Algebra comes from the Arabic word al-jebr, which means "reunion of broken parts".
TIL (Today I learn) where the word Algebra come from?
https://www.mathematics-monster.com/algebra.html
Albert Einstein and Algebra
The famous scientist Albert Einstein learned algebra from a young age.
His Uncle Jakob gave him books on the subject and called algebra "a merry science".
He compared algebra to hunting a little animal. You didn't know the name of the animal, so you called it "x". When you finally caught the animal you gave it the correct name.
https://www.reddit.com/r/todayilearned/comments/bfybqy/til_as_a_child_einsteins_uncle_jakob_introduced/
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“It is an under acknowledged truism that, just as you are what you eat, how and what you think depends on what information you are exposed to.”, p.13, Tim Wu, The Master Switch, 2010.
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Daniel Kahneman
Daniel Kahneman gave a talk @ Google talk, I think it is called Talk @ Google, it is on youtube.com, look for it if you are interested, any how, he said some thing like, there is no System 1, or, System 2; System 1, System 2, are notional idea - a teacher helper; for example, and he did not say, this, but he said some thing close to this, thinking about an object, or, an abstract concept (a notional idea, a fantasy, a fictional character, a made-up idea) as being an imaginary little demon helper and having personalities, instead of an abstract ideas (or an object whether it is real or not) as having properties x, y, and z; if you think of it as a (little demon helper) - and this is a thought experiment - it might be easier; try it out and see; subconsciously it might be easier, because from biological human evolution, we are already wired to make connection and to understand (little demon helper) with personalities with different roles in a social tribe; for teaching and learning purposes, it is a helpful teaching aid (crutch); (think about it, when you were growing up, you had two big demons helpers, called parents, or maybe it was only a mom, or only a dad, or a sister, or a brother).
I thought what you just read might be helpful for Teachers, Trainers, and the Coaches who teaches the Teachers how to teach, specifically abstract concept and ideas - notional things; because if you think of math as a hero journey, a field trip, and you, yourself as the hunter, going on an adventure to find a little animal, or a fisher fishing for gold fish, you don't know the name of the animal so you called it "X", or, you don't know the name of the gold fish; however when you find the little animal or catch the little gold fish, you can give it the correct name. That idea can be helpful. Basically they are solving a math problem, but you do NOT put it that way. They are hunting or fishing for a cute little animal that they don't know the name[;] so for now, let's call it "X", and when they find the (cute little animal), they can give it the correct name.
“The difference is that it is framed in a way that makes it about people, instead of about numbers, and as human beings we are a lot more sophisticated about each other than we are about the abstract world.” (p.160, author Malcolm Gladwell., his book, The tipping point: how little things can make a big difference, 2000, 2002, )
The parents (guardian or care-taker) would be in on this. Because this could be a problem if the little girl comes home and tells her Papa or her Mama that today in math class she went on a great adventure to hunt for (cute little animal) that she do not know the name of, and when she finds the (cute little animal), everyone in class can give it the correct name; well, most parents are not going to understand this; they are going to think, this isn't math; what are they teaching my child in math class; the fact that the little girl has to solve the algebra problem in order to find the correct name to the (cute little animal) is incidental to her thinking process.
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Malcolm Gladwell., The tipping point: how little things can make a big difference, 2000, 2002
p.160
But, as the psychologist Leda Cosmides (who dreamt up this example) points out, it is exactly the same puzzle as the A, D, 3, and 6 puzzles. The difference is that it is framed in a way that makes it about people, instead of about numbers, and as human beings we are a lot more sophisticated about each other than we are about the abstract world.
(The tipping point: how little things can make a big difference / by Malcolm Gladwell., 1. social psychology., 2. contagion (social psychology), 3. causation.
4. context effects (psychology), HM1033.G53 2000, 302──dc21, originally published in hardcover by Little, Brown and Company, March 2000, first paperback edition, January 2002, 2000, 2002, )
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Why learning is an adventure?
Because "you leave the safety of the world you know and enter the unknown.";
"In the Departure stage, you leave the safety of the world you know and enter the unknown."
https://scottjeffrey.com/heros-journey-steps/
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TIL Rice on a Chessboard
Oct 10, 2018
- Exponential Numbers (double the amount of rice of the previous square)
•••• ••• ••••
The Legend of Payasam at Ambalappuzha
Once upon a time, the king who ruled over the region of Ambalappuzha was visited by a travelling sage, who challenged the king to a game of chess. The king was well known for his love of chess and so he readily accepted the challenge.
Before the game started, the king asked the sage what he would like as a prize if he won. The sage, being a travelling man with little need for fine gifts, asked for some rice, which was to be counted out in the following way:
Give me one grain of rice on the first square of this chessboard, then two grains on the second square, four grains on the third square, eight grains on the fourth square and so on, so that each square contains double the amount of rice of the previous square.
Now the king was taken aback by this. He had expected for the sage to request gold or treasures or any of the other fine things at his disposal, not just a few handfuls of rice. He asked the sage to add other things to his potential prize, but the sage declined. All he wanted was the rice.
•••• ••• ••••
source:
► https://owlcation.com/stem/Rice-on-a-Chessboard-Exponential-numbers
► https://amiracarluccio.com/2018/02/27/ancient-indian-legend-the-rice-and-the-chessboard-storylearning-about-mathematics/
► https://www.mathscareers.org.uk/the-rice-and-chessboard-legend/
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p.265
“If a system is hard-to-detect or hard-to-measure at some instant and no substances can be added to an object, then these substances generating easy-to-detect and easy-to-measure field should be added to ambient medium, and the state of the object can be judged from the state of ambient medium” [1].
“If a system is hard-to-detect or hard-to-measure at some instant and no substances can be added to an object, then these [cute little demons] generating easy-to-detect and easy-to-measure field should be added to ambient medium, and the state of the object can be judged from the state of ambient medium”, [because the {cute little demons} are going to generate easy-to-detect and easy-to-measure field, which can be detect and measure from the state of the ambient medium] [1].
• Maxwell's demon, a tiny imp, agents, small smart demon, “small smart people”, “inanimate particles”, “tiny smart particles”, ...
• “small smart people” that could do anything a problem solver needed to do in the problem-to-solution transition.
• During TRIZ classes, Altshuller realized that the weak point of empathy is the strong tendency to reject any action that is unacceptable to the human organism.
