Vocab for INDV 101 at UA
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TICS 276 ELS Language and Conceptual Development series TRENDS in Cognitive Sciences Voxx Noxx Monthxxxx Full text provided by wwwsciencedirectcom SCIENCEltdDIRECT39 Number and language how are they related Rochel Gelman1 and Brian Butterworth2 1152 Frelinghuysen Road Rutgers Center for Cognitive Science Piscataway New Jersey 08844 USA 2Institute of Cognitive Neuroscience University College London 17 Queen Square London WC1N BAR UK Does the ability to develop numerical concepts depend on our ability to use language We consider the role of the vocabulary of counting words in developing numeri cal concepts We challenge the bootstrapping39 theory which claims that children move from using something like an objectfile an attentional process for responding to small numerosities to a truly arithmetic one as a result of their learning the counting words We also question the interpretation of recent findings from Amazonian cultures that have very restricted number vocabularies Our review of data and theory along with neuroscientific evidence imply that numerical concepts have an ontogenetic origin and a neural basis that are independent of language Introduction We have the intuition that our thoughts are inseparable from indeed dependent upon the words we use for them This is especially so for numerical cognition where the claim is that some basic aspects of numerical cognition depend crucially on language be it knowledge of the vocabulary of counting words or the recursive capacities of syntax and morphology This argument has been made from neuropsychology where arithmetical facts are held to be stored in a verbal format it has been made from neuroimgaing where numerical tasks appear to activate language areas it has been made from developmental psychology where counting words are claimed to be necessary for concepts larger than three or four and most recently it has been made from studies of Amazonian tribes whose language lacks counting words In this article we challenge all of these claims Number terminology Natural number and arithmetic Numerical terms and notation can serve a variety of non mathematical functions The numeral 4 can denote the position in a sequence a TV station a particular football player a sign of good luck and so on For all of these examples there are other ways to refer to the concepts in question A sequence can be represented by letters and a TV station can be called NBC or ITV The distinctive Corresponding authors Gelman R rgelmanruccsrutgersedu Butterworth B bbutterworthuclacuk numerical concept and the one that is the focus of this article is numerosity the cognitive equivalent of the cardinality as normally denoted by the natural numbers Each numerosityN has a unique successor N 1 which is why we can say for example that a set of 5 objects includes a set of 4 objects and so on 1 Of course it is possible to estimate or approximate an exact numerosity and to use continuous variables to do so For objects of the same type say apples the greater the number of apples then the greater the amount of apple stuff or weight of apples It would be possible to estimate the number of a set of apples or to compare the cumulative magnitudes of two sets of apples on the basis of these continuous variables However this does not mean that the mental representation of numerosities must be approximate or continuous In fact we know that humans are able to think of both an approximate and exact value for a given set See 2 in this series of papers and 3 General vs specific considerations However to identify the numerosity of sets larger than about 4 some kind of itembyitem enumeration is required 4 Typically this will involve counting using the familiar vocabulary of specialized count words one two but it could also involve sequenced hatch marks or mapping objects onto a set of known numerosity such as body parts 5 For sets with large numerosities we might not bother or be able to enumerate and instead rely on methods of approximation Therefore we need to distinguish possession of the concept of numerosity itself knowing that any set has a numerosity that can be determined by enumeration from the possession of re representations in language of particular numerosities The relationship between language and number in the brain At a broad level the functional relationship between number and language should be evident from their relationship in the brain Ever since Henschen s extensive case series in the 1920s it has been known that disorders of language and calculation abilities can occur indepen dently 6 Recent detailed case studies have con rmed that it is possible for previously numerate adults to have severely impaired language but relatively wellpreserved numerical skills 78 Perhaps the clearest reported case is the neurological patient IH suffering from semantic wwwsciencedirectcom 13646613 see front matter 2004 Elsevier Ltd All rights reserved doi101016jtics200411004 TICS 276 w dementia whose language comprehension was at chance in most tasks and whose production was limited largely to stereotyped phrases however he scored at or near ceiling on single digit and multidigit calculation 910 More sophisticated approaches de ne speci c roles for language in adult arithmetic For example in the Triple Code model 11 some operations such