Answer: climbing vine
When she is giving a tour to the parents who have kids that are going to Chapman, she says 'A child is like a climbing vine, with enough structure to cling to and the right gardener to tend them'. As she says this, she is in the outside of the school, twirling her cane round one of the vines to mime the quote - which would be very understandable since it is a 'school traditional phrase'.
The actual phrase means 'A child is a growing animal, with enough work to keep their minds on, and the right teachers to teach them.'