The personal computer companies, already experiencing a spectacular boom in an ailing national economy, have begun a megabucks scramble for what could be their most lucrative market yet: America's school systems.

At stake could be the way Johnny learns to read and write and do his arithmetic, as well as a multibillion-dollar market that could help the United States keep its rapidly dwindling world lead in advanced technology.

Yesterday, with the blessing of Education Secretary Terrel H. Bell, the Tandy Corp. announced the most ambitious program to date to move computers and computer understanding into schools, a multimillion-dollar offer to give training on its Radio Shack computers to every teacher in America.

Earlier this week IBM, a corporate giant but late comer to the burgeoning small computer field, unveiled a similar but smaller plan to train teachers in three states and donate $8 million worth of personal computers to schools and teacher training centers.

Both corporate moves came after another company, Apple Corp., failed in an effort last year to get congressional approval for a tax write-off in exchange for giving its computers to schools. That proposal crumbled under intense lobbying.

But perhaps more important than the corporate race for dominance is the question of whether American children are being trained for the computerized world in which they seem destined to live.

In joining Tandy officials at a news conference yesterday, Bell said, "Frankly, I don't think most of our society has fully grasped the significance of the computer as a force that will change our entire teaching and learning methods."

But he said he thought Tandy, whether for commercial or public interest reasons or both, had chosen the best approach by helping teachers "become computer literate" before large public investments are made in technology that is changing rapidly and often poorly understood.

In a Pac-Man and Donkey-Kong age of youngsters flocking into computer game arcades, Bell said the classroom problem often is the teacher keeping up with the pupil rather than the other way around.

Still, the number of computers in schools has risen from an estimated 92,000 a year ago to about 300,000. Tandy officials estimated schools would own 500,000 personal computers by the end of this year.

The computer boom has come so rapidly that training clearly has fallen behind job demand even in a recession era. Studies indicate that U.S. universities turned out only 50,000 graduates for 115,000 computer-related jobs last year.

At the same time universities as well as secondary and elementary schools are facing severe budget restrictions that make some educators wonder whether an expensive surge into high-tech for children is a wise idea.

Some educators note that a generation ago television was hailed as a teaching marvel that would change America's school system radically. The change has been modest.

Still others worry that the failure of a broad, nationwide move of computers into schools could lead to the creation of a generation of "computer elitists" with vast advantages over those who failed to get the benefits of modern technology while in school.

At present, the bulk of the schools that own computers are in wealthier urban and suburban areas that generally spend more money on education.