Where should kids be

Rob asks a question in the comments:

"If you start with a good instructional model what level of math should kids be doing at the end of sixth grade? (I am happy if you break it down by IQ group)"

I'll answer as well as I can, based on experience and reading:

2 Standard Deviations above normal (IQ 130, top 2%)...every single one of them is ready to be taught Algebra at 10, if they get 1 years of prior decent math instruction @ 9. And that's probably conservative.  All the ones who succeed here, though,  have encouragement, and support, and a teacher who actually understands math.

The 4 Stdv kids I've seen (of course, not many...probably a half-dozen) are able to (though they didn't all do it this way) teach themselves algebra from a tolerable book by themselves by 9 or 10, and most of the ones who did it did it at a college level (Algebra, Algebra II, Trig, Precalc all in 9 months).  I've seen it done solo-ish from a book at 8 (me tutoring occasionally), and also solo-ish off decent-to-poor software.

1 stdv kids (IQ 115)?  I'm assuming that Sudbury-attendance, or magnet school in a college town, or unschooling parents constitutes evidence of +1Stdv, and my experience matches Daniel Greenburg's.  Across-section of kids from 8-14 can do all of pre-algebra in 6 months, 1 total hour of instruction (over 2 days) a week, given interest. 

All this stuff is consistent, by the way with the Army research suggesting that IQ 80 takes about 5x as long to learn something as a IQ 120...normalizing via geometric mean, and stretching just a smidge...let's call it a learning speed difference of 1.6x per standard deviation.  2 standard deviations = 3+x learning speed.  3stdv=5+x.  4stdv = 10x

And...given the aretaevian multi-factor model...We need translation factors for interest and for teachers.

My best guess is that typical teachers' variance is worth about half as much as IQ variance: 1 stdv variance in teaching skill ~= 1.25x learning speed.  The data I've seen (90th %ile teachers are about 3x as effective of teachers as 10th %ile ones [in grade level advancement per year]) lines up pretty well with that.  I'll broaden this to constitute effective educational practice, because I think that in many cases, learning proceeds without any structure. 

The interest variant is HUGE, though...I'm inclined to place it at significantly larger than the value of IQ.  I've seen a(n unschooled) 2-3 stdv IQ kid (who was learning programming from me) walk in with weak algebra skills, and negotiate learning complex numbers from me.  In 2 hours of instruction, he learned enough complex numbers to follow up starting to learn of the C++ graphics libraries (solo), and come back in 2 weeks having built some fractal generators.   Interest (as economistically measured by willingness to throw hours off-assignment learning stuff) is worth large numbers.

Similarly, disinterest variance is HUGE as well.  Someone who is committed to not learning math, or not learning history is at a substantial disadvantage compared to someone without said attitude.  Unfortunately, I don't have a unit-wise measure here.  I'll say that I'm pretty convinced that "I want to learn it, let me find a teacher, or a book, and spend 20 hours trying, even if it's hard" is worth on the order of 2-3stdv of IQ.

And finally...there's cognitive development.  I have seen kids who were not cognitively ready for Algebra and formal operations.  I've seen some at 8, and some at 14 and some at 30.


My hypothesis (testable):

I believe that normal kids hit the level of formal operations necessary for algebra at around 10-12 (which lines up well with Piaget's number of 11).   After formal operations, I think that Algebra is a 1-year learning activity...

1 stdv of IQ is worth about 1-1.5 years of time to start, and about 1.6x learning speed.

Interest is worth up to +/- 3stdv of IQ.  Both bad teaching, and lots of failure can cause disinterest.

Effective educational practice is worth about 1/2 as much as IQ (per stdv), 1.25x learning speed,

Teaching proceeds at a fixed rate, and will slow down anyone ahead of the fixed rate.

Having no method (beyond a book) costs about 1stdv of IQ in making it harder (you learn about 0.6 as fast as you would with a well-targeted learning method, but for most (3+ Sigma) smarter folks, this is still faster than what any teacher is willing to teach at)...

My assumption is that self-efficacy and conscientiousness are also factors, that can at least match IQ in learning-speed for 2 stdv...but I don't know that the range is as large. 


Does that help?