Today I had another experience with living in “the mess“.
We started working with elliptic curves today which I’ve been waiting all term for!!! My current instructor was actually on the same campus as me for my second year, and my class even joined his for a few lessons. Elliptic curves were one of the things we talked about during these few lessons, and it is part of the reason I even wanted to take this class! However, because he knows I’ve done some of this before, he keeps pointing out in class that I’m not allowed to say much even thought sadly I don’t remember as much as I would have liked to.
Today we worked on a giant problem to try and derive a formula for adding points on an elliptic curve. We worked in pairs for over an hour and a half probably, but not 3 hours like the class big problem. The problem was really challenging, like pages of college level math that even our TA wasn’t sure about and my RC happened to come into class today and was very confused as well. It was a good challenging though because this way when we were done it was so much more satisfying.
Our instructor had a little chart up on the board signifying big steps of things we needed to solve and you would get a check mince if you almost had it, and a check for getting it. These were things like “equation, insight, x sub 3,…,” so they were really vague and kind of confusing until you got the check.
But I think it was important for us to derive the equation on our own because it helped us better understand why things worked while teaching us good learning skills with trying to test things to get stuff you want from it.
At the end we learned that it could really easily be simplified but to get there you had to start by getting a really messy long equation. At the end, in a quick summary, our instructor literally wrote on the board something to the extent of, “x^3 + m^2x^2+ [messiness]x +[messiness]= x^3+(-x1-x2-x3)x^2+ [nastiness]x +[nastiness]”. Today was our first day with elliptic curves, and this process was helping us learn more about how they work so we’re prepared in the future.
When you are able to work through all of the mess, that is when things become clearer and more satisfying. What if we had more moments in school that felt like a “mess”? Moments where teachers could give us one problem expecting it to take an hourish, so that we have time to really try and work through the messy details of a problem before we really learn much about it. What if we weren’t always just told, “Here is a formula, and now we will do some practice problems,” and instead we were told, “Here’s one way to do a problem with an accompanying example, now try to figure out a general formula so you can always solve this kind of problem”? To me I think that would be so cool and lead to interesting discussions that will help people better understand what they are doing!