As a tenure-track professor, my colleagues visit my class every semester to better get to know me. Aside from the fact that this is an evaluation with professional consequences for me, I find this to be a great practice when done correctly—in fact, I wish that tenured professors would make a habit of visiting each other’s classes. It is easy to become too familiar with one’s own teaching, and it is good to periodically have an outsider question your choices.
I recently had a visitor, and he was concerned about my homework assignments. It was a blessing that we had this conversation, because it reminded me about why I make some of the decisions that I do; I have internalized these, but it is good to make them explicit from time to time.
- I believe that any sort of upper-level mathematics that is worth doing is worth taking your time on. Upper-level mathematics is about ideas and problem-solving, not computations and memorization (for the most part). I believe that a well-written homework assignment should require that the student take time to complete it; it should require stops and starts, dead-ends, and sometimes flashes of inspiration. These problems cannot be churned out, factory-like, on a schedule. Because of this, I eschew the practice of giving proofs after every class period; rather, I prefer to give students a “cycle” (6 days—recall that my college/s is/are weird) so they can dwell, fail, try again, and ultimately succeed.
- I believe that the teacher’s job is to make him/herself expendable. I used to revel in the fact that my students would come to my office hours on a daily basis for help. However, I no longer think this is in the best interest of the student. While it does make me feel good to feel needed, I have (in the past) had a tendency to set up co-dependent relationships with my students, where the students never feel like they can do the mathematics outside of my presence. This, in my opinion, is the opposite of education, and I have been improving on this by leaps and bounds over the past eight years. My latest tool is cooperative learning, which has been considered a great success by my students. A second success is giving my students more flexibility in when the homework is done; again, I have found that giving students homework every cycle creates less dependence than making homework due every lecture—even if the amount of homework is the same in both cases.
- I believe in quality over quantity, if I must choose. I prefer to give fewer homework problems while expecting a higher quality (and giving a higher quality of feedback) than more homework problems of lower quality.
- I believe that mathematics homework is not just about mathematics; rather, this is an opportunity to improve writing skills, which will ultimately be used more than the mathematic skills for 99% of my students.
One of my main points is that I do not think that having homework due every lecture meshes with my values, but this does not mean that I do assign homework that is due the next lecture period. Rather, I reserve these homework questions for simple computational problems or problems that apply a definition.
These are ideas that I have come to in my years of teaching, although I make no claim that these are optimal. I prefer weekly/cycle homework to daily homework for proofs, but I feel like I could easily change my thinking on this if I hear a good argument. I would appreciate suggestions and other people’s rationales in the comments.