In his State of the Union address in January, President Bush stressed the importance of improving math education. He proposed to "train 70,000 high school teachers to lead advanced placement courses in math and science, bring 30,000 math and science professionals to teach in classrooms, and give early help to students who struggle with math."

But where will these teachers come from? And will the training of teachers be sufficient to increase the number of students choosing math and science careers? And why does all this matter?

Because mathematics is the foundation of the natural sciences. It is no coincidence that Isaac Newton, the man who formulated the law of gravitational attraction that revolutionized our understanding of the universe, was also the man who popularized the calculus. And the natural sciences, however pure, are what give us airplanes, cable TV and the Internet.

In the 2003 Program for International Student Assessment, a test that measures math literacy, American 15-year-olds performed worse than their peers in 23 countries, as well as those in Hong Kong. It's not hard to see why. According to the National Council of Teachers of Mathematics, 40 percent of the nation's middle school math teachers do not have the equivalent of an undergraduate minor in math. The average starting salary of a teacher is only $30,000, whereas the average starting salary for a recent college graduate in computer science or engineering is $50,000.

Short of following the British, who have proposed paying experienced math teachers more than $100,000, with a guaranteed minimum of $70,000, where will we find a way to attract the thousands of teachers George Bush wants?

New York State initiated an innovative program to bring teachers from Jamaica for two or four years to teach in New York schools. Jamaica, a developing nation where one U.S. dollar equals 65 Jamaican dollars, is nonetheless a stable, English-speaking nation with an unbroken democratic tradition; it stands poised to beat the United States in establishing the world's first Institute for Mathematical Methods in Counterterrorism. When teachers for the New York program were recruited on the campus of the University of the West Indies, recruiters found more experienced math and science teachers than they ever dreamed they would.

But you can have all the teachers in the world and still not inspire kids to learn math. My friend Autumn e-mailed me about her nephew, Joshua: "He's upset because he's asked several of the math teachers why math is important or what are certain formulas used for -- there has to be a use, correct?"

Autumn told her nephew about my work in counterterrorism and for the television crime drama "Numb3rs." Autumn reported, "He's told his math teachers about you as well, and about the show 'Numb3rs.' He's informing them that through something called lattice theory you are managing to fight terrorists -- all with math."

Mathematics is art, and should be appreciated for its beauty, not simply for its utility. But we cannot expect 11 year-olds to cherish totally order-disconnected topological spaces as much as professional mathematicians do.

As I first proposed in January 2005, television shows like "Numb3rs" (or "Medium") -- where the main characters are mathematicians -- could work with the National Council of Teachers of Mathematics to show kids how math is really used; the council and Texas Instruments are now working together to use "Numb3rs" to promote math literacy in schools.

Another way to inspire kids is to relate mathematics to something they see every day. In order to excite students and draw funding to his school, school superintendent Ronald Ross of Roosevelt, N.Y., has begun looking into the idea of creating a curriculum involving math and counterterrorism. What kinds of topics would students learn?

The opening line of the Oscar-winning movie "A Beautiful Mind" is "Mathematicians won the war." During World War II, the mathematics underlying cryptography played an important role in military planning. Winston Churchill admired Alan Turing, the mathematician who had mastered the German codes, recognizing him as the man who had perhaps made the single greatest individual contribution to defeating Hitler.

At Los Alamos, the lab that built the atomic bomb, Cliff Joslyn uses lattice theory to mine data drawn from thousands of reports of terrorist-related activity to discover patterns and relationships that were previously in shadow.

Lattice theoretical methods developed at MIT tell us the probability that we have disabled a terrorist cell, based on how many men we have taken out and what rank they hold in the organization. Lauren McGough, a Massachusetts high school student, tested the accuracy of this model by getting her classmates to pretend they were terrorists, passing orders down a fictitious chain of command, essentially confirming what the theory predicts.

High school students could learn algebra, trigonometry, calculus and logic while also learning concrete applications involving homeland security. No longer would students yawn and ask, "What is math good for?" Beauty could defeat both terror and boredom.

Whatever you may think of the State of the Union address, when it comes to supporting math education, we should all see pi to pi. President Bush is correct when he says that mathematics education in America must improve if the United States is to stay economically competitive, but the stakes are much higher than that. During the Cold War, the United States would not have tolerated a military gap between itself and its adversaries. Yet today, with 61 percent of all U.S. doctorates in math going to foreigners (15 percent to Chinese), we readily accept a "math gap."

