Williamson Co TN records of the school census for that county are online but not yet indexed at FamilySearch.org. I was extremely glad to find them as I'm always looking for more clues for my 2nd Great Grandfather William Clifford Cook and I knew he was on the 1838 tax list in District 25 of Williamson Co TN.
The School Census section of that film begins on image 934 of 2937
The School census start in 1838 with reports from each of the districts which list the name of the adult and how many children from their household are in the age range for attending schools. The range changes a bit over the first few years. In the beginning it was children over 6 and under 16 but by 1842 it appears to change to above 6 and under 21. The Districts weren't always in order so I had to read over them to seek out the 25th District which I wanted to check.
A check of the years 1838-1849 gives me the following results
My William C Cook is only on the 1841 School Census listed as having one child over 6 and under 16 as of 30th of June of that year. That one child in the age range for school is Wm C Cook's oldest daughter Minerva Cook. This may be her first year of school. Wm C Cook was on the Bedford Co TN Dist 10 tax lists for 1836 & 1837. He wasn't listed in Bedford Co TN on the 1838 list but rather in Williamson Co TN as I previously stated.
By 1844, there is no longer a listing for District 25 due to county boundary changes which seem to already be a bit "hazy" to begin with. Years 1850-1853 do not appear with the records but they do resume in 1854.
Below is my attempt at a transcription of Williamson Co TN Dist 25 School Census
1839 (nothing located for Dist 25)
Williamson County (Tennessee). County Court Clerk. (1998). School Census 1838-1939. Salt Lake City, Utah: Filmed by the Genealogical Society of Utah. imaged at FarmilySearch https://www.familysearch.org/ark:/61903/3:1:3Q9M-CS1W-9Q86-J?mode=g&cat=823595