Unfortunately, the mathematicians who made up the California Mathematics Content Standards seemed to assume that the young people of California are characters from Heinlein novels.

Here, for example, is the very first of the 25 items in California's Algebra I content standard (to put that into perspective, LA public schools students must pass Algebra I, Geometry, and, beginning this fall, Algebra II to graduate from high school). This is what California 8th or 9th graders are supposed to learn on roughly the day after Labor Day when they first begin Algebra I. (Although in many cases, they are 10th, 11th, or 12th graders who are trying to pass Algebra I for up to the fourth time.)

1.0Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:

Now, I'm sure most of you are saying, well, ho-hum, of course everybody knows the closure properties for the four basic arithmetic operations. And how can students move on to studying Abelian closure without being introduce as soon as possible to simple closure?

Unfortunately, I'm not a Heinlein character, so to be honest, my eyes glazed over when I read that standard. With some effort, I've finally managed to focus upon what the words are, so I've been able to move on to trying to find out through Google what they mean.

Dr. Anthony at Math Forum says:

Well, that's rather interesting ... but is this level of abstraction appropriate for the first thing taught to public schools students? In the Los Angeles Unified School District, less than one out of ten students will score 500 or higher on the SAT math test. What about the other 90+%?The idea of 'closure' is actually very simple. If you add together

two whole numbers, you will always get another whole number. If you

multiply two whole numbers, you will get a whole number as a result.

So we say that whole numbers (integers) are 'closed' under the

operations of addition and multiplication.

What about division? Well 12 divided by 2 is 6, which is a whole

number, so in this case we get a whole number result. But 12 divided

by 5 = 2 and 2/5, so now we have moved out of the field of whole

numbers. If we divide two whole numbers we cannot guarantee that the

result will still be a whole number. So the set of whole numbers is

not closed under the operation of division.

Positive whole numbers are closed under addition - you always get a

positive whole number in the result. But they are not closed under

subtraction, since, for example, 4 - 9 = -5 and -5 is not a positive

whole number.

To decide whether a set of numbers is closed under some operation or

other, look for cases where the result is no longer in the set you

started with.

In the case of real numbers, which include positive, negative,

fractional, and irrational (like sqrt(2)) numbers, the operations of

addition, multiplication, division and subtraction are all closed (apart

from division by zero which is not defined). But taking square roots is

not closed because if, for example, we try sqrt(-5), we no longer get a

real number as a result. In fact, we have moved into the realm of

complex numbers.

I suspect that the mathematicians who dreamed up these standards wish that they had been taught like this in high school. They wouldn't have been so bored if their courses had been geared at a much higher level of abstraction.

So, this is how they get their revenge on the assistant football coach who bored them so badly when he taught them Algebra I -- by making him try to explain, on a hot day in early September, the closure properties of the irrational numbers to high school freshmen who add and subtract on their fingers.

It's just another little victory in the endless war the right half of the bell curve is waging so successfully on the left half.

My published articles are archived at iSteve.com -- Steve Sailer

## 37 comments:

I suspect that the mathematicians who dreamed up these standards wish that they had been taught like this in high school. They wouldn't have been so bored if their courses had been geared at a much higher level of abstraction.Eh, but the vast majority of kids are going to forget all the mathematics they ever learn beyond basic arithmetic anyway... might as well teach to the smarter kids.

Besides, the thing about closure properties is probably actually easier than things like completing the square and other mainstays of high school algebra.

God forbid Steve write an article that doesn't mention football or sports in some way.

A high school diploma is a certificate that you can get up in the morning, get dressed and show up (or have parents who can make you do so), figure out how to behave just enough not to get thrown out, follow simple instructions, and have very basic literacy and numeracy.

Don't tell my teenagers, but all the rest is commentary.

I found the discussion of closure quite interesting, but I doubt it will beat out Britney in the popular mind any time soon.

21.0Students graph quadratic functions...22.0Students use the quadratic formula or factoring techniques or both...23.0Students apply quadratic equations to physical problemsThis Quadratic fellow must have been pretty important!

What is the whole point of all this theory?

I thought 1st algebra was about

X=2Y+200; solve for x.

