**Procedures or thinking?
**

*Part 5 of a series*

The vast majority of video games that claim to teach mathematics do not actually do that. Rather, what they do is provide a means for students to practice what they have already been taught. For the most part, the focus is on basic computational skills.

A good example is the first-person shooter Timez Attack. Mastery of the multiplication bonds (times tables in parent-speak) is an extremely useful thing to achieve, and the sooner the better. All it requires is sufficient repetition, and I know of no better way to achieve that than with an entertaining video game.

Such games are the low hanging fruit for the math ed video game designer, and like most low hanging fruit, it has pretty well all been picked, leaving game designers coming into the math ed space having to look elsewhere for a useful application of their talents. The good news is, since repetitive practice of basic computational skills is a tiny part of learning mathematics — albeit an important part, in my view (some educators disagree) — most of the fruit in the math ed orchard is still waiting to be picked. The bad news is, that fruit is a lot higher up, and thus more difficult to reach.

The difficulty hits you as soon as you decide to go for more than mastery (ideally to fluency) of already taught basic computational skills. Are you going to approach mathematics as a collection of procedures or as a way of thinking? These are not completely separate classifications; indeed, the latter is in many ways a broader conception than the former. But they do tend to cash out in very different forms of pedagogy. (Spoiler: instruction versus guided-discovery.)

This distinction is to a great extent relatively recent. Until the nineteenth century, mathematicians viewed the discipline as a collection of procedures for solving various kinds of problems. Originally, the problems studied arose in the world. Then, in due course, the focus widened to include more abstract problems arising within mathematics itself. Proficiency in math meant being able to carry out calculations or manipulate symbolic expressions to solve problems.

By and large, high school mathematics is still very much based on that earlier tradition, so few people outside the professional mathematical community are aware that in the middle of the 19th century, a revolution took place.

Working in the revolution’s epicenter, the small university town of Göttingen in Germany, the mathematicians Lejeune Dirichlet, Richard Dedekind, and Bernhard Riemann pioneered a new, broader conception of mathematics, where the *primary* focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. This was a shift in emphasis from *doing* to *understanding*.

For the Göttingen revolutionaries, mathematics was about “Thinking in concepts” (*Denken in Begriffen*). Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts.

Of course, during the course of this conceptual thinking, mathematicians still made use of procedures. What changed was the primary emphasis. The reason for the change? An increase in complexity, in science, technology, business, society, and, derivatively, within mathematics itself. In a simple world, a few well-practiced procedures can generally get you by. But when things get more complex, you need understanding in order to select from a variety of different procedures, to fix old procedures that no longer work, and to develop new ones.

I give this somewhat lengthy detour through recent mathematical history not because it has a direct bearing on how we teach K-12 mathematics. The one attempt to modify K-12 education to take account of the 19th century shift in mathematics as practiced by the professional mathematicians, the “New Math” movement of the 1960s, was so badly bungled that even a professional mathematician, Tom Lehrer, satirized it. (It was also hardly “new math” at the time, being already a century old.) Rather, I am stressing the distinction between math-as-procedures and math-as-thinking because it is now extremely relevant to the way we educate our next generation of citizens. The complexity of 21st Century life is such that ordinary citizens now need to upgrade their mathematical knowledge and abilities the same way the professional mathematicians did in the mid nineteenth century. The changes in society, and in particular technology and the way we do business, that were made possible by the newer, richer, and more powerful mathematics that developed in the 20th Century, now affect us all in the 21st.

I discussed the growing importance of “mathematical thinking” in a “Devlin’s Angle” column for the MAA back in 2010, and summarized those arguments in a more recent article in the *Huffington Post*. My purpose here is not to argue for any one approach to the design of video games to help students learn math. Heavens, the medium is so new, and there are so few games of any real educational merit, there is scope for a wide variety of approaches. Give me *any* video game that plays well and helps students learn math and I’ll applaud, whatever the pedagogy.

What distresses me is that the medium offers so much promise for good mathematics learning, it is a waste of time, effort, and money to focus on the lowest level — repetitive practice of the basic, procedural, computational skills. We’ve done that. Let’s move on.

Step 1 for the math ed video game designer today is, to recap, deciding whether to develop a game to help students master mathematics procedures or to develop powerful mathematical thinking capacities. As readers of my book will already know, I favor the latter, in large part because mastery of mathematical thinking capacity carries mastery of procedures along with it, just as the person who sets out to build a house will have to develop skills in bricklaying, carpentry, plumbing, and so on, along the way. But as I said a moment ago, the challenge we face in K-12 mathematics education is so great, and video games offer such potential, hitherto largely untapped, I’ll settle for any approach that works.

It’s your call which view of mathematics you take, pre-1850 or post. Both have strong track records. But you do have to make that call, as it will affect every design choice you make from then on. Engineers who set out to build a bicycle and then act as if they are building a car tend not to succeed, even though both are transportation devices. I’ll try to make this blog series helpful whichever way you make the call.