• In the Israeli teaching experience, it was found that students did not always use small smart people effectively. It seems that subconsciously some students were reluctant to place these small smart people in situations that would be life threatening to humans, such as in strong acids or extreme fields.
pp.283-284
This chapter discusses a problem-solving based on solving Agents. In contrast to all other TRIZ heuristics, this is the first TRIZ method to have been developed almost independently in numerous countries -- Russia, Israel, and then US. Agents in TRIZ originally were “small smart people” that could do anything a problem solver needed to do in the problem-to-solution transition. They were derived by Altshuller at the end of 1960s from Synectics, the American method of creativity activation. Ten year earlier, William Gordon, the author of Synectics, had suggested using personal analogy or empathy in the solving process [1]. The essence of empathy is that a persons “enters” into the object to be improved and tries to imagine the action required by the problem. During TRIZ classes, Altshuller realized that the weak point of empathy is the strong tendency to reject any action that is unacceptable to the human organism. This drawback is overcome with the help of “small smart people” in modeling [2]. A transition from the “small smart people” to “inanimate particles” was proposed by Solomon D. Tetel'baum about 15 years ago [3], but the idea was not supported by other TRIZniks who often used teams of boys and girls during their lessons. Due to emigration of some TRIZniks from USSR to Israel in the 1980s, this methodology became popular in the Middle East. In the Israeli teaching experience, it was found that students did not always use small smart people effectively. It seems that subconsciously some students were reluctant to place these small smart people in situations that would be life threatening to humans, such as in strong acids or extreme fields. Therefore, Genady Filkovsky, Roni Horowirz, and Jacob Goldenberg from the Open University in Israel replaced small smart people with inanimate particles [4].* This particles method is now used actively in the Israeli derivative of TRIZ simplification named SIT, where it represents almost half of these problem-solving activities [4,5]. However, some of the author's students have argued that they are more easily imagine various actions performed by the “small smart people” than by inanimate particles. This is all a matter of sematics and the term itself is not as important as the method. But we will use the neutral term agents. The experience of Russian, Israeli, and American specialists is summarized and generalized in the Agents Method described in this chapter.
* In general, this idea is not new in problem solving; even the famous antique Greek philosopher Demokrit used small particles for explanation of natural phenomena. The famous physicist James Clerk Maxwell used small demons (human-like beings) for resolution of scientific problems.
( Savransky, Semyon D., Engineering of creativity : introduction to TRIZ methodology of inventive problem solving / by Semyon D. Savransky., 1. engineering--methodology., 2. problem solving--methodology., 3. creative thinking., 4. technological innovations., 2000, pp.283-284 )
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Howard Rheingold, Tools for thought, 1985 [ ]
Maxwell's demon
pp.121-122
James Clerk Maxwell, yet another 19th century scientist, proposed a paradox concerning this elusive quality called entropy, which seems to relate such intuitively dissimilar measures as
energy,
information,
order, and
predict ability.
The paradox became in famous among physicists under the name "Maxwell's demon." Consider a container split by a barrier with an opening small enough to pass only one molecule at a time from one side to the other. On one side is a volume of hot gas, in which the average energy of the molecules is higher than the average energy of the molecules in the cold side of the container. According to the second law, the hotter, more active molecules should eventually migrate across to the other side of the container, losing energy in collisions with slower-moving molecules, until both sides reach the same temperature.
What would happen, Maxwell asked, if you could place a tiny imp at the molecular gate, a demon who didn't contribute energy to the system, but who could open and close the gate between the two sides of the container? Now what if the imp decides to only let the occasional slow-moving, colder molecule pass from the hot to the cold side when it randomly approaches the gate? Taken far enough, this policy would mean that the hot side would get hotter and the cold side would get colder, and entropy would decrease instead of increase, without any energy being added to the system!
([
Conversely, what if the imp decides to only let the occasional fast-moving, hotter molecule passes from the cold to the hot side when it randomly approaches the gate? Taken far enough, this policy would mean that the cold side would get colder and the hot side would get hotter, and entropy would decrease instead of increase, without any energy being added to the system!
Moreover, what if both methods is used. By obtaining information of each molecule as it approaches the gate, our all-knowing-imp let the colder molecule passes from hot to the cold side; and let the hotter molecule passes from the cold to the hot side. In this way, the demon is able to add more order (entropy) to the system.
That is one big if, isn't it. Because how does the all-knowing-imp differentiate or get the needed information about the molecule.
])
In 1922, a Hungarian student of physics by the name of Leo Szilard (later to be von Neumann's colleague on the Manhattan Project), then in Berlin (Germany), finally solved the paradox of Maxwell's demon by demonstrating that the demon does indeed contribute energy to the system, but like a good magician, the demon does not expend that energy in its most visible activity -- moving the gate -- but in what it knows about the system. The demon is part of the system, and it has to do some work in order to differentiate the hot and cold molecules at the proper time to open the gate. Simply by obtaining the information about the molecules that it needs to know to operate the gate, the demon adds more entropy (order) to the system than it subtracts.
(Tools for thought : the history and future of mind-expanding technology, Howard Rheingold, 1985, pp.121-122)
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(In Hawaiian legend, the Menehune were strong, skillful imps who would work all night without stopping.)
M. Mitchell Waldrop, The Dream Machine, 2001 [ ]
Bob Metcalfe (Robert Metcalfe)
p.372
This last task, especially, kept him in a more or less constant state of jet lag, which was how he happened to find himself in Steve Crocker's guest room one night during a swing through Washington, D.C., tossing and turning on the sofa bed. Desperate for something to put him under, he happened to catch sight of a thick blue volume on the bookshelf next to the bed: American Federation of Information Processing Societies Conference Proceeding, volume 37, fall 1970. Perfect. Metcalfe started reading “The Aloha System”, a paper by Norman Abramson of the University of Hawaii.
p.372
The Aloha system, he learned, was an experimental, ARPA-funded network that transmitted computer data via radio waves, instead of via the telephone lines used in the Arpanet. The University of Hawaii, he also learned, was natural setting for such an experiment: its campuses on the various islands were separated by large stretches of open ocean, which made for telephone connections that were noise, unreliable, and very expensive. Abramson's paper accordingly described a network in which the main IBM System/360 computer on Oahu sent packets of data back and forth to terminals out on the island campuses via radio. Serving as a front end to the 360 was Menehune, a small, packet-switching computer that handled the actual radio connections and that was similar in function to an Arpanet IMP. (In Hawaiian legend, the Menehune were strong, skillful imps who would work all night without stopping.)
