Cartwright studied mathematics at St Hugh's College, Oxford, graduating in 1923 with a first class degree. She was the first woman to attain the final degree lectures and to obtain a first. She briefly taught at Alice Ottley School in Worcester and Wycombe Abbey School in Buckinghamshire before returning to Oxford in 1928 to read for her D.Phil. Cartwright was supervised by G. H. Hardy in her doctoral studies. During the academic year 1928–9 Hardy was at Princeton, so it was E. C. Titchmarsh who took over the duties as a supervisor. Her thesis "The Zeros of Integral Functions of Special Types" was examined by J. E. Littlewood, whom she met for the first time as an external examiner in her oral examination for that 1930 D.Phil.