~ Not Quite in Defense of Number Cubes ~

§ What the hell is a number cube?

I don’t think about number cubes very often, but when I do I get angry. I can usually manage a few relatively peaceful months, and then something like this happens:

why do these high school probability questions keep calling a 6 sided dice a "number cube". That's dumb.

— Rhett Allain (@rjallain) October 20, 2015

And Rhett’s right: that’s dumb. We should all get up in arms about how dumb it is. Which we do. Anecdotally, there are two main reasons people get upset. The most common response, from People Who Know These Sorts of Things, is that the word “die” is too closely associated with gambling for curriculum companies to utter it in public — which is both ridiculous and believable. The other major complaint, from People Who Like Language, is that we already have a perfectly serviceable word for this item, so there’s no need to introduce a superfluous noun phrase that sounds jargony and lame.

But if I turn down the volume on my cynicism for a minute, I can think of at least two reasons why “number cube” isn’t the absolute worst.

§ Number Polyhedron

Number cubes are geometrically well defined in a way that dice aren’t. Any Platonic solid makes a good die, so if you want to talk about the cubic variety,¹ that requires some qualification. “Six-sided die” is probably your best bet, which is even clunkier than “number cube.”² “Cubic die” should work, but that’s not in common usage either — and doesn’t strike me as a major improvement. If we’d rather not modify “die,” our other option is to modify “cube.” Since we’re talking about a cube with numbers on it, “number cube” is actually pretty straightforward.

Look, I still don’t love it. I’m just saying that, if we want to SMP6 like responsible math educators, our two immediate avenues — limiting the scope of “die” or augmenting the nature of “cube” — are both a little awkward, and I don’t see a compelling reason to hate “number cube” more than “cubic die.” Certainly not to the lopsided extent that people really seem to hate “number cube.”

§ What the hell is an inning?

Here in the U.S. of A. we tend to use a lot of games as examples when we talk about probability.³ That makes sense: it’s a setting where most people have likely had some kind of tangible experience with a subject that’s notoriously counterintuitive. Of course experience is a highly subjective thing, which can mean trouble.

In college I took a probability course that included four students from China. Our first exam had a multi-part question about a hypothetical double-header between the Mets and Yankees. The minute our poor professor finished handing out the papers, he turned to see many frantic waving hands from the Chinese delegation. What is an at-bat? How and why does one become out in an inning? So the pitcher throws the ball, but then sometimes it isn’t a ball? When does the two-headed person enter the game?

By the end of the pre-test clarifying questions, we had covered a significant portion of the major league rule book⁴ and eaten up thirty minutes of exam time. All so we could (finally) get on with the business of calculating the probabilities of some compound events.

I’m a fan of context in math tasks, but there’s always the possibility for it to alienate a (potentially large) subset of students. If context improves a problem to the extent that it justifies its own risk, it offers a net gain. Since the notion of “die” is sensitive to experience — I can imagine a student (correctly) arguing that the probability of rolling a pair of threes is 1/400 — and already highly artificial, I see no real harm in replacing it with a slightly more neutral artificiality. The risk of die confusion, though small, doesn’t have much of an upside. Why not do away with it?

So yes, number cubes are still dumb, but we can probably ease up a bit. We have more pressing concerns. Somewhere there’s a two-headed shortstop running amok.

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