Goodstein then did research at Cambridge on transfinite numbers under Littlewood's supervision. After receiving his Master's Degree he accepted a post in pure and applied mathematics at Reading University in 1935. During the war years he had to cover teaching in a wide range of pure and applied mathematical topics, among them engineering, applied mathematics, analysis and group theory.
While undertaking this strenuous teaching load Goodstein carried out research which he submitted for a doctorate to the University of London in 1946. He was appointed professor at University College, Leicester, in 1948 and he remained there for the rest of his life. At Leicester :-
... he presided over an expansion from a mathematics staff of six to 23 at the time of his retirement in 1977, with a corresponding increase in student numbers.At Leicester Goodstein played a full role. He was Dean of Science at the critical time of the transition to full university status which occurred in 1957. From 1966 to 1969 he served as Pro-Vice-Chancellor to Leicester University.
Goodstein worked on mathematical logic, in particular ordinal numbers, recursive arithmetic, analysis, and the philosophy of mathematics. He was extremely interested in the teaching of mathematics and  lists 66 papers which he published on teaching mathematics at both school and university level.
His eleven textbooks were, according to :-
... characterised by their clear style and ingenious methods to elucidate difficult points. He was disappointed that his mathematical analysis text ... which presented a novel approach to elementary differential and integral calculus ... did not find favour.It is interesting to note that he was the first mathematical logician to hold a chair at a British University and he exercised an important influence on the development of this topic in the UK :-
A private and essentially shy person, Louis Goodstein will be remembered for great courtesy and generosity towards colleagues and students.We should also note Goodstein's excellent work for the Mathematical Association :-
... he was instrumental in arranging for the transfer of its library and later its headquarters to Leicester, and contributed nearly 70 notes as well as hundreds of reviews to the Mathematical Gazette, which he sustained at a high academic level during his editorship from 1956 to 1962.
Article by: J J O'Connor and E F Robertson