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(Excerpted from Stop Getting Ripped Off) About three in four U.S. 15-year-olds told international researchers in 2003 that they regularly received good grades in mathematics classes. It’s a good thing they weren’t going to school in Finland, Canada, the Czech Republic, or even Iceland. Every three years, the Organization for Economic Cooperation and Development (OECD) runs an international test of 15-year-olds around the world, so nations can compare their educational systems. All the world’s industrialized nations participate, along with several non-OECD members – more than 40 nations in all. In 2003, the focus was on math achievement. The U.S. teens scored highest for grade inflation – more U.S. teens said they got good grades than any other nation. But when it came to actual math skills, the U.S. landed near the bottom of the test. In addition to the countries listed above, students from the Netherlands, Latvia, Poland, and even Hungary outperformed their U.S. counterparts. The U.S. found itself nestled among a group that included Italy, Serbia, and Uraguay – and 25th among the 30 developed nations studied. The study also noted that the Czech Republic, which landed 13 spots above the U.S., spent only one-third as much to educate its students. The U.S. was said to have the poorest outcome per dollar spent on education. And in a statement that gave pause to many U.S. educators, the trendline was perhaps most disturbing: "The gap between the best and worst performing countries has widened," said Andreas Schleicher, who wrote the report and ran the study. Any number of international studies unearth much the same results; the nation that first landed a man on the moon is being left in the dust by developing countries in math and science studies. The news was so disturbing that President George W. Bush appointed a panel in 2006 to investigate the problem and propose some answers. The National Mathematics Advisory Panel published its findings a year later. Its complaints run the gamut; textbook publishers seem to confuse heft with utility, as children’s math textbooks can be 700-1,000 pages long, far longer those from nations with better test scores; teachers are ill-prepared for mathematics instruction. It all added up to a disturbing result: “Close to half of all seventeen year olds cannot read or do math at the level needed to get a job at a modern automobile plant,” the panel’s report concluded. “Barring some other special knowledge or talent that would allow them to earn a living as, say, a plumber or artist, they lack the skills to earn a middle-class paycheck in today’s economy.” The researchers zeroed in on one specific, and perhaps fatal, flaw in math education. U.S. students seem to fall off a cliff when the subject of fractions is introduced. The report said that about half of U.S. eighth-grade students could not solve a word problem that involved fractions. Absent the ability to work with fractions, U.S. math students hit their ceiling in math studies at a very young age. “The sharp falloff in mathematics achievement in the U.S. begins as students reach late middle school,” the report said. “Difficulty with fractions [including decimals and percents] is pervasive and is a major obstacle to further progress in mathematics, including algebra.” And, it should now be clear, in settling restaurant tabs and buying cars and homes.
**** Of course, it’s darn near impossible to learn something if your teacher doesn’t know it. Johnny will never be able to work with fractions if Johnny’s teachers can’t. As the presidential report was swatting away at U.S. students for not understanding fractions, other researchers were poking away at teacher education programs with disappointing results. The National Council on Teacher Quality surveyed 77 top teachers’ colleges to find out how well they were preparing future educators to teach math in 2003. Its findings: Teachers weren’t required to know math to enter their programs, and even worse, they weren’t required to know it when they graduated, either. Standardized tests are required for entry into these schools. One such test is called Praxis, sort of like the LSAT for law schools. Praxis does include math questions: elementary school-level math questions. In other words, aspiring 5th grade teachers need only prove that they could pass a 5th-grade math test! In only one of the 77 schools studied was there a requirement to prove proficiency in high school math. "Almost anyone can get in (to teachers’ schools),” the report said. “Compared to the admissions standards found in other countries, American education schools set exceedingly low expectations for the mathematics knowledge that aspiring teachers must demonstrate.” Once in school, education certification is controlled at the state level. The report found that 18 states had no requirements whatsoever that elementary teachers attend even a single math course. But the terms of graduation are even worse. When the Praxis II is administered, to prove the teacher has mastered coursework required to shape young minds, there is no separate mathematics quiz in any state. And because the questions are all scored together, it’s possible for teachers to answer every single math question wrong and still receive a passing grade! It’s not enough for teachers to successfully add and subtract, the report found. Nor is it enough for teachers to be “one chapter ahead of the students,” and know just enough to teach the material from a book. Teachers who will try to inspire youngsters to memorize times tables and follow the disciplined rules of division must have a deeper understanding of why multiplication tables work, or why algebra matters. Without those skills, they are unlikely to effectively teach good math lessons – in the same way that reading teachers must know more than how to read, but must understand phonics, word attack strategies, and numerous other literacy skills to teach reading. It shouldn’t be surprising that math skills are rare among elementary teachers. People attracted to teaching young kids tend to skew towards verbal and communication skills, and often end up attracted to coursework that augments those skills. In fact, society often permits excellent communicators to neglect so-called “right brain,” analytic studies. It does so at great peril to our society’s math skills; and to nearly every consumers’ pocketbook. Massachusetts officials set out to take on math teacher training head on in 2007 after this distressing result. Michael Klugerman , director of math training for a Massachusetts state program, regularly tests elementary school teachers before his classes. Of 200 teachers he tested in a recent year, less than half could answer questions such as "30 is what percent of 75?" and "what is 14 divided by 1/2?" “Elementary teachers are phobic about math," Klugerman told the Boston Globe. "I've seen that very much in my classes. There's a lot of anxiety about math." Parents have picked up on this deficiency, of course. There have been mini-revolts at school board meetings where parents demand better math instruction. There’s even been “Math Wars,” which pit fans of old-fashioned memorization against fans of “reform math,” which stresses conceptual understanding over drills. In California, parents can send their kids away to summer camp – Money Camp – that teaches them practical mathematics like how to compute compounded interest. But so far, these efforts have amounted to little more than sound and fury. All the school board sit-ins in the world over education philosophy won’t change a thing if half of all eighth graders can’t understand fractions.
