Devlin's Angle

February 1998

. . . Before This Decade Is Out . . .

Speaking at the Joint Mathematics Meetings in Baltimore, Maryland, in January, the Director of the US National Security Agency, USAF Lt. General Kenneth A. Minihan, called on the mathematics community to join with the NSA and other defense agencies to prepare to fight World War Four. According to Minihan, this will be the third war in which the outcome will depend in large part on the hidden role played by mathematicians.

World War II was the first major conflict in which mathematics played a major role, Minihan observed, citing how it was the breaking of the German Enigma code by British mathematicians that overcame the U-boat blockade in the North Atlantic, and the cracking of the Japanese purple code by American mathematicians that gave the Allies the entire German plan for defending the European mainland against a seaborn invasion from England.

The NSA is a direct successor of those wartime US codebreakers. Today, the agency is the largest single employer of mathematics Ph.D.s in the world, hiring between forty and sixty new math Ph.D.s each year. Their main task is to break foreign codes.

According to Minihan, the end of the Second World War was followed almost immediately by another major global conflict: The Cold War could justifiably be called World War III, Minihan suggested. Where World War II was measured in terms of battled won, he explained, World War III was measured by battles not fought -- at least, not in a physical way. It was a struggle for secrets, with spies the front-line troops and mathematicians providing the technical support.

In one significant respect, major mathematics conferences are different from other scientific gatherings. The astronomers, biologists, chemists, physicists, and whatever use their big meetings to announce major new discoveries. But, because of the nature of contemporary mathematical research, it is rare for a major new result to be announced at a large mathematics meeting. These days, when a mathematician makes a major discovery, he or she first shows the result to a small number of friends, who check the proof for correctness. This can often take days, sometimes several weeks. Only after the discoverer is sure that there are no mistakes does he or she arrange to present the result at a meeting. By then, news of the breakthrough has inevitably leaked out and circled the globe via the Internet.

But the absence of new mathematical results does not mean that there is nothing of note going on. With four thousand professionally active mathematicians gathered together, the January Joint Meeting provides an opportunity to check the pulse of the US mathematical community and to look for any significant new trends.

A visit by the NSA director can be viewed as highly significant. This was a first. Admittedly, with the meeting being held in Baltimore, it was a local gig for Lt. General Minihan. But that alone would hardly justify a major address from the agency's director. Minihan's mind was on World War IV. This next struggle, he said, would be fought in cyberspace.

As the world of the twenty-first century becomes increasingly dependent on the world wide web and succeeding technologies, mathematicians are going to play a crucial role in ensuring that this infrastructure remains secure and intact, Minihan asserted. To succeed, it would take the collaborative effort of both the NSA's large staff of researchers and the mathematics community at large. In an era where the communication media are public and open, only mathematics could provide the necessary security.

The importance the government currently attaches to mathematics was brought home by a second unusual visitor to the Baltimore mathematics meeting. For the first time ever, a US Secretary for Education made a major address at a Joint Meeting. Taking as his title "The State of Mathematics Education: Building a Strong Foundation for the Twenty-First Century, Secretary Richard W. Riley told a packed conference hall that it was time to bring an end to the current "math wars" in K-12 education, a battle that pits one teaching philosophy against another. Those involved in K-12 mathematics education should declare a truce and work together to find a mix of information and styles that work, Riley advised. It was important to "Recognize that different children learn in different ways and at different speeds," he remarked.

Professional mathematicians must make the education of the next generation of mathematics teachers a major priority, Riley continued, observing that, in the present era, "The basis of essential knowledge must be mathematics." College and university mathematicians should take a critical look at the way they taught their students -- the young mathematicians who would become the high school teachers of the future, Riley advised, asking the audience to "Remember [that] teachers teach the way they themselves were taught."

"Make the preparation of K-12 mathematics teachers a priority," Riley urged, adding, "The message [about the importance of mathematics and the priorities in K-12 mathematics education] should come from you." He called on university mathematicians to create new partnerships with their local communities, with museums, with high schools, and with local businesses, to help get the word out.

Another powerful call for help from the mathematical community to improve K-12 mathematics education came from a third plenary speaker at the meeting: Gail Burrill, the President of the National Council of Teachers of Mathematics. A twenty-five year veteran of the high school math class and a former recipient of a Presidential Award for Excellence in Mathematics and Science Teaching, Burrill is now at the University of Wisconsin.

After intriguing the audience with an example of the motivational mathematical problems she thought were required to inspire interest in mathematics among a greater proportion of today's young people -- yes, the solution required a calculator . . . and a whole variety of thought processes besides -- Burrill cautioned that the generation coming through the K-12 system today are very different from those who teach them at school or university.

Having been born into the information age, today's school and university students do not view calculators and computers as devices to aid thinking, Burrill observed. For them, the use of that technology forms an integral part of their thinking. "We do not know how they are thinking," Burrill told her audience. "They are doing something fundamentally different from us."

In order to help today's students develop their mathematical skills, therefore, we would have to present them with math problems that have meaning for them, she said, problems that fitted in with their world view. Problems that call for a much broader range of thought processes than the more narrowly defined skill-and-drill problems of yesterday's (and in some places today's) math class. Math problems far more like the kind of problems that arise in real life, in fact. It was, said Burrill, a difficult challenge, one that could only be met if the professional mathematics community took a lead.

The last time the United States made mathematics a major priority was in the aftermath of Sputnik. Startled by the Soviet breakthrough in space exploration, throughout the 1960s and 1970s, the National Science Foundation pumped millions of dollars into the preparation of hundreds of mathematics and engineering Ph.D.s and the support of mathematical research. As a result of that effort, Americans not only went to the moon, they led the world into the microcomputer age.

At the same time, increasing amounts of money were allocated to improve the state of K-12 mathematics education. There the results were far less dramatic. [Now for the part where I inject my own views, a move that invariably infuriates half my readers and brings cries of support from the other half.] One can argue endlessly about the significance of the various international comparisons of the measured abilities of K-12 math students across the board -- the USA generally scores poorly -- and equally endlessly about the causes. But regardless of that debate, for the sake of our young people who are the citizens of tomorrow, and for the sake of the society which will depend on them when they become adults, we must ensure that the mathematics education we provide those students is both stimulating and a match for their abilities.

The next few years will show whether the three calls for action heard in Baltimore turn out to be just rhetoric or if there is more to it. If there is, then January 1998 in Baltimore could turn out to be every bit as significant as President John F. Kennedy's 1961 promise to put a man on the moon and bring him back alive before the decade was out.

- Keith Devlin

Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. He has just finished work on a book to accompany a six-part television series, Life By the Numbers, which will be broadcast nationwide on PBS in April. The book, which has the same title as the television series, will be published at the same time by John Wiley and Sons.