On completing her studies at Cambridge, Maddison was awarded a scholarship which enabled her to spend the year 1892–93 at Bryn Mawr College in the US.

She used the latter to study at the University of Göttingen in the academic year 1893–1894, where she attended lectures by Felix Klein and David Hilbert.

Maddison attended lectures by Klein, Hilbert, and Burkhardt during her year at Göttingen where she played a full part in the exciting mathematical atmosphere of the department.

She was awarded a PhD in 1896 for her thesis On Singular Solutions of Differential Equations of the First Order in Two Variables and the Geometrical Properties of Certain Invariants and Covariants of Their Complete Primitives and in the same year appointed as Reader in Mathematics at Bryn Mawr.

Her paper On certain factors of c- and p-discriminants and their relations to fixed points in the family of curves which she published in the Journal of Pure and Applied Mathematics was based on her doctoral dissertation and she was awarded the Gambel Prize.

Her will endowed $10,000 in memory of President M. Carey Thomas to be used as a pension fund for Bryn Mawr's administrative staff.

Her will gave a large sum of money in the memory of M. Carey Thomas, who died in 1935, to be used as pension funds for non-faculty members of staff at Bryn Mawr.