I do think, though, that when we're paying attention to our own children on an individual basis it's important not to insist on them mastering topics in sequence. I think this is intimately related to the importance of learning by solving lots of problems. If they skip ahead to something that interests them more, fine: let them do problems to do that relate to what they're interested in. If an earlier skill really is essential, then they'll find they have to brush up those skills in order to solve the problem they're interested in, and really needing to understand something for one's own purposes is surely better motivation than just being told that this is what's next.
I agree with you on this quite a bit, except to the extent that we're talking about a math curriculum in a classroom, which is the scenario of EM (I'm not familiar with anyone using EM as a homeschool curriculum). Perhaps we ought to compare EM to Saxon, which I don't really know either, but I believe Saxon spirals with tons and tons of review, and might not have this issue that EM has over not expecting understanding. In a classroom setting, there's no reaching forward and then backward via intrinsic motivation. It's external, and it's not individualized. In contrast, I can envision a situation where a student could struggle with a topic, move on to see where it fits in the bigger picture, and then return to it with new zeal to master it, along with far more understanding that comes from context (just such a thing happened with one of my kids) - the mastery did occur, even though it may have been out of the usual sequence (which tells me that perhaps the sequence ought to have been reversed for his learning style). I don't get the feeling that that would be the case in an EM classroom at all.
I'm afraid that EM's allowing kids - especially non-gifted ones - to not understand essentially allows them to not learn things that are ultimately fundamental to their ability to do math. Here's some food for thought that I came across, though I'm not sure I can explain at this late hour why I feel as though it's relevant:
Mathematics involves three things: precision, stages, and problem solving. The awareness of these components and the ways in which they interact for basic stages such as the real numbers or the spaces of Euclidean geometry and the stages where algebra plays out are the essential components of mathematical proficiency. Perhaps the biggest changes in K�12 instruction that should be made to bring this to the forefront are in the use of definitions from the earliest grades onwards. Students must learn precision because if they do not, they will fail to develop mathematical competency. There is simply no middle ground here.
(from
ftp://math.stanford.edu/pub/papers/milgram/milgram-msri.pdf)
thanks for allowing me to think out loud - off to bed...
