Frustration!
I began by posing some quick problems just to check for understanding of the context. I asked how many candies were in 2 rolls, then in 5 rolls. Jack and Sunny both counted by tens to figure these problems out. Emilio too seemed to understand, although looking back, and knowing what he did later in the session, I’m not sure now.
I limit their use of tallies.
Today I asked them to sit at the same table, because I wanted to orchestrate a conversation about their use of tens in their solutions. I began by reminding them how sometimes they solved problems by making single tallies:
I said that today, I didn’t want them to use tallies like this. They could use the unifix cubes in sticks of ten or use numbers written on their paper or solve it mentally. My purpose was to push them to work with groups of tens. Allowing unifix cubes meant that if they needed to count by ones they could; but the structure of groupings of ten would at least be something they had to choose to ignore!
Jack sustains the progess he made last week.
As it turned out, Jack began the problem by drawing the rolls without candies:
He then decided to add the candies in.
Just as he was up to the next to the last roll, I asked him if he needed to show those candies in order to count them. This conversation was just like the one we had last week! He quickly said no and wrote “10� in the last roll. When I asked him later to write a number sentence or write numbers to show how he solved it, he wrote: 10, 20, 30, 40, 50, 60, 70. Progress!
It’s not clear what Sunny undertands about ten as a unit.
Sunny was slow getting started. She seemed to be confusing the idea of 6 rolls (with 10 each) and rolls of 6. She easily modeled the 10 loose candies with 10 single cubes. But for the rolls she had a stick of 6 unifix cubes and described it as “a roll of 6.� I clarified: “6 rolls of 10, not a roll of 6,� and she was off, modeling the rolls with 6 sticks of 10. There was some confusion about how to count the total of 6 sticks of 10 and 10 loose ones; she got 16 at first, but with a discussion in which I asked her to connect it back to rolls and candies, she counted appropriately. I emphasized in my revoicing of what she had done that she could count the rolls – 1, 2, 3, 4, 5, 6 – or count the candies – 10, 20, 30, 40, 50, 60. (Plus the loose ones, which no one has any trouble counting!) Success!
I am frustrated with Emilio!
Emilio solved the first problem and got 16 (adding 6 rolls and 10 candies). I asked him to solve it a second way, and he drew a stick of 6 and a stick of 10, and counted all to get 16. I asked him to talk with Jack about his strategy, and listen to how Jack solved his, but he didn’t talk and didn’t listen. I asked him if the problem was to hard for him, but he didn’t answer (instead, he concentrated on figuring out what time it was and when he could go home). I asked Sunny to explain her (terrific) direct modeling strategy, hoping he would see the difference between 6 rolls (sticks of ten) and 10 loose candies (individual cubes). It seemed like he looked everywhere but at Sunny and, more importantly, her strategy. At each step of Sunny’s explanation, I stopped her to ask Emilio a question, trying to get him to make a connection between the cubes arranged in sticks of ten and rolls of candy, trying to get him to make sense of the problem. It felt like he was resisting. It felt like he was deliberately not engaging.
Maybe. But I decided I didn’t want to assume that he was deliberately avoiding work. Perhaps it was his way of expressing boredom and confusion. So finally, as it was nearing time for our session to be over, I asked him if he wanted me to make him an easier problem. He said he did. So I turned his paper over and wrote “2 rolls, 10 candies, how many candies?â€? “12â€? he quickly replied. So I asked him to use cubes to show the rolls and the loose candies. “How many candies in one roll?â€? I asked him. He put his head down, said he was ready to go home, but, feeling resolute, I told him he couldn’t leave until he solved this problem. I was thinking about the fact that he had solved problems like this in the past. “How many candies in 2 rolls then?â€? I asked. “20â€? he squeaked out, with his head buried under his arms. “So, if you put 10 more candies with them … why don’t you represent those 10 candies with these cubes, any way you want.â€? He picked up about 4 sticks of 10 – just what I was pushing towards him in making my suggestion – and began to put them with the 2 “rolls.â€? “Now,â€? I pressed him, “show me the 10 loose candies.â€? It took a while but, finally, he produced a stick of 10, and put it with the 2 rolls. “How many?â€? I asked. “30,â€? he said, without even counting.
I don’t know if this was a power stuggle or a cognitive leap. What is the residue (to use a term I like a lot coined by Jim Hiebert and colleagues in Making Sense) of this interaction for Emilio? What did he take away from it? An understanding? A predisposition? A feeling that he CAN do math? A feeling of being forced to do something he didn’t want to do? The answer to that question — which I’m not at all sure of — is much more important than the fact that he answered “30″ in the end.
Posted in Case Study: Four Second Graders, Emilio, Jack, Sunny, Uncategorized | Comments (0)
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