If you start with a good instructional model what level of math should kids be doing at the end of sixth grade?I answered this once...in the spirit of attempting to answer the question Rob wanted me to answer, rather than following my strong impulse and ranting extensively about how this is the wrong question. This is the post I wanted to write, now that I've calmed down:
This is the wrong question. Even asking the question probably causes more problems than it solves.
How about these questions instead?
How much math do people really actually need?
When would the lack of math cause problems for someone if they didn't have it?
How could you generate interest in math?
What matters in education beyond when someone learns something?
If a person wishes to be an Architect/Engineer...they really should have a couple years familiarity with Calculus (Differential Equations is just a different way to write the ideas in Calculus)...which really means 4 years of comfort with Algebra, or 5-6 total years of math study (after correspondence theory of numbers -- count 1 number per object -- which I've seen missing as old as 7 or 8).
If someone wants to study any of the sciences (social or otherwise), they ought to understand statistics...which can be done after 2 quarters of college algebra. Probably also 4 total years of math to understanding (1 year to do, 1 year to settle for each of algebra and statistics). Physical scientists (and perhaps economists) should also learn the calculus, and statistics with calculus.
If you're not a scientist or engineer, how much use is math in your life? I say you need to be able to estimate whether your calculator is right, and do basic decimals (for balancing checkbooks), fractions (for cooking and doubling recipes, and for home repair with measurements), and percentages (sales). And honestly, every bit of this that you need can be done in 1 year, any time before they need it.
Auto mechanics? Need some ratios for gear ratios...some understanding of what a PSI is for tires...but not much else.
Lack of math causes problems when you need to use it for something else, or when you need to use it in life. Most of it is irrelevant until you (a) have a job, or sometimes (b) an allowance, though it doesn't matter much then, or (c) are trying to do further academics.
SHOW COST OF MISSING?
I like math-y games. Also reality. Cook with kids, and double (or 1.5x recipes). balance checkbooks, or give an allowance that they can keep track of. It's not a cost if they can do everything they want without it...so you get to try to build a life where one of the following holds:
Either math's prevalence is obvious, rather than hidden like it is in normal life when parents keep the math out of sight of the kids.
Or math is a source of games and puzzles that the kid would like to figure out, and can use math to do so.
Social proof: other people doing math, using math,or playing with math at a level above what the kid can do is usually a much better motivator than anything a teacher can do...
Overall..if someone's not going to be a engineer or scientist, I'm not convinced that their life is more improved by knowing Algebra than by knowing something about Art History or about juggling.
And that's me with substantial math-fu, a math degree, and who believes that a good understanding of the world requires both concepts of limit and infinity not encountered until Calculus, and a good handle on probability and statistics. I'd bet someone without such a math background might suppose that one could get by with even less math than I suggest.
Now the question...supposing that it's only useful to scientists and engineers, why do we teach it to everyone?
IQ, Interest, Skill at learning solo, and self-efficacy (how much you believe you can succeed on-topic) are the most important factors in learning...and few of those are impacted (much) by school.
If you don't have interest, game over...no learning. Sure you may get test-passing...but I've tutored those folks: "test-able" and "learned" are two entirely different things. This is especially true if test-prep is disallowed.
If you don't learn how to learn, solo, then you lose as soon as you leave school. Since school does/should only comprise on the order of 1/5-1/10 of a person's life, a focus on school learning is stupid. Proper schooling should primarily focus on issues of how to learn, more than specific content.
Self-efficacy may be the most interesting, because the costs to self-efficacy of teaching folks when they're not ready are large...and because most learning is normally distributed...you're likely to have some folks in a class that are not ready for the topic. Is it worth damaging their belief that they can succeed, in order to follow a curriculum?
IQ...well, it's there...and seems not to respond to interventions.
Why the heck does anyone ask questions as misguided as "what should a kid be doing at various grades?" Clearly the question is bass-ackwards, and suitable only for messed up approaches that uses the teacup model of education (You are a vessel, and I will fill you with information).
< / RANT >