Let me virtually shake your hand. I've always been successful teaching first from the concrete and then slowly abstracting - literally "pulling away" in Latin -

Let me virtually shake your hand. I've always been successful teaching first from the concrete and then slowly abstracting - literally "pulling away" in Latin - as that's how I understand abstraction - it is the common ground between concrete things and therefore it was discovered from those things. To then try and teach students first from the most "pulled away" concept possible to only in the very end try and meet them in the real world - I just never understood that approach. Like you said, the government curricula prescribe an approach that is backwards to what makes sense. In Plato's Meno dialog, Socrates teaches a kid the abstraction called "irrational numbers" by having the kid draw triangles on the sand and then try and find the square root of 2. By playing with a concrete example (sqrt(2)) the kid learned the abstraction (that there are numbers that can't be shown as a fraction). This backwards way of thinking we're stuck with today is so pervasive that one approach I was developing to teach monads - by first writing each instance separately and then later "pulling away" to try and see the bigger picture - was recently excoriated in the Haskell IRC channel for not teaching the actual abstraction. The path is not the destination. You can't pull away (abstract) from nothing.