When it comes to addition however, any attempt to learn addition by tabulation and memory would clearly be futile.
I certainly don't mean to suggest that one should study a theorem in a particular corner of mathematics from fundamentals up; rather understand it "one level down", to something previously interned in a similar manner (and since forgotten because one no longer had need of it) or to somemething axiomatic.
For example, say I come across an equation for the area of a right triangle for the first time. I've previously understood and accepted some other problem that I can reduce it to - area of a rectangle, say, or integration - so I can do so and actually understand why, and be able to reproduce it. Easier to learn, recall, and apply than rote memorisation in my opinion.
When we start a lot of things are interrelated and unless you a bare minimum is committed to memory, it's difficult to start.
It is akin to learning a language where one need to cram the basics building blocks i.e. words and grammar rules before applying them. I think idea of trying to understand before remembering ends up impairing learning at least in my case.
I have a two year old and I see this process in action. She will first cram things like counting and then start attaching meaning to it slowly.