If you work it out you know it in your bones, if you know how to push the buttons you know how to push the buttons. The neurons you need to make the calculation

If you work it out you know it in your bones, if you know how to push the buttons you know how to push the buttons. The neurons you need to make the calculations can be applied to other situations, if you don't have that pattern in your head because you only ever pushed the button then you can never use that pattern any where else. I'm speaking as one of those kids who could always do maths and that's how I do it, maybe there's another way but a lot of people I've spoken to who is the same agrees, with some exceptions. I view it as a skyscraper each level of learning requires the underneath level to be solid, if each level is a little wonky, then as you get higher and higher it can all come crashing down. Once you know how though,and have worked through the problems a few times, then use the calculator by all means. An example: addition is commutative, if you know your sums, then this is pretty obvious, so when you do group theory, commutation is obvious, because you know 1+2 = 2+1, and you wonder why this is an issue, if you don't know you're addition tables then commutation is a strange thing, you can see it's something thats useful but why's it a big deal. Same with the identity element, then you can move onto rings and fields with the same level of understanding. If you don't know the basics in your bones then each of these concepts gets a little more rubbery. Edit: or maybe look at it this way - if you're a writer and don't know the rules of grammar in your bones you'd be a lesser writer than some who does. Why would maths be any different?