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The hard thing about fractions is that you can’t really calculate them by “feel.” Simple addition and subtraction are fairly observable in the regular course of life: You’ve invited eight friends to your house, but three guests don’t like beer. One six pack and a four-pack of Smirnoff Ice should cover at least the first round of drinks. Even simple multiplication seems to be second nature. If you plan to make two sandwiches for lunch every day this week, a loaf of bread with at least twenty slices will do the trick. These kinds of problems don’t require much brain power. The answer comes almost instinctively. But let’s twist things, just a little. Back to those eight friends (nine, including you). You have three-quarters of a container of half and half in the fridge. You know everybody will want coffee. Do you have enough creamer? How much cream goes in a cup, an ounce? A tablespoon? Maybe two. But how many tablespoons are in a pint? Only bakers keep facts like that at hand. (It’s 32). You might then remember that the break room at work offers Land O Lakes mini-moos, and people generally seem to use two of them. Say luck is on your side, and you happen to have one in the cabinet next to your sugar bowl – and you see that each mini-moo is 3/8ths of an ounce. And so it begins. How much is two mini-moos? And then how many two-mini-moo servings are in three-quarters of a pint of half and half? At this point, most people would grab their coat and car keys, head to the grocery store, and just buy more creamer. Let’s just hope they check the sugar bowl before leaving. If fractions are the Waterloo for the majority of elementary school students today (and perhaps for you, too), we should back the equation up just a little farther in school. Pupils learn arithmetic in this order: addition, subtraction, multiplication, and then division. Fractions are just an advanced way of expressing division ( 3/8ths is really three divided by 8). But working with fractions ultimately requires division, and that’s where a lot of the trouble begins. I think the reason is obvious: the answer *isn’t* obvious. It might be obvious how many beers or slices of bread you need in the examples above, but it’s not obvious how much creamer. There’s only one way to get the answer to this creamer dilemma: You must work it out. You must sit down with pencil and paper (lucky people can do this in their heads) and calculate the answer. It requires discipline. It requires applying a multi-step prescription. It’s not trivial. And it really can’t be done with a cell phone calculator. Long division is hard. For example, to divide 3 by 8, you must add a place and a decimal point, multiply 3 times 8, subtract 24 from 30, write the answer of 3 and carry over the result of six, then multiply 7 times 8, enter the result of 56, write in the answer of 7 and carry over the 4 (when will this end??) and then multiply 8 times 5, write in the result of 40 and the answer of 5 and there you have it (0.375). Don’t worry if you didn’t stay with me. That’s the point. Fractions aren’t obvious. I’ll leave it to the footnotes to express the calculation for how many two mini-moo servings are in 12 ounces. But I’ll tell you that you didn’t need to waste the gas. In fractions, as in long division, there really is no shorthand way to get the right answer. You must take the long road. Americans aren’t too pleased when there isn’t a shortcut. In fact, we are the land of shortcuts. That’s why you can buy anything now and pay for it later. That’s why we play guitar hero over learning how to play guitar. And why college graduates expect their business degrees give them the right to a six-figure salary (by the way, why do we let 19-year-olds take classes in “Management?”) This trouble with fractions and with division in general has profound impact on our nation’s financial savvy. Fractions are the gateway to many other mathematical concepts. Chief among them: percentage. A population that has trouble calculating percents – and doesn’t even really understand how to work out percentages -- can’t possibly shop intelligently. All those 15 percent off sales are crippling us. |
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