How to design video games that support good math learning: Level 2.

The educational goal
Part 2 of a series.

Anyone setting out to design a video game to help students learn mathematics should start out by reading – several times, from cover to cover – the current “bible” on K-12 mathematics education. It is called Adding it Up: Helping Children Learn Mathematics, and was published by the National Academies Press in 2001. The result of several years work by the National Research Council’s Mathematics Learning Study Committee, a blue-ribbon panel of experts assembled to carry out that crucial millennial task, this invaluable volume sets out to codify the mathematical knowledge and skills that are thought to be important in today’s society. As such, it provides the best single source currently available for guidelines on good mathematics instruction.

The report’s authors use the phrase mathematical proficiency to refer to the aggregate of mathematical knowledge, skills, developed abilities, habits of mind, and attitudes that are essential ingredients for life in the twenty-first century. They then break this aggregate down to what they describe as “five tightly interwoven” threads:

Conceptual understanding – the comprehension of mathematical concepts, operations, and relations

Procedural fluency – skill in carrying out arithmetical procedures accurately, efficiently, flexibly, and appropriately

Strategic competence – the ability to formulate, represent, and solve mathematical problems arising in real-world situations

Adaptive reasoning – the capacity for logical thought, reflection, explanation, and justification

Productive disposition – a habitual inclination to see mathematics as sensible, useful, and worthwhile, combined with a confidence in one’s own ability to master the material.

The authors stress that it is important not to view these five goals as a checklist to be dealt with one by one. Rather, they are different aspects of what should be an integrated whole. On page 116 of the report, they say [emphasis in the original, image reproduced with permission]:

The most important observation we make here, one stressed throughout this report, is that the five strands are interwoven and interdependent in the development of proficiency in mathematics. Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands. … [W]e argue that helping children acquire mathematical proficiency calls for instructional programs that address all its strands. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics.

In my book, I describe in some detail how to incorporate these educational goals (actually, to be faithful to the NRC Committee’s recommendation, I should say “educational goal”, in the singular) into good game design for a video game that seeks to help children learn mathematics. In this post, I’ll simply distill from that discussion eight important things to avoid. Try using this list to evaluate any math ed video game on the market. Very few – and I mean VERY few – pass through this filter.

  • AVOID: Confusing mathematics (a way of thinking) with its (symbolic) representation on a static, flat surface. (cf. music and music notation.)
  • AVOID: Presenting the mathematical activities as separate from the game action and game mechanics.
  • AVOID: Relegating the mathematics to a secondary activity when it should be the main focus.
  • AVOID: Reinforcing the perception that math is an obstacle that gets in the way of doing more enjoyable things.
  • AVOID: Reinforcing the perception that math is an arbitrary hurdle to be overcome, or circumvented, in order to progress .
  • AVOID: Encouraging the student to try to answer quickly, without reflection.
  • AVOID: Reinforcing the belief that math is just a large bag of isolated facts and tricks.
  • AVOID: Reinforcing the perception that math is so intrinsically uninteresting it has to be sugar coated.

I’ll be referring to Adding It Up a lot in this series. I shall also discuss many things TO DO when designing a good video game that support good learning, not just what to avoid. As you might (and for sure should) realize, with two challenging goals, good game and good learning, designing a successful math ed video game is difficult. Very difficult. If you do not have an experienced and knowledgable mathematics education specialist on your team, you are not going to succeed. Period.

Game programmers who think that because they were good at basic math (and they have to have been to become successful programmers) they can design a video game that will provide good learning are deluding themselves.

It’s easy to underestimate the depth of expertise of professionals in areas other than our own. Let me stress this point from the perspective of a hypothetical math educator who knows how to program in html5 and decides to create the next Angry Birds.

S/he might well think, “I can write code that produces screen action like that.” Indeed s/he probably can; it’s not hard. But as any experienced game developer will attest, the coding is the easiest part. The huge success of Angry Birds is not an accident. It is a result of brilliant design on many levels. (See this article for an initial, eye-opening summary of some of what went in to making that success.) The expertise it took to be able to create that game was acquired over many years. The Helsinki, Finland based Rovio game studio built ten other games, picking up a ton of increased expertise and insights along the way, before they reached the design heights of Angry Birds.

To build a successful game, you have to understand, at a deep level, what constitutes a game, how and why people play games, what keeps them engaged, and how they interact with the different platforms on which the game will be played. That is a lot of deep knowledge. On its own, being able to code is not enough.

To build a game that supports good mathematics learning, requires a whole lot more.  You have to understand, at a deep level, what mathematics is, how and why people learn and do mathematics, how to get and keep them engaged in their learning, and how to represent the mathematics on the different platforms on which the game will be played. That too is a lot of deep knowledge. On its own, being “good at math”, or at least the relevant math, is not enough.

