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Traffic Jams In Your Brain
An anonymous reader writes "Carl Zimmer's latest foray into neuroscience examines why the brain can get jammed up by a simple math problem: 'Its trillions of connections let it carry out all sorts of sophisticated computations in very little time. You can scan a crowded lobby and pick out a familiar face in a fraction of a second, a task that pushes even today's best computers to their limit. Yet multiplying 357 by 289, a task that demands a puny amount of processing, leaves most of us struggling.' Some scientists think mental tasks can get stuck in bottlenecks because everything has to go through a certain neural network they call 'the router.'"
Have they tried unplugging it, waiting 30 seconds, and plugging it back in?
First!
So the claim is that our brain is a field-programmable gate array (for economy and flexibility and performance) that takes time to re-arrange to accommodate different sorts of tasks.
Sounds entirely sensible to me.
But distracted me too long to get first post.
Rgds
Damon
http://m.earth.org.uk/
Couldn't it just be that we do not really have direct access to the raw computational capacity of the brain? There are savants and people who have trained themselves tremendously who can do arithmetric like this, using memory tricks and such. Wouldn't that be more like a hack to "reach down" to utilize the low-level capacity of the brain? The brain is nothing like a man-made computer, but doesn't the "layers of abstraction" still apply? The brain can calculate 357 by 289, but it does not naturally "understand" what 357 or 289 is, or for that matter what the high-level instruction from "me" to "multiply" is.
Emotions! In your brain!
103173
I don't see what so hard about opening a calculator and typing some numbers.
Kids these days!
How about 357 * 289 being hard is because 7 is the average size of the short term memory, and you need to remember more numbers than that to arrive at 103,173?
doing complex multiplication isn't inherently necessary to staying alive. being able to discern who is who is.
289 x 357 is surprisingly the same result.
If it works for the Borg why not us?
... the way math developed and was taught is not the only way to teach "math", this is one thing that I've learned as I've grown up. And I'm still doing much research in this area.
There are better ways to teach people how to do those computations but it requires a conceptual understanding that there is not a "Set" way of thinking about "numbers" (really our alphabet for communicating distinction and differences) linking the way we naturally think with foreign languages developed by a narrow set of minds. see: Mayan numerals.
http://en.wikipedia.org/wiki/Maya_numerals
Notice how mayan numerals rerepsent themelves as geometric objects that are easily discerned at a glance versus our our highly compressed representational notation (1,2,3,4). Mayans knew that all numbers are made of distinct geometric distinctions and hence they used simple uniform geometric objects as representation to communicate numbers "at a glance", representation _matters_ to how we think about concepts and how we can use them and map them between systems of thinking that only SEEM different on the surface.
You have to understand the numbers can be rethought as natural ratio of shape and size in the real world, when we measure things in the real world we use arbitrary ratios of an object in regards to our own visual system.
For instance 357 by 289 can be broken down to
3.57 x 2.89
What you're trying to do is limited the # of elements by changing the ratio you have to see "lots of things" as merely representations of smaller scale things and things get a lot easier once you understand this principle.
The whole way math is taught is really fucked up and made for a narrow range of particular minds that function and "Get" how our mathematical system developed. If you begin studying the history of math, you realize that representation and HOW YOU THINK about how we mathematize nature matters a hell of a lot more then just throwing stuff other people figured out at kids in a symbolic format developed for a narrow subset of human minds.
Math is just a symbolic language to communicate our observation of distinctions and differences in regards to space, matter and time in the world.
Exactly. That's why I've always been baffled by the commonplace brain-computer comparison, when it's so very clear that what goes on under the respective hoods is completely different.
Knowledge is power; knowledge shared is power lost.
In the I, Borg episode of Star Trek TNG, they went a step further in this idea trying to shut down the collective showing them an unsolvable geometric problem. And as with this kind of traffic jams, we can always refuse to go thru this road, or even exit it if took too much time.
For fun I'll record in the slashdb record how I surprised myself and managed to correctly multiply 357 by 289 in my head. I don't have a history of being able to do such things, and it did take me a couple minutes, but I was pretty surprised to see bc confirm my answer.
