This research studies students' functional understanding of magnetic induction, the process by which changing magnetic flux induces electrical current in a circuit. Functional understanding includes the ability to reason both qualitatively and quantitatively about a concept, as well as to translate among multiple representations of the concept. Since this phenomenon has many applications, such as in power generators, microphones, seismometers, and automobile alternators, it is important that students gain some understanding of it from their physics courses. However, this research shows that over half of the students in an introductory physics course for engineers cannot correctly answer conceptual problems about induction on their final exam. There are a number of theoretical reasons why the topic of magnetic induction might be difficult for students to learn. It is comprised of multiple inter-related abstract quantities that are inherently three-dimensional and that are changing in time. In order to improve instruction on these difficult topics, it is necessary to study student learning in detail. Instructional materials have been developed and studied along with initial- and final-states of student understanding.