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(This is the first post in a four-part series about Common Core)

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Why is my kid’s math homework so weird?

Last year, Facebook lit up with posts from dumbfounded parents trying to help their children with math worksheets. Homework questions demanded that students turn an addition problem into a complicated multi-step process or write a letter to an imaginary child about using a number line to solve a subtraction problem.

These kinds of questions are intended to help classes adapt to Common Core, the unified standards for mathematics and English that have been adopted by 40-plus states and Washington D.C. But is there any evidence that solving these kinds of problems actually helps kids learn math? The answer is yes. Kind of.

In a 2010 paper for the Review of Educational Research, Christopher R. Rakes, Jeffrey C. Valentine, Maggie B. McGatha, and Robert N. Ronau conducted a meta-analysis of 82 studies about how to teach algebra, representing a total sample of 22,424 students. The studies looked at various strategies for improving instruction, including adding new technology tools, adopting different instructional strategies and using three-dimensional objects to make ideas easier to grasp. All the methods produced some improvements in student achievement, but one factor stood out.

“Interventions focusing on the development of conceptual understanding produced an average effect size almost double that of interventions focusing on procedural understanding,” the authors wrote.

So… what’s the difference between conceptual and procedural? That’s the issue at the heart of the debate over Common Core math. Rakes and his colleagues write that procedural math asks students to apply a particular set of steps, like the quadratic formula, to solve an equation. “In such a problem, students can find a solution without any understanding of the meaning of quadratic functions,” they write.

Conceptual lessons, on the other hand, try to make sure students really understand the underlying principles. “A conceptually focused question might ask students to graph v = x2 + 6x + 8 and explain how the x intercepts of the graph are related to the factors of the equation,” Rakes et al. write.

This is the idea behind questions that make students use different methods to get a result, express numbers graphically, or write paragraphs about how they found an answer. But if conceptual lessons were just about changing textbooks and worksheets, teaching would be a much easier job than it is. Another Review of Educational Research paper, by Robert E. Slavin, Cynthia Lake and Cynthia Groff, published in 2009, reviewed 100 studies of middle and high school math classes. “One surprising observation is the lack of evidence that it matters very much which textbook schools choose,” the authors say. Even curricula funded by the National Science Foundation didn’t make an impact on student achievement. What did matter? The strategies that teachers used in their classes. In particular, cooperative learning made for higher student achievement.

“The findings of this review suggest that educators as well as researchers might do well to focus more on how the classroom is organized to maximize student engagement and motivation rather than expecting that choosing one or another textbook will move students forward by itself,” the authors write.

But here’s the thing: Fundamentally changing how a classroom operates is hard. In 2003, Carmen M. Latterell and Lawrence Copes wrote a paper titled “Can We Reach Definitive Conclusions in Mathematics Education Research?” for the Phi Delta Kappan, including a teacher’s first-person account of one classroom’s experience with the sort of conceptual, cooperative lessons that the research suggests is most effective. Students were asked to find an equation that represents the fact that there are six times as many students as teachers in the school. Many at first chose the formula 6S=T until one student in one of the groups, Chris, questioned that conclusion.

“The group began to argue heatedly,” the teacher wrote. “I gave Chris a thumbs-up sign to keep the argument going and sent representatives from the groups to eavesdrop on the discussion. Soon the entire class was in turmoil.”

What followed was a messy, time-consuming discussion with the teacher functioning more to guide the conversation than to offer answers. By the end, the students seemed to have gained a better understanding of variables and equations.

Since that paper was written more than a decade ago, cooperative teaching methods haven’t exactly taken the country by storm. A fascinating story in the New York Times Magazine this past summer explored just how tricky it’s been to introduce this kind of math education in the U.S., despite efforts going back to the 1970s. To a Japanese education reformer visiting the country in 1991, the situation was mystifying: “the Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them.”

The question is whether Common Core will change that. And the answer, almost certainly, is that it depends how new teaching methods are implemented. We’ll explore that in the next post in this series.


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Review of Educational Research, Vol. 80, No. 3 (September 2010), pp. 372-400
American Educational Research Association
Review of Educational Research, Vol. 79, No. 2 (Jun., 2009), pp. 839-911
American Educational Research Association
The Phi Delta Kappan, Vol. 85, No. 3 (Nov., 2003), pp. 207-211
Phi Delta Kappa International