` I've been putting off posting my other Noci Notes stream because I seem to have misplaced them - which is sad, because I've found the next one decidedly more interesting and amusing than the previous ones... well, you'll see.
` In the meantime, I thought I'd post the notes for a book by NPR's Keith Devlin called 'The Math Instinct'. They don't cover all subjects in the book - in fact, almost everything related to biological growth, form and instinct being mathematically determined has been omitted.
` And still, I think almost anyone would find this to be interesting enough to see how far they could read it. Trust me, try it and see.
` On your mark, get set... go!
` Numerous psychologists around the world, starting with Karen Wynn, have discovered that babies know the difference between different numbers of objects. This even applies to babies that are only two days old!
` You can tell whether or not they perceive something as odd because they stare at anything that does not meet their expectations for a measurably longer time than what they were used to.
` First off, the baby’s attention was directed to an empty puppet theater. Then, an experimenter’s hand came into view from one side with a puppet on it. After the baby got a good look at it, a screen came up in front of it. Then, another puppet approached from the other side and went behind the screen.
` When the screen was lifted, two puppets are expected, and so, the babies didn’t seem to think there was anything unusual here. But, when the screen was lifted to reveal three puppets, or only one, the baby stared in confusion.
` So, a baby understands that one plus one equals two, not three or one. Similar types of puppet-screen experiments revealed that babies also understand that one plus two equals three, two minus one equals one, and three minus one equals two.
` Of course, the babies could have been reacting to, perhaps, the position of the puppets, not the number. But this was not so, as Etienne Koechlin discovered, as the experiment had the same results when the puppets were placed on a slowly-revolving table.
` Tony Simon did a very unusual version of this experiment – sometimes, he changed the objects behind the screen. Sometimes, the screen was raised to reveal puppets that were a different color, or even no puppets at all, but rather balls. However, as long as they were the correct number, the baby viewing this showed no surprise.
` So, apparently a child’s expectation of how many something should be develops before their expectation of color, shape or appearance. Apparently, babies up to one year of age do not balk when they see a red rattle ‘turn into’ a blue one – but when they expect one rattle and see two, they stare.
` Babies up to a year old also cannot distinguish between four and five objects, either, so the experiments had to be limited to three or fewer objects. Even babies a few days after birth can do this, according to Antell and Keating. They were videotaped as they looked at slides of two dots in various arrangements. After they became bored with two dots, the experimenters switched to three, and the babies showed more interest. The same thing happened when they were shown three dots in different configurations and suddenly they were reduced to two.
` Similarly, Ranka Bijeljac showed the same thing with sounds by having babies suck on an artificial nipple that triggers the sounds of nonsense words through pressure sensors. When a baby was listening to nonsense words of two syllables and begins to become bored with it, the computer would produce words with three syllables. At this point, the baby would suck more vigorously in order to produce more of them. The same thing happens when the baby is hearing three syllables and they change to only two.
` This could go on, back and forth many times, and it demonstrated that the baby was not reacting to the sounds of the words, but rather the number of syllables.
` Prentice Starkey did an even stranger experiment – he interested a six-to-eight month old baby in two slide projections, side-by-side. Either one would have a picture of two objects of some type, and the other would have a picture of three objects. At the same time, either two or three drumbeats were played.
` After a while, the baby began to link the drumbeats with the pictures in an amazing way – when two drumbeats sounded, the baby would look at whichever picture had two objects. When three drumbeats were played, the baby would look at the picture with three objects.
` Drumbeats, syllables, objects of any type... babies recognize twoness and threeness. Apparently, the babies’ brains have certain patterns of activity that makes them more receptive to two or three objects.
` Various kinds of birds, and mammals have been shown to have a similar number sense as babies... and yet human babies outperform them.
` Devlin points out that calculators were made by humans to do mathematics, but dogs were made by nature to do calculus, concerning finding the shortest path to a ball in a lake diagonally to the dog to chasing a ball in an arc.
