The other day I was working with some high school students on problems like this one:
I asked what I thought was a pretty straightforward question, "What does this mean?" [SPOILER: it wasn't as straightforward as I thought.]
One student said/asked something like, "Split the square root? Make it the square root of one over the square root of 64?" (Seriously, these were implicit questions asking, "Is this where I start?")
I responded, pointing emphatically at the problem, "But what does this mean? How would you say it in words?"
The other student interjected, "The square root of 1 over 64." And then asked, "We should rationalize the denominator, right?"
I tried again, "You're focusing on what you would do. I want to know what you understand about square root problems. What if it was the square root of sixty-four, then what would the problem be asking you to find?"
The first student, clearly getting frustrated, "I'd take a half of 64."
STRIKE THREE! YOU'RE OUT, COFFEY!
I didn't see any point to continuing to frustrate them or me with more questions. So instead, I tried to model my thinking about solving problems involving radicals.
When I was a kid, I used to bring a four-function calculator with me on long road trips (like the one shown here). There was no square root button on the calculator, so I would play What's the Square Root by typing in a guess, pressing the multiplication button and then equals, and use the result to see how I might need to adjust my guess. It was hours of fun!
If my question was, "What's the square root of two?" then I was essentially asking, "What number multiplied by itself is two?" So when I ask what this (again, emphatically point at the original problem) means, I'm thinking about playing the game and trying to determine what factor squared gives this (more pointing) product, the number inside the radical.
I went back to our simpler problem, "So this is asking me, 'What factor squared equals sixty-four." They both answered eight.
Then we went back to the original. "What does this mean?"
They said, "What number times itself gives one over sixty-four?"
"Yes, What factor squared is one-sixty-fourth? Now that we understand," I said, "we can begin thinking about what to do."
It's troubling to me that so many years after Pólya wrote How To Solve It, that students race past the first phase of the problem solving process - Understanding the Problem.Why is that? It can't be that we are focusing on the procedures and not the concepts. No, it can't be that, can it?