Tonight was my fourth evening of math with my kids. It was rather frustrating, but turned out good in the end. I pulled out the number line again, which had markings for 0, 1 and 2, as well as 1/2, 1/3, 1/4, etc. through 1/17 and then an annotation saying: “infinitely many more like these.” I said “the zero has lots of friends, but what about the one?” Mostly they grumped around and doodled for a while. Then they kind of stared at me blankly as if expecting me to answer my own question. They weren’t talking to each other at all, so I left the room and told them to work it out on their own. B called me back a while later saying that 1 had some more friends now. I came back into the room and he showed me that he had written 1/2, 1/3, 1/4, etc. through 1/17 between the 1 and 2. I prompted him to tell me what the difference between the 1/2 between the 1 and 2; and the one half between the 0 and 1 was. He said that they were the same, except that they were in different places. I asked him what he meant and he continued to insist that they were the exact same number, but that they show up in those two places. Next he told me that the 1/2 between the 0 and 1 is supposed to mean half of zero, while the 1/2 between 1 and 2 is half of one. Hmmm…something was going wrong here. I asked what half of something means. B said that it means you split a number up. A basically wasn’t participating. I was starting to get frustrated, so I took down a bunch of bananas. I asked them to show me one banana. Now A perked up and B shut off. A held up one banana. I told here to put it in its spot on the number line. She did. Then I asked them to show me two bananas. A picked up two bananas, and I prompted her to put it in its spot. Then I asked B if she was right that this was two bananas. He said that he didn’t know (!). Obviously he was frustrated. He knows what two bananas is. Finally he agreed that A was right with her two bananas. Next I asked for zero bananas. They both understood that with no problem. “Now, how many bananas do we put by this 1/2?” I asked. B claimed that we should put zero there. He was still thinking of it as half of zero. I’m beginning to understand that he was confused with the concept of half as a number (an element of the real numbers) and half as a scalar multiplier of the integers. (“half of zero is zero. Half of one is, well it’s half of one. Half of two is one,..etc., but what is one half? It’s a different kind of object…right?” that’s what I imagine was going on in his brain, but I’m not sure.) Finally I got out a knife: “Show me half a banana.” He was starting to get it now. He cut a banana in half. There was lots more frustration before the night was out, but eventually we got 5/3, “one and a half,” 4/3, and 2/3 on the number line in their correct places (!) and ended up with a bunch of cut up bananas. Smoothies for breakfast!
Ok, tonight was not tons of fun for any of us. We’ll give the number line a break for a while. It’s tough for me to work with the kids on something where I know the answer and they don’t. I need to get better at that. But next week I plan on moving them closer to doing some math research where I don’t know the answers.