Thomas O. Johnston, a lawyer representing both schools, argued that the state Education Department staff member who decided not to renew the schools lacked the legal authority to do so. The law gives that power to the education commissioner, he said.
In addition, leaders of the schools and the lawyer say letters explaining why their charters were not being renewed contained factual and statistical errors. They say they should be allowed to remain open, even if on probation, as other schools in similar situations have been.
D.U.E. Season, opened in 2005, was on probation, and its leaders say it was in the middle of carrying out an improvement plan that was showing results when they got the nonrenewal notice.
Renaissance Regional, open since 2010, got a warning letter from the state last June but was not put on probation. Its leadership says it also had put in place new initiatives that were improving performance.
Both say their students would be harmed by the schools' closing.
"We're absolutely not giving up," said Lorna Hassel, Renaissance Regional head of school. "Parents have really been behind us the whole way."
The nonrenewal letters, which an Education Department spokesman referred to for comment, describe both schools' academic performance as "dismal," and did not view their leaders and staff as capable of substantially improving student achievement.
Renaissance Regional's letter states that 44 percent of the students tested proficient or better in 2012-13 in language arts and 57 percent proficient in math, but the state performance report put those rates at 46 and 59. D.U.E.'s letter put its language arts proficiency at 26 percent and math at 45 percent; the performance report says the rates are 24 and 46.
However, Galloway Community Charter School had 49 percent proficiency in language arts and 56 percent in math, and the Environment Community Opportunity Charter in Camden came in at 46 percent proficient in language arts and 51 percent in math, according to their performance reports, and they were allowed to remain open on probation, the schools say.
The nonrenewal letters say Renaissance Regional is in the bottom 13.4 percent of schools for academic achievement, and D.U.E. is in the lowest-performing 5 percent.
However, the schools point to state performance reports that put Renaissance Regional in the 22d percentile for academic achievement - the same as Galloway. D.U.E. was in the ninth percentile - one point less than Environment Community Opportunity.
D.U.E.'s chief education officer and founder, Doris Carpenter, said that if her school closed, most of her students would end up in Camden's traditional public schools, almost all of which rank in the lowest 5 percent in the state.
"You're going to close our school to send our kids back to a school that is performing lower than us?" Carpenter asked.
Her school also offers students three meals a day, extended-day activities, and summer and weekend programs - all of which they would not get if the school closed, she said.
A petition to keep D.U.E. open has attracted about 2,300 signatures, Carpenter said. The almost-560-student school has a waiting list of about 200.
She shared a letter written by a fifth grader to Hespe. The boy said his father is in prison and his mother gave him up. The school, he said, has become his family.
"If the school shuts down, I would lose my family . . . people that care for me and love me. I would lose the people I love. I have lost enough. I don't want to lose them. Please save my school I am begging you," he wrote.
Renaissance Regional, which has about 160 students and 100 on its waiting list, has almost 1,800 signatures on its petition. Located in the Pemberton Township School District, it includes quite a few military families.
Parents spoke enthusiastically about the school's sense of community, its inclusiveness, and its positive effect on their children. It scored in the 80th percentile in the state for student growth, according to its performance report.
"We just really want the opportunity to show the changes we have implemented have worked," said Courtney Chmielewski, head of the home-school organization. "We want the opportunity to show them we can succeed."