In this series on common mistakes made by students in middle school mathematics, let’s look at this one:
Jessica’s response: .
We can agree that this is wrong, and we can probably tell the student what to do to avoid this mistake. What I’d like to pursue instead is the question: What is Jessica thinking so that her response follows from that thinking?
In other words, I’m not asking to name or identify the mistake, nor am I asking how to do this correctly. I’m asking what you think the model is that she is thinking from. Underlying my question is an assumption that Jessica isn’t just guessing randomly, but does indeed think from some kind of model, a model that has probably worked for her in prior situations, a model that she now applies in a situation where it is insufficient.
To approach this, I want to compare two other questions. The first one is:
How much is 3 apples and 5 more and then 7 more?
And the second one is:
How much is 3 apples and 5 oranges and 7 oranges?
When I ask this question, I get pretty consistent answers: “15 apples” for the first question, and “3 apples and 12 oranges” for the second one. Depending on the age of the kid I ask, the number part of it may be easy or hard, but I have yet to encounter kids who are confused about the apple or orange part of it. Kids are clear (or at least consistent) in what “5 more” means: it means 5 more apples. The “apple” part is simply implied. If it is made explicit, as in the oranges case, kids will keep them straight. Though they could have answered, correctly, “15 fruits”, they just never seem to do this unless pressed for a single number.
Back to – it is clear that for the teacher this resembles the apples and oranges case. The “apples” are the parts with “x” in it, and the “oranges” are the parts without an “x”. I wonder if for Jessica, it evokes the “and 5 more” idea. If so, my normal interventions when students like Jessica write are misdirected.