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.