Cosmas Student Work

The piece of student work I chose for this week comes from the summative assessment I gave at the close of my unit.  One of the topics covered on this assessment was exchanging pennies and nickels.  I taught students a grouping strategy to help them see this concept of “exchanging” more clearly.  They would have a row of change that looked like this (with circles around each):

 N   P    P    P    P    P

Students would first need to count the amount of money and write it (using cent notation) on the line given.  Then the directions would say to “Show this amount of money with less coins”.  Students were taught to start with the nickels, draw a line down, and redraw the nickel as is however many times appropriate.  Then, students were taught to draw a bracket around 5 pennies, draw a line down, and draw 1 nickel instead.  To show these pennies were “exchanged” they would cross off all 5 to show they were gone and to double-check their grouping of 5.  This would continue until no more groups of 5 pennies can be made. 

On this piece of student work, the student did something interesting with his counting and exchanging.  The assessment had 3 problems like this and on all of them he first counted how much money was there incorrectly.  He was always 1 cent too high. However, then when he would start his grouping and exchanging the end amount he showed would be equal to the start amount showing that he understood the process correctly.  The interesting part is he would write (for example) 13 ₵ when the example was 12₵, but then in his exchange would show 2 nickels and 2 pennies equaling the correct amount of 12 cents.  This tells me that the student did not double-check his work otherwise he would’ve found that the two amounts he was dictating were not the same.  Since his counting response is always 1 off this leads me to think that the student has some type of misunderstanding when counting up the coins.  I assume that he understands that a nickel is worth 5 and starts counting there (because of the way he exchanges 5 pennies for 1 nickel and because his answer is only 1 cent off, not 5).   However, I assume also that he does not understand to start counting by 1’s right away once there are not more nickels.  He may have started to do something like this, for example: 5, 10, 12, 13, 14.  He may not understand the idea of counting on by 1’s from 5 or 10 or whatever interval of 5 he leaves off with.  Also, he may just not be taking his time and accurately counting the amount of pennies.  One thing that would help this student would be to slow down and make some type of marking after each coin he counts to actively keep track of his thinking.  I would also like to ask this student to do an example with me or to look back at their work and explain.  During the unit the student seemed to have no trouble with this, however when the assessment task was fully individual he showed this possible misunderstanding of counting and exchanging pennies and nickels.