I had just understood, that the area of a circle formula is like Length times Width. But, hold on... a Radius is not the same thing as length. So, you do the same math expression for a rectangle, it will not work, but you do it for a square, it will. Because half the parameter of a square times its radius would equal the area. So this math works for a square, because it is equal, the same that it works for a circle, because it is equal. However, said area formula will not work for a rectangle. Because a rectangle's side lengths are not equal.

It is because of this, and is also the same that Quintic Roots cannot always be solved, that the P versus NP cannot always have a solution, and only some Quintic Roots can be solved. Because solving it would generalize a universal formula for all shapes, which is impossible.

However, many are not unsolvable. One can take known principles of geometry, and combine them like a Geometric Proof, and solve many NP difficult equations, if one graphs the shape and breaks it down into its composite parts. Like in a Geometric Proof.

After looking at it, the teacher's right. I didn't see the whole instructions. She's just teaching her pupil about where the tenths and hundredths and thousandths are.{}

Not wrong at all. After looking at the directions. She's just teaching about where the tenths, hundredths and thousandths are.

I mean, would anyone do that in real life? No... the kid's correct in all but one of his answers, too. So, probably for habit, it's best he didn't follow through with the activity. As sometimes these kinds of things can confuse students. It's mostly all mathematically correct, but with the narrow instructions the Teacher's grade is correct... but why would you do that? That's a question in itself. So... a little interpretive power, the student didn't follow directions. Though, he got almost all the answers right.

I mean, should you have such directions? Eh... you know... that's a trippy one. I just think the student grasps the concept of rounding, so it should possibly be an A [instead of a C].

You learn something new every day. But, do Middle Schoolers need that kind of information [placeholding the Zero to account for minute differences of number it would determine in calculations]? I'd say, it's just the teacher trying to teach them where the tenths, hundredths and thousandths are. What you're talking about is Junior in College Math. Or AP placement. This isn't something you'd learn in Middle or Elementary School. And isn't something you'd use in pure Mathematics. I watch people do Geometric Proofs--a good channel--and he doesn't even do that. Not that I see. That's kind of nitpicky in its own right, and probably something you'd only need to learn in engineering. Especially since most measurements are irrational, I can see how this'd be too much and a little too narrow.

I mean... I see it from both perspectives. It's hard to say what the right answer is, but if she's just teaching him about where the tenths, hundredths and thousandths are, I can see why she's doing it. But, you know... what you're talking about is good for me--as a 35 year old man interested in math--but for a Middle or Elementary Schooler, it might be a little too advanced for them to truly understand. But, she's right in the sense that she's teaching them where the decimal is. You have to learn your ABCs before you learn how to read Chaucer. It's simple as that. She's teaching ABCs. And I'd say after looking at it, she did nothing wrong. It's just teaching where the decimals are, which the child needs to know.

I mean... I just think it may be a little too harsh, though. As if she's expecting the student to know what you're saying--that's something you shouldn't learn until you're in college and taking Engineering. And even then, almost all measurements are irrational, so it shouldn't even be an issue, as you're just nitpicking at that point. It's not wrong for the reason you're saying, is what I'm saying. You're doing Advanced Engineering, when this child is just learning ABCs and spelling. That's way too advanced for a fifth grader, and possibly for even a Freshman in College.

[What people are calling "The Enlightenment"] is the exact opposite of the [E]nlightenment. Maybe we need a new [Enlightenment]? Don't you think?

[T]he introduction of Buddhism to the West is what caused it, like [one says] Nietzsche and Kant. The actual [E]nlightenment was what came before it. That's the beginning of Modernism what [they're] describing. 

Generally, The Enlightenment was established in the Philosophy of the Declaration of Independence, and based itself in reason and sound first principles. Fortunately, we still have its tradition preserved in the Constitution and Declaration, but in the phrase "We hold these truths to be self evident" [t]here's no admission of subjectivity, as "Self Evident" is not "Subjective" but rather based in first principles of something being of itself evident, which is how Geometry works. You read in Thomas Jefferson's letters, or Locke the cusp of the Enlightenment, which was philosophy based in sound and evident first principles, that were apparent to the learned, but maybe not the unlearned.

