Let me come back with a question. Why would anyone imagine that multiplication tables are something to be memorized? How about knowing and understanding them instead? They are interwoven with all kinds of patterns. Why miss all that?
I don't see why it has to be a binary proposition. It seems to me that the child should first be given a conceptual basis for understanding what, exactly, multiplication is. Then, as the children are being asked to learn individual multiples, discussion of the patterns presented is valuable.
The final step should be demonstrating mastery of the tables through memorization for speed and fluency purposes, which will be tested shortly, when multi-digit multiplication and long division are explored, then moving on into decimals, conversions of fractions to decimals, etc. The child who hasn't memorized their tables by this point will come across as "slow," or "not mathy," with all the consequences that entails. On the other hand, the child who has memorized them will not be wasting unnecessary brain power on many simple operations during this stage, will therefore learn more about the bigger concepts, and will enjoy this stage of math much more.
I have seen children flailing and failing in middle school math primarily because they never learned their multiplication tables in third grade.
