Cosmas Student Work

 

This piece of student work comes from a first grade student after an introductory lesson on frames-and-arrows.  The day this sheet was done, students were being exposed to frames-and-arrows for the first time so many common mistakes were occurring.  This student, however, got all of the answers in each frame correct but did some interesting things to get there.  When I taught this lesson I told students to always write the rule above each arrow so that they can constantly be checking their work by being reminded of the rule.  I also want students to follow this strategy so I can get a better idea of where misconceptions lie, or how mistakes happen. 

Looking at this piece specifically, the first two diagrams with rules “count up by 5’s” and “add 3” she completed fine.  When we look next at the last two diagrams which are “count back by 2’s” and “subtract 4” she does something interesting with her arrows.  When looking at “count back by 2’s” first, it is clear that the student completed the pattern correctly filling in 6, 4, and 2.  However, she writes the rule above each arrow as “+2” instead of “-2”.  Though she recognizes that the pattern of counting back means the numbers will decrease, she still dictates the rule as addition but doesn’t follow that.  The same thing occurs in the next example of “subtract 4”.  The student correctly finishes the pattern with 10, 6, and 2, but dictates the rule above each arrow as “+4” instead of “-4”.  When I asked the student why she wrote +2 and +4 she said, “Because that’s what the rule says”.  I then pointed out to her that the rules signal counting back or subtracting which she does correctly but that she has the signs confused.  The student agrees that her dictation was incorrect.

Reflecting on this piece of work I see three possibilities for why the student had this problem.  First would be that she has a misconception about what the + and – sign actually mean.  She may have the concepts of “add” and “subtract” solidified but may not understand which signs correspond.  This would lead her to be using them incorrectly as she does.  However, I can’t be sure that this assumption is correct because for this to be true she would be using the minus sign for addition problems which she does not.  My second thought on her possible confusion has to deal with which way the arrows are pointing.  We work extensively in my class on “hops to the right mean adding on” and “hops back mean taking away”.  With frames-and-arrows, all of the arrows move to the right which on a number grid or number line would suggest the numbers are getting larger.  However, if the rule is subtract or count back the numbers will get smaller regardless of the direction the arrows are moving.  This student may understand how to subtract 4, for example, but labels the arrows as addition because they are moving to the right and all of her prior knowledge supports that moving to the right means addition.  My third thought on this students piece is that the worked from the last frame backwards.  For example, she counted up by 2’s in which case her “+3” rule would make sense for the direction in which she was solving, but not in the context that the rule was given.