Stephen Hawking's parents lived in London where his father was undertaking research into medicine. However, London was a dangerous place during World War II and Stephen's mother was sent to the safer town of Oxford where Stephen was born. The family were soon back together living in Highgate, north London, where Stephen began his schooling.
In 1950 Stephen's father moved to the Institute for Medical Research in Mill Hill. The family moved to St Albans so that the journey to Mill Hill was easier. Stephen attended St Albans High School for Girls (which took boys up to the age of 10). When he was older he attended St Albans school but his father wanted him to take the scholarship examination to go to Westminster public school. However Stephen was ill at the time of the examinations and remained at St Albans school which he had attended from the age of 11. Stephen writes in :-
I got an education there that was as good as, if not better than, that I would have had at Westminster. I have never found that my lack of social graces has been a hindrance.
Hawking wanted to specialise in mathematics in his last couple of years at school where his mathematics teacher had inspired him to study the subject. However Hawking's father was strongly against the idea and Hawking was persuaded to make chemistry his main school subject. Part of his father's reasoning was that he wanted Hawking to go to University College, Oxford, the College he himself had attended, and that College had no mathematics fellow.
In March 1959 Hawking took the scholarship examinations with the aim of studying natural sciences at Oxford. He was awarded an exhibition, despite feeling that he had performed badly, and at University College he specialised in physics in his natural sciences degree. He only just made a First Class degree in 1962 and in  he explains how the attitude of the time worked against him:-
The prevailing attitude at Oxford at that time was very anti-work. You were supposed to be brilliant without effort, or accept your limitations and get a fourth-class degree. To work hard to get a better class of degree was regarded as the mark of a grey man - the worst epithet in the Oxford vocabulary.