The Subjunctive Thing

In yesterday’s post—part of my before-reality-sets-in idealistic lunacy—I briefly mentioned the subjunctive mood while talking about inferential statistics. That deserves a little elaboration. (I elaborated on it quite a bit in a paper (here, so you can read that if you wish. This is shorter.)

The subjunctive mood is the bane of many language students. One of the reasons is that in English, the subjunctive is becoming invisible. It still exists in a few places (“If I were to give you an A for that work, I would be doing you a disservice” is correct, if pompous) but even that construction is vanishing (“If she was teaching summer school, she couldn’t go to Hawaii” sounds increasingly OK).

One of the reasons to use the subjunctive is to express something contrary to fact. That is, I’m not giving you an A. She is not teaching summer school. It also expresses something you might do in the future, when you’re not sure of the outcome: If I were to give you a puppy, would you love me forever?

Aside. Note that we could also say, “If I gave you a puppy, would you love me forever?” In that sentence, gave is subjunctive, but it looks just like past tense even though it’s in the future. That’s one reason it’s hard to identify subjunctive in English. Note how the indicative If I give you a puppy, will you love me forever? seems different: it’s more an offer than a hypothetical.

Statistical inference is fundamentally subjunctive: we’re saying, if Belinda had no power and if she were to flip 20 coins over and over, how often would she get 16 heads? It’s a hypothetical question. In an orthodox stats class, you would hardly ever flip actual coins; but using George-Cobb-inspired randomization tests, that’s exactly what we do (in simulation at least) all the time. We take the contrary-to-fact subjunctive and make it real.

I claim that one of the things that makes inferential statistics hard is that the machinery is based on a strange, hypothetical, subjunctive, contrary-to-fact set of assumptions and procedures that none of us are well-equipped to understand for more than about 30 seconds at a stretch. So to the extent that we can alleviate some of the unreality, students will have a better chance of understanding what it’s all about.

Do I have evidence for this claim? I do not. I will at least get some insight into it next year. With a real class, I wonder if I will see any evidence one way or the other…

Randomization

If you’re not a stats maven, this may sound esoteric, but let’s see if I can express it well.

One of the things that’s hard to learn in orthodox statistics is the whole machinery of statistical tests. You can train yourself (or a monkey) to do it right, but it seems to be a morass of weird rules and formulas. Remember to divide by (n–1) when you compute the standard deviation. You have to have an expected count of at least five in every cell to use chi-squared. You can use z instead of t if df > 30. And then there’s remembering what tests to use in which situation. You wind up with a big flowchart in your head about whether the data are paired, whether the variables are categorical, etc., etc., etc. And as a learner, you lose sight of the big picture: what a test is really saying.

George Cobb wrote a terrific article explaining why this is all unnecessary. The short version goes like this: you can unify a lot of inferential statistics if, instead of the tests we now use (z, t, chi-squared, ANOVA…) we used randomization tests.

Here’s the basic idea, to which we will often refer as the “Aunt Belinda” problem. Your Aunt Belinda claims to have supernatural powers. She says she can make tossed nickels come up heads. You don’t believe her, so you get a dollar’s worth of nickels (20 of them); she speaks an incantation over them; you toss them all at once; and sixteen come up heads.

Does she have supernatural powers?

Continue reading Randomization

Motivation

I’m inspired to write this by a small group of what seem to be like-minded but more experienced math teachers around the blogosphere (whom I will reference as soon as I learn how). They are scary smart, young, and energetic, and can probably actually type. They sure as hell can write. And you seriously want your kid in their math class.

If they’re so great, why should I blog? Although I can dream that my unique contributions may help others, the real reason is to help me record what I’m thinking, what I try, and what happens in the classroom.  I’m in such a good situation, it seems to demand some sort of record.

Put another way: What happens when you have no excuses? I’ll try to live up to the trust implied by all the resources I get from the school community, and I’ll try to make something good out of a situation most teachers would give various limbs for.

I’m also hoping that by putting some of my plans out there—such as using standards-based grading, OMG, I can’t believe I actually put that in print—I’ll feel some obligation to follow through.