**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.

Thank you, Dr. Devlin, for this fascinating article. We’ve worked on using video (now apps) to teach math for a while. And there are quite a few “math games” that teach in a “drill and kill” fashion. You’re article opens up more possibilities, such as a video game that is a mystery where each clue needs to be solved before acquiring. You are absolutely right that designing simplicity is most complex!

A limit to this method might be that the possibility (curriculum-wise) of only low-level (K-8) topics will be taught. Even if K-8 math is taught extremely well, how can a video games of this kind teach higher level math (precalculus and calculus) – which is where even those who’ve done well in K-8 math struggle?

This is a good point. I discuss it in my book. Even for K-8, I don’t view video games as the primary educational vehicle. Rather, in terms of a student’s overall education, I see them as useful supplementary tools. For me, the primary actor is the human teacher. You can certainly make use of video games in teaching beyond K-8, but then you are talking about mathematics that is intrinsically symbolic, and the pedagogy has to be different. I think games the allow for user generated contents can be particularly valuable there. For an example of one way to use video games in university level mathematics instruction, check out the video http://www.youtube.com/watch?v=LYGwaI-haOM of Brianno Coller’s work at Northern Illinois University.

This is a dangerous spot. Some previous efforts at reform failed because they did not deliver the symbolic skills that people recognized as success in math. Manipulative support, alternative algorithms, good technology use or alternative representation … all have been derided because we never worked on the transfer. It’s okay for it not to be in the game, but I think the burden is then on the teacher to do/lead the generalization to symbolic.

The revision and extension of our quadratics game is looking at connections to symbolic representation. (iOS, http://itunes.apple.com/us/app/parabolax/id451650904?mt=8, free)

At some point in this series will you get to your view on the place of games in math education? Are they high engagement moments? Are they extra-curricular? Are they extra-scholastic, ‘tricking’ students into doing mathematical thinking without knowing it? Are they to eventually replace the current curricula?

Regardless, thanks for writing this series. I’ve been enjoying it and sharing it.

Yes, I intend to get to all those points. (Most of them are actually touched upon to a greater or lesser degree in my book.) In their Jiji game, MIND Research addresses the issue of mastery of symbolic mathematics within the game. At the higher levels, the symbols start to appear. The other option is to let the teacher focus on the symbolic math learning. There is no reason not to have both.

For the record I am not a fan of “stealth learning,” where you try to trick the player into learning. Trickery is not a good basis on which to build a teacher-student relationship, even if the teacher is a game designer who the student never meets. Besides, the trick won’t work. Kids will see right through it. (I did work on a commercial, stealth learning MMO for a few years, but we were never going to *hide* the fact that it was math-focused. Rather, we were just going to release the game as a game and let people make of it what they would. Sadly, like many large-scale projects, it did not result in a released product. But I learned a lot in the process.)

In some cases, a “math edu” game could play well and do well in the market simply by virtue of being a good game. Portal and Angry Birds are self-evidently physics-based games, and physics teachers have been using both in classes. There was no attempt to hide the physics — how could they? Equally, they did not present those games as “physics based.” They were simply digital games that happened to be built on basic principles of physics, which became hugely successful. The same thing will happen for math sooner or later. (Actually, Sudoku, which is an out-and-out math game, is a hugely successful example that exists already. There are other examples, but they are nothing like as successful, and hence not so well known.)

But for most math ed games, you likely won’t get the same kind of market success as Angry Birds. They could still provide significant educational benefit, however.

I’ve been following your series with interest, partly because it’s my kind of topic and partly because I’m just embarking on the home schooling of a small child who seems mathematically inclined. She also seems inclined towards self-education through computer play. She adores the activities (both reading and math) at starfall.com, which are designed for 4-6 year olds. I would go on at length about how positive our experience has been with Starfall but I’m afraid I would sound like a paid shill. She is not at all put out by their explicitly educational nature because she has no negative associations to get in the way.

