Here's the story of this spring semester, and how I made some courses for myself. This catches the reader up to my four-week checkup on progress.
Background and Design
After finishing my undergrad studies three years ago, I took a break from schoolwork to teach at a private school, tutor on the side, and pursue my other interests like playing violin. That went well, but lately I've been getting antsy, wanting to return to higher math in full force. I know I'm a little rusty, and I also know that there's some textbooks that I've always been meaning to get around to that will likely fall by the wayside if I were to attend school again now. So over winter break, I decided it'd be a perfect time to really go all in on self-teaching this spring semester, and design some full courses to manage myself.
Algebra. The first course would use the Algebra text by Dummit and Foote, designed to shake off my rust, since Algebra is probably what will be my specialization, and the first two parts of the text should be a problem-driven review. So this course would be focused on working problems, only occasionally skimming the main text.
Knot Theory. The second course would use The Knot Book by Colin Adams, a text which I'd been wanting to go through ever since my first course in knot theory five years ago, when we had used Knot Theory by Charles Livingston and I was thoroughly disappointed by how it covered the material. After doing searching for other texts at the time, I felt that Adams's would be a much better style for me, and I bought it immediately. I used it to teach the first two chapters of material to a high school student, but otherwise, it's simply remained on a bookshelf unread. This course would be focused on reading through the second half of the book, working the problems that interest me most.
\(p\)-Adic Numbers. The third course would be focused on understanding \(p\)-adic numbers, which I've long found interesting, but never gotten around to actually making sense of. When I tried to understand them three years ago, I was reading through Fernando Q. Gouvêa's book at Fondren Library. I recently visited Fondren again, found the book, and another by Alain M. Robert that seemed closer to what I needed. With two good texts, this course would focus on reading through the material, typically making up my own problems to try to make my confusions precise and resolve them.
Topics. The fourth course would be centered around all my extra math work. See, I get a lot of random ideas and problems to solve all the time, and I am very disorganized at dealing with them in any reasonable way. I typically work on whichever one happens to be on my mind, writing notes on random pieces of paper that get lost, if I write anything down at all. That... sucks. I also want to try to start staying in touch with new math papers as they're published, incorporating that material in with these other problems I'm working on. So resolving these two issues would be the focus of this course.
I remember about a decade ago, I asked a physics professor (Dr. Weathers) a question that reminded him of previous work, and he opened a file cabinet and quickly located a file containing a napkin with perfectly relevant notes on it. That moment always stuck with me as the kind of thing I'd like to be able to do some day, though I've never been one to deal well with physical organization. The trick for me would certainly be to require my notes to become electronic immediately as they are written, but I still don't have a good system of organization for the content even once it's saved.
I've tried to deal with all of these sorts of things in the past, and failure seemed swift with every attempt. I decided this time to really make it feel like school again, and in the first week, wrote up syllabi for each course, set assignment structures, test dates, everything. I also set weekly "class time" events when homeworks would be due and so on. This has worked really well at helping me form the habit of class for a few reasons:
I've often failed on previous attempts because I'd miss my own deadlines in the first week or two, and then have no good way to scale up to what I wanted. Having a syllabus week started [the habit][(remembering when the deadlines are)] while having necessary yet achievable, low-pressure tasks to complete.
Having a week of accomplishing something at each deadline gave me a better feeling for how much to include in homework assignments.
Putting my plans down in writing made it feel truly real and forced me to know if I was sticking to them. Previous attempts of just vaguely saying "I'll do stuff every week" made me able to wiggle around what I initially had in mind, and ease up on myself unnecessarily.
The other largest hurdles in course design in the past have been motivating homework and creating exams. I'm always motivated to work problems conceptually, but actually going through the effort of writing the answer down or trying to work a quantity of problems rather than my favorites have both been difficult. For exams, it always feels like I have to invest so much time in coming up with a good selection of problems, and it always pans out that I've chosen too many problems for the exam, so much of my effort was in a sense a waste.
This time, I realized that I could solve my homework and exam woes simultaneously with the same solution. When I choose my homework problems from the text, all those that I don't work on go into a test pool. When test day arrives, I run a python script that randomly selects problems from the pool and writes the exam for me, which I must try to complete in a limited time. Thus, my time spent choosing homework and exam problems is efficiently merged, and I'm strongly motivated to work the harder problems for homework, as I wouldn't want them for the exam. Finally, any problem may be chosen for the exam, so even the ones I find tedious to write out may have to be written out.
Fourth Week Checkup
I knew that however I wrote up my syllabi to begin with, they would probably have to be changed mid-semester. Some of them even explicitly stated which parts should be considered first for modification. To aid in my analysis, I made sure to time journal, recording in Google Calendar the time I spent working on each subject.
After four solid weeks of work, I found myself burnt out. (Acquiring a knee injury and having an almost-sick weekend certainly didn't help.) So, it was clear that my fifth week plans would be replaced with "take a break and make modifications".
The most stunning result of my analysis has been that I was spending about as much time per week on algebra as I was on knot theory. (They were 11 and 10 hours per week, respectively.) This is stunning to me because I knew I was spending way too much time on algebra, but felt that my time spent on knot theory was much less and more appropriate. (Why?)(why-time-lost)
My best guess for why the disparity between my expectation and reality exists is that my algebra work has been mostly about concepts I'm already rather familiar with, so most of my time is spent writing the proofs. On the other hand, my work in knot theory has been in less familiar territory, so most of my time is spent devising the proofs. Apparently, my ability to estimate time spent pondering is completely bunk.
So, I have decided to change to a maximum time limit rather than a minimum work limit. That is, I will strive for similar goals on homework each week, but after 5 hours of work in a week on a particular subject, I am not to work any more. This should work well with my current homework/exam system, since any problems I don't get to finish as homework problems are then put in the test pool, so I have the same incentive to work as before.
In order for this change to work, I'll have to change how I manage my time a bit. My plan is to set alarms to check on my progress as I'm working. One plan is to set an alarm for half the time I intend to work in a given session. That way, at the halfway point, I can evaluate how I've spent my time so far, and possibly change my efficiency for the final half. Alternatively, if I have many tasks to do in a session, I may set many smaller alarms, each for the amount of time I want to spend on each task. We'll see.