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

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Mine wasn't. They simply use doubles as an easy frame of reference, as most people intuitively do memorize those. At no point was he ever asked to memorize doubles.

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So when yours was given questions like Q1 in the first post of this thread, what do his teacher mean by 'double facts'?

Both of mine were given complete units on visualizing but then completely memorizing double facts, 5 facts, and 10 facts. Then, as you say, those memorized facts become the framework for double -1 etc...