Like most of us, I am still teaching a traditional developmental algebra course — two of them, in fact (beginning algebra, intermediate algebra). This traditional content includes standard topics, and one of them has been bothering me for some time. Can we drop the quadratic formula from our courses?
First, let’s come up with a list of good reasons to study the quadratic formula. Well, did you think of any? The most common rationale given is that “the quadratic formula solves ANY quadratic equation”. That is partially true; the formula CAN solve any equation that is strictly quadratic (but not equations that are quadratic in form). Let’s ignore that shortcoming, as perhaps it is not a significant gap. The importance of the quadratic formula, then, depends on how important it is to solve a strictly quadratic equation. Is it the only way? Of course not — completing the square can also solve any strictly quadratic equation … often with less computational effort. We also can use numeric and graphical methods to solve any quadratic equation. Another common rationale for the ‘QF” (as it is sometimes known) as it facilitates the use of the discriminant; whether the discriminant is worth the bandwidth depends on how we use it, and how the study of the discriminant contributes to the mathematical reasoning of our students. Some people use the QF to determine linear factors of quadratic expressions, which fits in to the ‘correct answer’ world view; I doubt if using the formula to factor expressions contributes to an understanding of equivalence. [However, I have to admit that our normal instruction of factoring is not really designed to produce understanding of equivalence.]
How about good reasons to NOT study the quadratic formula? Well, did you think of any? Quite a few of my students dislike the formula because they realize how likely they are to make a minor arithmetic mistake which results in catastrophic failure to solve the equation. Some of these students have a strong preference for completing the square — because it provides a logical sequence of steps that avoids many mistakes. We also have a mythology among our students that says success in mathematics depends on the mastery of formulas to generate the correct answers required in a class. Few of us concur with the importance of ‘correct answers’ in that myth, but many of us contribute to the myth by placing an emphasis on the quadratic formula. I would say that the use of the quadratic formula to solve an equation detracts from the mathematical reasoning that I am trying to develop in my students.
Of course, the ‘elephant in the room’ with us is the role of quadratic equations and expressions in general. Why are they important? We could spend several blog posts on that topic, and we might go there someday. For today, here is a brief summary: the quadratic equations are included to foreshadow some authentic uses in STEM courses later, so we include some puzzle problems that result in quadratics in the developmental course (rectangles of a certain area, projectile under the influence of gravity; we also use quadratic equations as a field test of other algebraic skills (factoring, radicals, complex numbers, etc). Very few processes (either in nature or in society) are essentially quadratic; the most common quadratic equations in valuable applications come from modeling data (such as fuel efficiency vs speed, profit vs production, etc).
Very few of the applications leading to quadratic equations have a value in helping our students become more sophisticated in mathematical reasoning, nor in problem solving in general. The solving of these problems is an exercise. Therefore, this exercise should develop something of value in our students … and this does not mean ‘correct answers’. Many applications are solvable by using square roots (like x² = 18), and that method can be connected to a series of related knowledge. If the problems involve a full-quadratic, numeric & graphical methods provide solutions to most with connections we can make to other knowledge. Resorting to the quadratic formula bypasses connections and understanding the process, and the QF stands isolated from other knowledge (for almost all of us).
How about a reality check: How many of us “reach for the formula” to solve a quadratic equation arising from a situation or problem that is worth solving? These problems often involve non-integer coefficients. We are likely to reach for the QF primarily when the solutions are complex numbers, where numeric and graphical methods are less accessible.
Unless we teach the Quadratic Formula in a connected fashion, richly connected to basic concepts of mathematics, I think we do more harm than good. Without those connections, the formula reinforces the myth of right answers. Mathematics is important in life; the quadratic formula has few contributions to make.
Join Dev Math Revival on Facebook: