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Midterm 2 Textbook Notes

31 Pages

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Psychology 2410A/B
J.Bruce Morton

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Week 7 Chapter 2: Babies Who Count • The human brain is endowed with an innate mechanism for apprehending numerical quantities, one that is inherited from our evolutionary past and that guides the acquisition of mathematics • In the first year of life, babies already understand some fragments of arithmetic Baby Building: Piaget’s Theory • The numerical competence of babies is now studied empirically, but before 1980, developmental psychology was dominated by constructivist views • Piaget, the founder of constructivism, thought that logical and mathematical abilities are progressively constructed in the baby’s mind by observing, internalizing, and abstracting regularities about the external world • Genes do not give organism abstract ideas about the environment, they just instill simple perceptual devices and a general learning mechanism that takes advantage of the interactions of the subject with its environment to organize itself • In the first year, children are in sensorimotor stage, and begin to notice some salient regularities • These babies progressively construct a series of refined and abstract mental representations of the world • In this view, the development of abstract thought consists in climbing a series of steps in mental functioning • Piaget believed that number must be constructed in the course of sensorimotor interactions with environment • It takes children years of attentive observation before they really understand what a number is • Piaget had seemingly collected proof of young children’s inability to understand arithmetic • For example, their lack of object permanence • Piaget found that number concept does not begin to be understood before the ages of four or five, before that children fail the number conservation task • Show them two rows of objects, both contain the exact same number of objects but one row is more spread out, children will report that the more spread out row has more • Do not realize that moving the objects around leaves their number unchanged- lack ability to conserve number • Piaget concludes that prior to the age of reason, children lack knowledge of the most elementary bases of set theory, which many mathematicians believe to be the foundation of arithmetic: younger children ignore that a subset cannot have more elements than the original set from which it was drawn (ie cannot be more roses than flowers) • Before the age of six or seven, child is not ready for arithmetic • According to Piagetian theory, it is best to start by teaching logic and the ordering of sets, because the notions are a prerequisite to the acquisition of the concept of number Piaget’s Errors • This aspect of Piaget’s constructivism was wrong • Young children are not devoid of genuine mental representations of numbers, even at birth • One just has to test them using research methods tailored to their young age • Piaget’s tests are flawed because they rely on an open dialog between experimenters and young subjects, children may not understand the questions they are being asked • When children’s minds are probed without words, their numerical abilities can be shown • It has been shown that results of Piaget’s number conservation test change radically according to context and to the children’s level of motivation • When the task is done with M&Ms, and the children are told to pick which row they want to eat (increasing children’s motivation to choose the row with the most treats), the majority of children will select the row with the most amount of candies, even if the alternative row was more spread out • It was also shown that children who were about 2 years old could perfect this task even with marbles (where motivation is low), but the older children fail to conserve number • This may be because the older children (3-4) interpret the experimenter’s question incorrectly, and think they are being asked to judge the length of the rows rather than their numerosity • Older children must find it strange that a grown up is asking the same question twice (is it the same thing, or does one row have more marbles?), so they assume that the second question has a different meaning than the first • They assume that the experimenter is asking the same question a second time because they expect a different answer than the first • This makes sense as understanding a sentence usually consists of going beyond its literal meaning and retrieving the actual meaning initially intended by the speaker • In an experiment, same as other number conservation experiments but with a twist- the experimenter looks the other way for a second and then a teddy bear is added to one of the two rows, then the experimenter turns back and says “oh no! The silly teddy bear mixed up everything!” • Now, when the experimenter asks which has more, the question seems more sincere (the adult actually doesn’t know which has more now) and is more likely to be interpreted literally • In this situation, majority of children responded correctly on the basis of number (rather than on the