“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.
I once assigned my liberal arts beginning programming students to read a science fiction story (Asimov's The Last Question, as I recall) and write two paragraphs explaining it.
You would have thought I asked them to help me bury bodies.
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?
What scale is the graph? how far apart is 1 and 2? Is the circle filled in?
If my students were fussing over details like that, Nome, I would be delighted. If only.
Obviously, we want a circle with its center at (0,0), to get the target subject into the crosshairs. That explains it.
While I can't begin to arrive at the graph (my last math class was 38 years ago, and I don't recall graphing anything in it), I can't see that the scale of the graph matters, nor how far apart 1 and 2 are - unless the units aren't equal. That would only affect the overall size of the graph, right? Not its internal consistency? And filled in - do you fill in graphs? I admit I would have said "it touches the y axis at 3", but I realize I wouldn't need to... just it would make it easier for me to visualize it.
I admit that the filled in part was a stretch. As to the scale - how could you reproduce the graph without it?
Wonderful tray image, Zeno.
Anonymous: "graph" is a technical term in mathematics. It's just a subset of the plane, without any extra information pertaining to which part of it we're looking at, or how far zoomed in we are. But the students probably perceive the question as "describe the particular picture you're looking at", which could then include the scale, etc.
(That is, they probably perceive it that way if they can get past the mental compartmentalization Zeno describes.)
I've just started analytical geometry, and I was actually considering skipping this bit.
They need to know what the equation of the circle is, and how to read off those parameters, but what need do they have of being able to complete the squares? Is one ever given the equation in expanded form outside of a mathbook?
The answer to your question, Sili, may be somewhere in the following comments, depending on what priorities you want to set:
Every algebra student encounters completion of the square as the clever technique that establishes the quadratic formula, one of the rare mathematical magic tricks that works without fail. (That is, of course, unless they are just handed the quadratic formula at the end of elementary algebra, which would be a shame.)
Completing the square naturally arises again in intermediate algebra, college algebra, analytic geometry, and precalculus as a tool for putting equations in standard form. Since putting equations of conics in standard form is not the most crucial task in the world (and, yes, most of the students' future encounters will be with equations already in that form), one may be tempted to downplay it. It's a tenable position.
Those students who continue on to calculus and differential equations, however, will find it useful on occasion to complete the square while evaluating certain antiderivatives (again, of course, unless the approach is "ask your TI-89") and in working with Laplace transforms and their inverses.
Since completing the square is a relatively simple technique with more than one application, I tend to give it the good old college try in elementary and intermediate algebra.
Wow, where did these people go to school? And when? From an early grade, my kids (now out of college but not by very many years) were getting write-your-reasons and explain-to-a-friend questions in math homework.
(Sometimes the assignments were pretty dumb, too, but they didn't compartmentalize the disciplines, anyway.)
Reminds me. I really must re-read The Way It Spoze To Be sometime soon. Students, even ones who can barely (or not at all) read, quickly learn just what Real School is like, don't they?
My students, when I was a Grad TA at a rather fine private midwestern university, in engineering, used to bitch constantly that I would correct their grammar. I didn't even take points off for it. Just corrected it.
Oh the rage! Luckily, the professor, my advisor, loved language, and loved me. So to hell with the little bastards.
In a previous life, I taught freshman calculus in college. On my final exam, I had the following question, worth 10 points (10% of their total course grade):
In one page maximum, explain what a derivative is and mention some of its uses. Your target audience is a high-schooler who's good at math but who never had calculus. You may use graphs or equations in your answer, but you are not required to.
It was an eye-opener. Many students were unable to define a derivative correctly. Almost none of them mentioned "instantaneous rate of change", and only about half said anything about tangent lines (and only half of those mentioned slope). Only two of them, out of 35, explained the slope of the tangent line as the limit of the slope of a secant line, which is how I had introduced the derivative in class.
Weaker students almost always defined the derivative as: "The derivative of x^2 is 2x. The derivative of sin x is cos x." And so on.
Uses of the derivatives fared much better. Several students mentioned optimization and curve tracing (two applications on which we had spent a lot of time). A few mentioned Newton-Raphson.
Without exception, every student who failed this question failed the course (and would have failed it, even if I hadn't asked that question). Some students had good marks on this question but still failed the course, almost always due to problems with algebra.
Completing the square is useful in tackling some Gaussian integrals which arise in probability, statistical physics and field theory. That's probably not the most common application, nor the one most relevant to Dr. Z's students, but it's the most recent instance I bumped into, so it sprang to mind.
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