Dollar for dollar, the best defense against our adversaries' weapons of mass destruction may be our allies in the Americas, armed with weapons of math instruction.

Improving math education is not merely a smart idea. It is a matter of national security. Algebra is one revolutionary Islamic concept we cannot afford to neglect or ignore.

George Habash, a militant and former secretary-general of the Popular Front for the Liberation of Palestine, once characterized terrorism as a "thinking man's game." Using mathematics, researchers at Stanford University's Center for International Security and Cooperation (CISAC) have made fighting terrorism a thinking man's game as well.

CISAC affiliate Lawrence M. Wein of the Graduate School of Business and CISAC Science Fellow Jonathan Farley are both applying mathematical models to homeland security problems, such as preventing a nuclear detonation in a major U.S. city and determining whether terrorist cells have likely been disrupted.

Wein, who teaches operations classes about different business processes used to deliver goods and services, has focused his research on bioterrorism and border issues. He has performed, he says, the first mathematical analyses of hypothetical botulism poisoning, anthrax outbreaks and smallpox infections.

"One overriding theme of my work is that all these homeland security problems are operations problems," said Wein, the Paul E. Holden Professor of Management Science. "Just as McDonald's needs to get hamburgers out in a rapid and defect-free manner, so too does the government have to get vaccines and antibiotics out and test the borders for nuclear weapons or terrorists in a rapid and defect-free manner."

In collaboration with Stephen Flynn of the Council on Foreign Relations, a nonpartisan research center, Wein recently has conducted research to improve security at U.S. borders and ports. Port security has received significant attention recently owing to the furor over Dubai Ports World's bid to manage six terminals at major U.S. harbors. The aim of Wein and Flynn's work is to prevent terrorists from bringing into the country a nuclear weaponbe it an atomic bomb or a so-called "dirty bomb," or conventional explosive packed with radioactive waste.

"Of all the problems I've studied, this is the most important because the worst-case terrorist scenario is a nuclear weapon going off in a major U.S. city and also it is the one the government has dropped the ball on the most," Wein said. "They have done a very poor job."

Instead of using the existing approach, where U.S. Customs actively inspects a minority of containers based on information from a specialized tracking system designed to identify suspicious containers, Wein and Flynn have recommended the government use a multi-layer, passive screening system for every container entering the country. Under their system, Customs would photograph a shipping container's exterior, screen for radioactive material and collect gamma-ray images of the container's contents. If terrorists shielded a bomb with a heavy metal such as lead to hide it from radiation detectors, gamma-ray imaging would allow inspectors to see the shielding and flag the container for inspection. Wein and Flynn believe this whole process would cost about $7 per container.

"Right now about maybe 6 percent of the containers are deemed suspicious and they will go through some testing and the other 94 percent of the containers just waltz right into the country without an inspector laying an eye on them," Wein said. "What we're proposing to do is 100 percent passive testing."

Wein's earlier work addressed a different threat: bioterrorism. In 2005, Wein revealed the nation's milk supply was vulnerable--a terrorist could potentially poison 100,000 gallons of milk by sneaking a few grams of botulinum into a milk tanker. Although the government and dairy industry have collaborated to intensify the heat pasteurization formula for milk, Wein is still pushing for additional botulinum testing, which he says would cost less than 1 percent of the cost of milk.

Wein also has used math to study smallpox outbreaks, the U.S. fingerprint identification system and U.S.-Mexico border security issues. Wein's congressional testimony on the fingerprint identification system in 2004 led to a switch from a two-finger system to a 10-finger system. His 2003 research on anthrax attacks resulted in a Washington, D.C., pilot program to use the U.S. Postal Service to distribute antibiotics throughout the capital after an outbreak. Seattle is now testing a similar program.

"In Washington, D.C., now, if there is a large-scale anthrax attack, postal workers will be the first to get their Cipro and, on a voluntary basis, they will go door-to-door distributing antibiotics," Wein said.

He said the common thread throughout his research is queuing theory, or the mathematical study of waiting lines, but he also draws upon mathematical epidemiology for his smallpox studies; air dispersion models for the anthrax model; supply chain management for the milk study; probability theory for the fingerprint identification system; and models for nuclear transport and detection for his work with containers.

From tainted lactose to lattice structures

While Wein is working on improving the government's counterterrorism systems, Jonathan Farley is working to figure out when terrorist organizations have been effectively disrupted. His mathematical model is designed to help law enforcement decide how to act once they have captured or killed a terrorist or a number of terrorists in a cell.

