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I recall calculating volumes of cones and thinking that it might be useful...and since then I've built a house and done innumerable renovation projects and never once calculated the volume of much of anything. Area is good enough for most home construction projects.
I didn't connect the dots that this emphasis had to do with physical engineering disciplines. It makes sense.
What I found most interesting about Vitalik's post was the the idea that changing our basic math curricula might result in more people having better intuition about AI or large data sets. Seems valuable.
I'd welcome more and better education in probability & stat at the high school level.
Part of the problem is that prob & stat education is pretty horrible all around, exactly because it was never emphasized in K-12.
That's why you still get trained scientists horribly misunderstanding p-value and Bayesian probability.
The classic example is that many trained doctors get this question wrong:
"A random person tests positive for a disease that occurs in 1 out of every 10,000 people. The test is 90% accurate. What's the probability that the person has the disease?"
For a long time I've thought the core math curriculum should build towards statistics rather than calculus.
When I took my first Engineering Statistics class, we used calculus to prove the underlying equations. Business statistics skips the proofs.
Mathematical statistics was one of my favorite courses and I generally think people should take more math.
The sequence I’d go with is something like geometry-> algebra-> discrete math and basic probability-> calculus-> advanced topics.
There are simplified calculus sequences that teach and prove the concepts without grinding through so much trigonometry. That’s what I would do for general track students and leave the more thorough calc w/trig for students pursuing math/engineering/physics.
I think i am like the trained doctors. But it's something I hope to rectify with my children. I try to talk to them about low level statistics frequently. Unforutnatley (fortunately?) they are getting to the point where I'm going to have to actually do some reeducation on my own part if I want to continue to impart meaningful knowledge.
The above question is a great place to start, honestly. It's both practical and leads to a lot of natural additional lessons. Like, after calculating the answer, a natural question to ask is, "Yeah, but people who seek to get tested usually aren't random, right?"
I fully agree with you on calculus but I think the purpose of geometry was different.
Geometry was the only course where formal logic was taught and I believe that was the case because applying it to familiar shapes and lines made it easier for students than if it were applied to unfamiliar topics like algebra.
That's probably right. And the history of geometry in education probably goes all the way back to the Greeks. That being said, I'm sure they were using geometry for their engineering back then too.
Yes, they didn’t have algebra, so they essentially framed as many mathematical problems as possible as geometry problems.
Oooo, I have many thoughts on this.
Historically, geometry and calculus were emphasized because they're crucial for the physical engineering disciplines... people designing actual bridges and submarines and aircraft and whatnot.
In modern times, where a lot more math is applied to AI, economics & finance, decision systems, these are a lot less useful. These are realms better served by discrete math, probability theory, and statistics. You still need a bit of calculus for optimization theory, but usually less than if you're dealing with physical systems.
Education hasn't fully caught up to modern times, which is why there's still such a heavy emphasis on geometry and calculus. (The calculus taught in high school also doesn't emphasize optimization, but rather calculation of areas and volumes, etc, which is closer to the physical modeling applications)