And the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and

And the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and other people believe that the procedural memorization does kids a disservice by divorcing mathematical thinking from the concrete world entirely. In my very limited teaching experience, I think the answer differs based on the student. At the individual level I don't think there's much controversy. Just align with the student's learning style. At scale, I have no idea and do not have the data/experience to have a well-formed opinion. I was perfectly happy to focus on mechanical mastery of the multiplication rituals well before I had any concrete reasons to use them. I know plenty of others didn't work that way.