I wonder if, in time, we will see a regression back to city-states once urban populations get big enough. Tokyo is basically its own country, and the same goes for SF, LA, and NYC.
I believe the limiting factor on country size is 1) communication ability, and 2) transportation (force projection) ability.
Roads were a major factor in the size of the Roman Empire, for example. City-states were common when there was no force regionally large enough to conquer the city. City states also needed to maintain farmland surrounding them, so they could remain fed.
In my experience, it's because high school math is taught equally terribly. No... more terribly, because the subject matter is more complex. Useless busywork and rote memorization abound.
See, on this point we actually agree. I was appalled at Physics 11 and 12 for example; once I hit first year calculus and all the stupid formulas we were applying and memorizing v=1/2at^2 for velocity of an accelerating object etc.. just fell out of simple calculus cases. But Organic Chemistry and balancing reactions, that needed to be exactly what it was.
The paper you linked had a musical example... and berated the fixation on music theory, and was a good read. But at the same time, theory is good too, and the history of music too. It is not bad to teach and test those, its bad to ONLY teach those.
But elementary school math? I'm just NOT seeing the issue you have. They are drawing things, and piling them up, and working with sequences, nearly everything they do at the beginning is based around patterns and symmetry. All the times tables are introduced gradually, and as sequences, and visually. The relationships established between numbers, grids of squares, piles of beads. It doesn't seem bad to me at all.
Yes, memorization of basic arithmetic facts kicks in grade 3 and 4 but I just can't get upset by that. Its a small but important piece. And even if they "fixed" the latter years education, I'm hard pressed to imagine a curriculum that wouldn't be facilitated by having single digit arithmetic as a basis skill to draw from. Just as I can't imagine a written language course that didn't require you to have at some early point memorized the alphabet and their canonical sounds. (Or in the case of a language like Mandarin, the basic set and the rules that govern the alphabet..)
Just as your document mocked painting in terms of theory, and rightly so, there is a need to be able to name colours taught alongside the freeform expression of fingerpainting. Does a child need to know that colour they smeared from here to there in a pleasing squiggle is blue to make that blue squiggle? No all they need is paint and imagination. But they still DO need to be taught that the color is blue to be able to communicate. And that has to be memorized. There is no deeper understanding of the names of colours -- you just have to remember which are called blue and which are called green, etc.
Your linked paper went into detail talking about the joy of discovering analytic geometry by drawing a rectangle around a triangle, but how would you teach this if your students hadn't previously memorized what a rectangle and triangle actually were? And how would you teach the names of shapes? They are occasionally descriptive... quadrilateral, triangle, parallelogram... but why is it canonically called a triangle and rarely a trilateral? And what the fuck is a rhombus or a trapezoid or a hexagon? And usually what is meant by a hexagon is a regular hexagon, god help the kid who tries to bisect an irregular hexagon into 6 equilateral triangles...
"A similar problem occurs when teachers or textbooks succumb to âoecutesyness.â This is
where, in an attempt to combat so-called âoemath anxietyâ (one of the panoply of diseases which
are actually caused by school), math is made to seem âoefriendly.â To help your students
memorize formulas for the area and circumference of a circle, for example, you might invent this
whole story about âoeMr. C,â who drives around âoeMrs. Aâ and tells her how nice his âoetwo pies
areâ (C = 2Ïr) and how her âoepies are squareâ (A = Ïr2) or some such nonsense"
Yikes. I've never seen something so banal in my own or my kids education. We can agree that's terrible. But I can also stipulate that my kids weren't exposed to it either... has anybody actually been taught that? Was it ever more than a failed experiment? Tried for a few years, found wanting, and then abandoned?
The upshot, in my opinion is that something like the area of a circle, just like my physics 11/12 formulas really SHOULDN'T be taught until after the kids have learned trigonometry, periodic functions, and calculus... because those are necessary to really understand the answer.
There's no reason to memorize the forumula though. Ever. And I'm not sure they are expected to now.
Your linked article also writes:
"Mathematics is the purest of the arts,"
I'd argue that philosophy (logic) is purer still. Mathematics itself is a construct of logic. (And for truly fun mind games, take meta-logic.)
