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?

Wrong.

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.

Design Overview - Puzzles

For Mathbreakers, our level design consists of puzzles, action segments, and boss fights. Many of our puzzles are strict, in that the player can't avoid solving them -- they serve to prove that you understand the mathematical concepts. Our high-octane action segments are what give a lot of life and spirit to our game, and they serve to reinforce mathematical concepts through videogamey repetition. They don't work as proof of understanding though, because they are not nearly as strict as our puzzles. Boss fights allow us to combine many different gameplay pieces into one amalgamation. This post will focus on the first element of Mathbreakers: the puzzles.

Designing Puzzles

Creating puzzles requires a deep understanding of the game's mechanics, so you can mix and match mechanics in a way that allows you to challenge the player. You can approach designing a puzzle problem-first, for instance: "The player has to understand how to find a common denominator to get past this puzzle." You can also approach it from an existing idea, such as: "How can I challenge the player mathematically by combining swarm enemies and floating number hoops?" In either case, you will be manipulating these game mechanical tools:

Simple number objects: Number spheres and cubes are the basic building block of all our puzzles. You can pick up these numbers, throw them at eachother to add (+), and load them as ammunition. When two number objects add to zero, they disappear. If they add to a different number, one consumes the other.

Items, weapons, and tools: All of these can be earned by the player and they manipulate numbers. The multiply wave is loaded up with a number as ammunition, and when activated it will multiply everything the wave touches by that ammunition value. The halving sword divides any number it touches into two equal parts. And so on!

Number walls: These are simple number objects but arranged in a way where they block the player's path until destroyed.

Number monsters: Also numbers, but they can hurt you. Some of them rush at you with spikes, others will zap you with a laserbeam.

Hoops: Walking through a hoop multiplies or divides the number(s) you're carrying. A hoop may be a times 3/divided by 3 hoop. This means if you walk through it one way, it multiplies by 3, and if you walk through it the other way it divides. It also affects all your ammunition in the same way.

Gem zones: Inside the zone, there is a rule or a pattern for gemification. For instance, all prime numbers in a zone will be turned into gems. Or, in another zone, all powers of two become gems.

The possibilities with just this small set are enormous, but we have a few more on the way.

Here's an example of one puzzle framework, created by mixing and matching mechanics. The actual number values of the pieces could be made more interesting, and the values of the hoops is left undefined.

Fast Pace Spikey Chase

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).