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Teaching Math: Abstract Equations Beat Reality

Students learn math better if teachers stick to abstract equations rather than trying to bring in real world examples.

[M]any educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train.

[…]

In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems. “They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”

Very interesting if incredibly counterintuitive. I was always reasonably good at math but it eventually reached the level, somewhere in second semester calculus, where it just became too abstract for me to grasp. It was just memorizing equations, which I could do, but it didn’t translate into actual learning in my case because I could no longer figure it out for myself, as I could with “real world” models. But, at least within the parameters of this experiment, this experience apparently isn’t the norm.

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Press Not Doing Its Job?

Elizabeth Edwards, who despite no public policy credentials other than having been married to a one-term senator and yet oddly seems to get op-ed space in the major papers whenever she requests it, has a rather strange editorial in today’s NYT whining about how the mainstream media is failing in its duty to inform the public.

The first several paragraphs make the silly argument that the press covers only the drama of the race and ignores the issues, with the effect that “voters who take their responsibility to be informed seriously enough to search out information about the candidates are finding it harder and harder to do so, particularly if they do not have access to the Internet.” This, frankly, is nonsense. There’s so much information out there that it’s virtually impossible for those who can’t devote full time to immersing themselves in it to read it all. And who are these people who are simultaneously starving for information about Joe Biden’s health care proposals and yet lack Internet access? Presumably, there are people who are poor and don’t work in a connected office who are interested in public policy. But there’s always the public library.

Interspersed in this is a more interesting, if not particularly novel, complaint: That the press decides who the legitimate candidates are.

What’s more, the news media cut candidates like Joe Biden out of the process even before they got started. Just to be clear: I’m not talking about my husband. I’m referring to other worthy Democratic contenders. Few people even had the chance to find out about Joe Biden’s health care plan before he was literally forced from the race by the news blackout that depressed his poll numbers, which in turn depressed his fund-raising.

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Who is responsible for the veil of silence over Senator Biden? Or Senator Dodd? Or Gov. Tom Vilsack? Or Senator Sam Brownback on the Republican side?

The decision was probably made by the same people who decided that Fred Thompson was a serious candidate. Articles purporting to be news spent thousands upon thousands of words contemplating whether he would enter the race, to the point that before he even entered, he was running second in the national polls for the Republican nomination. Second place! And he had not done or said anything that would allow anyone to conclude he was a serious candidate. A major weekly news magazine put Mr. Thompson on its cover, asking — honestly! — whether the absence of a serious campaign and commitment to raising money or getting his policies out was itself a strategy.

This is fair enough. Then again, Thompson was a more plausible contender than Dodd or Vilsack or Brownback for the same reason that Hillary Clinton and John Edwards and Barack Obama — are relative novices — were. It’s a Catch-22: Candidates with name recognition and decent poll numbers are deemed legitimate enough to warrant press coverage but without press coverage it’s very hard to build name recognition and poll numbers.

Then again, Mike Huckabee and Ron Paul managed to do so.

The press, while holding a certain public trust, is ultimately not a collective but rather a myriad of private businesses that together form a web. Do we really expect the Big 3 networks, already losing viewers at a rapid rate, to devote their 8-12 minutes of nightly political coverage equally among all declared candidates? Or to spend it on the eye-glossing details of Joe Biden’s health plan rather than the interesting kerfuffle of the day?

Similarly, newspaper circulation is declining in almost every market. Papers have more space to devote to features than the television networks and, sure enough, they provide more in-depth coverage. But how often are they supposed to print charts comparing the health plans of the various candidates? If they do so once, are they good? Or must they do so repeatedly to reach occasional readers or those who happened not to read that particular edition?

Here’s the thing: If the public displayed an appetite for these things, the businesses would cater to it. Instead, readers demand more comic strips, horoscopes, recipes, movie listings, gardening tips, “human interest stories,” “good news,” and so forth.

At the same time, though, the incredibly tiny minority of us who are interested in public policy have more ability than ever in human history to get that information in as much detail as we want, as often as we want, and at a time that is convenient to us. That’s a pretty good trade-off.

John Edwards, Joe Biden, Chris Dodd, Mitt Romney, and the others lost, not because the press didn’t cover them properly but because the public looked them over and didn’t see them as “presidential.” It’s probably true that most people couldn’t tell you much about the health plans of these guys. But, really, who cares? There was never much chance that these people would be president. Why waste your time reading their white papers?

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McCain Saved Money, Legally, on Charter Jets!

John McCain (who, as we all know, is the beneficiary of kid gloves treatment from the mainstream press while the two contenders for the Democratic nomination are constantly getting unfair coverage) has been targeted with another non-scandal scandal by his good friends at the New York Times.

Given Senator John McCain’s signature stance on campaign finance reform, it was not surprising that he backed legislation last year requiring presidential candidates to pay the actual cost of flying on corporate jets. The law, which requires campaigns to pay charter rates when using such jets rather than cheaper first-class fares, was intended to reduce the influence of lobbyists and create a level financial playing field.

But over a seven-month period beginning last summer, Mr. McCain’s cash-short campaign gave itself an advantage by using a corporate jet owned by a company headed by his wife, Cindy McCain, according to public records. For five of those months, the plane was used almost exclusively for campaign-related purposes, those records show. Mr. McCain’s campaign paid a total of $241,149 for the use of that plane from last August through February, records show. That amount is approximately the cost of chartering a similar jet for a month or two, according to industry estimates.

The senator was able to fly so inexpensively because the law specifically exempts aircraft owned by a candidate or his family or by a privately held company they control. The Federal Election Commission adopted rules in December to close the loophole — rules that would have required substantial payments by candidates using family-owned planes — but the agency soon lost the requisite number of commissioners needed to complete the rule making.

So . . . McCain backed a law that allowed him to do exactly what he later did — and which was legal before the law was passed, too — and he then did it. The FEC would have passed a regulation that changed the law (how, exactly, I don’t know because bureaucratic agencies lack the authority to override legislation) but, alas, it didn’t.

Like Roger Kimball, I don’t see what the story is here.

Josh Marshall agrees that there’s probably nothing wrong here but considers this “gaming” the system and using unnecessary “roundabout” to disguise what was going on.

MyDD’s Josh Orton calls the story a “blockbuster” which “exposes two more broken McCain pledges: to not to fly on corporate jets, and to not exploit his wife’s wealth for campaign advantage.” Amanda Terkel of Think Progress agrees. But McCain objection to corporate jets was that said corporations might thereby gain undue influence; surely, that’s not a consideration when the corporation is owned by his wife? And it’s rather different to use your wife’s airplane or another existing, durable asset than to accept millions in cash, no? The jet, after all, remains an asset whereas cash is spent.

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