[goku12]

There ought to be something similar for calculus too. While a detailed and formal treatment of the subject can be delayed till the age at which it's introduced now, I feel that the intuition and feel for calculus can be formed more efficiently at an earlier age.

If nothing else, it may help them understand where to seek solutions for the common problems they encounter. I started learning Electronics at a fairly young age using undergraduate level textbooks that I found lying around. The need for and relationships between concepts in calculus, logarithms and trigonometry were a recurrent problem for me.

PS: If anybody is wondering, those books were from an earlier generation engineer. They were very interesting, to say the least. All the circuits (amplifiers, rectifiers, oscillators, multivibrators, mixers, various RF Txr and Rxr designs, etc) were using vacuum tubes! Diodes, triodes, pentodes, thyratrons, magnetrons, TWTs, etc were used liberally in them. It had a description of an early form of the Instrumental Landing System (ILS). There were also descriptions of some early generation semiconductor devices and their similarity to vacuum tubes. I don't think ICs were in much use back then, because the book had no mentions about them.

I used to spend hours at a time with those books when I was a child. Later I graduated in Electronics engineering and went on to work on the avionics for a satellite launcher. Vacuum tubes were museum pieces by the time I was born. But I was the only one in my undergraduate class who had seen or knew anything about vacuum tubes, when we had lessons on CRTs, magnetrons, etc. I can't stress how deeply those books influenced my education and career. Sweet memories!

[smj-edison]

> The need for and relationships between concepts in calculus, logarithms and trigonometry were a recurrent problem for me.

Oh my gosh, this was me growing up! I loved tinkering with electronics and programming, but I kept bumping against my lack of knowledge wrt more advanced math topics. I usually hacked around it, or more often just switched to a different project.

Now that I'm taking calculus, I feel like I always have a corresponding application for each topic we cover. It's very exciting!

[goku12]

> Now that I'm taking calculus, I feel like I always have a corresponding application for each topic we cover. It's very exciting!

Mathematical concepts need not necessarily have any practical or real-life applications (pure mathematics), but it's a worthy pursuit on its own. However, every maths concept we learn outside of specialized mathematics courses have some application (applied mathematics). That's how these courses are designed.

It's funny when some people lament that much of the mathematics they learned at school or uni/college have no real-life applications! I wonder what they think about why they learned it. It also shows a big problem with math education and pedagogy. Many people are sailing without a destination in mind, hoping that they'll just hit some land in the future. The academic practice of mathematics has turned into the drudgery of endless symbolic manipulation.

But at the same time, people like you who lean into creativity, hit the limits of their mathematical knowledge early. For them, the problem isn't that the mathematics they learned have no applications. It's that they don't know where to look for the mathematics they need. Here is the failure of math education. If everyone was taught about the different branches of Mathematics, how they relate to each other and where they're applied, I can guarantee that many students will learn those on their own before they're formally introduced to them. That would enable the talented and interested students to shape their own destiny.

[smj-edison]

Agreed.

> Mathematical concepts need not necessarily have any practical or real-life applications (pure mathematics), but it's a worthy pursuit on its own.

I guess I always had a sense that this was true, but taking a formal proofs class has really opened my eyes to how true this is. I'm actually planning on doing abstract algebra because I've enjoyed proof writing a ton!

> It's that they don't know where to look for the mathematics they need. Here is the failure of math education.

This is a really interesting space imo, because I did try teaching myself calculus through 3blue1brown, who has super cool visualizations, but also isn't rigorous enough to apply it to complicated problems. On the other end, I also tried Khan Academy, but found it too abstract and hard to follow.

Perhaps that was just since I was 14 at the time, and now I'm better at symbolic reasoning. Yet, there's something so enticing about visualization, that I wish there was a way to have the rigour of set theory with the intuitiveness of visualization (that call is a big reason I love the work that dynamicland.org and folk.computer are doing right now).

[goku12]

> I'm actually planning on doing abstract algebra because I've enjoyed proof writing a ton!

That's interesting! I started with abstract algebra (on my own) because I don't know the proper sequence of learning it. I guess I will give proofs a try now.

> I did try teaching myself calculus through 3blue1brown, who has super cool visualizations, but also isn't rigorous enough to apply it to complicated problems. On the other end, I also tried Khan Academy, but found it too abstract and hard to follow.

> I wish there was a way to have the rigour of set theory with the intuitiveness of visualization

Agreed! Intuition and rigour are equally important in Mathematics. The hardest part is making these two meet at some point.

One such moment in my life was when I created a visual model of 3D vector calculus (divergence, curl, gradient, etc) and developed it far enough to explain all its rigourous treatment that I learned in my engineering class. Things just flowed from there. I suddenly gained the superpower to imagine up the explanations for phenomena like the skin effect, and then effortlessly derive their mathematical models.

