This past weekend I gave a talk entitled “Mathematics Through Narrative” at the NarraScope conference on narrative games at MIT in Boston. Much of what I discussed I had also put into a series of blog posts several months ago. (See part 1, part 2, and part 3 of those posts.)
However, in the talk I also presented a set of five theses on mathematics in games that did not appear in those blog posts. Here they are, with commentary.
Mathematics in games should…
- Be integrated into the story world.
- Be used to solve puzzles in-game.
- Feature, if possible, repetition.
- Allow the player to play with the mathematical concepts.
- Avoid symbolic manipulation as much as possible.
Mathematics in games should be integrated into the story world. This was the primary point I was making in part 1 of the series of blog posts from several months back, so perhaps just a quick recap is all that’s necessary here. First, we retain new ideas (both mathematical and otherwise) when we can situate them in a web of mental connections that we already have. (I don’t think this is a particularly controversial or outlandish claim.) Therefore, games that attempt to teach mathematics should integrate the math into the story world. They should not yank the player out of the story world and force him to solve mathematics problems that are otherwise irrelevant to that story world in order to advance in the game. To do otherwise turns the gaming experience into a homework set with a fancy wrapper around it. It also likely leaves the impression that mathematics isn’t relevant to real life. We can do better than that.
Mathematics in games should be used to solve puzzles in-game. This follows from the previous thesis. If we’re going to integrate math into the story world, how do we go about doing it? Well, lots of games from a variety of genres have puzzles that must be solved in order to advance in the game. And mathematics is a great source of ideas for puzzles. It’s also an underutilized one: There is a lot of mathematics out there, but I have seen very little of it actually appear in a game. I’m sure that’s because mathematics is perceived as inaccessible. In addition, game designers aren’t used to thinking about mathematical ideas as part of their puzzle-building toolkit, beyond some standard logic puzzles. However, there are examples of mathematical ideas – even advanced ones – appearing in games. Jessica Sklar has an article  on linear algebra in game puzzles. I also have an article  that discusses – among other examples – the Klein bottle puzzle in Trinity and Zeno’s bridge in Beyond Zork. Then there’s my A Beauty Cold and Austere, whose gameplay is all about solving mathematics-based puzzles. There is still a lot more mathematics out there that could be tapped to create engaging challenges for players, too. Combinatorial optimization problems in particular seem like a rich, underutilized puzzle source.
Mathematics in games should feature, if possible, repetition. I don’t think this is a particularly controversial claim. After all, it’s clear that we learn and retain ideas not just when they’re placed in a context but also through continued repetition. I put the qualifier “if possible” in there because the repetition goal can cut against other design goals. If a game is going for breadth, mathematically-speaking, then there’s not going to be space in the game to hit every mathematical idea multiple times. In addition, if a game just wants to feature a few mathematical ideas as puzzles or has more of an exposure-to-mathematics goal in mind, then repetition becomes less important. But if a game really has the intent to teach mathematics, then repetition must be one of the features the game designer has to keep in mind.
Mathematics in games should allow the player to play with the mathematical concepts. This is related to the previous thesis. One way to implement something like repetition without creating multiple puzzles that feature exactly the same concept (which, if not carefully done, could get boring for the player) is to create a mini-sandbox featuring the mathematical idea you want to get across. Then the player can engage the concept from multiple angles, seeing how it responds under different choices she makes. I’ll draw an analogy here with people: If you want to get to know someone, you’ll have to see how that person responds in lots of different scenarios (e.g., with friends, with family, in good times, in stressful situations), not just in the context in which you originally met them. The same is true with ideas – mathematical and otherwise. You can’t truly understand an idea until you look at it from multiple perspectives. One example in A Beauty Cold and Austere is the sequences-and-series machine. By playing with the various controls on the machine you can create sixteen different settings for the golden path leading out of the room, each of which represents a different sequence or series. Following the path for a particular setting takes the player to a number space corresponding to the limit associated with the underlying sequence or series. Some of these paths represents solutions to puzzles, and some just exist to allow the player to engage with the mathematics.
Mathematics in games should avoid symbolic manipulation as much as possible. An expression like is a concise way to represent the alternating harmonic series and its value, but it’s also a foreign language to the vast majority of players out there. As such, when the typical player encounters something like this expression in a game it is not inviting or engaging. Instead, it’s intimidating and off-putting. As game designers it is our responsibility to find ways to express fascinating mathematical ideas without triggering players’ phobias around mathematics, many of which are (I’m convinced) tied to those ideas’ symbolic representations. For example, when I gave my talk at NarraScope I displayed the alternating harmonic series equation above as an example of something to avoid. Then I demoed a couple of puzzles from A Beauty Cold and Austere, the second of which was the sequences-and-series machine. After playing with several of the settings to see what happened I picked a particular collection of settings and asked the audience whether we could predict where the golden path would lead. We agreed that these settings would not get us to infinity (unlike another setting) nor result in values that bounced around more and more wildly (also unlike another setting). Instead, these settings would create a path that would eventually settle down to a limiting value somewhere between 0 and 1. And sure enough, that’s what happened, with the game telling us that the actual value was the natural logarithm of 2. Then I pointed out that we had just recreated the ideas behind the alternating harmonic series equation I had displayed fifteen minutes earlier, without using any mathematical symbols at all. That produced multiple murmurs from the audience, including a “mind blown” comment from someone sitting near the front. (I’ll admit that that was my favorite moment of the whole talk.) The point is that we can avoid symbolic manipulation when representing mathematics in games, even if we have to get creative in how we do it.
So, these are my five theses on featuring mathematics in games. It’s a full ninety fewer than Martin Luther had. I also did not nail them to one of the classroom doors at MIT.
- Sklar, Jessica. “Dials and levers and glyphs, oh my! Linear algebra solutions to computer game puzzles.” Mathematics Magazine 79(5), 2006: 360-367.
- Spivey, Michael Z. “Mathematics through narrative,” Math Horizons 26(3), 2019: 22-25.