Over the past couple of weeks, Leo and I have worked through the entire Dragon Box Algebra 12+ app. Dragon Box builds a series of gamified math apps. I can’t speak to any of the others, but Algebra 12+ (hereafter, DBA12+) is about as fun as I could imagine making Algebra, unless, like me, you think that Algebra is just inherently fun on its own! Unfortunately, DBA12+ has, to my mind, some significant flaws.
Note that Leo is only 8 (closer to 9), and I’m not sure what the “12+” means — probably target age — so I hadn’t expected the thing to be self-usable by an 8/9 year old. So Leo and I worked every problem together, although I tried to do my best hands-off scaffolding by only intervening when he was absolutely stuck — although sometimes I was absolutely stuck as well! Another thing that I did was to provide running commentary, giving names, and sometimes explanations, to back up the algebraic concepts being demonstrated, or those he was employing in play. In my parent-child interaction research we called this “active language”, and we hypothesized that this matters…a lot! I’ll get back to this below.
First some of the great things about the game. The central conceit and motivation is well designed and fun: You are feeding a creature in a box to make it grow from an egg into a big scary (in a fun-scary sense) creature. (The box is your algebraic unknown — generally “X”, as will be eventually revealed.) The artwork is great, and the goal is motivating (at least to an 8/9 year old). The moldularization and “leveling up” of the algebra content is well done too. There are around 10 major levels, each with 20 problems. New capabilities, like factoring, are introduced usually at the major level breaks, and new equation-level structural complexities are introduced within levels. This works extremely well, and the problems are extremely well designed to very rapidly bring together and reinforce previously learned capabilities into more and more complex problems, without being either boringly redundant, nor taking too large steps. Leo (sometimes with my help around the edges) was able to “solve” pretty much every problem, although the latter problems sometimes took a lot of exploration — sometimes even blind trial-and-error.
One thing that DBA12+ does that is, one the one hand very clever, but on the other hand somewhat annoying, is to begin with entirely non-mathematical symbols, specifically, pictures of funny creatures, and then, as you level up, it introduces more and more mathematical notation. Sometimes this works, and sometimes it doesn’t. I think that the creatures are supposed to represent symbols that you might see in an algebraic expression, like A, B, M, N, and Y, and the box is always definitely X. But soon enough numbers are introduced (as dice-sides, initially, as with dots, and eventually as numbers), but there are operations that work on numbers that don’t work on symbols, like factoring, so you sort of have to figure this out by trial and error, because the animals could just as easily have been representing numbers!
There are some unfortunate quirks imposed by a combination of the gamification and the requirements of the leveling-up educational sequencing. For example, until very late in the game you are only allowed to add/subtract/multiply/divide through by “reified” terms that it offers you at the bottom of the screen. This blocks some paths that would make sense (to me at least), and which would potentially provide simpler solutions. In the next-to-last level of the whole game you are given an operator that will make (almost) any term in the current equation into a “reified” term, and only at that point can you do what we had been reaching for throughout the game to that point!
Another pretty painful aspect of DBA12+ is that, although there is a “help” button, it only gives you the complete solution, not hints nor steps that you might consider. So if you’re in the middle of a solution, it’s not smart enough to give you a hint from where you are; You just have to follow it back to its one-and-only-one “correct” solution. (Their solutions are actually often more complex than necessary. In about a quarter of the cases we were able to find solutions that are shorter than theirs!)
One significant problem is that what DBA12+ counts as a “correct” solution is sometimes a bit hard to fathom. A target number of steps is given for solving the problem, and if you go over those, you don’t get “full credit” (three stars). This has the positive effect of tending against “random” solutions where you just keep moving things around until you get something, but it also tends against a very useful strategy, which is to simply move things around until you get a solution, then you’ll know what the solution is, and can start over and work on a more efficient path.
Also, and very unfortunately, like many other games, once you “try”, whether you are correct or not, the solution vanishes into the next problem, so you don’t really get a chance to think about your solution, or their solution, or pretty much anything at all. There’s actually no way to think about your solution, because there’s no way to review it. This would be a nice addition: A way to look at the steps you took to get to wherever you find yourself, whether right, wrong, or in the middle of the solution, and esp. at the end!
