The number of public-school students from low-income families and single-prent homes has surged since 1970, bring severe social problems into the classroom and making it more difficult to raise academic achievement, a new report says.

At the same time, the number of middle-class children in public schools declined as birth rates fell and private-school enrollment increased, according to the report entitled ''Cheating Our Children.''

''Public schools stand the risk of becoming the schools of low-income, disadvantaged and poverty-level children,'' said C. Emily Feistritzer, the education researcher who compiled the report chiefly from government statistics. ''People who are highly educated and making money seem to be turning to the private schools.''

Because of the ''harsh reality'' in public schools, Feistritzer said it will be difficult to enforce more stringent high school graduation requirements adopted recently by many states. Instead, she said there should be ''more differentiation . . . based on ability levels.''

''That's not a popular notion now because we are so hung up on equity,'' Feistritzer sasid. ''But not all students can handle three years of hard science . . . . There ought to be a common core that all are exposed to, but recognition of different ability levels is desirable.'' Feistritzer is director of the private National Center for Education Information and publisher of several Washington-based newsletters. Last year, she prepared widely noted reports on training and quality of teachers and on school finance.

According to the new report, the proportion of school-age children from poverty-level homes rose from 14.8 percent in 1970 to 21 percent in 1983. The increase in this low-income category amounted lto 1.9 million children, compared with a decline of 6.1 million children from more prosperous homes.

Households headed by women accounted for almost 90 percent of the increase in numbers of poverty-level children, the report said. The number of school-age children in low-income households headed by women has reached 4.7 million, an increase of almosty 67 percent in slightly more than a decade.

Although blacks total about 5 percent of all school-age children, Feistritzer said they account for about one-third of those in poverty.

Since 1970, she said, enrollment in private schools, including nurseries, has increased by about 4 percent, while public-school enrollment plummeted by 11.4 percent.

Feistritzer cited several studies showing a strong relationship between family income and student achievement. She said children from broken homes also generally do worse in school than those from intact homes.