A couple of education profs out of University of Illinois maintain that math achievement of students in public schools, meaning schools run by school districts, is higher than the achievement of similar students in private schools. (Note: The article is in Education Week which is subscription only, so the link may not bring up the complete article.) The profs, Christopher and Sarah Lubienski, wrote a book on the subject, The Public School Advantage: Why Public Schools Outperform Private Schools.
The same study found that district and charter schools are about equal.
Is it true? As would be expected, pro-privatization scholars have disputed the findings. Here's the definitive answer on which side is right: Who the hell knows? Every education study is suspect because children aren't lab rats, every classroom is a universe unto itself, and different scholars can slice and dice the same data to come up with wildly different conclusions. So all "results" from educational studies should be taken with heaping helpings of salt.
That being said, every reasonable study I've seen has concluded there's about a dime's worth of difference between the achievement of similar students in district, charter and private schools, and that dime gets passed around to different types of schools depending on the grade level of the students, the subject matter and the nature of the study. This goes all the way back to the George W. Bush administration, where his Department of Education tried to prove private and charter schools were better than district schools and ended up with the conclusion that one scored higher in one area, another scored higher in another area, but the whole thing came out as a wash. Bush's folks were so frustrated with the study, they ran it again a few years later and got the same results.
However, one difference comes out in study after study. The lowest performing schools are conservative Christian private schools. Even the Bush-era studies agree. I'll leave it to you to reach your own conclusions about that.
The pro-privatization "education reform" movement perpetuates the myth that students do better in charter and private schools than they do in what they like to refer to as "failing government schools." These folks who love to pre-test, test and post-test students within an inch of their lives tend to "prove" their claims anecdotally, because the data they so lovingly compile just doesn't come up with results they like. Then they use their billionaire donors' money to help them repeat their anecdotes, misstatements and lies over and over until people think, if I've heard it said that many times by that many people, it's gotta be true. But as so often happens, the facts aren't on their side.