@JZ: Thanks for the kind words! Glad you’re enjoying it :).
I love your way of explaining things. I’ve read two articles so far, including this one, and they were both awesome. 
Love the metaphor and the whole philosophy. Thank you!!!
@Valter: Thanks for the comment, I’m happy if the analogy helps you convey how to develop intuition to your students!
Thanks Kalid.
Very very god metaphor! Surely it will increase my teaching performance. I teach Operational Research for a business course in a local University and my great challenge is to make my students to get the intuition about the problem nature before teaching them the “arrows” of simplex and so on.
Thanks for this insite. I have a daughter in HS who is having difficulty in math. I hope this site will help her.
Kalid, Very well said.
I just want to add one thought. What you said about maths applies equally well to other subjects.
Of late, I have started reading metallurgy books and many times I have wondered how wonderful it would be if somebody wrote a book on metallurgy like Kalid does on maths.
I believe that any science or engg should be and can be explained in simple words.
I remember your words: Learn the bow and one arrow; more arrows will follow.
@alex: Definitely, Wolfram’s talk has some good overlap with Lockhart’s lament.
I totally agree with you about the focus – I’ve seen dozens and and dozens of “proofs”, but seldom did they stick. I realized I had to (for myself) explain things in a way that actually worked, which meant focusing on intuition and how learning math changed the way I thought.
I really like Wolfram’s approach, and explicitly making the 1-4 steps known. Computation is such a small part of understanding, it’s the most mechanical and yet the most focused-on. I think the reason it’s emphasized is because many educators have never seen the inherent beauty, and focus on the part that can be measured (similar to someone who has never actually heard a song, assigning sheet music transcriptions and thinking that is the study of music).
The history of mathematics is actually where I go to look for insights :). For e, for example, it was discovered in the context of interest rates, so tracking where interest comes from, how it creates its own interest, etc. sheds a lot of light on the subject.
Thanks for this site… very inspiring!