What are the criteria? Basically, the game has to engage brain areas related to specific thinking skills as shown by research. Most people would say, okay, so with a math game you want to activate the computational areas of the left hemisphere. But just doing math itself does that. Great, so any game in which you do math is a cognitive math game. Not quite.
Take Timezattack, pictured above. In this online 3D adventure style game your avatar (the little green guy) explores a 3D environment in which you battle monsters by solving multiplication problems. The software tracks your rights and wrongs, gives remediation and so on all within the world of the game. You control your travel through the world, but there is a sequence to things and, from what I saw of the first level, a linear path to take (also left brainy, which is good, right?).
One thing that originally attracted my interest to Timezattack was the 3D play space, which engages the hippocampus and is probably responsible for the spatial attention gains noted as a result of playing action games, which support math ability. But the 3D is all front facing and linear, and for all the great graphics and battle format, this game boils down to dressed up drills. Not that these aren't effective in teaching math--they claim that after 30 minutes a week of game play for 8 weeks any student (barring dyscalcula) will know their times tables inside out. I believe them, and I believe the game is fun compared to other things you do in math class. But if it was really fun, kids would want to spend more than 30 minutes at a time on it, wouldn't they? It's designed for schools and from my limited game play it seems to belong there.
Built as a world of numbers, CM forces the player to use numbers to DO things. The pedagogical style reminds me of Minecraft, an open ended environment where you are an active, free willed component of the game and in which you become part of the environment. To survive in CM you not only have to solve math problems, you have to think in numbers. Such thinking engages not only the left hemisphere but the right, which is needed for insight problem solving, the kind of "aha" moments that make discovery exciting and gameplay addictive.
Just read these instructions for one of their puzzle frameworks they call Fast Paced Spikey Chase, taken from their blog:
In this puzzle, the player has access to no weapons. The player must guide the spikey monsters through the multiply hoops in a particular order and run them over the number walls. If the spikeys are -3/4 by the time they hit the wall, the wall will break and the player can move to the next part of the level. Getting them to -3/4 requires guiding them through the +/- hoop (-1/2), the /2 hoop, (-1/4), and then finally the *3 hoop (-3/4).
This kind of challenge requires all kinds of neural connections--timing, aim, calculation, emotion--resulting in an understanding of numerology that goes far beyond memorization. That's the cognitive component, and that's why I'm looking forward to the full release of Crystal Mathematics (even the name is cheeky) and eventually using it in my workshops.