In the car the other day, the Maker, currently in 4th grade, asked me what 100 divided by 3 is. (Sometimes when he asks me arithmetic questions, I'll just toss it back to him and ask him to tell me the answer, but I knew he'd been working on understanding division so I told him it was 33 and 1/3.) "And what's 100 divided by 4?" he asked. "Twenty-five," I said.
When it appeared that he wasn't going to ask anything else, I said, "And what's 100 divided by 5?"
That slowed him down for a minute. He suggested the answer was 30, but I pointed out that 5 times 30 was 150, so that wouldn't work. Then he suggested 29, then 28, and so on down to 26, each time adding up five of the number (in his head), attempting to find one that would add up to 100.
When he got to 26 and that one didn't work, he declared that the answer must be a fraction between 25 and 26.
I asked him how he knew the answer wouldn't be a number below 25, and after thinking about it for a minute, he started trying the "add the number to itself" approach with 24, then 23, and so on until he got to 20, at which point he declared this to be the answer.
Later, I asked him why he initially thought the answer had to be greater than 25, and he got a piece of paper to show me. He explained that he had gotten division confused with multiplication. If you have two multiplication problems, and one answer (i.e., the product) is greater than the other, yet one of the multiplicands is the same in both problems, then you know the other multiplicand must be greater than the corresponding multiplicand in the first problem. In division, it's kind of the reverse, since, as The Maker explained, multiplication and division are opposites. In other words, as the divisor gets larger, your answer (i.e., the quotient), gets smaller.
In this case, The Maker's reasoning was perfectly sound; he understood that there was a pattern in the division problems we were talking about, and he was systematically following the pattern to find the correct answer. The problem was that he was using the pattern for multiplication rather than the pattern for division. Once he worked his way to the correct answer for the problem he was working on, he realized his mistake.
A professor at Swarthmore College, Heinrich Brinkmann, is said to have been well-known for being able to find something right in what every student said. No matter how outrageous a student's contribution or question, he could respond, 'Oh, I see what you are thinking. You're looking at it as if...'" (See the book, Math Power: How to Help Your Child Love Math, Even If You Don't.)
What a wonderful trait for a teacher (or parent) to have. As Jo Baeler notes in her book What's Math Got To Do With It?, "unless a child has taken a wild guess, there will be some sense in what they are thinking - the role of the teacher is to find out what it is that makes sense and build from there."