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   Number Processing  CARLO SEMENZA  Department of Psychology, University of Trieste, Trieste, Italy ABSTRACT Neuropsychological studies on patients with brain lesions have contributed to contemporary understanding of numerical process-ing on equal footing with studies of healthy populations. Having carved the cognitive chicken at its joints via the traditional method of studying neuropsychological dissociations has thus paid hand-somely. Indeed, also thanks to fast-improving neuroimaging tech-niques, but mainly thanks to increasingly refined conceptual tools newly applied in clinical research, we are witnessing an unprec-edented progress in the interdisciplinary domain of mathematical cognition. Fascinating new data are emerging on how and where the brain represents numbers and performs calculation. The rela-tion that numerical cognition has with language is much clearer than it was just a decade ago. 21.1. INTRODUCTION Numbers affect many aspects of contemporary life. It would be a mistake, however, to think that numbers are just a product of advanced civilization. Even life in primitive conditions could not and cannot dispense with numbers. Only one language (Box 21.1) out of the thousands known, Pirãha, spoken in a limited Amazon region, has no words for numbers! Indeed, written representations of numerals can be found in bone engravings and on cave walls by Neolithic men, dating about 30,000 years ago. The first numerate societies, like the Sumerians and the Babylonians, started using numbers around 10,000 BC, and by 3000 BC they had developed a wide variety of number-specific cultural tools. C H A P T E R   21 Box 21.1 Numbers and cultures Different cultures use different numerical systems. There is only one language that we know of without number words (Pirãha, Amazon), but several languages are known that have very restricted systems. Examples of this kind come from Australian absrcinal languages, like Mangarayi (that has numbersup to 3), Yidini (up to 5), Hixkaryana (up to 5, plus 10: the word for 4 means “his brother twice over;” 5 is “half our hands,” 10 is “our hands completely”). Haraui (New Guinea) has an up to 4 system with addition: thus 3  2  1 and 4  2  2.While several cultures still keep track in counting using body parts (up to 74, examples are given in the main text), most culturesuse bases, that is collections of numbers from which other numberscan be derived by multiplication and addition. Thus, in base systems, numbers can be expressed as a multiple of a base plus anynumber that is smaller than the base. For example, in a language with base 60, the number 361 would be expressed as 6   60  1.Base 10 is the most popular, but there are several others and many cultures use a mixed base system. Thus, there are sys-tems with base 20, 12, 6, 60, 32 and so forth. However, most languages, for historical reasons, contain irregularities. One very regular number word system with base 10 is found in Mandarin Chinese. But look at English: the next number after “ten” is “eleven,” not “ten-one” as full regularity would dic-tate! A very endangered language, Resian Slovene, regularly alternates base 10 with base 20. In French base 20 intrudes in the base 10 system (“quatre-vingt,” 80  4   20) and there are mixed numbers like “soixante-dix-sept” (77  60  10  7). All the first 100 numbers in Hindi are irregular.Descriptions of number systems in languages worldwide may be found in the vast work of Bernard Comrie, the leading expert in the field. 219  Handbook of the Neuroscience of Language Copyright © 2008 Elsevier Ltd.All rights of reproduction in any form reserved. CH021 indd 219 CH021.indd 219 2/6/2008 2:36:01 PM 2/6/2008 2:36:01 PM  220 Experimental Neuroscience of Language and Communication  Elementary number processing is demonstrable in infants. For example, they show signs of renewed interest when three dots appear after they have been shown two for some time, and they seem to notice the difference if two puppets disap-pear behind a screen and then only one reappears when the screen is removed. Even animals of several species, not only those more proximate to homo sapiens, are now thought to register numbers in some representational form and enter them in simple mental computations. It is therefore evident that numbers and their manipulation reflect a basic ability developed in living organisms with natural evolution, and that culturally transmitted, number related, abilities are just the latest result of a long and complex evolutionary process. Number-specific tools represent numbers in variousformats: numerals (Arabic, Roman and so forth), number words, dots on dice and so forth. More complex tools are meant for use in calculation: arithmetical facts (e.g., 6   3  18), arithmetical procedures (e.g., addition, subtraction, multiplication, borrowing, carrying and so forth), and arith-metical laws (e.g., commutability: 5   4   4   5). The use of these tools involves the use of different skills in order to perform several different mathematical tasks. A huge theoretical and empirical effort has been made in order to understand how the cognitive system acquires these skills and deals with these tasks. Significant contributions have come from animal psychology, human experimental psychology in adults, developmental psychology, behav-ioral genetics and neuroscience. This chapter will summa-rize the results of this effort, emphasizing how traditional neuropsychology has handsomely contributed to models of numerical processing, on equal footing with experimen-tal studies on typical, non-brain-damaged populations. The observation of counterintuitive clinical phenomena, more-over, prompted research in unforeseen directions. 