basis of length), and the same children failed at the task when it was without the twist (the transformation was performed by the experimenter, not the teddy bear) • This shows that when the question is asked in a context that makes sense, young children get the answer right, they can conserve number • Some scientists think that failure on Piagetian tasks reflects the continuing maturation of the prefrontal cortex, a region of the brain that enables us to select a strategy and to hold it firm • Main point is that Piagetian tests are not good tests of when a child begins to understand the concept of number Younger and Younger • Piaget might have argued that the modified experiments made the children’s task too simple, in his view, children really mastered the conceptual underpinning of arithmetic only when they could predict which row had the most items on a purely logical basis • Resistance to misleading cues was part of Piaget’s definition of what it meant to have a conceptual understanding of number • Piaget might have argued that choosing the largest number of candies does not require a conceptual understanding of number, only a sensorimotor coordination that allows the child to recognize the greater pile and orient toward it • It has been found that animals can acquire “sensorimotor numbers” but not a conceptual understanding of arithmetic • In the 1980s, numerical abilities were observed in 6-month-old infants and even in newborns • Test of this rely on babies’attraction to novelty • Experiment- 4-8 month old babies were shown a screen with two large black dots (more or less spread out from trial to trial), over the course of a few trials, babies looked at the screen for less and less time. Then the slides were changed to three dots, and the baby immediately started to fixate longer at these unexpected images • This experiment was also shown in newborns • To make sure that it is really the change in number noticed by the babies and not any other physical modification of the stimulus- in one experiment, dots were aligned in a global figure so they provided no cue to number, they also varied the spacing between the dots so that neither their density, nor total length of line would suffice to discriminate 2 from 3 • An even better control- used colour photographs of common objects of all kinds, objects were small, large, aligned or not, and photographed from near and far, only their number remained constant • Experiment was also done using moving displays- babies could still notice the constancy of objects and extract their numerosity Babies’Power ofAbstraction • It remains to be seen whether this sensitivity to numerosity merely reflects the power of babies’visual system, or whether it betrays a more abstract representation of number • 3 questions regarding babies’number abilities: 1. Are they able to extract the number of tones in an auditory sequence? ­ Experiments done on auditory modality ­ Used sucking rhythm rather than gaze orientation to test it ­ Babies sucked on a nipple connected to a pressure transducer connected to a computer ­ After a few minutes of hearing three syllable words, babies sucking becomes non- vigorous ­ When word switches to two syllables, babies begin to suck vigorously, showing their ability to detect change in number of sounds ­ The number of syllables was the only parameter that can enable babies to differentiate the first list of words from the second (showing its actually a good test of number) ­ Has also been shown that at six months of age, babies can discriminate number of actions (puppet doing 2 jumps vs 3) 2. Do they know that the same abstract concept “3” applies to three sounds and three visual objects? ­ Experiment has shown that the answer to this is yes ­ Baby is seated in front of two projectors, the one on the right has 2 objects shown and on the left has 3 objects shown ­ Simultaneously, the baby hears a sequence of drum beats played by a central loud speaker ­ Results show that babies look longer at the slide that matches the sequence of sounds that it is hearing ­ This shows that babies can most likely identify number of sounds, and is capable of comparing it to the number of objects its looking at ­ This shows that numerical representation is not tied to a low level of visual or auditory perception ­ The child really perceives numbers rather than auditory patterns or geometrical configurations of objects ­ This may reflect an abstract module for number perception, implanted by evolution ages ago, deep within the animal and human brains 3. Can they mentally combine numerical representations and perform elementary calculations such as 1+1=2? ­ To study this, design that relies on infants’ability to detect physically impossible events ­ Infants display strong puzzlement when they witness “magical” events that violate the fundamental laws of physics ­ Babies express surprise when an object remains mysteriously suspended n air, or if two physical objects occupy the same location in space ­ Infants’surprise is demonstrated by a significant increase in the amount of time they spend examining the scene, relative to a control situation in which the laws of physics have not been violated ­ Using this design, experiments have tested