So what is the low hanging fruit on the conceptual tree? Timez Attack spawned such a large number of similar games, that I suspect a successful conceptual thinking game would also spawn others.

What I fear, though, is that the conceptual games will not gain purchase unless they also generate computational benefits.

But the conceptual thinking really is the generalization of concrete experience. So are these two game types really as distinct as you say? Maybe it’s more of a level up. Now that you’ve mastered multiplication, what’s true in general? How does it interact with other operations? How does it extend to other types of numbers?

Thanks for commenting. No, I’m not saying the games are distinct types. Math-as-a-way-of-thinking extends math-as-procedures. Mathematical thinking hits the road by way of the rubber of computation. But they two conceptions of mathematics do lead to different design principles for video games. Two examples of video games that focus on learning concepts are “Motion Math” and “Number Bonds”. Both are aimed at very young children, since that age group is where we find the low hanging fruit on the conceptual tree. It’s worth noting that both those games were developed by very experienced educators and they took a long time to design. The simplicity you see on the screen is deceptive.

This has been an excellent series and i look forward to reading your books. You would have an attentive and eager audience of future mathematicians and programmers if you happen to come to Dallas or if you would be willing to speak to Stanford’s Online High School (http://epgy.stanford.edu/ohs).

I just got done reading this whole article and I would like to say this was very informative. I am only a Math/CS undergrad, but for this spring/summer I was looking into potential creation of a video game that incorporates math concepts, but is yet fun, challenging, and meaningful at the same time. The norm today is that once a game becomes educational, it is bad and unimaginative. Kids will never play something that they can’t shoot something, put a sword through it, or even feel like they are learning anything. Your are absolutely right that creating such a game takes huge amounts of thought since the reason why people play video games is to relax and have fun. Today’s society doesn’t deem learning as either of those, which means video games have big hurdles to jump to become “educational”. Especially mathematics, which I would consider the most painful subject to mostly anyone that isn’t a math person.

Personally, I believe that making such a mainstream game cannot have the conventional computation skills unless they are a well thought out mechanic in the game. A good example of this is Math Blasters. I played this a ton when I was a kid, mainly because I knew I was good at math. But it never taught me anything I didn’t already know or could learn from a teacher. If I am to design a game, I want the math to be hidden behind the mechanics. A game that I see that has done this well is a recently released indie game called Waveform. I don’t know how much you follow the indie gaming scene, but essentially this game is about manipulating a sine wave’s amplitude and period to gather gems with various devices in game to change how the pathing works (black holes, mirrors to reflect the player, etc). Now a player that has never taken a high school algebra class or has just forgot it will never know this is a sine wave and all the game is doing is manipulating its properties. From this simple concept, the game has received fairly positive feedback from Metacritic ( http://www.metacritic.com/game/pc/waveform ), which is an achievement in itself.

If we really want to teach people complex math and get away from computational work, we need to start looking at hard concepts and hiding them behind the scenes. Can we teach elementary school children concepts in Abstract Algebra or Linear Algebra? Sure we can, but we need to be able to find a way to hide the fact that it is actually Algebra in the first place. I recently saw a speaker at a math conference at Lewis and Clark College where the keynote talked about how Linear Systems were embedded into puzzle games such as Myst and other point and click adventures. A player would click levers or buttons to move a certain interface, but coming up with the correct combination on your own proved to be very tedious. When you could write it on paper, the solutions took less time than just fidgeting with it, which was really cool. The only problem is, a person would never have the slightest knowledge that this was a linear system if they never made it into college mathematics… But, if these two examples say anything, it can be done, but it needs to be done right!

Repeats,

Thanks for the comments. I used to think that the math had to be hidden. After working on doing that for almost five years on a commercial project (which did not see the light of day), I am no longer sure that is the required. Here’s why. I agree that symbolic expressions (formulas, equations, etc.) have no place in a video games unless their appearance makes total sense in the context of the game, and hence does not interfere with the game-flow. But (for middle-school math) symbolic expressions are required only if you want to capture the mathematics on a static flat surface, like a clay tablet, parchment, paper, slate, blackboard, whiteboard, etc. Mathematics can be represented up-front, in a natural fashion in an interactive, dynamic medium like a video game. In fact, I would say, in a video game we can be even more explicit about representing math than we can with a textbook. In a textbook, the math is actually hidden behind the symbols. With video game technology, we can provide a direct link to the math, the same way a piano or a guitar provides a direct link to music.

Maybe this is what you are saying. If by “hide the math” you mean keep symbols out, i agree with you 95%. But the symbols are not the math. We may just be using the same words to mean different things here!

These games will not only provide fun for child, but can also teach them something as well. This will help to broaden your children’s horizons and set a strong foundation for their morals as they will be getting something out of it.