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**