Greenwood's parents emigrated from Samoa to Whanganui in New Zealand, shortly before Greenwood was born; they moved from there to Auckland when she was a child. She earned a bachelor's degree at the University of Auckland, and after some time in Australia became a secondary school teacher in Auckland.[2]
Returning to the University of Auckland for graduate study in mathematics, she earned a master's degree and then completed her PhD in 1999, under the joint supervision of David Gauld and David W. Mcintyre. Her dissertation was Nonmetrisable Manifolds.[2][3][4] She and three other students who finished their doctorates at the same time became the first topologists to earn a doctorate at Auckland.[2]
After postdoctoral research, funded by a New Zealand Science and Technology Post-Doctoral Fellowship, she obtained a permanent position at the University of Auckland as a lecturer in 2004,[2][5] later becoming an associate professor.[2] Beyond mathematics, her work at the university has also included advocating for the interests of Pasifika and Māori students.[2][5]
