F. Matthias Alexander (1869-1955) was an Australian who developed the educational process that is today commonly called the Alexander Technique - a method of helping people learn how to improve their posture, balance and coordination. The two men met around 1918 in New York City when Dewey had a series of lessons with Alexander. These, and subsequent lessons and conversations and correspondence, had a profound impact on Dewey. He felt that his lessons taught him how to stop and think before acting and enabled him to hold a philosophical position calmly once he had taken it or to change it if new evidence appeared.
A useful website can be found at http://www.alexandertechnique.com/articles/dewey