Thursday, November 14, 2013

Logical Leaps: When Math is Out of Context

When we first started building a math game, we thought it would be easier compared to other subjects since we wouldn't encounter any discrepancies in interpretation. (Imagine trying to tackle a history game!) After all, mathematics is supposed to be the universal language -- a common tongue spoken across diverse cultures with little variation (even though the level of literacy may differ). We figured that we would do all the fun, creative game design stuff and leave the math-y heavy lifting to our code. Mathematics = calculations, and therefore the computer can handle the execution because the answer can only be right or wrong. Right?


Here's the thing -- math is not just about performing calculations, and neither is it a universal language. We have merely come up with a near-uniform set of symbols to represent mathematics, refined over centuries of standardization. The "universality" of this system crosses terrestrial borders but, as far as we know, does not extend beyond human civilization.

Math itself is subjective. While calculations may produce consistent outcomes, there could be a myriad of different meanings and associations attached to the math depending on who's doing it. Those meanings and associations are probably not innate; rather, they're developed in the learning context. In the Western education system, that context is generally the symbols and calculations themselves, and students typically don't attach any other meanings to math until they start applying these concepts in the "real world." That could mean counting change as a cashier or leaving tips after a meal (I still mess that up constantly and could use a little help… level designers, are you listening?) Or, if you're a lucky duck like my surfer friends, math could mean optimizing surf times through complex calculations of wind velocities, tide changes, etc. (Although these same people are thoroughly perplexed by basic arithmetic, go figure.)

Dr. Keith Devlin's research has shown that children in developing countries that help their families run market stands could perform complex calculations on the fly with over 90% accuracy. When asked to do the same calculations on paper, that accuracy rate drops to about 40%. Conversely, Dr. Devlin gives an example of American students on a field trip to Mt. Diablo -- in a class full of trigonometry aces, not one could figure out the height of the mountain based on its distance from where they were standing. For each of these kids, math has been taken out of context.

Context gives math meaning. I'm not saying that mathematics must be attached to real-world applications in order to have meaning at all -- abstraction can be a beautiful thing and even a fun toy to play with -- but you'd have to be able to wrap your head around it first. The question is, how do we bridge that gap between the concrete and the abstract and vice-versa? When we say we want to build a game that teaches math, what we're really trying to do is help players make the logical leap when they transition from one context to another.

In the Mathbreakers world, we started off by making our numbers "tangible" -- whack a number with a Factor Hammer to get its prime factors, blast it with a Fun Times Wave to multiply it, chop it with a Halving Sword to literally produce two halves -- you get the idea. This approach removes the player from a typical symbolic context and lets them play with mathematics as though it's something tactile. We thought that was good enough (actually, we thought it was quite brilliant) until Dr. Devlin wisely pointed out that each gadget is but one way to teach a concept, and a single concept needs to be reinforced in many different ways before students can begin to grasp the abstraction.

Does this mean our math gadgets could essentially shape the meanings that our (young and malleable) players associate with the corresponding operations? While we would feel pretty proud if a child instinctively reached for her Halving Sword whenever she needed to do division, we're probably not helping her learn math by giving her just one or two tools for performing each operation. Every math gadget is really just a subjective interpretation of its inventor. We need to aggregate all sorts of different subjective interpretations of one concept in order to form an objective abstraction. If that's the case, no one can ever saturate the demand for gadgets that teach division or any other concept -- and our math toy box grows infinitely bigger.


  1. Yes, my child immediately got the sword correlation to "cutting" whole numbers into fractional parts. He thinks the sword is "totally cool" and hopes to see the sword "power up" in other levels.

    1. That's great feedback! The sword is in several levels right now, and we hope he has lots of fun with it. In case he's interested, he can also chop enemies with it, not just regular numbers. :-]