Yes, Jefferson imagined schools as places that teach a broad range of academic basics, such as reading, writing, arithmetic and history. But that's about the extent to which Jefferson would agree with our elementary and high schools.
For Jefferson, there were three stages of academia. The first was elementary or pre-grammar schooling. The second was grammar school ("for teaching Greek, Latin, geography, and the higher branches of numerical arithmetic"). And the third was the university.
But Jefferson -- as well as most of the other Founding Fathers -- couldn't ever imagine that public education would be controlled by the federal, much less a state, government. He believed that it should be run and funded by parents and those in local communities or wards.
In 1784, Jefferson proposed a bill in Virginia that outlined his thinking for a public education system. In it, he planned to divide "every county into small districts of five or six miles square ... and in each of them to establish a school for teaching reading, writing, and arithmetic."
As late as 1816, while still trying to enact his educational plan, Jefferson wrote to Sen. Joseph C. Cabell about the sheer folly of states running the educational system: "If, however, it is intended that the State government shall take this business into its own hands, and provide schools for every county, then by all means strike out this provision of our bill. I would never wish that it should be placed on a worse footing than the rest of the State. But if it is believed that these elementary schools will be better managed by the governor and council, the commissioners of the literary fund, or any other general authority of the government, than by the parents within each ward, it is a belief against all experience."
Jefferson argued for the founding of "each school at once under the care of those most interested in its conduct." Each elementary school was to be supervised by local parents and peers who were passionate to ensure children's education, which included the selection and oversight of teachers (tutors), as well as their salaries. And those who couldn't afford their children's tuitions would be covered, at least in part, by those who could assist them.