` Truly, only some math is about numbers. The whole of math is about patterns. Birds have solved the problem of sustained flight in one way, while dragonflies have in another way, and humans have in yet other ways.
` One fun little thing you can do, if you're bored on a rainy day, is to put a lobster in a dark container, confuse it with magnets, and put it in a random location in the ocean. Once on the bottom, the lobster will immediately take off in the direction of its home, as if it had a GPS system. Even on a cloudy day they can manage to do this, as lobsters have two magnetite-filled sensory organs in their heads.
` Other neato facts:
* Homing pigeons have similar organs in their beaks and so, with an anesthetized beak or a magnet strapped to its head, any such bird will become confused as to which is the direction it's been taught to fly.
* Birds also have a sense about which is the star that other stars revolve around, though it need not be the north star: Introduce some indigo buntings in a planetarium in which the stars rotate around Betelgeuse (rather than Polaris) and Betelgeuse becomes their north star.
* Tunisian desert ants wander aimlessly through the desert until they find food, then they go in a straight line back to their nest. If you move them, they will go back to the position relative to them that the nest would be, and then search fruitlessly for the hole.
` (I have recently come across a scientific article that declared that ants measure distance by how many steps they take - perhaps that is how the desert ants manage this!)
* Some birds recalibrate their internal compasses while migrating against the stars or the setting sun because magnetic south in nothern Alaska is actually due west, and further down, it’s closer to south. When some migrating thrushes were exposed to an overpowering magnetic field during sunset, they took off in the direction they would have assuming the magnetic field had been the natural one. During the next sunset, the thrushes were not interfered with and so the next morning they moved off in the correct direction.
* Salmon ‘migrating’ in a tank will swim south according to the sun, even if you change the magnetic field. However, on a cloudy day, they will always follow magnetic south. Therefore, they don't pay attention to the magnetic field until they can no longer use the sun.
* Monarch butterflies make an annual migration – however, they don’t live an entire year, so this takes three or four generations! Wherever the descendents of the spring migrants are, they suddenly head for a pine grove in the mountains west of Mexico City, all one hundred million of them!
` We know they navigate by ultraviolet light – a North American Monarch flying in full sunlight will stop if an ultraviolet filter is put between it and the sun. The butterfly also pays attention to the amount of daylight it is exposed to.
` If you put a Monarch in a chamber for a week with September-type sunlight – twelve hours light and twelve hours dark – it will take off in the direction it needs to get to Mexico at that time of year at that time of day.
` If the artificial daylight is from 7 to 7, it will go diagonally to the right of the sun if released in the morning. If the daylight is from 1 to 1, they will head off diagonally to the left of the afternoon sun, which is the correct direction to go if it were morning.
` Monarchs that were subjected to 24 hours of daylight simply flew towards the sun when released.
` Therefore, any Monarch has an instinct to fly in a certain direction no matter what time of year it is born - therefore, several generations inevitably trace a specific migration route. Of course, nobody yet knows how they know how far west or east they need to go to get to Mexico from a certain latitude.
` (My uneducated guess is that certain lineages only go in certain directions relative to the sun each year, so each lineage will trace a path to one particular state and then trace the same route back. The particular direction they go to get there and back would have to be hardwired genetically - however, in order for this to work, only two butterflies that are traveling in the same direction could mate, thereby keeping the populations separate.)
* Besides bats, which use echolocation, an owl can perform audio triangulation – its right ear is up to 50 percent larger than the left ear and is positioned 10 to 15 degres up on the skull. It has amazing three-dimensional vision, and when the sounds in both ears are equal, it senses that its prey is straight ahead.
* It is also interesting to me to learn that children don’t develop stereoscopic vision (taking no interest in stereograms) until three to four months of age. This seems to be because their eyes don’t get as far apart as those of adults until that point. (That makes sense, because the visual cortex is unchanging throughout life.)
` If a child (or other animal) is made to wear a patch over one eye during that time, it has major problems with depth perception.
