The title of this posting is the title of a chapter in “Making Learning Whole”, by David Perkins (2009), which I mentioned in my previous posting. I recommend it highly.
What is the “hidden game” in High School mathematics? What mindsets, approaches, techniques, etc. do those comfortable with the work asked of them rely upon, yet perhaps neglect to address or teach directly? What do people do when their first approach to a problem doesn’t quite accomplish what they hoped?
My list of candidates (so far) for the “hidden game” in mathematics includes:
- Persistence. If at first you don’t succeed, try it again a different way. Many problems require several attempts before I stumble across a useful approach. Don’t let the failure of one approach dissuade you!
- Patience. If your first approach did not work, sometimes it can take a while for a second approach to come to mind… and sitting there staring at the problem may not help an alternative spring to mind. Doing other homework or taking a walk can help, as can getting a good night’s sleep and trying again another day. Don’t let the sometimes slow pace of creative thinking frustrate you.
- Visualize the problem. Doodle it; draw diagrams with all information noted; close your eyes and picture yourself immersed in the situation… what thoughts come to mind as you do so? Might they help solve the problem?
- Talk to someone else about the problem and how you have tried to solve it so far. Talking about something I am wrestling with will often bring new ideas to mind, or as I have often said “I have some of my best ideas while my mouth is open”.
- Look for patterns in the problem’s illustration or mathematical formulation that resemble patterns you have seen before. Look for them on both a small scale and a large scale… for example, in geometry there are often figures within figures, and my eye can often fail to notice either the outside ones or the inside ones; in algebra, when faced with the sum of two squares you might ask if this situation could be represented with a right triangle, and if so, does the Pythagorean theorem apply?
- Rewrite and clean up the problem. This might mean organizing terms in a different order. Or either factoring things as much as you can, or multiplying out everything you can. Rearranging and changing the appearance of a problem can often lead to an insight.
- Revisit the statement of the problem. Look for anything explicitly given that you have not taken into account. Then look for assumptions or other implicit information you may not have made use of in your solution attempt. Is there some principle, theory, or formula that I can apply in this situation?
- Look for a related problem and see how it was solved. Even if the approach to its solution will not work for your problem, the approach may remind you of things you have not tried yet.
Finding a place to explicitly teach the above can be a challenge. Some students are not developmentally ready for the degree of “patience” or “persistence” required. I find these topics fit in best at “teachable moments” rather than at pre-planned points in the syllabus. But when the moment arrives, it is worth drawing the attention of the class to the fact that this is a useful technique that can help when you are stuck… part of the “hidden game” in mathematics.
MathMaine 
I recall you once envisioned developing what I would call a “stealth learning” game to teach math concepts through an entertainment vehicle. Have you run across anything that fits the bill yet?
I did indeed envision grand things, and quickly realized that I could not develop such a beast on my own – it would require a team similar to what is required for an animated movie.
Furthermore, there were a bunch of “educational” games coming out that sounded promising. All of the ones I have seen fell flat (in my estimation), and my sense of what needs to be taught has also evolved (surprise!).
In this day and age, I would lean towards either a “save the earth” from environmental catastrophe game (involving algebra and statistics through calculus and beyond), and/or a “become an economic powerhouse” game (involving mostly linear equations through linear programming, perhaps with some non-linear demand curves eventually).
Both would use the standard video game model of multiple levels with increasing sophistication/difficulty – but the problems to be overcome would not be posed as math problems (even though math would most certainly be needed to solve them).
Players would be immersed in a virtual environment (like Myst) and have to figure out what they wanted to measure, collect their own data, build their own models, and make their decisions. Initial levels could all be done in your head, but the need for modeling would gradually grow.
I have seen some efforts that head in sort-of that direction, but they keep getting derailed by the notion that they have to frame and pose the algebra problem for you… have you ever stumbled across an equation lying on the sidewalk?
Mafia Wars did take a step in the right direction (do you want to buy something now, or save up to buy something later that will give you a greater return?), however it did not seem to base its game on an “optimizable” model for the player as much as a social/advertising one for Zynga.