You are making sense, and I don't want to sound like an apologist for Everyday Math, which I don't know nearly well enough to take that role. I do want to write a bit about this point:
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.
Yes and no. Let me witter, but please bear in mind that I'm making a case that I think interesting, not necessarily one I am totally convinced is correct - I'm in the process of thinking about this.
It's really deeply embedded in how we teach maths that we expect children not to move on until they understand. But I really am very far from convinced that that's the best way to teach maths. (I should confess - you may be relieved to hear :-) - that I have no involvement in teaching children other than my own, although I teach adults for a living.) It's certainly not the only way to learn maths - e.g. if one goes to a research conference on maths, with highly technical talks being presented from 9 till 6, absolutely nobody in the audience will understand everything, and yet in order to be getting anything from the conference they have to be getting some understanding even though a full understanding depends on something they've missed. I remember a very hard transition from the school model of not feeling allowed to move on with the tiniest thing unclear in my mind, to a mode of learning like that.
One might respond that this mode of learning is only applicable to adult researchers, but my response to that would be to point at how children learn language. We don't introduce a child to their native language systematically, one word or grammatical concept at a time, refusing to go on until we're sure the child has understood up to this point. Even with foreign languages, we understand that immersion is the most efficient way to learn, even though if you analyse the things that people have to understand in order to become fluent in a language, it looks just as dependency-structured as maths does. When you learn a language by immersion you make lots of mistakes, all the time; you fail to understand basic grammatical constructs, you form false hypotheses about what words mean and how grammar rules work. Gradually by greater exposure you sort it out.
Of course, you have to have lots of exposure to native speakers to make this work: you can't put 30 children with one adult who's scared of speaking German and expect them all to end up learning German! Maybe this is the real reason why we can't transfer this to maths. I completely agree that there's a big gap between what works for one child at home and what works for a class following a curriculum. In fact I'm quite happy to stipulate that EM is (in most hands) horrible.
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.
For me this is all part of letting maths be a real subject, that can really do things and be intrinsically enjoyable in its own right, as opposed to for the extrinsic reward of a row of ticks.