* A rat can learn to press a lever about sixteen times in order to get food. If it doesn't press the lever as many as sixteen times it will receive a shock, so to be safe, the rat tries to press the lever more than fifteen times, though doesn't often go over seventeen.
* A raven can see the similarity between up to six randomly-arranged spots and another picture of spots of the same number but a different arrangement.
* Jackdaws can figure out how many pieces of food they have eaten when instructed to take a certain number of food pieces from various boxes and stop at that number.
* And of course, Alex the parrot knows the names that go with numbers!
* A canary raised in one area will hear a certain number of calls in birdsong, and will repeat it. Another canary raised in another area will repeat whatever number of calls are in the local dialect. They have a sense of how many calls and in what order they are in.
* Lionesses in a pride will retreat when they hear a number of roars that exceed their own numbers. Conversely, if there are fewer roars, they will stay and prepare to attack. A very useful thing....
* A chimp can match a half, a quarter, or three quarters of a glass of liquid with a half, a quarter or three quarters of a pie or apple - they have a sense of fractions that applies to different objects.
` Chimps can also tell the difference between six and seven chocolates, even when they are arranged in differently-sized piles.
` And of course, chimps can also do math using symbols and small sums.
` It would seem, though, that we humans do math differently than chimps, and usually, differently than the ways we are taught in school.
` For example, a twelve year old boy selling coconuts in the Brazillian city of Recife, having had six years of education, exchanged these words with researchers:
“How much is one coconut?” [asks the researcher].` Yes, this actually happened, according to a paper by Nunes, Schliemann and Carraher. Why did the boy not simply add a zero if he was multiplying a number by ten? Because; he was not using school math! Rather than using a trick, he simply added the number in his head.
“Thirty-five,” he replies with a smile.
[The researcher says], “I’d like ten. How much is that?”
The boy pauses for a moment before replying. Thinking out loud, he says: “Three will be 105; with three more, that will be 210. (Pause) I need for more. That is... (pause) 315...I think it is 350.”
` In fact, the children at these market stalls were 98 percent correct while doing business. When presented with identical word problems on paper, they were only correct 74 percent of the time. Using arithmetic symbols, they averaged only 37 percent correct. Apparently, they couldn’t remember the method.
` In the book How to Survive in Your Native Land, schoolteacher James Herndon was teaching a class of junior high students who were failing. One student was being well-paid to keep track of bowling scores.
Seeing a golden opportunity to motivate this pupil to do well in class, Herndon created a set of “bowling score problems” and gave them to the boy. The attempts was a complete failure. In the bowling alley in the evening, the boy could keep accurate track of eight different bowling scores at once. But he could not answer the simplest scoring question when it was presented to him in the classroom. In Herndon’s words, “The brilliant league scorer couldn’t decide whether two strikes and a third frame of eight amounted to eighteen or twenty-eight or whether it was one hundred eight and a half.”...` !!!!!
`...To a girl who admitted she never had any trouble shopping for clothes, he gave the problem: “If you buy a pair of shoes costing $10.95, how much change do you get from a twenty?” (In 1971 this price would have been realistic.) The girl answered “$400.15” and wanted Herndon to tell her if it was right.
` Exclamation points aside, I'm sure you'll agree that this sounds quite messed up.
` Apparently we all do math differently when we’re actually shopping, rather than tests that simulate shopping. When trying to determine fractions and unit prices, it’s not so easy if the fractions are complicated.
` Young children don’t seem to connect the concepts of counting or reciting numbers with an actual number of real objects (independent of the order in which they appear, etc.) until about age four - quite a strange thing to ponder if you've never thought of it that way before.
` Apparently, linking numbers of things with words has to be learned, and so while keeping track of amounts of things is an automatic human instinct, making names for different numbers is not. For additional evidence, all one needs to do is realize that some primitive cultures are removed from number usage.
