Three years later, looking back, Ms. Sprowal said she felt her son had been done an injustice. Matthew, who has had a diagnosis of an attention disorder, has thrived at P.S. 75. His second-grade teachers, Johanny Lopez and Chanté Martindale, have taught him many ways to calm himself, including stepping into the hallway for an exercise break. His report card last month was all 3s and 4s, the top marks; the teachers commented, “Matthew is a sweet boy who is a joy to have in the classroom.”

Matthew’s story raises perhaps the most critical question in the debate about charter schools: do they cherry-pick students, if not by gaming the admissions process, then by counseling out children who might be more expensive or difficult to educate — and who could bring down their test scores, graduation rates and safety records?

Kim Sweet, director of Advocates for Children of New York, said she had heard many such stories. “When we look at our cases where children are sent away from schools because of disabilities,” she said, “there are a disproportionate number of calls about charter schools.”

There is no more tenacious champion of charters than Ms. Moskowitz, whose students earn top test scores and who has plans to build a chain of 40 schools. She saw Matthew’s experience in a far different light, as her spokeswoman, Jenny Sedlis, explained in two voluminous e-mails totaling 5,701 words.

“We helped place him in a school that would better suit his needs,” Ms. Sedlis wrote. “His success today confirms the correctness of his placement. I believe that 100 percent of the time we were acting in Matthew’s best interest and that the end result benefited him and benefited P.S. 75, which now has a child excelling.”

Ms. Sedlis denied that Matthew had been suspended, and said he was not disciplined when he was kept after school.

“Practicing walking through the halls is the opposite of a punishment,” she wrote. “Just as in math, when a child does not get a concept, we re-teach. We don’t let the child fail. We ensure he gets it. We take the same approach with behavior. If a child is struggling, we re-teach. This is an example of when the school went out of its way to help Matthew be successful.”