Facts without meaning and facts with meaning

As a child growing up, I learned in the public school system designed around the metric of state test scores. Decades later, I have a vague understanding of the general complaint that teaching students to take tests is the wrong way to teach, but only today do I understand that complaint in a personal way.

Today, I am learning trigonometry by myself, in order to understand something deeper which requires knowing trigonometry. I have the following tabs open in my browser:

This is how I typically read a paper in a scientific journal (the first link above). I have the PDF open in one tab, and as I encounter new concepts that I want to know more precisely, I start opening search queries and more tabs in the browser^[1]. I'm learning about something called "rational trigonometry," but I don't know much about trigonometry itself. First I wanted to know about Euler angles, so I opened a few tabs. While reading those, I didn't know the names of the Greek letters used, so I opened a tab on the Greek alphabet. Then I encountered a mention of Euclidean distance which I wanted to know more precisely, so I searched on that, which then led me to "law of cosines" and sines and cosines, which are topics I vaguely remember learning about in high school, but which I almost completely ignored since it all seemed meaningless to me at the time.

It all seemed meaningless at the time

There were maybe a half-dozen kids in the one memorable high school math class who were interested in trigonometry. The other twenty of us were various degrees of not interested, and I think it's safe to say I was the least interested. None of it was interesting to me. There was a daily litany of facts delivered by a math teacher who intended to be educating us, but who was failing to reach most of us. In other classes, I was attentive and learned eagerly, but math was boring and inscrutable. It made no sense and I had long lost the ability to learn things which required any extra effort on my part. The classes where I excelled involved reading and writing which were no effort because they were fun for me.

The important point here is that the math was presented without a context within which I could use it. I was expected to memorize as much as I could, and pass multiple-choice tests and nothing else. There was a vague sense that I might need the skills when I got to college, but that wasn't very attractive at the time.

It's meaningful now

Today I'm learning intro trigonometry, and will spend several days learning it, because I want to. My motivation is internal: I'm convinced there is a more efficient way to do rotations in space than using quaternions, which are themselves more efficient than using Euler angles. I know this partly because I understand the general ideas behind Rational Trigonometry (developed by a math professor in Australia over the past couple decades), and I want to know more. What made this compelling to me? I learned that this 2019 paper in a peer-reviewed journal showed rational trigonometry requires half as many logic elements to compute rotations as compared to the normal state-of-the-art rotation algorithm. In other words, a piece of logic which is used billions of times in almost all computer animations was made more efficient. The strange thing is this: hardly anyone knows this. That's exciting to me. There's room for someone like me to learn and help create an advance which will affect a lot of fields of study.

So I'm learning trigonometry, and liking it, because it's finally meaningful for me. And this is the thing I was missing in high school. Meaning.

To break out of the study-to-the-test cycle, we need to find ways of teaching math which make it interesting to young minds. We need to focus on the beauty of math, not the memorizability.

Footnote:

^{^}I usually open a half-dozen or so tabs for a new concept by skimming through the search engine results looking for pages that seem most appropriate for my query. When I have several, I go through one by one and quickly skim through articles thus opened. If the contents are appropriate, I leave the tab open and move on to the next. This way I end up with all the "good" hits on a given topic. Then I start reading more carefully, but quickly jumping from page to page if one is mediocre, switching to another which might cover the material better. I always close a tab as soon as it is clear that it's not excellent, because later I bookmark all open tabs as a summary of a given topic which I can come back to in the future. I recently did an archive of all bookmarks, where I noticed that I have about 70,000, most of which are dedicated to math and physics research. I know this open-a-bunch-of-tabs-before-reading approach is slightly unusual because whenever I do this during team meetings, someone comments on how "that's a good idea, I think I'll try that." I guess other people open up a single tab at a time? This stage is necessary to get a full image, a 360-degree picture, of an idea in my mind. At this stage of the process, I aggressively use the Internet to learn new concepts. It's hyperefficient because I do this for hours pretty much every day of my life, and I've been doing this for years now. Sometimes I have nested pursuits, where one topic has subtopics, and I'll rapidly have a few dozen links open in my browser. I generally use three browsers simultaneously, because when I close each browser, all history and cache are deleted, and sometimes I have a browser full of tabs open for a day or two. But only that long; I prefer to begin with a blank slate as often as I can.

Clearhat