Well-Formed Math Problems

Sebastian Deterding, in his keynote on gamification:

I would argue that these are not only principles for good games, but for any human activity to be well-formed: we enjoy situations with clear, structured, unconflicting goals, clear, limited action spaces with choice, clear and fair rules, scaffolded challenges and complexity matched to our abilities, and clear, actionable short- and long-term feedback.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Dan, thanks for the link. Very interesting talk. Thought the bits on feedback, were spot on.

  2. Not just well-formed math problems, but a well-developed classroom or school, I would say. Particularly when thinking about striking the balance between autonomy and rules, the freedom of experimentation in a safe space and the goals towards which you experiment.

    Quite an interesting lecture.

  3. Some good points here — lots to consider from an educational perspective. Whilst I agree that learning should be enjoyable I’m not sure that it needs to be “fun” in the same way that a trip to an amusement park is. However, learning needs to be rewarding and relevant. With some subjects / topics learning also needs to be explicit — game “play” may disguise what actually needs to be made obvious to be effective in the long run. Only well designed games promote higher order thinking – many educational titles only offer low level skill practice. That’s not to say that there isn’t a place for well designed activities — or “games” — as long as the focus is clearly on the learning involved. Anyone who has read anything by Alfie Kohn knows the long term ineffectiveness of tokenizing education (via “rewards” etc).
    For another voice on game play I’d suggest a viewing of James Paul Gee interview at …

    or another one with similar focus here http://www.youtube.com/watch?v=JU3pwCD-ey0

  4. Sorry to post again – but I just came across this post which lists a variety of sites for game making. Might be useful for anyone who wants to become a producer rather than a consumer… or it simply might be fun.

  5. “We enjoy situations with… unconflicting goals.” Also, “the first act of a good story introduces a conflict.” What’s the relationship? Maybe a good story introduces conflict about the means, but not about the goal?

    Also, I think it’s true that we enjoy situations with unconflicting goals. And that worries me. I’m stuck on how to help students learn how to constrain a problem themselves. It makes sense to me that this is not your goal with the “perplexity problems” — more likely it’s the goal of your teacher-training workshops. Still, how does it affect students if the problems they encounter are always well-defined?

    It seems like it’s related to what Labaree points out in your next post (thanks for that):

    “The myopia problem is that you’re looking at something that’s presented to you… in the process of looking at it up close, the whole context disappears.”

    Going back to Jo Boaler: “Students come to know this about math class. They know that they are entering a realm in which common sense and real-world knowledge are not needed.” I’m worried that presenting problems fully constrained causes the myopia that Labaree is talking about, and increases the alienation between “school skills” and “rest of life skills”.

  6. I think this is one of the reasons that Modeling Instruction (modeling.asu.edu) is so successful. It’s not just inquiry – it’s structured inquiry. The modeling cycle takes the confusing and intimidating task of investigating the natural world and develops a structured way of asking and answering our own questions. It models the scientific process. This is crucial as most students until this point have only read/heard about science and have not had a chance to do it. Without the structure provided by modeling, they would be thrust into a task with no concept of how to proceed.

  7. Mylene:
    ““We enjoy situations with… unconflicting goals.” Also, “the first act of a good story introduces a conflict.” What’s the relationship? Maybe a good story introduces conflict about the means, but not about the goal?”

    These are two slightly different uses of the word conflict. In a story conflict means struggle, controversy… essentially a challenge of some sort.

    In game design conflicting goals refer to contradictory or disagreeing goals. An example would be if Mario lost points for finishing a level. When goals conflict the gamer is unable to escape punishment and incapable of doing everything right. That might be true to real life, but it sucks.

    Some great games present conflicting goals, such as whether to save BioShock’s “Little Sisters,” but it’s done very carefully. It’s typically a moral decision that drives narrative or branching paths and the game is balanced to be fun no matter which you choose.