Extending the children’s thinking.
As the kids were having their snack, I introduced the idea of true/false number sentences, based on ideas in Thinking Mathematically, by Tom Carpenter, Megan Franke, and Linda Levi.
First I wrote on a big piece of paper with a marker: “2+3=5â€? and asked them if it was true or false. Jack and Sunny said it was true; Emilio wasn’t sure, perhaps because, I figured out later, he wasn’t sure what “trueâ€? and “falseâ€? meant, especially in the context of math. Then we looked at: “2+3=6â€?. Catching on, Emilio decided it was false. The number sentence that generated the most controversy was “10 = 10â€?. Sunny declared it false, because, Sunny said, a number cannot equal itself. (So, you see, the reflexive property isn’t axiomatic for everyone!) Emilio agreed with Sunny. Jack declared it was true, because it was like 10 and nothing added was 10. I wrote underneath to clarify: “10+0=10â€?. Sunny agreed that this one was true but continued to disagree that 10 could be “equal to itself.” I promised we would revisit this debate and moved on to the first problem.
Emilio does something different.
As the children worked on the new Candy Factory problems, they fell into their usual patterns, with the exception of Emilio. I sat with him first to get him started. He read the problem to himself then I asked him rephrase it in his own words outloud. No problem; he remembered the quantities in the Candy Factory. I asked him how many candies in a roll; he said 10. So he understood the context and the problem parameters. “So,â€? I asked him, “how many candies does Dr. E. have?â€? His first reponse was 40 because, he spontaneously gave the reason, there’s candy in the four rolls. When I asked him about the 11 loose ones, he got 52, at first, because he added 10 on to the 40 (nice work!), and counted up somehow to get 52. (Being off by one in his counting reminds me of the counting mistake he made the first day we worked together. I’m not sure what’s going on.) When I asked him why he added the 10 on, he didn’t really say, and ended up solving the problem by counting up from 40 by ones. What a terrific solution! In contrast to his thinking last week, he did not seem to have any problem distinguishing groups of 10 from singletons; and once he understood the context, he had no problem applying his knowledge of multiples of 10.
I must admit, I am puzzled by how easily this strategy came to Emilio, considering the struggle last time we met. I wonder how much of his success, or lack of it, is based on whether he is preoccupied with something more important or perhaps just more interesting than the problem at hand. After all, when our attention is split, our capacity to reason is compromised, as this research suggests.
Jack uses numerals to represent his thinking.
Jack direct modeled by representing the groups of 10, showing each candy. But just as he has been doing, he counted the solution by 10s. He has shown that he doesn’t really need to do this and so I asked him to write a true number sentence that showed how he solved the problem. He wrote: 10, 20, 30, 40, 10, 1 — showing the quantities separately, and not how he combined them by counting tens. I asked him to write another number sentence using plus and equals like we had been doing, and he wrote 51 = 52 – 1; 51 = 53 – 2; 51 = 54 – 3; 51 = 55 – 4 — NOT, as I was hoping, 10 + 10 + 10 + 10+ 11 = 51. I wonder if it would make a difference if I ask him to show the rolls and candies with a number sentence? I want him to articulate (verbally or symbolically) how 51 is related to groups of 10.
Dramatization helps Sunny.
Sunny, like before, seemed to have trouble getting started. She confused rolls and candies, and at one point said there were 10 rolls, instead of 4. She also didn’t combine the rolls and the loose candies at first. Although her strategy wasn’t clear to me, I think she separated out the 11 loose candies from the 4 rolls. I thought that animating the situation for her, and in particular, putting her in the problem with me, might help her visualize the context. So I dramatized the problem with her as a character asking Dr. E. about the candies she had, just as Vicki Jacobs and Becky Ambrose reported teachers do. It worked. She decided that she needed 4 rolls of 10 and the extra 11, and counted them all to get 51.
Making connections.
At this point I decided to gather the children together. They had three different strategies but all of them had in common the use of tens in some way. A number sentence would be a good way to help tie together the ideas that were in each of these strategies. I wantged to create an interplay between representing thinking in abstract ways and their concrete strategies to stretch their understanding. I asked Jack to describe his strategy and as he was talking about the sticks of ten I wrote “10 10 10 10� to represent what he had drawn and to make a connection. Then on big paper, we followed through with: “10 + 10 + 10 + 10 + 11 = 51� to represent the entire situation. In my mind, I was thinking that building up to the number sentence from the concrete strategy to the numbers to stand for the quantities instead of pictures would help children build a connection. We’ll see.
The children write their own problems.
Emilio was pretty engaged throughout and even asked for a piece of paper to write his own problems. Everyone wanted to do this so I let the kids create some problems or number sentences.
Jack wrote a problem: There were 61 cats and then I found 41 more. How many cats? (What nice number choices!). He started to direct model by drawing a stick of 10 and I stopped him and told him to use numbers to solve it, because I knew he could. And he could. Here’s what he wrote: 10, 20, 30, 40, 50, 60, 1; and below it: 10, 20, 30, 40, 1. And so, I asked him, how can you use this to figure out how many in all? And he handily counted 10, 20, 30, 40, 50, 60 (and I was wondering what he was going to do when he started on the 40), 70, 80, 90, 100, 101. (He had forgotten an extra 1, which we talked about.) He loves writing these problems!
Sunny, at my suggestion wrote one true number sentence (10 + 10 = 20) and one false number sentence (35 – 6 = 30). It was interesting to me how she figured out the false number sentence, especially because any number (but the one correct answer of course) would make it a false number sentence. But she carefully solved the problem by counting back by ones to 29, and decided her wrong answer would be just one off – 30. I think there’s an aesthetic to this thinking, akin to a mistake rather than a wildly wrong answer.
Finally, Emilio wrote a bunch of addition and subtraction sentences involving numbers in the hundred thousands which were, he admitted, too hard for him to solve!
Posted in Case Study: Four Second Graders, Emilio, Jack, Sunny, Uncategorized | Comments (0)
No Comments »
No comments yet.
RSS feed for comments on this post.
Leave a comment
You must be logged in to post a comment.