Difficulties and breakthroughs

I had a couple of rough days working on some math homework last weekend. I found myself chasing my own tail for hours on single problems, getting stuck and not knowing how to un-stuck myself, arriving at what I thought was an answer only to come back to it later and realize I had some initial assumption wrong all along and having to restart the problem.
Whispering doubts started to creep around in my brain.

Directly after these couple days I took a new approach: When working on a problem I would surround the scratch paper with all the relevant definitions, equivalencies, helpful theorems and examples. I’d let my mind stew in all these patterns and once sufficiently marinated I’d find that avenues to solutions began to present themselves. I realized that due to the newness of all this information I can’t expect it to be at-hand yet in my problem-solving approach. And for solving these types of problems (logic arguments, proofs about number theory, etc.) it’s necessary to use conceptual tools that aren’t necessarily intuitive prior to repeated exposure (“tools” like: for integers , which might make easy sense when reasoning about, but having it written nearby in this explicit form helps for pattern-recognition.)

This helped trigger a minor breakthrough. I still spend a long time on problems and I’ll still sometimes work down an erroneous path for too long before realizing and backtracking. But when I’m right I’m more sure I’m right, and when I’m working towards being right, I feel more powerful and confident in the working-towards and less bumbly. One evening, I worked on a problem for some time and went to bed thinking I had a good solution. I woke up the next morning with a better solution running through my head. It was simpler, more elegant, and more direct. Looking back at the previous night’s work I realized I was grasping and a bit confused, while the morning’s solution was clear and confident.

I took this to be a heartening indication that the information is getting in my head in a useful way and its on me (where “me” is the conscious, intentful character played by my brain) to make sure I access and wield it.


Last night I found myself writing this text to my girlfriend:

I just realized that all this math homework I’ve been doing has been FUN. I’m essentially opting to do it over other recreational stuff without feeling any complaint about it.

If you know me or read this blog, you’ll know that I have a natural, contemptible inclination towards recreation over responsibility (otherwise-known as laziness). I guess it’s mostly a lack of discipline. It’s something that had me worried at the outset of these classes. It’s also something that I’ve always vowed to erode, grain by grain, wherever I can. I’ve had a bit more success with that this year than I have historically.

When I reread the text and realized the honesty in it I was surprised with myself. It occurred to me that I’m truly enjoying the challenge of learning, which is not an experience I often had in earlier academic experiences (there were a couple examples in my mid-20s - and those were the ones that gave me confidence that I could take these classes and succeed). I understood my hours-long “tail-chasing” on homework problems to be a fun and satisfying game of “can I figure this out? have I learned this thing?” rather than “can I get this homework handed in so I get a decent grade?”. The former is inspiring, the latter demoralizing. In this context, I also understood that the doubts I had last weekend weren’t about passing the class or getting a good grade or getting into grad school - they were about successfully learning the material for the sake of learning it, because doing so is fun and rewarding!

So I can’t say I’ve totally overcome my proclivity towards recreation, maybe more that I’ve found more useful ways to have fun recently. Spending three hours on a math problem sure feels like a better use of my time than spending two on a video game. But I haven’t yet determined that spending half of one on a blog post is a good idea, so I’ll cut this short now!

Next time, I hope to write a little about specific topics I’ve learned and any insights I’ve gleaned about them. By the end of this whole journey I’d like to be able to look back and have a record of mini-epiphanies :)