> 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.

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.

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.

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.

Yes the current kits are dumbed-down and soulless. I don’t know that anyone else makes decent ones nowadays.