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Saturday, December 24, 2005

U. S. Math Education: do we really lag behind the rest of the world?

We’ve often heard how poorly our grade school students (high school and down) stack up to those students in other countries, especially when we compare test scores. A careful look at the data paints a nuanced picture. Yes, our schools (or our schools plus parents plus students) could do better.
But, the current situation, while far from optimal, might not be as dismal as it seems. It appears that our relatively poor showing on the international exams may be influenced by several factors which include:
  1. Sometimes students from our regular comprehensive high schools end up being compared with those in other countries who are attending more specialized academies.

  2. Other countries don’t necessarily have the same socio-economic stratification within their citizens that we do. That is, there are some societies where the lower social classes are not citizens of their respective countries.

  3. Our lack of homogeneity is also a factor. For example, our white highs school students score comparably with white students from Europe. This might indicate that our lower scores may well be reflecting problems with our society rather than problems with our academic education.
Again, this is not to say that all is well. And, frankly, as someone who teaches incoming science, and engineering freshmen, I am appalled at how poorly prepared most of them are, especially those who enter with sterling grade point averages.
I also see big problems with parental attitudes; I’ll go into this a bit later. Nevertheless, the following article gives a balanced view of those distressing test results.


How We Measure Up

Is American Math and Science Education in Decline?

As if coordinated to provoke headlines, top executives at three of the nation’s leading technology firms recently issued bleak appraisals of the American education system, criticizing especially how American students are taught science and mathematics. Microsoft Chairman Bill Gates minced no words at a summit of the nation’s governors: until high schools are redesigned, he declared, “we will keep limiting, even ruining, the lives of millions of Americans every year.” The chief executives of Intel and Cisco Systems shortly followed suit, suggesting that America’s lackluster schools will increasingly force companies to look overseas for talent.
Of course, these concerns are hardly new. But the somber prognoses from the heights of high-tech have added high-profile urgency to recent press reports about the declining performance of U.S. students in science and math compared to other nations, and the potential rise of China as a technological and economic superpower. Leading U.S. media outlets have featured major stories on the consequences of China’s rise for America’s future, like the recent Newsweek cover story by Fareed Zakaria appealing for a “massive new focus” on science and technology in the U.S., lest America “find itself unable to produce the core of scientists, engineers and technicians who make up the base of an advanced industrial economy.” In such a media atmosphere, one could be forgiven for having concluded that the United States is drifting unawares into an educational backwater while the rest of the world paddles furiously past it.
The truth is more complex. Cross-national studies of scientific and mathematical ability, interpreted rightly, tell a complicated story, giving reason to question how well the tests measure America’s real educational standing in the world. The two tests cited most frequently in press reports are the Program for International Student Assessment (PISA) and the Trends in International Mathematics and Science Study (TIMSS). PISA, undertaken by the Organization for Economic Cooperation and Development (OECD), most recently spanned 41 countries and tested 15-year-olds on mathematical word-problems. The latest TIMSS, in 2003, comprised more traditional, textbook-style math and science problems and was administered to fourth- and eighth-graders in 25 countries by an international team of researchers based in Boston and Amsterdam. The Department of Education funded both studies in the U.S., with help from the National Science Foundation.
Both tests have repeatedly been invoked by sensationalists seeking to cast the United States as unprepared for the high-tech, global economy. When the latest PISA results were released toward the end of last year, for instance, the Christian Science Monitor ran with the headline “Math + Test = Trouble for U.S. Economy,” and concluded that the study’s emphasis on “real-life” math skills makes it an accurate and “sobering” predictor of students’ performance in “the kind of life-skills that employers care about.” Federal officials expressed concern about the test results, too. “If we want to be competitive, we have some mountains to climb,” said Deputy Education Secretary Eugene Hickok.
To be sure, the results of neither TIMSS nor PISA should send American educators and policymakers rushing to the champagne. In most math areas tested by PISA, the gap between the average U.S. student and the average student in the highest-scoring countries—often Finland, the Netherlands, Singapore, Japan and Hong Kong—was roughly equivalent to the gap between the United States and low-scoring countries like Uruguay or Mexico. Where 44 percent of Singapore’s students reached the TIMSS “advanced international benchmark,” only 7 percent of U.S. students did. And, in general, the longer students had remained in the U.S. school system, the worse they performed relative to their peers abroad.
The first question that must be asked of such broad results, however, is whether the tests themselves accurately represent the countries’ student populations. International surveys such as these are not given to every student in each participating country; the tests’ organizers pick out statistical samples that are supposed to represent each country’s entire student population. Even so, schools—especially in the United States—sometimes decline to participate in the tests, potentially skewing the sample. As far as accurate sampling is concerned, early incarnations of the tests were not encouraging. In the first TIMSS general achievement test, conducted in 1995, only 5 of 21 participating countries met the study’s guidelines for conducting representative samples. While most countries participating in the latest studies have dramatically improved their overall sampling, the United States remains a notable exception. Only 73 percent of U.S. students chosen to be sampled were actually tested, a figure below the “minimum acceptable” rate of 75 percent. In most other countries, that number was well over 90 percent. If the omitted 27 percent of U.S. students were even slightly above or below average, their exclusion casts serious doubts on the accuracy of the U.S. sample.
The studies also inevitably confront large differences between countries’ school systems. “In Cyprus, students taking the advanced mathematics test were in their final year of the mathematics and science program; in France, the final year of the scientific track; in Lithuania, the final year of the mathematics and science gymnasia; in Sweden, the final year of the natural science or technology lines; and in Switzerland, the final year of the scientific track of gymnasium,” Professor Iris Rotberg of George Washington University wrote in Science concerning the 1995 TIMSS assessment, which tested high-schoolers. “In contrast, students in several countries, including the United States, attended comprehensive secondary schools. The major differences in student selectivity and school specialization across countries make it virtually impossible to interpret the rankings.” In TIMSS, especially, tests are conducted by grade-level rather than by age. In elementary and middle school, where topics are often covered and learned over the course of a few weeks, the risk of comparing students at incommensurate stages of their education is great.
