We also didn't force memorisation of facts before moving on. I feel like memorisation and rapid fire recall is a separate skill which is more age-based, and forcing it would have taken all the fun out of it. So for four years one of my kids progressed through math knowing how to multiply and divide but not knowing the times tables, then BAM, third grade hit and it was all memorising Pi and memorising this and that and everything else. Sure, my five year olds can do third grade math, but they're not third graders.
I think the point you made about the brain being ready for different aspects of mathematics at different ages is very important. It seems that a parent or teacher could waste months pushing the wrong topic, that could be covered in weeks later, simultaneously making math more of a chore than it should be.
To a large extent, the process of solving a new type of challenging math problem involves using a lot of math that a student already understands. For example, using a lot of arithmetic to solve simple algebra problems. If a child is unmotivated to practice arithmetic on its own, but is motivated to use arithmetic to solve algebra problems, it seems to me they are better off doing algebra.
We used one of those grids for help, especially with division. Have you tried Kahn academy? It's pretty fun and they can follow their interest.
My daughter is using the multiplication and division boards in her Montessori pre-school. I think they are a wonderful manipulative for tactile and visual learners. I'm formulating a plan to have her calculate some squares and square roots on it, too (which I doubt the teacher will present on her own.)