In the comments from an earlier post you discussed your frustrations with funding. I’m curious how much money you need to take the next step in development and whether you’ve considered a Kickstarter campaign. There are all kinds of games there that have raised tens or even hundreds of thousands of dollars in small contributions from interested individuals. Combined with some clever viral marketing among parents and educators you might get enough money to keep moving. It isn’t going to pay for an epic virtual world but it might let you take a single-concept game all the way to polished completion. I’d kick in $25. $50 if I get a T-shirt.

For my own part, I’m interested in the distinction you make between conceptual math (“doing math”) and symbolic math. I did well in math until calculus, where I struggled a bit and stopped my formal math education. I now work as a software engineer in an environment where there is lots of calculus, linear algebra, and statistics going on around me and I feel the lack of formal background rather keenly. I often find that, even at this level, the mathematics can be readily understood (and even implemented in software) using intuition, story-telling and analogy. But only if you have a translator. An inability to read and manipulate the symbolic representation on one’s own is crippling.

As I think you said, middle school level math (say pre-algebra) represents a turning point in math education, where the symbol barrier gets suddenly steeper. But even if one passes that point fairly easily, the difficulty remains high. The symbols get more cryptic, the textbooks get harder and harder to follow on your own, and having a firm foundation in simpler principles becomes ever more essential. Point being, we don’t need games (or other new learning methods) merely to teach fractions or to bootstrap kids into algebra. We could really use them at higher levels, too. At those higher levels, the gap between conceptual understanding and symbolic understanding seems bigger to me. Do games offer a way to close that gap?

Anonymouse: I could write an entire book on the significance of this statement you make: “She is not at all put out by their explicitly educational nature because she has no negative associations to get in the way.” One day maybe I will. But not now. :-)

As to using video games to help learning higher-level, symbolic mathematics, I addressed that above. I guess I should add a further comment. I don’t know that video games cannot *on their own* help develop those abilities and skills — as opposed to being used in a motivational and evaluative way, as Brianna Coller does, for example. I don’t know because I never tried it. And I never tried it because I could not see how to do it. I don’t think video games can do that. But then, when I first heard about Facebook and Twitter I thought neither of them would get much traction, so I’m not much of a technology soothsayer! :-)

I would like to read that book. Could you write it in the next year or two while my kids are still young?

As for games that teach higher level symbolic math, I think they might be able to take on a very different form than games for younger children. For example, they might just be increasingly difficult puzzle games in which you have to manipulate the notation correctly to finish levels. Glorified worksheets with a plot. By this stage in one’s education, one can be expected to be actively engaged in understanding the underlying ideas rather than getting trapped in Benny’s local minimum.

Have you ever seen Manufactoria? (http://pleasingfungus.com/Manufactoria/) It is an incredibly effective teacher of a certain kind of abstract thinking.

Maybe that could work. Having never been involved in the development of such a game, I don’t know. My gut feeling is that user generated content or the Briano Coller approach are the ways to go, in part because I think the goal of higher math instruction *is* the symbolic stuff. BTW, I can’t get Manufactoria to play. I’m guessing it’s Windows only.

Thanks very much Keith for your kind words about Motion Math. We’re looking forward to reading the rest of this series, and to developing more video games in which children can directly play with numbers.

We’re curious: how important do you think it is that learning games include not only an interactive, but also a constructivist component? Sometimes we worry that our games, like most video games, limit the world of the player; ideally a learner should be encouraged to create their own examples, meanings, and applications of concepts. Do you think we should try to design games with constructivist ambitions, or is that best left to the classroom?

Jacob: I agree it would be good to have games that let students act creatively in the kinds of ways you suggest. Hence my remarks regarding user generated content in some of my earlier replies, though I made those remarks in reference to players in grade levels 9-12 and above. In terms of gameplay not in conjunction with a teacher-led class, this strikes me as harder, since you need to provide the player/student with good feedback on what they have done. Success in the game with player-generated content could be achieved by a hack that does not advance mathematical understanding.

My feeling is that there are so few good math ed games so far, it makes sense to keep the current focus on games that try to develop conceptual understanding (and skills mastery — I’m not knocking skills practice, just arguing that there is more to achieve than that) in the way exemplified by the three games I mentioned, including your Motion Math games.