Broad curricular differences have probably had a role in deflating U.S. scores. TIMSS and PISA use the same test in every participating country, and the material that makes it onto the test is selected through a winnowing process that leaves the tests considerably narrower than any single country’s general curriculum. Countries that include large amounts of material in their typical curricula are therefore at a disadvantage compared to those countries that focus their curricula more intensely on fewer subject areas. Regardless of its other merits or failings, the American strategy of repeated exposure to a broad range of subjects—American textbooks are the bulkiest in the world—is likely to lend itself to unduly poor performance on standardized tests, as full understanding of any single concept takes longer to develop.
Demographics and culture are also thought to confound the results of cross-national comparisons. In the United States, the tested students come from every socioeconomic rung, while other countries sometimes lack some rungs because of cross-border employment. For example, much of the labor force in Hong Kong (which is treated on the tests as an independent entity) is made up of tens of thousands of low-paid Filipino household workers whose children live and are educated in the Philippines; in light of the extensive literature tying socioeconomic indicators to educational achievement, this cross-border employment surely affects both countries’ scores. A similar situation obtains in other places with significant immigration and cross-border commerce, as Gerald Bracey points out in a 1997 article in the journal Educational Researcher. “Each morning thousands of Malaysians enter Singapore to sweep streets, pick up garbage, and do other low-level jobs. They return to their homes at night, relieving Singapore of having to educate the children of poor laborers,” Bracey writes. “If one reads the [domestic] educational research literature, one is struck by the lengths—the extreme lengths—that researchers go to to ensure that samples in their studies are comparable....The research community would never accept test results in this country that simply compared scores in an inner-city slum and an affluent suburb as if they were comparable,” he writes. The opposite circumstance holds in the United States: Students from all socioeconomic rungs are educated and scored on these tests.
Amid this deluge of confounding factors, the inference that the U.S. education system is going down the tubes is an unjustified logical leap. The United States is still pumping out tremendous numbers of new Ph.D.s in the sciences—more, in fact, than our economy can presently absorb, as there is a well-reported dearth of jobs for newly-minted science Ph.D.s. The same is true in engineering: According to a recent National Science Foundation report, the number of engineers graduating from U.S. schools will continue to grow into the foreseeable future, outstripping the number of available jobs. Of these new engineers and Ph.D.s, an increasing number are foreign-born—but increasing even faster is the percentage of those who decide to stay in the United States. Federal research funding for scientific research and development has consistently risen in absolute terms and as a fraction of discretionary spending—and industry research dollars have risen dramatically on top of that, to the tune of 7 percent per year in real terms—according to calculations by the Consortium for Science, Policy and Outcomes at Arizona State University. (Alarmist media reports often use GDP, against which research spending has fallen, as a comparative baseline.) And countries that have “outperformed” the United States in educational studies for many years—a number of European countries top this list—still fail to rival the U.S. in any measure of research productivity. When Bill Gates and others seem to appeal for school reform in the U.S., perhaps they are merely providing their companies with political cover and a post hoc justification for employing foreign engineers who, while not better educated than U.S. workers, are often significantly cheaper.
Nevertheless, there remains good reason to worry about what the global economy portends for those American students who really are badly educated. In only one other OECD country (New Zealand) are internal educational inequalities worse than in the United States, according to a recent analysis by researchers in England and Italy. Where these inequalities lie is no mystery. The gap in test scores between white and ethnically Asian students on the one hand and black and Hispanic students on the other is a well-known attribute of U.S. schools and is noted ruefully in nearly all cross-national studies. Two University of Pennsylvania researchers recently aggregated scores from a number of cross-national studies and found that white students in the United States, taken alone, consistently outperform the predominantly white student populations of several other leading industrial nations. “There is compelling evidence,” they write,” that the low scores of [black and Hispanic students] were major factors in reducing the comparative standing of the U.S. in international surveys of achievement. If these minority students were to perform at the same level as white students, the U.S....would lead the Western G5 nations in mathematics and science, though it would still trail Japan.” In PISA, for instance, white students performed above most European countries, whereas black students performed on par with students in Thailand. So while the performance of minority groups in the U.S. does refute the alarmist assertion regarding an across-the-board decline in U.S. schools, it does so in a particularly unfortunate way—namely, it suggests that some American minority groups will be shut out of high-paying jobs as companies look for better-educated workers overseas. Although the most recent TIMSS saw the white-black score gap close slightly, it is almost certain to remain shockingly large in the near future.
None of this is to say that other countries are not catching up technologically, nor that the United States is safe from competition in even a single technological sector. China is without doubt the most aggressive challenger. In the mid-twentieth century, Japan’s economy grew 55-fold over the course of thirty years through stringent government control; observers of Japan’s rise will remember the key role of its Ministry of International Trade and Industry, which employed many of the nation’s brightest stars and guided the economy on a carefully directed path of technological growth. China’s strategy has been similar, though its tremendous size has necessitated delegation of heavy-handed economic control to regional governments in what scholars have termed “local state corporatism.” It has simultaneously harnessed the power of markets in a way Japan did not. Regional governments lavish tax breaks on high-tech industries (many of them funded from overseas) and pump millions into China’s new universities—which are poised to graduate more Ph.D.s than the United States by 2010, according to some projections. Nearly all of China’s top leaders are scientists and engineers by training: President Hu Jintao is a hydroelectric engineer, Premier Wen Jiabao is a geological engineer. Their predecessors, Jiang Zemin and Zhu Rongji, were both electrical engineers. The technocrats steering China’s ship of state are working hard to modernize scientific education in their country.
But the United States need not worry—not yet. The U.S. is by no means in technological decline, though China and India will inevitably pose challenges in years to come. Although not a crisis, this competition should motivate the U.S. to improve its science and math education, especially for poor and minority students who might lose out in a globalized, high-tech economy. If sensationalists must take up a cause, it should be the plight of those students and not a hyped-up “threat” of China or the “impending decline” of technological innovation here at home.
The Editors of The New Atlantis, "How We Measure Up," The New Atlantis, Number 9, Summer 2005, pp. 111-116.