21.2. THE REPRESENTATION OF NUMBERS Knowledge of numbers seems somehow independent from other types of knowledge stored in our brain. After left parietal lobe damage, CG, a lady studied by Cipolotti  et al.  (1991) , was still proficient in language tasks and rea-soning and retained a normal IQ. Yet she could not deal at all, verbally or otherwise, with numbers above four! In contrast, the case reported by Cappelletti, et al  ., (2001) , affected by semantic dementia, was severely impaired in naming, read-ing and writing of non-number words as well as in several other verbal and pictorial tasks, while showing no compara-ble deficit with numerical material. Similarly, another patient was reported whose reasoning abilities were almost nil, while his calculation was excellent! In classic amnesia, involving disorders of episodic memory, number skills are instead pre-served. Patients with short-term memory impairment may often retain the ability to carry out mental calculation. Neuropsychological cases thus show how numerical rep-resentations and calculation abilities are stored in long term, semantic, memory in a highly modular way and can be selectively impaired. 21.2.1. Number Meaning in the Brain How are, then, numbers represented in the brain? Why do they differ from other domains of knowledge? The way quantities are appreciated is crucial in understanding how the brain represents number knowledge and needs to be described in some detail. Capturing the number of elements in a given collection and accessing the appropriate mental representation is per-formed via three basic tasks: subitizing, counting and esti-mation. Each of these tasks is independent from other tasks in the brain. Suppose one has to tell how many dots appear in a given array. Reaction times in this task appear to be about the same for each quantity inferior to five dots and increase exponentially thereafter. The ability to correctly appreciate such small quantities without serial processing is referred to as “ subitizing. ” The abilities of patient CG, who could not deal with numbers above four, were thus restricted to the subitizing range. For appreciating quantities of five and more, one has to resort to counting or estimation, abili-ties that CG did not retain after her brain injury. “ Counting ” basically consists of pairing each element of a collection to a number word. The last assigned word is the label for the entire collection. This latter property is called “ cardinality. ” The concept of “ ordinality, ” instead, refers to the ability to  judge which of two different quantities is larger. According to Gelman and Gallistel (1992) , cardinality is part of innate mathematical knowledge. Such knowledge would also include realising that: (a) each element must cor-respond to one verbal label only; (b) a conventional order rules verbal labels; (c) counting applies to any collection of objects; (d) changing the labeling order does not modify the numerosity of the collection. A separate ability involves sep-arating already counted objects from those that remain to be counted. Not all authors believe that counting rests on innate knowledge: the alternative view is that children deduct prin-ciples of counting from experience ( Wynn, 1998 ). Ordinality and cardinality are really different aspects of number representation and were shown to dissociate in neuropsychological patients. A selective impairment with number cardinality only has in fact been reported ( Delazer & Butterworth, 1997 ): the patient could correctly order any sequence of numbers, while unable to add 1 to a given number! He thus could say what number came after 6, but, rather paradoxically, he did not know the answer to 6   1! According to the position of Dehaene and co-authors (e.g., Dehaene & Cohen, 1995 ) patients with left hemisphere lesions may not be able to appreciate the exact meaning of num-bers and may not perform exact calculation. Some of them, CH021 indd 220 CH021.indd 220 2/6/2008 2:36:01 PM 2/6/2008 2:36:01 PM  221 Number Processing however, may nonetheless be able to provide good number estimates and perform approximate calculation. “ Estimation ” is a quick but less accurate process than counting. Several variables, besides objective numerosity, have been shown to influence estimation: the physical properties of the stimuli, their spatial disposition, practice in the tasks, visual clarity and so forth. Many researchers think that the internal repre-sentation of a number is analogical and its precision decreases as the number increases. But what does “ analogical ” mean in the case of number representation? The understanding of this concept started with Galton in the nineteenth century. 