Kalid Azad wrote books explaining mathematics in that way with his "Better Explained" series (https://betterexplained.com/). For me, that intuition is the seed in the consciousnes from which the rigor can then form a structure around.
When I applied it to concepts I was learning on my own -- monoids, monads, semi-groups, semi-lattices, partial orders, etc. what I found was that I'm often overthinking the intuition. The intuition for a specific idea is very, very precise. It is exactly as it is, and yet something so precise and clear seems difficult to get.
It helps to approach things from a lot of different angles until you "get it". It's not always about repeitition on manipulating symbols. The Soviet method for teaching math (and remember, the Soviet system was intended to raise enough mathematicians to be able to work with a planned economy) was to let their kids manipulate things with their hands in a concrete way. It was in a way more like Common Core, but you're playing with toys with your hands.
I can tell you that I picked up being able to add and subtract things at an early age, but I didn't really get into the deeper stuff until I was exploring monoids and groups.
I've met people who tell me, subtraction and negative numbers are difficult. They know how to do the operation arithmatically, even fluently, but they don't "get it". No amount of repetition was going to change that. I had a similar block when I came across the idea of instantaneous rate of change. I get the proofs, the idea of limits, and how it defines instantaneous rate of change, but it was the first thing that I came across that I couldn't get over the idea that this is an abstract idea, not concrete. I didn't know how to handle it. It wasn't until I came across Azad's way of explaining the _intuition_ on instantaneous rate of change that I focused on getting the intuition first before trying to develop rigor.
When I applied it to concepts I was learning on my own -- monoids, monads, semi-groups, semi-lattices, partial orders, etc. what I found was that I'm often overthinking the intuition. The intuition for a specific idea is very, very precise. It is exactly as it is, and yet something so precise and clear seems difficult to get.
It helps to approach things from a lot of different angles until you "get it". It's not always about repeitition on manipulating symbols. The Soviet method for teaching math (and remember, the Soviet system was intended to raise enough mathematicians to be able to work with a planned economy) was to let their kids manipulate things with their hands in a concrete way. It was in a way more like Common Core, but you're playing with toys with your hands.
I can tell you that I picked up being able to add and subtract things at an early age, but I didn't really get into the deeper stuff until I was exploring monoids and groups.
I've met people who tell me, subtraction and negative numbers are difficult. They know how to do the operation arithmatically, even fluently, but they don't "get it". No amount of repetition was going to change that. I had a similar block when I came across the idea of instantaneous rate of change. I get the proofs, the idea of limits, and how it defines instantaneous rate of change, but it was the first thing that I came across that I couldn't get over the idea that this is an abstract idea, not concrete. I didn't know how to handle it. It wasn't until I came across Azad's way of explaining the _intuition_ on instantaneous rate of change that I focused on getting the intuition first before trying to develop rigor.