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?

Me: What’s 91 divided by 7?

Her: [Draws hands on board.]

Me: What are these for?

Her: For counting.

My move was to nail the question down on a context and ask her the question again.

Me: Hold on. Let’s make up a division story for this question. Let’s say that 7 people are equally sharing 91 crackers.

Her: Can we change it to mushrooms?

Me: Sure.

And she starts counting on the hands. She hadn’t done this for smaller numbers, like 30 divided by 3. There she articulated that 30 divided by 3 is 10, because 3 times 10 is 30. That doesn’t seem to be on her mind right now, so I try to ask a suggestive question.

Me: [Draws 7 stick figures.] Here are the 7 people. They don’t have any arms though.

Her: Can you make one super tall and one super short?

Me: Not this time. They’re all the same armless height. Anyway, how many mushrooms can we definitely give to each person?

Her: 10.

Me: Cool, and that would take care of a bunch of the mushrooms. That would take care of 70 of the mushrooms. And how many left would there be for us to take care of?

Her: 21.

Me: Nice. So, how many more mushrooms can we give to each person?

And then she goes back to her hands and does a bunch of counting. I interrupt her and ask her whether we could give them each 4. She says no, after some thought. She says that it would have to be more than 2. It takes a little bit of thinking before she tries and confirms that 3 works.

I think that this picture, and this dialogue, captures an important step in learning multiplication and division, and how awkward it all is.

I’m very new to all of this, so I’d appreciate some comments. As is our custom on this site, here are a few prompts:

• Umm…how did that dialogue go? What worked? What could’ve gone better, in your view?
• I feel like there’s some wisdom here about how people learn division and multiplication that I’m not able to articulate particularly well. Maybe you can?
• How do you ween kids off of relatively slow and sloppy methods like counting?

Lots of good stuff going on here. But I don’t think I entirely understand where 1/8 came from, though I get how that gets turned into 5.8.

[I never know whether to include all the mistakes from a class set or just a few. I feel as if it's helpful to include more mistakes, but sometimes overwhelming. My solution today is to post one especially cool mistake largely, and the others smallerly. Let me know whether that works.]

The kid’s handwriting is hard to read, so I want to point you towards 9 times 13, near the top of this page.

I think that this is a great example of a mistake that you can feel fairly good about. Your thoughts, on any of his work?

The kid also answered the “How do you know?” question:

“Because 5 is half of 10, and 50 is the same number as 5, just with a 0, and 10 is the same number as 100 just with another 0.”

What does this mistake say about the way kids see numbers and multiplicative relationships?

I’m inclined to say that this is a classic working memory mistake. You’ve got a resources-heavy calculation being done in the student’s head, and you’ve got this 3 floating around in the problem, and it ends up in the ones place.

Agree? Disagree? Thoughts?

Any idea what’s going on here? In case fuzziness is an issue, the question is “What is the biggest multiplication that you know without thinking very much about it?”

Also, I anticipate getting some flack for the wording of this question being potentially confusing. I don’t disagree, necessarily, but I want to offer a partial defense of the question. First, I had been reading the TERC curriculum and they make a point of not saying “multiplication facts,” instead always saying “multiplication combinations.” I haven’t wrapped my head around what makes sense to me, so I punted on the question, figuring that I’d be there to help kids figure out what it meant. And I did, and everybody else offered answers that made sense. Most importantly, the question served it’s purpose: some kids wrote “8 x 8″ while others wrote “1,000,000,000,000,000,000 x 10.”

Explanations? What lessons are there about the way kids think in this work?

1. What exactly is the shortcut?
2. Why does this shortcut seem reasonable?
3. We’d all agree, I think, that 2 x 6 = 26 is not a result of this kid not understanding what multiplication is. I’ve made that mistake before, and I bet that you have too. So, why is it a  common mistake? What does this say about how a mind works while working on math?

[Any advice on how to tag this, CCSS-wise?]