21.2.2. The Number Line Since Galton (1880) , in fact, scientists were made aware that some people are conscious of an internal spatial represen-tation of numbers that mostly takes the form of an imaginary line. This “ number line ” is used in understanding numbers and, sometimes, in calculation. In modern psychology, the nature of the semantic representation of numbers is indeed controversial. While some people (e.g., McCloskey, 1992 ) assume a precise representation even for large numbers, others (e.g., Dehaene & Cohen, 1995 ) hypothesise an analog magnitude representationfor approximate quantity, that is number meanings are rep-resented in a continuous, spatially extended, scale. Vis-à-vis such controversy, the concept of “ number line ” is gaining newpopularity. A modern addition to Galton ’ s observation is, in fact, that even people who are not consciously aware of a mental number line may nonetheless use it. Dehaene et al  . (1993) showed that within a given interval, people who read left to right are generally faster at making judgements (e.g., odd/even judgements) about smaller numbers with the left hand but faster with their right hand for larger numbers. This effect (known as “ SNARC ” —Spatial-Numerical Association of Response Codes) may be weaker or inverted in speakers of languages with right-to-left writing! This suggests that an unconscious number line is there in most people and that its orientation depends on the direction of reading. In the case of left-to-right reading, smaller numbers are thus on the left and larger numbers on the right of the line. Interestingly, further evidence for this conclusion comes from patients with left sided neglect. Given the task of telling which is the midpoint of a numerical interval, they show the same displacement effect to the right they would show in line bisection (thus answering 6instead of 5 when asked the midpoint between 3 and 7). Order rather than magnitude may however be the critical property of the number line. A SNARC-like effect is in fact observed for other ordered sequences, like the letters of the alphabet or the days of the week. 21.2.3. Fingers Count (and So Do Other Body Parts) A spatially organized line is not the only way we help ourselves to mentally represent numbers. Our body helps us as well. Neuropsychologists proved indeed long time ago that dealing with numbers and dealing with fingers are linked abilities. Damage, or temporary inactivation ( Rusconi et al  .,2005 ), in the left parietal lobe leads to what is known as “ Gestmann ’ s syndrome, ” a condition whereby patients can- not calculate and cannot recognize fingers. Why does our brain process numbers nearby the represen-tation of fingers? This may be because mental representations of numbers are influenced by strategies used to keep track on how many items one has counted up to a point. Several devel-opmental and cross-cultural studies show that spontaneous finger-counting strategies are developed and used by children in almost all human cultures ( Butterworth, 1999 ). However, while fingers are invariably used in body counting, other body parts and body locations (e.g., between-fingers intervals) sometimes with repeated passages, up to 74, are used! The sequential order is not intuitive. For example, Yupno males (with respect to our own culture, political correctness may have different implications in this tribe of Papua New Guinea: female do not count or count less) count up to 33: they start from the little finger of the left hand, then they count fingers on the right hand, the left foot, the right foot, then the ears, the eyes, the nostrils and the nose, the nipples, the navel, the left testicle, the right testicle and the penis. South of them, natives of the Torres Straight also count up to 33, but start from the little finger of the right hand, then count the wrist, the elbow, the shoulder, go on with a mid-chest point and start on the other side with the shoulder, elbow and wrist down to the left hand fingers; then they go to the fingers of the left foot, then the ankle, the knee, the left hip, the right hip and down to the knee and the ankle to end on the little finger on the right foot. Use of body parts, typically finger counting, appears thus to be a universal strategy to deal with numbers. Using and practising prototypical finger counting may lead to long-term associations between fingers and digits. Long-established fin-ger-counting strategies have been recently shown to influencethe way numerical information is projected into physical space and perhaps mentally represented ( Di Luca et al  ., 2006 ). The existence of a neural network supporting finger move-ment representation has been advocated on the basis of neu-roimaging observations ( Pesenti et al  ., 2001 ). Joint activation of left pre-central and parietal areas would sustain finger counting and numerosity quantification. It would support early representations in childhood and, by extension, become the substrate of numerical knowledge and processes in adults. 