Greenwood completed a
PhD titled
Journeys into A Third Space: A study of how theatre enables us to interpret the emergent space between cultures at
Griffith University. Greenwood then joined the faculty of the University of Canterbury, rising to full professor. She is director of the Research Lab for Creativity and Change. She was appointed professor emerita in 2018. Greenwood taught drama in primary, secondary and tertiary education settings for twenty years. She has written a number of resources for teachers, and was involved with the Te Mauri Pakeaka project. This project, created by
Arnold Wilson, ran from the late 1970s until 1988, and was an educational programme aimed at making a 'third space' for Māori and
Pākehā cultures to come together in the creation of art. Greenwood was publications officer for the New Zealand Association for Drama in Education. She is also a playwright and poet. One of Greenwood's notable doctoral students is Professor
Jo Fletcher, in the School of Teacher Education at the University of Canterbury. == Awards and honours ==