SBG: One Dollop of Fear

I should also record what I’m afraid of. Here’s one that keeps coming back:

Suppose I really do try standards-based grading (SBG). I’ve been reading and lurking. It sounds really attractive. But in order to have SBG, you need S. You need a list of standards (or objectives or outcomes, whatever).

There are lists of these things for statistics, but they don’t list everything that I value. So I’m trying to figure out how to write that list. The problem is like—you know when you’ve been dreaming, and you wake up, and for just an instant it’s all there? But the moment you start telling about it, two things happen: the thing you’re not talking about dissolves, and then what you remember is the words, not the underlying ideas?

That’s what I’m afraid will happen about these other, less effable things that I care about.

At somebody’s suggestion (probably Meg’s) I have been keeping a document that’s a brain-dump list of whatever ideas I come up with, and it is purposely unordered. It’s allowed to be redundant. I don’t have to organize it. And so far, it has gone pretty well. I think the guts of a good list are in it.

In addition, I know that I’ll have ideas in the class and in collaboration with the class, and they’ll let me add or modify these standards on the fly.

What kinds of things am I thinking of? besides, you know, content, I want to include stuff we might call “habits of mind” and the like, such as:

Data Goggles. When appropriate, the student spontaneously looks for the data in a situation and does something useful with it (e.g., make a display).

House of Mirrors. The student consciously and explicitly uses multiple perspectives (graphical, tabular, model, formula, etc) to get insight into a situation through its data.

Could these be standards? By writing these down, do I lose the thousand other wisps of aspirations for my students? Will kids point-grub to get high marks on these? Experienced SBG-ers, any advice welcome.

We’re Idiots!

A great quote:

People will often make the case, “We can’t be that stupid, or we would have been evolutionarily wiped out as a species a long time ago.”  I don’t agree. I find myself saying, “Well, no.  Gee, all you need to do is be far enough along to be able to get three square meals or to solve the calorie problem long enough so that you can reproduce.  And then, that’s it.  You don’t need a lot of smarts.  You don’t have to do tensor calculus.  You don’t have to do quantum physics to be able to survive to the point where you can reproduce.”  One could argue that evolution suggests we’re not idiots, but I would say, “Well, no. Evolution just makes sure we’re not blithering idiots. But, we could be idiots in a lot of different ways and still make it through the day.”

—David Dunning

From an op-ed by Errol Morris in the New York Times

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…


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


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.