No, what I'm talking about is about the world of "math games" that really just consist of drills of basic facts, sussed up so that adding numbers blasts an alien or something. Math games, in other words, that practice very low-level skills and that don't work to develop problem-solving, thoughtful application of skills, multiple pathways, etc.... in other words, math games that turn math into a kind of spelling practice.

And then there's

*Dragon Box.*

*In Dragon Box, you drag cards around a two-sided playing field (looks like a tennis court, behaves like an _ _ _ _ _ _ _ ) to isolate the "dragon box", which is represented by a picture of a sparkling crate [screenshot from the PC version, because it's easier for me to use with Blogger]:*

In the picture above, I've just added a purple fly to the left side to cancel out the black fly -- totally allowed, although unorthodox -- and the computer is prompting me to deal out another purple fly on the right side. By merging the flies together, I can make a "galaxy" (zero) and isolate the box!

As you go through the levels, two things happen. First, the tasks get more complicated: you start having to flip the cards to their opposites before adding them (in this example, I was given the opposite card); the cards at the bottom include ones you don't need to use; the game adds representations of multiplication and division. (Yes, as with addition, the program forces you to distribute correctly, at least in the first 60 or so levels that I've worked.) Second, the program spirals in and out of representations that look much more like traditional algebra:

Notice the

**c**and**-c**cards?
And this brings me to a big-picture musing. My son, in third grade (and not yet doing official Algebra) loves this game. He's learning a set of rules, without worrying too much about the conceptual underpinnings. In recent decades, progressive teachers have moved away from this model: everything should be explained when it's introduced, so that it feels natural. And yet, I'm not so sure. If we wait until after Jonah has played 200+ levels of this game and does the right moves every time to explain the underlying algebra--what have we lost?

My experience in learning mathematics was that often self-contained systems only made sense later. And I don't think that's a terrible thing. The sum and difference identities for trig functions? The first important thing for 99% of students is just to know and be able to apply those identities--and whom do we serve if we spend the first twenty minutes of the period in a lengthy derivation that leaves students angry and confused? And anyway, some of the proofs I've seen -- using matrix multiplication and rotation formulas, for example, as my own

*UCSMP*book does -- aren't necessarily that convincing.
Of course, I know that it's possible to motivate these formulas in lots of ways. My own class on sin(

*A*+*B*), for example, started with the example of sin(*A*+ π/2) ≠ sin*A*+ sin π/2 because of all the cool ways we could see that the "identity" isn't true; then, graphing on the calculator, we came up with a better answer that we could justify using the unit circle. But then I went ahead and taught the "right" rules, and we spent the rest of the period learning how to apply them. The*following*day was when we went back and came up with some proofs (using geometry, actually).
There are lots of questions, besides "why is it true?", that engage a student in understanding a formula:

- What happens when we swap the variables around or change them in some other way?
- What versions of this formula correspond to cases we already know? Does this formula yield the results we've come to expect?
- What about extreme cases?

For example, when I teach Hero's formula in Geometry--which, despite the fact that the proof is beautiful, I do

**without**proof, we discuss the following:- Does changing the order of the sides change how the formula works? The area of the triangle?
- What if a = b = c, so that the triangle is equilateral?
- What happens if a + b = c, so that the triangle is degenerate?
- Why are there four terms under the square root sign? What are the units of the variables and of the answer?

For the vast majority of my honors geometry students, these questions provide a lot of thinking--a lot more food for thought than the kneejerk "Why is it true?"

Obviously, we do a terrible disservice to kids when we say "Math is a collection of rules; your job is just to learn them." The thing most of my strongest students

*like*about math is that there's so little to memorize. But I think it's okay to say "Here, we're going to learn to play with this set of rules for now, and then we'll figure out why they work or what they apply to." At least once in a while.
Agree? Disagree? Let me know. But if you have kids -- and even if you don't -- get DragonBox. What a blast!