I saw this post by Vitalik:
And it made me think: there are certainly some benefits to how different mathematics concepts change your understanding of the world, but I agree that I didnt gain too much from trigonometry and I haven't had much occasion to use it in my life. I certainly wish I had spent a lot more time on statistics. What do you think?
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)
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.
The study of measuring triangles, one of the most ancient and foundational subjects in math, right above geometry itself. This guy i found on youtube reoriented the subject ~15 years ago to remove the irrational functions and use pure highschool algebra instead. He then applies the resulting trig concepts to all manner of advanced topics. Very interesting, clear and thorough.
I was trained as an engineer. Technically, I don't use 95% of the things I learnt during my courses. Yet, the problem-solving skills I learned along the way are probably helping me a lot in my current work.
So, no, I don't regret taking courses on general relativity or fluid mechanics. They were indirectly useful. And they've helped me understand the world at a deeper level, and that's worth a lot.
The specific course matters less than the broader skills you learn.
So, for that reason, I'm not going to argue in favor of or against trigonometry, cryptography, etc., with our favorite ETH armchair philosopher...
Also, with purely utilitarian thinking, we can probably get rid of a lot of degrees.
But would that be a sad world...
I do wish more students had an attitude of "learning this is cool" as opposed to "how will I use this"
I like Star Trek's vision of humanity: in the future, when material needs are met, we explore the universe for discovery's own sake.
I didn't read it as a case of getting rid of degrees so much as adjusting the most commonly taught path of math for US schools.
Curiosity gets you very far, especially in the era of AI, but we still seem to have standard ideas about what people should learn (eg the US inexplicably is awful at teaching finance and accounting to highschoolers). I was homeschooled, and so not subjected to very much of the "standards" and yet the bled into my own education because my parents, not being intellectuals or particularly curious people, just kind of looked at what everyone else was learning and said, let's do that.
I suppose if we are thinking about school as a project that is supposed to help young people have the skills that make them useful in the world (and possibly to inspire them by giving a little taste of how big and interesting the world is), we might consider abandoning many subjects in favor of more practical ones.
Homeschooling my own kids, I can basically do anything (especially in Texas), it's interesting that I tend to follow the paths where curricula have already been developed. And it certainly takes more effort and thought to go off the beaten path.
Oh hell no! Trig is the key to Calculus. It literally makes Calculus makes sense. WTH it's not required in high schools before college blows my mind but there seems to be this mistake consensus that it's not necessary. Then people get to Calculus and flounder hard.
I used to believe this until I started following a math teacher on YouTube. He makes me realise how much of our time was wasted at school with empty formulas without ever actually clearly touching on their relevance in the real world...
I won't claim I fully get all the stuff I'm watching but there is something fascinating about being able to visualise and conceptualise a sin, cos function in your mind.
Not sure I fully agree with that. A lot of students feel like these things are irrelevant to the real world, but it's only because they can't see the relevance yet.
Sometimes, the relevance isn't taught till later. Actually, some of the relevant applications would be too difficult to them, so imagine complaining that the easy thing is irrelevant when you can't even do it; and would get destroyed by the real world relevant stuff.
YES!
it's important to grasp how trig works, I think. memorizing any of the interplay is probably diminishing returns, tho.
we should understand concepts of periodicity and plane projection... but just enough to open us to other concepts.
doing trig problems on paper maybe isn't the win it used to be. hard to know.. I rarely refer to those challenges, but I remember hsin and such are valuable ways to perform calculations on hanging wires
He probably mentioned only what he was interested in. What if you are interested in computer graphics, need to do a rotation, and need to understand why it did what it did?
Also what about ham radio amateurs and anyone interested in waves? I am in both categories.
Also let's say your second hobby is about drones and you want to know how signals are encoded, you need trigonometry.
Also when someone beats you at mental arithmetic or you are being ridiculed for using your fingers to count, it can help to throw cos(π) to divert attention (just did that yesterday).
Yeah, I’ve never really used trig in real life either :) But I do think math should focus more on stuff we actually use.
I did college-level calculus but never have to apply what I had learnt then.
I think the usefulness of Maths depends on what you choose as your profession
Whoever built the great pyramid in Egypt had no known access to modern trigonometry
But it was built....
But yeah they probably just laid some ropes out and had a good eye 😂😂
https://twiiit.com/VitalikButerin/status/2102026491043492146
I think the problem is that trig gets taught as "triangles", when it's really the math of waves and rotation. That's why it feels useless while you're quietly using it all day:
@Scoresby, since you built a house: the one place it shows up directly is roof framing. Rafter length = run ÷ cos(pitch angle). For a 6/12 pitch (26.6°) over a 12 ft run, that's 13.4 ft, not 12. Framers skip the math because the speed square and rafter tables have it baked in.
So I'd agree with the conclusion but not the reason. Most people don't need to compute trig. But "periodic things can be broken into sine waves" is one of the most useful ideas in engineering, and statistics won't teach you that.