Comparing public and private schools

I am still trying to make sense out of this study:

A study finds that U.S. public school students do as well on math tests as those in private schools when student backgrounds are taken into account.

Christopher Lubianski and Sarah Theule Lubianski, of the University of Illinois used data from the 2003 National Assessment of Educational Progress, comparing the scores of 340,000 students in 13,000 schools, the New York Times reported. The number of students taking the test increased tenfold that year.

The Lubianskis used statistical techniques to adjust scores for family income and school and home circumstances. They found that the average raw scores of Catholic school fourth-graders were 14.3 points higher than public school students and 3.4 points lower when adjusted.

On this scale, an academic year is about 10 points.

What I don't get is how they adjusted for school circumstances. They were supposed to be measuring differences in school circumstances, not concealing them. I cannot find the paper online.