"I don't believe are well-supported by the literature in educational psychology, cognitive psychology, or mathematics education."
Wouldn't that be a huge PLUS for me?
"One serious problem is the assumption we've made that the abstract building of the real numbers from the counting numbers starting with the Peano postulates and working "up" in abstraction from the allegedly concrete natural numbers to the ostensibly highly abstract real numbers is the way things really developed historically (and that kids' thinking does and/or should follow this order)."
Who starts with the Peano postulates? Who said that? What does that mean? See my second reason of my rebuttal to your next paragraph.
"I think Davydov and his colleagues have made a pretty solid case that this isn't how things actually work, and that in fact from a measurement, rather than a counting perspective, the real numbers are actually less abstract than the natural numbers."
There are three major problems with your statement.
The first is glaringly obvious, and that would be if Davydov were correct then I and all the many like me that got math so well in the traditional route were just a fluke, since we didn't do it "correctly". Do you happen to have a theory of learning that might account for that? And how do you suppose that the majority of actual mathematicians do not agree with this at all? Are you supposing that we don?t know what makes us tick?
The second fault is a common fallacy that you, Kirby and Bill are falling victim to. You are discussing curriculums in mature mathematical contexts which have entirely NOTHING in common with that of a child. Cantor sets. Peano postulates. That is hogwash. And both of those mathematicians would say so.
And finally, your idea of learning is wrong. You don't start with the abstract and build to the concrete. You have been involved with kids enough to know how PERSONAL reasoning and abstraction is. It builds from within, not from without and is entirely whimsical. And without context (which exists only in our head) there is nothing to build on. If you do it in that fashion you will get a semantical empty shell of a mind that you will not be able to pass off as authentic. A fate you are probably more than familiar with and that causes you great discomfort.
There are three parts to learning, syntax, context and concept. Syntax is the language, the notation, the medium of communication and it is the only external piece of those three parts. The human brain (in fact most all brains) has an uncanny ability to recognize syntax. I mean, even a preschooler can recognize the syntactical pattern of a fraction even though they don?t have a clue as to its significance. It seems to be a given that neural networks are masters at pattern recognition. It?s like the iPhone commercial, whatever the pattern, there?s a neuron for that.
Context however, is internal and it is what that fraction becomes in our head and this context is decorated with aspects the likes of which you couldn?t imagine. Well you could if you would take the time to ponder it and accept just how much it takes to be educated. In fact, most of Kirby?s writing is basically a regurgitation of aspects. And these aspects do not come out of thin air, they come from examination. If it is ?the number 5? then it is the examination, whether consciously or subconsciously, of everything ?the number 5? that decorates this context. We call that ?attention to detail?.
And finally, there is concept and concept involves the rules with which we compare and manipulate our highly decorated context as a means to a goal. Unlike context which shares an interface layer with syntax, concept is entirely internal and very personal. Concept is the biggy and it is not something you can't teach directly. That appears to be the talent section and the only way I know of getting it to improve is with a lot of work. Without it, you lose all that decorative detail in context because without concept there isn't any reason to discern between something like a 4 or a 5 (fourness and fiveness). There simply is no point.
You need all three of these parts to grow steadily in sync. We don?t parse language like computers parse programs, we make sense of it, we will figure the syntax out on the fly, and we can only do that if we have enough context (experience) to do that. And in order for us to form concepts and manipulate that context it must be highly decorated with aspects. All three of these things are very important and drive each other and any curriculum that raises one above the others has a serious flaw.
This does not mean that there are not portions of these curriculums that do have usefulness. And the best curriculums incorporate many things, but they stay within the bounds of keeping these three parts progressing in sync. In other words, they follow a traditional escalation of sophistication of reasoning. Other than keeping those in sync, there is no particular order. As long as you keep those in sync then any more discussion of ?order? is like trying to answer the question ?What came first, the chicken or the egg?? So now you know why I can look at a curriculum so quickly and tell you whether it is valid or not. I look at its balance in those three parts. And as an example, I gave Discovering Arithmetic points for validity. Its fault was not in these three things. It failed on authenticity, not validity. It was simply not algebra. I overlooked some of its excursions like fractals because I think excursions are good as long as that isn?t all there is. I said it had merit as a pre-algebra book, but it was in no way algebra and I don?t know what the hell the authors intended to come between that text and their much more authentic Discovering Advanced Algebra text. Discovering Algebra Again? And yes, I think having students discuss things in groups is NOT effective because that isn?t even teaching. I?ll explain the flaws with constructivism at a later time but generally it has to do with the learning cycle involving these three parts. It is simply too inefficient to keep up. That is why teachers exist.
Reread my puzzle analogy. That is as close to a perfect analogy as you can get as to how WE complete the minds of our children. If, while you are putting your puzzle together (that would be your son) and you are working on an easy section such as the edges, and you happen across a couple algebra pieces that go together, then put them together by all means. That's what you do with a real puzzle. Just don't go off looking for all the algebra pieces at that instant because you will find that they are spread throughout the puzzle and you are better off putting the puzzle together as it comes to you.
What does this mean? It means if you are engaging your son in mathematical activities and he asks something like "does one times a number always equal the number?" then by all means engage him, but remember this, that question comes from the concrete examples that he has been given and his personal examination of those examples. Not from thin air and not cause you told him. And questions (epiphanies) like those are the absolute key to what you seek. You cannot put them in groups of 5 and get them to ask those questions, and you cannot teach them about Peano postulates or Cantor sets and expect them to ask those questions. It takes time and as a parent you have an advantage in engagement over the teacher of 20 kids.
Since all kids do not have the luxury of parents that take the time to truly raise and educate them, then I think the best route to altering the outcomes we see in public education, and history, is to focus only on math, reading and PE in the early grades. I mean, if you want a revolution of understanding of mathematics then stop looking for some holy grail that doesn?t exist and get down to business. The notion of increasing the length of the day has some merit but let?s be serious. These are kids and you can only get so many hours out of them. The science and social studies can come from the discovery channel for a few years.