When will everyone realize the wisdom of the greatest 20th century American philosopher, Barbie.

"Math is hard"

It sure is.

But I'm not so sure that showing kids hard math is such a bad idea. It seems that math is something that smart people from any background background can master. Sure, the less bright students will be lost, but the few who are smart will be easier to discover in more g-loaded classes.

I remember a friend's young daughter complaining about the Elements of Set Theory and her elder brother advising her not to worry "Just memorise it for the moment and then you can forget it - it never gets used."

So, this is how they get their revenge on the assistant football coach who bored them so badly when he taught them Algebra I -- by making him try to explain, on a hot day in early September, the closure properties of the irrational numbers to high school freshmen who add and subtract on their fingers.You know, maybe we shouldn't be using "assistant football coaches" to teach important academic subjects.

Just a thought.

Excellent posting. I'd add only that in this case it isn't just a war against the slow half of the bell curve, but a war against those without the math gene. I'm pretty bright in a general kind of way but fall instantly to sleep (from confusion, inadequacy, and lack of interest) whenever math topics come up. In the old days, I managed to get through high school and even through a fairly fancy college. These days -- well, if they're serious about demanding algebra 2, would I be able to graduate from a California high school at all?

And here's another question: Of what possible use are courses like Geometry and Algebra 2 to people who will never go into math-and-science fields? I'm having a full and enjoyable life, and I'm managing to bring in a middle-class income, and I never need to do anything math-y beyond basic arithmatic. Which means that the years that I spent in math class beyond basic arithmatic really were the needless torture they felt like.

Why do we do this to kids? It seems like the purest kind of sadism.

There is a lot of silly stuff they teach in algebra class, and that I had to teach in my tutoring center, that comes from New Math of the '60s and the Bourbaki group's approach to math teaching. Both are highly formalized and deductive.

If you ever had to learn what the commutative law was, it's their fault. You don't need that until you study linear algebra or abstract algebra, which few are going to do anyway.

About closure, here's how I helped my students get it: to do a basic arithmetic operation, you need two numbers. You can't say, "What's 5 plus?" So, imagine you're taking these two numbers from a bag -- does the answer also come from that bag? If so, then the operation is closed for that bag. (That helps tie it in with the term "closure." You never leave that bag.) If you have to go to a different bag to get the answer, it's not closed.

If you have a kid in algebra, or are a teacher, I've found that picture helps a lot.

I took Algebra I in 8th grade before even going to high school, so I don't see that requirement as being too onerous.

What are these kids supposed to be learning during their four years of high school?

Too abstract for that level.

Philosophical induction is how the mind works. Hit a four-year-old with "The properties of furniture are various" and he will not understand what you mean - until and unless you point out to him real, physical, tactile chairs, tables, stools, etc. and name them, and then say: "Stuff like this that we're using for sitting, lying down, or putting things on is called furniture. Now see how the stool is taller than the chair? Furniture comes in different shapes, doesn't it?" Then on to define "various," "properties," etc. Finally the original sentence - "The properties of furniture are various" - will mean something to the kid. It will have content.

You have to start with the concrete and move gradually up to abstractions (abstractions are abstracted from the concrete).

Now hit a 16-year-old with "The closure properties of whole numbers..." It's pure, unadulterated gibberish - until and unless he knows how to add and therefore can begin to understand mathematical patterns.

Starting with the highest level of abstraction (even with simple content) and then working your way down to examples is the pattern of deduction. Pedagogically, deduction is a disaster: a conclusion stated prior to examples is meaningless and makes the subsequent examples meaningless. The attraction of it is that it's the general method of mathematical proof. But you

proveonly what you already know! Tolearnsomething requires philosophical induction. Start with concrete instances and then gradually introduce the abstractions that organize them.Yes, group theory and real algebra (not what they call in high school "algebra") is probably going to be too advanced for the average LA high school student. But how many kids will even be able to recall after they finish high school that the solution to a quadratic equation is "

xequals negativebplus or minus the square root ofbsquared minus four timesatimesc; divide all that by two timesa?" My father, a civil engineer, required me to know that sort of stuff, but most kids won't recall it and won't need it after high school. How about summing geometric series, using the binomial theorem, or remembering what they're taught about conic sections in high school algebra?Given that math is the most difficult subject for a lot of students, the question of what math they should teach and who they teach it to in high school is something worth serious debate. I think we all would agree that a capable student should have the opportunity to pursue higher mathematics if they choose to do so. But how much of what is taught in high school does an average or below average kid really need to know? Wouldn't a lot of these kids benefit from a more concrete mathematical regimen emphasizing basic arithmetic skills, solving proportionality problems, elementary pre-calculus statistics and probability theory (of which they current teach very little in high school), and so forth?

