kitchen table math, the sequel: how to talk about spirals

Saturday, July 28, 2007

how to talk about spirals

I am just now getting to the Singapore/Saxon thread (and I STILL have not read the 888 lesson - which is why I haven't weighed in - SORRY! I'm frantic about my book, stuck on a question about stereotypies & environment & the brain, etc.)

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 )
Imprecise topics
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
American Educator

(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.


Exo said...

In Soviet schools the content with building-on spiraling was set across a number of years. Lets say, biology would start in grade 5 and it will be Botany (2 times a week), then Zoology (grades 6-7, twice a week), then Human Anatomy and Physiology (grade 8, twice a week), then General Biology (grades 9-10, twice a week). Of course, topics were studied in logical order according to evolution theory, and concepts of cells, tissues, organs, functions were coming up again and again. It was allowing for understanding of complexity and patterns in living things and processes.
Here I have to teach general bio (Living environment) to students who have no idea of plants stuctures, animals, or human's anatomy and functions; I have to mix it all together with emphasis on molecular biology (that requires knowlege of chemistry)and genetics, that heavily connects with evolution. Many of my honor students don't know that insects pollinate the plants!

When I was in school, by the time we had general bio, we knew evolution of plants, organization of animal kingdom, anatomy of humans, and were up to organic chemistry in chemistry class and nuclear physics in physics class that was well alligned with molecular bio and gene engineering. Oh boy... And when in vet school, we took the entrance exam in bio (covering everything in curriculum from Botany to General bio) and didn't have a general bio class ever again.

BTW, from my trip to Ukraine: the school curriculum is still in place the way I had it - still logically built, but spread over 12 years now. However, there is no school that would have students sitting 8 periods per day. The maximum number of periods students can have is 6 in grades 6-12, and 5 in grades 1-5. A period is 45 minutes. The school year starts September 1st, and ends May 25th, with final exams in higher grades scheduled from May 28-to June 12.

le radical galoisien said...

It is in part due to the flexible module system, isn't it? Sometimes the prerequisites aren't really cohesive ...

My current American high school generally has it to take one science subject per year, and thereby acquire the entire credit for that subject in one year. The next year, another science subject is taken, and so on.

I assume this is standard for most American schools. At first, I thought it rather interesting compared to my old schools in Singapore, but now I am questioning their effectiveness. After all, I have not done any real biology for so long, none of the material is fresh in my head, even though I am very highly interested in oncology and so forth. The same applies to mathematics in terms of matrices, which were taught to me three years ago and were only re-introduced recently in calculus class following the AP exams.

It is this that makes me miss the Singaporean system and especially since I worry that I am falling behind my Singaporean friends.

Perhaps the whole problem is the credit-based system. Credit units should not be used until university. Having to accommodate classes to the nature of credits at a secondary level can make the interaction of knowledge between those classes ineffective.

Catherine Johnson said...

Just noticed your comments! (And am too tired to read at the moment!)

Tomorrow I'll read, pull these up front, and take a look at my Singapore books to see how they spiral place value across the 3 years they teach the subject.

I have to look through my Saxon books one of these days, too.

But that's hard ---

The principal in Tested says Saxon books have a "recursive" structure.

I'd never heard that before, but it "clicked."