject: Why every Canadian must factor quadratic expressions
Dear Colleague,
I am so pleased to have a chance to share my thoughts on what every Canadian needs to know. As we are both math teachers, I will focus on a particular piece of the math curriculum that is often feared by students and criticized by skeptics– the factoring of quadratic expressions.
There is arguable justification for downplaying the usefulness of this particular piece of mathematics learning. I would ask when you or I have used the quadratic formula to solve a problem in our respective lives. I certainly never have, yet I find this to be an essential piece of knowledge. This is because understanding this process is not an end in itself, but way into understanding various mathematical ideas. When we factor a quadratic expression, we are looking at a collection of numbers and figuring out how they work together. We have to look for patterns in the expression as a whole (is this a square or a difference of squares?), relationships between numbers (do these two multiply to a number that these two add to?), or relationships between images and mathematical language (if this number is big enough, then the parabola never crosses the axis, rendering this unsolvable). When we learn about quadratic expressions, we learn to interpret and solve problems. We learn to dig into what is in front of us to find a conclusion.
Why is this important to Canadians? In a democracy, we are often faced with choices. As a group of people voting, we can usually expect to have decisions driven by values. Sometimes though, we are faced with the responsibility to vote on something more intellectually involved. A recent example in BC is the pair of referendums held in 2005 and 2009 to change our election system to a Single Transferable Vote model. An understanding of the issue would require some engagement with the moderately complicated mathematics behind the system. Given the tendency for people to dismiss such engagement with claims like, “I’m no good at math,” I worry that too many people voted according to how public interest groups endorsed each side of the referendum, without looking any closer. As in the example of a science student given by Smith and Siegel (2004), “without such understanding, the student can’t be said to know the law at all, because she has no reason for thinking it true (other than that the teacher said so)” (p. 563). Without comprehension, what should be an informed democracy is instead one where we vote once every four years or so and let the elected officials direct us for the rest of the time. There is no active engagement with the information and choices before us.
Shulman (1986) suggests that “the teacher need not only understand that something is so; the teacher must further understand why it is so” (p. 9). This is in the context of a teacher deciding how to approach subject matter to be taught, but it applies as well to making informed decisions that affect our lives and the lives of those around us. Math instruction in general, and working with quadratics in particular, help students gain the tools to engage and understand concepts that they may be asked to decide upon as a group.
Sincerely,
Shaun Stewart
References
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15(2), 4-14.
Smith, M. U. & Siegel, H. (2004). Knowing, believing, and understanding: What goals for
science education? Science & Education, 13(6), 553-582.