It sounds like you think, because only 10% of the kids are getting over 500 on the SATs, we should keep the standards low, so that every kid can graduate high school? Why should this be a goal, when US students consistently underperform other developed and undeveloped countries - ignoring our failures won't help these kids compete.

Furthermore, basic algebra is part of what makes a successful person who can manage his or her own finances. Sending kids away from high school without a basic understanding of numbers, into a world where they will be pitched credit cards, adjustable rate mortgages, and payday loans, is just a failure.

i am well within the right half of the bell curve and i am young enough to remember attempts to teach math this way. it is relatively useless abstraction that a "mathematician" with 20 years teaching high school algebra must have dreamed up to make their life a little less boring.

Algebra I is a hard class for the smart kids. I remember my college prep class at a 95% white school back in the day (the 1960s). Half the class flunked. I got an A but had to apply myself. Expecting ordinary high school students to pass Algebra I (much less Algebra II) is nutso.

I agree with Steve. Ideas of "closure" are just not very interesting unless you are going to do abstract algebra and move on to study general groups, fields, etc. Not only the vast majority of students, but even the vast majority of *teachers* lack the cognitive ability to appreciate abstract algebra.

Completing the square may seem like a drag, but at least it's possible to write a word problem where such algebra has a practical application. I defy you to write a word problem that shows practical application of closure properties.

(Here let me try:

Latrella is designing a computer architecture for a processor that needs to be able to add and subtract, but not multiply or divide. What numbers need to be representable in the computer's hardware? What types of approximation errors might result?)Remember, at the time the standards were written, and continuing, the standards essentially needed to be one standard for all. As we see from the application of the Texas standards and the NCLB, teaching goes only to the minimum level of the standards and associated exams. Smart kids in otherwise not high performing schools get screwed by this more than any other group. Higher standards can help protect particularly those kids.

BTW - is it unreasonable to think that the holder of a high school diploma should be able to do a significant portion of basic algebra?

Ideas of "closure" are just not very interesting unless you are going to do abstract algebra and move on to study general groups, fields, etc.I disagree. The closure concept is the ONLY way to introduce negative, rational and real numbers in a reasonably intuitive way. We're talking about high-school students here: they're not expected to come up with formal definitions or proofs.

anonymous:It sounds like you think, because only 10% of the kids are getting over 500 on the SATs, we should keep the standards low, so that every kid can graduate high school? Why should this be a goal, when US students consistently underperform other developed and undeveloped countries - ignoring our failures won't help these kids compete.If there is a true "Algebra" standard, and if real [as opposed to fake] tests are administered to enforce the standard, and if the results of those tests are published, then what they will show is that only low single-digits' worth of Blacks & Hispanics will be able to cut the mustard [just off the top of my head, I'd say maybe 3% to 6%, although that might be a little high - it might be more like 1%].

So then The Powers That Be will face a dilemma:

1) Preserve the integrity of the exam, but flunk out 95% or more of all Blacks & Hispanics.

2) Retain the exam, but water it down to the point that it becomes meaningless*.

3) Ditch the exam altogether [ipso facto ditching the "Algebra" standard with it].

*Cue the typical jokes [substitute "Algebra" for "Math"]:

Teaching Math In the fifties: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. What is his profit?

Teaching Math In the sixties: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price, or $80. What is his profit?

Teaching Math In the seventies: A logger sells a truckload of lumber for $100. His cost of production is $80. Did he make a profit?

Teaching Math In the eighties: A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20 Your assignment: Underline the number 20.