[globular-toast]

They shouldn't teach calculus like they taught it to me and my peers. Basically we just one day started "differentiating" equations. We learnt a completely mechanical process. Like how to chop an onion, except it doesn't actually feed you or taste delicious.

It took me a while to realise the point. It's all about rates of change. They should start with that. No need to bother with the maths, just look at graphs and be like "that's a steeper slope than that", and, ooh, that one's sloping in the opposite direction. This is a fundamental intuition that's so useful to have. Most people don't understand that braking is acceleration. They just don't have the mental model that lets them see fuel burn and braking as opposite things. The sooner this intuition is there the better. Then teach the maths.

[goku12]

Agreed on all counts.

> They shouldn't teach calculus like they taught it to me and my peers. Basically we just one day started "differentiating" equations. We learnt a completely mechanical process.

I had a similar experience and it did ruin the fun in Calculus for me. It took me a long time to derive a bare-minimum mental model that I was satisfied with. It was at this point that I could 'feel' (imagine) how the general second-order linear differential equation (of two variables) works, without the need to 'calculate' or derive anything. This equation is the fundamental model for countless phenomena in the universe. It's such a shame, because that equation is easy to explain in words, without doing a single step of derivation.

Don't get me wrong. Formalism and rigor do have very important roles in Mathematics. But ignoring intuition and emphasizing formalism doesn't get you anywhere. Intuition isn't always right, but it shows you the 1000ft view of the problem when it does get it right. Formalizing the solution gets easier from there.

I have noticed that even professionals are taken by surprise when I convey the descriptive explanation. It shows how badly these things are taught. (I don't know if this is the situation everywhere.)

> The sooner this intuition is there the better. Then teach the maths.

Yes, that is exactly what I was suggesting. However, that 'intuition' is also part of Mathematics. Many practitioners call it the 'Mathematical sense', as opposed to common sense. You might have seen a rare few gifted individuals who find the correct answers to unintuitive and confusing problems (like the infamous Monty Hall problem) in their first try. They're employing this mathematical sense while the others revert to common sense. Who knows? Even you may be using it and surprising others without realizing it.

Unfortunately, our educational systems have reinforced this misconception that Mathematics is all about manipulating numbers and symbols (for many, even the idea about symbols are missing). This is a very sad situation that just sucks the life out of mathematics. A long essay (book) by Paul Lockhart, named 'A Mathematician's Lament' explains this problem splendidly.

PS: Funnily enough, I always struggled in and hated mathematics! Others were so good at applying long sequences of operations to get to the answer, while it was Chinese to me! (No offense intended here). But I was good at science. I relied on countless diagrams, tables, concept graphs, signal flow graphs, etc in place of equations and formulae to achieve this. I just converted them to equations and formulae whenever I needed to reproduce those. I thought, "Who needs mathematics when you can reason your way to the answer?"

It was close to the end of my formal education that I realized that every reasoning that I had done in my life was proper Mathematics! I had strong autistic traits and following numerous steps in sequence and in parallel was near impossible for me. But where I made up for that was in spatial intelligence. I had created book after book of Mathematics described in a visual language that I could digest. I didn't really hate mathematics. What I hated was the way in which it was taught and represented.

Learning Mathematics has become a whole lot easier and enjoyable after realizing it and embracing the fact that I needed my own ways of doing it. But honestly, I wish that so much time wasn't wasted in needless frustration.

[in_cahoots]

My father introduced this book to me when I was around 10. I will forever be grateful.

https://www.worldofbooks.com/products/calculus-the-easy-way-...

[goku12]

Thank you for your life account and the suggestion! Smart dedicated parents play an unseen and underappreciated role in the success of their children.

My parents didn't know any of those - they were not in STEM professions. I think my questions exceeded their capability when I was quite young. That forced me to seek out resources and professionals. But they didn't let their modest earnings or lack of specialist knowledge prevent me from persistently pursuing interests. The lifestyle they provided me was a richly fertile ground for ideas and growth. I'm also extremely grateful to my parents.

[cultofmetatron]

> There ought to be something similar for calculus too.

mathacademy.com very thourough and highly effective.

[WalterBright]

The tube circuits I studied were marvels of clever engineering.

[goku12]

Yes! Vacuum tubes seemed much more predictable than semiconductor devices. I don't remember exactly why, but perhaps it was that the tubes had more linear behavior than the SSDs. Regardless, I'm also happy that I learned about those amazing things. It's such a shame that they were dropped in all cases where SSDs replaced them.

There were some efforts to create ICs with vaccum tubes of micrometer sizes. At those sizes, they don't need heating or vacuum hermetic sealing like the traditional large tubes. But they retained one big advantage over SSDs - they have superior fault resistance and survivability against radiation. That makes them prime candidates for space and nuclear applications. This was like 15 years ago. I thought that they would hit the market for sure by now. I wonder what happened to them.

[WalterBright]

Basic integration and differentiation would fit right in with algebra class. You can teach it in about half an hour.