...[W]hen a member of the Vedda tribe of Sri Lanka wants to count coconuts, he collects a heap of sticks and assigns one to each coconut. Each time he adds a new stick, he says, “That is one.” But if asked to say how many coconuts he possesses, he simply points to the pile of sticks and says, “That many.” The tribesman thus has a type of counting system (or perhaps more precisely, a system for representing quantity), but it doesn’t use numbers.` And this is interesting – we may have begun counting digits (numbers) on our digits (fingers and toes), which probably explains why ten-base number systems are so common. The left parietal lobe - the region of the brain that is linked most with controlling the fingers - is the most intensely activated region of the brain while we are doing arithmetic.
‘ Or take the Warlpiris, an Aboriginal tribe in Australia. Their native language permits them to count up to two, after which everyhing is simply “many.” The fact that the members of those tribes have no difficulty learning how to count in English shows that it is noy that they are unable to count. Rather, their native language harks back to a time when counting simply went: “one, two, many.” (Other “primitive” peoples count “one, two, three, many.” But never “one, two, three four, many.” The cut-off point for a universal number sense is three.)
In addition to the evidence from the neuroscience laboratories, clinical psychologists have also found a connection between finger control and numerical ability. Patients who sustain damage to the left parietal lobe often exhibit an unusual condition known as Gerstmann’s syndrome, in which sufferers lack awareness of their individual fingers. For instance, if you were to touch a patient’s finger, he or she would be unable to tell you which finger was being touched. Sufferers also are typically unable to distinguish left from right. More interestingly, from our point of view, people with Gerstmann’s syndrome invariably have difficulty coping with numbers.` Devlin suggests that our ancestors used their fingers to represent and even manipulate numbers (like the sticks standing for coconuts), because they were convenient. Maybe for a time, humans could only do arithemetic on their fingers. Once they became unnecessary for that, the muscles became disconnected from abstract mathematical thought.
` I've always found interesting the fact that you cannot picture in your mind what a four or a five or a six is. You may have an idea, but, as numbers are not solid objects (rather than mental ones) this is not possible. So, as Devlin says, you can think of five pennies (five being a descriptive word) but not of a five. “We can’t touch it or smell it. But we can think about it, and we can use it.”
` A five does not really look like ‘5’. That is just one way we humans can write it out - of course there have been many others since the dawn of number-writing: The first record of marking down numbers was apparently discovered by Denise Schmandt-Besserat, archaeologist from the University of Texas in the 1970s and 1980s.
` The ancient Sumerians of 3300 to 2000 B.C.E., she discovered, used ‘small clay tokens of different shapes, including spheres, disks, cones, tetrahedra, ovoids, cylinders, triangles, and rectangles.’ Apparently, they were to mark quantities of different items, from jars of oil to animals to wares and foodstuffs.
` The Sumerian businessman or trader put his clay tokens on a flat sheet of wet clay and folded it up into a pouch. In order to do business, however, he’d have to break the pouch open.
` Even more irritating, he’d have to do the same thing if he forgot what his balance was. To remedy this, these people began pressing their tokens into the outside of their pouches, creating a clear record.
` Eventually, the Sumerians realized that they could just keep their records on a flat clay tablet using this system – there was no need to carry any tokens around!
` Later on in history, recognizable pictures became abstract symbols. And so, this became the earliest-known number system.
` (Because I'm posting this on a blog of many subjects I shall note that, like biological evolution, one system arose for practicality's sake, and yet the forms of that system - the pouch and tokens - went out of circulation once it evolved to the point that there was a more efficient innovation - a plain slab with reed-marks in it. Yet, that innovation would probably not have taken place if it hadn't been for this differently-working, dissimilar-looking precursor.
` This is because human technology does not often arise completely spontaneously, rather it involves a certain kind of speedy, human-driven evolution that mostly involves primitive forms and artificial selection. Complete breakthroughs often require 'transitional forms', even though we are also capable of technological ideas 'from nowhere'.)
` The number system used today, from India... well, I find this to be somewhat humorous:
It was introduced by Arabic traders and scholars in the seventh century, and as a result is generally referred to as the “Hindu-Arabic number system,” or more simply as the “Arabic system.” It is one of the most successful conceptual inventions of all time.` Ha ha ha ha! The European White Males were awfully behind, weren’t they?