Teaching Math In the nineties: A logger cuts down a beautiful forest because he is selfish and inconsiderate and cares nothing for the habitat of animals or the preservation of our woodlands. He does this so he can make a profit of $20. What do you think of this way of making a living? Topic for class participation after answering the question: How did the birds and squirrels feel as the logger cut down their homes? (There are no wrong answers.)

Teaching Math In 2005: Un ranchero vende una carretera de madera para $100. El cuesto de la produccion era $80. Cuantos tortillas se puede comprar?

i am well within the right half of the bell curve and i am young enough to remember attempts to teach math this way. it is relatively useless abstraction that a "mathematician" with 20 years teaching high school algebra must have dreamed up to make their life a little less boring.I think a lot of writers of mathematical textbooks are trying to explain the foundations of their subject to students so that they can provide a rationale for all this other stuff they teach in those textbooks that students won't actually be using in real life (or, if they could use it, will not be able to remember it).

I find it somewhat distressing that so many people have trouble with algebra, but I would also have to admit it would be better that average kids gain experience with elementary statistics instead of learning esoteric skills like determining the asymptotes of a hyperbola or finding the coefficients of a term using the binomial theorem, skills that they won't be able to recall even six months after getting through algebra and will almost certainly never use. How many middle-aged adults do you meet who can even factor trinomials? How many adults have you met that have ever needed to factor a trinomial outside of high school or college algebra?

Let kids who enjoy algebra take it and let those who don't focus more heavily on statistics and probability theory. Math is a subject that could be improved by offering more than a single track for all students. This would fit roughly with Steve's proposal for multiple versions of the high school diploma: testing out in algebra (provided a high score) would be a more prestigious alternative to testing out in elementary statistics. My motto for mathematical education reform is "diversity is strength!"

The United States is going increasingly insane.

I'm sure the data exist. If no one else has it, the armed forces surely do. What are the reasonable minimum IQs for learning genuine algebra I, geometry, and algebra II? Does anyone know? Does anyone have a reliable source?

Excellent post above on "teaching math".

Along those lines: In 1987 this story was fresh; In 1997 this story was familiar; In 2007 this story, along with the iSteve commentary, is boring.

Steve, your hometown Los Angeles is dropping off functional America's radar. And your California stories increasingly register as missives from a boring, brown version of the USA. In another ten years you'll need to move to another state in order to maintain an audience for your local color stories. Because the truth is that "hot-blooded" Latin culture, with the corrupt plutocracy sitting on top of the dysfunctional brown masses, is intensely boring. There is little innovation in art, and no innovation in science. And there is only a sideline role in world events.

Have you considered that if you were writing in Spanish and headquartered somewhere in Latin America that your blog would fall on deaf ears? That it wouldn't "compute"? In a Latin-American country you wouldn't have the Reconquista to generate sparks. Your math/science topics would generate mass clickaways. You'd be reduced to easy-to-understand Latin-themed movie reviews.

No, Steve you'll eventually have to move in order to keep your blogging hobby interesting. You could go up to the Bay Area and start regaling your readers with the Asian-American phenomenon. With tales of going to the mall on the weekend surrounded by nothing but Asian faces in Silicon Valley. Or would that be boring too? It does seem that the network show bookers can't maintain audience with Asian guests. But I'm sure you can figure something out.

Here's something I don't think has been said before, though it is implicit in Steve's many posts on this topic: those engaging in the Standards of Achievement arms race have a flawed premise as their motivation.

They believe that any kid can learn any material so long as they are properly taught it and its prerequisites.

Starting from the end and working backwards, the prerequisites part is motivation for the standardized tests in NCLB. If kids don't know 6th grade math, you can't age-graduate them to 7th grade then expect them to make it. Fair enough, but there is an obvious error.

Some kids aren't smart enough for 6th grade math. At least at the typical age for 6th graders. So, while you can hold them back until they do understand the material, it is obvious this approach doesn't scale. 7th graders driving to school and 11th graders that can buy beer make for disruptive classmates.

Finally, even if one were solve the problem of what to do with markedly older kids, it should be obvious every person has a limit to which they can intellectually grasp. At the extremes this is obvious, Differential Equations is the realm of engineers and mathematicians. But everyone has a limit, and for some it may lie within the regular high school curriculum.