infant’s number sense by showing events that can be interpreted as numerical transformations (one object plus another), and tested whether infants expect the precise numerical outcome of two objects ­ Experiment- 5 month olds watched as hand place Mickey Mouse on stage, then screen comes up, then they see a hand with a second Mickey Mouse place it behind the screen (children now infer that there are 2 Mickeys behind the screen) ­ Then the screen comes down and reveals only 1 Mickey! ­ The time infants spent fixating on the impossible situation (1+1=1) was measured and compared to fixation time for the expected outcome (1+1=2) ­ On average, infants looked longer at the impossible event than the possible one ­ Same experiment was done with 2-1=2 and results were the same ­ Devils advocate- maybe babies know that 1+1 cannot equal 1 but they still do not know the answer to it, but experiment was done where 1+1=3 and babies fixated on this for longer than 1+1=2 ­ Babies know that 1+1 makes neither 1 nor 3, but exactly 2 ­ Similar results were shown in monkeys ­ Another experiment prevented children from building a precise mental model of the objects’location and identity to see whether they can still anticipate their number ­ Objects are placed on a slowly rotating turntable that keeps them in constant motion even when they are hidden behind the screen, it is therefore impossible to predict where they will be when the screen drops ­ Results show that babies still find the impossible events surprising, hence their behaviour does not depend crucially on the expectation of precise object locations ­ Another experiment has shown that if two Mickey Mouse toys are placed behind the screen, infants are not shocked to discover two red balls when the screen drops, as long as no object vanishes or is created, the operation is judged to be numerically correct and yields no surprise reaction in babies ­ Shows that child’s number sense is sufficiently sophisticated to avoid being deceived by object motion or by sudden changes in object identity The Limits of InfantArithmetic • While young children’s numerical abilities are real, they are strictly limited to the most elementary of arithmetic • Their abilities for exact calculation do not seem to extend beyond the numbers one, two, three, and perhaps four • It is expected that babies are unable beyond some limit to discriminate a number n from its successor n + 1 (this is what is observed beyond number four) • They are however expected to recognize numbers beyond this limit when they are contrasted with even more distant numbers (ie babies may not know that 2+2 is 3, 4, or 5, but they should be surprised if they seen a scene suggesting it is 8) • Second limitation- in situations where adults would automatically infer the presence of several objects, babies do not necessarily draw the same conclusion • Babies do not consider the fact that quite different shapes and colours alternatively come out from behind the screen as a sufficient clue to the presence of several objects (ie if they see red ball and green truck alternatively popping out of the screen, will not be surprised if the screen reveals just the red ball) • The only clue that babies seem to find conclusive is the trajectory (path) followed by objects • When same experiment is repeated with two separate screens with a space between them, if an object alternatively pops out from the right screen and the left screen, with out seeing it cross over through the space between them, babies infer the presence of two objects, one behind each screen • Information about the spatial trajectories of objects thus provides a crucial cue to numerosity perception • Spatial information about the location of discrete objects in space and time is critical to set up the representation of number in the baby’s brain, but it is not needed once this representation has been activated • Babies numerical inferences seem to be completely determined by the spatiotemporal trajectory of objects • If the motion that they see could not possibly be caused by a single object without violating the laws of physics, they draw the inference that there are at least two objects • Only location and trajectory matter • The reason that babies think it could be possible that a truck could turn itself into a ball is because they have seen a tiny piece of red rubber transform into a pink balloon (for example) • This is why babies’default hypothesis is that there is only one object out there, and they maintain this hypothesis until there is clear proof to the contrary, even if they witness curious transformations in object shape and colour • Human number sense uses at least 3 laws: 1) An object cannot simultaneously occupy several separate locations 2) Two objects cannot occupy the same location 3) Aphysical object cannot disappear abruptly nor can it suddenly surface at a previously empty location, its trajectory has to be simultaneous • Even very young babies understand these laws with a few exceptions (shadows, reflections, and transparencies) • The tight link between discrete physical objects and numerical information endures up to a much older age, where it eventually has a negative impact on some aspects of mathematical development (ie 