as number comparison and subtraction depend on manipulations over the analogue magnitude code whereas arithmetical facts are stored in the Verbal code Addition and multiplication which are held to depend more on stored facts than subtraction and division will therefore be more vulnerable to language disturbances than subtraction and division However IH was as unimpaired in retrieving stored facts 10 The Triple Code model also distinguishes exact calcu lation which is held to depend on understanding exact numerosities from approximate magnitudes which are the responsibility of the Analogue Magnitude code Evidence in support of this comes from patients who can handle approximate quantities but not exact calculation 12 It is not clear from presentations of this model how these two codes analogue and verbal interact in the process of arithmetic Neuroimaging studies make it clear at least to us that the crucial brain systems involved in numerical process ing are in the parietal lobe some distance from any classical language areas In a valuable metaanalysis Dehaene and colleagues distinguish the bilateral parietal brain areas that have been found to be more active in estimation and approximation tasks horizontal segment of the intraparietal sulcus and the posterior superior parietal sulcus from the area more active in exact calculation tasks and therefore more dependent on language left angular gyrus However the left angular gyrus is not a classical language area although active in verbal workingmemory tasks 13 It has been known at least from the time of Gerstmann 14 that lesions to this area can cause impairment to exact calculation without concomitant language disorder see 15 for a review One neuroimaging study has found that activity in Broca s area is depressed relative to rest during numerical tasks suggesting that numerical and linguistic processing are even in opposition 16 Even if the intimate relationship between number and language is not re ected in adult neuroanatomy a relationship might nevertheless be a requirement for the development of the neural basis of number Developmental perspectives The strong Whor an claim of lingusitic relativism is that language shapes the development of numerical concepts It is rarely clear from the defenders of this position whether they believe that it is the concept of numerosity itself that depends on language or whether it is concepts of particular numerosities that are entailed Both Mix Huttenlocher and Levine 17 and Carey 3 assign to language a causal role in people s acquisition of concepts of natural numbers and their properties As Carey offers an extensively developed account we focus on hers here www5ciencedirectcom TRENDS in Cognitive Sciences Volltlt Noltlt Monthxxxx Causal dependence on the list of counting words A crucial assumption for Carey is that infants numerical abilities involve two different mechanisms one for small sets of 4 or less and another for larger sets Parallel individuation of objects with its builtin limit of 3 4 serves the small number range 318 20 An accumulator process that converts discrete counts into analogue quantities serves the larger number range The demon strations of behavioural discontinuities between small and larger numbers in infants studies justify this dichotomy 18 20 but there are also failures to nd the presumed limit in the small number range 21 23 Carey s account of infants parallel individuation of objects introduces an unusual use of the notion of object les and their function Object les for representing numerosity The function of an object le is to point to a visual object and integrate the perception of the properties of that object such as size shape color and so on 2425 If a child believes that the word two referred to a particular set of two object les it would presumably be useable only in connection with the two objects they pointed to It would be a name for that pair of objects not for all sets that share with that set the property of twoness A particular set of pointers cannot substitute for is not equal to another such set without loss of function because its function is to point to a particular pair of objects whereas the function of another set of pointers is to point to a different pair There is no reason to believe that there is any such thing as a general set of two pointers a set that does not point to any particular set of two objects but represents all sets of two that do so point Any set of two object les is an instance of a set with the twoness property a token of twoness but it can no more represent twoness than a name that picks out one particular dog eg Rover can represent the concept of a dog Being able to abstract away from the particularities of the transient representation of items of current attention to represent their numerosity seems to be a precondition for making use of this system to bootstrap concepts of number rather than a consequence of using it It is most unusual to use a processing constraint the number of objects whose features can be bound in a transient store as the basis for the development of a system of knowledge Nevertheless let us grant the case for sake of argument that there is a discontinuity in infants numerical discriminations and that the small number range uses Carey s version of an object le mechanism Language rst Carey s argument goes forward as follows once