There is nothing deep about the concepts of addition and subtraction. Tell a young kid you have two different piles of a number of objects. Combine them into a big pile and count how many are in it. Now they've mastered the concept of addition. Take a pile of a certain number of objects. Remove a certain number from the pile, how much do you have left? By gum, the concept of subtraction has been mastered. The CC processes are tricks to do the calculations more quickly. And since we have calculators that can do that anyways, who cares?

Things get more complicated with fractions. One part that trips people up is how dividing a number > 0 by a fraction > 0 and 1 leads to a number greater than what you started with? (Assuming positive numbers). Say you have a medicine of 8 oz and you must drink 1 oz each day, how many days does it take to finish it? 8 days from 8 divided by 1. Now take the same 8 ounces and you have to drink 1/8 of an ounce a day - how long? Now the correct answer matches your intuition and it makes sense that you'd come up with something larger. THAT is an example of concepts, not calculation tricks.

My favorite example of a mathematical concept, something to introduce to students after they know simple arithmetic, is the method that a young Gauss came up with to quickly add the integers from 1 to 100. It's easy to understand, clever, can be easy to show how to generalize up to any number, and it begins to show the difference between arithmetic and math.

But that' a big reason why we're arguing about this in the first place. If the first exposure kids have with math (arithmetic) is taught in such a way that they're put off by the subject then they're going to lose all interest before they even get to the real math.

More important than breaking things up is getting an intuitive understanding of what you're doing in the first place. If you can do something fairly simply in a subject intuitively, but it's taught in a way that introduces many more steps that remove you from the intuition then your interest-level is going to go from 60 to "why bother" in five seconds. The old method has less steps to clutter things up. Want to be able to do arithmetic quickly? Use a calculator; it's what we invented them for. Math is far more interesting than arithmetic anyways.

To go slightly off-topic, I'd love to know if they have a method to evaluate whether or not their curriculum actually works on the student body at large. Schools spend a lot of effort evaluating teachers, but isn't the curriculum you tell the teachers to use at least as important as the teachers themselves? I don't know if there is any amount of evidence that could convince them if the curriculum was deficient. "Oh look, we completely changed how everything is taught and the scores are going down. Obviously the problem is the teachers! "

These methods may come in handy at some point, but in my opinion they're horrid when introducing students to simple arithmetic. Make sure the students have mastered the fundamentals first and only then perhaps introduce them to some parlor tricks.

"In this house we obey the laws of thermodynamics!"

-Homer Simpson

From the Pentagon's document, under Zombie Threat Summary (section 4.6 vii)...

2. (U) Of note, where normal carniverouse zombie commonly groan the word "brains" semi-comprehensibly, VZ's [vegetarian zombies] can be identified by their aversion to humans, affinity for plants and their tendency to semi-comprehensibly groan the word "grains".

So then why can we not say, "The consciousness problem exists. A functional solution to consciousness does not exist. It does not exist because it is not computable."?

For what it's worth I also believe that the mind, including consciousness, is completely the result of what is functionally a computer. But I don't think it's possible to identify the essential qualities of what makes a working algorithm yield consciousness. That isn't to say that we can't come up with some self-learning machines that could yield consciousness. But we may still not be able to identify what it is that makes some of these achieve consciousness and not others.

The only way that conciousness could be non-computable would be if there is a supernatural element to it. Otherwise, the fact that it exists means it must be computable.

Nonsense. Just because something exists and is not "supernatural" doesn't mean that it must be computable. Take the halting problem for instance. There is no Turing Machine that is able to take any possible TM and input and determine whether the inputted TM will eventually halt or go into an infinite loop when run with the given input. This is true even though every TM must either halt or go into an infinite loop for any possible input. There's nothing supernatural about it.

Believing that no gods exist does not necessitate the belief that science (or anything else for that matter) can come up with "all of the answers".

This is quite controversial, mavericky science because it's very hard to test -

If it's not testable, then by definition it is not science.

hard to test != untestable

I oppose the death penalty in practice. I also support the idea of the death penalty, in theory.

Hmmm. Given your nickname I thought you'd be more willing to smite the dragons.

There isn't. Our bodies are machines, no more no less, and ultimately science will solve every riddle they pose.

Great. And how many complex machines do you know of that are able to run indefinitely without breaking down?

Also, I love how you present your wild speculations as obvious inevitabilities.

Solving the following riddle will reveal the awful secret behind the universe, assuming you do not go utterly mad in the attempt. If you already happen to know the awful secret behind the universe, feel free to skip ahead.

Let's say you have an ax. Just a cheap one, from Home Depot. On one bitter winter day, you use said ax to behead a man. Don't worry, the man was already dead. Or maybe you should worry, because you're the one who shot him.

He had been a big, twitchy guy with veiny skin stretched over swollen biceps, a tattoo of a swastika on his tongue. Teeth filed into razor-sharp fangs - you know the type. And you're chopping off his head because, even with eight bullet holes in him, you're pretty sure he's about to spring back to his feet and eat the look of terror right off your face.

On the follow-through of the last swing, though, the handle of the ax snaps in a spray of splinters. You now have a broken ax. So, after a long night of looking for a place to dump the man and his head, you take a trip into town with your ax. You go to the hardware store, explaining away the dark reddish stains on the broken handle as barbecue sauce. You walk out with a brand-new handle for your ax.

The repaired ax sits undisturbed in your garage until the spring when, on one rainy morning, you find in your kitchen a creature that appears to be a foot-long slug with a bulging egg sac on its tail. Its jaws bite one of your forks in half with what seems like very little effort. You grab your trusty ax and chop the thing into several pieces. On the last blow, however, the ax strikes a metal leg of the overturned kitchen table and chips out a notch right in the middle of the blade.

Of course, a chipped head means yet another trip to the hardware store. They sell you a brand-new head for your ax. As soon as you get home, you meet the reanimated body of the guy you beheaded earlier. He's also got a new head, stitched on with what looks like plastic weed-trimmer line, and it's wearing that unique expression of "you're the man who killed me last winter" resentment that one so rarely encounters in everyday life.

You brandish your ax. The guy takes a long look at the weapon with his squishy, rotting eyes and in a gargly voice he screams, "That's the same ax that beheaded me!"

IS HE RIGHT?

-John Dies At The End

Having technical skills doesn't necessarily mean that someone is smart, especially when it comes naturally to them.

Sure it does. If someone's a natural mathematician (like Euler or Ramanujan for instance) or physicist or (to a lesser extent) programmer then they are naturally smart. These topics engage the intellect. Being a natural people-person is an innate skill that does not require any proper definition of intelligence, and they don't need to appeal to the intellect much at all to be successful at it.

That isn't to say that being good with people isn't an important skill; it is vitally important. But being good at it does not always require or engage smarts.

Meh. Having people-person skills doesn't necessarily mean that someone is smart, especially when it comes naturally to them. Someone without natural people skills and are able to apply their intellect to gain them are very intelligent, however. But a lot of people out there don't have to think about it much. Must be nice.