3 or 4 year old will count each piece of the broken fork as a separate fork) • Up to a relatively advanced age, children cannot help counting every single object as one unit Nature, Nurture, and Number • The first year of life is when the baby’s brain possesses maximal plasticity • During this period, they absorb an impressive amount of knowledge each day and cannot be considered a static system whose performance is stable • The numerical abilities discussed in this chapter should be situated within a changing framework • Newborns can readily distinguish two objects from three, and two sounds from three sounds, the newborn’s brain apparently comes equipped with numerical detectors that are probably laid down before birth • Abrain module specialized for identifying numbers is laid down through the spontaneous maturation of cerebral neuronal networks, under direct genetic control with minimal guidance from the environment • We probably share this system with many other animal species • At the present time, the connection between 2 sounds and 2 images or 3 sounds and 3 images has only been shown in 6-8 month olds, no younger • Its possible that learning, rather than brain maturation, is responsible for the baby’s knowledge of numerical correspondence between sensory modalities • The environment can be highly unpredictable and ambiguous- some objects generate more than one sound, others no sound at all, therefore it is unclear that they would support any form of learning • It is therefore highly likely that the babies’preference for a correspondence between sounds and objects stems from an innate, abstract competence for numbers • The 1+1 and 2-1 experiments were performed only with babies who were 4 and a half months age at youngest- this lapse of time may have been sufficient for the baby to empirically discover these elementary mathematic equations • In this case, Piaget would be partially right- babies have to extract rules of arithmetic from their environment- although at a much younger age than Piaget thought • This knowledge may however be inborn, and become manifest as soon as the ability to memorize the presence of objects behind screens emerges, around 4 months of age • One arithmetic notion that they may be lacking is the ordering of numbers • No noticeable ordinal competence is found before the age of 15 months • Younger babies seem unaware of the natural ordering of numbers, simply see numbers as adults see colours • The concepts of smaller and greater are among the slowest to be put into a baby’s mind • They probably arise from the observation of the properties of addition and subtraction (greater number are reached by adding, smaller ones by subtracting) • In conclusion, babies are much better mathematicians than we thought 15 years ago Chapter 5: Small Heads for Big Calculations • This chapter examines the calculation algorithms of the human brain • Mental arithmetic poses serious problems for the human brain • An innate sense of approximate numerical quantities may be embedded in our genes, but when faced with exact symbolic calculation, we lack proper resources Counting: TheABC of Calculation • In the first six or seven years of life, children learn to calculate, select the best strategies for calculation, the majority of which are based on counting with or without words/fingers • According to one view, children are endowed with unlearned principles of counting; they do not have to be taught that each number has to be recited once, or that number words have to be recited in a fixed order, etc. • Another view insists that counting is a result of imitation, and starts off as rote behaviour devoid of meaning, and children progressively infer what counting is about by observing other people count • The truth is some aspects of counting are mastered easily, and seem to be innate, while some are acquired through learning and imitation • Children seem to understand counting quite early on with no teaching (children as young as 2 can count how many times Big Bird appears in an episode) • Astudy has shown that 3 year olds can identify and correct very subtle counting errors • By four years old, children have mastered the basics of how to count and they can generalize counting rules to novel situations • Some believe that children are genetically endowed with these counting abilities- reciting words in a fixed order is probably a natural outcome of the human faculty for language • In terms of the one-to-one correspondence (only counting each item once)- this probably has been selected for in terms of its survival benefits (ie a rat visiting each arm of a maze only once in order to minimize exploration time) • As adults, we know why we count, counting is a tool that serves a precise goal- enumerating a set of items, and we know that the final numeral is the most important • According to one view, children do not appreciate the meaning of counting until the end of their fourth year (ie if you ask a 2 year old to bring you 3 toys, he will bring a random amount, even though he can already count to 10) • The meaning of counting eventually settles in at age 4 and the preverbal representation of numerical quantities probably plays a crucial role in this process • At a very young