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Saturday, March 15, 2008

Fractions

That’s fractions, not fractals. (There is, actually, a significant and non-coincidental connection between the words, but that would be something of a digression.) I’m noticing two recent and diametrically opposed views on fractions — views that I want to discuss here.

First came a report in USA Today on January 23, entitled “Fractions should be scrapped” It quotes Dennis DeTurck, an “an award-winning professor of mathematics at the University of Pennsylvania”:
“Fractions have had their day, being useful for by-hand calculation,” DeTurck said as part of a 60-second lecture series. “But in this digital age, they’re as obsolete as Roman numerals are.”
DeTurck points out that the arguments against him “always boil down to: ‘What would we do in cooking and carpentry?’”

The article goes on to say that “DeTurck does not want to abolish the teaching of fractions and long division altogether. He believes fractions are important for high-level mathematics and scientific research. But it could be that the study of fractions should be delayed until it can be understood, perhaps after a student learns calculus, he said. Long division has its uses, too, but maybe it doesn’t need to be taught as intensely.”

In response, “Penn State mathematician [George] Andrews says he believes DeTurck’s ideas will ‘unfortunately’ gain traction because of the misguided belief that math education can somehow be made easy: ‘Math is hard. The idea that somehow we’re going to make math just fun is just a dream.’” It’s too bad the article is so thin, but that’s USA Today for you.

Then there’s an article in the Boston Globe two days ago. It’s far too long to quote here, but I can provide some excerpts:
Schools could improve students’ sluggish math scores by hammering home the basics, such as addition and multiplication, and increasing the focus on fractions and some geometry, a presidential panel recommended Thursday.

“Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra,” the panel, appointed by President Bush two years ago, said in a report.
So, what’s the truth? I have to admit that the phrase “panel appointed by President Bush” immediately makes me suspicious. And the conflating of fractions with decimals and percents makes me more so. But let’s continue:
Because success in algebra has been linked to higher graduation rates and college enrollment, the panel focused on improving areas that are the foundations of algebra. Average U.S. math scores on a variety of tests drop around middle school, when algebra coursework typically begins...

“Students don’t know how to translate fractions into decimals or into percentages and they can’t locate fractions on a number line,” said panelist Tom Loveless, a senior fellow and education expert at the Brookings Institution, a Washington-based think tank.
OK, I have to admit, that bothers me too. But let’s move on:
In general, U.S. math curricula ought to be streamlined, according to the report.

“There is I think a tendency in American curricula to cover too many things too shallowly,” Larry Faulkner, the panel’s chair and the former president of the University of Texas, said.

The report takes a diplomatic stance when it comes to taking a position on the best methods to teach math to kids.

In recent years, there has been a dispute over whether children should learn a sequence of basic skills in math, including multiplication tables and some memorization, or should understand the theory behind math problems and come up with solutions on their own.
The report says both quick and effortless recall of facts and conceptual understanding of math are beneficial.

In addition, the back-to-basics camp has tended to favor “teacher-directed” instruction, in which teachers do all the explaining, while the opposing side has backed “student-centered instruction,” in which students have the main responsibility for learning math — often through working with peers.

The panel found students can benefit from both styles.
Well, yes! It’s hard to know why this has to be an either/or situation. Of course we need to do both:
“You need some element of discovery to allow kids to secure concepts in their minds, and you need to be able to have a reasonably efficient approach to be able to cover the material,” Faulkner said.
So it’s not clear just what the controversy is, nor what it says about fractions. Science teachers tend to emphasize decimals more than fractions, and students who are calculator-dependent are certainly more comfortable with fractions. Sometimes I fear that I take an elitist position, something like this: “Students in honors math classes need to understand fractions.” That’s surely true, but it’s not at all clear whether students in non-honors math classes need to do that. Maybe some of us are still living in the 19th Century. Maybe most people don’t need fractions. More later...

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