Fractions of tears

"Is the following statement reasonable or unreasonable? Please explain your answer.

At Fairchild Elementary, 1/2 of the children brought their lunches from home on Monday. The rest got their lunches in the cafeteria on Monday. 2/3 of the children at Fairchild Elementary ate lunch from the cafeteria on Monday."

This was the first problem on a sheet of word problems for a fifth grader I worked with yesterday. It was her homework assignment and since I'm a Homework Helper she needed my help with it. When I asked her if she thought the statement about 2/3 of the students was reasonable, she said yes. I asked, "Are you sure?" and with that cue she said, "It's unreasonable." So I asked her why she knew it was unreasonable.

She had no idea. She didn't even recognize it as a multiplication or division problem, and told me their teacher hadn't taught them how to multiply fractions yet (which I knew wasn't true). I asked her if there were 100 kids at the school, and half of them brought lunch, how many of them brought lunch. Her math was perfect: 100/1 divided by 1/2 = 50. But she couldn't tell me how many kids ate lunch in the caf if 50 kids brought lunch. Several times she insisted 2/3, because that's what was on the sheet.

We went over it several times. I told her about the magic of two: that you would always be even if you dividing by two, that two, in this case, equaled half, and halves are always even. I folded a piece of paper in half and described the sheet as "whole" and described the two halves on either side of the fold as "half," pointed out how they were absolutely even. I asked another HH for help. She came over and drew two boxes, coloring half of one of them in and coloring in 2/3 of the other one to demonstrate that 1/2 was smaller than 2/3. The girl nodded but I could tell this drawing didn't explain the concept either.

"Two thirds had lunch in the cafeteria," she would say. She even showed me how many kids were 2/3 of 100. She couldn't make the connection. It reminded me of the scene in Brave New World where Huxley describes the test run on kids, whispering facts to them while they slept. The kids would remember the facts, but only in the way they were whispered, so that while they could repeat something like, "Fifty is half of one hundred" they couldn't answer a question like "How many times does 50 go into 100?."

I realized that the student was crying. I totally understood why because I was that girl once, and I've forgotten more about fractions and word problems than she knows yet. The way I felt when I was in her position was this: "Here's this person who is explaining fractions/algebra/trigonometry to me and I don't get it. I know the explanation should make sense but I don't understand it." That was frustrating enough, but I would start feeling like the tutor was equally frustrated, and would get mad at me and think I was stupid. Most of the time I knew I wasn't stupid, but times like this made me doubt my intelligence: I should understand this, everyone else is... And that thought would oh-so-quickly turn into shame. No wonder math homework made me cry, as it did this girl yesterday.

Like my tutors did with me, I refused to give her the answer. It was her homework, and I'm old-fashioned enough (or went to strict enough schools) that I think a kid should do the homework herself. Ultimately the girl wrote on the sheet, "Unreasonable. 2/3 is bigger than 1/2" and I didn't push her on it. "Let the teacher deal with it," I thought, "It's the right answer for the wrong reason." Just like most of my geometry homework was.