21.2.4. Number Words Are Special: The Number Lexicon Cardinal (and ordinal) numbers are also represented in a speaker ’ s language. Neuropsychological studies show that number words can be damaged in a very selective fashion ( Dohmas et al  ., 2006 ; Semenza et al  ., 2007 ). For example, one patient made, in oral production, lexical substitutions CH021 indd 221 CH021.indd 221 2/6/2008 2:36:01 PM 2/6/2008 2:36:01 PM  222 Experimental Neuroscience of Language and Communication of number words but not of other words. Conversely cases were found with phonological impairments in production that spared numerals. In these cases the problem seems to be located in the activation of the lexical system from the semantic system. While leaving many unanswered ques-tions, these cases are nonetheless important because they show how number words are in several respects different from other collections of words. 21.2.5. On Knowing About Nothing: The Elusive Number Zero The most mysterious number word, and elusive number-related concept is zero. How does our mind appreciate and manipulate the concept of nothingness in the numerical domain? There is good reason to think that this is not an easy task. While adding a null quantity or subtracting it from a given quantity can be easily represented (e.g., by visual imag-ing), it is less obvious which representation may be invoked in multiplication or division by zero. Zero as an operand, unlike any other operand, makes any quantity disappear! The idea of a collection with no members or, in mathemat-ical terms, of an empty set, is not difficult to grasp. Yet it took a long time to efficiently represent zero within a numerical system. Zero was not part of the ancient formal number sys-tems. Mathematicians in Babylon, classic Greece (including Archimedes!) and Rome, ignored zero altogether! Working independent from Europe, the Maya are now believed to have preceded everybody by developing a system represent-ing zero in the first centuries AD. Only in the early seventh century, in the Indus valley, did zero develop a meaning in its own right. The notion took about another 150 years to be adopted by Arabs, whose traders passed it to the Europeans only in the twelfth century! This historical delay, even more surprising when confronted with the extraordinary mastery of other mathematical principles, reflects the intrinsic difficulty of manipulating a null quantity in arithmetic. The same story holds within individual development. Young children, once they have learned to deal with positive integers, and have learned both the word “ zero ” and the corresponding Arabic symbol, take a considerable time to appreciate that zero is a numerical value corresponding to nothingness. Neuropsychologists, on their part, showed that the notion of zero remains somehow segregated in the brain. Patients have been described, for example, showing how even the same type of knowledge, that is, the 0-rule in multiplication, may be correctly retrieved or not depending upon the arith-metic context. In describing these phenomena, Semenza  et al  . (2006b) suggested that processing of zero in arithmetic is likely to be mediated by shallow, context-bound rules and procedures, whose binding with conceptual knowledge is rather marginal. Indeed, though automatic and routinized pro-cedures may efficiently support calculation in full-functioning individuals, they may become isolated and neglected pieces of knowledge in the context of disturbed cognitive systems. 21.3. NUMBER MANIPULATION:  TRANSCODING Written and spoken words are obviously not the only way to represent numbers. Various codes for numbers are used and are indeed independently represented and processed in our brain. Patients have been reported who could not use the alphabetical code but could flawlessly use numbers in the Arabic code. The reversal case has also been documented. Processes dealing with numbers are often “ transcoding ” processes, in that they transform a number from a given representational format into another format. Thus, in thecontemporary Western culture (other codes, e.g., the Romancode, have been used or are used in other cultures), a task like reading aloud presupposes transcoding Arabic or alphabetically written numbers into spoken number words; in writing on dictation the reverse transcoding process is at work. Other common transcoding tasks are, for example, number repetition or number transformation from one code into the other (e.g., “ two ” into “ 2 ” or vice-versa). How does the cognitive system perform such tasks? Patients committing revealing patterns of errors in these tasks have been described. Naturally, errors mostly occur when transcoding complex numbers. Complex numbers are built according to a sort of lexical syntax. “ Number syntax ” has been shown – on the basis of neuropsychological dis-sociations – to be independent from language syntax and defines the set of acceptable structures formed by combin-ing the basic lexical elements. The semantic component in the system allows attribution of meaning to each sequence. The most interesting findings thus concern two main types of transcoding errors: “ lexical ” and “ syntactic ” errors ( Deloche & Seron, 1982 ). These errors often co-occur and are sometimes observed in