Correspond to two other biscuits

I awoke with a gasp, mid-turn, sweating under the covers. The bedroom was cold, a half-open window letting in the December air, and my short breaths sent little clouds into the night which dissolved before my eyes. The cruel memo was burned into the back of my mind. I saw it clearly.

”The end of semester is upon us. Prepare the final midterm. Next week. Before the vacations.”

That was a week ago. Maybe I could have been ok, if I’d shifted less around or denied everything, but the students found out somehow. They always find out. I had to tell them something. I panicked. With a hollow and tinny voice, like a purposeless man in a candy shop holding two lollipops, I told them the midterm would be on Friday. Tomorrow.

I thought I’d have time. That I could come up with a few exercises before then, or at worst steal some from other textbooks. But the textbooks were shallow uninspired things. The students would know their problems hadn’t come from me. They would ask questions, always questions, probing for any weakness.

The thought was unbearable. No. No, I’d have to make up my own exam. Inspiration would come.

That was then. A week ago. In the long ago. This was now. I had nothing. No exercises. No exam. Nothing. I’d tried, I really had, but each problem idea was worse than the last. Have them calculate the distance between a point and a hyperplane? Too abstract, they’d never make it. How about doing some intersections of lines and planes? No, no, no, that was completely trivial, it’d never do.

At some point I’d gotten up and sat at the couch in my living room, hunched over the low table before it. There seemed little point in going back to bed. I got some pieces of paper and a pen and in what little moonlight that made it past the buildings towering over my window I made failed attempts at points and lines and planes.

Then it came.

Two points. Six vectors. Two planes, defined by different means.

Oh yes. Yes. It felt right. Convert between the two forms, tell me sweet little lies, and bases; what if there were two, side by side in orbit, around the fairest sun?

I sharply inhaled and briefly ran a hand over my chest up my neck and bit a finger before turning turning returning once more to the page and imagined a linear system, then another. One overdetermined at first sight but not at the next, the other seemingly involved and generous yet not.

There was yet something missing. Then a mad thought.

Oh no. I couldn’t. I couldn’t. Not linear applications. I’d only defined them two days before, I couldn’t possibly include them here?

Could I?

Teasing my way around familiar letters, I traced an opening parenthesis, hesitated once, twice, then wrote coefficients and variables in quick succession until none remained, close parenthesis. Show me its kernel. Open up its image. Gaze upon its properties. Behold.

A moment passed in silence before I ruined it and moaned softly at the exam draft before me. I picked it up as if to convince myself it was real and set it down again. And I looked upon it and I knew it was good.

Dawn came later, with coffee and dairy products in contrasting colors as clichéd metaphors in waiting, then a dash to a photocopy machine, pull on your finger, another finger, then cigarette. The students followed shortly.

One by one they read the copy I gave them. Two by two their shoulders slumped. Sitting closely by, I knew and envied them. To see such beauty for the first time.

Some of the students cried softly. At the beauty of the exercises, I imagined at the time. Later, while grading the results, it became clear this was not the case.

Advertisement