` Once the Arabic system was available for representing any positive whole number, it was easy to extend it to represent fractional and negative quantities. The introduction of the decimal point or the fraction bar allows us to represent any fractional quantity (3.1415 or 31/50, for example). Introduction of the minus sign “-“ extends the range to all negative quantities, whole or fractional. (Negative numbers were first used by sixth-century Indian mathematicians, who denoted a negative quantity by drawing a circle round the number; European mathematicians did not fully accept the idea of having negative numbers until the early eighteenth century.)
` Another interesting thing is the fact that people can have brain disorders that allow them to read symbols for numbers, but not words! Other people can read words, including words for numbers, but not number symbols if it is more than one digit.
` Writing both language and numbers really do involve different areas of the brain:
` In an extreme case, a woman named Donna had surgery to her left frontal lobe and was left unable to read or write words, even though she can read and write numbers. She can only remember half the letters of the alphabet and is unable to write her name (at least, legibly) even though she is able to write out arithmetic problems perfectly.
` And yet... language leaves a fingerprint in reciting math facts such as the multiplication table. Apparently, in order to answer the exact math problems in a different language than one is taught in, one takes about a second longer in answering because one needs to translate the problem into the other language and back.
` According to Stanislas Dahaene, when an exact answer was wanted, the subjects’ frontal lobes were more active. When asked for an approximate answer, which did not take any longer no matter which language it was posed in, the parietal lobes were more active instead.
` This is because we 'talk though' math problems in our heads using the same language (and accent!) that we learned them in.
` This concept helps to explain another link between language and numbers; the fact that children who speak Western European languages (such as English) have a harder time learning their numbers than Chinese and Japanese children.
` In China, 4 is si and 7 is qi – the words for numbers are all very short and simple. Their way for making bigger numbers is simpler as well: Insted of saying ‘eleven, twelve, thirteen,’ they say ‘ten-one, ten-two, ten-three.’ Twenty is ‘two-ten’ and twenty-one is ‘two-ten-one’. In French, one says quatre-vingt-dix-sept for ninety-seven. In Chinese, on the other hand, you simply say ‘nine-ten-seven'. Whereas Germans say; vierundfünfzeig, the Chinese say, essentially; ‘five-ten seven,’ which is intuitively even simpler than ‘fifty-four’.
` It is no wonder that Chinese children can count to forty when they are four, while American children can usually only count to 15 by that age and generally make it to 40 a whole year later.
` Also interestingly, said children show no differences in learning numbers until up to 12... this is because each number is a different word. Learning 'eleven' and 'twelve' and then proceeding onto numbers that end in ‘teen’ (rather than all of them ending in ‘teen’ seems to be the source of the confusion – the pattern isn’t as consistent. This explains the number of children who say ‘twenty-nine, twenty-ten, twenty eleven, twenty-twelve....’ and then what? Twenty-teen? That, of course, makes no sense, and I always used to think this all was quite confusing.
` Also, Chinese number-words agree more closely with printed numbers than English number-words: When you see the number 25, it is super-easy to think ‘two-ten-five.’ In English, we don’t have the word ‘two-ten’, so kids have to memorize that a two in the tens column is called ‘twenty’.
` Oh, and hearing numbers in one's head also relates to Devlin's very interesting friend who can calculate the square root of six-digit numbers in his head:
A few years ago I attended a luncheon in which Benjamin was giving a demonstration of his arithmetical skill. Just before he was due to begin, he asked the organizers to have the air conditioning turned off. While we were waiting for that to be done, Benjamin explained that the hum of the system would interfere with his calculations. “I recite the numbers in my head to store them during the calculation,” he said. “I have to be able to hear them, othewise I forget them. Certain noises get in the way.” In other words, one of Benjamin’s “secrets” as a human calculator is his highly efficient use of linguistc patterns – the sounds of the numbers as they echo in his mind.` Of course, the memorization of numbers also can be improved when they mean something to you – dates, ID numbers, pi, any number that might have meaning to you can help you remember other numbers!