A professor at the University of the West Indies who will chair the Department of Mathematics and Computer Science there next year, Farley is on a one-year science fellowship at CISAC. In 2003, he co-founded Phoenix Mathematical Systems Modeling Inc., a company that develops mathematical solutions to homeland security problems.

He is using lattice theory--a branch of mathematics that deals with ordered sets--to determine the probability a terrorist cell has been disrupted once some of its members have been captured or killed.

"Law enforcement has to make decisions about what resources they should allocate to target different cells," Farley said. "The model should provide them with a more rational basis for allocating their scarce resources. ... It will inform you when you're making decisions about how much time and effort and how much money you're going to spend going after a particular cell."

While at Stanford, Farley hopes to unearth the perfect structure, mathematically speaking, for a terrorist cell--or in other words, a cell structure that is most resistant to the loss of members.

"If it's possible to determine the structure of an ideal terrorist cell, you can focus on a much smaller number of possibilities, because it makes more sense to assume the adversary is going to be smart rather than stupid," Farley said.

Farley has suggested it is possible Al-Qaida and other terrorist organizations already may have figured out the perfect structure for a terror cell by trial and error.

"I don't expect Osama bin Laden to be reading lattice theory in his caves in Afghanistan," said Farley. "But if it follows from the mathematics, perhaps heuristically, the terrorists will have come to the same conclusion--that this is the best way to structure a terrorist cell."

Although Farley acknowledges his model is not a panacea for terrorism, he hopes it will help reduce guesswork that might be involved in pursuing terrorists.

"It's not that I think mathematics can solve all of these problems," Farley said. "Because it can't. But it's better to use rational means to make decisions rather than guesswork."

John B. Stafford is a science-writing intern at Stanford News Service.

You're being watched...Congress will soon hold hearings on the National Security Agency's domestic spying program, secretly authorized by President Bush in 2002. But that program is just the tip of the iceberg.

Since 9/11, the expansion of efforts to gather and analyze information on U.S. citizens is nothing short of staggering. The government collects vast troves of data, including consumer credit histories and medical and travel records. Databases track Americans' networks of friends, family and associates, not just to identify who is a terrorist but to try to predict who might become one.

Remember Total Information Awareness, retired Adm. John Poindexter's effort to harness all government and commercial databases to preempt national security threats? The idea was that disparate, seemingly mundane behaviors can reveal criminal intent when viewed together. More disturbing, it assumed that deviance from social norms can be an early indicator of terrorism. Congress killed that program in 2003, but according to the Associated Press, many related projects continued.

The Defense Advanced Research Projects Agency runs a data-mining program called "Evidence Extraction and Link Discovery", which connects pieces of information from vast amounts of data sources. The Defense Intelligence Agency trawls intelligence records and the Internet to identify Americans connected to foreign terrorists. The CIA reportedly runs Quantum Leap, which gathers personal information on individuals from private and public sources. In 2002, Congress authorized $500 million for the Homeland Security Department to develop "data mining and other advanced analytical tools." In 2004, the General Accounting Office surveyed 128 federal departments and agencies to determine the extent of data mining. It found 199 operations, 14 of which related to counterterrorism.

What type of information could these mine? Your tax, education, vehicle, criminal and welfare records for starters. But also other digital data, such as your travel, medical and insurance records - and DNA tests. Section 505 of the Patriot Act (innocuously titled "Miscellaneous National Security Authorities") extends the type of information the government can obtain without a warrant to include credit card records, bank account numbers and information on Internet use.

Your checking account may tell which charities or political causes you support. Your credit card statements show where you shop, and your supermarket frequent-buyer-card records may indicate whether you keep kosher or follow an Islamic halal diet. Internet searches record your interests, down to what, exactly, you read. Faith forums or chat rooms offer a window into your thoughts and beliefs. E-mail and telephone conversations contain intimate details of your life.

A University of Illinois study found that in the 12 months following 9/11, federal agents made at least 545 visits to libraries to obtain information about patrons. This isn't just data surveillance. It's psychological surveillance.

Many Americans might approve of data mining to find terrorists. But not all of the inquiries necessarily relate to terrorism. The Patriot Act allows law enforcement officers to get "sneak and peek" warrants to search a home for any suspected crime - and to wait months or even years to tell the owner they were there. Last July, the Justice Department told the House Judiciary Committee that only 12% of the 153 "sneak and peek" warrants it received were related to terrorism investigations.

