We begin our work together.
The four children we are working with are Sunny, Daniella, Jack, and Emilio. I decided to spend the first session finding out about what they could do, what they understood, how they expressed it — both orally and in writing — how confident they were, and so on.
Not knowing exactly what to expect, I start with fairly small numbers.
The first problem was a Join Result Unknown (13 jelly beans, get 8 more). The children either counted up from 13 to solve it or direct modeled it with cubes or pictures. Nobody used any derived or recalled facts.
Emilio used a counting up strategy but his answer was off by one. I noticed when he counted up he started with the number 13 and counted it as one of the 8 he wanted to add on. He ended up with 20. When he heard that the other children got 21 for their answer, he changed his answer from 20 to 21.
I wanted to explore this counting mistake with Emilio. So I asked Sunny to explain her direct modeling strategy using cubes. My idea was that Emilio would be able to see the 13 as a separate set and that the 8 to be joined to the set should be counted beginning with 14. No such insight occurred! But I believe nonetheless that it will be useful for Emilio to continue to attempt to relate counting strategies to more concrete direct modeling strategies so that he can figure out how counting works.
The second problem we worked on was a Separate Result Unknown (28 pennies, lose 13). Daniella solved this problem by writing 28 – 13, vertically:
She separated the tens and ones into two columns, and subtracted the ones first (got 5) then the tens (1). Her use of this algorithm made me curious about what base-10 concepts she understood. I began listening for evidence of base-10 understanding among all the children and noticed that, even though the problem involved double-digit numbers, none of the children had used base-10 concepts in their strategies.
I therefore created a third problem, on the spot. After ascertaining that they all knew about and liked soccer, I posed this problem: “You’ve got 3 big bags of soccer balls. Each bag has 10 balls in it. You’ve also got 2 loose balls. How many balls do you have?� I used hand gestures to indicate the bags were BIG and repeated the problem to be sure the children heard it. They set to work. Everyone but Emilio was direct modeling the problem by drawing all the balls individually:
No use of tens! Emilio wasn’t doing anything, so I repeated the problem for him. “Oh,� he said. “10 plus 10 is 20.� I was so pleased with this insight that I emphasized to him (and for the benefit of the others) that he didn’t even need to draw any pictures to figure it out. It prompted Jack to remember that he too knew that 10 plus 10 was 20.
Still the children struggled. Sunny wasn’t sure whether to add or subtract the two loose balls. I told her she had to decide for herself what made sense. She subtracted because, she said, the two “loose� balls could roll away. (Interesting point!) Emilio ended up with 30, then when I asked him how he was going to count the 2 loose balls, he changed his answer to 31. And Jack ended up with 28, because one of his bags had the wrong number of balls in it.
Wow. There was so much to talk about, but it was time to go, so I made a note to myself to return to problems like this one next time.
Posted in Case Study: Four Second Graders, Daniella, Emilio, Jack, Sunny, Uncategorized | Comments (0)
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