Linda Fisher reflects on the kind of mathematical thinking she saw the students doing during the lesson and the depth of the mathematical arguments they made in the class discussion.
LINDA FISHER: When I was looking at it, on the very first problem, remember I asked why they were going to spend so much time on the 6/9ths and the 2/3ds, because kids in that classroom did really well on that problem, but it sort of sets the stage for later. The two people I was observing both said, “Well, the 2/3ds has nothing to do with the picture. That doesn’t make sense at all.” And they just went to the reducing algorithm. Then, because there was extra time, they go “Oh well, it might.” And they started to see that relationship between it. But their first response was “That picture just doesn’t apply to the 2/3ds at all.” So that was really fun.
I really liked the real deep mathematical arguments that they made to convince each other. I think the process allows things like mathematical connections, like the boy who came up with “Well, it’s a fractional model.” When they looked at the second example of Valerie and Cindy, and they said it’s ¼ and ¾ . That’s a really nice mathematical connection to make. But I was even more impressed with how they were really talking about the mathematics of proportions and ratios, when they were discussing why the answers were wrong. So if the person had 8 cups of cream, or chocolate, then there would have only been 1 cup of cream, and that doesn’t make sense because there’s 1 cup of cream to 2. So they took those answers and did real mathematical justifications. Or when it said that the answer was 2 cups of chocolate, they go, “No, because that would mean that there were 7 cups of cream, but the cream is supposed to be smaller.” So they really have that sense of proportionality in mathematical justification, when they made their answers. That was really nice to look at. Then when they went to the next problem after doing Cindy and Valerie, one of the students drew a 3x3 box and filled it all in. Because the one for Cindy and Valerie, they were given the box. I really wanted to butt in and say, “How did you know what size you needed to make your little grid to fill in?” You know, how they figured that out. I just think that there was a real lot about mathematics of proportions that came out in their strategies. Different ways that they went about doing it. I saw one student with a nice equation of (2x3) + (3x1)=9. Students who drew it all out. Just nice pieces of work. One of the things that happened for me was because Hillary and Carolyn said that there were a lot of mistakes in student work, but when they were having the class discussion, they were so articulate about the correct mathematics. So that I think that there’s something significant that happens when they’re trying to discuss an idea that’s different, maybe, than their thought process that happens when they’re just like working on a piece of paper. I’d like to explore that at a later time. They just seemed to approach the amount of sense-making they need to do in that class discussion differently than the amount of sense-making they do when they’re just doing a piece of paper.
LINDA FISHER: In my years teaching mathematics, I've learned that kids need to have their misconceptions confronted head-on. With re-engagement, we thought, let’s take that and pose them as dilemmas for kids to think about – get them talking about why some of these common things don’t make sense. That way, we can bring a focus on the mathematics and the concept, rather than solely on the solution and the answer.
When I work with teachers, they want to know what to do in terms of remediation. Teachers usually confront student mistakes by going back to a clean slate and start at ground zero. But there’s something profoundly different about reteaching than teaching it for the first time. When you go back to do a re-teaching leson, you don’t want to start as if people have never learned things. You want to get students to let go of why what they’re doing doesn’t work. Teachers need to find a way in to facilitating the conversation, to helping students see why what they’re doing doesn’t work. That’s one kind of reengagement: having kids reach the conclusions.
One of the critical things is that kids have a lot of mathematical ideas that teachers don’t see. My “hidden” agenda is to get teachers to be able to read student work and make sense of what they’re not understanding. Everyone with the wrong answer doesn’t need the same kind of help. In learning to classify student errors, we want teachers to tease out what are the different kinds of help, and why are they different, because they’re based on different mathematical ideas.