If someone could come out with a great game that involves student created work, having a mechanism that ensures good, informed feedback, that would be a valuable addition. But apart from games designed to be used in some sort of class setting with a teacher monitoring results, the only feedback mechanisms I can think of have clear drawbacks.

One obvious danger you’d want to avoid is any kind of crowd sourcing feedback mechanism, where some players could ridicule others for their efforts. Even done anonymously, that could have huge negative consequences.

I am sceptical about the phrase “serious gaming”. The idea that students like playing games, I agree with, as well as puzzles. We should however be careful not to model too much on existing games. Portal was mentioned. This is a first class game with great graphics. As Keith says: the game was there, the principles were secondary, students learn. When making curricular applications you can’t just slap a game on a concept you want students to learn. And even if it could be done, the bad graphics and gameplay makes NO comparison with existing games. Don’t call it gaming, call it “play” or something.

That being said, they should be designed by educationalists who know the classroom. Otherwise, it would just be “luck”. This is a field that is better developed than “using video games in mathematics education: they offer the possibility of direct manipulation of mathematical concepts”. Unfortunately, some people don’t know as they have only been looking into this with the advent of tablets. Tablets that don’t come with java and/or flash (this is why manufacturia probably doesn’t work). It would be sad if lots of work of the last decade(s) would remain unknown because of this.

Agreed on all points. Thanks for writing. Way back in January, in my second posting in this series, I argued for the need for bringing many different experts into the design process. Part of what makes the design, build, and distribution (with ongoing support) of good edu video games so time consuming — and usually so expensive — is the need for a substantial team.

Oh, and that penguin JiJi isn’t functional at all. For the 1d form. x+x=2x but it is also an area. I’m not sure whether these games cater for such a flexible skill. And especially the transfer from visual to symbolic often is problematic.

Lots of questions!

What do you think will be the role of trial and error in math learning games? Similar to the “Benny’s rules”, students could try various things until they hit on the solution that works. How can I tell what exactly a child is learning by playing something like the Jiji games? You recommended three games, but what was the true basis of the recommendation? What is “good” about “dynamic” or “medium native”? Is it just that those traits help retain a child’s attention or they have some more fundamental cognitive role? Is there any such thing as over-reliance on dynamism?

The Jiji games present symbolic content at higher levels. What about symbolic interaction? The mechanical engineering education video (Northern Illinois University) said they had students write programs to drive the cars which would require the students to think abstractly about the problem they’re solving. What might be the equivalent of that for younger children and lower level math?

What are your thoughts on the “Logo” language for exploring geometry? You mention two categories – “math as procedures” and “math thinking”. How about “math as exploration”? Perhaps, in some environments, children may discover things that we don’t know yet? What might be environments/games conducive to that? What about robotics kits like the lego mindstorms? If we look at math as abstractions of reality, wouldn’t it be even better to base the math education on reality itself versus simulations of it? How would you compare, say, the tools used in Montessori schools to, say, the Jiji games (which are the best thought out design I’ve seen so far in that category as far as I can recall).

How to handle transfer is a difficult call. It’s a difficult issue however you come at it. In my own work we have set our goals on providing materials that play well on their own as games, but are designed so a teacher can leverage and/or use them in class to explore concepts and develop good conceptual thinking, leaving the teacher to address transfer.

When I started to use Motion math, Zoom and Hungry fish in my classes I could see progress in students´ thinking. Instead of just doing, they began to think mathematically while they were playing the games.

I can add that they are very motivated.

I agree that it is my job to help them to the next level by using symbols. It´s the same thing as when I want them do draw a picture when they solve complex problems. I don´t leave them at that stage. They need the symbols as well, but first they need to be confident in the concept before they get to the abstract level.

Thank you for the tips about jiji and number bonds. I´ll look into those games to see what they have to offer.

Happy Easter from Sweden

Birgitta Wikström

Hello,

I’ve been working on educational games for about 4 years now, and share most of the opinions you’ve expressed in this series. Thanks for writing this up.

