Language and number: a bilingual training study

The role of language in mathematical development: evidence from children with specific language impairments

A sample (n=48) of eight-year-olds with specific language impairments is compared with age-matched (n=55) and language matched controls (n=55) on a range of tasks designed to test the interdependence of language and mathematical development. Performance across tasks varies substantially in the SLI group, showing profound deficits in production of the count word sequence and basic calculation and significant deficits in understanding of the place-value principle in Hindu-Arabic notation. Only in understanding of arithmetic principles does SLI performance approximate that of age-matched-controls, indicating that principled understanding can develop even where number sequence production and other aspects of number processing are severely compromised.

Effect of Language Switching on Arithmetic: A Bilingual fMRI Study

The role of language in performing numerical computations has been a topic of special interest in cognition. The “Triple Code Model” proposes the existence of a language-dependent verbal code involved in retrieving arithmetic facts related to addition and multiplication, and a language-independent analog magnitude code subserving tasks such as number comparison and estimation. Neuroimaging studies have shown dissociation between dependence of arithmetic computations involving exact and approximate processing on language-related circuits. However, a direct manipulation of language using different arithmetic tasks is necessary to assess the role of language in forming arithmetic representations and in solving problems in different languages. In the present study, 20 English-Chinese bilinguals were trained in two unfamiliar arithmetic tasks in one language and scanned using fMRI on the same problems in both languages (English and Chinese). For the exact “base-7 addition” task, language switching effects were found in the left inferior frontal gyrus (LIFG) and left inferior parietal lobule extending to the angular gyrus. In the approximate “percentage estimation” task, language switching effects were found predominantly in the bilateral posterior intraparietal sulcus and LIFG, slightly dorsal to the LIFG activation seen for the base-7 addition task. These results considerably strengthen the notion that exact processing relies on verbal and language-related networks, whereas approximate processing engages parietal circuits typically involved in magnitude-related processing.

Ordinality and inferential abilities of a grey parrot (Psittacus erithacus)

A grey parrot (Psittacus erithacus), able to label the color of the bigger or smaller object in a pair (I. M. Pepperberg & M. V. Brezinsky, 1991), to vocally quantify < or =6 item sets (including heterogeneous subsets; I. M. Pepperberg, 1994), and separately trained to identify Arabic numerals 1-6 with the same vocal English labels but not to associate Arabic numbers with their relevant physical quantities, was shown pairs of Arabic numbers or an Arabic numeral and a set of objects and was asked for the color of the bigger or smaller one. The parrot's success showed he (a) understood number symbols as abstract representations of real-world collections, (b) inferred the relationship between the Arabic number and the quantity via stimulus equivalence, and (c) understood the ordinal relationship of his numbers.

Language-specific effects on number computation in toddlers

A fundamental question in developmental science is how brains with and without language compute numbers. Measuring young children's verbal reactions in France (Paris) and in England (Oxford), here we show that, although there is a general arithmetic ability for small numbers that is shared by monkeys and preverbal infants, the development of such initial knowledge in humans follows specific performance patterns, depending on what language the children speak.

Learning by strategies and learning by drill—evidence from an fMRI study

The present fMRI study investigates, first, whether learning new arithmetic operations is reflected by changing cerebral activation patterns, and second, whether different learning methods lead to differential modifications of brain activation. In a controlled design, subjects were trained over a week on two new complex arithmetic operations, one operation trained by the application of back-up strategies, i.e., a sequence of arithmetic operations, the other by drill, i.e., by learning the association between the operands and the result. In the following fMRI session, new untrained items, items trained by strategy and items trained by drill, were assessed using an event-related design. Untrained items as compared to trained showed large bilateral parietal activations, with the focus of activation along the right intraparietal sulcus. Further foci of activation were found in both inferior frontal gyri. The reverse contrast, trained vs. untrained, showed a more focused activation pattern with activation in both angular gyri. As suggested by the specific activation patterns, newly acquired expertise was implemented in previously existing networks of arithmetic processing and memory. Comparisons between drill and strategy conditions suggest that successful retrieval was associated with different brain activation patterns reflecting the underlying learning methods. While the drill condition more strongly activated medial parietal regions extending to the left angular gyrus, the strategy condition was associated to the activation of the precuneus which may be accounted for by visual imagery in memory retrieval.

