National schools evolved after the sucsess of British Schools which were started in 1810. The National Society for the Education of the poor was formed in 1811, and there were 17,000 schools by 1851.
Schools provided before 1902 for mass education were sponsored either by the Church of England through the National Schools Society and therefore called National Schools, or by non-conformists through the British & Foreign School Society, and thus called British Schools. Such schools were all largely maintained by voluntary effort, but from the 1830's could attract Government grants. Eventually, government began to impose administrative requirements on those schools receiving grants, such as inspection, higher performance, compulsory attendance, and a compulsory leaving age. Ultimately the 1902 Education Act brought all these schools under state control, to be administered by the appropriate local authorities.
More information on the history of Peterborough schools:
http://www.peterborough.gov.uk/leisure_and_culture/libraries/local_studies_and_archives/online_info_on_collections/a-z_guide/q-z/schools.aspx
In the 1851 census (Peterborough District 8) there is no mention of the school. Ann Chappel, 76, an annuitant lives 2 doors away from the Blacksmith. Frances Phillipp, 65, has occupation school mistress, she is a widow living with her son John 33, a gardener. Elizabeth Clark 17, unmarried and living with her parents (Luke Clark, 50, Cattle Dealer & Mary A Clark 49), is described as a governess.
In 1861 Census the National School is 2 doors away from the Blacksmiths, but there is no one living on the premises, and no one with the profession teacher in Peterborough District 19 which covers Dogsthorpe and Garton End.
In the 1871 census (Peterborough St Paul District 23) the National School is simply marked as an uninhabited building 2 doors away from the blacksmith. There is Mary Bailey 42, a school mistress and grocer, living in Garton End. She was born in Peterborough, but her 2 children aged 5 and 3 were born in Sutton Surrey. Susan Plant, 15, is living in Dogsthorpe Lane, New England, with her father a brickmaker and the rest of her family, and has the occupation teacher at school.
In the 1881 Census (Peterborough District 32) Mary Bailey is still living in Garton End, at the grocers shop next door to the Comet. She is no longer head of the household her husband George has joined her, his occupation is shopkeeper, no mention of school mistress.
In the 1911 Census
Jessie Brown 15, daughter of the landlord at the Bluebell Inn is described as a School Teacher working for the County Council.