Monday, December 12, 2005

Embedding a Klein Bottle in 4-space

My wife is leaving for a 3 week trip to India and decided to give me my Christmas present early. What I got was a glass model of an immersed Klien bottle. On the left, you see some models and on the right, a drawing made by Mathematica software. She got the idea for the gift from Dus7's blog and went to Classy Glass to get it.
Ok, so what is a Klien bottle? And, why is it important, and why did I use the word "immersed"?
First, a Klien bottle is an example of what is known as a "2-manifold"; that is, a very nearsighted ant that lived on the surface of the bottle could not distinguish the area immediately around it from what it would see if it lived on an ordinary plane (the kind of plane you did geometry on in high school). Furthermore, for those who have had some calculus, there is a way of "parametrizing" the Klien bottle so we could do calculus on it.
To obtain the Klien bottle from a section of a plane, consider the following figure:

Take this rectangle and glue the upper edge to the lower edge as shown. One obtains a long open "tube". Now if one were to glue the remaining "circle" edges to each other in a normal way, would would obtain a surface that looks like the skin of a donut; this is called a "torus".

But if we glue the circles in the reverse way (as indicated by the arrows; this is similar to what one does when one makes a Mobius Strip) one obtains the Klien bottle. Go ahead and click on the link as it gives better drawings than I give here.