... 57 + 14 is 71 thus 1071, then add those 2 0's back on to get 107100. Then throw that pesky round 100K off to the side to be remembered later, and we have 7100 - 11 * 357. Which is 7100 - (10 + 1) * 357, so 7100 - 3570. Since half of 7100 is clearly 3550, then 3530 is the partial. now 1 more 357 to subtract from that. I went with 3530 - 400 is 3130, but have to add back the 43 to get 3173. Now just add back that big ol round 100K and we have 103173.
:) Of course I expect other people may have more digit storage memory than I do, and thus can just do it the standard simple way...
I went with the route of breaking it down to 357 * 300 - 11 * 357. 357 * 300 is 35700 * 3. Even that is pretty hard so I figured 35700 * 2 = 71400, then + 35700 .
See, no sweat
have you heard of associative processing/memory? That's CS concept with PoC that we can construct machines like human brain, but again, they could analyze face in fraction of second, but multiplying is gonna take some time...
Our memory gets things wrong all the time. We can scan a crowd and associate a face with someone who looks similar. When we multiply a number in our head, we're trying very hard to get the exact, correct answer. Perhaps the brain is just a lot better at fuzzy problems than those demanding strict correctness.
It might also be an input problem. Our number system is a very effective way to communicate math, but it may be a very foreign and sub-optimal way for the brain to process math. Maybe we need a new way to represent things that provides a better balance.
It's worse than you think. I got the 825 instead of the 655 and now I can't get voice to work.
And don't get me started about where to put the USB key for a firmware upgrade.
-Matt
--- Need web hosting?
People are great at recognizing faces. Computers are great at computing.
Don't fight for your country, if your country does not fight for you.
I am not expert, and this is just from a brief conversation I had in an elective class many years ago with a neural science professor but I asked how it is the brain does things in an instant that would likely take a powerful micro computer most of a day, while simple multiplication is often quite difficult for me to do in my head.
The reason he gave is that the brain works usually in a in precise manor. You have lots of different groups of neurons that your relatively plastic brain has wired up to do things like recognize certain patterns. If enough of those go high other parts of your brain proceed as if there was certainty. That works well for evaluating how hard the sterling wheel is pushing back and deciding how much more to stimulate muscles to contract. When you doing something like math though there is only a very specific correct symbol. They parallelism of the voting system breaks down and your brain how to check that all or almost all of those networks agree.
Repeal the 17th Amendment TODAY! Also Please Read http://www.gnu.org/philosophy/right-to-read.html
I think this is a case of having a hammer and everything looking like a nail. Yes, we have computers. Yes, we have math. After decades, our best technology barely functions at the level of a small rodent.
Either life functions at several orders of magnitude higher levels of computation, or it's fundamentally different, more like a huge analog ball of yarn that furthermore encodes itself.
The Chinese used "GPU"s instead of "CPU"s for their supercomputer record - and that is supposedly "unfair". How is it anywhere close to fair then comparing the diverse capabilities of a brain and a computer?
357 TIMES 289? That's easy. It's uh...uh.... hold on...
errr.........
BLUE SCREEN OF DEATH! *Beeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeep*
Here's an analogy to illustrate the category error people make when comparing the human brain to a computer:
"A Sony Walkman can record and play music in realtime, fast-forward and rewind, and store an hour's worth of music. These tasks require a 75 Mhz processor and 100 megabytes of memory on an iPod Shuffle. Therefore, a Sony Walkman has a 75 Mhz processor and 100 megabytes of memory."
Makes me wanna scream and shout.
RETURN without GOSUB in line 1050
In the USSR, the army joined you.
Time for that 7600 series you've always wanted. That or go with a nice ISR. You'll get used to the Cisco stamp on your forehead. :-P
We had a family friend (he has passed now) who could go to a railroad crossing with a train going 60 miles per hour down the track and correctly add the 7 digit (or more sometimes) numbers on each train car as the train passed.
He said that he would not "add" the numbers but allow for them thus coming up with a total more through allowing the right answer than by math manipulation like we would have to do consciously.
