## Case 2 – Ann

#### Grade 6, October

When do I borrow? Is this where I cross out something? What if there is a zero in the column I want to take from? When do I stop borrowing?

##### 1. In Ann’s case 2, examine the methods presented by Jason, Bert, Holly, and Joe. Apply their methods to this problem: 83-56.

**Jason’s thinking: **

- 83 – 56= ?
- First subtract 50 from 83.
- 83 – 56 = 33
- Then subtract 3 (which is part of the 56)
- 33 – 3= 30
- Finally subtract the remaining 3 from the 30
- 30- 3= 27

- Jason started on the left with the tens column.
- Then tried to make the number “friendlier”, by making the ones column a zero.
- For Jason, going from 83- 3= 80 and 33-3= 30 was easier than 33- 6= 27

**Bert’s thinking:**

- 83 – 56= ?
- 83 – 10 = 73
- 73 – 10 = 63
- 63 – 10 = 53
- 53 – 10= 43
- 43 – 10= 33
- 33 – 3 = 30
- 30 – 3= 27

- Bert subtracts 5 groups of 10, then subtracts 30 to bring his ones place to zero, then subtracts 3 more to get 27

**Holly’s thinking:**

- 83 – 56 = ?
- Take away 3 from both to make 80- 53
- 80 – 10= 70
- 70 – 10 = 60
- 60 – 10= 50
- 50 – 10= 40
- 40 – 10= 30
- 30- 3= 27

- Holly’s thinking is similar to Bert’s although she starts off with his last step.

##### 3. What is it that children need to be able to understand about number in order to solve math problems in these ways?

- If there wasn’t regrouping [as in 83-56] then “when the number is bigger” you don’t switch, and you add the answer to the final number instead of subtracting it.

##### 4. In case 4, Emily writes that one of her second-graders, Ivan, invented a solution to a subtraction problem that was both new to her and confusing to figure out.

What Emily observed:

- 52 – 28 = ?
- Ivan’s first answer was 36
- His friend got 24 and explains why Ivan is wrong
- His friend: “20 away from 50 is 30, but you can’t take the 2 away from the 8; you have to take the 8 away from the 2 because the 8 is on the bottom.”
- Ivan: “but you can’t take the 8 aways from the 2 because the 8 is too big.”

Ivan’s Thinking:

- You take the 20 away from the 50 and get 30.
- Then you take the 8 aways from the 2 which is -6.
- Then you take the -6 away from the 30 and you get 24.

##### 5. Does Ivan’s method resemble any of those presented in Ann’s case 2?

##### 6. Emily declares this to be a time when she was learning mathematics by interacting with her students. What is your reaction to that?

Everyone you will ever meet knows something you don’t – Bill Nye

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