the rst three or four count words are memorized they can be treated as separate from the nonprecise quanti ers of some few many The meaning of the count words is induced from the fact that they map to different sized sets of object les and they are used in a xed order with each successive word coming to represent a larger object le set This allows for the integration of the infants numerical representations of small numbers and their TICS 276 w short list of memorized count words Language serves as the mechanism for creating a restructuring of the non verbal notions of number and according to Carey does so as follows A child comes to recognize the ordering of the referents of one two three and four because a set of two active object les has as a proper subset a set of one object le and so on The child infers that addition applies to the things referred to by these words because the union of two sets of object les yields another set of object les provided the union does not create a set greater than 4 This is the foundation of the child s belief in the successor principle every natural number has a unique successor The crucial bootstrapping comes when the child realizes that it is possible to go beyond the narrow limits of parallel individuation of objects to sequences of objects that are not all within the span of immediate apprehension This move is prompted by the use of count words for sets of more than 4 objects That is the child having established the correspondences for N 4 the word one refers to the state of the object le system where there is just one object the word two refers to the set of two objects which is one more than one and so on makes the inference that other syntactically equivalent words ve six and so on refer to numerosities greater than the immediate span of apprehension This account entails that meanings are rst assigned only to the words that have been mapped onto states of the object le system So when the child is unable to give reliably on request four ve or six objects it has to be assumed on this model that he or she knows nothing about the mapping and that these words will refer indiscriminately to numerosities greater than 3 However there is evidence that although children at this stage of development are unable to give exactly what number is designated by ve they know that only numberchanging manipulations of the target set will require a change of number word 2223 Although young threeyearold children fail a Give N task when N is greater than 3 they can succeed on prediction and checking tasks with set sizes ranging from 1 5 23 The implication is that young children understand how numbers work before they have fully mastered the mapping from particular numbers to particular numerosities Carey s position makes clear predictions about the numerical capacities of children growing up in cultures where there are few or no counting words and especially where there is no linguistic means for creating ever larger number names One prediction is that the children will not develop true understanding of the natural numbers because it is the system of counting words that is crucial Other semiotic means for repre senting number such as using bodyparts 526 tally ing or drawings in the sand DP Wilkins PhD thesis Australian National University 1989 are not mentioned as enabling the development of numerical concepts beyond 3 or 4 the limit of the object le system of parallel individuation We concur with Harris 2728 that it is a mistake to dismiss these alternative re representation systems www5ciencedirectcom TRENDS in Cognitive Sciences Volltlt Noltlt Monthxxxx 3 Amazonian mysteries The Tououpinambos One of the earliest accounts of the numerical abilities in people with restricted number vocabularies comes from the English philosopher John Locke 29 who wrote Some Americans Ihave spoken with who otherwise of quick and rational parts enough could not as we do by any means count to 1000 nor had any distinct idea of that number He was not referring to the founders of the USA but to the Tououpinambos a tribe from the depths of the Brazilian jungle whose language lacked number names above 5 A system of number names was not in Locke s view necessary to have ideas of larger numbers because he says we construct the idea of each number from the idea of one the most universal idea we have By repeating this idea in our minds and the repetitions together by adding the idea of of one to the idea of one we have the complex idea of a couple And so on Thus he says concepts of numbers are independent of their names Indeed the Americans can reckon beyond ve by showing their ngers and the ngers of others who were present Even the idea of in nity he proposes is simply a consequence of understanding that we can repeat the procedure for adding one as many time as we wish Possessing a system of number names is useful for keep track and communicating but not necessary for having the ideas of distinct numerosities and their in nity Locke s point was that number names conduce to well reckoning by enabling us to keep in mind distinct numerosities That is the possession of a system of number names can be helpful in learning to count and to calculate but is not necessary for the possession of numerical concepts Recent reports about the Piraha 3031 and Munduruku Amazonian Indians 31 provide informative test cases of Locke