Reading, Writing, and Rigor

My career as a serial extortionist was triggered by an act of theft — more specifically, by an honor student’s appropriation of another student’s words on a homework assignment.

To tell this story properly, I should back up a bit and describe my earlier, non-extorting self.  When I was a naive young assistant professor, I was convinced that if I wrote up detailed solutions to the homework problems I assigned, students would eagerly read them and absorb not only habits of effective problem-solving but also habits of clear writing.  I know for a fact that the students appreciated the extra effort I went to in writing up the solutions; they praised me for it in their end-of-term evaluations.  There was only one problem: over the course of years, it became clear to me that hardly any of them actually read my solutions.

THE PAPER CHASE

I tried appealing to students’ self-interest.  I described to them the scene in the movie “The Paper Chase” in which the hero, a young law student, breaks into a library so that he can see his professor’s own law-school notebooks; he thinks that if he can get a glimpse of how his professor’s mind works, he can do a better job of giving the professor the kind of answers he wants to hear.  After describing the scene, I would tell my students, “You don’t have to break into a library to learn how my mind works or to find out what I regard as a perfect solution; just read my solutions and use them as models of your own!”

I didn’t realize just how ineffectual these exhortations were until the incident of plagiarism I mentioned earlier.  Two students in my honors calculus class handed in identically-worded solutions, in violation of my stated policy that students can work together on solving homework problems but must write them up independently.  It was pretty clear to me who had plagiarised whom (one of the students was handing in work a full letter-grade higher than the other), so I asked the presumed culprit to stay after class so that we could talk.

“Do you know why I want to talk to you?” I asked.

“Yeah, I do,” he answered unhappily.  And indeed he did.

“So why did you do it?” I asked, after he’d confessed.

“I always work with Mindy,” he said (not her real name), “but she always gets higher grades than me on her proofs, even though our proofs use the same ideas.  So I thought I should just use her words.  It was stupid, and it was wrong, and I won’t do it again.”

I appreciated his show of remorse, but I wanted to delve more deeply into his underlying problem, namely, how one goes from knowing why something is true to explaining it clearly to others — that is, how one goes about writing a proof.  “Have you tried reading my posted solutions? That is, have you tried copying from me, the way you’re supposed to?”

“No,” he admitted.

And that’s when I realized that desperate measures would have to be taken.

WHEN BRIBERY ISN’T ENOUGH

Let’s back up a bit, again, to discuss the unavailing measures I was already taking. One of my teachers in graduate school, Elwyn Berlekamp, offered $1 for every new mistake found in his book (Don Knuth and other authors do it too). This kind of incentivized quality control system appeals to me, and I’ve used it in my own teaching for many years.  Students in my classes can get small amounts of money (and small amounts of course credit) for finding mistakes in the course web-page, or the syllabus, or the textbook (even though I’m not the author).

I do this because math is a subject that requires, among other things, the habit of close reading.  If you can only see tiny details, and can never see the big picture, you’ll never be the kind of mathematician who creates new ways of seeing — but if you can’t see tiny details when you choose to, you’ll be no kind of mathematician at all.  Many students come to college with little practice in the art of close reading; they’ve been rewarded for skimming, and that’s what they know how to do best.  So when they see a box like the one shown below (a theorem about limit laws from James Stewart’s calculus series), they skip over the preconditions at the top (“Suppose that c is a constant and the limits exist”) and just remember the five formulas.

Then on homework assignments or on exams they apply the formulas in situations where the preconditions aren’t satisfied, and they’re surprised when they learn that their answer was wrong. (See Endnote #1.) “But I used the formula from the book!”

Part of the trouble is that many students have been conditioned to think that formulas are what’s important, not the words that surround and contextualize them.  But just as importantly, these students have learned to read too quickly and have forgotten how to read any other way.  They skip the “fine print”, even when it’s a vital part of what they’re reading.

So, rewarding close reading with money from my own pocket (usually between 50 and 100 dollars a year) sends a message to the students that attention to detail is something I value.  I’ve done it for over a decade.  Occasionally I’d even slip in a mistake on purpose and challenge students to be the first one to find the deliberate error.  But dollar bills and bragging rights weren’t enough of an incentive to convince many students to read my solutions when I posted them online an hour or two after collecting the homework. And dollar bills and bragging rights, and a desire to get high marks for his solutions, weren’t enough to coerce Student X into reading the posted solutions.

BENEVOLENT BLACKMAIL

That’s why I now lock up my homework assignments. Not the solutions; the questions. That’s the desperate measure I resorted to. It’s all done in the interest of helping the students learn, but there’s still an element of Nice little GPA you’ve got there. It’d be a shame if you got a zero for the next calc assignment because you didn’t even get to find out what the problems are. Call it blackmail, extortion, coercion, whatever: it seems to work.

