Our last session together.
Today I wrote a mix of problems for the children to solve that would give us some insight into what the children had learned about base-10 concepts and their use in problem solving. I worked with Jack and Sunny, and Luz worked with Emilio and Daniella.
Jack.
Jack seems to have made a great deal of progress. He solved the first problem (8 rolls of candy, 10 in each roll) today by drawing a rectangle-like representation of each roll. At my suggestion, he wrote “10″ above each one. He finished off by drawing the extra 12 candies individually. As he counted them however, he pointed out the extra 10, for a total of 90 and 3, oops, 2 more:
I also asked him to write a number sentence and he wrote “10, 20, 30, 40, 50, 60, 70, 80, 90, 92″ (as before). This strategy is significant because he no longer depends on representing the individual units (each candy) to construct 10. Ten is a unit for him!
In the middle of our session he spontaneously showed me this strategy for subtracting:
It’s an interesting contrast with the rest of his work today. It was as if he was trying to re-call the steps involved in subtraction with regrouping rather than re-construct the steps based on what made sense to him. I didn’t address this mistake with him, because I wanted him to continue to work from what he understood.
I asked Jack if he could solve the second problem (30 pencils, 29 more pencils) in his head; he thought for a moment, said no, and proceeded to draw this:
This time I asked his to write a number sentence using plus and equals to show how he solved it. He wrote “10+10+10+10+10+9=59.” As he was writing the tens, I asked him how many tens in 50. “Five,” he said. He understands the place-value relationship between 50 and 5 10s.
Jack’s solution for the third problem (45 beads, 10 beads per necklace) suggests that his new knowledge of ten as a unit may be somewhat fragile. When I checked in with him, he had written on his paper:
The problem seemed to be solved. He seemed to think the answer was 4. Excellent! But as I questioned him about what he had done and why he had done it, his answer changed … to 5 (pointing to the remainder), then to 40 (the number of beads in 4 necklaces). I continued to ask him questions to help clarify his thinking and to emphasize the context of putting beads on necklaces, and the relationship between beads and necklaces. We finally arrived back at his original answer of 4 total necklaces.
Turning the open number sentences, I again asked Jack if he could solve the problems in his head. “Yes,” he said, for 30+40=__. He counted on by tens from 30 to get 70. I skipped 25+20 in order to see what he would do with another problem that involved only multiplies of 10. “How about 60-20?” I asked. “80,” he replied. I drew his attention to the minus sign (he knew what is was). “So if it’s plus,” I said, wanting to reinforce his mental strategy, “the answer is 80. What if it’s minus?” Jack easily counted back by tens to get 40 and likened the problem to 6 take away 2.
Nice job Jack!
Sunny.
Sunny’s strategies were more concrete than Jack’s, but I noticed that the language she was using to describe these strategies suggested an emerging understanding of base-ten and place-value concepts.
For instance, for the first problem (8 rolls of candies, 10 candies per roll), she direct modeled the problem using unifix cubes in sticks of 10. But when she described her solution she said, “It’s 80, because 8 tens is 80, when you count by 10 8 times, it’s the number 80.” I can see in this explanation that Sunny is making a connection between “counting by tens” a certain number of times and multiples of ten.
The emergent nature of her understanding of the base-ten structure of numbers was apparent in her strategy for adding 30 pencils and 29 pencils. Again she direct modeled the quantities. But beyond this, Sunny made little use of base-ten concepts: she counted up by ones from 30. I think she is just arriving at understanding 30 is 3 tens and that applying this knowledge in constructing a solution such as counting on by tens is somewhat beyond her. However, if I were to continue to work with Sunny, I would continue to give her addition and subtraction problems with double-digit quantities to help her develop strategies that made more use of base-ten concepts and processes.
Sunny also did something for this problem that I don’t think I understand. As she was modeling it, she included 2 extra cubes. I asked her what the 2 extra cubes were for. She didn’t say clearly. I pointed to each quantity she had made and asked her what part of the problem it was. She pointed out Jack’s pencils, and Emilio’s pencils and realized that the 2 cubes didn’t have a referent in the problem.
I had to leave at this point (appointment with the dentist) and left the children in Luz’s hands.
Posted in Case Study: Four Second Graders, Jack, Sunny, Uncategorized | Comments (0)
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