Part of my shtick as a Geometry teacher is the Rubik’s Cube. I have a 3^3 and a 5^3 in my room; I’d had a Void as well, but it’s wandered off (perhaps due to my own doing). The students will distract themselves with the cubes; some of them will ask me to do it. The 3^3 takes me under two minutes; the 5^3, around ten or so (parity issues notwithstanding). So far, none of my students can get farther than a single side.
Several of them, naturally, have asked me to teach them how to solve the cubes. I’ve given a few side lessons here and there, and doing so has allowed me to really think about how the Cube relates to mathematics and problem-solving. The Cube, in any of its sizes, cannot be solved all at once, which is what my students seem to want to do. The solution involves breaking a complex mass of chaos into manageable chunks and then solving those chunks.
For instance, one basic algorithm for solving the 3^3 is:
- Choose a side to begin. This can be any of the six colors. The middle cubes on each side are fixed in location.
- Solve that side, making sure that all of the edge pieces are lined up appropriately. More often than not, my students will arrange the colors on one side correctly, but will have ignored the other sides.
- If you have done step 2 correctly, one-third of the cubes will now be in their correct places. Now solve the middle tier of the cube.
- The last tier of the cube is the most difficult, of course. First, place the four corner pieces correctly. This involves switching one or two pairs of corners.
- Next, rotate the corners so that their orientation is correct.
- Once this is done, rotate the side pieces until they’re correct. The algorithm I know rotates three of the four pieces while keeping the fourth in place.
- Finally, once the edge pieces are in place, change their orientation as needed.
There are other algorithms. The speedsters who do the cube in a matter of seconds, blind-folded, use more efficient methods. But this method is the one I’ve always used, since the 80s, and it is an easy one to illustrate how to systematically bring order to chaos.
Now that I have my time down, this is also a particularly good one to have during an introduction to the class, talking while I’m doing it about the nature of finding patterns and making sure to stay focused on only part of a task, so that it doesn’t become overwhelming.
I was thinking about this recently, in terms of larger project. I’ve been thinking about how to better prepare and present my thoughts on mathematics, education, and so on. I realized I’ve been trying to tackle everything at once, or to only move around a few little segments. I haven’t set myself to a systematic, consistent approach, breaking down the large task into coherent chunks.
I’m now in the process of doing exactly that. The Cube is easy in that regard, because it’s easy to see what the finished product will be. But that’s certainly part of the goal here: Decide on what the goals will be, then break down the project and proceed.