The government needs to control education so that standards and effective teaching is accomplished. This issue is best left to the state and local governments because they have the most intimate and best knowledge about the schools they are working with. The federal government tends to mess things up in schools, though.
The federal government should have certain standards of ethics and practices set forth for the entire country. It currently achieves this through requirements for grants and funding mostly. States are allowed to run their own schools, and no federal funding or interference is really necessary. However, if the federal government offers a $40,000 annual grant to have a PE class mandatory for all high school students, there will likely be a PE class in high school.
I think the Government should be able to set standards and funding levels for the nations education system, I think allowing those to be set by individual school districts would bring about a very uneven education system with some parts being dramatically better then others and others being dramatically lower then others.
Yes, the government should have control, to an extent, over the nation's educational system. There would be more equal funding and treatment for schools and it would be better for students. Public schools are already funded from taxes anyway, so the government should have some influence if they could do a good job.
Education and the government should not be intertwined in the way they are. For years, people have been insisting that more government intervention is necessary, and as the laws and other legislation stack up, things have only been getting worse. Recent examples of the government's harmful intervention that not only encroach upon our individual freedoms, but are also entirely ineffective. As a student still in school I can tell you that the new common core standards have only made things harder for everyone.
Education should be a provision of the states and different systems and opportunities should incentivize districts to be accountable to their local populations. Federal running of education tries to meet specific needs with broad measures which are wasteful and ineffective at best. States and districts however should be heavily involved in their local educations and push to find ways to improve standards and develop competent and able citizens.