Student C was a girl. Her problem was, Tom has 2 pennies in his bank. He collected 9 more pennies. How many pennies does Tom have in his bank now?
First she looks at her fingers, thinking about the problem. I think she may have realized that the answer was going to be larger than 10, which is more than she can directly model on her hands. I think she decides she doesn’t need to completely count this problem out on her fingers. So, she finishes the problem in her head and writes on the paper 9+2=11. She explained her thinking by showing nine fingers and then counting 10 on her pinky and then counting 11 by starting over on her thumb.
This student does not draw a model or use the blocks. So, I know she is past the direct modeling stage. She seemed to use simultaneous double counting when she counted 1,2 on her pinky and thumb, which actually represented 10 and 11 in the math problem. She solves the problem by using her fingers as anchors but mostly holding on to the numeric values, showing that she has quotity.
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The CGI problem type she was given is active, joining, result unknown.
Student D was a girl. The problem she was given was Sam had 11 stickers. He gave 5 stickers to his sister. How many stickers does Sam have left?
This student draws the stickers with tally marks. Then she crosses out five of them by counting each one until she gets to 5, she counts the tally marks that are left and says 6.
This student used direct modeling to solve this problem because she needed to physically make each sticker, using one to one correspondence to be able to accurately keep track of the amount she was manipulating. This shows that she has not yet developed quotity.
She used a direct modeling strategy to solve this