When kids aren't able to "make a case" for what they are doing as far as algorithms go, what can they really do? One can train a parrot to talk but a parrot can't string thoughts together, just "parrot" back words. Kids who memorize a procedure but can't explain why it works are kind of like the parrots. So many parents still want to push their own kids toward speed in recall over understanding how mathematics works but, in the end, many of these kids eventually end up facing mathematical brick walls when their memorization bank is full. Mathematical understanding enables kids to climb over or dig under or go around these walls. Algorithms for some kids lead to understanding but many, many more need a chance to "mess" with the mathematical connections behind algorithms in order to understand how they work. Encouraging a variety of ways to get to a result and requiring a mathematical justification for the result will reach more kids.