relative isolation from each other, reflecting independent “ lexical ” and “ syntactic ” processing mechanisms. “ Lexical errors ” consist of the incorrect production of one or more of the individual elements in a number (e.g., 4 instead of 8, 57 instead of 58, 2506 instead of 2406). “ Syntactic errors ” are instead violations of the order of magnitude that spare the correct lexical elements: they are made when the power of ten is wrong with respect to the target. Syntax errors may thus result in longer numbers as in “ 510,028 ” instead of “ 528 ” or “ 50,028 ” instead of “ 528, ” or in shorter numbers (when deletions of one or more zeros occur, as in “ 25 ” instead of “ 205 ” ). Different cultures adopt different number syntax systems. Thus the nature of number syntax errors may vary among patients who speak different languages. For example in French, whereby base 20 intrudes in a base 10 system (Box 21.1), patients make syntax errors that cannot be observed in English. CH021 indd 222 CH021.indd 222 2/6/2008 2:36:01 PM 2/6/2008 2:36:01 PM  223 Number Processing  However the distinction is not that simple. Granà et al.  (2003) showed that there is an important difference between “ lexical ” and “ syntactic ” zeros. One patient made errors exclusively on this latter type. While lexical zeros are semantically derived zeros, like those in tens (e.g., the zero in “ 30, ” and the first zero in “ 20,104 ” ) that srcinate from a numerical concept, syntactic zeros are syntactically produced as the result of syntactic rules (e.g., the zeros in “ 13,004 ” that have just place value). Syntactic zeros are thus more dif-ficult to manipulate. One still debated point is whether or not number trans-coding and calculation necessarily imply semantic mediation via a central abstract semantic representation and whether asemantic, code-specific routes exist besides the semantic one (different models have been proposed on this subject: forreviews see Noël, 2001 , and Cipolotti & van Harskamp, 2001 ; examples are illustrated in Box 21.2).  21.4. CALCULATION Calculation needs more than just numbers. Its different components, shown to dissociate in neuropsychological cases, include arithmetical signs, arithmetical number facts and rules, calculation procedures, approximate calculation and conceptual knowledge. 21.4.1. Signs, Facts and Rules Some very rare patients were described who could not understand and name arithmetical signs (  ,   ,   , : and so on), but perform otherwise correct calculation according to their misidentification! Patients do also exist who can no longer tell the result of single-digit, over-learned addition (e.g., 4   2) or multiplica-tion problems (e.g., 5   6) listed in the so called arithmetical Box 21.2 Number processing models Figure below integrates McCloskey’s (1992) and Cipolotti and Butterworth’s (1995) number processing models. In McCloskey’s version, number transcoding happens exclusively via abstract, modality neutral, semantic representations that are used for all calculations; transcoding implies understanding. In Cipolotti and Butterworth’s version, transcoding may be performed via non-semantic direct associations: thus “3” in the Arabic format may directly become “three” in the alphabetic format and understanding the meaning of the number three may not be necessary to perform the task. Arithmetical operations are stored separately. Number size is represented as base 10 units.The triple-code model by Dehaene and Cohen (1995) (see second Figure) specifies the anatomical location of its components. Number size is represented in a logarithmically compressed form. There are no separate stores for arithmeti-cal operations. Transcoding and calculation may be performed without semantics. For instance, one digit multiplications may be retrieved directly as over-learned “facts.” Multi-digit opera-tions may be accomplished with the aid of visual images. Cipolotti, L., & Butterworth, B. (1995). Toward a multiroute model of number processing: Impaired number transcoding with preserved calculation skills.  Journal of Experimental Psychology: General , 124 (4), 375–390.Dehaene, S., & Cohen, L. (1995). Towards an anatomical and func-tional model of number processing.  Mathematical Cognition ,  1 , 83–120.McCloskey, M. (1992). Cognitive mechanisms in numerical process-ing: Evidence from acquired dyscalculia. Cognition ,  44 , 107–157. Analogue-magnitudeCode ComparisonApproximate calculationIntraparietal sulcus bilaterally Visual Arabic numbercode Parity judgementsMulti-digit operationsFusiform gyrus bilaterally Auditory verbalnumber code Addition and multiplicationtablesCountingLeft angular gyrus Perceptual SystemsAction Systems Calculation system arithmetic signs,arithmetic facts, procedures,conceptual knowledge Verbal numeralcomprehension (spoken and writtennumber words)Arabic numeralComprehensionVerbal numeralproduction(spoken and writtennumber words)Arabic numeralproduction Semantic system abstract numberrepresentationsTranscodingnon-semantic routesfor example: 3   threethree   3 /three/    3and so forth CH021 indd 223 CH021.indd 223 2/6/2008 2:36:02 PM 2/6/2008 2:36:02 PM
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