I wrote this blog post last year and somehow never finished it:
My youngest son is in the fourth grade. He is a good student and loves math, but I wasn’t sure if he was truly understanding fractions. I decided to ask him how he would solve 5 x 3/8. He started by trying to use an algorithm (which he did not understand) but luckily saw the mistake and was able to reason about the answer to determine that the algorithm did not produce a reasonable answer (The entire video is at the bottom of this post).
After he was unsuccessful with the algorithm he just looked at me and said, “but 9 is more than 5” with a puzzled look on his face. He knew something went wrong but he wasn’t sure how to figure out what. I asked him if there was a different way he could represent the problem to make sense of it and he said he could draw a picture or a diagram. I said, okay what would that look like. He struggled a little with trying to visual 3/8 so I asked him what 1/8 would look like. He went to a number line. However, when he labeled his number line, this is how he labeled it:
I asked him to show me where 1 was and he quickly erased a line and was able to tell me that one was the same as 8/8. I asked him to show me where zero was and he corrected the number line to look like this:
He then was able to plot 3/8 on the number line like this:
When I asked him what two 3/8 would look like, he was stumped. I said, where would you put the dot to show another 3/8? He was able to produce this number line and we discussed that this was two groups of 3/8. He proceeded to extend the number line to show five groups of 3/8 which landed him at 15/8.
I asked him if he had a way he could teach this to someone else who didn’t know how to complete this problem and he produced this scenario and drawing:
Skylor: “There was a chocolate chip cookie and eight children. They only had one chocolate chip cookie and they all wanted it and they needed an equal share. So they broke the cookie into eight pieces and then let’s say five of the kids wanted to save theirs for later because they had just had food and they weren’t that hungry.
So only three children had their cookies at that time. And then I will shade in the cookie pieces that those children ate. Those pieces would be ⅜ of the cookie. But if there were five cookies, then you would take this and there would be five of these (five cookies all split up so the children could have their pieces of the cookies).
Five of these ⅜ would be ate at one time. So that would be…let me just draw that…so that is five split up cookies. And then three children had their share of the cookies (that represents ⅜ of the one cookie). There was five cookies so then there was that much. These were the pieces that were ate, but the total of those pieces that were ate were 3, 6, 9, 12, and 15.
Me: “So 5 x ⅜ equals…”
Skylor: “So five times ⅜ equals fifteen eighths.”
We know how important it is to allow students to make sense of a mathematics, yet many times we feel rushed to teach. If I would have tried to “teach” Skylor the algorithm for this problem, would he have been able to internalize the concept?
If I would have switched to teaching mode and given him steps to complete or a specific diagram to draw (my diagram) what might have happened?
Will Skylor be able to make sense of the algorithm now?
As a Math Recovery Intervention Specialist, one of the most important things I learned was the importance of wait time and the practice of questioning instead of telling. When we stop leading students thinking and start allowing them to make sense of their own learning (MP1) it is amazing what they can accomplish.