OH this is so good. Thank you for "confessing" where you are in math with your daughter. I agree with all your shared, but agreeing isn't the same as action! When my son turned 15 I became so anxious because he hadn't crossed into algebra yet. But you know what, he isn't having a hard, frustrated time working on it! I still have my younger children doing their basic operations time-drills, but making the jump to preparing them for algebra and geometry seems to be a trouble spot for me. I so agree about the ability to do the process doesn't mean perception is present.
When you exemplified 7 + 2 doesn't mean the readiness for 9-2 or 17+2 it made me think of Common Core. As a 47 year-old adult, I love the idea of breaking down 17 into: 10 and 7, in order to mentally add the 7 to the 2, because I already know 7 + 2 = 9. But I've been playing with arithmetic for forty years! I really began to notice my comprehension of "numberness" only in the last ten years! No one was really brining it to my attention before then, so I am not saying students couldn't be awakened to it sooner, but not as soon as it being pushed now, nor in the way it's being pushed.
I feel like there is this idea that we are to be competing in these "subjects" with other cultures and nations. We can't compete with Japan or Singapore when their whole culture is arranged and focused differently than ours. We'd have to transform more than our curriculum to get their "results". Which, what even does that mean when we are referring to human beings?
I read that Leigh Bortins, the founder of Classical Conversations, had one son out of four who just couldn't seem to get past algebra. And she wouldn't let him either. He made a "C" all three years. She wouldn't move past it till he mastered it. But he apparently didn't hate math because he applied to an engineering college. They called Leigh to say he received a full ride scholarship! She thought they had made a mistake in the names or something . But they said because of his WRITING and his reading list, they wanted him. When she asked about his math, they said they would improve his math. Apparenlty he's done fine too! Amazing! Really goes along with what you shared. Colleges aren't just looking for "scores" or credits. You have to have actually comprehended what you were supposed to have learned, and you have to have a work ethic and the ability to submit to learning.
I don't want my children to hate learning. I love how you showed the link between your daughter loving math and therefore loving learning. The love of learning is certainly disunited when we harm one area of learning. - Lord have mercy, I don't want to do that to my kids, or myself as the teacher-learner.