While looking into something else tonight, I came across the following:
From the beginning, accordingly, Everyday Mathematics was designed to take advantage of the spacing effect. An explicit attempt was made to ensure multiple exposures to important concepts and skills, spread over two or more years. As the First Grade Everyday Mathematics teacher�s manual states, �If we can, as a matter of principle and practice, avoid anxiety about children �getting� something the first time around, then children will be more relaxed and pick up part or all of what they need. They may not initially remember it, but with appropriate reminders, they will very likely recall, recognize, and get a better grip on the skill or concept when it comes around again in a new format or application�as it will!�
From
http://everydaymath.uchicago.edu/about/research/distributed_practice.pdfI quite possibly missed something here, because I can't figure out how they made the leap that "spacing" review and practice (is this the common meaning of "spiraling"?) must be at the expense of mastering the skill the first time around (except that they're hoping that not requiring mastery will avoid anxiety about math in the student?) Why can't there be both? Something about the quote makes me feel slightly ill.
Both what? Mastery the first time and later review and practice? But if a course is designed in the expectation that (most, or all) children will master a new concept the first time it is introduced, there would be no benefit in revisiting it later. I suppose you could say that it's useful to do a very brief review of something just to check it's still there, but if a topic really has been mastered, then by definition it'll still be there and you wouldn't want to spend more than a few minutes on checking that. I take this quote to mean that EM is designed so that
most children will
not master a concept the first time round, but just take a large bite of it, and get the rest the next time round. I know this is unpopular with people here because their children get these concepts the first time round and that's what they see as they "right" way to do things, but bear with me a moment...
... thinking about it, what we might call pulsed learning is indeed how my DS6 learns mathematical concepts that are
really at the level that's appropriate for him. (I've seen this recently with various things from proof and logic, e.g. the idea of decidability and the Halting Problem, proof trees, Goedel's theorems.) He reads about it a bit, talks about it enthusiastically, wants to try things out relating to it, has obviously some some correct ideas and some misconceptions. At this stage, he is not always up for discussing the misconceptions; if I try to correct one, he may pay attention for a moment and then fade out. I've learned not to push it, but just to say "that's not quite right, we'll talk about it some time". Then it stews for a while; he may carry on talking about the topic at the same level, or he may stop talking about it altogether. A while later - weeks, months - he'll pick up the book or start talking about the topic again, and off he goes again, enthusiasm intact, to do the next chunk.
Now, this kind of learning provokes anxiety in me, but I have come to think that that's probably a fact about me and my miseducation - like most of you, I always expected to get things completely first time at school, and never met anything challenging enough that I couldn't at that stage. However, what it is very like is doing research, and indeed, I think, solving any genuinely challenging problem, grasping any really stretching concept. Isn't it the school situation of first-time mastery that's unusual, really?
So I think it's very likely that a spiralling approach has something going for it. Trouble is, the idea that a course designer can lay down in advance what an appropriate amount of material to present so that everyone will be challenged enough that they don't get it all, and noone will be so overwhelmed that they give up, seems implausible. Certainly I can't predict that even just for my DS - we just provide the material and see what he takes.