age (as shown in chapter 2) children have a mental representation of numbers, and if a young girl is playing with 2 dolls, they have the number “two” represented in their head • If for some reason this child decides to count the dolls, she will realize that the last number of the count, “two”, represents the entire set • After many occasions like this, the child will infer that whenever once counts, the last word arrived at has a special status- it represents the numerical quantity that matches the one provided by the internal accumulator • Children eventually learn that counting is the best way of saying how many Preschoolers as Algorithm Designers • With the help of counting, children find ways of adding and subtracting numbers without requiring any explicit teaching • The first calculation algorithm children find out for themselves is adding two sets by counting them both on their fingers • Initially, young children find it difficult to calculate without using their fingers • After a few months, children discover a more efficient algorithm- ie adding 2 plus 4: first they count up to the first set “one two” and then they move forward as many steps as specified by the next set “three four five six” • Eventually, they learn that 2+4 is the same as 4+2 and count with the big number first to speed up the process (four…five six), this is called the minimum strategy- this principle is in place by 5 years of age • Children quickly master many addition and subtraction strategies, and learn to carefully select the strategy that seems most suited for each particular problem • Ie for 8+4, they may remember that 8+2= 10, so they might decompose 4 into 2+2 and then simple count- ten, eleven, twelve • Children evaluate the algorithms they use, determining which work and which do not, and which are time efficient • If you ask a child to solve 8-2, they will count backwards (seven, six), but if you ask a child 8-6, they will apply the algorithm of counting forwards (seven is one, six is two)- the child knows to do this because they will have figured out the rule that if it is greater than half the starting number, counting forwards is the winner, and if it is less than half the starting number, counting backwards is the winner! MemoryAppears on the Scene • For a 7 year old child to add two numbers, calculation time increases in direct proportion to the smaller addend, showing that the child is using the minimum algorithm • Ie computing 5+1, 5+2, 5+3 takes an additional 4/10ths of a second for each successive one, at 7 years, each counting step takes about 400 milliseconds • These results have even been shown in college students (except the size of the time increment was much smaller) • This is because 95% of the time, students would directly retrieve the results from memory, but that 5% of the time where memory failed, they would have to count at the speed of 400 milliseconds per unit • This proposal was challenged by the realization that students’response time did not increase linearly with the size of the addends, large addition problems like 8+9 took disproportionately longer • Another hypothesis came- the time to add two digits was best predicted by their product or by the square of their sum • It was found that the time to multiply two digits was essentially identical to the time taken to add them • Therefore, it was proposed that young adults hardly ever solve addition and multiplication problems by counting, instead they retrieve results from a memorized table • Accessing this table takes an increasingly longer time as the operands get larger • The accuracy of our mental representation drops quickly with number size • Order of acquisition may also be a factor, because simple arithmetic facts are often learned before more difficult ones with larger operands • Another factor is that because the frequency of numerals decrease with size, we receive less training with larger multiplication problems • The hypothesis that memory plays a central role in adult mental arithmetic is now universally accepted • During preschool years, children suddenly shift from an intuitive understanding of numerical quantities, supported by simple counting strategies, to a rote learning of arithmetic The Multiplication Table: An Unnatural Practice? • Any student executes tons of elementary calculations daily • Over a lifetime, we solve more than ten thousand multiplication problems • Yet, our arithmetic memory is at best mediocre • It takes a well-trained young adult more than 1 second to solve a problem like 3x7 with error rates of 10-15%, and for a problem like 8x7, it takes more than 2 seconds and error rates are over 25%, WHY? • There are forty-five addition and thirty-six multiplication facts to memorize (this doesn’t include multiplication by 0 or 1), why is it so difficult for us to store them if we can remember hundreds and thousands of other facts so easily? • The answer is because of the particular structure of addition and multiplication tables- they are closely intertwined and full of false regularities, misleading rhymes, and confusing puns • If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative, it weaves multiple links among unrelated data • Associative links allow for the reconstruction