The types of talk is important

It is clear that having students talk during math class is important. The types of conversations students have are just as, if not, the most important part. Students may be talking about math and about solving problems, but this talk may not be of any use to helping them construct the appropriate knowledge they need to make the necessary connections to acquire the conceptual understandings. This can be clearly illustrated in the excerpt below taken from a teacher’s blog, noting the ineffectiveness of student conversation about multiplication. In this example, students were talking about mathematical content but not in the way that allowed them to learn anything new.
October 21th, 2008
Today I wanted to get into multiplication strategies. I read a book entitled “Amanda’s Amazing Bean Dream” which is all about a girl, named Amanda, who loved to count everything and finally realizes that using multiplication can help her to count faster. A lot of my students have trouble with the whole concept of multiplication and I thought this would be a good place to start with getting their conceptual knowledge caught up with their procedures in multiplication. I read the book and we discussed different multiplication scenarios in the book. I used different open questions like asking particular students to look at the pictures and tell me a multiplication story and why it is multiplication. They, most, were getting good at that, and then I broadened it to telling me a multiplication story involving the classroom. Up until this point, I thought they all, finally, knew what multiplication meant. Until, they could not give me a multiplication scenario that involved them or the classroom. Oh, oh!
So what did I do? My mind went to Boaler’s book, specifically, to the exerpt by Sarah Flannery, about the importance of conversations in math. I was letting students talk about math, multiplication in this situation, but not in the right way. All they were doing really, was giving back to me what I had asked for, a simple multiplication sentence that involved a picture in a book which, on every page were similar. There were rows of books on so many shelves, how many books? There were rows of windows on so many floors, how many windows? There were 4 wheels on a bicycle, so many bicycles, how many wheels? There was no thinking needed. They could simply regurgitate the answer that I wanted.
So, I put the book down and we began talking. We talked about the book, but it wasn’t about the pictures of rows or columns, it was about the math that was going on there. We talked about how Amanda loved to count and how she did not want to use any other method of counting, only the way she already knew. We talked about counting and the different things that we counted each day, money, points in a game, step to the bus, minutes until the bell rings and so on. From there we talked about how as numbers started getting bigger Amanada was faced with the challenge of counting these larger numbers faster. We then talked about their experiences with wanting to count faster or using larger numbers. They came up with things like estimating how much money we spent on school supplies (this one made me happy as they were using one of their problem solving strategies – think of a similar problem), how many meals we have eaten in our life times and how many days we have been alive. (these last two made me happier as it gave me ideas for the next day) Then we talked about how Amanda finally realized that multiplying numbers together did make counting easier. This lead into a discussion about how in order to multiply you must have equal groups of something. I then asked the students to think of a multiplication question/situation that could be used in the classroom and asked them to write in on a piece of paper.
* Each and every one of those students was able to come up with an appropriate question that involved multiplication.