How do I inspire my students to work through problems on their own?
How do I inspire my students to work through problems on their own?


By giving students assignments that they truly work on their own, with minimal hand-holding, then assigning the appropriate grade. Some students will be upset enough with their less-than-desirable grades such that they become "inspired." The other students who fail to keep up with the pace of the course will earn the grades which truly reflects their lack of ability/skill/etc. 

Sitting down with the under-performing students, and explaining that they may benefit, for example, from retaking the necessary prerequisites may be in order. It's not easy to tell a junior/senior that they should probably go back and retake a sophomore-level class, but that's life, and it's better for everyone in the long run.

Although I've not been in my current faculty position for very long, I've already seen the consequences of dumbing down the assignments, and the effect that it has on the quality of the follow-on courses, both in student capabilities and my ability to teach the things that need to be taught at the appropriate pace. 

The way I see it, you can either make your assignments easier, or be tough: I have (somewhat reluctantly, at first) grown to prefer the tough route.