One of the nicer talks that I attended at the Joint Mathematics Meetings in DC was given by Jeff Suzuki, on “A History of College Algebra in the United States During the Nineteenth Century“. Suzuki’s talk focused on equation-solving, and he noted that factoring hadn’t been a significant equation-solving tool throughout that era. Equations were solved by other means (e.g., for quadratics: completing the square, or the quadratic formula; for higher order equations, roots might be found/estimated by bisection or other iterative approximation schemes).
This reminded me of one of my pet peeves from teaching algebra. Modern elementary algebra texts teach four ways of solving quadratic equations: graphing (and finding x-intercepts by inspection), factoring, completing the square, and the quadratic formula. Most current books aimed at the “college algebra” market emphasize factoring. (One of my daily reads, jd2718, has an interesting take on the role of factoring in HS algebra.)
What irked me when last I taught college algebra were the “applications” of quadratic equations. (Too) Many of the applications amounted to numerical exercises: “Find two numbers whose product is 45 and whose difference is 4” was a typical example.
The intent is that a student will introduce one or two variables, representing the two numbers as either x and x+4 or as x and y, and eventually arrive at the equation , either directly or via a system of equations.
From here, we solve the related equation , by finding two numbers whose product is -45, and which add to +4: that is, two positive integers whose product is 45 and whose difference is 4.
The usual technique for achieving this is to list all integer products which equal 45, and find the factorization by inspection. But of course this amounts to a direct solution of the original word problem, without recourse to any algebra whatsoever.
Granted, there are some advantages to this algebraic method, which become clear when you consider situations with non-integer solutions. For example, if we want to find two numbers whose sum is 6 and product is 10, trying a brute force attack is unlikely to work. Setting up the problem algebraically, and reducing it to solving , allows us to apply another paradigm — completing the square — to eventually find the two numbers and .
I suppose that there just might be some slight pedagogical value in having students see that solving their quadratic equation is equivalent — literally and explicitly — to solving the original word problem. But this amount of circularity always struck me as a bit daft, and I feared the day when an eye-rolling student in the back of the class was going to point out that the Emperor was underdressed.