Math games are harder to design for me since a lot of research about math misconceptions is focused on symbols. It is hard to strike the balance of adding just enough symbols to allow transfer to the material schools are used to. I was curious to hear what your opinion was on the usage of symbols in this game: http://games.cs.washington.edu/Refraction/

Thanks!

-Francois

Francois, I love Refraction. I have no problem using symbols as names, as in that game. What I find great about that game is the way the player has to manipulate the fractions in a meaningful way. The reasoning is with the actual fractions, not with symbolic representations of them. Great title too. Of course, it comes from a world class team! Thanks for writing.

I had to show this particular article, “How to design video games that support good math learning:

Level 6 profkeithdevlin” with my best pals on twitter.

I personallyonly needed to distribute your great posting!

Thank you, Mel

Seems like you actually know plenty pertaining to this specific issue and that shows by

means of this specific article, labeled

“How to design video games that support good math learning: Level 6 profkeithdevlin”.

Thank you ,Magaret

I came across your work in the latest American Scientist. It looks promising – especially considering that you’re going beyond the drills and cute bunnies.

My first question is, why do you limit the platforms to portables? Is it because those are more suited to what you’re doing? I’m an old salt, rooted in desktops (before that, mainframes), who recently made the switch to laptops.

Money! We had limited initial funding, so focused on the best platform to reach the largest sector of our initial target audience. Our aim is to produce Android and Web-based versions for the initial audience, then build out to older age groups. (Though to my mind there is a significant loss in going away from touch screens, since they allow the player to manipulate directly the mathematical concepts, just as the pianist or the guitarist feels the music in their fingers.) Our startup business strategy is to use the iOS version to help us raise additional investments.

I read your article in “Best Math Writing 2010″, which I had for some time – and thought the name was familiar. That’s going to lead me to “Intro to Mathematical Thinking”.

In it you mentioned mathematician’s notebooks – much like Beethoven’s scores – with things crossed out, paths leading to dead ends; and that when we see the proofs – or hear the symphonies – we see the final, close-to-perfect result of a lot of struggle.

So I’d really like to see Euclid’s notebooks.

A good proof is sort of like a chess game: you wonder why he started here, where he’s going, then all of a sudden, it’s “check”, = QED.

Speaking of proofs, have you read about Mochizuki’s proof of the ABC Conjecture? So far, nobody understands it, and Mochizuki isn’t talking. I hear that if the conjecture is really proved, it would clear up a lot of other areas.

Would it be unmathematical to assume the proof is valid, then go to the other areas that seem to depend on it, and see where that leads?

Excellent and very insightful series – please do share more on the topic, if time permits.

I see your point about the symbolic barrier, and the ability of people to perform exceptionally better in a more natural setting that doesn’t involve symbols. I feel that there are two ways to attack that problem – by avoiding the symbolic barrier to make players feel comfortable and spend more time in the game, or by introducing symbols in a game with simple enough rules thus inducing acceptance for the symbols into the realm of their natural setting. I am somehow biased towards the latter, and in my most recent game (Sumurai, iOS and Android), I have taken that approach. I have avoided most of the 8 mistakes you mention (in contrast to my earlier games, where I am guilty of most).

However, I have decided to use symbols (arithmetic and numbers) directly and invite people otherwise uncomfortable with symbolic manipulation to see how easy and fun it can be (by providing a gripping gameplay).

I agree that it may be hard to get people to play games with more symbolic load. However, there may just be a path to be treaded which involves starting with very basic symbols (like basic arithmetic, as in Sumurai) and working up to more complex symbolic manipulation on the same lines.

Sumurai requires only basic arithmetic, and a few simple rules, but the levels of complexity are astronomical within that space (akin to other intellectually stimulating games like Sudoku, Chess, the Rubik’s cube and Go-moku) . The game is about numbers and operators being moved around in a triangular grid to reach a valid state. Most people who tried the game have found it intriguing, challenging and mostly addictive – in spite of not being very symbolically comfortable. Perhaps there is some potential for symbolic games to regain acceptance through efforts of this nature.