Between this link and the traitlet docs on github and just playing around, I finally figured this out:
w = widgets.Dropdown(
options=['Addition', 'Multiplication', 'Subtraction', 'Division'],
value="Addition",
description='Task:',
)
def on_change(change):
if change['type'] == 'change' and change['name'] == 'value':
print("changed to %s" % change['new'])
w.observe(on_change)
display(w)
Overall this looks a lot richer than the deprecated interface, but it could definitely use more examples.