pp.372-373
Now, on the surface, Metcalfe could see, this was a matter of straight substitution: anywhere Arpanet had a wire, Alohanet had a radio link. Beneath the surface, however, when you got down to the nitty-gritty of how the packet transmission were regulated, the differences were much more interesting. On the Arpanet, where the bits flowed through telephone lines, an IMP with packets to send could wait for a break in the traffic, so to speak; that way, the packets never collided with one another. But on the Alohanet, where the bits were carried by staticky, interference-prone radio waves, a terminal with packets to send had no way of knowing what the traffic was like. It could transmit back and forth to Menehune (with luck), but it probably couldn't even HEAR what the other nodes in the network were sending. So, since waiting would be pointless, Alohanet allowed each terminal to fire off a packet to Menehune whenever it needed to, regardless of what the others were doing. If the terminal heard Menehune acknowledge receipt of that packet, then fine. But if it didn't--meaning that another terminal's packet had arrived simultaneously and turned the bits into gibberish--the first terminal would just back off, wait for a random interval of time, and then transmit its packet again. Since the second terminal would also be retransmitting, but with a different random interval, both packets now had a reasonable chance of arriving unscathed.
p.373
Beautiful! thought Metcalfe. It was control witout control: the terminals were completely free agents, unregulated and unsynchronized by Menehune. And yet the packets got through anyhow.
p.373
Or did they? Alas, wrote Abramson, the system's beauty came at the price of instability. Using a branch of mathematics known as queuing theory, he argued that Alohanet couldn't use more than about 17 percent of the total capacity in its radio channel without causing a kind of chain reaction. Push it past that point, and each collision of packets would trigger the transmission of replacement packets, which would increase the probability of more collisions, which would generate more replacement--on and on until every packet was statistically guaranteed to hit another packet. The system would grind to a halt.
p.373
By now, recalls Metcalfe, he was wide awake: this couldn't be right. As he wrote about it later, “The Abramson paper ... made two assumptions about the computer terminal user behavior that, on Steve Crocker's sofabed late at night, I found totally unacceptable. Abramson's model assumed that there were an infinite number of terminal users, and that each of them would go on typing whether or not they received answers to earlier inputs.”
p.373
He would give them theory. He would do this queuing analysis RIGHT. “That night”, he remembers, “and in the weeks to follow, I worked hard on the less-tractable mathematics of Aloha channels with a few users, each of whom would insist on receiving a response to [his] input before typing a new one. I worked so hard, in fact, that Xerox sent me to work with Professor Abramson for a month--in Hawaii!”
p.373
By October 1972, says Metcalfe, just in time for the Arpanet “coming-out” party in Washington, he had produced a paper showing that an Aloha-type network could indeed be made stable under much heavier loads than Abramson had believed.
(Waldrop, M. Mitchell.; The dream machine : J. C. R. Licklider and the revolution that made computing personal / M. Mitchell Waldrop., 1. Licklider, J. C. R., 2. microcomputers--history, 2001, )
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Benedict Carey, How we learn, 2014 [ ]
p.196
These kind of stories always remind of the Grimms' fairy
tale “The Golden Bird”, in which a young man on a mission to
find a magic bird with golden feathers falls in love with a
princess, whose father the king will grant her hand on one
condition: that the young man dig away the hill that stops the
view from his window in eight (8) days. The only complication?
This is no hill, it's a mountain, and after seven (7) days of
digging, the young man collapses in defeat. That's when his
friend the fox whispers, “Lie down and go to sleep; I will work
for you”. And in the morning, the mountain is gone.
(Carey, Benedict., How we learn: the surprising truth about when, where, and why it happens/ Benedict Carey., 1. learning, psychology of., 2. learning., BF318.C366 2014, 153.1'5--dc23, 2014, )
____________________________________
finess
brute force
outsource
How do you add 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?
or to put the problem another way
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ???
+-+-+-+-+-+-+-+-+-+-+
0 | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
1 |×| | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
2 |×|×| | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
3 |×|×|×| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
4 |×|×|×|×| | | | | | |
+-+-+-+-+-+-+-+-+-+-+
5 |×|×|×|×|×| | | | | |
+-+-+-+-+-+-+-+-+-+-+
6 |×|×|×|×|×|×| | | | |
+-+-+-+-+-+-+-+-+-+-+
7 |×|×|×|×|×|×|×| | | |
+-+-+-+-+-+-+-+-+-+-+
8 |×|×|×|×|×|×|×|×| | |
+-+-+-+-+-+-+-+-+-+-+
9 |×|×|×|×|×|×|×|×|×| |
+-+-+-+-+-+-+-+-+-+-+
10 |×|×|×|×|×|×|×|×|×|×|
+-+-+-+-+-+-+-+-+-+-+
1 2 3 4 5 6 7 8 9 10
If you decide to use the brute force method, then you might do the add operation in the following order:
0 + 1 = 1
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
10 + 5 = 15
15 + 6 = 21
21 + 7 = 28
28 + 8 = 36
36 + 9 = 45
46 + 10 = 55
If you decide to do the finess method, then you might address each number, breakdown and rearrange the problem to look like the following:
As the first (1st) step
0 = 0
1 = 1
2 = 1 + 1
3 = 1 + 1 + 1
4 = 1 + 1 + 1 + 1
5 = 1 + 1 + 1 + 1 + 1
6 = 1 + 1 + 1 + 1 + 1 + 1
7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
then if you look carefully, and start to think a bit (asking yourself some questions, like are there ways that can I rearrange the 1's such that the triangle would become a square like rectangle, without adding more 1's or subtracting 1's from the current account), you might notice that the sequence of 1 + 1
1
1 + 1
1 + 1 + 1
1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
is a like triangle, with a slope going up on one-side, and a verticle drop off, like a cliff; you can represent (reformat, rearrange) the triangle as follow:
+-+-+-+-+-+-+-+-+-+-+
0 | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
1 |×| | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
2 |×|×| | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
3 |×|×|×| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
4 |×|×|×|×| | | | | | |
+-+-+-+-+-+-+-+-+-+-+
5 |×|×|×|×|×| | | | | |
+-+-+-+-+-+-+-+-+-+-+
6 |×|×|×|×|×|×| | | | |
+-+-+-+-+-+-+-+-+-+-+
7 |×|×|×|×|×|×|×| | | |
+-+-+-+-+-+-+-+-+-+-+
8 |×|×|×|×|×|×|×|×| | |
+-+-+-+-+-+-+-+-+-+-+
9 |×|×|×|×|×|×|×|×|×| |
+-+-+-+-+-+-+-+-+-+-+
10 |×|×|×|×|×|×|×|×|×|×|
+-+-+-+-+-+-+-+-+-+-+
then if you can notice that,
0 = 0
1 = 1
2 = 1 + 1
3 = 1 + 1 + 1
4 = 1 + 1 + 1 + 1
5 = 1 + 1 + 1 + 1 + 1
6 = 1 + 1 + 1 + 1 + 1 + 1
7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
can be flip upside down like so
10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
6 = 1 + 1 + 1 + 1 + 1 + 1
5 = 1 + 1 + 1 + 1 + 1
4 = 1 + 1 + 1 + 1
3 = 1 + 1 + 1
2 = 1 + 1
1 = 1
0 = 0
pp.41-42
Problem solving
A problem does not have to be presented in a formal manner nor is it a matter for pencil and paper working out. A problem is simply the difference between what one has and what one wants. It may be a matter of avoiding something, of getting something, of getting rid of something, of getting to know what one wants.
