This is a topic that is of great interest to me, but I am having a hard time finding some credible sources. I know that there is some debate nowadays in some states about fair/equal funding between rural and urban schools, but it's mostly just news articles rather than books or journals.
Thank you in advance!
As education isn't explicitly mentioned in the constitution, each state set up its own system and structure. As a result, rural schools in the 1950's vary wildly, depending on where in the country you're looking.
One pattern that's interesting to explore, though, relates to the Works Progress Administration (WPA) in the 1930's. Many of the east coast and midwest WPA school construction projects resulted in large, multi-story high schools that served as an architectural testimony to public education. West coast projects were more likely to be one or two story buildings with a larger footprint (more of an "open campus" design.) It's not uncommon to see imposing Georgian Colonial style high schools in the middle of small towns. The typical design encouraged a semi-standardized approach to the school experience; students' days were broken into a handful of subjects, generally covering a liberal arts curriculum, with each subject taught by a different teacher, more likely to be a white woman than not. The high school may have had seperate entrances for boys and girls and definitely had separate bathrooms. By the 1950's, attending high school was more the norm than not. Segregation was also the norm - hallways were typically monochromatic.
Meanwhile, the concept of a high school "dropout" was only beginning to emerge. For many students, school was over when they were done. The idea of graduation as an expected norm wasn't really shared across genders, races, and social classes until the mid-60's. It wasn't uncommon for a student to simply stop going because their parents needed them to work or to care for siblings. White girls were most likely to finish high school, but again, it varied by community. Black students, especially those in Southern rural communities, were often denied access to a high school experience.
Younger students' experiences could range from a one-room schoolhouse with the same teacher for 8 years to a grammar (4 or 5 years) and intermediate (3 or 4 years) experience with a new teacher each year. The later was more likely if the community had been targeted for several WPA projects or if they had made a commitment to public education. But like high school graduation, going to school every day wasn't necessarily expected. This would shift as some state-level school funding was linked to population and attendance and states tightened child welfare laws.
Funding and tax structures also varied across the country. Some rural communities expected local families to board and pay the teacher(s) and taxes only covered physical costs related to school. However, in most rural communities, the nature of school funding was tightly linked to the idea of consolidation. The 1950's marked the apex of the creation of school "districts." Which, again, varied wildly across the country (i.e. the rise of districts in rural Long Island, NY is dramatically different than the rise in rural Virginia.) Unsurprisingly, one of the biggest advocates of rural school consolidation were school bus manufacturers, especially the International Harvester Company, who advertised their buses in their catalogues, knowing that many school board members were farmers.
The best sources for understanding the evolution of rural schools are probably the National Rural Education Association, especially their annual reports and histories of education with a focus on a particular area of the country or state.