Following the award of his doctorate, Begle taught for the year 1940-41 at Princeton. He was then awarded a National Research Council Fellowship to spend the year 1941-42 back at the University of Michigan. In 1942 he was appointed as an Instructor at Yale University where he spent the next nineteen years. He was promoted to Assistant Professor in 1944, then to Associate Professor in 1949. His early publications were in topology: Homology local connectedness (1941); Locally connected spaces and generalized manifolds (1942); Intersections of contractible polyhedra (1943); Regular convergence (1944); and Duality theorems for generalized manifolds (1945). He continued to produce papers of significance such as A note on local connectivity (1948), Topological groups and generalized manifold (1948), A note on S-spaces (1949), A fixed point theorem (1950), and The Vietoris mapping theorem for bicompact spaces (1950). In the last mentioned paper, Begle generalised the Vietoris homology theory of compact metric spaces to the bicompact Hausdorff spaces by considering their finite open coverings instead of the metric.
However, his interests steadily turned away from research and towards teaching. Charles E Rickart writes :-
Ed and I first taught Algebra to Army ASTP students, and the Calculus to Yale freshmen, spending hours discussing the problems of that teaching. Out of this grew Begle's elementary calculus text, which was unique at the time in that it contained serious mathematics written not for colleagues but for the students themselves ...This textbook was Introductory Calculus, with Analytic Geometry which Begle published in 1954. He writes in the Preface:-
This text differs from most others in this field in that it treats calculus as a branch of mathematics rather than as a mere adjunct of the physical and engineering sciences. ... We start with a list of axioms and show how the theorems of the calculus are derived from these axioms. ... Our aim in presenting calculus in this fashion is to give the student more of an understanding of the basic concepts of the subject than is usually done in an introductory course. ... The mathematical techniques employed [in science and engineering] have become so numerous and varied that no student can be expected to master more than a few of them without a framework of theory around which they can be organized.D T Finkbeiner begins a review of the textbook as follows:-
With new calculus books appearing each year, it is the rare instructor who has the time, the inclination, or the opportunity to examine all of them. Equally rare is the book which makes a real contribution to understanding of calculus at the first year level. In the reviewer's opinion Begle's book does make such a contribution and should be examined carefully by all instructors of beginning calculus.Begle also played major roles in what one might refer to as the 'administration of mathematics.' In 1951 he was elected secretary of the American Mathematical Society and undertook the duties associated with this important position with great diligence for the next six years. In 1958 two conferences were held under the sponsorship of the National Science Foundation both with the aim of making recommendations on how the school mathematics curriculum might be improved. The outcome of these conferences was the setting up of the School Mathematics Study Group (SMSG) to undertake a study of curriculum development. Begle was offered the position of director of the SMSG which he accepted with enthusiasm. The project was based at Yale, where Begle was based, and he was able to quickly bring together many of differing opinions all of which he channelled in a positive way with his skilful chairmanship. Donald E Richmond writes :-
Ed guided without dominating. His exceptional integrity ensured that SMSG was administered with scrupulous honesty. ... Throughout, Ed dictated no solutions but strove to harmonise the opinions of all. ... no curriculum was to be imposed on any school system.In 1961 Begle left Yale to become both a professor in the Department of Mathematics and a professor in the School of Education at Stanford University. However, he became more and more involved in mathematical education, particularly with his work as director of the School Mathematics Study Group.
Brian Thwaites was involved in school mathematics curriculum development in England. He wrote :-
Not being a pure mathematician, I had not heard of Professor Begle in his younger days as a creative researcher in his own subject. But as soon as I and a few others in England began to gather together in the late 1950s to talk about the deteriorating situation of mathematics and especially about the need to reform the content of school mathematics, the name of Begle sprang out of nowhere, so to speak, apparently heading every list and every communication which came to us from other countries. ... So by 1961 he and his work with the SMSG were already well known to us. ... And that was the year of his visit to this country as a participant of - and (little did he suspect) a guide of incalculable value to - the Southampton Mathematical Conference 1961. ... how, you may ask, have we in the UK viewed Professor Begle's achievements as head of that (to us) vast organisation of the School Mathematics Study Group? First he led the way to team authorship, a startling innovation for conservative Europeans, but one which we adopted with fervour and moulded to our own fashion. ... Second, the sheer drive and administrative skill with which he prosecuted the SMSG's activities were abundant causes for astonishment and admiration on this side of the Atlantic. ... Third, ... groups all over the world would constantly be referring back, or across, to what was going on in Stanford. Whether or not there was invariable agreement is beside the point: what mattered was the sense that something very definite was happening under Ed's leadership. Fourth, ... one felt that there was no sense of competition in him; rather that he was carrying out some preordained strategy whose merits would ultimately be tested by the experience of millions of children and their teachers.Among the many notes Begle wrote on mathematical education, let us mention: The School Mathematics Study Group (1958), Comments on a note on "variable" (1961), Remarks on the Memorandum 'On the Mathematics Curriculum of the High School' (1962), A Study of Mathematical Abilities (1962), The Role of Axiomatics and Problem Solving in Mathematics (1967), and SMSG: The first decade (1968). In  this picture of Begle is painted:-
He was often seen pacing the University corridors with dark eyes shrouded in thought, a tall man with a full white beard who liked to walk alone. Edward Begle was an independent man. His voice will be missed in the School of Education where a well-reasoned paradigm might resolve a tangled faculty discussion. It was the mathematician in him that made him impatient with untidy thought.The authors of  also paint this picture of Begle's home life:-
In their home, Edward Begle and his wife, Elsie, were generous hosts. The big, handsome house on Bryant Street in Palo Alto always offered a warm reception, reflecting the kind of family life the Begles had. The dark, oak-paneled study where Edward Begle did most of his writing was lined with pictures of the seven children: Cornelia, Sarah, James, Emily, Elsie, Edward, Douglas.Begle received many awards for his contributions to mathematical education including election as a fellow of the American Association for the Advancement of Science (1960) and the Award for Distinguished Service to Mathematics made by the Mathematical Association of America (1969). Part of the citation for this award reads :-
With the assistance of many individuals and components of the mathematical community ... he has conducted a national experiment, unprecedented ... in its combination of depth, scope and size. He has done so with character and courage, with good judgment and balance, with understanding and endurance, and in a continual searching for the first rate. He is a mixture of Welshman, New Englander, American, mathematician, teacher, and sachem, and we are all in his debt. The mathematical and scientific part of American life in the middle third of this century may well be judged to have been outstanding; if the history of it is ever properly written, E G Begle's role therein will clearly have an exceptional place.He served on the Board of Directors, the Finance Committee, and the Research Advisory Committee of the National Council of Teachers of Mathematics. In addition he served on the Committee on the Undergraduate Program of the Mathematical Association of America, was chairman of the Conference Board of the Mathematical Sciences, and was a trustee of the American Mathematical Society. He served on the United States Commission on Mathematical Instruction, from 1962 to 1966 and again for a second term from 1970 to 1975. He was Chairman of the Commission from 1963 to 1966. He served on the Executive Committee of the International Commission on Mathematical Instruction from 1975 to 1978. He also received the Rosenberger Medal from the University of Chicago (1971).
Begle died of emphysema at the age of 63 while working on a review of research in mathematics education.
Article by: J J O'Connor and E F Robertson