Error

I'm frustrated.

I'm smack dab in the middle of grading the latest homework set in 280, and so far it ain't pretty.

They're having profound difficulty in proving a few propositions with which students have had little trouble in the past, things like "the operations of union and intersection are commutative" and "the operation of set difference is not associative." The second it trickier than the first, no doubt: it requires not only the recognition that a counterexample will suffice and is in fact called for, but moreover the construction of said counterexample.

The former proposition is a fairly straightforward one, or so it's seemed before. For example, all one need do to prove that A ∪ B = B U A is use the fact that x ∈ A ∪ B if and only if x ∈ A or x ∈ B, and then use the "commutativity" of the word "or." Generally students overlook this method proof because it seems to easy; this time around I get the feeling that some students are overlooking this method of proof because they've simply not put any thought into their work.

Indeed, several of the students' papers bear evidence of quick completion, as though they've put maybe a half hour's, or at most an hour's, worth of work into the homework set: there are obvious omissions that seem to spring from haste, errors in transcription that belie similar speed, and signs that very few students are drafting and redrafting their work before it's stapled and slid under my door.

What's most frustrating is the fact that the students have had two weeks' time to meet with me and ask about any uncertainties they might have had about the homework. Judging from the quality of the submitted work, uncertainties they must have had, indeed uncertainties galore.

Why not ask?

I don't bite.

Not hard, anyway.

And not at first.

And several students ought to know this. Several of the people in this class (seven, if my mental tally's right) I've had in some other class before. Of those roughly half of those are doing very well (three, with a fourth coming round quite well in the past couple of weeks), while the other half are struggling mightily.

My long-time friends, you know how you are: why've you not yet felt the need to darken my office door? I know you're having a rough time of it, and there's no shame in that. It's a brutal class, hard as nails and heavy as a hammer. MATH 280 is the stone that stands at the halfway point from the major's entry to its exit, with jagged letters chiseled into its rock-hard face, something about abandonment of faith.

Yet like Dante on his trip through Hell, you'll not go unaccompanied. I'll be your Virgil, if you'll let me, and your classmates, for a while at least, will join you on your way.

If you'd asked, what might I have said?

1. Write, rewrite, and rewrite. Chances are your first attempt will be an ugly one: it'll be clumsy, oafish, and bear little resemblance to English, math, or anything in between. Your first draft of a solution (if it's anything like one of mine) will be barely-readable scrawl; it'll be something like pure thought, a bubbling spring of formulas and figures that needs to be bucketed and brought to boil before it looks anything like a coherent paragraph or proof. Only in the drafting and redrafting of your solution will you work away your initial errors and misstatements. If you're only writing one draft, you're almost certainly writing it wrong.

2. Picture this. If you can't get your mind around the definitions and the quantifiers, draw a picture. Draw a graph. Draw a Venn diagram. Draw an arrow diagram. Draw a concept map. Create some sort of meaningful visual representation of the problem with which you're faced, even if the picture you draw means nothing to anyone but you.

3. For example... If a proposition makes no sense to you, consider what it might look like for small cases or small sets. If a claim's made about all natural numbers n, what happens when n is 1? When n is 2? 3? If a claim is made about all sets, what does the claim say when the set is empty? When it's got one, two, or three elements? If a claim asks about a function, might you not try to see what it says when your function is constant, linear, bounded, or continuous? While examples alone don't often serve as proofs (the exception being when a counterexample establishes the falsity of a universal claim), examples go a long way to helping gain intuition and insight into a problem, and once intuition's gained, a proof is often close behind.

4. True or false? Right or wrong? If you've been asked to decide whether a given proposition is true or not, you can do far worse with your time than play around for a little while, considering both sides and trying to come to a conclusion as to which side makes more sense. This is where (1) and (2) can serve you well: sometimes the right picture can suggest a statement's truth, or an easy example demonstrates that the statement's not true at all because a counterexample's lying close at hand.

5. Wash, rinse, repeat. If the proposition you've been asked to prove is very much like one you've seen before, you might as well try to prove the new claim using a proof very much like the proof you've seen before. Why not? Similar problems call for similar solutions, and often a slight reworking of an old proof will yield the verification you're looking for.

And if you ask, what might your colleagues say?

I've asked a few of them just that. A couple of hours ago I e-mailed the six students who I feel are at the top of the class right now, students who've submitted consistently strong homework sets and exams during the first half of the semester, students who've demonstrated some sort of je ne sais quoi so far this term, and I asked them, if they would kindly oblige, to provide us with a few tips for their classmates on meeting the challenges posed by our course's concepts. I hope that they'll indeed oblige.

While I wait for them to write me back, I hope that those of you who are struggling will take a moment to read and reread the advice I've given above. If you're floundering and feel like you're about to go down for the first, second, third, or fourth time, you're not alone: we're here to help, and the only real mistake you can make is ignoring the hand we're holding out to you.

In other (though, as it turns out, not wholly unrelated) news, I've just tonight had a chance to read an article I've been meaning to get to for quite a while now, David Bartholomae's "The study of error," which appeared nearly 30 years ago in the journal College Composition and Communication (October 1980, 253-269). Like most good papers it's made me want to read several others to which it refers, but its own take-home points are many; as it's nearly midnight and I've got another long day tomorrow (peer review in MATH 480!), I'll limit myself to one, but I'll make it a big one.

Bartholomae focuses on "intermediate systems" of meaning evident in so-called beginning writers of college-level prose, witnessed by "intentional structures" that arise in these writers' compositions: certain sorts of errors crop up when students have successfully constructed idiosyncratic but in some way internally consistent grammars that help them govern their writing. As I understand it, while the gap between the writer's text and a conventional text may be great, an understanding of the writer's intentions (gained through interviewing the writer about her writing, for instance) can lead to an understanding of the writer's personal grammar and therewith and understanding of her understanding of the world.

What does this mean for math?

Consider the following passage from Bartholomae: "They [beginning writers] are not, that is, 13th graders writing 7th grade sentences. In fact, they often attempt syntax whose surface is more complex than that of more successful freshman writers. They get into trouble by getting in over their heads, not only attempting to do more than they can, but imagining as their target a syntax that is more complex than convention requires."

Now consider a line from a paper ("Math and metaphor: using poetry to teach college mathematics") I just submitted to the WAC Journal: "Even when asked to use 'their own words,' [beginning math] students' papers are overburdened with jargon, passive phrasing, and misused terminology that has a 'mathy' ring to the students' ear. The writing is stilted and unconfident."

I can't help but notice how well "syntactically complex" matches up with "mathy": just as beginning writers in the traditional sense often sink because they dive into the deep end, beginning writers of mathematics might sink because they think they've got to sound more like mathematicians than mathematicians really sound. (Two examples spring to mind, both common with 280 students: first, students will often say "n number of elements" in place of the more correct but less technical sounding "n elements"; cf. "7 number of elements" versus "7 elements." Second, consider this one, which I saw quite frequently today: students will often say "there exists an x in the set X" instead of simply "x is in X," or better still "x ∈ X.")

So what's to be done? Bartholomae prescribes close reading with the students, in order to encourage them to catch their errors and learn to perform more effective self-editing.

And so close reading I shall ask the students to do. Those of you who've had a struggle with this most recent homework assignment, please expect me to be calling you onto my carpet in the next few days: I'm going to ask that you stop by so that we can go over one or two of your problems together. It's high time we did so, for both of our sakes.