Descartes was famous for saying, "I think therefore I am" but the whole point of that, was to establish a true set of knowledge. Like Descartes was a Mathematician, and framed the Quadratic Equation in its modern incarnation. He was not afraid to claim there's objective knowledge. 

I'd actually say Nietzsche and Kant were the beginning of Modernism, as Modernism is the exact opposite of the Enlightenment, which it opposes that there's any true knowledge beyond the material. And Postmodernism objects to there being knowledge at all. 

The Enlightenment was man first discovering there was knowledge, and then giving that fire to the people, like Prometheus. I mean, the entire Enlightenment was of course created by the Masons, who founded that there was knowledge. That's why The French Revolution deified Reason. 

I don't know... something smells fishy about this.

I see lots of fireflies, they just have a short gestation period. About two or three weeks, that's all they're out for. Usually between June and July. Most insects are only spawning for a short period. Dragon Flies are no different. First comes the Horse Flies and House flies around Mayday and they peak around Memorial Day, and then the Dragon Flies come out and gobble up lots of them from July into August, that lasts about two months. The Mayflies are also usually only out around August and September--ironically because they're called "mayfly" but aren't out very long. Ticks also come out for a short period, about two months. I've observed that for close to seven years now. The Fire Flies aren't out for very long, and most people just aren't outdoors during the summer anymore, so they miss them. Now House Flies have a long gestation period, and so do Mosquitoes. But horseflies are usually dormant in Late August or Early September.

There are really two things being done in Calculus equations, which we've understood primarily for a long time--as pi was understood using hexagons and area, which intuited to the curve of the circle, and brought us the area of a circle formula.

The first thing being done in any Calculus equation, is bits of blocks being calculated in a repeating pattern, to draw out a ratio of the curve--which will be calculated further through the rates of change and differentials--and these blocks will intuit a pattern, which divide by the entire shape, and then result in the correct answer.

The second thing being done in a calculus equation, is intuiting the area beneath a curve. Such as a x^2 parabola, is always 1/3 of the two dimensions. And in more difficult equations, the differentials and rates of change combine, to form into a curve, which then get intuited by the the first described process, to receive an area beneath the curve. As, this area can be used for any quadratic function, and its area, which possess innumerable uses as a form of logic, but always forming correct uses. 

There’s tiers of intelligence. I’m just going to go over them.

1.Reactive. It just moves according to stimuli.
2.Interactive. It interacts with things in its environment.
3.Communitive. It understands how to socialize with other lifeforms.
4.Extra-communitive. It forms into social hierarchies and communities and forms social behaviors.
5.Existential. It has an awareness of self. And possibly a sense of humor.
6.Extra Existential. It not only has an awareness of self, but also has an ego—will look into a mirror is an example of this.
7.Creative. It has catharsis, and needs creative activity to keep it interested. Elephants and Dolphins are the only animals known to have this so far. Not even Chimpanzees.

Now above this, only humans have.

8. Reason. The ability to put things together, and build upon them to higher principles, and find things through putting things together in new ways.

9. Judgment. The ability to correctly identify right and wrong.

10. Logos. The ability to not only judge and reason, but correctly interpret patterns and understand them the right way.

11. Prophetic. The ability to construct patterns, and predict them. Which is what Calculus does, actually. You need a Prophetic Intelligence to understand it. (I don’t mean “Prophet” as in a Bible Prophet, but rather someone who can understand and predict behaviors and trends before they happen, by analyzing previous data. Or an ability to form conclusions with incomplete data.)

Some people have high IQs but lack certain kinds of intelligence, and some people have lower IQs but have the higher cognitions. It’s really not about IQ, but your capacity. And unfortunately, having a High IQ doesn’t always correlate with having higher forms of cognition. It really only measures Reason.

I’d also say, some people only come to tier 6. So, there are some animals that are more intelligent than humans. But, humanity has a moral worth the animals don’t. I don’t know how to quantify that, but it’s not necessarily intelligence that makes us human, but God’s image being borne in us. That’s the only thing I can say… and why humanity has a moral worth higher than all other life forms on Earth.