Here’s the work of a 4th Grader named Jaden. He has a lot of interesting ideas for finding the area of a rectangle. What do you notice in his work? What do you wonder?

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When I asked teachers this question as part of a Desmathmistakes activity, there were a lot of interesting responses. While all sorts of observations about student work are valuable, it can be especially valuable to transform our observations about student thinking into some next step. (Researchers look at work as an end in itself. When teachers look at student work it’s almost always to evaluate it or to figure out what to do next in class. We’re doing the latter here.)

Here were three of my favorite responses to the activity, with thanks to (in order) Mary, K, and Cindy.

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In case you’re curious, here is everybody’s rectangles:

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Finally, on twitter Kristin Gray is thinking in a different direction:

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Kristin’s idea is for a string of area calculation problems that all total to to the same area, but are partitioned in different ways:

Some meta-questions: What were people thinking about during this activity? What were they doing? Were they learning something? Could they be learning something?

Jump into the comments if you have some thoughts about Desmathmistakes Experiment #2.

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1. Isn’t this an interesting multiplication mistake?

2. I used to ask “how could we help this student?” or “why do you think this student made this mistake?” I still think that these questions are valuable to ask when looking deeply at student thinking. But, when teaching, the better question seems to be not “what mistake did this student make?” but instead “what could this student know that might help her?”

In this case, I’d say that this student could use more versatile ways of breaking numbers apart more than any sort of reflection on the errors of her ways.

Every once in a while people get in touch with me because they don’t like that this site is focused on mistakes. I think this is probably what they’re getting at.

In this case, the mistake (or whatever we call it) isn’t about what the student wrote, but what he said.

At the end of class, I asked my 3rd Graders to write a story problem for 13 x 2 and hand it in. As he was leaving, a boy handed me this slip and apologized for it.

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“Why are you apologizing?”

“Because my story is for 2 x 13, not for 13 x 2.”

Commentary:

The big lesson here is that the order matters in multiplication, as it does with addition (for most young kids 9+2 is much easier than 2+9) and as it does for algebra (4 + 2x = 10 is not the same as 10 = 2x + 4). Each of these problems has a different flavor for people who are beginning to get comfortable with these types of problems. Saying that two problems are “the same” is a substantive mathematical claim, and it needs to be taken with the seriousness that all mathematical claims require.

Here’s the breakdown of student thinking about double-digit multiplication that I’m seeing as we begin our unit in my 4th Grade class.

Direct Modeling:

Directly Modeling

 

Direct Modeling With Composition Into Groups:

Composing

 

Breaking The Numbers Apart With Addition:

Breaking The Numbers Apart

 

Breaking The Numbers Apart 2

 

Breaking The Numbers Apart With Arrays:

Arrays Arrays - Breaking Numbers Apart

 

Use of Standard Algorithm:

Standard Algorithm Standard Algorithm - Wrong

 

No Real Strategy, But Knowledge Of Multiplication by Multiples of 10:

Not Sure

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You can’t say that the kid is incapable of understanding what the box means here. Still, in the space of one line, it slipped through her fingers.

Is this connected to the way kids inconsistently treat exponents? I’m struggling to articulate a general principle, but it goes something like “Operations defined in terms of others are strongly associated with their parent operation, to the point that students often perform the parent in place of the derivative operation. As a result, students should always be introduced to a new operation in its own context, not in terms of other operations, whenever possible.”

Thoughts?