The FBI has used Patriot Act powers to break into a judge's chambers and to procure records from medical clinics. Documents obtained by the American Civil Liberties Union recently revealed that the FBI used other new powers to eavesdrop on environmental, political and religious organizations.

When Congress looks into domestic spying in the "war on terror," it should ask a series of questions:

• First, what information, exactly, is being collected? Are other programs besides the president's NSA initiative ignoring traditional warrant requirements? Are federal agencies dodging weak privacy laws by outsourcing the job to private contractors?
• Second, who has access to the data once it is collected, and what legal restrictions are set on how it can be used or shared?
• Third, who authorized data mining, and is its use restricted to identifying terrorists?
• Fourth, what is the collective effect of these programs on citizens' rights? Privacy certainly suffers, but as individuals begin to feel inhibited in what they say and do, free speech and freedom of assembly also erode.
• Fifth, how do these data collection and mining operations deal with error? As anyone who's tried to dispute an erroneous credit report can attest, once computer networks exchange data, it may be difficult to verify its accuracy or where it entered the system. Citizens who do not know they are under surveillance cannot challenge inaccurate information that may become part of their secret digital dossier.

What will Congress do to ensure that the innocent remain so?

This week President Bush will seek to focus the nation's attention on border security and immigration reform. But the president's proposals won't protect Americans from our gravest cross-border threat: the possibility that a ship, truck or train will one day import a 40-foot cargo container in which terrorists have hidden a dirty bomb or nuclear weapon.

The Bush administration maintains that it has a smart strategy to reduce this risk. A new 24-Hour Rule requires that importers report the contents of their containers to customs inspectors one day before the boxes are loaded on ships bound for the United States. The Department of Homeland Security's National Targeting Center then reviews the data, checking against other intelligence to determine which boxes may pose a threat. Although the containers deemed high risk are inspected at cooperating foreign ports or when they enter the United States, the rest--more than 90 percent--land here without any perusal.

We have two concerns about this strategy. First, it presumes that the United States government has good enough intelligence about Al Qaeda to reliably discern which containers are suspicious and which are not. But our inability to thwart the attacks in Iraq demonstrates that we lack such specific tactical intelligence. And supporting customs inspectors, who must make the first assessment of risk, is not a priority for the intelligence agencies. Inspectors must rely on their experience in spotting anomalies--a company that claims to be exporting pineapples from Iceland, for example.

Second, determined terrorists can easily take advantage of the knowledge that customs inspectors routinely designate certain shipments as low risk. A container frequently makes 10 or more stops between its factory of origin and the vessel carrying it to American shores. Many of the way stations are in poorly policed parts of the world. Because name-brand companies like Wal-Mart and General Motors are widely known to be considered low-risk, terrorists need only to stake out their shipment routes and exploit the weakest points to introduce a weapon of mass destruction. A terrorist cell posing as a legal shipping company for more than two years, or a terrorist truck driver hauling goods from a well-known shipper, can also be confident of being perceived as low risk.

So what needs to be done? A pilot project under way in Hong Kong, the world's largest container port along with Singapore, offers one piece of a potential solution. At an estimated cost of $7 per container, new technology can photograph the box's exterior, screen for radioactive material, and collect a gamma-ray image of a box's contents while the truck on which it is carried moves at 10 miles per hour.

Terrorists can defeat radiation sensors by shielding a dirty bomb with dense materials like lead. But by combining those sensors with gamma ray images, the Hong Kong system allows inspectors to sound the alarm on suspiciously dense objects. Inspectors would need to analyze enough of the scans--perhaps 20 percent to 30 percent--to convince terrorists that there is a good chance that an indistinct image will lead a container's contents to be sent for more reliable X-ray or manual examinations. Images of container contents would then be reviewed remotely by inspectors inside the United States who are trained to spot possible nuclear weapons.

If terrorists were to succeed in shipping a dirty bomb, for example, the database of these images could serve as a kind of black box--an invaluable forensic tool in the effort to identify how and where security was breached. That information could help prevent politicians from reacting spasmodically and freezing the entire container system after an attack.

Such a program could significantly reduce the likelihood that terrorists will smuggle plutonium or a dirty bomb through American ports. But it still would not stop a terrorist from importing highly enriched uranium, which can be used to construct a nuclear weapon. Lengthening the time that a container is screened for radiation would help, and this could be done without increasing waiting times if additional monitors were added to the Hong Kong system near the gate where the trucks must already stop for driver identification checks. Better still would be for the Department of Homeland Security to make the development of new technology that can recognize the unique signature of highly enriched uranium an urgent priority.

