Output vs. exploration and discovery
My husband tends to show the math shortcuts: adding columns before my 8yo is really clear on place value. Times tables. Multiplying numerals with "0" in them.
I tend to place more value on exploration. Let her work through basic multiplication so she is understanding the "four-ness" of 2x2, for example.
DD1 likes both approaches, but I hesitate when dh goes straight for the memorization and technique. She loves what he teaches her. I'd like her to take the path of deeper understanding.
I know the best thing is in the balance: at some point, those "shortcuts" can be mighty useful. Can they?