Mathematics in Games, Part 1

When my wife and I were buying our first house many years ago our real estate agent Rita quickly found out that I was a math professor.  She gave a response I have heard frequently over the years: “I’m not very good at math.”  At some point during the process it came time to calculate our monthly mortgage payment.  Rita handed me a calculator and said that I could come up with the answer better than she could.  I had been teaching the amortization formula for determining mortgage payments in a class that semester, so I took the calculator and gave it a try.  After several keystrokes I handed the calculator back to Rita with the answer.  She looked at it and said, “That’s not right!”

Question: Who should you believe here?  The math professor, or the real estate agent who said she wasn’t very good at math?

Well, the math professor knew who you should believe: the real estate agent.  Sure, I had been teaching the amortization formula, but it’s a complicated formula.  I might have forgotten a small detail when put on the spot like that, or I might have hit a couple of buttons in the wrong order, or I might have been unaware of how an unfamiliar calculator handles order of operations.  There are lots of little things that all had to go exactly right in order for me to produce the right answer.

What Rita had, on the other hand, was years of experience as a real estate agent.  She had a very good idea of what the payment should be for our loan amount and interest rate.  She needed the calculator to determine it exactly, but her domain knowledge was more than enough to realize that whatever answer I had come up with could not possibly be correct.  I, on the other hand, was a first-time home buyer who had never had to calculate a monthly payment on a loan of that size when the answer had a real impact on my life and finances.  I had no good way of knowing that my answer wasn’t right (at least, not without repeating the calculation multiple times as a check).

Let me drive home the point: Rita, the self-professed “not good at math” real-estate agent, was the more trustworthy expert on a math-heavy question concerning home buying than the math professor.

I think when people say “I’m not good at math” what they mean is that they are not good at the kinds of symbolic manipulation (mainly, arithmetic and algebra) that generally comprise a course in mathematics at the elementary school and junior high levels.  But mathematics isn’t just symbolic manipulation.  For example, it also includes estimation, as Rita displayed in my story here.  And I think that there are plenty of people – like Rita – who say they’re not good at math but who have, in certain domains, developed strong mathematical skills through experience.

Keith Devlin tells a story with a similar moral in his Mathematics Education for a New Era: Video Games as a Medium for Learning.  He talks about a research study undertaken with some young children (between 8 and 14 years of age) selling fruit in a street market in Brazil.  When asked, while working in the market, to do complicated arithmetic calculations to determine the price of several coconuts, the children got the answer correct 98% of the time.  (And they did this in their heads – often making use of some rather clever arithmetic shortcuts in the process!)  When presented, in their homes, with verbal word problems dealing with market stall sales transactions, the children got the answer correct 74% of the time.  However, when asked to solve written symbolic arithmetic problems that were mathematically identical to the market sales problems they only got the answer correct 37% of the time.

I tell these two stories as lead-ins to something I’ve been thinking about lately: Context often matters when it comes to doing mathematics.  And I think it matters with respect to teaching mathematics.

One thing I’ve been interested in the past couple of years is teaching mathematics through games.  My impression of most games that I’ve seen that really attempt to do this is that they repeatedly pull the player out of the context of the game in order to solve a series of math problems.  For example, the online game Prodigy (which my oldest son was really into during first grade) does this.  Prodigy is an RPG (role-playing game) aimed at elementary school kids.  As a player, you explore the game world and battle a series of monsters, earning experience and better equipment, as well as leveling up and acquiring new skills.  The math part appears in that every time you attempt an attack, the game throws a math problem at you.  In order to complete the attack successfully, you must solve the problem.  One thing the game does really well here is to ramp up the difficulty level with the player, so as you gain more experience you’re presented with harder and harder math problems.

However, having to solve a math problem has nothing internally to do with the game world you’re exploring in Prodigy.  As a result, you lose some of the sense of immersion in the game every time you attempt an attack.  In addition, the math problems are externally-imposed obstacles that you must overcome in order to get back to the real fun of the game.  I fear that an unintended side effect of games like Prodigy is to reinforce kids’ sense that math is an obstacle – a chore.  I also fear that they reinforce the impression that math is irrelevant to what really matters.

I also wonder how well the mathematics that a Prodigy player is supposedly learning is really sticking with him or her.  The questions are multiple-choice, for one.  (This is probably a good move from a gameplay standpoint, but this format doesn’t really require the player to think through the problems.)  Even more troubling is the fact that the mathematics is so separate from the rest of the game.  The more we can connect a new concept with things we already know (i.e., the more we can place new knowledge in a context, like with my real estate agent and those Brazilian kids), the more we retain it.  Contrariwise, new knowledge thrown at us that is not embedded in any larger context doesn’t tend to remain with us.

I really don’t mean to hit Prodigy too hard here.  I do think it’s one of the better kids’ math games I’ve seen.  Also, my son had a lot of fun playing it, and I’m glad he was spending time with it rather than some of the other games out there.  But I want to think about how we can do even better at embedding mathematics in games.

I have written a couple of my own games that feature mathematics – A Beauty Cold and Austere and Junior Arithmancer.  They’re text-based and feature no graphics, so that makes their appeal somewhat limited compared to a game like Prodigy.  Perhaps in my next post I’ll talk about what I think I did right in those games, mathematically-speaking.