With goals and content in mind, I can now focus on how to best get the students to learn the material. One aspect of this is homework.
This is a proof-based course. My theory is that there are three things that need to happen if you are going to learn how to successfully do proofs:
- You must read a lot of proofs.
- You must write a lot of proofs.
- You must analyze the proofs you read.
The third point will largely be done in class, since I do not think I can expect students to know how to analyze proofs. I have several ideas for formats that will allow the students to read and write a lot of proofs:
- I might have students evaluate their own homework. Students would give me a photocopy of their homework, but keep the original for themselves. I would create a solution key/rubric. They would use the rubric to evaluate the homework outside of class; perhaps students could comment on the “differences, omissions, and additions” of their proofs compared to mine, and comment on how important these differences/omissions/additions are. Students would email me their evaluation, noting the strengths and weakness of their proofs. I would spot-check their work by using the photocopied homework.
- I might allow students to resubmit unlimited attempts on homework problems to me. Problems would have two possible grades: “Near-perfect” and “Incomplete.” Students would resubmit until they received a grade of “Near-perfect.” I would provided detailed comments on their proofs to help them with the next draft.
- I might have students evaluate other students’ proofs as part of their homework. I would create a packet of 3-5 student attempts at proofs. Students would be expected to contribute to class discussions on the proofs.
- I might have “homework committees.” This idea comes from from Patrick Bahls. Here, a committee of 2-3 students would look 1-2 selected problems from the homework assignment. This committee would look at all of the student solutions that were submitted, categorize the different approaches that students used, and discuss the relative strengths, weaknesses, and validity of each approach. The committee would give a short summary of what students did in class.
I think a combination of these approaches would work well to get students to read, write, and analyze a variety of proofs. I am leaning toward a combination of the first three approaches. I am planning on giving 3-5 problems that the students will self-evaluate each “cycle” (6 school days=1 cycle). Students would additionally get 1-2 problems that students would be allowed to revise as many times as needed. I would use these revisable problems to create the packets for students to evaluate. On the fourth approach, I am in agreement with Patrick that the homework committees might create more overhead than I care to handle.
I am strongly considering following Patrick’s lead and teaching the class LaTeX. I would then require students to use LaTeX on the revisable homework, which would make their revisions easier.
The one point that have not settled on: I would like students to give presentations. I have not yet determine how this should relate to the homework. I welcome input on how I should organize the course—on the subject of presentations, or any other aspect of homework.