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What Does ‘Understanding’ in Mathematics Really Mean?

What Does ‘Understanding’ in Mathematics Really Mean?

As math (as we have come to know it) experiences reform under Common Core standards, many advocates argue that the new way of doing math helps ensure students actually understand the material.

Says The Atlantic, “[o]ne distinction popular with today’s math-reform advocates is between ‘knowing' and ‘doing. A student, reformers argue, might be able to ‘do' a problem (i.e., solve it mathematically) without understanding the concepts behind the problem-solving procedure.”

To alleviate this perceived lack of true understanding, new math under the standards requires students to show their work, in theory indicating that they have not relied “too heavily” on procedure and instead truly grasp the material.

"The underlying assumption here is that if a student understands something, he or she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: What constitutes a satisfactory explanation?” The Atlantic asks.

After spending time in a middle school where 10 percent of weekly instructional time was dedicated to teaching students how to explain how they solved math problems, The Atlantic took a look at some of the more difficult problems students were expected to explain.

"For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious,” the article said.

The Atlantic found that “[d]espite the goal of solving a problem and explaining it in one fell swoop, in many cases observed at the middle school, students solved the problem first and then added the explanation in the required format and rubric. It was not evident that the process of explanation enhanced problem solving ability.”

This kind of expectation, it says, can be damning for “certain vulnerable types of students.”

"Consider students whose verbal skills lag far behind their mathematical skills—non-native English speakers or students with specific language delays or language disorders, for example. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain—whether orally or in written words—how they arrived at their answers.”

For these students, then, “understanding” takes on a very different meaning but is not reflected in the standards whatsoever.

"Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math?”

"Measuring understanding, or learning in general, isn’t easy. What testing does is measure “markers” or byproducts of learning and understanding. Explaining answers is but one possible marker,” but perhaps shouldn’t be treated as the end-all-be-all, The Atlantic says.

Read the full story.

Article by Nicole Gorman, Education World Contributor


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