JoaLDG: Negative Reinforcement (or, Sin and Redemption in Mathematics)

In his book Innumeracy John Allen Paulos, the veteran professor and exegete of mathematics and statistics, recounts a story about some pilots and their instructors.

This phenomenon[, regression to the mean,] leads to nonsense when people attribute [the regression] to some particular scientific law, rather than to the natural behaviour of any random quantity. If a beginning pilot makes a very good landing, it's likely that his next one will not be as impressive. Likewise, if his landing is very bumpy, then, by chance alone, his next one will likely be better. Psychologists Amos Tversky and Daniel Kahneman studied one such situation in which, after good landings, pilots were praised, whereas after bumpy landings they were berated. The flight instructors mistakenly attributed the pilots' deterioration to their praise of them, and likewise the pilots' improvement to their criticism; both, however, were simply regressions to the more likely mean performance. Because this dynamic is quite general, Tversky and Kahneman write, "behaviour is most likely to improve after punishment and to deteriorate after reward. Consequently, the human condition is such that ... one is most often rewarded for punishing others, and most often punished for rewarding them." It's not necessarily the human condition, I would hope, but a remediable innumeracy which results in this unfortunate tendency.

He goes on to give two paragraphs' worth of further examples about movie sequels, music albums, baseball players (what is it with Americans and baseball?), and stock markets, but the lesson for would-be pedagogues is clear: don't treat your subject with statistical rigour and suffer the fate of all pseudosciences that came before.

The contrary view alluded to in the first quoted sentence, namely, the ascription of intentionality and significance where there are only the capricious mechanisms of probability, is, I believe, a symptom of a whole another and different ontology of the physical world than the scientist/naturalist's, one which is surely mistaken. And it is one which is far too vast to take on in the screed of a single evening. So we'll consider ourselves admonished on two levels, namely with regard to our evaluation of our methods, and with regard to our concepts of praise and blame, and press on.

(By the way, every living person in the developed world, I kid you not, should read one of Paulos' books. It doesn't really matter which one, they're all more or less the same. Innumeracy is good and A Mathematician Reads the Newspaper is too. Go read!)

Not to put it in too maudlin terms, but I think I believe in positive reinforcement. As a very general principle this is connected to the Nietzschean optimism that (on our weblog's happier days, if I have succeeded) is our spirit here. (I keep saying that's our spirit, anyway.) If one "doubts with well-founded suspicion" that good things are possible, I want the courage to build a world, speaking literally or of the world as a metaphor for my ontology, which those good possibilities populate. As a principle of pedagogy it is maybe a reaction to something, some grousey grouch in my past, or maybe a recognition that the undergrads of today are the colleagues of tomorrow -- put another way, that the aims of a class are largely but not entirely about the students' command of the syllabus material and I am judged not merely by the standard of review for an educator-as-mechanism.

On the face of it, there's something very suspicious about negative reinforcement. I don't insist on being loved or liked -- I'll settle for respected -- and I don't repudiate Burrhus Skinner (entirely). It's just unclear how this negativity is going to get any desired result, particularly when many students are already very anxious about math, or about their academic position. What intermediate steps have failed that we need to take out this big club with its dangerous and indeterminate consequences? I fear that its advocacy comes from exasperation, but my perverse optimism tells me it could still lead to good things.

Because there's exasperation to be had. I read an article, "Teaching Freshmen to Learn Mathematics," whose author, Steve Zucker, (a professor of mathematics), took the following extreme (?) position. "We shouldn't," he argues, "overlook the power of negative reinforcement." He goes on to describe two mathematical errors often committed (in the past -- he tells us that their frequency has since dropped off) by his Calculus II students which he calls the "ultimate sin" and the "penultimate sin."

(They are, for the interested and mathematically inclined, respectively (a) the belief that a series, say the harmonic series, converges if its terms vanish; and (b) the computation of a limit of a sequence whose terms are given by some expression in n, like (1 + 1/n)ⁿ, by selectively letting instances of n go to infinity -- in this case, inside and then outside, concluding that the limit is 1, when, as everyone knows, it is in fact e.)

He says that the demonstration of these sins by his students argues some very dire things about the relationship of the student to the course and even to the instructor. (It's clear at any rate that the students don't understand what they've been taught if they make such errors, but naively we might wonder about the value of introducing eschatological language to describe these mistakes.) He informs his students that commission of the penultimate sin on any submitted work will immediately earn zero credit for the problem, and commission of the ultimate sin will earn negative credit.

I have to admit that I'm a little envious. I'm pretty sure I couldn't get away with that.

Anyway, since we're all here to hear about linear algebra, and not about calculus, which no one understands anyway, I have a couple of candidates for confusions that drive me up the wall. Following precendent, I would like to propose, as an "opening bid," the:

• Penultimate linear algebraic sin -- phrases like "the vector is linearly independent" or "the vector is linearly dependent." I would also like to take on the concept of a "redundant vector," which to my view is simply pedagogical lemonade encouraging a nonexistent intentionality, but them lemons are in the textbook, with a footnote about how "redundant vector" isn't actually a linear algebraic concept but the author has found it "useful" in teaching, if you can believe that, so I'll save that one for another post.


• Ultimate linear algebraic sin -- any sentence of general type, "a basis for the kernel of matrix A (or another space) is the span of such-and-such vectors."

I'm not wedded by iron chains to these as my two least favourite things to read on students' papers, but it's hard to take a shot at things like co-ordinate chauvinism or implicit inner products which are, among other things, too far advanced errors to strenuously indict. (We'll see if I still feel the same way about co-ordinate chauvinism in the next two weeks, when eigenvectors and similarity of matrices are the topics of discussion.) The two cited transgressions show, roughly, that the student is confused about the relationship between vectors and vector spaces: the role of linear combinations in general and their significance to the concepts of independent sets and spanning sets (and bases) in particular.

And besides, these not-silly-mistakes (as Zucker would put it) -- especially the ultimate linear algebraic sin -- show up with depressing regularity. So I think I can summon a roughly analogous frustration to our poor calculus instructors -- though, mind you, I'm just grading these papers, not teaching the class. So how do I feel now about negative reinforcement?

I'm giving it a try. As I said I don't think I could get away with negative credit for problems, and anyway it would be a completely unfounded move for someone who doesn't even have any interaction with the students to tell them why. I mean, I can't even explain to them verbally what's wrong with what they've written, but commission of the ultimate linear algebraic sin is cause for loss of two points from five, no matter what the problem was about, and no appeals, damn it. It's not right for this nonsense to get written and for me to say nothing: that doesn't do anyone at all any favour.

How's it working? It's a little tough to evaluate, as I said, not being the instructor. I don't recall reading an instance of that sin lately. It drives my blood pressure up a notch to see so I'm pretty sure I'd remember if it had been there -- and, hey, I have been privately calling it the ultimate linear algebraic sin for a couple months now. Forgetting that would be like forgetting another fall of man. On the other hand, the students haven't been asked too many times to give a basis for the kernel of some matrix, and when they do, now that I think about it, they've been saying things like "the kernel of A is the span of such-and-such vectors," and not even using the word basis, as though they'd like for me to draw an inference, just answered like they would have in the first weeks of the class.

I...

I think my plan may have had some unintended consequences. Like apostasy.

Negative reinforcement, huh --

At this stage I'd like to remind everyone that it's not my fault, I didn't do anything, it's just regression to the mean. Somehow.

I'm going to abruptly end this post on the pretext that it's already long enough and, umm, further data must be collected to continue the discussion.