Category: Elementary School

Categories
Division Numbers & Operations in Base 10

91 mushrooms, 7 people

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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?

Looking forward to your thoughts.

 

Categories
Decimals Division Fractions Numbers & Operations -- Fractions Numbers & Operations in Base 10

44 crackers shared equally with 8 people…

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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.

Remainder4

 

Remainder3

 

Remainder2

 

Remainder1

 

Remainder6

 

[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.]

Categories
Division Numbers & Operations in Base 10

Lots of correct ways to do division

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Div1 Div2 Div3 Div4 Div5

 

No mistakes here, but I’m sort of blown away by the variety of division techniques I saw in yesterday’s problem set. Thought that it was worth sharing.

Categories
Multiplication Numbers & Operations in Base 10

9 times 13 is 121

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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?

Categories
Elementary School Equality Operations & Algebraic Thinking

7 = 1

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7 = 1, huh?

Check out my post over at the other blog for what I did about this. Comment here or there, whichever you prefer.

Categories
Data Measurement & Data

The average is whichever thing has the most

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I don’t have a picture for this, but every single one of my 4th Graders thought that “average” meant “the most common thing.”

(a) Where do they get this idea from?

(b) Is it a big deal misconception?

(c) How do you create a need for something besides “most common”?

 

(I think I have my own answers for (a) and (c), but I’m more curious to know what you guys all think.)

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Categories
Numbers & Operations in Base 10 Place Value

Place Value is Hard

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I think that this might be my favorite conversation since I started hanging out (as a math assistant) with 3rd Graders. File this one under “place value.”

The question that this kid was grappling with was to write “8,000 + 500 + 20 + 6” in standard form. I’ve forgotten most of the details of the conversation, but all of the interesting stuff is right there on the page.

First, note how she first writes 25,03 to that first question, at the top of the image. And though it’s harder to see, she’s done that for 60,47 as well. When I came over to her, I thought that she was just confused about the convention, and I offered a correction.

Then she started working on the “8,000 + 500 + 20 + 6” problem. First she wrote 26, and then she tried to figure out how much more it was. She ended up with what you see in the image, at which point I realized that there was something about the way she was thinking that I hadn’t anticipated. She was chunking the number into 26 and everything else.

She really struggled to figure out what to do with 8,000 and 500. The boxes around the numbers are part of my attempt to push her to tell me what each numeral represented (“5 whats? Shouldn’t this just be 8 plus 5 plus 2 plus 6, so 21?”).

But there was no way that was going to work. The fact that she was chunking “26” together means that she doesn’t really get place value for 10s either. Those of you with more experience will hopefully help me out in the comments, but I’d imagine she sees 26 as a single number, not as composed of any parts.

This was confirmed when I asked her what the 2 in the 26 meant. She thought I was nuts. She said that 26 is the number right after 25. I repeated the question, and she thought it was ridiculous.

To me, this really speaks to the value of activities that defamiliarize place value for students. See Anna’s Ba-na-na or Christopher’s Orpda for activities that do this.

 

Categories
Counting Multiplication Numbers & Operations in Base 10 Place Value

“5 is the same as 50, just with a 0.”

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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?

Categories
Multiplication Numbers & Operations in Base 10

More, More Numbers and Operations in Base 10

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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?

Categories
Multiplication Numbers & Operations in Base 10

More Numbers and Operations in Base 10

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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.”