We had the problem “If you are traveling at 40 miles per hour, how long will it take you to go 10 miles?”
Walter: 4 minutes.
JH (me): How did you get it?
W: Divided the 40 by the 10.
A quick look at my face told him that this would not do. After a while he wrote, “15 minutes.” I wanted to check his understanding.
JH: If you were going 50 miles per hour, how far would you go in 24 minutes?
W: (quickly): 36 miles.
JH: How did you get that?
W: Subtracted 24 from 60.
He still hadn’t gotten it. I tried again.
JH: If you were going 50 miles per hour, how far would you go in 30 minutes?
W: 25 miles. 30 minutes is half an hour, and half of 50 is 25.
It sounded as if he knew what he was doing at last. I thought he would have no trouble with the 24 minutes problem. But it took a long time, with some hinting from me, before he saw that 24 minutes was 2/5 of an hour, and therefore that he would go 2/5 of 50 miles, or 20 miles, in 24 minutes. Would he have discovered it if I had not paved the way with leading questions? Hard to tell.
Most teachers would have assumed, as I would have once, that when he got the 15-minutes problem, he knew what he was doing. Even the skeptical would have been convinced when he gave his explanation about the 30-minutes problem. Yet in each case he showed that he had not really understood what he was doing, and it is not at all certain that he understands yet.
John Holt : How Children Fail