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!

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I’m so glad someone in education gets it! Thank you for being that person 🙂 I’ve always said, why ask questions in a way that tricks kids. Chase is so smart but it will never show in his work or exams simply because of the wording or set up of the questions.

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I remember a teacher at the high school level saying that he deliberately made tests tricky. That drives me crazy!

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It’s ridiculous. It defeats the whole purpose 😦

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Wow! Are these methods new? I don’t remember being taught Math like that. For me it was as easy as: 1/4 of 1/4 is 1/4*1/4 which is 1/16. So we have 1/4-1/16, which you solve by using simple math fraction rules: common denominator…etc. and we arrive to 3/16. No more complications. I know this reasoning might be hard for a kid. But I’m 100% sure those graphics weren’t involved at all in my fraction lessons at school 🙂

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Yeah, it’s new. The idea is that you don’t teach algorithms. Instead, you lead kids to construct their own understanding. There are obviously many pitfalls along the way!

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There is a huge difference between rote learning of a numerical operation, however represented, and learning it in context. Humans need context to experience meaning. Its the difference between learing to calculate and learning mathematics.

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Good point. That’s certainly the goal but teachers often don’t have the time to allow all kids to get there. It’s that rush to “mastery” for test-taking purposes, usually.

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Real understanding can’t be rushed. IMO, all this so called “accountability” via testing has been way overdone.

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Yes!

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Thanks for your post. I’ve shared the article with the district math specialists. We are going to use it as a springboard to discuss how to provide better resources for our classroom teachers, so that instruction around this standard develops understanding and not further mis-understanding.

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That’s wonderful! I wish I had known much of this years ago myself. I can see how weak my skills were in this area.

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