Which branch of government has the biggest role in shaping school policy and practice?
Under the Constitution, the federal government is actually granted very limited influence over education policy. However, under the due process and equal protection clauses of the 14th Amendment, the federal government is allowed to pass some rules to regulate education policies. It was under the 14th Amendment that we made our greatest education policy changes, such as desegregation in public schools, which was a result of the Supreme Court case Brown vs. Board of Education of Topeka. Congress also has the authority to pass legislation affecting education policies under the "Commerce" and "General Welfare" clauses of the constitution. It's under these clauses that Congress has passed many statutes that have been applied to education policy, such as Title IX of the Education Amendments of 1972. Under Title IX, Congress also passed The Equal Access Act of 1984, granting all students--young, poor, and minority--equal access to education. It's also under Title IX that such acts as the Elementary and Secondary Act were passed, an act President Lyndon B. Johnson pushed for to try and breach some of the educational gaps between poor and middle class children ("The Role of The Federal Government in Public Education"). The act allocates funds to schools, plus establishes high standards for schools without establishing a national curriculum. It is under this act that President Bush pushed for his No Child Left Behind Law.
However, while major educational policy changes certainly can be initiated at the federal level, it's really the state and local governments that have the most power over implementing educational policies. Of course things vary per state, but many states leave educational policy making to the legislative branch of the state government. Such policies can include school funding, school health and safety standards, curriculum and graduation requirements, how to handle certain crimes like assault in schools, reporting child abuse, reporting student drug or alcohol use, and school labor laws. Most states will also create state boards of education who are responsible for creating legislation related to "certification of school personnel, curriculum, pupil attendance and transportation, and special education" ("The Role of the State Government in Public Education").
Hence, while the federal government under the legislative branch has passed some legislation that has heavily affected education policy, it is the state government legislative branches, especially the state boards of education, that play the biggest role in most education policies.
In most circumstances, each State's government has the power to set and maintain education standards. They are able to make decisions about education with minimal interference from the federal government. From there, most states are separated into school districts or zones. These districts are given guidelines and rules that they must follow, but have free will from there.