When I engage preservice or inservice teachers in analyzing this passage, I want to make sure that they note that Randy Harris provides a space for Devon's solution method, even though there is an "easier" way to divide this rectangle into eights (there are 8 rows in the rectangle). I've often seen middle and high school teachers "cut off" this type of explanation by saying something like, "it would be easier to use rows instead of columns to begin" in an effort to "help" the student see the "easiest" way of completing the problem. Devon's solution indicates the depth of his understanding about fractions. So, I want to engage teachers in considering the validity of Devon's method and how Randy Harris valued that method.
One thing that I notice consistently when facilitating this case with teachers is their reliance on computational procedures to complete the opening activity. I think that this case has the potential to help teachers think about the mathematical content in a deeper, more connected manner. Therefore, when I facilitate this case, I think that it is important to focus the teachers on using the rectangles before doing any computation.
When I read this, the first thing that I thought was that this Devon kid was pretty precocious. His line of thinking ("So 1½ columns would make an eighth") seems similar to the way I tend to think, and it's been a long time since I've thought that about a seventh grader. Then he hits the wall when asked to convert it to percent. I was impressed by Randy's prodding questions that led Devon eventually to come up with the right idea. I'm not sure I would have taken that much time and effort, but it definitely paid off. It shows the value of patience and persistence from the instructor, which is something from which I can learn as a teacher.
What struck me the most about this case was a more general thought about reflecting on one's instruction. When reading this, it appears that Randy has a class full of hard-working, well-behaved, exceptionally bright students. I personally would love to see the video of this case and see what other things were going on in the classroom—what other obstacles Randy is overcoming. It makes me wonder what I would discover about my own instruction if I were to record one of my classes and then write it up. Some days when things seem to be hectic and it doesn't feel like class goes so well, perhaps writing up what learning actually did take place might help me see the real progress students are making. Of course, doing this would also point out some potential improvements in my role, I'm sure. This type of reflection on teaching is not very common for those of us formally trained in pure mathematics. To me, this case emphasizes the value of recording and writing up a lesson for the instructor's own analysis.
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This case is taken from Smith, M. S., Stein, M.K., & Silver, E. A. (2005). Improving instruction in rational numbers and proportionality: Using cases to transform mathematics teaching and learning (Volume 1). New York: Teachers College Press.