The whole thing was sort of spooky to behold... here we were writing down the numbers of each car and he effortlessly knew the running total. It was if he allowed his unconscious part of his brain to observe the number, add it to the running total without interfering with the process mentally and then his conscious mind would retrieve the answer from the unconscious mind at the end of the train or after 20 cars have passed or other terminating choice.
And in the end, the love you take is equal to the love you make
Being able to recognize friend or foe would have had great survival value in human evolution, as would the ability to identify a prey animal in a dark forest. Being able to do arithmetic in an instant would not have the same value.
You can scan a crowded lobby and pick out a familiar face in a fraction of a second, a task that pushes even today's best computers to their limit. Yet multiplying 357 by 289, a task that demands a puny amount of processing, leaves most of us struggling.
Isn't it how you define processing? For a computer it don't make any difference if you process 5*5 or 123565*435456, because the internal representation of the numbers and the hardware paths are basically the same for both tasks. But our brain is a highly specialized device, specialized on tasks that help us to survive in the good old jungle.
So the task to multiply 357 by 289 takes only for our computers a puny amount of processing, because our computers are specialized on this tasks, but for our brain in takes a high amount of processing, because it is specialized for different tasks.
There are for some reasons FPGAs out there which will improve the speed for a specialized task by 10 or 100 times.
http://www.mueller-public.de - My site http://www.anr-institute.com/ - Advanced Natural Research Institute
So -- now we know the mechanism behind Nerd Sniping.
Look for a book called "FingerMath." It teaches how to use your fingers like an abacus. After you get used to it you can stop moving your fingers and just kind of "feel" the calculations. No, I never practiced it enough to get good at it. But it is a pretty good book.
..seems to have the same traffic jam effect.
Any sufficiently unpopular but cohesive argument is indistinguishable from trolling.
What use was it in our evolutionary development to be able to multiply two three digit numbers? That hasn't been something important to humans on a whole for, at most, two or three thousand years, and we could probably make an argument that it has really been important for more than a few hundred years.
On the other hand, how important has it been to be able to recognize faces in a crowd? Extremely so, at least since we start running around in groups. Gotta know who's friendly and who's not, right?
So the brain evolved to solve the problems that actually helped survival, and arithmetic was kinda low on the list.
On the other hand, anyone with reasonable intelligence can train themselves to do arithmetic problems in their head. There are relatively easy techniques, some of which have already been mentioned in the discussion. Mostly it's a matter of learning how to break problems down into easily manageable pieces.
Sean.
It's 3 steps each time here, for each column involved. Very easy to do (I do this with my little 7 yr. old niece all the time to help her understand multiplication & how it works, while I help her w/ her homework).
E.G.-> 357x289 breaks down into 3 sets of multiplications of the "100's columns x 10's column x 1's columns" as follows:
A.) You must "bust up" 357 into its constituent parts (100's column (300), 10's column (50), & 1's column(9)) elements and start multiplying them against 289's 1's column (which is 9 in this case), first:
1.)
300 x 9 = 2700 (300 from 357 x 9 (1's column in 289))
50 x 9 = 450 (50 from 357 x 9 (1's column in 289))
7x9 = 63 (7 from 357 x 9 (1's column in 289))
2.) Add their results = 3,213
---
B.) You must "bust up" 357 into its constituent parts (100's column, 10's column, & 1's column) elements and start multiplying them against 289's 10's column (which is 80 in this case):
1.)
300 x 80 = 24,000
50 x 80 = 4,000
7 x 80 = 560
2.) Add their results = 28,560
---
C.) You must "bust up" 357 into its constituent parts (100's column, 10's column, & 1's column) elements and start multiplying them against 289's 100's column (which is 200 in this case):
1.)
300 x 200 = 60,000
50 x 200 = 10,000
7 x 200 = 1,400
2.) Add their results = 71,400
---
LASTLY, ADD THE BOLDED RESULTS OF EACH ABOVE? You'll have your answer = 103,173
APK
P.S.=> This is how I teach kids how to multiply larger numbers, & it tends to work. It's VERY IMPORTANT though to stress that the 100's, 10's, 1's columns are what they are though - representations of LARGER numbers than just their single digits (beyond the 1's column): Once you get a kid past that? They're great @ it...
