Rest assured your acquaintance isn't standing on particularly solid ground - it's difficult to prove or disprove this particular claim.

The first challenge is the nature of American schools. Because education isn't in the Constitution, it's up to the states to decide what and how the children within its borders are prepared to be future employees, voters, citizens, and adults. To that end, free Black children were attending well-resourced (albeit segregated) schools in Boston while enslaved children in South Carolina were legally barred from learning to read (which didn't stop them, to be clear.) And some places, Native or Indigenous children were being forced to read scripture and to break ties with their families. So, it's difficult to make a broad statement about American schools and performance.

That said, even when states were legally prohibited from keeping Black children out of White schools, some states and districts found ways to keep their schools segregated. The most notorious example was the creation of segregation academies in several southern states by white parents as a way to avoid desegregated schools. The most widespread response was what's referred to as "white flight." White families with the means to move left the cities in droves, which lead to an expansion of suburban districts which enforced enrollment through residency checks.

One key thing to keep in mind is that American schools were built for the success of its white sons. It's not uncommon for people to cite literary rates in the early 1800's - America was a remarkably literate country, if you looked at white men and boys. In effect, white boys had a head start on what it means to be educated in this country. White girls had limited access until teaching became seen as an acceptable career for a young, unmarried woman and schools became a place for girls and boys in the mid-1800's. Concurrent to this history, their parents did everything in their power to keep students of color out of the better insulated, better built, and better resourced classrooms their white children used.

By the time of Brown v. Board in the 1950's, there were generations of Black children who had been denied a full, rich liberal arts education because of the actions of white parents. To be perfectly clear, though, when they had that access, Black students excelled and at times exceeded white students' performance. The most striking example of this is Dunbar High School, the first public all Black high school in America.

Located in DC, Dunbar (originally known as M Street High School) was staffed by highly trained, highly skilled Black Americans - many of whom couldn't get a job in their chosen field because of racism. Students studied Latin, Greek, Sciences, and Mathematics, not unlike students at the White high schools. When the city gave citywide exams, M Street students consistently performed as high as and for several years, higher than students at white high schools.

All of which is to say that there is compelling evidence that desegregation had a net positive impact on student achievement in places where both the letter and spirit of the law were upheld (Reber, 2010; Guryan, 2004). However. In the 1990's, court rulings made it easier for districts to back out of desegregation and integration efforts and without that external pressure, schools began resegregate. This data project provides a number of displays that speak to how segregated schools have become again.

The second problem is the only measure, the US National Assessment of Educational Progress, that looks at student's skills and abilities across all 50 states was offered for the first time in 1969. There's no way to compare student achievement before segregation to after on a national level.

There isn't one book (yet) that gets to your question, but it's coming. Nikole Hannah-Jones, a NY Times reporter, is working with historians to write the history of segregation in American schools. In the meantime, a few resources you may find helpful:

Guryan, J. (2004). Desegregation and black dropout rates. American Economic Review, 94(4), 919-943.

Reber, S. J. (2010). School desegregation and educational attainment for blacks. Journal of Human resources, 45(4), 893-914.

Rothstein, R. (2017).The color of law. New York: Liveright.

Williams, H. A. (2009). Self-taught: African American education in slavery and freedom. Univ of North Carolina Press.

Stewart, A. (2013). First class: The legacy of Dunbar, America's first black public high school. Chicago Review Press.