Yes, I mean both mastery and review. Or perhaps a more precise way to say it would be understanding and practice. I think practice is important even where there has been understanding - with some kids, even though they understood the first time, they may not remember it well until they've used it again and again in different contexts. But that's not the same as not understanding it and then being asked to use it in different contexts in the hope that the light bulb will suddenly turn on. What if the light bulb never goes on - a whole lot of learning built on whatever the misunderstood concept was is also going to fail.
I have two problems with the EM approach as set forth in that quote. The first one is about anxiety about math. I can't think of a better way to create more anxiety than to constantly introduce concepts that the students are not likely to understand, or aren't expected to understand. Self confidence is generated by actually completing challenging tasks, not by being allowed to slide by without learning what has been presented.
My second issue with it is more fundamental. A couple of my kids are definitely big-picture-before-the-details kids. Without a context, there are no memory hooks on which to hang information presented. From one angle, EM's approach may sound like it make sense, that they'd learn the info better the next time around when it's presented in a(nother) context. However, kids like mine are less likely to remember anything from the prior experience with the information precisely *because* they didn't understand it (except perhaps they'll recall the uneasy feeling that goes along with not understanding), so the time spent on the information previously may well have been wasted at best, and couterproductive at worst, as it builds a lack of self-confidence with the topic.
I can see the importance of providing a big picture first - I think that can be very helpful for some kids. But I do not think that's what EM does with its spiraling or that that has anything to do with the way EM is set up (just from various things I've read, years may go by before a big picture is evident in EM, if ever). In my opinion, purposely and continually introducing things that the students are not expected to master provides a convenient excuse for a teaching method/explanation/instruction, etc. that just didn't work. If the child didn't understand the first time, then the topic would be better approached differently during the next lesson or during extra help sessions, not six months later in the middle of a new topic. (that's me, the expert on teaching LOL.) So much in math depends on understanding what one has previously learned - I wonder what happens in the meantime before the topic comes back around again.
I think the EM approach is likely to be even more problematic for an average or struggling student than one who understands math easily.
It might be that I really don't know EM well, but this is my take on that quote I found last night. To me, the way you described your ds's learning of new and challenging topics is not analogous to a curriculum applied in the classroom to a broad range of students for a subject in which it is often critical to build on a sound foundation before moving to a new topic. I guess that is the gist of what bothers me - I don't see how a student can learn a new topic if the new topic requires knowledge that they failed to learn during the previous topic. I'm not sure if I'm making any sense...time for more coffee
I haven't yet had the coffee and here I am already editing. I can envision a situation in which the student doesn't understand the bits until it's presented in the larger context (the big-picture-first kid) but that still leaves the first go-round as a waste, at least for that person. For that person, the "backwards from traditional linear curriculum" is not the same thing as spiral learning.