If you are a game developer who happens to have both kinds of expertise, then go ahead and build a game on your own. But I have yet to meet such a person. For the rest of us, the answer is clear. You need a team, and that team must have all the expertise you will require to do a good job. If that team does not include, in particular, an experienced, knowledgable, math education specialist, then you are not a good engineer. You are an amateur.

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11 Responses to “How to design video games that support good math learning: Level 2.”


  1. 1 Sean Gubelman (@gubester) February 1, 2012 at 9:06 am

    Great topic! This is my first reading in the series, and I’m excited to read more. :)

    I’d like to make educational games in the future– just getting started here, but I always thought that math education was the most accessible of the disciplines. Thinking about it now– that’s probably not the case. ;)

    • 2 Keith Devlin February 1, 2012 at 5:36 pm

      I’d say history and social studies are the easiest. Sid Meier’s Civilization and Pirates were pure entertainment games but powerful educational tools. Of course, any game is educational. The question is, what is the player learning? Mathematics – the real stuff – is certainly doable, but not easy, IMO.

  2. 3 B.F. February 2, 2012 at 5:01 pm

    In what way would you say Civilization and Pirates are educational tools? Certainly, they have educational merit and can inspire an interest in history, but do they actually teach useful information? While there are some historical facts in these games, there are also a whole lot of non-facts. If we are encouraging students to learn history from games like these, we run the risk of them believeing Ghandi was a warlord of the Indian Empire (the role he serves in Civilization).

    The distinction I see here is between an “educational tool” and a game simly with “educational merit.” As you stated, any game is educational, so in order for something to be considered a tool, it must be useful in accomplishing some particular task. It seems to me that a game that focuses on a small task but does it very well (extending the tool analogy, think of a staple remover) would be more useful than an ambitiously complete but poorly designed game (http://www.vagabondish.com/wp-content/uploads/2007/04/wenger-85-swiss-army-knife.jpg).

    • 4 Keith Devlin February 2, 2012 at 5:18 pm

      B.F. Thanks for the comment. I think you are confusing the transmission/acquisition of information with education and learning. (A common error in the standardized-test-crazy USA.) The former is trivial, the latter is the challenging part. At its heart, education is about helping people acquire different ways of thinking. Once they have achieved that, soaking up any necessary facts is a piece of cake. James Paul Gee talks about this at length in his book “What Video Games Have to Teach Us About Learning and Literacy”, where he talks about “entering a semiotic domain”, and in my video games book I use the phrase “becoming an X-er” to refer to the same key aspect of learning.

      In the case of history education, the primary goal is learning to think like a historian. It’s not acquiring or knowing various historical “facts”. Once you can think like a historian, acquiring and making use of the requisite information is trivial.

      It is because video games are so well-suited to help people acquire these crucial, specialized forms of thinking that they provide such an excellent medium for promoting good learning.

      That is really the principal message in both Jim’s book and mine, and it is the essence of video game education – though almost all so-called educational video games on the market so far fail miserably in that regard. The two Sid Meier games I mention allow players to run through simulations of events, and gain experience of how things unfold, with decisions leading to consequences. The simulators used by the airlines and the military don’t have to embody real terrain (though they often do) to be effective at training pilots and soldiers. In fact, the whole point is to provide the learners with enough ability to cope with novel situations that they have never encountered before. It’s about ability, not facts.

      BTW, I agree with you about the value of small, digital, “appliance”-type games, though I think we should develop both kinds of games, appliances and VW games. Simple apps can help you come to understand a concept, but you still have to learn to use that concept and combine it with others, so you need a framework for appropriate use of that app. A VW game is one way to do that, but not the only way.

  3. 5 B.F. February 6, 2012 at 2:17 pm

    I thought about your reply over the weekend and I think I agree with almost everything you are saying. The types of games you describe really are the ideal we should be striving toward; that said, I don’t think we are anywhere near that level of design sophistication with educational games. Even great examples like Civilization only present superficial views of historical recreation. True, you can make decisions that have lasting repercussions throughout the history of the game, but the motivations behind these decisions are skewed.

    For example, in Civilization 4, one might choose to play as the Egyptian empire with the leader, Hatsheput, because she receives a boost to spirituality and creativity. With these boosts in mind, the player might choose to pursue a cultural win by utilizing their happy and satisfied populace to build lots of Wonders and diplomatically avoiding unnecessary conflict. This is a perfectly viable game strategy and it vaguely mirrors historical events but as an academic study, it completely flips around the cause-and-effect dynamic. Hatsheput didn’t build because she was deemed creative, she was deemed creative because she proved herself to be a prolific builder during her rule — even if the events are similar, the motivations (arguably, one of the most important aspect of studying history) are completely different. Like Raph Koster points out in his book “A Theory of Fun for Game Design”, players will mostly ignore the semantic representations in games and react to them based on their in-game function. This is generally a good thing, since it means we aren’t in danger of kids becoming sociopathic killers just from playing Grand Theft Auto, but it also means they will not retain many pertinent real-world connections from video games either.