So, the question, paraphrasing what was orginally put by Charles Murray is, what makes everyone so certain the high school curriculum is within the mental abilities of 100% of the US population aged 18-20? And what makes everyone so certain making the curriculum harder necessarily makes students smarter?

And so it is here where the IQ deniers meet their Waterloo. They cannot admit to IQ or a bell curve for its distribution. Thus they are left with the flawed model in which all children are capable of learning all the same material. Thus everyone must be capable of college (and a tier one at that). Thus the only jobs worth caring about are those of the college graduate.

The closure concept is the ONLY way to introduce negative, rational and real numbers in a reasonably intuitive way.Huh? This isn't intuitive at all. The number line is intuitive: a representation of having a credit or debit of something.

You can't explain the abstract by means of the even more abstract.

You can't even explain the concrete by means of the abstract.

What you can explain is the abstract by means of the concrete.

What you can "intuitive" I call a mental Rube Goldberg machine. A giant apparatus of levers, pulleys, and flying balloons employed to ring a doorbell.

(Very akin to complex theories of how to teach reading - which end up producing more illiteracy than would otherwise occur if McGuffey Readers were used.)

The "closure property" is intuitive PRIOR TO knowing what negative numbers and the like are? Wow. That's like saying the limit in calculus is intuitive before being introduced to whole numbers. You've got it backward.

It's true that very few people need to know the math that's taught in schools, including Algebra, let alone Geometry, Analytic Geometry (which is what Algebra II consisted of in my day), or trigonometry. This is true even for most kids who will go on to college.

Statistics, even if it were predominantly qualitative, would be useful to teach. Especially the fundamental statistical truth (which still eludes many people) that there is a difference between "some" and "all" and "most" and "none". I.e., if I say that women tend not to have as much aptitude as men for math, I mean most women, not all women. And the mention of a single counter-example - one women who is very adept at Math - does not change the truth of that statement.

But why study Algebra? How about in order to exercise their minds. For many students in the mid-range, even if they'll never use it, the study of it may make them smarter, at least within the limits of their innate intelligence.

With standards like these, it is no wonder that SAT prep classes, where they actually teach you to solve algebra problems, are so popular.

Huh? This isn't intuitive at all. The number line is intuitive: a representation of having a credit or debit of something.Actually, the number line uses a basic conceptual metaphor known as "Motion Along A Path" in cognitive-science-of-mathematics jargon: this is why it's more intuitive than a formal closure-based approach. But even then, you need closure in order to conceptualize what "subtracting a negative number" means.

In other words, the closure principle is what has allowed us to "explain the abstract by means of the concrete" in the first place. See "Where Mathematics Comes From", which relies on this very example in order to show the importance of "closure" to mathematical cognition.

But why study Algebra? How about in order to exercise their minds. For many students in the mid-range, even if they'll never use it, the study of it may make them smarter, at least within the limits of their innate intelligence.I'm not opposed to all algebra for average students. Obviously, a lot of elementary algebra is quite useful in solving real-world problems, but I think a significant portion of it is just a waste of time for the non-mathematically inclined student.

For bright kids capable of learning at an accelerated pace on their own, I would recommend courses algebra, Euclidean geometry, trigonometry, statistics, probability theory, formal logic, calculus and, after calculus, perhaps a brief course or two in something more exotic like non-Euclidean geometry, graph theory, game theory, elementary number theory, or group theory.

I made it through Algebra I & II, Geometry, Trig & Physics, mostly with Bs. I also did well on the math portion of the SAT. None of this made it possible for me to do Calculus or even pre-Cal at the college level. Why? Did my Bs really indicate competency in the 5 math courses I took in high school or did they merely represent the fact that I did my homework & my test grades weren't so terrible that they brought my average down to a D?

Going through the material of each class wasn't enough for me. I'm not completely innumerate and I did my homework. Why didn't all that effort = enough mastery of the subjects that taking a slightly more complicated version in college wasn't beyond me.