of memories n the basis of fragmented information • Associative memory is useful because it allows us to retrieve a large magnitude of memories that seemed lost only from a vague reminiscence, it also allows us to use analogies and apply old knowledge to a novel situation • Its disadvantages are that in domains like a multiplication table, where various pieces of knowledge must be kept from interfering with each other, it breaks down • Practically all multiplication errors belong to the multiplication table (ie for 7x8, instead of 56 people are likely to say 63 or 48, but hardly anyone will say 55, even though it is only 1 digit off the correct answer) • TheAutomatization of arithmetic memory starts at a young age, as early as seven where upon seeing two digits, their sum is automatically added • Astudy has shown that in the third grade, most students already know many additions by heart, but as they start to learn multiplication, the time they take to solve an addition temporarily increases, showing how interference effects memory Verbal Memory to the Rescue • Educators have long realized the huge potential of verbal memory • Reciting multiplication tables out loud is a widely used practice • Japanese multiplication table is made up of little verses called “ku-ku”, all of their multiplication memorization is done out loud • When arithmetic tables are learned verbatim, calculation becomes tied to the language in which it is learned in school • This could be because learning arithmetic tables is so labor intensive that it may be more economical for a bilingual to switch back to the mother tongue for calculation, rather than relearn arithmetic from scratch in a new language • The role of verbal arithmetic can also be seen when you find it hard not to name numbers aloud when performing complex calculations • Reading errors often occur where we say 5x6 is 56 or 2x8 is 28, and these errors are thought to occur during multiplication retrieval • This shows that reading and arithmetic memory are highly interconnected procedures that make us of the same verbal encoding of numbers • Verbal memory is not the only source of knowledge to be exploited during mental calculation, when memory fails, the brain falls back on other strategies like counting, serial addition, or subtraction • For instance, 8x9= (8x10)-8= 72 • The brain also uses shortcuts, results whose magnitude departs considerably from the truth are rejected in less time than it would take our brain to actually compute the calculation, suggesting that the brain estimates the size of the result automatically Mental Bugs • In order to calculate multi digit calculations fast, the brain is forced to ignore the meaning of the computation it performs (ie why do we carry over the one?) • Common mental subtraction errors include, leftward shift of all carry overs that apply to the digit 0 (you have to take from the digit to the left of 0, make the 0 10, then take from the 10 and make the 10 9) • These bugs may be due to the fact that no textbook ever describes the correct subtraction recipe in its full generality, children are expected to study examples, analyze the teacher’s behaviour, and derive their own conclusions • Textbooks do not address every possible type of problem, so many students apply rules incorrectly in subtraction problems because they do not know otherwise Pros and Cons of the Electronic Calculator • Human brains are functional in calculating formal operations, but far from optimal • There is an alternative- the electronic calculator, which is cheap, omnipresent, and infallible • The question is, should our pupils still have to spend hundreds of hours reciting multiplication tables, or would it be wiser to give them early training in electronic calculators and computers? • In many countries, the abacus and finger counting are widely used to do mental math, examples of alternatives to the rote learning of arithmetic • Some opinions say that if children as young as six or seven are allowed to use a calculator, they eventually attain a less intimate knowledge of number than reached through the practice of mental calculation • Author of the textbook believes that by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them concentrate on meaning • By studying a calculator’s results, children can discover that subtraction always yields a result smaller than the starting number, and thousands of similar facts • The mere observation of a calculator’s behaviour is an excellent way of developing number sense • Can make a young child find math cool, rather than hate it • Electronic calculators hold the promise of initiating children to the beauty of mathematics, a role that teachers often do not accomplish • Not saying that the calculator should completely eliminate all rote mental arithmetic- it would be ridiculous to use a calculator to compute 2x3 • However, the vast majority of adults never perform a multi digit calculation without resorting to electronics, division and subtraction algorithms are quickly disappearing from everyday life • At the very least, calculators in school should lose their taboo status • Mathematics curricula are not perfect, their sole objective should be to improve children’s fluency in arithmetic, not