Just a quick update on my previous post on gamifying the problem-solving process in high school algebra. I had a little time to spare (on an airplane flight, of all things), so I decided to rework the mockup version of the algebra game into something a bit more structured, namely as 12 progressively difficult levels of solving a linear equation in one unknown.  (Requires either Java or Flash.)   Somewhat to my surprise, I found that one could create fairly challenging puzzles out of this simple algebra problem by carefully restricting the moves available at each level. Here is a screenshot of a typical level:

After completing each level, an icon appears which one can click on to proceed to the next level.  (There is no particular rationale, by the way, behind my choice of icons; these are basically selected arbitrarily from the default collection of icons (or more precisely, “costumes”) available in Scratch.)

The restriction of moves made the puzzles significantly more artificial in nature, but I think that this may end up ultimately being a good thing, as to solve some of the harder puzzles one is forced to really start thinking about how the process of solving for an unknown actually works. (One could imagine that if one decided to make a fully fledged game out of this, one could have several modes of play, ranging from a puzzle mode in which one solves some carefully constructed, but artificial, puzzles, to a free-form mode in which one can solve arbitrary equations (including ones that you input yourself) using the full set of available algebraic moves.)

One advantage to gamifying linear algebra, as opposed to other types of algebra, is that there is no need for disjunction (i.e. splitting into cases). In contrast, if one has to solve a problem which involves at least one quadratic equation, then at some point one may be forced to divide the analysis into two disjoint cases, depending on which branch of a square root one is taking. I am not sure how to gamify this sort of branching in a civilised manner, and would be interested to hear of any suggestions in this regard. (A similar problem also arises in proving propositions in Euclidean geometry, which I had thought would be another good test case for gamification, because of the need to branch depending on the order of various points on a line, or rays through a point, or whether two lines are parallel or intersect.)