Unless I have some weird learning disability, I think I'm a good example to study regarding the mismatch between high school performance & what can be expected at the college level specifically with math b/c a course in study skills can't fix this. Also, my IQ is certainly above average & I've always performed well on the math sections of standardized tests.

I think part of the problem is grade inflation even at good schools. I know I wasn't objective enough to think that the average of my test scores was how I should really assess my ability.

Then there's the practical, concrete math that most kids who can pass Algebra I can probably do well. Business math is pretty difficult yet I think it's only an elective if taken in high school. Give a concretely numerate kid a year of business math, a semester or a year of bookkeeping/accounting along with some algebra and that kid can go to work right after graduation or at least make informed decisions about credit and education loans.

I know it can be helpful to challenge students to solve difficult problems but some education theorists believe that there's even an optimal level of frustration - too easy, the student is bored, too difficult, the student feels defeated. Then you have oddballs like me who get some of the math but not all of the math. Did I have a high fever at age 5 that damaged the portion of my brain responsible for doing matrix problems yet left me capable of understanding asymptotes? It's brutally unfair that understanding some of it is no better than understanding none of it except of course to boost your standardized test scores as long as you know personally that your math score has no real world value and that taking a math course despite supposed, yet not actual math aptitude will lead to disaster.

Math is evil!

Subtracting a negative number is just a funny way to look at adding. This understanding comes prior to any high "concept" from the "cognitive science of mathematics," or philosophy of mathematics.

Philosophy of mathematics is something mathematicians, who know their subject, dispute over after the heat of the day has passed. It's a way of elegantly organizing what they have learned. It doesn't help newbies learn, sorry.

We’re not really discussing maths. We’re discussing how the mind learns. It isn’t identical to how a subject is scientifically organized.

A subject (such as maths) is scientifically organized "top-down," or deductively. Mathematicians have arrived at abstractions, and they use these to organize the field.

BUT a student (any student, bright or mediocre) who is new to the subject is in a different position. He learns "bottom-up," or inductively. High abstractions cannot have any meaning to him, until they can be connected to more concrete material that he understands.

Here is a simplistic example for the sake of clarity: How do you teach a child who doesn’t already know it that 2 + 2 = 4?

The good mathematician/good mathematics education major might answer in this fashion: "First, the child must learn the concept of sets; the sets of whole numbers; the commutative property; and the concept of integers. I recommend that he read the chapter on sets first."

But the good teacher would answer this way:

"First, give the child two oranges, and hold a different pair of oranges in your hand. Next, have him count the oranges in his hand (2), and then the ones in your hand (2). Finally, give him your oranges, saying, 'My two added to your two is how many?' and have him count all the oranges (4)."

The same distinction between "good science" (or a good education major paper) and "good teaching" applies, mutatis mutandis, to all subjects and all age groups. (But NOT all IQ levels - nothing can be taught to a congenital moron.)

The simple is not explained by means of the complicated. The abstract is not explained by means of the more abstract.

Mathematically, you are (as far as I can make out) correct about the concept of the closure property: this concept organizes known facts elegantly. The point is, students who don’t know these facts will not learn them by being exposed exclusively to this piece of esoterica. A teacher will have superior results if he gets down on the ground level with his students and sees through their eyes. They have to WORK UP TO the closure property, not DOWN FROM it.

I defy you to write a word problem that shows practical application of closure properties.Gallian, Contemporary abstract algebra, 5th edition, page 274:

Theorem “15.2: Let phi be a homomorphism from a ring R to a ring S. Then Ker phi = { r elment of R such that phi(r) = 0 } is an ideal of R.”

“Problem 2, page 277: Prove Theorem 15.2.”

……

”… cognitive-science-of-mathematics jargon …”I.e., College of Ed. jargon, as opposed to real math jargon.

lazlo ludlow said:Along those lines: In 1987 this story was fresh; In 1997 this story was familiar; In 2007 this story, along with the iSteve commentary, is boring.Steve, your hometown Los Angeles is dropping off functional America's radar. And your California stories increasingly register as missives from a boring, brown version of the USA. In another ten years you'll need to move to another state in order to maintain an audience for your local color stories. Because the truth is that "hot-blooded" Latin culture, with the corrupt plutocracy sitting on top of the dysfunctional brown masses, is intensely boring. There is little innovation in art, and no innovation in science. And there is only a sideline role in world events.This is the most insightful post that I've ever seen at iSteve.

