On the heels of the series of posts on multiplication, I’d like to look at particular problem areas in middle school mathematics that show up year after year. Kids make mistakes or have difficulties, each in their own unique way, yet at the same time you start to see patterns that are independent of the particular kid or the particular teacher the kid had before he showed up in your classroom. I am going to focus on problem areas that I think relate to multiplication.
As a general note, let me emphasize that one big reason for focusing on problems and mistakes is that you can typically learn more about how the student is thinking from a mistake than from a correct answer. When a second grader puts down “35 – 3 = 5” you can make some powerful assumptions as to how the student is thinking; in contrast, “35 – 3 = 32” tells you far less.
One common thing I’ve seen seventh graders do repeatedly, that is, different students, year after year, is and replacing it with . If Jonah does this, what is he thinking? And let me be clear, I mean this as a completely serious question, not as a pejorative. “He’s not thinking!” is not a useful answer here. What is the thinking from which this particular step flows forth as a reasonable – if not correct – thing to do. The construct may show up in different ways, but a common one where I see the mistake is in the context of solving an equation like :
2x 


= 
x 
+ 
5 


x 

x 



2 

= 


5 
The student decides to subtract from both sides, which is a perfectly good strategy, since it will have the on the right side cancel altogether. On the left side, there is some canceling going on too. The students sees and and figures that what cancels is the part. So, what’s left must be 2….
The final result, makes no sense, so Jonah must have given up the expectation that math make sense, settling for doing the steps he knows the teacher wants to see.
What do you think Jonah is thinking when he combines to come up with ? It doesn’t seem farfetched to assume he is somehow thinking of the and the as being stuffed together in a bag, so that if you take the out you have left. The very same thinking would explain the elementary school mistake of , where the kid thinks of the 3 and the 5 in 35 as stuffed together in a bag and then taking out the 5, leaving 3 behind. Of course we know, and the kid will learn soon, that the 3 and the 5 in 35 have special roles, with the 3 standing for thirty (3 tens). Similarly, in Jonah’s situation, we know that in the has a special role in that it is supposed to be multiplied: stands for . Jonah’s misunderstanding seems to be centered around not appreciating that the and the are multiplied rather than added together. It would be fine for Jonah to think of as things stuffed together in a bag, but what Jonah would have to picture in the bag is an and another rather than an and a . The notation can in fact be consistently read as “two Xes” just like can be read as “two oranges”. And we know that “2 times x” and “two Xes” and “x and x” and “x + x” all amount to the same thing – and these are all ideas that have to be learned somewhere along the way. We weren’t born knowing it, and it is pretty clear that having been told it is not the same as having learned it.