I read the Whitenack and Yackel article first where they were highlighting a classroom where the teacher had set up an environment where children discussed and defended mathematical ideas with each other. The teacher was able to facilitate the discussion to allow this to happen and you could see the process of sense-making in math really unfold for these children. In all of these articles it always seems so effortless in the classroom. As I prepare to teach my own math lesson, I wish I had more resources available to me about how some of these teachers started working to build this sort of environment. I wonder how much of it comes naturally and how much teachers have to help foster to make children comfortable with and eager to share.
While reading about fractions in Young Mathematicians the students had so many different ways to approach the problem. When I read about the "which cat food is cheaper" I immediately solved it the way that Helaina and Lucy did and didn't really think about it much further. The process skills in this classroom are a lot deeper though, these students were really pushing to find different ways to explore the problem. The two boys constructed a ratio table and figured out their answer that way, then two other girls came through and did it the same way with different numbers to find out the answer as well. I wouldn't have thought that this conversation could have led to so many different ways to discuss equivalent fractions but the teacher was able to steer the conversation from method to method to discuss different ways to represent the same idea. This teacher seemed to demonstrate the approach of getting as many students involved in talking about the math as possible through the use of many different solution ideas and the "pair talk" which I'm assuming was where students talked in pairs about a certain math idea so everyone had a turn to express their thoughts.
The big idea discussed in chapter 4- the whole matters- seems so obvious that it feels odd to me that this was never brought up when I have taken prior math classes ever. At some point of course I realized it, but it was never consciously in the front of my mind. Yet in Carol's class the students are free to explore this idea which really will help ground fractions with different denominators in their minds. Fractions are always in relation to one another. Again, in this case I thought it was so critical that the teacher had the students present so many different examples of the problems and were allowed to explore the idea in depth without her just giving an explanation that might not have made sense to them. This seems to be an integral part of having content make sense to students and to allow them to develop the process skills necessary to really "do" math.
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Anne, I also wondered how this community of learners was established and what limitations or framework was implemented in order to promote successful and rich discussion. It seems to me that this process would be a lesson in itself. Perhaps creating classroom lists helping children begin responses to peers such as "I agree with Joe because.." or "I think Joe is trying to say..." etc. This would allow the students to build upon each others ideas while engaging in healthy and appropriate dialogue. I also agree that this would be a necessary practice to implement in the beginning of the year. Hopefully we will be able to look into this further in future classes..
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