When I post a new homework assignment (in the form of a PDF that I upload to the course web-page), the file is locked with a password, which is an extraneous non-mathematical word (like “turtledove”) that can be found in the solution to the previous assignment.  So, at least in theory, before you can get started on the nth homework assignment, you need to read the solution to the n−1st homework assignment.

Some of you (the kind who instinctively attend to boundary cases) are wondering, What about the case n=1?  That is, what password unlocks the 1st homework assignment?  It can’t be the solution to the 0th homework assignment!  Quite right; instead, the password that unlocks the 1st assignment is hidden in the course syllabus.

I didn’t use to stress the importance of reading the syllabus, until I had my share of incidents in which a student missed an exam (“I forgot there was a midterm”) or, after failing to hand in any homeworks all semester, complained that he didn’t realize that homeworks counted towards 25 percent of the final grade (“Homework only counts 10 percent in other courses I’ve taken”).

My system of locking up homework assignments can be gamed in various ways (for instance, one student can share the password to another), but on the whole, I find students rise to the challenge of trying to read my solutions, at least when they don’t have lots of exams coming at them all at once. I suspect that shame plays a role; students don’t want to admit to their peers that they couldn’t find the “Easter egg”.

Every semester, there’s at least one student (I’ll say “he” since it’s usually a male student) who says “Professor, I just read the whole syllabus twice, and I’m sure you made a mistake; there’s no extraneous non-mathematical word in there.” I ask the student “Did you try reading it aloud?” The student invariably says “No.” I say “Try it again, aloud.” The student always reports back, sheepishly, that he found the extraneous word.  That student has just learned something extremely important: that he is capable of thinking that he’s reading every word when in fact he’s not.

This is a lesson that I was lucky enough to learn at an early age, from a picture very much like this one:

If you’ve never seen it before, skim over it quickly, and then read it aloud — one, word, at, a, time. (See Endnote #2.)

Once you’ve learned that you aren’t always reading text as closely as you think you are, you’re on the road to training yourself to be able to read closely.  And close reading is intimately tied to the kind of “close thinking” that math requires, especially the more advanced, theoretical kind of math.  You need to be able to pass back and forth flexibly between the micro and the macro.  You need to see both the big picture and the minutiae, both the forest and the trees.

I ask my students at the end of each semester whether they like the whole hidden-Easter-egg thing, and they always say they do.  They appreciate the fact that I nudge them to do something that they recognize is in their best interest.  One time I asked “Don’t you feel that instead of using these sorts of tricks I should just treat you like adults?”, and one student replied “But we’re not adults — we’re college students!”

NOT QUITE THERE YET

I’m still tweaking my extortion scheme.  It isn’t foolproof; students who are in a hurry (and what student isn’t?) can game my system by training their brains to do a modified kind of skimming, with a limited sort of lexical analysis that distinguishes between words that are mathematical (“limit”, “integral”) and words that aren’t (“turtledove”, “ambassador”).

One piece of feedback I’ve received from some students is that I’ve gotten the timing wrong.  Students have told me that they feel that the best time to read the solutions to an assignment is not right after they’ve handed it in (when the problem is freshest in their mind) but rather a week later, when they get their solution back with comments from the grader.  So maybe the password that unlocks the problems for the nth assignment should be hidden in the solutions to the n−2nd?

Thanks to Sandi Gubin.

Next month: Swine in a Line. (For a preview, see my Barefoot Math videos on the game.)

ENDNOTES

#1. Here’s one example (similar to, but simpler than, the sorts of problems that typically trip students up in this way): Say I ask my students to compute the limit of 1/x − (1+x)/x as x goes to 0.  Some students will reason as follows: “By formula (2) from Stewart,

;

but since these two limits are undefined, the answer is undefined as well.”

This is incorrect reasoning, because one of the preconditions of formula (2) is that the two limits must exist.  If they don’t, all bets are off!

(I suppose a seriously confused student might write , but this hasn’t happened. Yet.)

For this specific problem, it’s easy to convince a student that “The limit does not exist!” is the wrong answer; if instead of applying formula (2) one simplifies the expression 1/x – (1+x)/x, one gets .

What’s really frustrating for a teacher is when a student uses incorrect reasoning but nonetheless obtains the right answer; it can be hard to convince the student that such a solution deserves less than full credit.

#2. The version of this puzzle that I saw as a child used the word “SPRING” instead of “SPRINGTIME”, and it was in a rectangle instead of a triangle; if any of you know which kids’ puzzle book I might have seen it in, please let me know!

#3. Something else I saw as a child, in a similar vein, was a version of what nowadays is often called The Three Minute Test: does anyone know where it comes from?