Just by using the "distributive property" of mathematics in multiplication (which allows you to "break up" larger numbers into smaller & more manageable ones to work with)... apk
The brain isn't a digital computer
Check out On Intelligence. It discusses a theory of how the cortex maps to the body and to the outside physical world using the same processes for action and interpretation.
The real problem is that the internal representation of numbers is logarithmic. It has exact resolution only for a limited range. With practice or talent, you can make small numbers into sparse matrices of these exact numbers in different digit positions, and work those out (probably the auditory cortex will do the job) at far higher speeds. You will still be limited to 4-7 ops per sec while relaxed, possibly being able to strain yourselv to reach 15-25 ops per sec if you really concentrate to the point of forgetting to breathe.
We can easily add two logarithmic numbers and get a new logarithmic number. Precision in a problem, though.
See, with concrete numbers, we are working in the wrong radix (8 or "e" would be better, by far) and really messing things up badly because we need to construct representational objects in short term memory. These are exact, but cannot be parsed any faster than spoken words, and there are problems in using them for calculations, which will require slow lookups and step by step processes for most people. With proper training, you will refactor the problem in parallell with solving it, which speeds it up a great deal. With repeated use, you will form circuits that are going to accelerate processing. But you will still probably not get much faster than 10-30 digits per second and 3-4 ops per second.
Hacks that give you access to the representation can solve the problem as noted above.
Hacks that give you access to the wetware can probably overload some brain regions into doing the math for you with parallell computing. I would hazard a guess at a temporal or insular location for such processing, but wouldn't know, as I haven't seen any fMRI studies on where savants put the circuitry to do it.
Asking why we can't do three-digit multiplication quickly even though our brains is complex is sort of like asking why a toaster can't tell you ratios of voltages even though it has resistors in it. It's the difference between what a machine does and how it works. Brains are fabulously complex, but one thing they weren't built for is three-digit multiplication. Does the brain "know" how to do multiplication really really fast? Yes, of course, there are all kinds of things going on in the brain that involve multiplication. Does it know how to do it with numbers that come in through the ears, and spew the answer out through your mouth? No, brains weren't built to do that. They were, however, built (so to speak) to do much more complicated (but different) things, like recognizing threats and understanding spoken language.
I don't know how good the router analogy will turn out to be, but it's not exactly breaking news that some things need attended, more-or-less serial processing, and that mental arithmetic is one of them. The things that don't need as much attention are things that are evolutionarily old and more or less built-in. Extremely overlearned tasks can fake it sometimes. Guys like Hal Pashler and Stan Dehaene are always making progress into understanding how and why these things work, but the idea of processing bottlenecks in cognitive function is very old. The router analogy is probably a bad one, because it's unlikely that the brain's router lives in any very specific place. It's more likely a property of how the brain adapts to tasks it wasn't designed for.
Thanks for sharing your insight. I have always been very frustrated by my experiences with math education. You seem to be one of the few people who understands that the representational notation isn't the math. The math is an abstract thing. It's "out there in the ether." I have a hunch that many people (myself included!) learned how to manipulate the representational notation without having any but the most vague concept of what the representational notation represents.
Music education has this exact same problem. Teachers teach students how to code and decode symbolic notation while students fail to develop any idea of what this notation would actually sound like if brought to life. That's how you end up with the characteristic "school band" or "school orchestera" sound: it's out of tune, the rhythms aren't quite right, and the sound is generally sloppy and uninspiring. This is not because students are inexperienced. It's because they're receiving poor instruction.
What's the solution? Change the way teachers teach? The problem is that many teacher-education students feel successful learning the way they have been taught, and will probably continue to teach the way they have been taught.
So what you really have to do is set up an organization to find people who agree with you. Then this organization concentrates on training other people to be early childhood and parent educators, so that the next generation of children will have even more and better teachers. It's a long term Benet Gesserit sort of thing, but I don't know how else to reform education.