preserve a ritual Innumeracy: Clear and Present Danger? • Many children reach adulthood without having really understood when to apply mathematics knowledge appropriately • Lacking any deep understanding of arithmetic principles, they are at risk of becoming little calculating machines that compute but do not think • Innumerates are prompt in drawing hazardous conclusions based on reasoning that is mathematical only in appearance • As early as preschool,American children lag way behind their Chinese and Japanese peers • Some educators view this learning gap as a potential threat toAmerican supremacy in science and technology • The designated culprit is the educational system, its mediocre organization, and the poor training of its teachers • Same problem in France, a problem was presented to them “twelve sheep and thirteen goats are on a boat, how old is the captain?” and a large proportion of them answered 25 because 12=13=25! • Innumeracy has much deeper roots than just the school system, it reflects the human brain’s struggle for storing arithmetical knowledge • The unfortunate counterpart to the automatization of mental calculation is that people jump to conclusions without considering the relevance of the computations they perform • Automatization can sometimes inhibit the prefrontal cortex, which is involved in implementing and controlling non-routine strategies • Because the prefrontal cortex matures slowly, children and adolescences are most vulnerable to arithmetical impulsiveness • It can be hypothesized that innumeracy results from the difficult of controlling the activation of arithmetic schemas distributed in multiple cerebral areas • Number knowledge does not rest on a single specialized brain area, but on vast distributed networks of neurons, each performing its own simple, automated, and independent computation • We are born with an “accumulator circuit” that endows us with the intuitions about numerical quantities • With language acquisition, several other circuits that specialize in the manipulation of number symbols and in verbal counting come into play • Innumeracy occurs because these circuits often respond autonomously and in a disconcerted fashion Teaching Number Sense • Our brain has modularity, this means it compartmentalizes knowledge within multiple partially autonomous circuits • In order to become proficient at mathematics, one must go beyond these compartmentalized modules and establish a series of flexible links among them • The expert calculator juggles mentally with number notations, moves fluently from digits to words to quantities, and thoughtfully selects the most appropriate algorithm for the problem at hand • Schooling plays a crucial role because it helps children draw links between the mechanics of calculation and its meaning • Our schools often do not meet this challenge • Schools are often content with teaching meaningless and mechanical arithmetical recipes to children • Teachers discourage things like finger counting, but it has actually been shown that finger counting is an important precursor to learning base 10 • Children are discouraged from using strategies (ie to solve 6+7, first think 6+6=12, and one unit after is 13), even though this is the strategy that many adults use • Pupils feel that they are supposed to do what the teacher does, even if they cannot make any sense of it • The formalist school of mathematical research, founded by Hilbert and French mathematicians called Bourbaki, said that mathematics had a firm axiomatic base, and their objective was to reduce demonstration to a purely formal manipulation of abstract symbols, thought children should be familiar with the general theoretical principles of numeration before being taught the specifics of our base-10 system • This type of thought leads to failure, this school says that 6% of children are “mathematically disabled”, but this is not possible as there is no single area in the brain that controls mathematics alone • The “mathematically disabled” are more likely students who got off tot a false start in mathematics, and then they decide they will never be able to understand it, and every time that study tries to do a math problem, they have anxiety and phobia surrounding it • Intuitionist schools of mathematics try to help children realize that mathematics has an intuitive meaning, which they can represent using their innate sense of numerical quantities • Rely on using schemes like temperature, distance, objects, to help understand number • The brain is not content with abstract symbols, concrete institutions and mental models play a crucial role in mathematics • This is why the abacus works so well forAsian children, it provides them with a very concrete and intuitive representation of numbers • In the US, the national council of teachers of mathematics is now deemphasizing the rote learning of facts and procedures and focusing instead on teaching an intuitive familiarity with numbers • Schools are slowly starting to use a more concrete approach to teaching mathematics by using unit cubes, ten bars, dice, board games, etc. • Education psychologists in the US have also demonstrated empirically the merits of an arithmetic curriculum that stresses concrete, practical, and