Brain mappings of the arithmetic processing in children and adults

Despite the increasing number of experimental mapping showing that human arithmetic cognition is supported by widely spread neural circuits; the theoretical reasoning about these data remains mostly metaphorical and guided by a connectionist approach. Although neurons at distinct areas in the brain are assumed to take charge of different duties in the solution of the experimental task, the results are always discussed by hypothesizing some association between the different areas without questioning any difference of behavior at the level of the neurons at each of these areas. Here, the brain is assumed as Distributed Intelligent Processing System (DIPS) formed by collections of loosely interacting specialized agents (neurons), each agent specializing, for example, in data collection (sensors), problem solving (associative neurons), data communication (interneuronal systems) and in acting upon the surrounding environment (motorneurons). A new technique for EEG brain mapping is proposed and used to study arithmetic cognition in elementary school aged children and adults. Factor analysis showed three distinct patterns of neuronal recruitment for arithmetic calculations in all experimental groups which varied according to the type of calculation, age and sex.

Agrammatic but numerate

A central question in cognitive neuroscience concerns the extent to which language enables other higher cognitive functions. In the case of mathematics, the resources of the language faculty, both lexical and syntactic, have been claimed to be important for exact calculation, and some functional brain imaging studies have shown that calculation is associated with activation of a network of left-hemisphere language regions, such as the angular gyrus and the banks of the intraparietal sulcus. We investigate the integrity of mathematical calculations in three men with large left-hemisphere perisylvian lesions. Despite severe grammatical impairment and some difficulty in processing phonological and orthographic number words, all basic computational procedures were intact across patients. All three patients solved mathematical problems involving recursiveness and structure-dependent operations (for example, in generating solutions to bracket equations). To our knowledge, these results demonstrate for the first time the remarkable independence of mathematical calculations from language grammar in the mature cognitive system.

Number Comprehension by a Grey Parrot (Psittacus erithacus), Including a Zero-Like Concept

A Grey parrot (Psittacus erithacus) that was able to quantify 6 item sets (including subsets of heterogeneous groups, e.g., blue blocks within groupings of blue and green blocks and balls) using English labels (I. M. Pepperberg, 1994a) was tested on comprehension of these labels, which is crucial for numerical competence (K. C. Fuson, 1988; see also record 1987-98811-000). He was, without training, asked "What color/object [number]?" for collections of various simultaneously presented quantities (e.g., subsets of 4, 5, and 6 blocks of 3 different colors; subsets of 2, 4, and 6 keys, corks, and sticks). Accuracy was greater than 80% and was unaffected by array quantity, mass, or contour. His results demonstrated numerical comprehension competence comparable to that of chimpanzees and very young children. He also demonstrated knowledge of absence of quantity, using "none" to designate zero.

Beyond the flesh: some lessons from a mole cricket

What do linguistic symbols do for minds like ours, and how (if at all) can basic embodied, dynamical, and situated approaches do justice to high-level human thought and reason? These two questions are best addressed together, since our answers to the first may inform the second. The key move in scaling up simple embodied cognitive science is, I argue, to take very seriously the potent role of human-built structures in transforming the spaces of human learning and reason. In particular, in this article I look at a range of cases involving what I dub surrogate situations. Here, we actively create restricted artificial environments that allow us to deploy basic perception-action-reason routines in the absence of their proper objects. Examples include the use of real-world models, diagrams, and other concrete external symbols to support dense looping interactions with a variety of stable external structures that stand in for the absent states of affairs. Language itself, I finally suggest, is the most potent and fundamental form of such surrogacy. Words are both cheap stand-ins for gross behavioral outcomes, and the concrete objects that structure new spaces for basic forms of learning and reason. A good hard look at surrogate situatedness thus turns the standard skeptical challenge on its head. But it raises important questions concerning what really matters about these new approaches, and it helps focus what I see as the major challenge for the future: how, in detail, to conceptualize the role of symbols (both internal and external) in dynamical cognitive processes.