Fortunately, music educators have this organization. It's the Gordon Institute for Music Learning, the only research based music teacher education organization. If you want music lessons for yourself or a child, see if you can find a GIML certified instructor in your area.
If there is a similar organization for math education, I'd be happy to throw my (admittedly small) weight behind them. Anyone know of anything?
Spoon not. Fork, or fork not. There is no spoon.
My Linksys is fueled by coffee. I don't have these issue......
What was I saying again?
BRAINS DO NOT WORK THAT WAY!
We can scan a crowd for a face in a second maybe because our brain is trained to see the food or the danger in our environment. It is something every animal can do. Math is not necessary for survival.
is none of them have actually been taught to do math other then 'here is the problem, here is the 100 dollar calculator your parents were told to buy. sit here and do problems 1 through 30 while i surf the internet and other stuff.'
ask a Japanese or well to do Chinese kid the same problem and they will be able to solve it if not in their head, on a scrap piece of paper.
We dont have the skills we dont use.. as in.. it is almost a survival skill to scan a room or place for friendly faces but you could probably go a while without needing to multiply large numbers in most cases.
As an example.. when I was 20ish, I got a job at a 7-11 that was surrounded about 6 blocks deep by apartments. It was insanely busy 24 hours a day. After about a week, I started memorizing common groupings of items with their taxes. It started with a pack of smokes and a 12-pack of beer (which was like 8.49 back then). From there it mushroomed and in about 2-3 more weeks I could look at what people had in their hands, do the mental gymnastics and tell everyone in line what they owed including tax. First person with cash out got served. People with checks..you go over there with the pen for later, people who dont believe me got rung up on the register over there where I dont really ever recall a mistake. I think its more about necessity than capability.
Or maybe it's because the brain does not do computations, and shouldn't be compared to computers at all.
4096R/EF7BAFA6 79E1 DF98 D09D 898F 9A11 F6F0 DDDC 23FA EF7B AFA6
Does this imply that if one hears "global warming warnings" 10 times/day vs "don't worry" once per day things are ten times worse than we thought? Go ahead, mod this down - you know you want to. It'll make you feel better, do it! Mod me into oblivion, you'll feel better in the morning and won't have to think for yourself! Just listen to the talking points and toe the line!
As previous posters before me have observed - it just depends how one was taught math, and whether they tried breaking away from the canonical thinking.
Therfore, what is 357 times 289?
(357 * 2) * 100 + (357* 90 - 357)
I always find it easier to arrange my calculations according to basic math laws then you just breeze through them in your mind.
I was helping my friend's child with homework - basic stuff that involved simple multiplications.
The child could not believe that 9*9 (which was too difficult to do in mind), was same as (9*10) - 9 (which was much easier to calculate).
With better explanation and understanding than at school, the child really started trying harder with maths (started trying harder... but still finds it a pain, heh)
jsut my 2p..
The reason a human can quickly and easily scan a crowd to recognize a familiar face is not at all due to the processing or computing power of the brain. Rather, it's a byproduct of the way the brain works combined with how humans develop after they are born. From birth, the brain is constantly being exposed to stimuli, and trying to build associations between those stimuli. That's why a baby is not capable of scanning a crowd and recognizing a familiar face the day its born; but fast forward several months, and it can. It didn't get better at computation, it simply had more neurons available that were hooked up in the right way for that task. It learned how to do that through a combination of stimuli, feedback, and association. That's essentially what the brain is ... a machine that takes in stimuli, and evaluates feedback.
It's entirely conceivable that you could take a baby, and create a specialized series of experiences which would effectively train its brain to solve very complex mathematical problems. Imagine a 3 year old who could only do the most basic human things (eat, sleep, excrete), but also take complex math problems as input, and produce the results nearly instantaneously. Once the brain knows "how" to solve a problem, it can usually do so very quickly... for example, scanning a crowd. The problem is teaching the brain how to solve the problem naturally.
