Progress!
The third problem (11 packages of 10 cookies each) proved interesting and productive. It was a challenge for most of the children.
Jack makes an advance.
Emilio and Jack both started out by adding up the numbers to get 26 for their answer. Jack then decided that wasn’t the correct answer (not sure what changed his mind; he didn’t say why), and easily direct modeled the problem by drawing groups of tens:
As he was drawing all of this out, I asked him if, instead of drawing each cookie in each package, he could represent the cookies in the package by writing “10� in each one. He said that no he couldn’t; it would be too hard. But I noticed when he counted the total he counted by tens, so I again I asked him if he could represent the cookies by “10� instead of drawing each one out. I pointed out to him that he had just counted each group by tens. It seemed to make sense to him this time so I gave him a new but related problem to solve, encouraging him to use this new strategy. The problem was 14 packages of cookies, 10 in each package, and 10 extra cookies. This is what he drew:
He said he forgot to use the strategy we had talked about (and only remembered when his hand started getting tired!), but because he had so easily solved this problem I was sure he could use the more abstract counting approach. So again I posed a new but related problem: him how many cookies would be in 12 packages. When I came back, he had this:
He agreed that this strategy was faster, as well as easier on the hand.
Emilio misinterprets the problem and I try to get him to listen to Daniella’s strategy to change his mind.
Emilio had trouble getting started on this problem. It’s not clear to me why. His initial answer was 26. He told me he got it by adding 11, 10, and 5. I asked him why he decided to add them altogether and whether they were all cookies or packages, but he gave no clear answer.
Daniella, like Jack, direct modeled the entire situation by representing each cookie, but she confounded packages of cookies with single cookies:
Because she had accurately represented the packages of cookies and Emilio had not, I decided to ignore her confusion about the 5 extra cookies for the time being and called Emilio over to compare what he was doing with what Daniella was doing. The first difference he saw was in how each of them had represented the package:
His was more realistic. Daniella saw that he had 6 cookies in his packages and she had put 10 in each of hers. With some prompting from me to address Emilio and not me, she was further able to tell Emilio why she had drawn her packages this way. Emilio decided to start over, and at my suggestion, gathered a bunch of sticks of unifix cubes in tens. He ended up with 34 sticks of 10 arrayed in front of him but didn’t solve the problem before it was time to go.
Sunny thinks really hard.
Like Jack and Daniella, Sunny started out drawing the packages of cookies with each individual cookie represented. I encouraged her to use the cubes in sticks of ten instead, thinking that the ten-to-one structure might support a more sophisticated strategy. She decided she wanted a bunch of sticks of four. I wasn’t sure where she was going with it, but decided to let her create them (and helped her). She then gathered sticks of ten and used the sticks of four to stand for the packages. The number of cubes in each stick was probably irrelevant; perhaps it was the long rectangular shape that reminded her of a package. The she “put� 10 cookies – a stick of ten – in each package, like this:
and counted the total beautifully by tens. During all of this she mentioned that she was thinking so hard she couldn’t even think of anything else. I think it was a keen observation because at the end I had to remind her of the 5 extra cookies. She included them but counted them as tens, as Daniella had at first.
Looking forward.
With encouragement, then, Jack, Daniella, and Sunny were able to move from counting by ones to counting by tens. Jack was able to represent groups of ten by something other than a collection of 10 things – a real advance, if he sustains it. I’m not sure what Emilio can do or how much he understands of problems like this. He has solved them in the past. His focus today seemed split so I think these problems didn’t get his full attention. In fact, he started out wanting to do his spelling homework!
Next time, we’ll work on more problems like the first and third problems and I plan to continue to push the children to represented sets of 10 with the numeral “10�. This may be facilitated by using smaller numbers. We’ll see.
Posted in Case Study: Four Second Graders, Daniella, Emilio, Jack, Sunny, Uncategorized | Comments (0)
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