> All the additional Common Core techniques seem to require some kind of visual representation,
Yes, because humans are not very logical or abstract thinkers. We are used to visualizing things. It's a lot easier to help students build the abstract and logical thinking required for mathematical literacy if they can fall back on visual representations when they need to.
> Whether it ends up helping students is moot; it helps put money into a lot of lobbyists' pocketbooks.
I mean I value helping students higher. And as you pointed out it can be drawn out.
> IMO the better teaching method is give them one way to get the answer. Once they've mastered that, then start looking at additional ways to reason about it.
Except that that doesn't work well. The point of giving them many different ways to reason about it is so that they can get the core concept and actually learn the concept, rather than just memorizing "the one way".
Yes, because humans are not very logical or abstract thinkers. We are used to visualizing things. It's a lot easier to help students build the abstract and logical thinking required for mathematical literacy if they can fall back on visual representations when they need to.
> Whether it ends up helping students is moot; it helps put money into a lot of lobbyists' pocketbooks.
I mean I value helping students higher. And as you pointed out it can be drawn out.
> IMO the better teaching method is give them one way to get the answer. Once they've mastered that, then start looking at additional ways to reason about it.
Except that that doesn't work well. The point of giving them many different ways to reason about it is so that they can get the core concept and actually learn the concept, rather than just memorizing "the one way".