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applied mathematics vs mathematics
Source: Slide 15 of a PowerPoint presentation, accompanying a June 2016 OECD report, “Equations and Inequalities.” Click here for a larger version.

Abstract, pure math — solving disembodied equations filled with x’s and y’s —  can often seem boring. Creative math teachers commonly try to come up with concrete, real-world examples to motivate students and make math relevant to adolescents.

But a new report from the Organization for Economic Cooperation and Development (OECD) finds that applied-math instruction, or the way it is actually taught in classrooms, may not be serving students well. Furthermore, it found that teachers were often using a watered-down, applied-math approach in classrooms of low-income students, while giving higher income students much more exposure to pure math.

“We’ve known all along that children from wealthier backgrounds tend to do better in math, and children from poorer backgrounds tend to do not so well,” said Andreas Schleicher, director of the education and skills department at the OECD, during a public webinar on the report. “But for the very first time, we’re actually also seeing that the exposure to that kind of conceptual understanding is actually quite significantly related to social background.”

In the report, “Equations and Inequalities:  Making Mathematics Accessible to All,” published on June 20, 2016, researchers looked at math instruction in 64 countries and regions around the world, and found that the difference between the math scores of 15-year-old students who were the most exposed to pure math tasks and those who were least exposed was the equivalent of almost two years of education. The research was based on how students answered survey questions that accompanied an international test, called the Programme for International Student Assessment, or PISA.

The result was surprising for two reasons. First, the PISA exam itself is largely a test of applied math, not equation-solving. For example, one question asks students to calculate the length of a revolving door entrance that doesn’t let air get out. And yet the students with more pure math instruction were better able to handle this and other PISA questions.

“Our analysis is [that] when students have really understood the foundations, they can extrapolate. They can apply that knowledge in another context,” said Schleicher. “However, if they only teach students tips and tricks, how to solve small everyday problems, they know how to solve those problems, but they’re not good at transferring that knowledge to another context.”

It’s also surprising because many veteran educators recommend using real-world applications of abstract math concepts as a motivational tool. And the OECD doesn’t disagree. But real-world examples aren’t enough. Students still need to learn the broad concepts and the mathematical notation. In South Korea, for example, students get a big dose of both applied and pure math instruction and they score among the top 10 in the world.

Schleicher says many teachers who take a more applied approach aren’t giving their students complex, multi-step problems that require problem-solving and deep thinking.

“The problem comes when students are asked to mechanically learn simple mathematical procedures and are then given lots of practical problems to apply these,” he said. “And there is a lot more of this kind of instruction going on in classrooms than you imagine.”

Diane Briars, past president of the National Council of Teachers of Mathematics and the former math director of the Pittsburgh public schools, said the inequity problem in the United States has shifted. More than 25 years ago, many low-income students weren’t even given a chance to take algebra in high school, and instead took applied math classes on how to read schedules, take measurements and balance a checkbook. Today almost all American high school students take algebra. But the quality of instruction isn’t the same for low-income students.

“We’re teaching a lot of kids to memorize rules, without understanding math concepts,” Briars said.

Consider how dividing fractions is taught. Instead of thinking through what it means to divide a fraction by a fraction, students, especially low-income students, are often immediately taught gimmicks, such as “Yours is not to reason why, just invert and multiply” or “Keep, change, and flip.” (Translation: keep the first fraction as it is, change the division sign to multiplication, and flip the numerator and denominator of the second fraction.)

Briars added that the new Common Core standards are aimed at boosting conceptual understanding, and that’s one reason teachers are asking students to draw all those crazy pictures that are lampooned in the media.

“There’s been a lot of push back in the media against these pictures and diagrams. The feedback in the media is, ‘Why don’t you just give them the rule?’ ” said Briars, “This report speaks to that. No, don’t just give them the rule. They need that conceptual understanding.”

In the OECD study, researchers looked carefully at survey questions on how often students said they encountered pure math tasks at school, such as solving an equation like 2(x+3) = (x + 3)(x – 3). They were also asked how often they encountered applied mathematical tasks, such as calculating how many square meters of tiles you need to cover a floor, or how long it would take to get from one place to another using a train timetable. Disadvantaged students tended to report having more exposure to the applied tasks. Students from wealthier families tended to say they had more exposure to solving equations. OECD analysts explained that this exposure to equations indicated that the students had been taught pure math that emphasized conceptual understanding.

Then the researchers compared these survey results with the 2012 PISA test, and found a high correlation between the type of math instruction students received and their math scores. Even after accounting for the fact that better-performing students may attend schools that offer them more mathematics instruction, the exposure to pure mathematics was related to higher performance, both in the United States and on average across other OECD countries.

This OECD report builds upon earlier research from William Schmidt of Michigan State University who found that schools are exacerbating socioeconomic differences by giving rich kids different instruction than poor kids. In that study, Schmidt focused on math content, and found that wealthier students were studying more topics, such as quadratic equations, than poor students. Now this OECD study argues that it’s not just the topics, but also the way they’re being taught.

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