Sunday, April 17, 2011
A bag of tricks
I remember when “Massha” enrolled in my algebra class. Recognize the name? She's a character created by Robert Lynn Asprin in his Myth Adventures fantasy series. He introduced her in the third volume, Myth Directions, well before the long string of pun-obsessed novels became rather labored.
My student lacked the bright orange hair or heroic girth that characterized Massha, but she reminded me of Asprin's creation because of the way she preferred to do algebra. The fictional Massha was described as a “mechanic”—or even “no-talent mechanic”—by other characters in the novels because she used amulets and other physical trinkets to cast her spells. She possessed no actual magical talents, but relied entirely on magical devices.
The Massha in my class drove the point home time and again whenever I tried to explain a procedure and she would counter with a memorized algorithm, as if that pre-empted any further discussion. It was an occasional irritant, but she was entirely sincere in her approach. She had found success in treating math as a collection of miscellaneous tricks and she resisted any attempt to explore below the surface. As far as our Massha was concerned, that was just a dangerous distraction.
A particularly clear case arose while we were discussing the solution of rational equations. These are nothing more than equations that contain rational expressions. For example,
is a rational equation. One of the glorious principles of algebra is that you can do just about anything to one side of an equation as long as you do the same thing to the other side. In the case of a rational equation, you can use this principle to eliminate all of the denominators. Just multiply both sides of the equation by the least common denominator of all of the rational expressions! For the given example, the least common denominator is x(x − 1). Multiplying both sides results in massive cancellation and simplification:
Since algebra students seem to regard operations involving division with more trepidation than anything else, I can usually engage their enthusiasm for a process that destroys all denominators. It has its moment of messiness, but the results are clean and rational (no math joke intended). I leave it as an exercise to the reader to complete the demonstration that x = −2.
Massha was not content with my demonstration. She wanted to rephrase things in her own way, which I'm normally inclined to encourage, as it indicates the student is assimilating the knowledge. What she said, however, disturbed me:
“Do we have to show our work and cancel things or can I just do it the way I learned it? I was taught that you compare each term to the LCD and give the term the part of the LCD that it's missing. Is that all right?”
I looked back at the problem on the board. Massha knew the LCD was x(x − 1). She would consider 3/x, observe that it lacked the x − 1 and “give” it that factor. At the same time, presumably, she would drop the factor x, which it did have in common with the LCD.
“And you drop whatever it already has in common with the LCD?” I prompted.
She nodded her head. “Just do that to every term,” she said. “It's faster!”
“Since this is our first encounter with rational equations in this class, I want everyone to show a step-by-step justification for our simplifications. Later on, when we return to rational equations in terms of applications, I'll let people reduce the amount of work they show. For now, though, always write down the LCD and show the step of multiplying through by it and reducing.”
Massha scowled at me.
“But I already know how to do this!” she complained.
“I am confirming that your short-cut is a valid algorithm,” I replied, “but I will be holding everyone to the same standard of completeness in presenting solutions.”
Massha was unhappy much of the semester. She had a keen memory, had had algebra before (so why was she in my class?), and retained quite an array of solution gimmicks. I was happy for her (sort of), but her view of mathematics had been reduced to the rote application of algorithms. It was enough for her to do well in class, but she chafed at every requirement to justify her solutions. She would have found a kindred spirit in the algebra student who had been in one of my previous classes. When I finished demonstrating the derivation of the quadratic formula, that student rolled her eyes and said, “Oh, Mr. Z, are you explaining things again? Why didn't you just give us the formula?”