Sunday, April 17, 2011

A bag of tricks

Stop making sense!

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!”

Massha had a magic amulet from her bag of tricks. She had memorized a procedure that saved her from writing the messy cancellation step because it algorithmically led her to the same result without actually justifying it. The juice had been squeezed out of the algebraic process and the dried husk preserved the result if not the rationale.

I pondered.

“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?”


Kathie said...

What, pray tell, would "Massha" have done if the denominator on the right-hand side of the equation hadn't conveniently been the product of the two denominators on the left-hand side? What if the denominator on the right had been, say, (x-2) x (x-3)? Not that I've solved such an equation yet, so I can't tell by inspection whether it would compute or not, but you get my drift ;-)

Zeno said...

She would look into her bag of tricks to see if she had something that fit that particular case. If you have enough tricks, you don't need general principles!

Improbable Joe said...

That's fine, as long as you're not planning on going any further than Algebra. If you're going further, and especially if you're going into an field where the math is applied in real situations, you need to understand the principle. Real life is a lot messier than the odd-numbered problems in the book.

Kathie said...

Oops, I posted my earlier comment just before logging off for dinner here in the EDT, so while eating I realized that the alternative problem I posed would lead to a cubic equation; didn't have a chance to get back online till now to correct my error. The denominator of the right side of the equation needs to be first order too, right? "Massha" has to be dragged figuratively kicking and screaming to the realization that a more generalized solving method is more elegant than a patchwork of special cases. Grrr....

horseplayer said...

A workmate told me to quiz one of the dimmest laborers on the crew. Ask him the sum of an extremely large multiplication problem. Approx. 5 secs. later he had it.(his did roll up while performing). Division-natch. Square root-natch. Harder problem took approx.10 secs. I can say i knew a least one idiot-savant in my life!

Billy C said...

At least her trick was valid.

Sometimes trying to help my son with math I would watch him perform an inappropriate operation, and try to correct him.

"That's how we do it at school," he'd say.

"Well, that works for another kind of problem but this one is a little different."

"That's how we do it at school."

"Sure, and it'll work, say for Exercise 23. But notice how Exercise 25 is a little different."

"That's how we do it at school."

"But notice you get the wrong answer."

"That's how we do it at school."

What do you do to help the student who has only a few tricks and insists on using a familiar but inappropriate one?

Zeno said...

That's a familiar problem, Billy C. A running theme in my math classes is "context." Whenever we apply a mathematical principle, I ask whether it's justified in the context in which we want to use it. This most infamously occurs when students want to "distribute" the power in (x + y)^2. Oops! Invalid! I stress that every gimmick, shortcut, procedure, algorithm, formula, or theorem comes with its own set of prerequisites or limitations. If those aren't satisfied, we can't proceed with what we wanted to do.

Anonymous said...

I tend not to stress procedure. Teach it, encourage its use, but not require it in homework or exams.

HOWEVER, answers must be justified. Justification is often most easily given by showing the work you did to find the solution. I allow for other ways to justify an `inspired' solution, however, as long as it demonstrates some mathematical rigor.

Of course, justifying that you have found *all* the solutions is a bit trickier if you are finding them using rote or intuitive methods.

Sili said...

I think I'll print this out and give it to my kids.

They not *that* bad, but they really don't like to show their work.