Yet it's also trivial - it's a tautological conclusion of everything which Steve Sailer writes about.

At some point, Sailer is going to have to follow his ideas to their logical conclusion, and ask himself whether the war has been lost - whether there are no more battles which could possibly be won, and whether it isn't time to start planning a strategic retreat into hiding.

On the other hand, the problem is that this time there isn't anywhere to hide.

The Puritans, facing annihilation from the Catholics [both Roman & Anglican], could flee to Massachusetts.

The Jews, facing annihilation from the Tsar & Hitler [and, in limited cases, Brehznev & his successors], could flee to New York & Chicago.

As could the Moravians, the Amish, the Armenians - the Irish, Protestant & Catholic alike, facing starvation in the potato famine - and countless other groups who fled persecution [& starvation] and found succor in the New World.

But this time there isn't any "New World" to flee to. Everywhere is known. Everywhere is established. Everywhere is Old World now.

Today there's an article about a new secessionist movement in Tennessee, but as I posted in another thread, the last time that was tried, it ended in the greatest carnage the world had ever seen.

And ironically enough, just yesterday, the Washington Times had an article about 20,000 Confederates who fled to Brazil in the aftermath of the Civil War.

Anybody care to flee to Brazil these days?

There just isn't anywhere to run to this time.

And demography is destiny - the people who make the babies make the future.

Right now, our future looks very black and very brown, and it's not clear that "functional America" can survive that much dead weight.

Personally, the more I look at the numbers, the less hope I have for any pretty resolution to this mess.

And I know that the future is a notoriously difficult thing to predict, but right now, all of the most likely futures for this country look very ugly to me.

Alanf said:

> They believe that any kid can

> learn any material so long as

> they are properly taught it and

> its prerequisites.

It seems to me that these algebra standards are one more example of a general trend : people tend to think that other people are just like them, on the inside. Lesbian feminism studies professors think that other women are just like them on the inside, and wearing skirts and heels just as some sort of capitulation to fear. White nationalists think that other white folks would feel the same way as them, if they were truly honest with themselves. And, most notably, NPR-listening, Volvo-driving, Starbucks-sipping, weekend-away-vacationing, enlightened-view having, high-achieving white professionals believe that ghetto blacks and barrio latins have an NPR-listening, Volvo-driving, Starbucks-sipping, weekend-away-vacationing, enlightened-view having, high-achieving professional inside of them, just yearning to get out - if only society would just locate and punish the residual Bull Connors lurking about creating institutional racism, and we would all learn to be more "inclusive".

Hey Mike, I don't have the art gene. Can I skip art appreciation? Most Americans do just fine without learning any history, art, or literature. Culture is a hindrance in most corporate jobs, whereas they at least expect you to be able to crunch numbers on a spreadsheet.

Ian:It seems to me that these algebra standards are one more example of a general trend : people tend to think that other people are just like them, on the inside.Freud called it projection, more or less.

And as difficult as it is for a moron to imagine what it might be like to be a genius, lately I've been wondering whether the opposite might be even more difficult - whether educated folks [IQ 120, 130, 140, etc] might have no idea at all what life is like for someone with an IQ of 80, or 70 [which would encompass literally tens of millions of these minorities].

What would it be like to be unable to perform simply Algebra?

What would it be like if you were so stupid that you were [more or less] incapable of being taught how to solve a pair of equations like

x + 2y = 2

2x + y = 1

During the immigration debate this past spring, I actually spent some time trying to imagine just that - what it would be like if your mind were trapped in a brain which couldn't perform the higher intellectual functions.

I sure do hope that I'll never be forced to experience it, but I suppose we'll all succomb to dementia eventually.

Today there's an article about a new secessionist movement in Tennessee, but as I posted in another thread, the last time that was tried, it ended in the greatest carnage the world had ever seen.USA 2020 or so = Yugoslavia 1990's.

T. Jefferson, America's original limo liberal: "The tree of Liberty must be refreshed from time to time with ..."

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