s conclusion The Pirah and Mundurukii The Munduruku language uses the count words for 1 2 and 3 consistently and 4 and 5 somewhat inconsistently 31 The Piraha do not even use the words for 1 and 2 consistently 30 How would members of these groups perform on various nonverbal tasks involving numeros ity The amazing result was that both groups succeeded on nonverbal number tasks that used displays represent ing values in one study as large as 80 The ndings on the Munduruku are especially note worthy because an elegant research design was used that incorporated the fact that some Munduruku adults and children are bilingual in Portuguese In the study 31 there were groups of adults and children who were monolingual and groups who were bilingual The chil dren s groups were divided into younger lt5 yrs and older as well as those who had had language instruction and those who had not Finally there was a control group of French adults The various groups were asked to point to the more numerous of pairs of dots whose numerosity could be as large as 80 All the groups showed the effects of number size and number difference in number comparison tasks 32 There was no effect of language or schooling amongst the Munduruku The data for the number difference effect for the Munduruku groups and the TICS 276 w French adult control groups were extremely close Although the Piraha study did not have as many subjects and conditions the results were comparable They too were able to engage in a variety of comparison and non verbal arithmetic tasks despite their lack of any clear number word vocabulary The Piraha solved the problems in ways that overlap extensively with those used by English and Henchspeaking individuals 32 The key claim of defenders of the 39language thesis is that language is necessary for mental representation and manipulation of numerosities greater than 4 In the Munduruku study exact addition and subtraction prob lems using sets of objects were tested According to the theory the participants who were bilingual and therefore knew the counting words of Portuguese should have performed like the numerate French controls or at least more like the Hench controls But this was not the case Both adults and children performed exactly like the monolingual speakers 31 The Munduruku continued to deploy approximate representations in a task that the French controls easily resolved by exact calculation 31 Why we may ask did the bilingual participants not use their Portuguese counting words Munduruku culture differs from Western culture in innumerable ways and it certainly uses numbers far less often than we do It remains possible that one or more of these many differences were responsible for the differences in per formance and not just the lack of a counting vocabulary This evidence from cultures with very limited number vocabulary does not convince us that differences in performance can be explained in terms of language rather than other aspects of culture see also Box 1 Of course it remains possible that Piraha and Munduruku have few number words because numbers are not culturally important and receive little attention in everyday life Causal dependence on the recursive capacities of language Bloom proposed that children s initial counting is embeddedin natural language as a result of their learning relevant distributional facts 33 As they learn more and more count words they infer that there are more count words With enough experience they infer that natural numbers are discretely in nite Hauser Chomsky and Fitch 34 offer a statement as to how this might work This is that the recursion involved in the mathematical idea of discrete in nity derives from a recursive capacity Box 1 Questions for future research o How do varieties of language especially varieties of number naming systems promote or inhibit acquisition of basic numerical concepts o Is there a critical or sensitive period for acquiring numerical concepts 0 How do developmental language disabilities affect the acquisition of arithmetical skills Can there be a selective deficit of arithmetical development 0 Why do mathematically literate individuals continue to use the non verbal approximate numerosity system 0 How does the neural network for numerical processing develop from infancy to adulthood www5ciencedirectcom TRENDS in Cognitive Sciences Volltlt Noltlt Monthxxxx that is the foundation of and unique to human languages Carey 3 makes the same inprinciple assumption We are puzzled about this claim Because the mental magnitudes that represent larger numbers are additive the recursive in nity of magnitude is already entailed Instead we are making the case that understanding recursive in nity is not derived from language at all To illustrate this in one study 35 children aged 5 years to 8 years 6 months written 86 were asked to participate in a thought experiment about the effect of repeated additions or countingon from what they said was a very large number Many of the younger children showed that they were still nding the language to express themselves as was the case for DA 59 When asked whether adding would yield a higher number she replied yes because You still put one and they get real higher Some of the older children were explicit about the possibility of inventing count words to satisfy the successor principle for natural numbers For example AR 73 said