Words versus numbers
“You can't do that!”
My student's emotions were an admixture of horror and disbelief.
“No, really! You can't do that! This is a math class!”
Oh, really? I guess I had lost track of that. I inquired as to the basis for the student's convictions.
“Why do you think I can't give you an essay question to answer?“
The student goggled in disbelief at my question.
“Because that's what we do in English class, not in math!”
Another student chimed in.
“Yeah. Math is about numbers and calculations—not about words!”
I've had this conversation a few times now, mostly in intermediate algebra and precalculus. It tends to occur when I hand out a quiz or exam with the following kind of problem:
Rewrite the equation x2 + y2 + 4x − 6y + 9 = 0 in standard form and graph your results. Describe your graph in words.It's a perfectly ordinary problem that occurs after the students have learned about the basic conic sections and the technique of completing the square. Upon rewriting the equation as
(x + 2)2 + (y − 3)2 = 4,
I've tried amplifying the prompt in an attempt to make it less intimidating:
“Think about how you would describe your graph over your phone to a friend so that your friend could graph it without having seen it.”
(These days I have to add the warning that it's no fair to just send the friend a quickly snapped image of the graph.)
Lots of students leave that part of the problem blank and move on. Others tentatively write “circle” (miffed that I didn't just ask for the name of the conic section and anxious that I used the plural “words”) and nervously move on.
And then there's the handful of students who write dissertations like this:
Subtract the 9 from both sides to isolate the variables. Look at the coefficient of the x term, which is 4, take half of it and square it. Add that to both sides. Change x2 + 4x + 4 into (x + 2)2. Now look at the y term...Wow. A complete procedural guide to deriving the answer (though seldom as coherent as the mocked-up example above). Where did I ever ask for that? (I'm sure they get a prickly feeling that something must be wrong when they overflow the tiny space I allowed for their answer and they continue their discourse on the back of the page.)
Why do so few of them offer the brief and straightforward response that “The graph is a circle of radius 2 with its center at (−2, 3)”? Wouldn't that suffice to fully inform their imaginary friend at the other end of the phone conversation?
Hardly anyone is pleased when I unveil the answer. The typical reaction is exasperation:
“Is that all you wanted? Why didn't you say so?”
I thought I did.
“You're just confusing us. This isn't English comp!”
My students are like fussy eaters who get upset if their corn touches their mashed potatoes. Food should reside in carefully demarcated regions and college curriculum should reside in strictly disjoint sets. (They're not like my kid brother, who regarded his dinner plate as an artist regards the palette whereon he mixes his colors.)
Eventually, however, I break down my students' reservations and most of them start scooping up the relatively easy points I assign for complete one-sentence answers to simple prompts. By the end of the semester they are rather less startled by questions that require a written response. They still don't, for the most part, like them, but they can do them.
Then the school term ends and I have to start all over again with a new batch. And I know what words will be coming out of their mouths.
