Ellen graduated from Oberlin College in 1878.
A year after graduating she became principal of the women's department of Adrian College, Michigan.
From 1879 to her 1916 retirement, she taught at Wellesley College, where she became head of the mathematics department in 1888.
Not only was Ellen a noted mathematician, she was also known for her knowledge in astronomy. She was the author of many math textbooks. "The Sycamore Trail" 1929, a historical novel in which she described "Teachers as Trail Makers" and "Wild Turkeys and Tallow Candles" 1920, which was an account of life in Granville, Ohio.
Hayes was a controversial figure not just for being a rare female mathematics professor in 19th century America, but for her embrace of radical causes like questioning the Bible and gender clothing conventions, suffrage, temperance, socialism, the 1912 Lawrence Textile Strike, and Sacco and Vanzetti. She was the Socialist Party candidate for Massachusetts Secretary of State in 1912, the first woman in state history to run for statewide office. At that time, of course, women were not allowed to vote for her. She did not win the race, but did receive more votes than any Socialist candidate on the ballot, including 2500 more than their gubernatorial candidate.
Hayes was concerned about under-representation of women in mathematics and science and argued that this was due to social pressure and the emphasis on female appearance, the lack of employment opportunities in those fields for women, and schools which allowed female students to opt out of math and science courses.
In her will, Ellen left her brain to the Wilder Brain Collection at Cornell University. Her ashes were buried in Granville, Ohio.