    So, what does this mean for aspiring educational game developers? The game industry is still very young and developing a set of standard practices and procedures for making games. Even so-called industry experts miss their mark surprisingly often. One must be intimately fluent with design, mechanics, art, sound, narrative, pacing, and so much more to even build a non-educational game. Even if a game developer is able to do this, they may end up working with an educational expert that makes unreasonable demands on the design of the game. For example, any mechanic that cannot be executed quickly and naturally should not be in a game requiring timing (imagine if you had to enter a long, complex string of commands to cast a simple fireball in World of Warcraft), but I have met many educators who insist on pushing complex operations (like fraction arithmetic) into action-oriented math games. Depending who is paying the bills, the game designer and educator are both forced to make compromises to each other and the final product generally ends up lacking in both educational and entertainment value. And speaking of bills, most schools will not invest money and class-time into any product that does not directly and quantifiably address common core standards and standardized testing which makes it economically unfeasible for developers to invest time into solving these design challenges.

    I think with time, educational game designers will eventually figure out creative solutions to the challenges inherent in the medium. Educational administrators and investors will also learn and mature in their approach to what educational games can and should be. Until then, perhaps the best approach for educational game developers is to work on mastering simpler, more well-understood concepts (say, Behaviorist approaches to arithmetic) before attempting more complex and complete endeavors. The industry must learn to crawl before it can run.

    • 6 Keith Devlin February 6, 2012 at 6:57 pm

      Good, now we are on the same page. :-) Now it’s my turn to agree with you.

      There is good news and bad news. I’ll focus on mathematics education, since that is what I am familiar with.

      First, the good news. I spent almost five years working with experienced game developers at a large commercial game studio to see how to blend challenging middle-school mathematics into video games that play well as games. We learned that it is doable, but expensive. Because of the cost, the studio ultimately decided not to pursue that project. (By their standards, the ROI did not justify continuation, and they put their talented game developers and engineers onto more lucrative projects!)

      So a group of us created our own company and, using the experience we had acquired, started from scratch and designed several mathematical apps that could embed challenging mathematics into video games, where solving the problem required conceptual understanding and good problem solving skills, not just the basic computational skills that you find in many so-called mathematics educational video games. (They are really basic skills practice games. Nothing wrong with that. A useful thing to have available. But that’s all they are, and it is not education.)

      Armed with a number of fairly basic but functional Flash-built prototypes, we set out trying to secure some funding (both federal grants and private venture funding – yes, we have a revenue model) to build robust beta release versions and get them out there so we, and teachers, could see how kids reacted to them and what results they produced. Now these are early days, and I know plenty of entrepreneurs who had a long haul before they managed to secure funding. But what was dismaying was the almost universal reaction to our prototypes: “There is no mathematics there.” Now I’ll tell you, we have levels on our games that took me hours to solve. When you peel off the game mechanics, our games are nothing but math. What those funders meant is, “I don’t see any mathematical symbols.” And they are right, there aren’t any (discounting a few whole numbers here and there used as names).

      That’s the whole point! We used the video game environment itself to represent the mathematics. Symbols are required to represent mathematics on a flat, static surface – the sand, a clay tablet, a slate, parchment, paper, a blackboard, a whiteboard, etc. A video game offers a rich, dynamic, interactive medium in which mathematics can be represented directly, without the need for symbols. We have developed video-game-native representations of mathematics.

      The familiar symbolic notation for mathematics is just an INTERFACE to mathematical thinking — which is something that goes on in your head. Video games provide an alternative interface. But, just as the Macintosh (later Windows) interface was initially regarded as “not real computing” compared to the more familiar (and hard to master) command-line text interface of the computers that preceded the arrival of Steve Jobs, so too our natural, easy-to-use-without-instruction video-game interface to mathematical thinking has been dismissed as “not real mathematics”.

      And for exactly the same reason! People got used to the fact that computers were operated by typing in commands, and people have gotten used to the fact that mathematics is done by writing and manipulating symbols. But neither of these are or ever were facts. They were states of affairs, resulting from the limitations of the available technology.

      Incidentally, I like the Macintosh analogy not only because it is really accurate, it also keeps me optimistic in the face of one failed funding pitch after another. (BTW, wearing jeans and a black turtleneck has so far not helped me.)

