Looking for overt mathematical formalism in all the wrong places (or, Thoughts about game design and the ETS Assessment Games Challenge)

A colleague at the university sent me a link to the ETS Assessment Games Challenge, a game design competition that encourages the use of video games to teach—and assess learning of—mathematics skills. There are two particular learning progressions that they want to see turned into game experiences: equations and expressions and functions and linear functions. Each of the two progressions provides a five-level model for learning, and each level has an explanation of its corresponding skills and misconceptions.

This intrigued me as an interesting context for study during my Summer of professional development. The challenge exhibits several positive traits. I like math, and I like game design, and I like teaching, and I like writing software. Furthermore, there's built-in expert evaluation through the judging format, which is great for a scholarly approach. Thus inspired, I spent a few minutes yesterday looking through my cabinet of board games for inspiration in game mechanics.

In my casual analysis, it seems that there really aren't a lot of games about expressions or about linear functions. Many games can be modeled using them, of course, but there aren't many about them. For example, there are many games where you have to amass resources to purchase items. Let's take Donald X. Vaccarino's Dominion as an example, in which you may wish to buy a province for eight gold. If I have a hand of five cards with gold values, call them c₀ through c₄, then I could pose the question as to whether this equation is satisifiable:

c₀ + c₁ + c₂ + c₃ + c₄ ≥ 8

Of course, I'd actually do it this way:

but that's just being pedantic.

When I play Dominion, however, I don't put it this way. It's a word problem—Do I have enough gold to afford a province?—and the sum happens in my head without formal representation. Playing games with my five-year-old son, know that he can do simple two-value sums (on a good day), but the notation doesn't seem to offer him any help in the game. My point is that you could apply this formal notation as part of the game of Dominion, but it's not necessary, and so from a game design perspective, it should not be there.

There are lots of games where algebraic thinking is quite useful as well. I had the chance to play Bernd Brunnhofer's Stone Age last weekend, an excellent Euro-style worker placement game. During the game, you collect four different kinds of resources. One way to earn points is to use these to purchase building tiles, many of which have costs like "spend four resources of four different types." In the game, they represent this with a custom syntax. (Sorry for the poor image quality.)

Once again, I could represent it more formally, where r_i is the number of resources spent of type i.

Solve for all r_i. Hey, I'm learning equations and first-order logic! Well... not really. It's just another case where I can model it as a mathematical formalism, but it doesn't gain me anything when the complexity of "spend four resources of four different types" fits neatly in my head. One might argue, then, that we have to raise the complexity to something that doesn't fit in the head, so that the mathematical formalism is necessary for managing complexity, but I fear that this will very quickly then become digital homework.

As I turned my eye towards games with specific equality relationships, rather than inequalities, I came up with only two good examples. David Sirlin's Flash Duel (although the pertinent part is actually Reiner Knizia's En Garde) and Cribbage. (Whoah... Cribbage has a Board Game Geek Entry?! And it was designed by Sir John Suckling in 1630?! I love the Internet!) Let's take Cribbage as an example, specifically the play portion. Here, players alternate playing cards and adding them to a running total. If you can make the total 15 or 31, you score two points. When my opponent leads with a card, my first question is "Is there a card that would make the total 15?" Let's say she played a 7. Then, unlike the cases above, I feel like this is rather cleanly represented as

7 + x = 15

Solve for x, and I know that if I have an eight, I should play it. Novice players will think only one step ahead, but as you gain skill at the game, you learn to foresee what your opponent might have. For example, you never lead with a five, because:

5 + x = 15

Here, x=10, and there are more tens in the deck (counting Jack, Queen, and King as ten) than anything else. There's yet more strategy to the play as it relates to the show, but we'll leave that as an exercise to the reader.

Playing Flash Duel is similar, and even more explicitly mathematical. If my pawn is on space 14 and my opponent's pawn is on space 8, then I think, "14-8=6, and so if I have two cards in my hand whose sum is six, then I can pull off a dashing strike." When I play with my five-year-old son, he occasionally says, "Can I do some math on this one?" This means that I put my finger on his space while he moves his pawn as if he were playing cards from his hand, to determine whether he can pull off an attack or not.

I've been playing cribbage for as long as I can remember. It's one of my family's favorite games. In fact, one of my most prized possessions is the board my father made for me upon my college graduation.

Do I enjoy math because I learned cribbage, or would I have loved math without it? That's a great question that's impossible to answer. A real research question can be made out of this, though: would a child learn equations better by overlaying them onto a cribbage game interface? That is, if one took cribbage as-is and displayed formal notation as I did above, would one learn the concept of variables and equality faster than if one did it by lectures and worksheets? I think that's a great question, but it is undeniably a research question, not the kind of thing one should bank a game design on.

There's no denying the mathematical nature of board games and video games, and I don't doubt for a moment that players learn mathematical thinking by playing these games. The ETS Assessment Game Challenge is clearly directed at more traditional assessments of mathematical reasoning, as expressed through conventional notation. It seems that there are two ways one could move forward with this. The first is to design a game where the math notation is overt, where the explicit mathematical representation is a necessary part of the game. This is an interesting design challenge. The other way to move forward is to design an assessment that can get at the tacit knowledge students get by playing games, to show that a student has learned the idea of equality, for example, without having need of traditional formal representation. This is a high-risk proposition, as transfer would be difficult to show (a research problem) and you'd also have to convince the judging board that your design actually matches their solicitation.

I don't know how much further I will personally pursue the competition. It certainly got me thinking, and I am eager to see what comes out of it whether I have an entry or not. Writing this essay has given me the opportunity to articulate some of my thoughts about the competition, and maybe it will be useful to you, dear reader, in your own attempts to create game-based-learning activities. In the meantime, if you know of any games that use mathematical notation as an integral part of gameplay, I'd love to hear about them in the comments.

EDIT:
Found one! Number Scrabble
and another! Equate
(Actually, my wife found both. Thanks!)