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Calculus Student Descending a Staircase

William Mueller
Department of Mathematics
University of Arizona
Tucson, AZ 85721

Stately, plump, Tyrone Incandenza came from the stairhead, bearing a textbook on which a table of integrals and a calculator lay crossed. An ungirdled T-shirt, bearing the logo of a school he did not attend, was sustained gently around him by the mild morning air. Halted, he peered down the stairs, letting a slow exhalation of vectors escape from his nostrils and sum with the breeze. In a barely audible perturbation of the imperturbable cool that he had constructed for his exit into the sun, he intoned: What happened?

Ineluctable modality of the calculus exam. I have been evaluated. Close my eyes, bright morning sun, thought through my eyelids, signatures of all the things that I am here to read. Yellow, orange, red at the edges. Contours of the infinite warp of the multi-dimensional. In which direction should the insect head to escape from the heat source as quickly as possible? Depends: Where does he want to go?

See now. Look at these others. Down the steps prudently, squinting, unsteady. Sun, maybe, or they did worse. Does that mean I did well? Watch it: Why aren't there handrails around this building? |x| / x . Doesn't tell you how to get from one step to the next, though, does it? Not that falling would be so bad. Quickest descent. Maximize sympathy for me at the critical point where my derivatives finally kinked and snapped, hurling me into the discontinuities of the outside world. Hands on me, all around me, human contact at last, drawn by my low potential. Fitting, really.

I studied all night. Could have gone out but I didn't. Stayed in, under the lamplight. The well-known posture of the bon élève, as seen on TV. The fantastically challenging Lagrange Channel, 14.3. He's the guy in the garret with the candle, making himself sick, isn't he? Multiplier, he thought. He wished. Head bent over one foot out like that: l . All those contours like a whirlpool, no wonder. Then two functions going at once. Reading about it I couldn't get my bearings. Then: newsflash! The anxiously anticipated blue box. You pay attention to those. I could tell you the channel, the time, the complete listing. Make them seem important that way but then they're watching some other channel. "Interpret the Meaning" on PBS, which who would even think to watch? A consortium of avant-garde filmmakers watches you, back through the screen. Supported by a grant from the National Science Foundation and viewers like you. Like me? The meaning? Is that what they think it is, all those words swirling around the blue boxes? Drown if you leave the raft in the lake Dad yelling so don't.

I mean I did the homework wrote it out sheet after sheet came to class listened tried to read that book. Came to his office all that way not exactly convenient asked questions and I did get it mostly when he showed me. Evens and high numbers if you could get him to do those too sometimes don't say anything look thoughtful but not too bright. Spent more time on this than any other class. Still—this. Disaster. It's like nothing I did gave me a clue. And I paid for this. I mean, a lot of money. Give me something I can use. Show me the money. Just makes me mad, really. Really mad. Spent so much time burning up being angry no time to just do it. There was no time.

What did they want? How did they expect me to behave? All about behavior. Everyone in my group had all those ideas. Give 'em what they want, get through. Lab mice: turn this way jump over this don't touch that button not ready for it gives you a shock—for your own good. I don't know who figured it out. Those two up front of course all eager to please. Rather fail than look so stupid getting all the answers. Rest of us—looking for anything. Guy in front of me telling anyone who will listen just memorize the blue boxes. Sure C. Girl in my group, eyes all frantic like someone with a religious problem: read these other books we used them in high school had this brilliant teacher got all As. Also you'd think they'd never put the high numbers on an exam too long model this interpret that but then there's always something like that on there. You'd think look at old exams but he keeps changing them like he can't decide what he's doing from one semester to the next. I don't know. Think he says relate to other problems. But what if I don't have any idea? Think? Relate? React just survive try to keep an eye out for what they're going to throw at you next. Everything's a trick. Solve it or you drown. End up sorting out the discontinuities working at the lakeside Whataburger in hell.

Laughing? Those two? They can't have done wellsit in the back, look bored, talk about guys mostly. Never pick me for group even when I sit right there look available. They must be delirious—giddy. Old-fashioned, that word. Bumpershoot. Vector. Saw it in that library book digging for those other books girl said from high school old binding all rotted peeling The Calculus open it up and there it is vector from like, what, 1800 something. Dot products, cross products, all of it. The thing's so ancient is what I don't get so why so hard? Less advanced then. Primitive. Josiah Willard Gibbs. I'd be getting an A with a name like that. Probably all jowls and whiskers, he was. How hard to invent a vector, anyway? Put an arrow on it. Gentlemen, I have two brand new products to show you! A × B = |A| |B| cos q , big card on an easel. Smattering of polite applause. Peels it off to show the next one: A ´ B = |A| |B| sin q , and you spin like this. Wild standing ovation. Easier then of course: formulas. You didn't have to understand what you were doing to get the answers.

But they always want something new. Why does it have to be different? What else can we stir into this, they think? Like those Boy Scout camp soups everyone brings a can. Always came out mud couldn't eat it. Makes you feel important like you've contributed something was the theory. So they look for something that isn't already floating in the pot: I don't think calculus has addressed the declining social skills of American youth—let's make them work in groups and express themselves. Grade school gym: square dance with the girls. All the teachers with their hands folded in a steeple up against their mouths holding their breath waiting to see what would happen. Of course we just got by did the steps hated it and they got whatever they wanted anyway. Social skills? Don't they know that nobody would be caught dead square dancing?