      Of course, it is going to be a much more difficult task to get people weaned off the idea that mathematics is about manipulating symbols, because it has been done that way for over five thousand years, whereas Steve Jobs only had to overturn thirty years or so of command line programming.

      Meanwhile, I continue to search for that one elusive funder who really “gets” it when I say that mathematics is not about manipulating symbols, it is a way of thinking, and you can mainline that way of thinking into a person’s head using a video game just as United Airlines can mainline how to fly a plane by putting their trainee pilots into a flight simulator.

      Incidentally, the potential payoff of this approach is that many more kids would become proficient in mathematics. I present hard evidence in my math ed video games book (Chapter 1) that practically everyone can become proficient in at least basic, everyday mathematics if you remove what I call the symbol-barrier. It turns out that it’s not mathematics that people find difficult, it’s the (unnatural!) symbolic representation. [Nothing I am saying here is new, BTW. All I did was read the research literature.]

      It’s not that the video game industry does not yet know how to do good educational video games. They do – at least in mathematics. They just don’t yet know that they know!

  4. 7 Jessica February 6, 2012 at 8:45 pm

    OK, well you mentioned Tom Lehrer’s New Math so I had to come by and make a few points.

    (1) I agree – game play and content need to be equally stressed. However, I think there are two steps to the educational process: one is the “math makes me happy” emotional learning – a much easier starting place – and the other is the actual math content. Civilization was the “history and systems theory makes me happy” pre-learning learning. Kids who do this walk into additional learning – whether in a classroom or independently – with a completely different cognitive preconception, “Now I’ll finally find out more about the Bronze age!” rather than “There are ages? The Bronze what?”

    The content learning part, I am completely biased, but I just don’t understand why math and science aren’t taught within the historical context of discovery. To use Nicole Lazarro’s framework, it’s “fiero” – the hard win – when you are able to finally undersand how to describe a new way of investigating. Why do we give students a superficial knowledge of a bunch of rules? Why don’t we give them context? Math is a language (as you’ve said) – why don’t we teach it as such? Then again, I think discrete math should be begun in 2nd grade, so I’m an outlier.

    (2) Long ago (mid-1990′s) I was pitching two games: one was Risk crossed with environmental remediation. This was targeting girls – before that was even vaguely ok – in the sense that territory was aquired (“yang”) but it was done by healing (“yin”), and I wanted to reach 8-12 year olds. (Note: not targeting girls in the sense of science in a pink box.) At the time, the indusry hot targets were pre-K and over-12 boys-only, and I didn’t know enough as an entrepreneur to adapt my product. The other product was a Chemistry-ready game, which I thought of as “Mario the Plant Manager” – where the winning tricks had players becoming familiar with behaviors such as tossing OH- onto H+, and other pre-10th grade Chemistry things. Needless to say, I didn’t get that off the ground either.

    (3) My son did early grade school math with Learning Company’s Math Rabbit. Fractions were so fun *I* did them, with my MS in science. It was playing with space. Learning Co was bought by MECC (oregon trail) which was bought by SoftKey… by then they killed most of the product line. I *think* this was the one: http://www.amazon.com/Reader-Rabbit-Adventure-Jewel-VERSION/dp/B00005IB4S/ref=dp_cp_ob_sw_title_0

    • 8 Keith Devlin February 6, 2012 at 9:07 pm

      Jessica: Agreed on all counts. I must be doing something wrong if no one has hurled an epithet at me yet… Perhaps I should say that maybe, just maybe, possibly, conceivably, Khan Academy isn’t “just the greatest thing you’ve ever seen.” There, that should get some folks at my throat. (Apologies to Sal, whose a great guy who has helped a lot of people, for free. This is not aimed at him, but everyone who claims that his material can do everything the nation needs except make the toast. And they get really, really annoyed – “very peeved,” in my native tongue – if anyone says otherwise, especially an experienced math educator. So there.)

      But I digress. In my book, and in this blogpost, I advocate working from the NRC’s five threads, outlined in Adding It Up. And when you cash them out, you get the kinds of design goals Jessica is referring to.

  5. 9 Climeguy February 9, 2012 at 1:51 pm

    Just started reading this… don’t know if anyone mentioned it… Adding it Up is free online at

    http://www.nap.edu/catalog.php?record_id=9822

  6. 10 L February 10, 2012 at 2:01 pm

    This sounds very intriguing. I am with you on the difference between the symbols (language), the actual ideas they are intended to describe, and more importantly, the type of thought processes a mathematician goes through. If you can get a kid to think in a mathematical way, you have won. It seems like most students are essentially illiterate in the symbolic language, and yet we continually use that language when attempting to communicate mathematical ideas & thought processes.

    p.s. I don’t want to read the book, I just want to try the games. Are any of the prototypes out there?


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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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