there is no end to the numbers Because you seepeople making up numbers You can keep making them and it would get higher and higher 35 There are several language experts who hold that despite the fact that some speakers of languages have restricted number terms they can easily acquire them Dixon quotes Kenneth Hale an expert on Warlpiri a language with terms for one two few many the English counting system is almost always instantaneously mastered by Warlpiris who enter into situations where the use of money is important quite independently of formal Westernstyle education 36 p 108 A potent example of the rapid uptake of the idea of discrete in nity comes from Saxe 5 who studied the Oksapmin of New Guinea a group who used use a xed number of sequential positions on their body as count words There came a time when some of the men were own out to work on plantations and received money for their labor Within 6 months the Oksapmin had introduced a generative counting rule 5 It is hard to see how such rapid learning of a new vocabulary for abstract objects like numerosities could proceed so quickly if the learners did not already possess the concepts nally although the Piraha do not use numerals in their everyday life Everett personal communication reports that it is easy to teach their children to count in Portuguese if the pronunciation rules are adjusted to t the phonetics of Piraha and the teaching is embedded in the everyday task of stringing beads Conclusions Cognitive development re ects neural organization in separating language from number Indeed the onto genetic independence of the number domain has been argued vigorously by the authors of many previous publication looking at both normal 43537 and abnormal 38 40 development of numerical abilities It would be surprising if there were no effects of language on numerical cognition but it is one thing to hold that language facilitates the use of numerical concepts and another that it provides their causal underpinning That very young children s knowledge of count words is TICS 276 w incomplete is far from surprising They constitute a serial list of sounds there is nothing about the sound one that predicts that the sound for two will follow and so on In addition the young child has to master the extensive coordination requirements of counting Locke put the matter elegantly more than three hundred years ago Children either for want of names to mark the several progressions of numbers or not having yet the faculty to collect scattered ideas into complex ones and range them in a regular order and so retain them in their memories as is necessary to reckoning do not begin to number very early nor proceed in it very far Acknowledgements BB39s research in this area is supported by the Commission of the European Communities HPRNCT 2000 00076 Neurornath and MRTN CT 2003 504927 NUMBRA and by the Leverhulrne Trust F07 134U RG39s work is supported by NSF No SPR9720410 References 1 Butterworth B 1999 The Mathematical Brain Macmillan 2 Feigenson L et al 2004 Core systems ofnumber ends Cogn Sci 8 307 314 3 Carey S 2004 On the Origin of Concepts Daedulus 4 Gelman R and Gallistel CR 1978 The Child s Understanding of Number Harvard University Press 5 Saxe GB 1981 The changing form of numerical reasoning among the Oksaprnin Indigenous Mathematics Working Paper No 14 UNESCO Education 6 Henschen SE 1920 Klinische und Anatomische Beitrage zu Pathologie des Gehirns Nordiska Bokhandeln 7 Rossor MN et al 1995 The isolation of calculation skills J Neurol 242 78 81 8 RemondBesuchet C et al 1999 Selective preservation of excep tional arithmetical knowledge in a demented patient Math Cogn 5 41 63 9 Cappelletti M et al 2002 Why semantic dementia drives you to the dogs but not to the horses A theoretical account Cogn Neuro psychol 19 483 503 10 Cappelletti M et al 2001 Spared numerical abilities in a case of semantic dementia Neuropsychologia 39 1224 1239 11 Dehaene S and Cohen L 1995 Towards an anatomical and functional model ofnurnber processing Math Cogn 1 83 120 12 Lerner C et al 2003 Approximate quantities and exact number words Dissociable systems Neuropsychologia 41 1942 1958 13 Paulesu E et al 1993 The neural correlates ofthe Verbal component of working memory Nature 362 342 345 14 Gerstrnann J 1940 Syndrome of nger agnosia Disorientation for right and left agraphia and acalculia Arch Neurol Psychiatry 44 398 408 15 Cipolotti L and Van Harskamp N 2001 Disturbances of number processing and calculation In Handbook ofNeuropsychology Berndt RS ed pp 305 334 ElseVier 16 Pesenti M et al 2000 Neuroanatornical substrates of Arabic number processing numerical comparison and simple addition A PET study J Cogn Neurosci 12 461 479 www5ciencedirectcom 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 TRENDS in Cognitive Sciences Volltlt Noltlt Monthxxxx 5 Mix KS et al 2002 Quantitative Development in Infancy and Early Childhood Oxford University Press Carey S 2001 Cognitive foundations of arithmetic Evolution and ontogenesis Mind Lang 16 37 55 Carey S 2001 On the possibility of discontinuities in conceptual development InLanguage Brain and Cognitive Development Essays in Honor ofJacques Mehler Dupoux E ed pp 303 321 MIT Press Carey S and Spelke E Bootstrapping the integer list Represen tations of number In Developmental Cognitive Science 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