There are three-types of problem:
• The first type of problem requires for its solution more information or better techniques for handling information.
• The second type of problem requires no new information but a rearrangement of information already available: an insight restructuring.
• The third type of problem is the problem of no problem. One is blocked by the adequacy of the present arrangement from moving to a much better one. There is no point at which one can focus one's efforts to reach the better arrangement because one is not even aware that there is a better arrangement. The problem is to realize that ‘there is a problem’ to realize that ‘things can be improved’ and to define ‘this realization as a problem’.
The first type of problem can be solved by vertical thinking. The second and third type of problem require lateral thinking for their solution.
(Edward de Bono, Lateral Thinking: a textbook of creativity, 1970, 1977, 1990, )
• The second type of problem requires no new information but a rearrangement of information already available: an insight restructuring.
the equation (add operation) can be seen as this:
10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
x--------> 9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
|
| x------> 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
| |
| | x----> 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
| | |
| | | x--> 6 = 1 + 1 + 1 + 1 + 1 + 1
| | | | 5 = 1 + 1 + 1 + 1 + 1
| | | x--> 4 = 1 + 1 + 1 + 1
| | |
| | x----> 3 = 1 + 1 + 1
| |
| x------> 2 = 1 + 1
|
x--------> 1 = 1
0 = 0
Diagram Α α (Greek letter alpha)
Α (Greek letter alpha upper case )
α (Greek letter alpha lower case )
having seen this, or if someone show this to you, then you can put the elements - that was taken apart - back together
if you take 10 + 0, you would get 10, and
if you take 9 + 1, you would get a 10, and
if you take 8 + 2, you would get another 10, and
if you take 7 + 3, you would get another 10, and
if you take 6 + 4, you would get another 10, and
if you take 5, the last number, 5, and you add the verticle column
like so 10 + 10 + 10 + 10 + 10 + 5 (represented here horizontally)
you should get the numerical answer 55 (the answer, or, solution)
below is an intermediate step that might help to give you better understanding
0 + 10 = 10,
10 + 0 = 10, and
---------
10 10
10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
0 = 0
+-+-+-+-+-+-+-+-+-+-+
10 |×|×|×|×|×|×|×|×|×|×|
+-+-+-+-+-+-+-+-+-+-+
0 | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
1 + 9 = 10,
9 + 1 = 10, and
--------
10 10
9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
1 = 1
+-+-+-+-+-+-+-+-+-+-+
9 |×|×|×|×|×|×|×|×|×| |
+-+-+-+-+-+-+-+-+-+-+
1 |×| | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
2 + 8 = 10,
8 + 2 = 10, and
--------
10 10
8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
2 = 1 + 1
+-+-+-+-+-+-+-+-+-+-+
8 |×|×|×|×|×|×|×|×| | |
+-+-+-+-+-+-+-+-+-+-+
2 |×|×| | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
3 + 7 = 10,
7 + 3 = 10, and
--------
10 10
7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
3 = 1 + 1 + 1
+-+-+-+-+-+-+-+-+-+-+
7 |×|×|×|×|×|×|×| | | |
+-+-+-+-+-+-+-+-+-+-+
3 |×|×|×| | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
4 + 6 = 10,
6 + 4 = 10, and
--------
10 10
6 = 1 + 1 + 1 + 1 + 1 + 1
4 = 1 + 1 + 1 + 1
+-+-+-+-+-+-+-+-+-+-+
6 |×|×|×|×|×|×| | | | |
+-+-+-+-+-+-+-+-+-+-+
4 |×|×|×|×| | | | | | |
+-+-+-+-+-+-+-+-+-+-+
here is another way to represent Diagram Α α (Greek letter alpha):
x------------------10-------------------x
| x----------10-----------x |
| | x--10---x | |
| | | | | |
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ???
| | | |
| x------10-------x |
x--------------10---------------x
10 + 10 + 10 + 10 + 10 + 5 = 55
now, the critical thinking dilemma; some things are never easy;
critical thinking (problem-solving method, how do I fix it);
dilemma (difficulties, hard stuff, until you figure it out, then it might become easier);
until someone show you the method (tips, tricks, techniques) or tell you that there is another way to solve
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ???,
or there is more than one way to address
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ???,
then, you are unlikely to look for it (another way, the other way, alternative, more than one way), or, work on it;
you might naturally (fall back) stick to your default method, the way (to do add operation) that was shown to you in school by your parents, teachers, relatives, friends, or trainers;
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ???
0 + 1 = 1
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
10 + 5 = 15
15 + 6 = 21
21 + 7 = 28
28 + 8 = 36
36 + 9 = 45
46 + 10 = 55
is the default method of how most people might add
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ???
