OVER THE PAST 20 years, the governments of Japan, West Germany and the Soviet Union have dramatically upgraded their education programs in mathematics and science. All three did so because they realized that economic success -- and, in the case of the Soviets, military success -- demanded a work force knowledgeable and skilled in science and technology.

American schools, lacking central direction or even encouragement from Washington, have headed the opposite way. The traditional course requirements in math and science for both a high school diploma and a college degree have been stopped. Not surprisingly, enrollments in the so-called "hard courses" have plummeted. And now the Reagan administration, while seeking increased economic productivity and a strengthened national defense, threatens to make a bad situation worse.

Japanese leaders often point to the rigor of their education system as a key to their economic success. Mathematics instruction in Japan has "a more rapid pace" than in the United States, and "a much higher proportion of students take advanced courses," according to a recent report by the National Science Foundation and the Department of Education. Japanese students take three years of geometry and a year of trigonometry by 9th grade. Calculus, probability and statistics are taught in high school. In the latest International Assessment of Education Achievement comparing students from 12 industrialized nations, Japanese students ranked first in math.

In Japan, an engineering degree is considered a ticket to business and social success the way a law degree is here. About 20 percent of all bachelor's degrees and 40 percent of master's degrees in Japan are awarded to engineers. In the United States, the figure for both is 5 percent.

In talking about economics, everyone seems to have a neat chart or graph, a la the "Laffer curve." Thus James Fallows, writing about the U.S. economic slump in The Atlantic magazine, found a union leader who had the solution: He drew a graph illustrating the increase from law schools and the decline in the percentage of engineering graduates. When the two lines crossed, he concluded, that's when American Productivity began to sink.

There's actually some evidence that, by inference, backs up this notion. Each year, about 1.7 percent of British students and 1.6 percent of Americans become engineers. Productivity for the two nations grew 51 percent and 39 percent, respectively, between 1963 and 1977. About 2.3 percent of West German students become engineers, and industrial productivity is up 114 percent there durin the same period. In Japan, 4.2 percent became engineers, and productivity went up 197 percent.

Still, the most impressive educational transformation is that of the Soviet Union. In November 1966, the Central Committee of the Soviet Communist Party resolved to prepare its young people for the coming "scientific and technological revolution." School curriculum reform, unlike ours, was turned over to the nation's most prestigious scientific and educational institutions -- the Academy of Sciences and the Academy of Pedogogical Sciences.

"In the last decade, the Soviets have made simultaneous quantitative and qualitative gains without equal in the history of their education system," said Isaak Wirszup, a professor of mathematics at the University of Chicago, in a letter to former President Carter. With support from NSF, Wirszup had carried out an extensive study of the Soviet education system. He concluded that in "content and scope," the math and science program required for all Soviet students "places them far ahead of every other nation, including the United States."

Today, about 98 percent of Soviet students graduate from secondary school. By contrast, the U.S. high school graduation rate has remained at 75 percent for the last decade.

Moreover, Soviet requirements are far more stringent. By the end of secondary school, a Soviet student has taken five years of algebra, 10 years of geometry, two years of calculus, five years of geography, five of biology, five of physics, four of chemistry.

In the United States, only 9 percent of high school graduates take even one year of physics, 16 percent a year of chemistry and 45 percent a year of biology. Each year, about 5 million Soviet secondary students graduate with two years of calculus behind them. By contrast, only 105,000 American high school graduates have had one year of calculus.

Some NSF experts say Soviet officials have pushed too hard and created resentment among educators and parents, especially in more backward, rural areas. And the Soviet economy, lacking incentives for innovation, can't make effective use of its well-trained technicians.

But NSF officials corroborate Wirszup's main conclusions. "In only 10 years," he said, "the Soviet Compulsory program for all students covers the equivalent of at least of at least 13 years of American schooling in arithmetic, algebra and calculus, and does so much more thoroughly and effectively."

American schools have the Soviets to thank for the last big science push, that following the 1957 launching of Sputnik. The governement, through the National Science Foundation, funded the development of new, more modern science curricula and supported summer refresher courses for science and math teachers. Gifted students were encouraged and supported to go into scientific fields.

But that booster was soon spent. By the mid-1960s, the pendulum swung back away from the "hard courses" to the humanities and more "relevant" electives. High schools dropped most of the graduation requirements in advanced math and science, and so did colleges. A majority of high school students don't take a single math or science course after 10th grade. Students can now graduate from many of the nation's finest universities without even taking a survey course in science.

Since 1965, the federal government's main concern in education has been students who were left out -- the "disadvantaged," the handicapped, those with little or no command of English. The budget figures tell the tale. This year's Education Department budget, pre-Ronald Reagan, allocated $3.26 billion for Title I of the Elementary and Secondary Education Act to pay for "compensatory education" for poor children. Another $1 billion would help support education for the handicapped. An additional $138 million would pay for bilingual education. For gifted students, the figure was $6 million.

In his last budget, Carter decided to increase the science education budget at NSF to about $116 million, a modest amount by Washington standards. Now the Reagan administration, in its budget-cutting zeal, has decided to elmininate entirely the science education program.

What this does is strip the science education business of its investment capital, a sure means of creating stagnation. Take just one example: the video-disc.

Science education leaders have been awed by the potential of the "intelligent video-disc," a tiny videotape linked with a computer. With such a device, the inner workings of a cell or a chemical experiment, all illustrated before them. The computer quizzes the student, explains further when necessary and allows him to move on as fast as possible. Lab equipment is expensive, and a school board than can barely pay to repair a leaking roof can hardly afford to upgrade it science program. The videodisc may be a way to make education better for less money -- to make it more productive.

We pay a political and social price for our scientific illiteracy. How, after all, can voters intelligently choose between a pro- and an anti-nuclear candidate if they can make no sense of how nuclear power works?

Science, its fruits and poisons, have "so permeated our lives," says John W. Hopkins, a biology professor at Washington University in St. Louis, that "those citizens who are unable to handle basic scientific information will no longer be full participants in society." British novelist C. P. Snow argued during the 1950s that modern society had two cultures -- one science, the other the humanities -- each uncomprehending and somewhat distrustful of the other. Since then, at least in Britain and the United States, that gap has grown wider.