The problem with math isn't symbology or notation per se, it's that it's not evolved to take advantage of modern technology. It's an artifact of pencil-and-paper being the medium of choice for expressing ideas. We can do much better.
When you are looking at an equation, you are looking at the purest distillation of an idea. Underneath that equation sits countless layers of abstraction and reasoning. Why can't I peel back these layers and see them on my computer or iPad? The equation is the iceberg tip peeking out of the water onto the paper, I want to see underneath. Let me feel around and slowly and confidently fill in the places where my understanding is fuzzy. Prevent me from moving back up to the final equation until everything underneath is fleshed out and solid in my mind. And if bits start to fade, let me quickly react to snap them back into focus.
This is what we are doing mentally anyway when we flip back and forth in a math text referencing previous proofs and equations. Math needs it's "hyperlink."
Anything I say here has been said better and more convincingly by Bret Victor:
Though I will argue that at least in his Kill Math writings he's throwing the baby out with the bath water in a certain sense. Let's keep the symbology as a optimal way to encapsulate our knowledge for the amazing mental leverage it gives us, but give us a way to move up and down the ladder of abstraction in order to facilitate understanding.
When you are looking at an equation, you are looking at the purest distillation of an idea. Underneath that equation sits countless layers of abstraction and reasoning. Why can't I peel back these layers and see them on my computer or iPad? The equation is the iceberg tip peeking out of the water onto the paper, I want to see underneath. Let me feel around and slowly and confidently fill in the places where my understanding is fuzzy. Prevent me from moving back up to the final equation until everything underneath is fleshed out and solid in my mind. And if bits start to fade, let me quickly react to snap them back into focus.
This is what we are doing mentally anyway when we flip back and forth in a math text referencing previous proofs and equations. Math needs it's "hyperlink."
Anything I say here has been said better and more convincingly by Bret Victor:
http://worrydream.com/KillMath/ http://worrydream.com/LadderOfAbstraction/
Though I will argue that at least in his Kill Math writings he's throwing the baby out with the bath water in a certain sense. Let's keep the symbology as a optimal way to encapsulate our knowledge for the amazing mental leverage it gives us, but give us a way to move up and down the ladder of abstraction in order to facilitate understanding.