Devon's visual approach to this problem was an excellent example of how students think. What was most valuable about this case was the emphasis on visual representation—the use of the grid to solve the problem—versus remembered mathematical algorithms and computation. It emphasizes the need to meet kids where they are in order to connect them to the formal computational rules we use and apply to comparing fractions, decimals and percents.
Studying such cases has aided in transforming my teaching from a more traditional procedural approach to a conceptual approach that engages students in the learning process. The problem tasks in these cases are very rich and the presentation of the cases promote reflection on one's own teaching and student learning.
I was impressed about the number sense Devon portrayed while giving his explanation on the fraction 3/8. I was also impressed by the diverse set of explanations that came from the different students in the case study. So often students are taught one way to solve a problem (usually the most efficient way) and different ideas are easily dismissed. This teacher did an exemplary job of incorporating multiple perspectives that led students on different tasks. He pushed students to think differently and their conceptual understanding of the relationship between decimals, percents, and fractions was incredible.
This was my first experience studying cases and I found this to be an amazing experience. I am early in my career and am always trying to find ways to increase student conceptual understanding of mathematical concepts. This workshop gave me an opportunity to see the perspective of other middle and high school teachers, mathematics educators, and mathematicians. The different perspectives brought to the group of what is important in a case and will affect my instruction in the future. I am looking forward to continued work in studying cases.
Devon's approach to solving the problem was very impressive to me. He started with a simple geometric division of rectangle into 4 equal parts, then divided fourth into two equal parts to get eighths of the original rectangle. This allowed him to shade three eighths of the original rectangle. I was equally impressed by the teacher's recognition that Devon's solution was interesting and a good one for the other students to see. His questions to Devon that prompted Devon to complete the other parts of the problem were excellent, and very much tied to the process that Devon had used to do the first part of the problem.
My reaction to looking at this case and one other from the materials is very positive. I believe that they can be used both to demonstrate the kinds of problems and the pedagogical moves that can elicit deep mathematical thinking from students. I would like to use this case with a group of mathematics educators and mathematicians, very much in the same way as we used it in the workshop. I would also like to use these materials with current middle school teachers who are taking courses with me. Finally, I would like to consider developing a case related to a university faculty member teaching teachers.
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.
