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Lecture 6

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Psychology 2410A/B
Adam Cohen

WEEK 6: HOW THE MIND CREATES MATHEMATICS Babies Who Count • If our working hypothesis is correct, the human brain is endowed with an innate mechanism for apprehending numerical qualities, one that is inherited from our evolutionary past and that guides the acquisition of mathematics • In the first year of life, then, babies should already understand some fragments of arithmetic Baby Building: Piaget’s Theory • According to Piaget, logical and mathematical abilities are progressively constructed in the baby’s mind by observing, internalizing, and abstracting regularities about the external world • In this view, the development of abstract thought consists in climbing a series of steps in mental functioning, the Piagetian stages, that psychologists many identify and classify • Piaget believed that number, like any other abstract representation of the world, must be constructed in the course of sensorimotor interactions with the environment • Piaget collected proof upon proof of young children’s inability to understand arithmetic • Object permanence • Number conservation • Piaget seemed to indicate that the number concept does not begin to be understood before the ages of four or five • Piaget readily concludes that prior to the age of reason children lack knowledge of the most elementary bases of set theory • They seemingly ignore that subset cannot have more elements than the original set from which it was drawn • According to Piagetian theory, it is best to start by teaching logic and the ordering of sets, because these notions are a prerequisite to the acquisition of the concept of number Piaget Errors • This aspect of Piaget’s constructivism was wrong • Young children have much to learn about arithmetic, and obviously their conceptual understanding of numbers deepens with age and education - but they are not devoid of genuine mental representations of numbers, even at birth • When candy was used, a majority of children selected the larger of the two numbers, even when the length of the rows conflicted with number • Performance on number conservation tests appears to drop temporarily between 2 and 3 years of age • The wording of the questions, and the context in which they are posed, mislead children into believing that they are asked to judge the length of the rows rather their numerosity • We constantly reinterpret the sentences that we hear, by performing complex unconscious inferences concerning the other speaker's intentions • First even a young chid is capable of interpreting the same exact question in two quite different ways, depending on context WEEK 6: HOW THE MIND CREATES MATHEMATICS • Second, Piaget notwithstanding, when the question is asked in a context that makes sense, young children get the answer right - they can conserve number Younger and Younger • Resistance to misleading cues, it seems, was part and parcel of Piaget’s definition of what is meant to have a conceptual understanding of number • In the 1980s, numerical abilities were observed in 6-month old infants even in newborns • Scientists relied on babies attraction to novelty • It is in this way that researchers have been able to demonstrate that, very early in life, babies and even newborns can perceive differences in color, shape, size, and more to the point number • In the first few months of life, babies appear to notice the constancy of objects in a moving environment and extract their numerosity Babies’ Power of Abstraction • Syllables is indeed the only parameter that can enable babies to differentiate the first list of words form the second • Very young children, therefore, pay equal attention to the number of sounds and to the number of objects in their environment • At six months of age they will discriminate numbers of actions • Seems likely that the baby can identify the number of sounds - even though it varies from trial to trial - and is capable of comparing it to the number of objects before its eyes • The very fact that a child only a few months of age applies a strategy as sophisticated as this implies that its numerical representation is not tied to a low level of visual or auditory perception • The simplest explanation is that the child perceives numbers rather than auditory patterns or geometrical configurations of objects How Much is 1 Plus 1? • We now know that the failure of children under one year in Piaget’s object- permanence task is linked to the immaturity of their prefrontal cortex, which controls their reaching movements • The demonstration is irrefutable: Babies know that 1+1 makes neither 1 or 3, but 2 • Elementary numerical computations can be performed by organisms devoid of language • 2.5 month old infants are not in the least confused by objects motion • They still find the impossible events 1+1=1 and 2-1=2 surprising • Hence, their behaviour does not depend crucially on the expectation of precise object locations The Limits of Infant Arithmetic • While young children’s numerical abilities are real, they are strictly limited to the most elementary of arithmetic WEEK 6: HOW THE MIND CREATES MATHEMATICS • In the first place, their abilities for exact calculation do not seem to extend beyond the numbers one, two, three, and perhaps fear • Babies however, most likely possess only an approximate and continuous mental representation of numbers • This representation probably obeys the distance and size effects found in rats and in chimpanzees • We should therefore expect babies to be unable, beyond some limit, to discriminate a number n from its successor n+1 • This is indeed what is observed beyond number for • In situations where an adult would automatically infer the presence of several objects, babies do not necessarily draw the same conclusion • 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 • Thus, the baby’s numerical module is both hypersensitive to information about object trajectory, location, and occlusion, and completely blind to changes in shape or color • At least three laws are exploited by the human number sense • First, an object cannot simultaneously occupy several separate locations • Second, two objects cannot occupy the same location • Finally, a physical object cannot disappear abruptly, nor can it suddenly surface at a previously empty location; its trajectory has to be continuous • Very few exceptions, the most prominent being caused by shadows, reflections, and transparencies Nature, Nurture, and Number • First year of life is when the baby’s brain possess maximal plasticity • Newborns readily distinguish two objects form three, and perhaps even three form four, while their ears notice the difference between two and three sounds • The plan required to wire up these detectors probably belongs to our genetic endowment • More likely, a brain module specialized for identifying numbers is laid down through the spontaneous maturation of cerebral neuronal networks under direct genetic control, and with minimal guidance from the environment • At present only in 6-8 month old babies has the connection between two sounds and two images, or three sounds and three images been demonstrated • It remains possible to maintain that learning, rather than brain maturation, is responsible for the baby’s knowledge of numerical correspondence between sensory modalities WEEK 6: HOW THE MIND CREATES MATHEMATICS • A rudimentary numerical accumulator clearly enables infants as early as six months of age to recognize small numbers of objects or sounds and to combine them in elementary additions and subtractions Small Heads For Big Calculations Counting: The ABC of Calculation • Counting, initially it is just a rote behaviour devoid of meaning • Only later of they learn to segment this sequence into words, to extend it to larger numerals, and to apply it to concrete situations • After years of controversy and tens of experiments, seems to stand somewhere between the all innate and the all acquired extremes • Some aspects of counting are mastered quite precociously, while others seem to be acquired by learning and imitations • Children seem to understand, quite early on and without explicit teaching, that counting is an abstract procedure that applies to al kids of visual and auditory objects • As early as three and a half years of age children know that the order in which one recites is crucial, while the order in which one points toward objects is irrelevant as long as each object is counted once and only once • By their fourth year, children have mastered the basics of how to count • Though children rapidly grasp the how to of counting, however, they seem to initially ignore the why • Children do not appreciate the meaning of counting until the end of their fourth year • Although the mechanisms of counting have largely fallen into place, children do not seem to understand what counting is for, and they do not think of counting when the situation commands it Preschoolers as Algorithm Designers • The first calculation algorithm that all children figure out for themselves consists in adding two sets by counting them both on the fingers • Initially, young children find it difficult to calculate without using their fingers • Words vanish as soon as they have been uttered , but fingers can be kept constantly in sight preventing one from losing count in case of a temporary distraction • Refinements are quickly found • Most children realize that they need no recount both numbers, and that they can compute 2+4 by starting right from the word two • To shorten calculation even further. they earn to systematically start with the larger of the two numbers • This is called the minimum strategy • As children count on their fingers, years before going to
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