Robert Woodhouse attended school in North Walsham. In 1790 he was admitted to Caius College, Cambridge where he became a fellow from 1798 to 1823. Woodhouse was Lucasian professor of mathematics from 1820 to 1822, then Plumian professor of astronomy and experimental philosophy from 1822 to 1827. He was made a Fellow of the Royal Society in 1802.
Woodhouse was interested in the theoretical foundations of the calculus, the importance of notation, the nature of imaginary numbers and other similar topics. He wrote an important work Principles of Analytic Calculation in 1803 defending Lagrange's calculus method based on series expansions. This support of Continental methods was aimed at his fellow professors at Cambridge but it had little effect. Peacock, however, considered his work of major importance.
Woodhouse, therefore, failed to have much influence as a reformer in mathematical studies at Cambridge but he wrote several widely used elementary texts which, during his lifetime, brought him more fame.
Woodhouse's other works include History of the Calculus of Variations (1810), Treatise on Astronomy (1812) and a work on gravitation published in 1818.