I actually don't. I had to think about it for a split second.

Sure and what did you think? You certainly didn't think 9+9 is 18 and 18+9 is 27 and 27+9 is 36 and 36+9 is 45 and 45+9 is 54, which is what you get from "an understanding of multiplication" without having any memorized answers. You seem to believe that you haven't memorized these things but if you are not counting on your fingers to add numbers together then you HAVE memorized these things, and you probably memorized them so long ago that you can't remember not knowing them. Certainly, there are tricks that make mental math easier, but those tricks still rely on you having memorized a sufficient set of single digit addition and multiplication problems.

I don't buy your logic that because someone has to think about it for a second, then they suddenly won't know what's going on.

It's not a second. If it's a second then they have enough stuff memorized to do the arithmetic. It seems that you've never worked with a student struggling with basic arithmetic such that it takes them 15-30 seconds to do 15+27, if they do it all without giving up. By that time the teacher is on a different problem and the student is falling further behind.

Using a tool often does not mean you don't understand how and why multiplication works.

Using a calculator before you understand how and why multiplication works means that you are likely to never understand. You seem to be under the impression that memorizing of tables is used *in place of* teaching how the operations work. Children are taught to add using tokens on a desk, dots on a line, and marks on a page, then they are pushed to memorize the single digit answers. Multiplication, similarly, starts with making rectangles of tokens and then moves onto paper before memorization of the answers.

The basic material is important. Memorizing a table or results, however, not so much

In arithmetic the only thing more basic than single digit addition is the numbers themselves.

The problem with rote learning is not in the memorization of simple arithmetic facts. The problem with rote learning in mathematics comes when you start doing mathematics instead of arithmetic.

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JimFive