[jmeagan]

****Or perhaps you remember that 3*7 is 21, so 6 * 7 = (3+3) * 7 = 3*7 + 3*7 = 21 + 21 = 42 etc.******

OMG, do you really think people would do problems that way, and be proficient?

Just like you do not have to understand the working of an internal combustion engine to drive a car, you do not need to understand all the concepts to be proficient in math for everyday life. However, you do need to know the basic procedures, which are easily learned by rote.

[cartan]

****Or perhaps you remember that 3*7 is 21, so 6 * 7 = (3+3) * 7 = 3*7 + 3*7 = 21 + 21 = 42 etc.******

OMG, do you really think people would do problems that way, and be proficient?

Why not? I certainly do, and have done it already as a little kid. Of course, you don't do it algebraically like that. Look exactly what is written there, then you will see that it is just an algebraic way of doing this:

•	•	•	•	•	•
•	•	•	•	•	•
•	•	•	•	•	•
•	•	•	•	•	•
•	•	•	•	•	•
•	•	•	•	•	•
•	•	•	•	•	•

=

•	•	•
•	•	•
•	•	•
•	•	•
•	•	•
•	•	•
•	•	•

+

•	•	•
•	•	•
•	•	•
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•	•	•
•	•	•
•	•	•

=

21

+

21

=

42

You can do that sort of thing easily and quickly in your head, but only if you practice “helping yourself” like that when you're stuck.

[metmom]

People should be able to do something so basic.

It’s not like functions or derivatives.