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According to Susan Lehmann's Institutions in Santa Cruz County -- 1850-1950:
The first educational institution in the County was Santa Cruz Mission where Indian neophytes were taught the Catholic religion and children of Spanish settlers learned to read, write and keep accounts. By 1847, a small contingent of English speaking families has come to Santa Cruz and, although Spanish was still the official language, they wanted their children to learn reading and writing in English. One such settler, Mary Amney Case, who arrived with her husband and child in 1847, held classes in her home in 1848 and 1849, thus becoming the first English school of record in Santa Cruz County. Following California statehood in 1850, more families with school age children came to the area and local churches were pressed into service as school buildings. Methodist churches in the city of Santa Cruz and Watsonville served this purpose until 1857 with teachers being paid by collecting funds from the parents of students. With an ever increasing number of children, residents of the county soon campaigned for a public school with its own school house. According to the history of the Santa Cruz County school system written by Margaret Koch in 1978 and published by the Santa Cruz County Office of Education, the first one room school house in the city was built in 1857 when a $400 lot was purchased by community subscription on Mission Hill.
It seems that the first school (as opposed to lesson) was the Mission Hill School.