# Analysis Of Placement Within Carpenter's Framework: Student A

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Student strategy: Student A starts by drawing 11 circles, which is what I saw. When asked to explain what he did, he said that he first drew 2 circles and then 9 more. He then added them together by counting them all to get 11 pennies, which is the answer. When solving the problem, he adds them by counting from 1 to 11, and he seems to have drawn 2 circles and then 9 more, following the time sequence. However, I never saw him count 2, 9, and then all of them, I only saw him count 1 to 11. Placement within Carpenter’s framework: Student A falls within the ‘direct modeling’ framework. Student A had a one to one correspondence with the circles and the numbers in the problem. He drew 2 circles to represent 2 pennies and then drew 9 circles …show more content…
After he heard the problem he mentally held the number 9 and then counted on. Using his fingers to count to 5, at the same time he counted on from 9, adding 5 more, to get to the answer of 14. Student B was double counting as he solved this problem.

Placement within Carpenter’s framework: Student B falls within the ‘counting’ framework. Student B was able to mentally hold the number 9, which shows that he has quotity. He then counted up from there while simultaneously counting on his fingers the number 5. Student B also ignored the time sequence of the problem by starting with the larger number and then counting on.

Student strategy: Student C uses his fingers to count on from the larger number in the problem and then he wrote the algorithm for the problem. After he heard the problem he mentally held the number 9 and then counted on, using his fingers to count to 2. While counting on from 9, he added 2 more to get to the answer of 11. Student C was double counting as he solved this
Student G had a one to one correspondence with the blocks and the numbers in the problem. He counted out 7 blocks to represent 7 trolls and then counted out more blocks until he had 11 blocks, which was how many trolls the girl in the story problem wanted to have. He then added all the blocks up starting with the first block, showing that he does not have quotity. However, Student G added a group of 4 blocks, which he may have counted on from the 7 blocks to reach 11, but it was hard to tell. When he had 11 blocks, he counted them all again to see if he had 11 blocks, but he kept the 4 blocks separate, knowing that the was the answer to the problem. I feel like he may be on the verge of moving onto the ‘counting’ framework, but isn’t quite there. He also followed the time sequence of the story problem by first grabbing 7 blocks, then grabbing some more blocks until he reached a total of 11

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