The goal of the sim is to develop a comfort level with polynomials, and bring an increased competence in solving traditional polynomial problems such as presented on national tests. This will be accomplished in part through a rethinking and presenting of the mechanics behind the math symbols traditionally used.
Polynomials Sim - Set Up
While some sims will have stories and scenarios, this sim will keep things a bit more abstract. The player/student will manipulate objects in order to solve presented math problems.
It is possible that different skins could subsequently be used to make the interactions more contextual, such as using, as examples, farmers with fields, friends at a party, or money in a bank, depending on player interest.
Polynomials Sim - Basic Screen Interfaces and Interactions
The sim will start off very simply and get increasingly complex. Instructions will only be minimally used.
The first level would be very easy, and consist of a single challenge. As with all levels, it will show both the traditional math symbols at the top of the screen and the workbench view below. In the workbench mode, the player/student would drag down a box until the two lines were equal. The player would hit [enter] when they believe they have solved the problem for x.
(The interface would also have other features, such as allowing the player to move around the right blue column via the other gray box, not to add or subtract value but to arrange the table in various size rows (here, 1 X 10 could be turned into; 2 X 5; 3 X 3 remainder 1) if they saw fit.)
The second level would then be slightly more complex.
(The player could also drag the red column to the right side of the equation and subtract from the right side that amount.)
The next level could add the next level of complexity. Here, a squared relationship can be introduced. The player can drag the corner to change the value of x, which will also change the non-squared X as well. The player is understanding visually and kinesthetically what these symbols really means.
The next level of complexity will be adding multipliers, which can be visualized through stacking. The player may be able to choose between how they wanted the material to be visualized.
Other levels of interactivity would be introduced. The sim can also highlight patterns, such as a lens showing if everything divisible by 3, for example. Various math steps would have kinesthetic analogies. In the sim, the role of the symbols can start taking on a higher role, and traditional technique for solving problems presented.
Finally, in more complex levels, there will not just be one solution to each challenge. The players will feel as if "their" solution is unique, and they can take pride in it.
Polynomials Sim - Conclusion
This sim will provide a visual and kinesthetic take on traditionally symbolic content. It will support, but not overlap, with traditional methodologies. By having a rigorous level structure, players will get a bit more ability, as well as a bit more complexity, each incremental level, providing motivation. Finally, it will increasingly show the role of symbolic manipulation to have efficiency in dealing with greater complexity.
There is a caveat, of course.
As with so many school related subjects, the better we collectively get at teaching them, the more limited, brittle, and useless the current curricula will seem. Even more tragically, this stick is often used to punish the innovative media, not the original program (see Schools: What if we are at a false peak?).
In this case, the problem is that math isn't real. Math itself is a pedagogy - a perfect self-contained microworld to (hopefully) eventually put imperfectly onto our imperfect world with imperfect results but to make better plans and decisions. Ultimately, it is this perspective that must be embraced to really teach math, rather than the self-referential quixotic rathole of today's school tracks.
However, as long as schools and tests are the context of math today, this would be one approach I would use to 'teach' it.