Exact and approximate arithmetic in an Amazonian indigene group

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic.

The semantics and acquisition of number words: integrating linguistic and developmental perspectives

This article brings together two independent lines of research on numerally quantified expressions, e.g. two girls. One stems from work in linguistic theory and asks what truth conditional contributions such expressions make to the utterances in which they are used--in other words, what do numerals mean? The other comes from the study of language development and asks when and how children learn the meaning of such expressions. My goal is to show that when integrated, these two perspectives can both constrain and enrich each other in ways hitherto not considered. Specifically, work in linguistic theory suggests that in addition to their 'exactly n' interpretation, numerally quantified NPs such as two hoops can also receive an 'at least n' and an 'at most n' interpretation, e.g. you need to put two hoops on the pole to win (i.e. at least two hoops) and you can miss two shots and still win (i.e. at most two shots). I demonstrate here through the results of three sets of experiments that by the age of 5 children have implicit knowledge of the fact that expressions like two N can be interpreted as 'at least two N' and 'at most two N' while they do not yet know the meaning of corresponding expressions such as at least/most two N which convey these senses explicitly. I show that these results have important implications for theories of the semantics of numerals and that they raise new questions for developmental accounts of the number vocabulary.

Six does not just mean a lot: preschoolers see number words as specific

This paper examines what children believe about unmapped number words - those number words whose exact meanings children have not yet learned. In Study 1, 31 children (ages 2-10 to 4-2) judged that the application of five and six changes when numerosity changes, although they did not know that equal sets must have the same number word. In Study 2, 15 children (ages 2-5 to 3-6) judged that six plus more is no longer six, but that a lot plus more is still a lot. Findings support the hypothesis that children treat number words as referring to specific, unique numerosities even before they know exactly which numerosity each word refers to.

Learning complex arithmetic—an fMRI study

Aim of the present functional magnet resonance imaging (fMRI) study was to detect modifications of cerebral activation patterns related to learning arithmetic. Thirteen right-handed subjects were extensively trained on a set of 18 complex multiplication problems. In the following fMRI session, trained and untrained problems (closely matched for difficulty) were presented in blocked order alternating with a number matching task and a fact retrieval task. Importantly, left hemispheric activations were dominant in the two contrasts between untrained and trained condition, suggesting that learning processes in arithmetic are predominantly supported by the left hemisphere. Contrasting untrained versus trained condition, the left intraparietal sulcus showed significant activations, as well as the inferior parietal lobule. A further significant activation was found in the left inferior frontal gyrus. This activation may be accounted for by higher working memory demands in the untrained as compared to the trained condition. Contrasting trained versus untrained condition a significant focus of activation was found in the left angular gyrus. Following the triple-code model [Science 284 (1999) 970], the shift of activation within the parietal lobe from the intraparietal sulcus to the left angular gyrus suggests a modification from quantity-based processing to more automatic retrieval. The present study shows that the left angular gyrus is not only involved in arithmetic tasks requiring simple fact retrieval, but may show significant activations as a result of relatively short training of complex calculation.

Approximate quantities and exact number words: dissociable systems

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann's syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.

The role of language in acquiring object kind concepts in infancy

Four experiments investigated whether 9-month-old infants could use the presence of labels to help them establish a representation of two distinct objects in a complex object individuation task. We found that the presence of two distinct labels facilitated object individuation, but the presence of one label for both objects, two distinct tones, two distinct sounds, or two distinct emotional expressions did not. These findings suggest that language may play an important role in the acquisition of sortal/object kind concepts in infancy: words may serve as "essence placeholders". Implications for the relationship between language and conceptual development are discussed.