Since Tennessee began taking over low-performing schools four years ago, one question has persisted: What is the endgame?

More specifically, at what point does the special state-run district known as the Achievement School District, or ASD, release control of schools back to local districts? And when will the ASD cease to exist — or will it live forever? In either case, exactly who should have authority over the ASD and its schools?

The questions are among several to be explored Wednesday when state lawmakers convene for their annual summer study session on education, designed for a deeper look at some of the more complex school-related issues that the legislature passed on earlier this year.

During this year’s session, lawmakers filed a flurry of bills to limit the ASD. The proposals didn’t go anywhere, but leaders of two House education committees pledged a closer look this summer during the off-season.

“What we’re trying to do is sit down and really listen to both sides,” said Rep. Mark White, a Memphis Republican and chairman of a key subcommittee. “This is not about who likes the ASD, who doesn’t. It’s about what are the issues, and how can we work together.”

Questions about the ASD’s future are particularly of interest in Memphis, home to all but two of the state-run district’s 33 schools, most of which were formerly with Shelby County Schools. Representatives of both the local and state districts will be in Nashville for Wednesday’s discussion.

Here are the main questions lawmakers will take up:

What progress has been made? The main source of debate is whether the ASD is succeeding in its mission to transform the state’s lowest-performing schools to top performers. Critics say the academic gains have not been significant enough to justify further expansion of the state-run district, as highlighted by its schools’ continued poor academic performance. Schools taken over by the ASD in 2012 are still among the state’s lowest-performing schools. And a Vanderbilt study released in 2015 suggested that, so far, the ASD hasn’t made any statistically significant gains overall, meaning that students at ASD schools are performing mostly at the same low levels that they likely would have had they remained with their local district.

Proponents point to gains in science and math in 2015, and predict that the greatest academic gains are yet to come. They say that progress is not reflected in ASD schools’ test scores, but in the test scores of all priority schools’ scores statewide, which have improved overall since the ASD’s inception. Even Shelby County Schools Superintendent Dorsey Hopson has said that the Innovation Zone, his local district’s heralded school turnaround initiative, wouldn’t have made gains so quickly without the sense of urgency inspired by the ASD. “Some of these discussions we’re having, they were not happening 10 years ago,” said Robert S. White, the ASD’s chief of external affairs. “The reality is people are paying attention to (low-performing schools) now.”

When and how will ASD schools return to their local districts? According to state law, schools must stay in the ASD for at least five years. If they’ve improved consistently and climb out of Tennessee’s bottom 5 percent, plans can proceed to return those schools to their local district. But specifics of the return process are murky. The ASD’s first six schools are getting closer to the five-year benchmark, and Shelby County Schools and Metropolitan Nashville Public Schools — the only two districts impacted thus far — are hungry for answers about next steps.

Will ASD schools that consistently underperform be closed? State law includes provisions for intervening in low-performing schools in local districts, but not for schools in the ASD that continue to be low performers. Critics of the ASD want to know if the state eventually will step in again on those schools.

What oversight does the ASD have? Ultimately, the state has authority over the ASD. Whereas all other Tennessee schools are controlled by elected local school boards, the  Achievement School District answers to state officials. Lawmakers created the districts and can make changes to how it’s operated  — the reason for the summer discussion in the first place. This state-controlled design is based on the idea that taking out the local level of government oversight would give charter schools more flexibility to meet the needs of their students. But since its inception, critics of the ASD have questioned how the state-run district can be held accountable to the public without regular meetings open to the public and without an obvious role for community members in decision making.

How will the ASD expand? In 2017, the ASD won’t add any new schools because of this year’s rocky rollout of TNReady, the state’s standardized test for math and reading, which determines which schools are eligible for ASD takeover. But in the future, it’s unclear if the ASD will continue to focus on Memphis, or if it will shift to Nashville and Chattanooga, the cities with the next highest concentration of bottom 5 percent schools. Also up for debate is whether the ASD should be allowed to expand at all until the district’s current schools show more dramatic test score gains.

Are communities better off with schools under ASD control? ASD leaders maintain that community engagement is key to their organization, and many of their schools have health and counseling services meant to help both students and their entire families. But the district still battles the perception that its work is unwanted in local schools, even though parent surveys are favorable in the schools they’ve taken control of, according to ASD leaders. Still, each year, the state’s announcement of new school takeovers is met with protest from parents and teachers who say the state is undermining local control and using their students as guinea pigs in turnaround efforts.