Now, back to the glass model and the drawing. Notice how the bottle seems to "intersect itself" in a circle (where the glass tube goes from the "outside" into the "inside"? This is unavoidable if one tries to put this bottle in 3-space.

But, if one goes up one dimension, one can avoid this self intersection.

Consider the following two figures.

Here, we pretend that we are in 4 dimensional space, with coordinates (x, y, z, w). In the first figure (the shorter tube) we assume that the two circle boundaries of the tube lie in the plane (x, y, 0,0). The rest of the tube lies in the hyperplane (x, y, 0,w). That is, for all points in the first figure, the z coordinate is "0". Label the first circle boundary (closest to you) by C3 and the second one by C1. Now the second tube lies in the hyperplane (x, y, z, 0); that is, all of the points in this tube have "w" coordinate "0". The three circle that I've shown lie in the plane (x, y, 0, 0). The circle closest to you is called C3, and the one just slightly to the right of that is called C1. Now the circles labeled C1 and C3 are the same circle as they have the same coordinates in the (x, y, 0, 0) plane. So, we can join these two tubes in 4 space to obtain a surface that has no "exposed" edges; we call such a thing a "closed" surface. Notice that these tubes do not intersect each other except for these two circles as all of their points differ in their z and w coordinates. And, when you join these two tubes up, you get a Klien bottle! This is because of the way the tubes connect at circle C1.

Sunday, December 11, 2005

Why is math so hard?

I am near the end of the semester and hope to make a few more entries in this blog over break. One of the things I hope to do is to talk a bit about the Nash embedding theorem.

I am teaching a numerical methods class and gave a take-home exam. One student included this photo in his work, though he crossed out the word "you" and wrote "I". This reminded me of a quote from the mathematician Ronald Graham:, "Our brains have evolved to get us out of the rain, find where the berries are and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." (source: )
This link goes to an interesting discussion which I've reproduced here:
Will There Be Anything Left To Discover?
Is the great era of scientific inquiry over? Have all the big theories been formulated and important discoveries made—leaving future scientists nothing but fine tuning? Or is the real fun about to begin?By JOHN HORGAN and PAUL HOFFMAN