+-+-+-+-+-+-+-+-+-+-+
0 | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+
1 |×| | | | | | | | | |
+─+-+-+-+-+-+-+-+-+-+
2 |× ×| | | | | | | | |
+───+-+-+-+-+-+-+-+-+
3 |× × ×| | | | | | | |
+─────+-+-+-+-+-+-+-+
4 |× × × ×| | | | | | |
+───────+-+-+-+-+-+-+
5 |× × × × ×| | | | | |
+─────────+-+-+-+-+-+
6 |× × × × × ×| | | | |
+───────────+-+-+-+-+
7 |× × × × × × ×| | | |
+─────────────+-+-+-+
8 |× × × × × × × ×| | |
+───────────────+-+-+
9 |× × × × × × × × ×| |
+─────────────────+-+
10 |× × × × × × × × × ×|
+-+-+-+-+-+-+-+-+-+-+
1 2 3 4 5 6 7 8 9 10
this is natural, because one of the first math problem we all learn intuitively, is that 1 + 1 = 2, and then we learn 1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6,
1 + 6 = 7, 1 + 7 = 8, 1 + 8 = 9, 1 + 9 = 10, so on and so forth,
why am I saying all this? why am I making such a big production (big deal) about adding numbers together? Because there are different way (other ways) to look at things; different ways in how to add, subtract, multiply and divide numbers together
Let's take a look at an example.
This is going to be an add operation.
Let's us see if we can represent the add operation in a table format;
Thee would the first time learner might get a bigger (more comprehensive) picture of what add operation is about.
So the first row in the table label ‘x’, and
the first column in the table label ‘y’, then
you add row (x), to column (y), to fill in the table
x + y, or, y + x (this is because (x + y) is equal to (y + x))
in add operation and, multiply operation, the order - which number come first, second, third, or fourth - does not changed the outcome of the [final] solution ─
(x + y) has the same result as (y + x), (a + b + c + x + y + z) has the same solution as (z + y + x + c + b + a). As a comparison, in cooking with heat, the order of the ingredients has high likelihood in influencing the outcome of the dish. ([ heat is an ingredients or, the heat dynamics is an ingredients ])
+ 0 1 2 3 4 5 6 7 8 9 10 x
x------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9 10
1 | 1 2 3 4 5 6 7 8 9 10 11
2 | 2 3 4 5 6 7 8 9 10 11 12
3 | 3 4 5 6 7 8 9 10 11 12 13
4 | 4 5 6 7 8 9 10 11 12 13 14
5 | 5 6 7 8 9 10 11 12 13 14 15
6 | 6 7 8 9 10 11 12 13 14 15 16
7 | 7 8 9 10 11 12 13 14 15 16 17
8 | 8 9 10 11 12 13 14 15 16 17 18
9 | 9 10 11 12 13 14 15 16 17 18 19
10 |10 11 12 13 14 15 16 17 18 19 20
y
above, what you have is a look-up table for the add operation for all the number combination from 0 to 10;
if you can get the first time learner to - in a way - memorize this addition table, the way we memorize the multiplication, would that make adding two numbers together easier to learn and understand.
the add operation, 1 + 1 = 2, or, 314 + 256 = 570, is usual presented as one off operation; but instead, what if we represent the numbers and the add function, as a group of add operational problem (combination), using single table format (a look-up table); what can we see; can we get better insight, better understanding; can we see (notice) any pattern or trend when the add function is represented in this way; ...
+ 0 1 2 3 4 5 6 7 8 9 10 x
x------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9 10
1 | 1 2 3 4 5 6 7 8 9 10 11
2 | 2 3 4 5 6 7 8 9 10 11 12
3 | 3 4 5 6 7 8 9 10 11 12 13
4 | 4 5 6 7 8 9 10 11 12 13 14
5 | 5 6 7 8 9 10 11 12 13 14 15
6 | 6 7 8 9 10 11 12 13 14 15 16
7 | 7 8 9 10 11 12 13 14 15 16 17
8 | 8 9 10 11 12 13 14 15 16 17 18
9 | 9 10 11 12 13 14 15 16 17 18 19
10 |10 11 12 13 14 15 16 17 18 19 20
y
Next using the add operation template, we substitute the add function with the subtract function, first, using (x - y), and then, using (y - x)
x - y
(x-y) 0 1 2 3 4 5 6 7 8 9 10 x
x------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9 10
1 | -1 0 1 2 3 4 5 6 7 8 9
2 | -2 -1 0 1 2 3 4 5 6 7 8
3 | -3 -2 -1 0 1 2 3 4 5 6 7
4 | -4 -3 -2 -1 0 1 2 3 4 5 6
5 | -5 -4 -3 -2 -1 0 1 2 3 4 5
6 | -6 -5 -4 -3 -2 -1 0 1 2 3 4
7 | -7 -6 -5 -4 -3 -2 -1 0 1 2 3
8 | -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
9 | -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
10 |-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
y
y - x
(y-x) 0 1 2 3 4 5 6 7 8 9 10 x
x------------------------------------
0 | 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
1 | 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9
2 | 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8
3 | 3 2 1 0 -1 -2 -3 -4 -5 -6 -7
4 | 4 3 2 1 0 -1 -2 -3 -4 -5 -6
5 | 5 4 3 2 1 0 -1 -2 -3 -4 -5
6 | 6 5 4 3 2 1 0 -1 -2 -3 -4
7 | 7 6 5 4 3 2 1 0 -1 -2 -3
8 | 8 7 6 5 4 3 2 1 0 -1 -2
9 | 9 8 7 6 5 4 3 2 1 0 -1
10 | 10 9 8 7 6 5 4 3 2 1 0
y
so one of the pattern that we see is that the two subtract operation table (x-y) and (y-x) are the reverse of each other along the ‘0’ diagonal.
now, you have a look-up table for all the combination of subtract operation from 0 to 10.
number line(s)
<---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--->
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
<---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--->
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
<---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--->
10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
<---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--->
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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pp.42-43
Or consider the normal mathematics curriculum, which continues relentlessly on its way, each new lesson assuming full knowledge and understanding of all that has passed before. Even though each point may be simple, once you fall behind it is hard to catch up. The result: mathematics phobia. Not because the material is difficult, but because it is taught so that difficulty in one stage hinders further progress. The problem is that once failure starts, it soon generalizes by self-blame to all of mathematics. Similar processes are at work with technology. The vicious cycle starts: if you fail at something, you think it is your fault. Therefore you think you can't do that task. As a result, next time you have to do the task, you believe you can't so you don't even try. The result is that you can't, just as you thought. You're trapped in a self-fulfilling prophecy.