Professors, though. You'd think they'd know something. Supposed to. Really, what is up with them? You need all this to succeed, they say. Modern World Today is always the title of the speech. But really? Fat guy all sweaty future boss punching a cigar at me give me the figures Incandenza and if you want to express yourself do it at home. That's what I'm afraid of. Do these people even know? That whole list inside the front cover like an Act of Congress. Academic politics. Get re-elected. Sell books, support the business that supports you. Kids, pay your taxes. Years of lobbying and committees, then the compromise that pleases no one: Be it enacted that today's student of calculus should learn to square dance. And I'm supposed to trust these people?

I try anyway have to do this get a good life it's the way it is can't change it get through it then forget about it never look back. They said this guy was supposed to be good. Like finding a mechanic to work on your car won't rip you off. Guy in line at registration upperclassman hair all over the place said no way he's way too hard stay away get an old guy cause they just like to do the same old things every semester everything's on file. But I thought well but the computers sounded good those I can see practical and then went to his office talked to him so enthusiastic I believed him. Prepare you for real work job and I thought good that's what I want great cause I don't even know what's real at this point. Finally actually learn something good. And I was willing to work but now I don't even know how. Too much stuff that wasn't what he promised. Cheated. Everywhere you go you get cheated.

That stadium wave in class first day everybody up and down what was that supposed to teach us for example? What am I going to do with this I thought. Everybody smiling waiting for the joke he says break into groups talk about it write down the formula. Work together, figure it out, and hey, have fun! Our group didn't get it though that girl just wanted to write down the answer said she knew it and we were all too afraid to say anything then didn't know anyway didn't even know quite what was going on still smiling waiting for the punch line. Next day: C. Serious, then, no joke. I didn't know what to expect after that, right from the start. Afraid he was going to bring in his collection of 78s and ask us to start square dancing.

Now the end. Exam was so quiet serious. Kind of scary honestly he just sits up there with his hands folded watching serious like we've been bad. You're no fun anymore I thought. He thought. Not like class questions groups computer explorations he was so totally into it in class. Doesn't really count though does it cause tests whole different thing no social skills for example. Over half our grade when you count the final so what's with all the dog and pony show? Suddenly it's different: Do it now. Sit quietly figure it out no more "lively engagement of ideas" unless they figure that's what's going on in your head at that point. Is that what they think is going on? Sugarcurves, dancing in my head. Would be nice. Actually more like shock: panic, then blank.

That one with the intersecting surfaces. Where is the max on the curve? Which section was that from? None. Made up. One of his "real life" combinations use your skills. I could see it on the computer. Obvious. But it wasn't a max of   f or g , so I couldn't do critical points, second derivative test, any of that. Which, you know, I can do. They ask all these funny questions instead of just letting you show you know how to do it. Two equations equal. Big implicit mess. Plot it. Parametrize. ...Oh. Maximize with t . Ach! I could have done that. Hard, though, to think of that. Course I just did. Numerically could've too an approximation but he says that's real life he'll accept that. The thing is I don't. Seems wrong somehow I want it exact.

Then that one with the vector fields. Just pictures, no formulas. Try to figure out how to calculate it instead but can't. Just wasted time. Problem was just like that computer project wrote that up put it in words he liked it B+ got it pretty much but what am I supposed to write on a test? What did I say? Around the loop was zero, in the first one but not the second. That's right, isn't it? Said that but didn't know how to show it. Big blank probably got a zero there. Spent all my time going around in a loop. Maybe I'm too conservative. There's the joke.

That sun. Promised bright future, calculus was the Yellow Brick Road. I was counting on it. Glossy book cover like a Pepsi ad all those happy people. At orientation shaking your hand smiling selling it to you how can you not believe it? Of course that's what you want brighter future why else would you be here pay all this money? Nobody plans to work at Whataburger. If they'd just given me something to hold onto told me how to get there tricks of the trade anything. Not sure I can figure it all out myself. Scary not to know. What do I do with communication skills? See those two from the back of the room sometime please help me Ñ ´ up with me ò ò close your eyes now sweet promise of continuous dreams. Not what I bargained for exactly but that's what I got.

The bottom. Where do I go now? Everyone off on their own tangent. Positive div. Time rate of change, flux outward. Tells you all about what's inside the loop but nothing about what's outside. What do I know? Follow the gradient. Or is it opposite the gradient? It's everywhere: sun, bird's path, each of us on our way road up and down. I see it but I can't quite get it can't quite hold onto it and say once just finally once with assurance yes yes yes that is the parameter that is changing here stop it hold it. I want that. Connect, then put it in order. How much do you have to study to get that? Your whole life? Is that the lesson?


[1] James Callahan, David Cox, Kenneth Hoffman, Donal O'Shea, Harriet Pollatsek, Lester Senechal, Calculus in Context, New York: W. H. Freeman (1995).

[2] Sheldon Gordon, Out of the Mouths of Babes: Student Questions and Comments in Reform Courses, Primus VII(1), March (1997).

[3] Deborah Hughes-Hallett, Andrew Gleason, et al., Calculus, New York: Wiley (1994).

[4] William McCallum, Deborah Hughes-Hallett, Andrew Gleason, et al., Multivariable Calculus, New York: Wiley (1997).

[5] Arnold Ostebee and Paul Zorn, Calculus from Graphical, Numerical, and Symbolic Points of View, Philadelphia: Saunders College Publishing (1993).

[6] Wayne Roberts, project director, Resources for Calculus, Washington D. C.: Mathematical Association of America (1993).

[7] David Smith and Lawrence Moore, Calculus: Modeling and Application, Lexington, MA: D. C. Heath & Co. (1996).

[8] Frank Wattenberg, Calculus in a Real and Complex World, Boston: PWS Publishing (1995).