Posts Tagged 'symbolic representation'

How to design video games that support good math learning: Level 6

Show me the action!
Part 6 of a series 

Whether you view mathematics as a collection of procedures or a way of thinking (see my last post), math is something you do. Or cannot do, as the case may be.

When I meet people for the first time and tell them my profession, they frequently reply, “I never could do math.” What they never say is, “I don’t know math.” Everyone, whether mathematically able or not, realizes that math is not stuff that you know, but an activity you do.

Of course, sleeping, sitting on the beach daydreaming, and watching TV are also activities, but they are passive activities. I am using the word “activity” in its stronger sense. That stronger sense certainly includes mental activity. As a simple rule of thumb, you know something is an activity in my sense if doing it makes you tired. By that metric, math is one of the most strenuous activities I know — and I’m one of those people who spend their weekends cycling over mountain passes for seven or more hours at a stretch.

What is the most efficient way to learn how to do something? We all know the answer, and we did so long before Nike turned it into a commercial slogan: Just do it!

If you want to learn to ride a bike, drive a car, ski, play tennis, play golf, play chess, play the piano, and so forth, you don’t start out by attending a lecture or reading a book. Those can be useful supplements when you have reached a sufficient level of proficiency and want to get better. But learning from a lecture or a book require interpretation and assimilation of incoming information (a static commodity), and that in turn requires sufficient prior understanding. No, what you do is start to do it.

Very likely you don’t start out doing it unaided. You seek guidance, from a parent, relative, friend, instructor, professional coach, or whatever. And in the course of helping you learn, that person may well give you instructions and advice. But they do so in the course of you performing the activity you are trying to learn, when what they say makes sense and has immediate, recognizable value.

With everyone, it seems, in agreement that mathematics is an activity, and given our collective experience that mastering an activity is best achieved through doing it, we have to ask ourselves how mathematics education has come to be dominated by the math textbook?

Though there is an argument to be made about the self-interest of textbook publishers, the fact is that mathematics instruction has been delivered through textbooks since the subject began. Archimedes’ Method, Euclid’s Elements, al Khwarizmi’s Al-Jabr, Leonardo of Pisa’s Liber abbaci, and on throughout mathematical history, the symbol-heavy, written text has been the primary vehicle for storing and disseminating mathematical knowledge.

Why? Because putting words and symbols on a flat surface was the only technology available for the task!

But video games — or rather, video game technologies —  provide us with an alternative. The digital framework in which a typical video game is embedded is dynamic and interactive, and can provide the experience of moving around in a 3D world. In other words, video game technologies provide platforms or environments suited (by design) for action. Which makes them ideal for representing and doing mathematics (an activity).

The task facing the designer of a video game to provide good mathematics learning experiences is to represent the mathematics using the natural affordances of the medium. This means putting aside the familiar symbolic representation. My own experience, having been doing this for over five years now, and working with others doing the same thing, is that it is initially very difficult. People have been using symbolic representations or one form or another for several millennia and that has conditioned how we think of mathematics. But it is worth making the effort, because the potential payoff  is massive: it will circumvent the Symbol Barrier, which I discussed in the third post in this series.

In addition, by representing the mathematics in a medium-native fashion, we will minimize, and in some cases eliminate, the degree to which “doing the math” detracts from the game mechanics. For some students — the ones with a natural affinity to mathematical thinking — this is not a big deal, since they will gain satisfaction from solving the mathematical problem, but for many students, advancement in the game will be the main driver.

I should stress that what I am advocating is not watering down mathematical thinking to a “video game version” of mathematical thinking. At a conceptual level, it is the same thinking; only the representation is changing. Once the student has mastered mathematical thinking presented in “video-game language”, a teacher could use that experience as a foundation on which to base instruction in the symbolic representation of the same concepts and thinking.

That last step is an important one, in part because mastery of symbolic mathematics is what is required to perform well in standardized math tests, and regardless of your views on the educational value of such instruments, they are currently a fact of life for our students. But there are two other reasons why it is important to transition the students to symbolic mathematics. First, mastering multiple representations greatly assists good conceptual learning, and the abstract symbolic representation, by virtue of its abstractness, is particularly powerful in that regard. Second, the symbolic representations make it much easier to apply mathematical thinking to a wide variety of new problems in novel domains.

I’ll pursue these ideas further in subsequent postings. In the meantime, let me leave you with three examples of video games that present mathematics in a medium-native fashion: Motion Math, Number Bonds, and Jiji. Notice that in each case the mathematical concept involved is represented in a medium-native, and dynamic fashion. The player interacts directly with the concept, not indirectly via a symbolic representation, in the same way that a person playing a piano interacts directly with the music, not indirectly via a symbolic musical score.

To my mind, this is one of the most significant, and potentially disruptive benefits of using video games in mathematics education: they offer the possibility of direct manipulation of mathematical concepts, thereby circumventing the symbol barrier. Achieving this direct connection to the concepts is not easy. Those three games may look simple. Indeed, to the player, they are simple, and that is the point! But I know for a fact that all three took some very smart folks a lot of time and effort to produce. That’s usually the case with any tool that looks simple and works naturally. Designing simplicity is hard.


I'm Dr. Keith Devlin, a mathematician at Stanford University, an author, the Math Guy on NPR's Weekend Edition, and an avid cyclist. (Yes, that's me cycling on the Marin Headland.)

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