(Norman, Donald A., The psychology of everyday things, 1. design, industrial--psychological, aspects, 2. human engineering, copyright © 1988, 620.82 Norman, pp.42-43)
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[pp.83-84]
self-fulfilling prophecy
As Merton defined it, a self-fulfilling prophecy is an initially false perception of a situation that evokes new behavior that makes the originally false conception come true (Merton 1948).
(Richardson, George P., Feedback thought in social science and systems theory, copyright © 1991 by the University of Pennsylvania Press)
(Feedback thought in social science and systems theory / George P. Richardson (1991), 1. social science--methodology., 2. feedback control systems., pp.83-84)
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[p.79]
vicious circle
Myrdals' "principle of cumulation"
(Richardson, George P., Feedback thought in social science and systems theory, copyright © 1991 by the University of Pennsylvania Press)
(Feedback thought in social science and systems theory / George P. Richardson (1991), 1. social science--methodology., 2. feedback control systems., p.79)
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[pp.191-192]
... In order to be able to put things in general context, you do not have everything you know in your mind at all times, so you retrieve the knowledge that you require at any given time in a piecemeal fashion, which puts these retrieved knowledge chunks in their local context. This means that you have an arbitrary reference point and react to differences from that particular perspective of the local context, not the absolutes.
(Taleb, Nassim (2004)., Fooled by Randomness, 2nd edition, paperback)
(Fooled by Randomness: the hidden role of chance in life and in the markets / Nassim Nicholas Taleb, 1. investments, 2. chance, 3. random variables, 123.3 Taleb, pp.191-192)
____________________________________
([ when people are in (negative state), they do not know they are in a (negative state), and they are unable (disable, disability) to get out of the (negative state) ])
([ when you are (trapped in a self-fulfilling prophecy), you are in a (negative state) - a dark room with no light what so ever - or in a reverse polar opposite - in a room filled with light with no darkness what so ever ]) ([ if you are in a room that is - a dark room with no light what so ever - or in a room filled with light with no darkness what so ever - you would be blinded by the absolute darkness or the absolute light. It is the shading of gray - the combination and integration of darkness and light - that would enable you to see. ])
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Tim Ferriss, Tools of Titans, 2017 [ ]
p.212
Tony Robbins
state -> story -> strategy
This is because you started in a negative state, then attempted strategy but didn't succeed (due to tunnel vision on the problems), and then likely told yourself self-defeating stories (e.g., “I always do this. Why am I so wound up I can't even think straight?”). To fix this, he encourages you to “prime” your STATE first. The biochemistry will help you proactively tell yourself an enabling STORY. Only then do you think on STRATEGY, as you'll see the options instead of dead ends.
(Tim Ferriss, Tools of Titans, 2017, 081 Ferriss, )
____________________________________
Douglas Hofstadter & Emmanuel Sander, Surfaces and essences: analogy as the fuel and fire of thinking, 2013
pp.438-440
pp.438-439
Let's return to the story of the solution of “the” cubic equation (the reason for the quote marks will emerge shortly). It all took place in Italy ── first in Bologa (Scipione del Ferro), and a bit later in Brescia (Niccolò Tartaglia) and Milan (Gerolamo Cardano). Del Ferro found a partial solution first but didn't publish it; some twenty years later, Tartaglia found essentially the same partial solution; finally, Cardano generalizing their findings and published them in a famous book called Ars Magna (“The Great Art”). The odd thing is that, as things were coming into focus, in order to list all the “different” solutions of the cubic equation, Cardano had to use thirteen chapters!
p.439
Nowaday, by contrast, the whole solution is covered by just one formula that can be written out in a single line, and which could easily be taught in high schools. What lay behind such diversity, of which we no longer see any trace today?
p.439
The problem was that no one in those days accepted the existence of negative numbers. For us today, it's self-evident that the coefficient in the third term of the equation x^3 + 3x^2 - 7x = 6 is the negative number -7.
p.439
It jumps right out at us, since we are completely used to the idea that a subtraction is equivalent to the addition of a negative quantity.
p.439
We could rewrite the as follows: x^3 + 3x^2 + (-7)x = 6 . For us who live five centuries after Cardano, these two equations are trivially interchangeable. The conceptual slippage on which their equivalence is based is so minute that we don't even perceive it at all.
p.439
But for the author of the vast tome on the third-degree equation, the concept of negative seven simply didn't exist. For him, the only legitimate way to get rid of a subtraction in a polynomial (that is, a term with a negative coefficient) was to move the misfit term to the other side of the equation, thus yielding a different but related equation ── namely, x^3 + 3x^2 = 7x + 6 ── all of whose coefficients are positive.
p.439
From a contemporary viewpoint, what Cardano did is comparable to someone who invents thirteen kinds of can-openers, each one working for just one type of can. It was a great feat, but what was lacking was an umbrella formulation, laying bare the hidden unity lying behind all this apparent diversity. That is, what was missing for “the” cubic equation was its universal can-opener. But this goal was unthinkable until someone recognized that all these different equation were, in fact, just one equation.
p.440
Obviously, what was needed was a new conceptual leap, this time extending the category number to include negative numbers.
p.440
The idea of giving such equations solutions had been considered but was always rejected (at least in Europe). Cardano himself understood that “fictitious” numbers (as he referred to them) could satisfy such an equation, but he rejected the idea with disdain. To him, the concept of negative three , being in no way visualizable, was an absurdity, somewhat like the concept of an object that violated the laws of physics. Such an idea might be stimulating to the mind, but it had to be recognized as absurb, because there was no way of actually realizing it in the world. Since Cardano was unable to associate negative numbers with any kind of entity in the real world, he labeled them “fictitious” and discarded them.
p.440
Nonetheless, his successors ── most especially Raffaello Bombelli Bologna ── were powerfully driven to find the elusive unity in Cardano's troubling diverse set of thirteen chapters, and in the end they wound up accepting the notion of negative numbers on the same level of reality (or nearly so) as positive numbers. This move yielded an enormous simplification in the solution for the cubic equation, as the thirteen families were gracefully fused into a single family, and the thirteen recipes associated with them were also fused into a single, compact recipe.
p.440
─
“”
p.440
Nonetheless, the welcoming of negative numbers into the category of number was not immediate or universal. Even 250 years later, the English mathematician Augustus De Morgan, a central figure in the development of symbolic logic, was still resisting, as this passage from his 1831 book On the Study of Difficulties of Mathematics shows:
“8 - 3” is easily understood; 3 can be taken from 8 and the remainder is 5; but “3 - 8” is an impossibility; it requires you to take from 3 more than there is in 3, which is absurb. If such an expression as “3 - 8” should be the answer to problem, it would denote either that there was some absurdity inherent in the problem itself, or in the manner of putting it into an equation...