` Having practicality and meaning are actually what push people to do math... hence the Brazillian children who were so good at doing math in their head while selling fruit even though they weren’t so good at math problems on paper. The same is true for shoppers, shippers, carpenters and bowling-scorers - including highly uneducated ones!
` This seems to be common to everyone – even though we have the potential to do all kinds of math, we have trouble when it looks like ‘meaningless symbols’. We follow the rules we are given, and that’s it, we won't necissarily have a very clear concept of what we're doing. Why? Because people have trouble doing things that have no meaning.
` In fact, Devlin gives an example (discovered by Lauren Resnick and Wendy Ford) of a boy who learned to consistently work a formula of basic addition... except, he wound up writing the carry number below the line and moving on. So he kept applying that rule and got the wrong answers whenever he had to carry! Therefore, 87 plus 93 comes out 11.
` Other than that, he had no other issues with arithmetic. This is because children get the idea that even if they don’t understand why you're supposed to do math in one particular way, you do it that way anyway without questioning it, even if it doesn't seem right to you.
` So, it has to be suggested, we need to find a more meaningful way to teach math to children, and find out which parts of their daily lives require arithmetic to accomplish. Perhaps someday some revolutionary educator will find some really amazing method that always works!
` I myself hated math until I reached third grade, when I had a nice teacher - just being around her encouraged me to do a week's multiplication sheet in only one day! Then in fourth grade, I could barely remember how to do all that stuff (due to summer break and lack of encouragement), and by fifth grade, I utterly stopped doing math problems because the teacher discriminated against girls and wouldn't teach me.
` For years I didn't actually participate in learning math, although I certainly was capable of finding out which math problems could be solved with which type of directions and simply following the directions, over and over. No matter how many times I repeated the steps for different problems, I never learned them.
` And, because I didn't go to school, I could get away with the fact that I couldn't perform any type of somewhat complex math problem in front of a human being. (In recent years, I've simply been depending on MAL to figure this stuff out for me! Pathetic, huh?)
` Since my stress levels have been relatively low in recent months I have managed to 'pick up' addition by actually writing it out and re-remembering how it was that you 'carry the one' and all that stuff. (It's a lot harder doing that on paper than in my head!) Soon, I managed to master (on paper) subtraction, multiplication, short division, and - for the first time ever - simple algebra!
` All by myself!
` It's only just now that I'm getting into the 'fun stuff' of my own accord, because you know, math is a really fun thing for me to do - simply thinking about patterns makes me high. ...This is probably because the unpredictability in my own life has mostly been coupled with the thought "am I going to experience extreme pain now or not?"
` I also sometimes tire of the feeling I get from challenging my own assumptions, however exhilirating this may be. And, like anyone else, I appreciate something that makes sense and follows a particular order. I think I partly understand why some people are addicted to dogma - it must hurt them far more than it hurts me to change their assumptions!
` Anyway, my sudden interest in math stems from the fact that I've always had this (correct) conviction that symbol-based math is this handy-dandy, itty-bitty little thing that you can use to express any kind of pattern that you see, and it's written out the same for anyone in the world, no matter what language they speak!
` Is that not cool?
` Well, I need to get going on a brisk walk. I'm not sure what I'll do during this walk, but you see, that's another thing relating to a need for predictability. I feel pretty good right now simply because I know I'm going to go on a walk in the tangible future and it's going to make my muscles stop screaming for activity:
` It doesn't matter that I no longer have 'walking errands' to do every single day, I simply must go on a walk, which often makes me prone to bursting into bouts of karate (despite the fact that I tend to forget things when 'Sensei Lou' isn't around) or even trying to invent a new dance.
` On the other hand, if I'm not sure whether or not I'll be going on a walk in the foreseeable future because I can't find an 'excuse', I'll feel pretty miserable inside, even if I wind up going on a walk anyway. Well, if keeping my mood up involves predicting that I'll go on a walk each day, even when there's nowhere in particular to walk to, so be it!