Charles and Ruth had six children and Ruth soon made use of her training as a school teacher, becoming an enthusiastic teacher for her own children. Until she was seven years old Ellen was taught at home by her mother. She not only learnt to read and write, but she also learnt some astronomy and botany. Her grandparents also were strongly committed to education, and for the times we are talking about, it was certainly most unusual for a girl to be brought up in an atmosphere created by both parents and grandparents equally encouraging education of the boys and girls in the family.
At age eight Ellen was sent to Centreville school. After completing her own school education, Hayes taught in a country school while she put aside money from her wages to allow her to attend College. After teaching for five years, she entered Oberlin College, Oberlin, Ohio in 1872. Hayes' choice of Oberlin College was a fairly natural one. The College, founded as Oberlin Collegiate Institute in 1833 to train school teachers, accepted both men and women from the beginning and it was the first College in the United States to have this coeducational policy. It had been called by its present name of Oberlin College for more than twenty years before Hayes began her studies there.
She spent some time preparing for her degree studies which she began in 1875, graduating with a B.A. in 1878. She took a range of subjects at Oberlin College; mathematics and science were her major topics but she also studied arts type subjects such as history, English literature, Greek and Latin.
After graduating Hayes taught for a year at Adrian College in Adrian, Lenawee county, southeast Michigan which had been founded in 1845. Then in 1879 she was appointed to Wellesley College. This private women's college in Wellesley, Massachusetts, had only opened four years before Hayes was appointed there and it was remarkable for a women's college in the United States since it was the first to have scientific laboratories.
Her most important pieces of original scientific work were done in astronomy. In 1887-88 Hayes observed at the Leander McCormick Observatory of the University of Virginia. She made observations of one of the minor planets (Minor Planet 267) and calculated its orbit. She also made observations of a comet in 1904 which were published in Nature in May 1904.
Hayes remained at Wellesley College from her first appointment in 1879 until her retirement in 1916. It was a career that saw Hayes become professor and Head of the Mathematics Department in 1888. Then, in 1897, when Wellesley College started up a new Department of Applied Mathematics she became its first Head. In 1904 the Department of Applied Mathematics was extended and became the Department of Astronomy and Applied Mathematics, with Hayes still as its Head. During her years at Wellesley College, Hayes wrote a number of fine textbooks: Lessons on Higher Algebra (1891); Elementary Trigonometry (1896); Algebra for High Schools and Colleges (1897); and Calculus with Applications, An Introduction to the Mathematical Treatment of Science (1900).
Louise Brown  was a student of Hayes. She wrote:-
It was in her teaching that [clear thinking, quiet enthusiasm and rare culture] had the remarkable effect of stimulating mental activity that surprised even her students themselves. My courses with her included Calculus, Celestial Mechanics, Logic, and Astronomy. Emphasis was placed on the nature and significance of natural law, on the need of clear definition of terms, the nature and use of evidence, conditions under which authority must be relied upon or rejected. These principles were later embodied in one of the best of her publications entitled "How Do You Know?" Her power as a writer, like her power as a teacher, was due to her clearness of thought and rare command of English. These qualities served her as well also in her public addresses in Boston and elsewhere, where her audiences responded with spontaneous enthusiasm to the quiet personality on the platform.However Wellesley College, however, did not find Hayes' views on education and politics easy to live with. She championed the rights of women to education, the rights of women to vote, and also trade union rights. These were not views which pleased the establishment and they led to threats being made against her and also to her arrest at one point. Her political views were left-wing and she regarded herself as a socialist but not a Communist. In fact she disagreed with many of the doctrines of Communism. Whitman  writes of Hayes' radical views:-
A dauntless radical all her days, in the eighties she was wearing short skirts; in the nineties she was a staunch advocate of Woman's Suffrage; in the first two decades of the twentieth century, an ardent Socialist. After her retirement, and until her death in 1930, she was actively connected with an experiment in adult education for working girls. Fearless, devoted, intransigent, fanatical, if you like, and at times a thorn in the flesh of the trustees, who withheld the title of Emeritus on her retirement, she is remembered with enthusiasm and affection by many of her students.Getting views across required a sympathetic press and certainly the press were not sympathetic to Hayes. She was far too determined a fighter to let that stop her, however, so after she retired she started up her own newspaper Relay which was published monthly. In her own words:-
... the Relay plans to camp in a hut by the side of the road and to keep a lamp or two burning - in the hope of being a friend to wayfarers and especially to the limping Under Dog.
Article by: J J O'Connor and E F Robertson