We're going away for the night, so I may not get to it now, either, but I've begun reading the Comments thread.
Like instructivist, I appreciate this parsing of the difference between the Singapore spiral and the US spiral:
Singapore's Framework is an Additive Spiral that Builds Topic Content Grade-by-Grade; NCTM's Framework Is a Repetitive Spiral Approach that Covers Similar Topics Across Grade Bands
Spiral approach by content strand (additive)
Specific for each grade (K-6)
Clear, specific topics
Mathematically logical sequence
Spiral approach by content strand (repetitive)
Within broad grade bands (K-2, 3-5, 6-8 )
Absence of boundaries and inclusive
The fluency idea has raised a question in my mind.
What's good about spiraling-with-mastery is the fact that after studying the same material 3 years in a row people remember it for the rest of their lives:
Studies show that if material is studied for one semester or one year, it will be retained adequately for perhaps a year after the last practice (Semb, Ellis, & Araujo, 1993), but most of it will be forgotten by the end of three or four years in the absence of further practice. If material is studied for three or four years, however, the learning may be retained for as long as 50 years after the last practice (Bahrick, 1984; Bahrick & Hall, 1991). There is some forgetting over the first five years, but after that, forgetting stops and the remainder will not be forgotten even if it is not practiced again. Researchers have examined a large number of variables that potentially could account for why research subjects forgot or failed to forget material, and they concluded that the key variable in very long-term memory was practice.*(see below *) Exactly what knowledge will be retained over the long-term has not been examined in detail, but it is reasonable to suppose that it is the material that overlaps multiple courses of study: Students who study American history for four years will retain the facts and themes that came up again and again in their history courses.
Practice Makes Perfect -- But Only If You Practice Beyond the Point of Perfection
by Daniel Willingham
(When I mentioned this finding to my sister-in-law, who is a federal prosecutor, she instantly said, "That must be why law school is 3 years.")
I'm wondering whether learning to the point of fluency, as opposed to mastery-defined-as-percent-correct, would alter this finding.
The article is about overlearning, which the fluency people suspect is the same thing as fluency. However, I don't know whether the subjects in the "50-year retention" studies studies had overlearned the same material in each of the three years they studied it.