I was thinking of procedural and conceptual learning and then turned to my eldests math learning arc. She started in a montessori preschool and went to a montessori school for her primary years. There she learned math concepts almost exclusively by use of manipulatives. where I learned multiplication by rote memorization of times tables, she used what they called 'cube chains'.

She spent HOURS working with those chains, skip counting to huge numbers (9 cubed, 10 cubed) playing with patterns and learning in this beautifully sideways manner.

Then she went to a public school where she was cruelly introduced to math minutes. Now, where in montessori, she might have had hours or days to immerse herself in a task or concept, she was expected to work in 3 minute chunks and complete things as fast as possible, rush in fact.

The result was awful that first year, and in fact the teacher gave me 'remedial' work for my daughter to do over the summer. ugh.

Then something happened, she 'got' something and things started to fall into place after that transition year. She hasn't looked back since.

If i see procedural and conceptual learning on a spectrum, one on the left and one on the right, i had been measuring them as equal and opposite, and unidirectional with the ideal starting at one extreme with the goal being the other.

Now I dont think that's true. at the montessori school, a student was given many concrete tasks, but not always given the tools to extrapolate to the abstract, at the public school the same student was given timed activities. Neither situation was inappropriate, what was missing was the motion, from one type of learning to the other, and that was reflected in her horrible transition.