Cosmas Student Work

The sample of work I chose this week is a worksheet from the Everyday Math curriculum my class uses about counting up and counting back on the number grid.  With my first grade students we have been working a lot on counting our hops up and back on the grid using their fingers so the concept is definitely familiar. 

This day, I conducted a whole group lesson specifically about counting on the number grid and this sheet was given for independent work after the lesson.  As I was walking around I noticed this student understood what each problem was asking him to do (count up or back), referring to the number grid to count, and filling in his responses.   However, each response was 1 off either too high or too low.  At first I thought maybe he was making the common mistake of saying 1 before he even made a hop making the final answer 1 too low  (Ex: start at 27, count up 10.  Student puts their finger on 27 and says 1, hops to 28 and says 2, etc.).  However, when I asked him to show me how he arrived at his answers he did something really interesting.  He did not do what I had anticipated.  He correctly started his hops, but when he arrived at the end of a row (10, 20, 30) he did two things: he either didn’t count the first number after the sweep (11, 21, 31) and moved on to the next number (12, 22, 32) leaving his answer 1 too big OR he counted the sweep AS a hop resulting in his answer being 1 too small.

In the photo included of his work his answers were erased and corrected after talking it through with me and modeling again how we make a sweep when counting on the number grid.  Once we worked together on this he completed #6 on his own and correctly.  This leads me to believe two things about his thinking.  First would be that in a whole class setting he didn’t receive the individualized attention he needed to understand fully how to maneuver the number grid.  There is a large class number grid next to the smart board where I model how to count on and each student uses the back page of their math journal for an individual copy.  I observed him complete this process in class  a couple times, but he may have just been going through the motions as I did them instead of making a conceptual understanding of how the “hop” helps us count up and back.  I think one way to push his thinking further or to make this concept come to life and make sense would be to incorporate this number grid into more of an addition problem and having him use manipulative on the grid itself.  From getting to know this student, I know that at home he works with his parents on simple addition problems and having a student start at a point and move 10 spaces to see where they end up on a grid is essentially an addition problem phrased differently.  I think this student would benefit from seeing what he is adding on to with some type of counters. 

Second, my observations and this sample of work leads me to believe that this student may not understand that counting on the number grid is the same process of adding on as counting on our fingers or with counters would be.  He may be actively paying attention to the hops he is making but not understanding that each hop is how we add 1 to a number.  If he understood that this is just another way of counting to add on or take away maybe he would have been able to self-correct that starting at 24, counting up 10, and ending at 33 doesn’t quite make sense because that only adds 9.  I believe that using the number grid as a reference and instead counting with fingers or counters may be more beneficial for this particular student to understand.  Also, I think that practicing the way we sweep our finger down a number grid would help with this student too.