Comment by mbbbackus
4 months ago
I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice.
He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport.
Also, he inspired me to relearn a lot of fundamental math on MathAcademy.com which has been super fun and stressful. I feel like I have the tetris effect but with polynomials now.
Sounds really cool.
It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda see the whole graph connected by the data that flows through them.
There's a lot of interesting discrete math that can supercharge programming at different levels of scale. What's pretty cool is that it reveals a layer of understanding when I watch my toddlers learn math from counting.
One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.
I would love to learn about some of these anti-pattern proofs if you have any examples or references you can share!
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> rotated shapes, slot machines, or origami
Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out!
Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not.
I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!
I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources that were more like "here's how we discovered this, what we used to do before, why it's useful, and here's some scenarios where you'd use it."
I have so wanted such resources for years. I have found some and should make a list.
The first time the difference between understanding some math, and understanding what the math meant, was after high school Trig. The moment I started manually programming graphics from scratch, the circle as a series of dots, trigonometry transformed in my mind. I can't even say what the difference was - the math was exactly the same - but some larger area of my brain suddenly connected with all the concepts I had already learned.
While ordering the "Mathematica: A Secret World of Intuition and Curiosity" I came across these books, which looked very promising in the "learning formal math by expanding intuition" theme, so I bought them too:
Field Theory For The Non-Physicist, by Ville Hirvonen [0]
Lagrangian Mechanics For The Non-Physicist, by Ville Hirvonen [1]
The Gravity of Math: How Geometry Rules the Universe, by Steve Nadis, Shing-Tung Yau [2]
Vector: A Surprising Story of Space, Time, and Mathematical Transformation, by Robyn Arianrhod [3]
[0] https://www.amazon.com/dp/B0CN7HMTJN
[1] https://www.amazon.com/dp/B0CN7HMK38
[2] https://www.amazon.com/dp/1541604296
[3] https://www.amazon.com/dp/0226821102
Excited to read each (based on their synopses & ratings), and if I will get compounding fluency across both math and physics between all five books.
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If you're interested in how vector calculus developed, and who was instrumental, all the way from Newton/Leibnitz to Dirac or so, by way of Hamilton, Maxwell, Einstein and others, then Robyn Arianrhod's 'Vector' is brilliant.
But be warned, it gets progressively harder, along with the concepts, so unless you're conversant with tensors, at some point you will have to put on your thinking cap.
The reviews on Goodreads – including my own – are worth reading to get a flavour: https://www.goodreads.com/book/show/202104095-vector
I really want to try MathAcademy.com. How quickly do you think someone doing light study could move from a Calc 1 -> advanced stuff using that site? In my case I could put in at least 30 minutes to an hour a day.
I can't speak to the advanced stuff but here's my stats on Fundamentals I:
Total time on site (gathered from a web extension): 40h 30m Total days since start: 32
Total XP earned: 1881
Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.
Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).
Hard to say but this should give you an idea [0].
At that rate, less than a year is reasonable.
[0] https://www.justinmath.com/what-is-the-highest-sustainable-d...
Would you say the book ventures more into the practical side of learning this stuff or is it closer to the tone of this article? I found this article hard to gain anything from. A lot of just motivational cliche statements and nothing really groundbreaking or mind altering. If the book is better at that, I'd love to read it. If it's stories and a lot of fluff, I'd rather skip. So I'm curious what you are getting from it and how practical and applicable it feels to you?
Agree. The article turned me off as well. No specific example, felt like an ad.
Yeah, I quit reading it because it didn't talk about the book, it felt like a meta article.
>>It's like an imagination sport.
Honestly speaking I think this is a wrong way to teach people to think about Math. Math is just one of those things which feels hard because people struggle to hold long trials of manipulations in their head. Especially if they are manipulations to something very large, evolved slowly over hundreds of steps. People are not coming short, its just how the human mind works.
IMO, the right way to teach Math is to teach people that its just base axioms, manipulation rules. And after that its how you evolve the base axiom using rules. People need to be taught how to make one valid change at a time. Of course this means tons of paper work and patience. But that is what Math actually is. Its taking truth and rules, to make new ones.
Im teaching this to my kid, and she often goes like this is it?? its really just laborious paper work??
Im using this method and LLM help at times these days to learn Algorithms and Data Structures. When you start working things from base conditions and build from there. A lot of Algos that otherwise seem like the domain of novel inventions just seem to follow from the manual steps you just worked, and then translated into a program.
When you remove all the fluff, Patience and Paper work is all there is to Math.
The author (and Grothendieck, liberally quoted in the book) disagree with you.
I think the reason you disagree is that it sounds like you’re teaching your child to be good at math class (a perfectly valid and good thing to do). Being good at math class requires being good at rational/logical thinking and computation. It also has only glancing similarities to anything that the author would recognise as mathematics.
>>It also has only glancing similarities to anything that the author would recognise as mathematics.
Nah, these are the same things. Trying to make Math look like is for people who are 'geniuses' i.e people with massive capabilities of holding large thought trials and changelogs in their head is how you arrive at making people look stupid doing math and eventually make them hate the subject.
Math is paper work. Approach it that way and all of a sudden doing a 100 page proof is within everyones reach. If you ask people to hold a 100 page proof in their head, and more importantly make changes to that in random places and fix the entire changelog trial, probably 2 - 3 people on earth will be able to do it, and you will just convince everyone else its not for them.
I have a hunch that big mathematical breakthroughs in history have happened around and after renaissance era due to paper getting cheap and ubiquitous. There is only that much you can do in your brain alone.
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I actually heard about this book very recently, and it's coming up soon on my (never-ending) reading list.
Happy to hear you're enjoying it, makes me even more confident that I should read it :)
+1 for Math Academy. I’ve been using it daily for over a year now (started October 2023). I summarized my experiences after 100 days here in case it helps anyone: https://gmays.com/math
Thanks for sharing this. I was debating buying the book.
This sounds like a book I needed for one of my early comp sci classes in college. It was called something like Think Like a Programmer: An Introduction to Creative Problem Solving. Maybe it was this, maybe it was something like this.
I mean to say, just applied scientific thinking is important. Even if you never get into pure math or computer programming, applying concepts like "variables", "functions" or "proofs" is universally useful.