The piece of student work I chose to analyze this week comes from one of my second grade students named “Carrie”.  This artifact is a worksheet that was given as homework to students and challenged them on their number grid knowledge.  The top portion of the sheet asks students to “think about the number grid and fill in the blanks” with questions about what happens when you move right, left, up, and down on a number grid.  On this top portion, Carrie correctly identified that when we move to the right, numbers get bigger by 1 (and smaller by 1 to the left), and that when we move directly down, numbers get bigger by 10 (and smaller by 10 directly up).  Then, the bottom half of the sheet has six different sections of a number grid with “cut-outs” where some numbers are filled in and some are left as blank squares.

              The first one looks similar to this:

76	77

              In this grid, Carrie correctly filled in a 67 upward, 78 to the right, and 87 directly down.  She seems to have no trouble with this small arrangement.  However, once the grids become a little bit larger OR include numbers in the hundreds she begins to make mistakes.  The grid that I found particularly interesting looks similar to this:

 

424				427

                            

                            

 

              This time when Carrie fills in the two boxes above 424 they read “324” and “224”.  She recognizes that the numbers will be smaller since the spaces are directly up, but she decreases them in the hundreds place instead of the tens place.  This is interesting because she already identifies that she knows numbers move in increments of 10 when moving up and down but she doesn’t show this here.  She does the same thing in the box above 427 and it reads “327”.  This makes me wonder two things.

              First, I wonder if she has a misunderstanding of place value when the number involves hundreds.  Considering the first example I believe she shows knowledge of tens and one’s place there, but may be confused when there are three digits involved which is which.  She may be confused also about what changes when you add to the hundreds, tens, or one’s place in a number. 

              Second, it makes me consider whether she understand the basic principle of counting by 10’s as it applies to all numbers.  She may be able to count by 10’s on a number grid with the usual “10, 20, 30 40”, but may not understand that we are still counting by 10’s if we say “6, 16, 26, 36”.

 I think one way to clear up this potential confusion would be to have a conversation with this child and ask her about adding or subtracting 10 in a sequence like we do when we just count by 10’s.  Then see if she understands that by adding 10 to any number we can still say we’re counting by 10’s even if the number doesn’t end in a 0 like the normal pattern follows.  I think Carrie may understand how to manipulate a typically number grid of 1-100 that we often see, but not how to make sense of it.  I say this because she clearly shows that she knows the directional “rules” of a number grid since she indicates them on the top portion, but demonstrates she may not be able to put that knowledge into effect.   I also think in the future having different starting points for the usual patterns of counting by 2’s, 5’s, or 10’s will benefit Carrie and her understanding that not all these sequences have to start with 2, 5, or 10 in order to make sense.  Offering her different entry points will be greatly beneficial to solidifying the actual meaning of the patterns we find on a number grid.