intuitive mental models of arithmetic • Aprogram called “RightStart” has been implanted in inner city kindergarten classes that uses games to help teach children mathematics, results have shown that it works phenomenally • Most children are pleased to learn mathematics if only one shows them the playful aspects before the abstract symbolism Chapter 10: The Number Sense, 15 Years Later Numbers in Babies • The adaptation technique is ideal for studying the brains of young children, who cannot yet to mental arithmetic but may already have a number sense • The brain-imaging adaptation technique is almost identical to the behavioural habituation technique, used to demonstrate surprise reaction to numerical novelty in babies • Babies are shown a constant number until they adapt (habituate) and then the number is changed to another one, the adaptation technique looks at the cortex of the baby and allows us to identify which brain areas are involved in this • The first number adaptation experiment with children was performed with 4 year olds, they had received any training in arithmetic, but their parietal lobe had already demonstrated the same numerical reaction as that observed in adults, with an increase in activation whenever a novel number was introduced • Response in the parietal lobe is particularly evident in the right hemisphere, underlying children’s nonverbal intuition of number before any education in arithmetic • Shows that at a young age, brain is already divided into specific streams dedicated to number and shape, parietal cortex for change in set’s numerosity, and ventral visual cortex for change in shape • Author did this study on 3-month olds using EEG, after habituation to 4 ducks, babies’brains reacted electronically to the presentation of 8 ducks • Completely different brain responses occurred when shape changed, showing that even at a few months, the brain is already organized into two distinct streams for shape and number • Results also showed that it was the right parietal cortex responding to numerical novelty, and the left ventral visual cortex reaction to the object novelty • Another experiment showed that newborns only a few days old could notice the numerical relationships between audio and visual stimuli (looked at the display that corresponded the number of sounds heard) • Number is one of the primary attributes that allow us to make sense of the outside world, right from birth The Special Status of Numbers 1,2, and 3 • Number sense hypothesis includes that “subitizing” is the capacity to identify 1,2, or 3 items at a glance • But it is wrong to say that subitizing is essentially a form of precise approximation • If this were true, then telling 1 from 2 should be just as easy as telling 10 from 20 (as long as they were separated by the 1:2 ratio • Experiment showed that this is not true, in trying to tell the difference between two displays of dots, results were that performance with decade number s was dramatically worse than with numbers in the subitizing range (1,2, and 3) • In the whole range of numbers being tested, 1, 2, and 3 provided different results from others • This shows that a distinct process deals with the subitizing range of numbers, conclusion that has been supported by brain imaging research • This indicates that our number sense is a patchwork of multiple core processes • The current consensus is that we have two systems for representing a number of objects without counting • Small-number system, called “object tracking” system only represents sets of 1, 2, or 3 items, this gives us an exact mental model of what happens when one objects moves in or out of a small set, gives us exact understanding of the arithmetic of 1 2 and 3, this system breaks down when number of objects exceeds 3 or 4 • The approximation system can represent any number, large or small, and allows us to compare them or to combine them into approximate operations, gives us immediate intuition about the continuity of numbers • The numbers 1 2 and 3 are simultaneously represented in both systems • There is no discontinuity in our mental representation, entire range of small and large numbers is represented on the approximate number line • Both of these number systems have been shown to be already available during the first days of life, and their combination play a crucial role in the acquisition of arithmetic • In many experiments, infants succeed only when the numbers are small enough to be subitized • Experiment- if you put 2 crackers in one box, and 3 in another, infant will know to reach for the box with 3, but if you put 2 in one box, and 4 in the other, infant will just reach at random • The explanation for this is that more than 4 events saturate the infants’memory until it collapses • Sequential presentation prevents the use of the approximation system, leaving the child with a limited sense of the numbers 1, 2, and 3 How Does Subitizing Work • Contrary to what we once thought, it is not independent of our attention • One glance at a set seems enough to effortlessly recognize that it contains 1, 2 or 3 objects • This is an illusion however, sets that are presented when our mind is occupied elsewhere a
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