Cosmas Student Work

Over the past two days in my classroom we have been working on “change to more”/ “change to less” problems and their appropriate number models.  As anticipated, this was a mildly difficult task for me to teach and for students to understand.  We started with change to more and using the specific language of “add 5” and “plus 5” to introduce the + and = symbol in creating number models.  We went through this process step by step by having a start number, a change of “add X” and an end number to create the number model.  Students used their number grids to start at a number, and manipulate the “hops” or added amount to arrive at their ending number.  This was fairly easy for students to grasp because manipulating the number grid and getting bigger is something they are very familiar with.  However, once we moved to change to less, students had difficulty hopping backwards specifically doing a backwards sweep. 

              The piece of student work I will be looking at is an example of the difficulty encountered when doing change to less problems and counting back on the number grid.  This sheet was given during independent work time and at first the student did every example as a change to more and added on completely ignoring the “minus” clue and all the examples we had done in guided practice together.  Once I explained again that minus means take away, and when we take away we move back on the number grid the student started over.  I sat with him for the first problem which required him to start at 15, and “minus 4”.  He accurately ended at 11 and wrote the appropriate number model with a subtraction and equal sign, so I moved on to other students with the intention to return in a moment.  When I came back I noticed something strange.  He had some correct responses where he didn’t have to complete a reverse sweep (ex: from 11 up a row to 10) and others were about 10 off.  I asked him to show me how he completed these problems.  One example was start at 22 and “minus 6”.  He began correctly at 22 and his fingers followed this route, “22, 21, 30, 29, 28, 27, 26”.  As I watched him do this I realized that as he needed to make a sweep from 21 to 20 he instead swept to 30, a bigger number.  When I asked him if it made sense to go from 21 to 30 he agreed it didn’t but wasn’t sure where he should’ve gone instead.  We went over the process of counting back together until he showed me independently he understood the concept.  The mistake he was making leads me to believe he simply had a misconception not about how to count back but which way to go about manipulating the number grid.