re: (Not Common Core)-- Why are they teaching math this way?

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If you want to introduce a number line to show 4+3 = 7 simply to provide basis for more complicated calculations, feel free, but in the mean time the kid needs to be able to flat out memorize the answer is 7 to function. That is my problem, supplement vs replacement. I am all for supplement.

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We agree much, much more than we disagree. I take it one step further than you did in this statement and say that there's nothing wrong with testing their ability to apply the number line. Memorization should come after understanding the concept. Many people know that 6x6 is 36 but don't understand why. Then when 12x5 comes around, they must pull out a pen and paper or calculator. I think the more that the basics are enforced, the more it pushes one's ability to perform mental math beyond what is generally memorized ("times tables to ten" as an example). While what is needed to function in everyday society is basic, rapid repetition of such facts, I think that can be attained much later much more easily while the concept of application, if taught early, makes the memorization later much easier. And, in 1st grade, knowing 3+4 isn't useful to your everyday life. As a teenager in the store buying pre-bagged apples for your Mom when she told you to get 7, it is.