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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pic3pic1

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

pic4

 

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.

Natasha had $8.72. She spent $4.89 on a gift for her mother. How much money does Natasha have left?

  • I gave this question to my 4th Grade class. (11 kids, one absent.) It was December, I had seen them do a variety of subtraction work. I knew that a lot of them could handle subtraction using something like the standard algorithm — though certainly not everyone — and I was wondering whether a money context would be easier or harder for them. Would you predict that $8.72 – $4.89 would be easier or harder than 872 – 489?
  • What approaches would you predict kids to take for this money problem? What mistakes do you expect to see?

Take a look below, and then report back in the comments:

  • Which student’s approach surprised you the most?
  • Assume that you’ve got time in the curriculum to ask students to work on precisely one question at the beginning of class the next day. What question would you ask to address some of the ideas you see in their work below?

Student 1

4thGradeSubtAssessment_020

Student 2

4thGradeSubtAssessment_002

Student 3

4thGradeSubtAssessment_004

Student 4

4thGradeSubtAssessment_006

Student 5

4thGradeSubtAssessment_008 - Copy

Student 6

4thGradeSubtAssessment_010

Student 7

4thGradeSubtAssessment_012

Student 8

4thGradeSubtAssessment_014

Student 9

4thGradeSubtAssessment_016

Student 10

4thGradeSubtAssessment_018

 

From Bedtime Math:

Big kids: The record distance for a thrown boomerang to travel is 1,401 feet.  If it traveled exactly 1,401 feet on the return trip too, how many feet did it travel in total?  Bonus: Meanwhile, the longest Frisbee throw is 1,333 feet – about a quarter of a mile! How much farther from the thrower did the boomerang travel than the Frisbee?

From the submitter, who sends in the thinking of two of his students:

(1) first student, having doubled the boomerang distance in the earlier question, now doubles the frisbee distance  and calculates (2801 – 2666) feet.
(2) Second student gets an 100 board and spends a short time calculating 100 – 33 = 67. Then thinks for a long time during which I’m sure he is going to say 67 + 1 = 68, but never quite does it. I stay silent until he announces: 667. No clue where the extra 600 came from. He wasn’t willing to write down or draw anything to explain his thinking.
Interesting!  I’m inclined to put the first student in the “extending the thinking you’d do in one model to a less familiar situation” category and the second student in the associational mistake (same link) category.

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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.

array

This child made it clear that

  • She knew that an array was a rectangle
  • That this was technically a rectangle
  • These super-long folks were not arrays, or at least she didn’t think they were, because they didn’t look like a rectangle
  • The 2 x 17 was an array

To what do you attribute this perception? (You can check your answers in the back of the book.)

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