In the digital age, a teacher's role is not simply to present knowledge, but to navigate it. The overwhelming majority of the information that an undergraduate gains in her degree is available for free on the internet or in libraries.
A digital age instructor's job is to guide and motivate students' paths through this information – to provide the vision and context necessary to develop expertise in a field. Introductory courses make up the bulk of students' collective experience with mathematics and statistics, so any expertise to be gained in those one or two courses needs to be a self-contained package.
For example, rigorous proofs serve introductory students very little; the practice of rigor and constructing proofs has little value until upper-division courses. Introductory students learn by doing the tasks that are actually relevant to the course: examples. As such, I prefer to relegate much of the proof work to optional out-of-class readings. The extra instructional time for guided, step-by-step examples makes the material more accessible. It also provides more opportunities to fill the fundamental gaps from high school mathematics that will otherwise prevent understanding. For the few that do continue in a mathematics or statistics major, I feel that what they may lack in experience with proofs is more than compensated by a stronger foundation in the introductory material.
This focus on accessibility extends to my policies on assignments and office hours. Assignments should be vehicles for students to struggle through a set of practice problems and receive formative feedback. However, logistics of providing quality feedback aside, that doesn't work for everyone. Assignments need to have grades attached so students will have extrinsic motivation to completing them, but these same grades penalize mistakes on something that should be practice.
I want assignments to be important and challenging enough to take seriously, but not so much as to tempt plagiarism. In the past, I have solved this by booking extra office hours on the days before assignments are due, and telling my students that I will give them entire solutions to assignment questions. I've found that on these office days, a group of 5-12 students would come to my office with their assignment hang-ups, but that they could answer each others' questions with only moderate guidance from me. Some of these students likely sat in to get their solutions from the rest of the office group, but that's still better than copying written assignments verbatim.
Finally, I try to explicitly declare the 'take-home messages' by including them in my lessons. That is, the few ideas that I hope students will remember long after the final exam is over. These messages include general strategies about the mathematical sciences such as “every hard problem is merely a collection of easy problems”, and George Box's quote “all models are wrong, some are useful.”. If my students are to retain anything from their time spent on coursework, I hope it's something of value and general applicability rather than memories of referring to tables of integrals and probability.