W
hile working my way through What Is Mathematics, I came across a particular formula -- in this case, for determining the sum of a geometrical sequence. I understood how the formula was arrived at, as I could follow the arithmetic, but the form itself left me puzzled. I felt like I should have an intuitive sense for why it was structured in the particular way that it was, but the text gave no overarching explanation. The math worked, and that was what mattered for this particular explanation.
I wasn't entirely satisfied with this, so I started chipping away at the formula to see if I could get a real sense for its structure. When I started this project of furthering my math education, I told myself that I would make sure that I fully understood every step that was being taken, so that I wouldn't get in too far over my head. I spent an hour or two trying out different ideas, and finally stumbled upon a very neat encapsulation of concepts that undergirded the formula, that allowed me to see the problem in an entirely new light. It was a wonderful feeling -- an illuminating discovery, taken on of my own volition, that informed my understanding of the problem, and of the related concepts I had been studying. It felt like real learning.
This ability to truly take my time with this form of education is an unforeseen blessing. In school, deadlines hover over every page. There is a continual pressure to skim, to guess, to skip and hope. But with my current work, the only deadline is the literal one -- I can spend as much time as I like, soaking in the ideas, and moving forward slowly but confidently. I'd like to make this particular brand of self-education a larger part of my life. I think, really, that everyone owes themselves this, to be increasing their own education until their death bed.