In southern states within the 1960's what did 'white' schools teach the kids about the racial issue?

Okay I've got this, here's all the decisions that were taken before and after the Civil Rights of 1964.

In May 17 1954, Thurgood Marshall The Supreme Court rules on the landmark case Brown v. Board of Education of Topeka, Kans., unanimously agreeing that segregation in public schools is unconstitutional. The ruling paves the way for large-scale desegregation. The decision overturns the 1896 Plessy v. Ferguson ruling that sanctioned "separate but equal" segregation of the races, ruling that "separate educational facilities are inherently unequal." It is a victory for NAACP attorney Thurgood Marshall, who will later return to the Supreme Court as the nation's first black justice.

In April 20 1971
The Supreme Court, in Swann v. Charlotte-Mecklenburg Board of Education, upholds busing as a legitimate means for achieving integration of public schools. Although largely unwelcome (and sometimes violently opposed) in local school districts, court-ordered busing plans in cities such as Charlotte, Boston, and Denver continue until the late 1990s.

...and in June 23 2003
In the most important affirmative action decision since the 1978 Bakke case, the Supreme Court (5–4) upholds the University of Michigan Law School's policy, ruling that race can be one of many factors considered by colleges when selecting their students because it furthers "a compelling interest in obtaining the educational benefits that flow from a diverse student body."

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Schools in the 1960s didn't really make them aware of the social situation of that time, they will instead focus on the rest of the Declarations and articles of the Civil Rights, as well as history.

There wouldn't be any black kid in a 'white' school because, as I said earlier, schools would select their students and leave black children. Those went to other schools separated from the white's schools.