What is the big idea behind Gears?

Demo video, 75secs. The BrainQuake Gears puzzle was originally released as the standalone math learning game Wuzzit Trouble. Video by BrainQuake.

One of the design principles we adopted for developing the BrainQuake puzzles was to create the mathematical equivalent of musical instruments. I talked about this approach at a presentation I made in 2014 at the National Board for Professional Teaching Standards’ annual Teaching & Learning Conference, held that year in Washington D.C. The 60sec abridged extract below introduces the key idea.

A 60sec (abridged) clip from the 20-minute video “Orchestras for Learning”, an edited version of the talk given by Dr. Keith Devlin (a visiting Professor at Princeton University on leave from Stanford University at the time) at the “Teaching & Learning 2014 Conference” of the National Board for Professional Teaching Standards, held in Washington D.C. Video by Keith Devlin.

The full, 20-minute video can be viewed at https://vimeo.com/89583763. In it, I describe the thinking behind approaching math learning the way music is usually taught—by providing the learner with one or more musical instruments to play. (Imagine how hard it would be to learn the piano or the guitar if you first had to master musical notation. The abstract symbols of musical notation acquire meaning when you are able to play with an instrument and learn what it is those musical notes represent. We never confuse music with the symbols we use to represent it on paper.)

Yet for thousands of years, math learners had to master the symbolic notation of mathematics before they could even start to learn the abstract mathematics they represent. No wonder many fell by the wayside. Indeed, unlike music, many people do confuse mathematics (a mental activity, a way of thinking) with the symbolic language we use to represent it on paper. Fortunately, modern technology allows us to teach math to (relative) beginners the way music is taught to (relative) beginners.

By designing and building first the Gears puzzle (released initially as the standalone learning game Wuzzit Trouble), and then the two additional “instruments” Tiles and Tanks, BrainQuake started out on a path to building the “orchestras of learning” I referred to in the title of my NBPTS presentation. Those puzzles “break the symbol” barrier (a strategy I discussed in the April 11 post on this blog) the same way a piano or a guitar, say, break what would otherwise be a symbol barrier to learning music.

Actually, my use of the word “orchestras” was way too grand, but I wanted to give my presentation a catchy title, and besides, at the time I gave my talk the crucial independent research that proved it would not require an “orchestra” of learning games, just a fairly small ensemble, had yet to be completed and peer-reviewed. As it turned out, just as attaining some mastery of one or a small number of musical instruments is enough to give you a workable understanding of music, the same is true for mathematics. Though school and college math curricula give the impression that math is a vast subject with hundreds of concepts, methods, definitions, procedures, and the like, underneath all of that detailed minutiae are just a few key ideas. Cover those, and the rest can follow. (In this regard, the analogy with music continues to hold, though as with all analogies it isn’t perfect and will eventually break down.)

Among the key ideas that underly mathematics at the early stages are number sense, familiarity with abstraction, mathematical thinking, algorithmic thinking, and general problem solving ability. I talked about BrainQuake’s focus on developing those key capacities in the April 17 post.

When you first launch the Gears puzzle, it looks like a simple device to provide practice in elementary arithmetic. And in fact, it does do that; but that is purely incidental. Just as a piano does (a lot) more than provide a device to practice musical scales, so too the Gears puzzle does (a lot) more than provide a device to practice basic addition and multiplication. (Check out the image at the start of the April 11 post to see the math corresponding to a relatively simple two-cog puzzle.)

I’ll leave you with a second image that should give you some idea exactly how the Gears puzzle is designed to help learners understand and master mathematical thinking. (I could provide analogous images for Tiles and Tanks, and any future puzzles we develop. In fact, you may like to come up with your own images for Tiles and Tanks.)

The drive cogs of the BrainQuake Gears puzzle are analogous to a piano keyboard. Given a symbolic musical score (notes positioned on a staff), pressing the keys on a piano the appropriate way will result in the music the notation represents being bought to life and experienced (and the tune will be played). Analogously, given the collection of symbolic items positioned on the large wheel, rotating the small drive cogs the appropriate way will result in the mathematics the items represent being brought to life and experienced (and the problem will be solved). Different scores/items will result in different tunes/math. (The analogy breaks down slightly, in that the Gears puzzle shows at most four “keys” of the “keyboard” each time. A ukelele would be a closer analogy in that respect, but worse in others.) Image by BrainQuake.

When we talk about learning mathematics by “playing games” we are using both common meanings of the word “playing”: playing a video-game and playing mathematics on a math-instrument.

– Keith