DeMorgan's comment is reminiscent of what a seven-year-old girl once said to one of us when she was a participant in experiments on subtraction errors. To explain why she'd written “0” at the bottom of a column containing the numerals “3” and “8”, she said, “If I had three pieces of candy in my hand and I wanted to eat eight, I'd eat the three I had and there wouldn't be any more left.”
p.440
Despite the passage of several centuries, the sizable age gap, and the immense amount of mathematical sophistication separating our two commentators, their reactions still share a common essence.
p.441
Only at this point De Morgan happy, admitting that the idea “-2 years will pass” is equivalent to the idea “2 years have passed”. He thus does accept the idea that a numerical value can be negative, but not that a length of time can be negative. All this would lead one to think that De Morgan had no qualms about negative numbers within pure mathematics, even if he didn't think they applied to the real world ── and yet, a little later in his book, when he deals with the quadratic equation, instead of considering it as one single, unified problem, he breaks it up into six different families of equations, insisting (in perfect Cardano style) that all three of its coefficients must be positive! De Morgan thus finds that there are six different quadratic formulas, rather than one universal one. And all of this nearly 300 years after Cardano!
(Surfaces and essences: analogy as the fuel and fire of thinking, Douglas Hofstadter & Emmanuel Sander, 2013, )
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That selection process is perception. “I am a very big believer”, Hofstadter told me, “that the core processes of cognition are very, very tightly related to perception.”
── Kevin Kelly, 1994,
from the book, Out of Control,
p.18, filename: ooc-mf.pdf
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Michael Lewis, The undoing project, 2017 [ ]
p.153
Avishai Margalit
“I'm waiting in this corridor,” said Margalit. “And Amos comes to me, agitated, really. He started by dragging me into a room. He said, You won't believe what happened to me. He tells me that he had given this talk and Danny had said, Brilliant talk, but I don't believe a word of it. Something was really bothering him, and so I pressed him. He said, ‘It cannot be that judgement does not connect with perception. Thinking is not a separate act.’”
p.153
He said, ‘It cannot be that judgement does not connect with perception. Thinking is not a separate act.’
p.343
People didn't choose between things, they chose between descriptions of things.
p.343
“choice architecture”
The decisions people made were driven by the way they were presented. People didn't simply know what they wanted; they took cues from their environment. They constructed their preferences. And they followed paths of least resistance, even when they paid a heavy price for it.
(Michael Lewis, The undoing project, 2017, p.153, p.343 )
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Taking cues from the environment has a big picture relationship to social referencing. “... showing off his social referencing skills. Social referencing, which is what we are doing when we look to another's emotional reaction to help us assess a novel object or situation, is an important milestone that kicks in during the first year of life, and it is one of the most fundamental ways that humans connect to each other.” (p.167, Frank Moss, The sorcerers and their apprentices, 2011)
____________________________________
People do not make judgement between things, they make decision based on the explanation of things. It is in the description and explanation that you can influence and shape perception, and through perception, people make cognitive judgement. What concerns me is not the way things are, but rather the way people think ‘they are’.
Harold Innis (Harold Adams Innis)
◇ The shift in perception redefines “knowledge.”
(a.) Acting involves changing our behavior,
(b.) Reframing involves changing our thinking, and
(c.) Transforming involves changing our perceptions.
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Evolutionary Psychology index
The Fallacy of Fitness Maximization
Evolutionary Psychology: A Primer
Leda Cosmides & John Tooby
Co-Directors
Center for Evolutionary Psychology
University of California, Santa Barbara
•••• ••• ••••
Two major categories of social conditionals are social exchange and threat -- conditional helping and conditional hurting -- carried out by individuals or groups on individuals or groups. We initially focused on social exchange (for review, see Cosmides & Tooby, 1992).
We selected this topic for several reasons:
Many aspects of the evolutionary theory of social exchange (sometimes called cooperation, reciprocal altruism, or reciprocation) are relatively well-developed and unambiguous. Consequently, certain features of the functional logic of social exchange could be confidently relied on in constructing hypotheses about the structure of the information-processing procedures that this activity requires.
Complex adaptations are constructed in response to evolutionarily long-enduring problems. Situations involving social exchange have constituted a long-enduring selection pressure on the hominid line: evidence from primatology and paleoanthropology suggests that our ancestors have engaged in social exchange for at least several million years.
Social exchange appears to be an ancient, pervasive and central part of human social life. The universality of a behavioral phenotype is not a sufficient condition for claiming that it was produced by a cognitive adaptation, but it is suggestive. As a behavioral phenotype, social exchange is as ubiquitous as the human heartbeat. The heartbeat is universal because the organ that generates it is everywhere the same. This is a parsimonious explanation for the unversality of social exchange as well: the cognitive phenotype of the organ that generates it is everywhere the same. Like the heart, its development does not seem to require environmental conditions (social or otherwise) that are idiosyncratic or culturally contingent.
Theories about reasoning and rationality have played a central role in both cognitive science and the social sciences. Research in this area can, as a result, serve as a powerful test of the central assumption of the Standard Social Science Model: that the evolved architecture of the mind consists solely or predominantly of a small number of content-independent, general-purpose mechanisms.
•••• ••• ••••
If the human mind develops reasoning procedures specialized for detecting logical violations of conditional rules, this would be intuitively obvious. But it is not. In general, fewer than 25% of subjects spontaneously make this response. Moreover, even formal training in logical reasoning does little to boost performance on descriptive rules of this kind (e.g., Cheng, Holyoak, Nisbett & Oliver, 1986; Wason & Johnson-Laird, 1972). Indeed, a large literature exists that shows that people are not very good at detecting logical violations of if-then rules in Wason selection tasks, even when these rules deal with familiar content drawn from everyday life (e.g., Manktelow & Evans, 1979; Wason, 1983).
