Helping students to construct understanding of physics concepts and develop good problem solving skills is a central goal in many physics classrooms. This thesis examines students' problem solving abilities and explores strategies to scaffold students' learning. In studies involving analogical problem solving between isomorphic problems, we evaluate introductory physics students' abilities to learn from provided solved problems and to transfer their learning to quiz problems involving the same physics principles but different surface features. Findings suggest that providing solved problems after students have attempted to solve quiz problems without help is a good way to scaffold students' analogical problem solving. Categorization of problems based upon similarity of solution provides another angle to evaluate and scaffold students' ability to reflect on problems' deep features. A study on categorization of quantum mechanics problems reveals that faculty overall perform better than students. However, unlike categorization of introductory mechanics problems, in which categories created by faculty are uniform and based on fundamental principles, quantum mechanics categorization is more diverse and based on concepts and procedures. In addition, we also explore possible strategies to help instructors improve their teaching of problem solving and to assess student difficulties more efficiently. Investigating how teaching assistants (TAs) design problem solutions, we find much room for improvement as TAs don't necessarily notice all solution components recommended in the research literature. Another study involving comparison between different assessment tools reveals that carefully designed multiple-choice questions can reflect the relative performance on the free-response problems while maintaining the benefit of ease of grading, especially if the multiple-choice question choices are weighted to reflect the different levels of understanding that students display.