ponder on

ponder on (something)

To give great consideration to something; to ruminate or think about something very deeply. We sat pondering on various ways to improve our business model when Janet suddenly came up with a brilliant idea. Well, you're review has certainly given me plenty to ponder on.
See also: on, ponder

ponder (up)on something

to think on something; to consider something. (Upon is formal and less commonly used than on.) Ponder upon this awhile. See what you come up with. I need to ponder on this.
See also: on, ponder