The Wason selection task provided an ideal tool for testing hypotheses about reasoning specializations designed to operate on social conditionals, such as social exchanges, threats, permissions, obligations, and so on, because (1) it tests reasoning about conditional rules, (2) the task structure remains constant while the content of the rule is changed, (3) content effects are easily elicited, and (4) there was already a body of existing experimental results against which performance on new content domains could be compared.
For example, to show that people who ordinarily cannot detect violations of conditional rules can do so when that violation represents cheating on a social contract would constitute initial support for the view that people have cognitive adaptations specialized for detecting cheaters in situations of social exchange. To find that violations of conditional rules are spontaneously detected when they represent bluffing on a threat would, for similar reasons, support the view that people have reasoning procedures specialized for analyzing threats. Our general research plan has been to use subjects' inability to spontaneously detect violations of conditionals expressing a wide variety of contents as a comparative baseline against which to detect the presence of performance-boosting reasoning specializations. By seeing what content-manipulations switch on or off high performance, the boundaries of the domains within which reasoning specializations successfully operate can be mapped.
The results of these investigations were striking. People who ordinarily cannot detect violations of if-then rules can do so easily and accurately when that violation represents cheating in a situation of social exchange (Cosmides, 1985, 1989; Cosmides & Tooby, 1989; 1992). This is a situation in which one is entitled to a benefit only if one has fulfilled a requirement (e.g., "If you are to eat those cookies, then you must first fix your bed"; "If a man eats cassava root, then he must have a tattoo on his chest"; or, more generally, "If you take benefit B, then you must satisfy requirement R"). Cheating is accepting the benefit specified without satisfying the condition that provision of that benefit was made contingent upon (e.g., eating the cookies without having first fixed your bed).
When asked to look for violations of social contracts of this kind, the adaptively correct answer is immediately obvious to almost all subjects, who commonly experience a "pop out" effect. No formal training is needed. Whenever the content of a problem asks subjects to look for cheaters in a social exchange -- even when the situation described is culturally unfamiliar and even bizarre -- subjects experience the problem as simple to solve, and their performance jumps dramatically. In general, 65-80% of subjects get it right, the highest performance ever found for a task of this kind. They choose the "benefit accepted" card (e.g., "ate cassava root") and the "cost not paid" card (e.g., "no tattoo"), for any social conditional that can be interpreted as a social contract, and in which looking for violations can be interpreted as looking for cheaters.
From a domain-general, formal view, investigating men eating cassava root and men without tattoos is logically equivalent to investigating people going to Boston and people taking cabs. But everywhere it has been tested (adults in the US, UK, Germany, Italy, France, Hong-Kong; schoolchildren in Ecuador, Shiwiar hunter-horticulturalists in the Ecuadorian Amazon), people do not treat social exchange problems as equivalent to other kinds of reasoning problems. Their minds distinguish social exchange contents, and reason as if they were translating these situations into representational primitives such as "benefit", "cost", "obligation", "entitlement", "intentional", and "agent." Indeed, the relevant inference procedures are not activated unless the subject has represented the situation as one in which one is entitled to a benefit only if one has satisfied a requirement.
Moreover, the procedures activated by social contract rules do not behave as if they were designed to detect logical violations per se; instead, they prompt choices that track what would be useful for detecting cheaters, whether or not this happens to correspond to the logically correct selections. For example, by switching the order of requirement and benefit within the if-then structure of the rule, one can elicit responses that are functionally correct from the point of view of cheater detection, but logically incorrect (see Figure 4). Subjects choose the benefit accepted card and the cost not paid card -- the adaptively correct response if one is looking for cheaters -- no matter what logical category these cards correspond to.
•••• ••• ••••
source:
http://cogweb.ucla.edu/ep/EP-primer.html
http://www.sscnet.ucla.edu/comm/steen/cogweb/ep/EP-primer.html
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Malcolm Gladwell., The tipping point: how little things can make a big difference, 2000, 2002
pp.158-159
It's probably the result of the way evolution has structured our brain. For instance, anthropologists who study vervets find that these kinds of monkeys are really bad at picking up the significance of things like an antelope carcass hanging in a tree (which is a sure sign that a leopard is in the vicinity) or the presence of python tracks. Vervets have been known to waltz into a thicket, ignoring a fresh trail of python tracks, and then act stunned when they actually come across the snake itself. This doesn't mean that vervets are stupid: they are very sophisticated when it comes to questions that have to do with other vervets. They can hear the call of male vervet and recognize whether it comes from their own group or a neighboring group. If vernets hear a baby vervet's cry of distress, they will look immediately not in the direction of the baby, but at its mother ── they know instantly whose baby it is. A vervet, in other words, is very good at processing certain kinds of vervetish information, but not so good at processing other kinds of information.
The same is true for humans.
pp.159-160
p.159
Consider the following brain teaser. suppose I give you four cards labeled with the letter A and D and the numerals 3 and 6. The rule of the game is that a card with a vowel on it always has an even number on the other side. Which of the cards would you have to turn over to prove this rule to be true? The answer is two: the A card and the three card. The overwhelming majority of people given this test, though, don't get it right. They tend to answer just the A card, or the A and the six. It's a hard question.
pp.159-160
But now let me pose another question. Suppose four people are drinking in a bar. One is drinking Coke. One is sixteen. One is drinking beer and one is twenty-five. Given the rule that no one under twenty-one is allowed to drink beer, which of those people's IDs do we have to check to make sure the law is being observed? Now the answer is easy.
p.160
In fact, I'm sure that almost everyone will get it right: the beer drinker and the 16-year-old.
p.160
But, as the psychologist Leda Cosmides (who dreamt up this example) points out, it is exactly the same puzzle as the A, D, 3, and 6 puzzles. The difference is that it is framed in a way that makes it about people, instead of about numbers, and as human beings we are a lot more sophisticated about each other than we are about the abstract world.
(The tipping point: how little things can make a big difference / by Malcolm Gladwell., 1. social psychology., 2. contagion (social psychology), 3. causation.
4. context effects (psychology), HM1033.G53 2000, 302──dc21, originally published in hardcover by Little, Brown and Company, March 2000, first paperback edition, January 2002, 2000, 2002, )
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