That said, the mechanical process of algebra is an important tool--one that can be honed by repetitive training. Having those physical patterns of grouping, distribution, cancelling, ingrained into your brain can make more abstract explorations easier. It's like learning how to walk in order to backpack the Wonderland Trail, or practicing strikes in martial arts to gain a better understanding of partnership. To that extent, games like these can be a fun and useful part of exploring math.
((3+7)*x)/(3+7+x) = 5
They'd grow confused on how to start. Should they combine the 3 and 7 in the numerator? In the denominator? Or should the multiply both sides by (3+7+x)?
Of course, it doesn't matter which one they choose. Any of those will get them a step closer to solving the problem. However, since they don't have the mathematics confidence to just play with the problem, they'll become paralyzed with the various, equally good options. On the other hand, if they were more comfortable with the symbolic manipulation parts, they'd spend less time worrying about trivialities and more time focusing on parallel resistors.
Here you have the same thing. In math you have syntax, and semantics. The other domain from semantics might be abstract, or it can be purely mechanical, but it can also be connected to reality. If your operations map to nothing that makes sense to you, the operations are just mechanical; you solve equations in some way because you know it's right but you don't understand why. If your operations map to other domains that you understand (also if they are abstract domains in your imagination), you can understand why the operations work like they do, and you know why you have to solve them the way you do it.
Maybe the equations don't have a specific meaning per se; but if they don't have any meaning for you, there is no way you understand what you are doing when you solve them.
For example:
a x = b (text) --> x is unknown, it is the right one if f(x) = a x equals f'(x) = b; both functions are programs you can compute and play with
from there you go mechanically to:
x = b/a (text) --> x is unknown, it is the right one if f(x) = x equals f'(x) = b/a
while in the first step it was hard to tell much about x, now we can see that it is trivial to guess which is the right x; x must be b/a
This is the first mapping from the domain of symbols to another domain that I could think of. There must be more natural mappings that can be used like this.
Seriously: there is something to be said for the claim that mathematics is the search for beautiful tautologies.
Like javelin throwing vs hunting, running vs outrunning a predator, or painting vs making a portrait using paint because that is the only way to do it, there is a difference between being doing math and using math to reach a goal.
I think it would be very nice if one managed to give kids, even those with little mathematical talents, a glimpse of that difference.
I think this is a brilliant idea, and it seems to be well executed. I don't have the necessary hardware to run it, so I haven't played with it, but it looks to be a wonderful game based on algebraic manipulations. I, along with everyone else, expect and hope that it will engage players and allow them to learn the rules and skills of such manipulations.
And probably that's a good thing. Let me try to explain the underlying reasons for my sense of unease, as best I understand them.
Firstly, I am concerned that this will merely enhance the sense that math is simply arbitrary manipulations with neither meaning nor motivation. Many of the kids I tutor can do the manipulation, but don't get the point, and never connect it with reality.
Next, some of the kids I tutor can't do the manipulations without making stupid errors, and I can't help but feel that even after practising with this, they will still make stupid errors. Link that to the apparent meaninglessness, and there's a recipe for frustration.
Thirdly, this doesn't help to connect the creation of equations with the physical problem to be solved, and it doesn't help interpret any final answer. These are the steps that the kids I deal with simply can't do.
Finally, as someone commented, this isn't intended to be the whole and entire course, and it's supposed to be just one tool to help one stage, and to be built on and leveraged by the teachers. I've lost count of the number of wonderful tools and ideas that I've seen whither and die because the teachers can't make use of them. In some cases the teachers don't really understand them, but I would hope that fate would be avoided by this.
So in summary, I think this is a wonderful tool, and it has the potential to be a fantastic aid to learning. I am deeply uneasy about the further divorcing of algebraic manipluation from any sense of meaning, but I look forward with interest to see if it can be used in a meaningful way.