Like it or not, definitions are VERY important in mathematics. I agree that "understanding" helps make the memorization process easier but to progress in math you have to be able to re-create "tight" definitions of math terms.When someone is certain that math acquired by school methods is real math, then the above seems true.
I don't think it's too off the wall to say that school math has done more damage to math understanding than anything else. When math is pulled out of context, it seems the only way to understand it is by very difficult school methods. (Whether one does them in school or on their own.)
I understand math *much* better far removed from school than I did in school. Experiencing it in context has helped my understanding hugely. My daughter was able to absorb an understanding first before she tackled the formal stuff. It worked out beautifully for her. She has a much better understanding of how numbers work than I did at her age.
In school, I learned how to identify types of problems and apply the solution we were learning. That's rote learning. I didn't really understand what was going on beneath. I enjoyed it though. It was like puzzles. I got very good grades in math and went onto engineering. What I found after college was that real world problems didn't resemble book problems at all. Real life equations are messy. What you need is an understanding of what was beneath the drill. Without that, the problems are baffling. (Fortunately I'd found my forte in software engineering so didn't have to tackle them by hand.)
Educators *hope* that understanding comes from rote. For some it does. For some it's just puzzles. For lots and lots of people it just makes them feel stupid and they end up hating math.
Math shows us new things about the real world. Without the real world to describe, without someone with questions needing answered, math has little meaning except for puzzlers who love to play with numbers.
I don't recall ANYONE suggest that any kids be "dragged" through textbook learning. I did suggest that a bright kid might enjoy teaching himself at his own pace just by using textbooks.And that is the type of thought many have at the beginning of unschooling.
The world is larger than textbooks. There are much *better* ways to learn than textbooks. It's helpful for new unschoolers—who are already too familiar with textbook learning—to see all the other ways that kids can learn.
Seems to me that if someone show the potential to be a good musician you do him a disservice if you don't at least expose him to the world of mathematics.Again, this is new-to-unschooling think. It seems like if unschoolers are not saying "Show kids the textbooks and make them available" that they're being handicapped in learning math.
Textbooks are good for learning school math. It's a very limited, artificial math disconnected from the real world.
Until someone can see how math gets learned as a side effect of doing, it's helpful for them to stay away from textbooks. Puzzles (online and books), games, building, comparing, measuring. There's a whole real world kids should be playing math in before even opening a textbook (if they ever do).
That sounds, of course, like no unschooler should ever hand a kid a math text. It doesn't. It means that until someone can see how math is learned more profoundly other ways, textbooks will get in their way.