-cheers
The actual strategy is detente first, and then a full alliance with Iran throughout the Middle East and North Africa. It has been on display since before the beginning of the Obama administration. During his first presidential campaign in 2008, Mr. Obama used a secret back channel to Tehran to assure the mullahs that he was a friend of the Islamic Republic, and that they would be very happy with his policies. The secret channel was Ambassador William G. Miller, who served in Iran during the shah's rule, as chief of staff for the Senate Select Committee on Intelligence, and as ambassador to Ukraine. Ambassador Miller has confirmed to me his conversations with Iranian leaders during the 2008 campaign.
In a way, it would be a relief if true; the knowledge that there is at least some rationale to this spinelessness, however disgusting, would have a shred of merit.
There is no accountability.
That's not important. Really.
#OccupyResoluteDesk's confession that he has no strategy may be the first un-bent thing he's said in my immediate recollection. My suggestion is that he rely upon Nancy Pelosi for advice, and do the precise opposite of whatever she says.
Because not everyone needs to do that as much as others.
Well, everyone needs to do it several days a week in math class for the remaining 8 years of school left after they start learning arithmetic, as they learn algebra, analytic geometry, polynomials, pre-calculus... you know, grade school math classes that everyone does.... so there's THAT. Not to mention where it shows up in science
Arguably you need it more in school, FOR school, than you do as an adult. Although it's pretty valuable too if you want to do any STEM post high school, and STEM is something school SHOULD be preparing kids for, even if most of them don't go that route.
Further in my experience, the kids that have trouble with high school math are frequently hobbled because they can't manipulate basic arithmetic efficiently, and too much of their time and concentration is spent adding, multipliying and dividing coefficients that they don't have anything left to do the actual math. They can't keep up. Homework is a huge chore -- because they are spending hours on arithmetic... 8x - 4= - 4x; they spend their time not on the simple algebra manipulations... but 8+4 = ?, and then 12/3 = ?
Because not everyone who does it 'manually' does it at the same speed. Because some people use tools.
And either way 15 minutes of homework turns into 2 hour marathons and they don't even learn anything because too much time and energy was diverted from learning algebra that it becomes like learning chop wood, but having to carry each log for 20 minutes before you can swing at it. 2 hours of practice, and you've only actually swung at the log 6 times.
You can just memorize a few and then observe simple, basic patterns.
So now your argument isn't that we need to rote memorize the entire multiplication tables, we just need to rote memorize part of the multiplication tables? No shit sherlock.
Pretty much nobody would memorize 100 separate multiplication facts.
The 0, 1, and 10x tables... nobody "memorizes those" as they are just:
0x? = 0
1x? = ?
10x? = ?0
Then 11x? = ?? (not even in the single digit tables, but its so easy you might as well remember this too)
The 2x table is 100% overlapped with addition. If you know 4+4 you know 2x4. So nothing new to remember there either.
Then you can reduce what is left to a diagnonal matrix. Nobody has to remember 9x3 if they know 3x9, etc. The only 9x table fact that need to know is 9x9. The only 8x facts they need are 8x8 and 8x9. The only 7x facts they need are 7x7, 7x8, and 7x9, etc.
The size of the remaining diagnonal matrix is 1+2+3+4+5+6+7 = 28 lousy multiplication facts should be memorized along with a handful of trivial rules. That's the most anyone needs to even TRY to commit to memory to have instant recall of the complete set.
The names of the letters of the alphabet are just as "arbitrary" and there's 26 of those.
Sure if you forget 7x8 one day, recalling 7x5 and counting by 7s to get to 7x8 is perfectly fine, but if your having to do stuff like that all in high school, your seriously handicapping yourself.
You want to graduate and forget everything you knew about history, arithmetic, physics, and chemistry that's fine. You can probably "get along fine" with grade 5 literacy, and the ability to use a calculator for basic arithmetic. Millions do. But that's hardly a good thing.
And you shouldn't knock nihilism, unless of course, you can actually find something wrong with it.
True nihilism would lack any such self-awareness. You've made a funny.
Politics: A strife of interests masquerading as a contest of principles. The conduct of public affairs for private advantage. -- Ambrose Bierce