A spirited debate, conducted via e-mail, between two acclaimed science journalists: John Horgan, author of the controversial book The End of Science, and Paul Hoffman, former editor of Discover magazine and past president of the Encyclopaedia Britannica.
HOFFMAN: The past decade has brought a spate of books sounding the death knell for a host of subjects. Francis Fukuyama served up The End of History and David Lindley The End of Physics. But your more sweeping work The End of Science (1997) attracted a lot more attention and controversy—and with good reason. The idea that science may have had its run—that we've discovered all we can realistically expect to discover and that anything we come up with in the future will be pretty much small-bore stuff—left people either intrigued or outraged. With today's seemingly frenetic pace of scientific discovery, however, how can you say that the whole enterprise is coming to an end? The scientists I know, far from preparing for the undertaker, are ebullient about the future of their field.
HORGAN: Sure, scientists are keeping busy, but what are they actually accomplishing? My argument is that science in its grandest sense—the attempt to comprehend the universe and our place in it—has entered an era of diminishing returns. Scientists will continue making incremental advances, but they will never achieve their most ambitious goals, such as understanding the origin of the universe, of life and of human consciousness. Most people find this prediction hard to believe, because scientists and journalists breathlessly hype each new breakthrough, whether genuine or spurious, and ignore all the areas in which science makes little or no progress. The human mind, in particular, remains as mysterious as ever. Some prominent mind scientists, including [Time Visions contributor] Steven Pinker, have reluctantly conceded that consciousness might be scientifically intractable. Paul, you should jump on the end-of-science bandwagon before it gets too crowded.
HOFFMAN: Don't save a seat for me quite yet, John. Take the human mind. I agree that we are not close to an understanding of consciousness, despite the efforts of some of the best minds in science. And perhaps you're even right that we may never understand it. But what is the evidence for your position? You've criticized scientists for having faith—a dirty word in the scientific lexicon—that our era of explosive progress will continue unabated. Isn't it at least as much a leap to think that the progress will abruptly end—particularly since the trajectory of discoveries so far suggests just the opposite, that supposedly unanswerable questions eventually do get answered?
HORGAN: My faith is based on common sense, Paul, and on science itself. As science advances, it imposes limits on its own power. Relativity theory prohibits faster-than-light travel or communication. Quantum mechanics and chaos theory constrain our predictive abilities. Science's limits are glaringly obvious in particle physics, which, as Steven Weinberg describes [in the Visions issue], seeks a "theory of everything" that will explain the origin of matter, energy and even space and time. The leading theory postulates that reality arises from infinitesimal "strings" wriggling in a hyperspace of 10 (or more) dimensions. Unfortunately, these hypothetical strings are so small that it would take a particle accelerator the size of the Milky Way to detect them! I am not alone in fearing that string theorists are not really practicing science anymore; one leading physicist has derided string theory as "medieval theology." Paul, here is persuasive evidence of science's plight.
HOFFMAN: Yes, but who is to say that all these scientific theories won't ultimately be replaced by ones with greater explanatory power? Galileo and Newton thought their laws of motion were the cat's pajamas, explaining everything under the sun and many things beyond, but 2 1/2 centuries later a Swiss patent clerk toppled their notions of space and time. Obviously, Galileo and Newton did not foresee what Einstein found. I think it's ahistorical to assert that in the future there will never be an Einstein of, say, the mind who will be able to pull together a theory of consciousness. And even if it's true that some of the big unanswered questions of science may never be answered, a lot of new and exciting science could still come from overturning truths that we now take for granted. Robert Gallo, the AIDS researcher, once told me that at the end of the 1970s, he was at a conference where a prominent scientist confidently summed up the truths of biomedicine— such truths as: epidemic diseases are things of the past, at least in so-called developed nations; a widespread outbreak of infectious disease is impossible unless the microbe is casually transmitted; the kind of virus found in animals known as the retrovirus doesn't exist in man; and no virus causes cancer in humans. By the end of the 1980s, these four truisms had hit the dustbin. Or take a more recent example: the newfound plasticity of the human brain. Until a year and half ago, it was a dogma taught in every medical school in the country that the adult human brain is rigid, that its nerve cells can never regenerate. Now we know our brains do have the ability to generate new cells—a discovery that may not only open up a new understanding of the brain but also lead to novel treatments for a host of brain disorders. HORGAN: Here's the big question we're dancing around: Can we keep discovering profound new truths about reality forever, or is the process finite? You seem to assume that because science has advanced so rapidly over the past few centuries, it will continue to do so, possibly forever. But this view is, to use your word, ahistorical, based on faulty inductive logic. In fact, inductive logic suggests that the modern era of explosive scientific progress might be an anomaly, a product of a singular convergence of social, intellectual and political factors. If you accept this, then the only question is when, not if, science will reach its limits. The American historian Henry Adams observed almost a century ago that science accelerates through a positive-feedback effect. Knowledge begets more knowledge; power begets more power. This so-called acceleration principle has an intriguing corollary: If science has limits, then it might be moving at maximum speed just before it hits the wall.
HOFFMAN: Of course, I accept that science has limits—and may even be up against them in some fields. But I believe there is still room for science, even on its grandest scale, that awe-inspiring discoveries will continue to be made over this millennium. The mathematician Ronald Graham once said, "Our brains have evolved to get us out of the rain, find where the berries are and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." Sounds reasonable, except when you consider that it could be similarly said that our brains didn't evolve to invent computers, design spaceships, play chess and compose symphonies. John, I think we'll continue to be surprised by what the brains of scientists turn up.
HORGAN: I hope you're right, Paul. I became a science writer because I believe science is humanity's most meaningful creation. We are here to figure out why we are here. The thought that this grand adventure of discovery might end haunts me. What would it be like to live in a world without the possibility of further revelations as profound as evolution or quantum mechanics? Not everyone finds this prospect disturbing. The science editor of the Economist once pointed out to me that if science does end, we will still have sex and beer. Maybe that's the right attitude, but there aren't any Nobels in it. No matter how far science does or doesn't advance, however, there's one wild card in even the most pessimistic scenario. If we encounter extraterrestrial life—and especially life intelligent enough to have developed its own science—then all bets are off.

Friday, December 09, 2005

Relativity from the Daily Kos

I doubt that regular readers of this page will learn anything here, but this would be a good place to send your not so scientifically inclined friends to if they want to get a brief glimpse of the basics of Relativity Theory and of the Theory of Quantum Mechanics:

And speaking of basics, I can recommend the following books to those who want to learn more about the topological aspects of the mathematics of space-time (not at the research level):

And if you care to play some of the games that the book recommends, you can go here (play tick-tack-toe on a "flat torus" or a "flat klien bottle"