Finally, we must find ways to ensure that terrorists do not breach containers before shipments arrive at loading ports. Sensors should be installed inside containers in order to track their movements, detect any infiltration and discern the presence of radioactive material. Where boxes are loaded, certified independent inspectors should verify that companies have followed adequate protocols to ensure that legitimate and authorized goods are being shipped.

Taken together, these recommendations will require new investments and an extraordinary degree of international cooperation. But increased container security will not only help the United States prevent terrorism, it will also help all countries reduce theft, stop the smuggling of drugs and humans, crack down on tariff evasion and improve export controls. What's more, such a program would require an investment of just one one-hundredth of the capital that could be lost if we shut down the global container shipping system after an attack.

Container security is a complex problem with enormous stakes. American officials insist that existing programs have matters well in hand. But we cannot afford to take these perky reassurances at face value while the same officials fail to embrace promising initiatives like the Hong Kong pilot project.

One of the less publicized but perhaps most important matters before President Bush on his recent trip to South Korea for the annual Asia-Pacific Economic Cooperation summit was that of relations between the United States and the host country. Although South Korea has long been a close ally of the United States, relations between the two have cooled in recent years, and the alliance has come under great strain. Bush's trip did not set a new direction for the alliance, which has been drifting for the past few years.

There is a way to reverse this cooling in relations, I believe -- to promote U.S. strategic interests in the region, including denuclearization of North Korea; to retain U.S. influence there; and to strengthen a long-standing alliance. What is needed is an effort to widen the "North Korea problem" from one of nuclear weapons to one of unification.

Controversy over the fraying U.S.-South Korea alliance focuses almost exclusively on cultural or emotional issues. In the United States there are some who feel that South Koreans are insufficiently grateful for the steadfast U.S. support to South Korea, particularly for the American lives lost in defense of the South during the Korean War of 1950-53 and for the extensive economic and military aid since. Others feel that rising anti-American sentiment in South Korea reveals the naivete of a younger generation of Koreans who are insufficiently worried about the North Korean threat.

But the problems in the alliance are not a result of emotion, naivete or ingratitude. Indeed, even if none of those emotional and cultural issues existed, the alliance would still be in dire need of revision. To find the best path forward for both the United States and South Korea, we need to focus on the real issues.

The main factor straining the alliance is the unresolved Korean War and the continued division of the peninsula. This has created differing long-term strategic concerns for the United States and South Korea.

For South Korea, the key issue is not North Korean nuclear weapons -- it never was. South Korea's overriding concern is how to resolve the issue of national unification and integrate North Korea back into the world's most dynamic region, whether or not there are nuclear weapons. All other South Korean foreign policy issues take second place.

In contrast to Korea's regional issues, U.S. concerns are global and military. For at least the next several years, the United States will be mainly concerned with countering potential terrorist threats. Distracted by the overwhelming focus on anti-terrorism, homeland security and other issues, the United States has viewed its Korea policy as a narrow extension of its anti-terrorism policy, focusing almost exclusively on denuclearizing the North. These different strategic priorities have led to severe strains between the two allies, despite the desire of both to maintain a close relationship.

The United States can improve its position in East Asia, as well as solidify its alliance with South Korea, by widening its focus beyond North Korean denuclearization and coming out strongly and enthusiastically in favor of Korean unification. Although the United States rhetorically supports unification, it has been noticeably passive in pursuing policy to that end.

Such a policy shift would achieve many U.S. goals and would strengthen our alliance with South Korea in the process.

First and foremost, denuclearization is far more likely to occur with a change in North Korea's regime and a resolution to the Korean War than it is without resolving that larger issue. Until now the United States has put denuclearization first, without making much progress. Folding the nuclear issue into the larger issue would provide far more leverage on both questions and potentially create new or broader areas for progress.

Second, such a policy would provide grounds for agreement between U.S. and South Korean policymakers from which they could cooperate and work together, rather than against each other. Exploring the best path toward unification will require both economic and military changes in the North -- changes that will provide the United States with more flexibility to rebalance its own forces in the region.

Finally, it would put the United States in a solid position to retain goodwill and influence in Korea after unification -- something that is far from ensured today, when many South Koreans are skeptical about U.S. attitudes and policies toward the region. If the United States is seen as a major source of help for unification, it is far more likely that the post-unification orientation of Korea will be favorable to Washington.

This would be a major policy change for the United States, but given the importance of the region and of the Korean Peninsula, it is the best path to follow.