Life inside an impossible Escher figure

When the M.C. Escher inspired puzzle video-game Monument Valley came out last year, I knew I had to check it out. The more so when it started getting rave reviews and winning awards. But with so many other things to hold my attention, I never managed to get round to it. The recent decision of the creators to make a version available for free prompted me to finally take a quick look. Not that it had been expensive. Rather, a tweet about the new free version happened to come when I had an hour or so of free time on my hands.

That free hour got immediately swallowed up, as did many more hours after that. I was hooked from the getgo.

Anyone who is intrigued by those impossible figures that Escher made famous, even those of you with little or no interest in puzzle (video-) games, will surely be captivated by Monument Valley, where the solution to many of the puzzles involves orienting the figure to create an illusion of a continuous object. For when the player views the object as continuous, characters in the game that traverse the figures can move along it. Impossible chasms that prevented a character’s progress suddenly disappear as you rotate the entire figure just the right way.

It’s not a learning game. I don’t see a player learning any new mathematics. But what it does is provide a rich, immersive experience of complex geometrical spaces from the inside. As a player, your task is to assist the princess on her quest, which involves finding her way through a fantasy world of Escher-like structures, the geometry of which you can sometimes change as you progress. By projecting yourself into the princess, you get a sense of what it would be like to live in such a world.

And a beautiful world it is. The creators, based in the UK and operating under the name UsTwo, have crafted a series of truly gorgeous fantasy worlds, which you encounter one after another. It is not so much a game as a collection of interactive pieces of art where you play with, and experience, geometric shapes.

In fact, it is the artistic creation that the developers bring to the work. The idea of taking Escher worlds and turning them into a puzzle game goes back to a 2007 video game called Echodrome, designed for the Sony Play Station 3 by the Japanese designer Jun Fujiki. By all accounts it was fiendishly difficult, and never broke out beyond a small group of hard-core puzzle aficionados.

Monument Valley shows the huge difference presentation can make. If you want to hold people’s attention, you often need to think carefully about the medium. The message on its own may not be enough. That holds in the math class or the math lecture hall as much as in a video game.

Regular readers of this MAA blog or my other blog profkeithdevlin.org will know that I have a long-standing interest in video games, particularly so as an educational medium, where I am professionally active as a player, a learning researcher, and an entrepreneur.

In fact, much of my career has involved looking for ways to use different media to make mathematics accessible to as many people as possible. I have authored many “popular mathematics books”, written for newspapers (MAA compilation of some of my articles here), worked on television programs (including A Mathematical Mystery Tour, BBC-tv 1984; Life by the Numbers, PBS 1998; and NUMB3RS, CBS, 2005-2008), and of course there is my regular Math Guy radio gig for NPR, which started in 1994. More recently, in 2012, I launched the first ever math MOOC on Coursera (the seventh session just ended). I even made a foray into using music, song, and dance, with the 2007 show Harmonius Equations. To me, video games are one more medium to carry mathematical content.

In fact, when it comes to K-12 mathematics, video games are in many ways the most effective medium we currently have to provide good math learning, as I tried to articulate in a book I wrote in 2011, and a presentation I gave at the big Teaching and Learning 2014 conference in Washington D.C. last year, a 20-min video summary of which is available here.

Until recently, there was relatively little research available to put any flesh onto educators’ beliefs/hopes/suspicions that video games could yield good math learning outcomes. That is starting to change. (Reports from two classroom studies, one of I was involved in, are due to be published in the International Journal of Serious Games this month. Preprints are available here and here.)

Certainly, the results obtained in those two papers raise more questions than they answer. (Moreover, pending further, and substantially larger, studies, the results themselves have to be viewed as tentative.) What we are seeing is that, for mathematics in the K-8 range, significant learning outcomes can be observed after a video-game intervention of as little as two hours play spread over a month or so. (Some measures show an increase of 20% over a comparison group.)

I’d seen reports earlier that made similar claims, and dismissed them as product- marketing masquerading as research. It was only when the first of the two particular studies I cited above came out in late 2014, carried out by Prof Jo Boaler’s research group at Stanford University’s Graduate School of Education, using my own math learning video game Wuzzit Trouble as the intervention, that I sat up and really took notice.

In fact, I did more than that. Together with a research colleague from Tampere University in Finland, Prof Kristian Kiili, who, like me, has founded a math-learning video game company, and who was spending the year at Stanford, we carried out our own study. (The second of the two papers I cited.) Kiili was developing a fractions learning game, Semideus, and wanted to see how well it could serve as an evaluation tool. So we repeated essentially the same study Boaler’s team had done, with Wuzzit Trouble as the intervention, but instead of a written pre- and post-test (which the Boaler team used), we used Semideus. The results were very similar to those obtained in the previous study. (With the added twist that this time we found transfer — in a game context — from the whole number arithmetic of Wuzzit Trouble to the fractional reasoning of Semideus.)

Something is going on, that’s for sure. But what? It did not take long to come up with a fairly long list of possible factors. Among the many things that a (well designed) math learning game can offer, all which are known to have a positive impact on learning, are:

• Breaking the Symbol Barrier – human-friendly representation (not the traditional abstract symbols of math textbooks).
• Focus on developing number sense and problem solving ability.
• High level of engagement.
• Instant feedback (both positive and negative).
• Steady flow of dopamine – known to have positive impact on memory formation and consolidation.
• Learning through failure – in a playful, safe environment.
• “Failure” treated – and regarded – as “not yet succeeded”.
• Constant sense of “I can do this on the next try.”
• Lots of repetition – but at the demand of the student/player.
• Student/player is in control.
• Student/player has ownership.
• Growth Mindset – good games encourage and develop this. (This is the important notion Carol Dweck is famous for.)
• Fluid intelligence (Gf) – games require and develop this. (Loosely speaking, this is the ability to hold several pieces of information in the mind at the same time and reason fluidly with them.)

I have written about many (not all) of these factors in my series of video game learning articles in my blog profkeithdevlin. (See also the many writings and videos on games and learning by Prof James Paul Gee.)

My current guess is that all of these factors, and likely others, are at play in those dramatic learning outcomes. The only way to find out for sure, of course, is to do more research. A lot more. Prof Kiili, now back in Finland, is already hard at work on that, as am I and some of my colleagues at Stanford. And we are by no means alone. The field is wide open. Stay tuned. (Even better, get involved.) Truly, it’s an exciting time to be involved in mathematics education.

Meanwhile, I have to sign off. Monument Valley is calling.