For most practical think on your feet applications, multiplying numbers should be done by rounding them to close approximations. So you have 357 and 289. 357 is close to 350. 289 is close to 300, the difference of rounding is good because you're going up in one and down in another. So now you're looking at multiplying 300x350. 100x100 is 10,000, so 3*3 =9, so 300x300 is 90,000. now you just have the trivial matter of 300x50. 3/2=1.5, so you're looking at 10.5 * 10,000. So you have the answer is 10500. With the aid of a calculator, I got 103,173 so it isn't far off. It is real easy to think about this way.
If you think on your feet and you try to do stuff like,"Carry the 1, remember a number in a short term memory cache, and then do another multiplication and try and remember again, by the time you do your new calculation, you probably forgot your short term." I'd hazard a guess(this is all conjecture from now on) that brains like to use the same variable over and over again to store data, and when you start doing a complex calculation it could rewrite the same memory location. I don't know about your brain, but I budget between processing and memory. If I'm constantly thinking about stuff, my memory goes and I need an external aid. If I'm just memorizing stuff, my processing goes. Try doing 357x289 and use a notepad to record your states and it is easy(memorization). Alternatively you could use a calculator and do 357x9 then memorize the result, followed by 357x80 then memorize result followed by 357x300 and memorize the result. Then you use your memory to add the three numbers in your head. If you can be pure memory or processing, the task doesn't get difficult.
God spoke to me.
... I don't have one. :P
Ant(Dude) @ Quality Foraged Links (AQFL.net) & The Ant Farm (antfarm.ma.cx / antfarm.home.dhs.org).
You don't need so many more registers, you simply need a better algorithm ... behold ... even you can remember three numbers as you work ..357x289:
Working number W and carry C and working answer A. You can even write down A and not need to remember it, just use 2 registers
{W} 7x9= 63 {W} therefore C = 6 and A = 3
{W} 5x9(45)+7x8(56)(101)+6 [C] = 107 {W} therefore C = 10 and A = 73
{W} 3x9(27)+5x8(40)(67)+7x2(14)(81)+10 [C] = 91 {W} therefore C = 9 and A = 173
{W} 3x8(24)+5x2(10)(34)+9 [C] = 43 {W} therefore C = 4 and A = 3173
{W} 3x2(6)+4 [C] = 10 {W} therefore A = 103173
So long as you don't mind giving your answer backwards when you write it down, you can do all this in your head, very easy shit.
Those teachers in primary school that taught you long multiplication were just simply ... ignorant and incurious.
I learnt how to provide the 28 repeating decimal for x/29 (where x less than 29) with just a SINGLE register. There's loads more really simple tricks, but we have drones as teachers. Credit to "Vedic mathematics"
Having been born with occipital lobe damage and agenesis of the corpus collossum, meaning the wiring that usually connects the right and left hemispheres of the brain was never there, which, when discovered, late in life, led two experts to believe I had become too mentally disabled to live independently much less practice law or do any gainful work, I find this fascinating. Once they realized that tests just done put my verbal abilities in the 97th percentile way past middle age and while being treated for major depression, this was congenital and long before I had been a National Merit Scholarship finalist and graduated in the upper third of my selective law school class, practiced law including trials and appeals, etc., they began to understand and explained why, despite trying hard to go into engineering, I never could do higher math--or subtract, for that matter, and I have been having "senior moments" since first grade 65 years ago. . Especially in the very young, the brain has amazing powers to rewire and preprogram itself that even the neurologists and computer experts are only beginning to think about understanding and using outside the brain. One of the questions nobody who buys into the current orthodoxy will address is how and why this level of redundancy and adaptability would and could ever have evolved and become widely distributed since the occasions for it to be beneficial would and do arise so rarely and, in a primitive world, would have left those like me at grave survival disadvantages. Furthermore, do any of those of you with better computer science backgrounds know of any computer that can, on its own, create multiple partitions, directories, and subdirectories, some with and some without cross-access to deal with overloads and other problems that would otherwise destroy data etc., as the human brain does in dissociative identity (multiple personality) disorder in response to childhood trauma, and particularly those that require the brain to deal with contradictory authoritative messages? I know some experts in and some people with this condition. Some don’t believe it is real, but anyone who has seen dissociation at that level knows that it is.
