By Juli K. Dixon, posted October 12, 2015 –

In my previous post, we explored how using errors as springboards to learning is influenced by the ways in which mathematics is taught. When mathematics is taught with depth, even the errors are different. The example that I used to make this point involved dividing fractions. The error emerged when a context was provided for fraction division and that context was modeled in the solution process. The error involved the interpretation of the remainder. The context was linked to the error. The same error would likely not occur when using a procedure to divide decimals.

Now consider fraction subtraction. This, too, should be addressed through the use of context when teaching fractions for depth. Consider the problem 6/7 – 1/2. Before reading further, use the following stem to write a word problem to support this expression, but do not perform the subtraction using an algorithm or in any other way.

Drew brought 6/7 of a candy bar in his lunch box. . . .

How did you finish the word problem? Does your word problem read something like this?

Drew brought 6/7 of a candy bar in his lunch box. He ate 1/2 of what he brought at lunch. How much of a candy bar does Drew have left?

Is your word problem similar? Draw a picture to represent the situation you provided. Does 6/7 – 1/2 model your situation? The one described here is illustrated below.

First, 6/7 of the candy bar is represented. Next, 1/2 of the 6/7 is marked as eaten. How much of the candy bar is left? According to the picture, 3/7 of a candy bar is left. Does this provide the answer for 6/7 – 1/2? It does not. So, what happened? Was the picture drawn wrong? The picture clearly represents the action of the problem. The issue is the word problem itself. This word problem is not modeled by 6/7 – 1/2.

Now consider the following word problem:

Drew brought 6/7 of a candy bar in his lunch box. He ate 1/2 of an entire candy bar.  How much of a candy bar does Drew have left?

How would the representation of this problem be different from the previous representation? In this case, the picture would need to show how 1/2 of a candy bar could be eaten if there is only 6/7 of a candy bar to start with, as shown below. When 1/2 of a candy bar is eaten from 6/7 of a candy bar, 5/14 of a candy bar is left. This is modeled by 6/7 – 1/2. So, what did the original word problem model? It modeled 6/7 – 1/2 (6/7). Notice the difference between the original illustration and the second illustration. The second illustration models finding the common denominator of 6/7 and 1/2, an important aspect of the operation that is missing from the first illustration.

If your word problem was similar to the first one, you are not alone. In-service teachers in a graduate mathematics methods class were asked to complete a similar task and they wrote problems very much like the first one provided here (see Dixon et al. 2014 for a discussion of this study). Students are not the only ones who will make new mistakes when mathematics is taught on the basis of rigorous, focused, and coherent standards. We, as teachers, must be able to create contexts for computations in the same ways that we expect students to do so. This will require a focus on making sense of mathematics for teaching. Many teachers likely have not been taught in ways they are now expected to teach. As we make sense of mathematics for teaching, we may encounter making our own new errors. We will need to use our errors as springboards to learning. Working in collaborative teams will likely make this process more productive for all involved.

What other errors should we anticipate when teaching mathematics based on rigorous, focused, coherent standards? To learn more about common errors related to fraction computation, see “The Whole Story: Understanding Fraction Computation” (Dixon and Tobias 2013).

Your Turn

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References

Dixon, Julie K., J. B. Andreasen, C. L. Avila, Z. Bawatneh, D. Deichert, T. Howse, and M. Sotillo Turner. 2014. “Redefining the Whole: Common Errors in Elementary Preservice Teachers’ Self-Authored Word Problems for Fraction Subtraction.” Investigations in Mathematics Learning 7(1): 1–22.

Dixon, Julie K., and Jennifer M. Tobias. 2013. “The ‘Whole’ Story: Understanding Fraction Computation.” Mathematics Teaching in the Middle School 19 (October):156–63.

Juli K. Dixon, Juli.dixon@ucf.edu, a professor of mathematics education at the University of Central Florida in Orlando, is interested in mathematics content knowledge for teaching as well as communicating and justifying mathematical ideas.