The difference between game and drill

So in my last post I opined that the optimal mathematics game in the Tiny Games spirit should “incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom.” That led to some confusion.

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Let me try a do-over:

During a game, when the primary action of the players is indistinguishable from doing traditional homework or test problems, it is a gamified drill.

Gamified drills are not always bad. However, they’re not the sort of thing I’d say counters the notion “that students must learn and practice the basic skills of mathematics before they can do anything interesting with them.” They are what Keith Devlin calls a “1st generation educational game”.

There’s lots of gamified drills. It’s easy to do: just take what you normally would do in a math problem review and tack on a game element somewhere (for me it’s usually Math Basketball). To be integrated the primary actions of the players will require using mathematics in a way that is linked with the context of the game. That 1-2 Nim requires understanding multiples of 3 is inextricable from the game itself and not interchangeable the same way Math Basketball can be easily switched to Math Darts.

Tiny Games, mathematics edition?

So there’s a Kickstarter project closing today which has me wondering about mathematics potential.

Tiny Games: Hundreds of real-world games, inside your phone.

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The concept here is to have games suited for different settings that can be described in only a few sentences.

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Could one make an all-mathematics variant — mathematical scrimmages, so to speak? The only games I could think of offhand in the same spirit as Tiny Games were some Nim variants and Fizz-buzz.

1-2 Nim (for two players): Start with a row of coins. Alternate turns with your opponent. On your turn you can take either 1 or 2 of the coins. The person who takes the last coin wins.

Fizz-buzz (for a group): Players pick an order. The first player says the number “1”, and then the players count in turn. Numbers divisible by 3 should be replaced by “fizz”. Numbers divisible by 5 should be replaced by “buzz”. Numbers divisible by both should be replaced by “fizz-buzz”. Players who make a mistake are out. Last one in wins the game.

Anyone have some more?

EXTRA NOTE: One condition I’d add is the games need to work as games and not as glorified practice. “Challenge a friend to factor a quadratic you made” meets the “Tiny” but not the “Game” requirement.

EXTRA EXTRA NOTE: Dan Meyer asks “Aside from the counterexample that follows, what are the qualities that make Fizz-Buzz and Nim gamelike and not, say, exerciselike?” In both cases the games incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom. Even though Wikipedia claims Fizz-buzz was invented for children to “teach them about division” (?), my first encounter was from The World’s Best Party Games. (This still doesn’t totally answer the question, I know. A related question is: what is the difference between a puzzle and a math problem?)