Although anecdotes are usually badly reported samples of size one, I'll answer with one. Trying to diagnose his problems with multiplication, I tried giving my grade school son variants of the same problem, written in "stacking" form, e.g. $123 \times 456$, $456 \times 123$, $1230 \times 456$, $4560 \times 123$, $1230 \times 4560$, etc. My hope was that after the first he would realize that rest could be deduced from its result with minimal computation. Instead I discovered that he had not learned to multiply well by "stacking" (he doesn't write neatly, and his columns are like the one in Pisa, so to obtain the final sum he adds digits from different columns; usually each row of the answer is correct, but the final sum is wrong). He did in fact perceive what I had hoped - he recognized that his five answers (four of them incorrect) were inconsistent, e.g. that the first two had to be the same and his answers weren't, and so forth, but this recognition didn't help him decide which answer was correct or where his mistakes were! The moral (for me at any rate) is simple (although perhaps this is just confirmation bias in operation) - for a student who doesn't have operational mastery of the basic rote operations it's silly to worry much about interpreting the operations being performed. Approaches that emphasize such interpretation are important and useful, but they can't substitute for learning some effective and efficient algorithm for reliably performing multiplications correctly. Rather they should accompany and follow the exercise/practice/drill that is their necessary precursor.