This week’s quotes for teachers:
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
There are many things you can do with problems besides solving them. First you must define them, pose them. But then, of course, you can also refine them, depose them, or expose them, even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found new goal they do lead to. It’s called play.
Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings.