I think the heart of the issue is whether it's more useful to teach children how to multiply two abstract numbers together as a kind of "mathematical procedure" that they need to memorize, or whether it's more useful to teach children that if they measure two sides of a square with a measuring tape, they can "multiply" the measurement and that the result is now in "square inches" rather than regular inches. 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.