L

ike most other children who grew up in the '80s, I owned a Rubik’s Cube that I couldn’t solve. Sure, I could complete one face of the cube. Occasionally, I would luck into completing most of a second face. Then, frustrated that any further manipulation would ruin all of my hard work, I did what I assume most kids my age did — I peeled off stickers and swapped them. The resulting cube would impress my younger brother at first, but after noticing the slightly-skewed stickers with wrinkled edges, he too learned my secret to “solving” the cube.

By swapping stickers, my brother and I would unwittingly make solving some faces impossible. It took me a while to realize that two same-colored stickers should never share an edge on the cube, or that having more than four same-colored stickers on corner pieces was a bad idea. Information like that eventually helped me swap stickers more efficiently, but it was never enough to help me actually solve the cube.

So, I eventually gave up. My father’s confidence in my budding genius probably never fully recovered.

Fast-forward 30 years. I am manning the AoPS table at the annual national MATHCOUNTS championships. At the table next to ours are representatives from Rubik’s. They’ve got cubes, and they’ve got instructions on how to solve them, and I have plenty of time. This was my chance at redemption!

### Terms and Symbols for Set Up

The pamphlet of instructions began with some useful terms and symbols. Edge pieces always have two different colors. I learned that the hard way 30 years earlier. Corner pieces have 3 different colors. I remembered that, too. Center pieces have one color, and they don’t move relative to each other. That fact had escaped me as a child. The yellow face is always opposite the white face, for example.

Next come more terms and symbols. There are 12 possible quarter-turn moves, named by the face you are turning (Right, Left, Back, Front, and the more awkwardly named Down and Up faces). A standard turn is clockwise, and a counter-clockwise turn is denoted by an “i” for inverted.

### The First Phase

OK, got it. On to solving the cube. The first few stages are pretty intuitive. With a little practice, I could confidently solve one third of the cube quickly and without memorizing any special sequences. It’s simply a matter of trading one piece for another until your cube looks something like this:

As a child, I never really considered that the colors on each face always match the color of the center square. You don’t move the centers, you move the pieces around them. I was mildly embarrassed that I didn’t think of this earlier, but also happy to be learning something.

### The Second Phase

The next step is to solve the middle third of the cube. This involves memorizing two sets of sequences that allow you to swap pieces from the top layer to the middle layer (without peeling off stickers!) while keeping the solved face intact. I learned to orient the cube properly and memorized both sequences (U, R, Ui, Ri, Ui, Fi, U, F and Ui, Li, U, L, U, F, Ui, Fi) until I could solve two thirds of the cube without peeking at the instructions. However, I never really understood why these turns worked and was a little skeptical about my newfound skills. Unlike the previous steps, these sequences were starting to feel like magic. Still, I pressed on.

### The Third Phase

The next stage of the instructions include three 6-move sequences and an 8-move sequence that sometimes gets repeated. Each of these sequences is paired with a particular cube orientation. At this point I was just trusting the algorithms blindly and trying to build some muscle memory as I practiced the various sets of twists and turns.

### The Final Stage

Completing the final stage involves memorizing a 13-move sequence of turns and two 12-move sequences. Missing a move or turning something the wrong way nearly always forced me to go back to the beginning. Nevertheless, I eventually managed to solve the cube several times without the instructions. After two days of playing with the cube and memorizing the sequences, I got my times down to what I felt was a respectable 2 minutes. (The current world record is less than 3.5 seconds!)

Unfortunately, my victory over the cube was pretty hollow.

Here’s the thing. I achieved the ultimate goal of solving all six faces of the cube, but I had almost no understanding of what I was doing. The sequences I memorized didn’t have any meaning to me. You would think that anyone who could solve all six faces could quickly solve exactly two faces. Nope, not me! You could take a solved cube, make 4 or 5 random turns, hand it back to me, and I’d barely be better off than if you mixed it up completely.

The only thing I really learned by “learning” to solve the Rubik’s cube is that I hadn’t learned anything at all. That turned out to be a pretty important lesson.

Figuring out one way to solve the Rubik’s cube gave me a glimpse into the mind of a child who is taught math as a set of memorized algorithms without ever learning the “why?” behind them. Too often, students memorize a set of steps that allows them to arrive at a correct answer without ever really thinking about what they’re doing, why it works, or how to adapt and apply it to other situations.

This was not a new realization for me. Early in my teaching career, I tried something with my 8th grade classes that would solidify my teaching philosophy. I created a page of 6 questions and asked my students to answer every one without writing anything except the answer. All of the work had to be done in their heads. Afterwards, I asked that they flip their papers over and write a brief explanation of how they thought about the problem. Then, we discussed their explanations.

One question asked students to compute 7×106. Nearly everyone got it right. However, in some classes, every single 8th grader explained how they mentally applied the standard multiplication algorithm, keeping track of digits as they performed the familiar steps. “First, I imagined the 7 under the 106. Then, I did 7×6=42. I imagined the 2 at the bottom, and put the 4 over the 0,” and so on. Kids mimed with their hands where the invisible digits would go in each step.

When I explained that they could have just done 7×100=700, then added on 7×6=42 to get 742, many were stunned. Whole classes of kids had never even considered using a method other than the one they had been taught in 3rd grade. Many asked, “Are we allowed to do that!?”

I couldn’t believe it. These were students who could use the distributive property with constants and variables, but who never considered breaking up a product to make it easy to compute mentally. Their grasp of multi-digit multiplication was as shallow as my grasp of the Rubik’s cube.

My experiences with the cube and with my students gave me valuable insights into the difference between knowing and understanding. Everyone seems to agree that educators should “teach for understanding,” but not everyone seems to agree on what that means.

When math is taught well, students are asked to confront uncertainty and overcome it on their own. This encourages real understanding — along with all sorts of other valuable traits like patience, curiosity, and resilience. But, parents watching a child struggle through tough problems often wonder why students can’t just learn the “easy” way. In other words, why can’t their child just memorize a set of steps that will always give them the right answer?

### Getting the Right Answer is Important, but not Enough

Knowing how to perform complex computations quickly and accurately is no longer a particularly useful skill. It should be obvious to anyone carrying a cell phone that knowledge is not nearly as important as it was 50 years ago. I will never know as much as I can look up on my phone. I’ll never be able to compute as quickly, either. Being able to recall information, apply formulas, and perform repetitive algorithms quickly and accurately is a domain that is increasingly dominated by machines. What’s useful now is the ability to apply what we know. That requires understanding.

While repeating an algorithm can lead to understanding for some students, it’s rarely the best way. For most students, a far more efficient path to understanding comes from discovering the “why?” behind an algorithm before they ever see it. Students who are encouraged to figure out the “why?” can make connections, create their own algorithms, and apply what they’ve learned in a variety of situations.

On the other hand, students who learn math the way I learned to solve the cube are left with a series of  meaningless steps that are hard to remember and easy to mess up. And at least when I mix up steps while solving a Rubik’s cube, I can tell I’ve done something wrong; the colors don’t match! Students don’t get such obvious clues when they make mistakes on math problems.

The consequences of my shallow understanding of the Rubik’s cube are trivial, but the consequences for students who have only learned to compute the “easy” way can carry on for years. If an easy way doesn’t contribute to understanding, math gets a lot harder in the long run.
I have long since forgotten the sequences of moves required to solve a Rubik’s cube. There was a time when I knew them, but I never understood them, so they were easily erased from memory. One day, maybe I’ll get around to truly figuring out the cube. When I do, I’ll have to write a pamphlet to help others understand it.