When I was a child, I was curious about the process of discovery, and of how we come up with ideas. I was also, from time to time, confused by how we are required to think about math problems in school. Like any other student (including mine to this day). I remember being upset in elementary school at a math problem about a parcel tied with three strings, each running over four sides of the box in a plane perpendicular to the others, and somehow there was supposed to be one knot that tied all three strings together... I don't think so. So I didn't get the officially "correct" answer to that problem. And I found strength in gaining the certainty that I never would get that "correct" answer, because the problem was ill-posed. When I was trying to do the best possible job learning something, I always felt that I needed to assimilate the concept I was studying, that I needed to make it mine. And sometimes, in order to do that, I would need to come up with my own way of doing things. In one instance, my ninth-grade math teacher got very upset and gave me the hairdryer treatment in front of the class because I had solved algebra homework using a different method from the one he had taught us (which I hadn't understood) and because he thought some tutor had solved it for me instead of me trying to learn what he had taught us. The method I used was correct, if more complicated. It was also entirely my own. I had tried to understand the teacher's method and it made no sense to whom I was at the time, so I had to do something... In seventh grade, we had been told about careers in scientific research, and that struck me as the best (the only at the time!) way to cultivate that interest in discovery, in the context of a job, since most of us have to have one and I had no idea what I wanted to do. (Actually, I still don't, but I've managed to love what I was doing for most of my life so far and for that I'm very grateful.) Then puberty and high school happened and I forgot about research. Until I met physics professors in college who were teaching us not to memorize anything, because they didn't, and sure led by example. Then it all came rushing back! Problem solving! Research! Yes! In terms of major, I still liked programming computers as much as doing physics and was undecided. But after two years of college the CS Department told me I wasn't good enough at math to major in CS so I became a physics major and learned quantum mechanics instead (somehow I still did OK with all the math there... go figure).
So here I am now. Still wondering how we get ideas and how discovery happens. Will you join me?
WARNING -- Too much head scratching may be detrimental to hair retention.
WARNING -- Correlation isn't causation.