It's interesting that you commented about music. I, too, thought about music. Just last night, I did a music test for an orchestra, that I paid hundreds of dollars to play in, just for fun. Think about that: pay and still do a test. Not surprisingly, after hours of practise, I still failed. The conductor was disappointed, I'm sure, and I was too.
I'm about to send him a message on why I was dissatisfied with the test, and I think that it has to do with what you are getting at. I think that most skilled musicians seem to have this innate sense of timing, tuning, and overall ability that mere mortals like me don't have. The thing that is so deceptive about their skills is that they must also practise. Therefore, they assume that because somebody does not play well, he did not practise, or practise enough.
To really put this into perspective, I practise for hours, but I often instantaneously forget what I learned. I'm just so frustrated. Once I start practising it again, it is easier to pick up, but it takes a lot of practise to pick it up again. Also, when we rehearse as a group, my abilities are much better, because I can take cues off of the conductor and the band, but during the test, I had no cues. I literally had silence, and was still expected to keep track of timing. 1 thing that I found so unusual about the test was that the silence was very distracting, since I am used to playing with the band, or playing with a metronome.
For the most part I agree with you, in that a lot of failures are due to poor instruction, but I also believe that the students don't practise enough. I can't blame them, because it takes skill to practise.
I'd like to read your thoughts on all that I wrote.
I'm going to check out your link.
You have a great sig.
testing out my trending skills
"What's the solution? Change the way teachers teach?"
Yes, but a curriculum has to be based off the research not just "changing teaching". i.e. a curriculum developed off the research itself. The whole point is that you have to teach people how to observe the world first and understand the process of "mathematization", this is the key thing, imagine you are observing a ball bouncing straight up and down in the world and you want to express and copy the relationships this ball bouncing is communicating to your senses.
This is where the observation process begins by looking at something and going over the process of conceptualization of abstractions, the whole process of conceptualization (translation) from our natural observations into any kind of abstraction we find illuminating to our particular mind (anything we want/prefer). This is the kind of stuff that is needed.
The idea that our observations of things and motion (and other things as well) must be conceptualized in a particular format is the great sin of mathematical education. Of course that's why I'm doing the research.
Then once one has the process of how to observe down and the process of conceptualization of abstractions, this can be taught finally to teachers. Since it's ultimately about observing the world and having a keen eye for how things are related to one another.
Did it mentally (no pen no papers), didn't time it but wasn't that long: I did (300*357 - 10*357) - 357 and got the correct answer :)
Me happy at 37 years old :)
When you learn to play an instrument, you are actually learning to play two.
The first and most important instrument is your brain. This is called your audiation instrument. This is how you learn to hear sounds which are not physically present.
The second instrument is your violin/flute/voice/drum. This is called your execution instrument.
Your sense of timing (rhythm) is based on your ability to move. Your brain "imagines" your body moving at a constant speed. That is how your brain knows time. Athletes have a better sense of timing than musicians because they know how to move.
The good news is that your ability to play your executive skills instrument may not reflect the potential strength of your audiation instrument.
If you need help finding a good teacher I can see if there is anyone in your general area, if you want to give me a city or zip.
Spoon not. Fork, or fork not. There is no spoon.
The reason arithmetic is slow is because of the abstraction inversion.
You're manipulating data that can fit into a few bits by moving semantic mountains.
How fast would a computer do arithmetic if you, say, represented numbers as PNG pictures, and cobbed it together with shell scripts invoking image processing utilities?
Would that show that your CPU has a bottleneck?
Thanks for the feedback. I'll have to look for more information on audiation.
Thanks for the offer. I'm really low on cash right now, otherwise I'd take you up on the offer. Also, the conductor offered to help me out once per week. I took him up on the offer. Maybe I'll get in touch with you again, in the future, but I'm supposed to go to China in January, to teach English.
The good thing about Slashdot is that there is such a wide variety of experts. Thanks again.
testing out my trending skills
I would like to have a neural network map in a Sim "game" that displays on my laptop. When the brain jams occur I can then build highways and bypasses for the more important traffic. Either that, or a firmware upgrade. Oh wait, the firmware is down in a sub neural net!