In a previous post, I expressed my concern about the use of some models and processes for representing multiplication of decimals and fractions. A recent issue of Teaching Children Mathematics provides a better description of why my students and I were both confused by the representations being used by some classroom teachers. As we plowed through classwork and homework, kids were increasingly anxious as I tried to inject some meaning into the process. “My teacher says to color these boxes!” the kids would exclaim in desperation. “We aren’t doing it that way!” When asked what the boxes meant, the kids had no idea.
In an excellent article by Webel, Krupa, and McManus, the authors describe the confusion that results from using visual representations without context. They emphasize that the visual should clearly depict the differing sizes of the two fractions. The authors express concern that visuals may be reduced to a series of steps which are no more helpful than blindly following an algorithm.
As I’ve taught students in 4th and 5th grade, all three of these pitfalls have been evident. Here are some graphics to demonstrate the authors’ perspective (and my students’ anxiety).
Let’s say we want to solve this problem (borrowed from the article): I have 1/4 of a gallon of orange juice in my fridge. Then I drank 1/4 of it. What fraction of the gallon of OJ is left in the fridge?
Here are three possible visuals for this problem.
A and B are very similar and bring back bad memories! Use a grid to color across and down. The double-shaded area represents the answer. Ta-da! Only it doesn’t answer the question! In the OJ problem, we need to know what’s LEFT. C is a superior representation of the problem but a teacher must guide kids into how we arrive at 16 squares in the grid. That discussion follows logically when you focus on the problem, which has 2 distinct units.
For fragile learners, A and B have great appeal. Kids can memorize the coloring process without difficulty. Unfortunately, they are not even close to understanding what multiplication represents. The “talking it through” process can be very frustrating because they know they’re BEHIND and want to speed along to catch up. And as we know, THIS ISN’T THE WAY MY TEACHER DOES IT!