Another aspect of DBA12+’s obscurity with regard to “correct” solutions is there there are certain forms that it is expecting as “the correct answer”, and others that are formally correct (because, of course, you’re only allowed to correctly apply algebraic rules), but which are somehow, to DBA12+’s mind, not simplified enough … or something. Because there are no words in the thing, except for the work “Yuk”, when it doesn’t like your solution, it can’t explain to you why it doesn’t like certain solutions. So you are sometimes — quite often, in fact — left guessing as to what it didn’t like about your solution, and just trying different things randomly to try to get to a solution it likes. In a similar vein, when there’s just a singleton box left on either side of the equation, it considers you to have completed the problem, and tries to “eat” everything on the other side, like it or not! I would have preferred, for all of the above reasons, a button called “Try” (or some-such name or appropriate icon) which, when pressed, the dragon comes out of the box and tries your solution, and if it doesn’t like it, you get to continue working, or back out, etc. And it would be great if, when it didn’t like something you left it, it told you why.
This leads me to my main criticism of DBA12+, which is that there are no explanations at all at any time, no explanations of why it doesn’t like your solution, and no naming of operations. To my mind, this significantly diminishes the educational utility of the product. If I was designing this app, as new operations were introduced, I would provide “active language”, giving names to the operations, as I was over Leo’s shoulder, like factoring, dividing or multiplying through, and so forth. And I would explain the constraints on when you can and can’t do these in terms that made sense to him in the context, using the vocabulary we had built up. Unless you have someone who knows Algebra pretty well sitting over your shoulder, there’s just no way you can make sense of some of the constraints that are either inherent in algebra itself (like that you can’t add fractions that don’t have a common denominator), or those imposed by DBA12+, like you can only do integer arithmetic (including rationals via fractions), and can only factor certain things and not others, or those leading it to disliking certain solutions that it simply thinks are “Yuk”y.
If, when new capabilities are introduced it would give them names, like factoring or distribution, and then these were maybe flashed on the screen whenever you used them, it would enable so much more depth of understanding of what’s going on. For example, you might be able to tell that factoring a common term out of a parenthesized expression is closely related to factoring a number, or that the “through” operations (dividing through, etc.) are related in terms of their application across all the terms. It could also use these terms in hints (e.g., “Maybe consider factoring?”), and in explanations for Yukky (Yuky?) results (e.g., “This isn’t in its simplest form!”)
After we went through the whole game, I had the opportunity to test Leo’s understanding of algebra in a natural experiment that happened to arise when we were talking about the earth’s polar regions. He asked why “they chose” -40 to be the same in Fahrenheit and Centigrade” (His hypothesis was that the folks who invented temperature, or perhaps just centigrade, lived in a polar region! 🙂 ) Aha! A perfect opportunity for some algebra! (The benighted reader can easily work this out for him or herself by taking the equation: C=5/9(F-32), or whatever version of that you like, asserting the fixed point: C=5/9(C-32), and then solving for C.) I used the DBA12+ conceit by making the C into a box, and asking him to solve the equation, just like in DBA12+. There are, of course, many paths to get to C=-40, but the DBA12+ way, where you can only do integer arithmetic, and can’t distribute composite terms (like 5/9) is to start from C=(5(C-32))/9, multiply through by 9, yielding 9C=5(C-32), distribute the 5 to get 9C=5C-(5*32), isolate: 9C-5C=-(5*32), do the math (on both sides): 4C=-(160), simplify out the minus sign (could be done earlier, or by distribution, or just drop the parens): 4C=-160, and then divide through by 4 to get just C=-40.
Notice how nice it is to express this solution path in a combination of words and symbols, and how if we were going to have a conversation about it, we’d be diminished to grunting and pointing if we didn’t have names for things! Active Language provides these names and ways of speaking about activity, and through this process forms the foundation for reasoning. Human language is possibly our only unique feature, and it is probably the most ennobling and enabling! Unfortunately, by trying to force absolutely everything into fun gestures and graphics, DBA12+, and similar “educational” games, diminishes their topic — algebra in this case — to mere grunting and pointing.