Exact and approximate arithmetic in an Amazonian indigene group

Symbolic arithmetic knowledge without instruction

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations, and their performance suffers if this nonsymbolic system is impaired. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is required. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children's difficulties learning symbolic arithmetic, and they suggest ways to enhance children's engagement with formal mathematics.

Numbers within Our Hands: Modulation of Corticospinal Excitability of Hand Muscles during Numerical Judgment

Developmental and cross-cultural studies show that finger counting represents one of the basic number learning strategies. However, despite the ubiquity of such an embodied strategy, the issue of whether there is a neural link between numbers and fingers in adult, literate individuals remains debated. Here, we used transcranial magnetic stimulation to study changes of excitability of hand muscles of individuals performing a visual parity judgment task, a task not requiring counting, on Arabic numerals from 1 to 9. Although no modulation was observed for the left hand muscles, an increase in amplitude of motor-evoked potentials was found for the right hand muscles. This increase was specific for smaller numbers (1 to 4) as compared to larger numbers (6 to 9). These findings indicate a close relationship between hand/finger and numerical representations.

The equality of quantity

Do disparate dimensions of magnitude share an underlying mental representation? Two recent papers offer suggestive evidence that participants' discrimination thresholds are identical across domains. Brannon, Lutz and Cordes showed that six-month-old infants' area discriminations match their number discriminations. VanMarle and Wynn demonstrated the same pattern for six-month-olds' discrimination of temporal duration. These parallels across infants' responses to number, area and time raise questions about the fundamental nature of quantity processing.

Nonverbal representation of time and number in adults

A wealth of human and animal research supports common neural processing of numerical and temporal information. Here we test whether adult humans spontaneously encode number and time in a paradigm similar to those previously used to test the mode-control model in animals. Subjects were trained to classify visual stimuli that varied in both number and duration as few/short or many/long. Subsequently subjects were tested with novel stimuli that varied time and held number constant (eight flashes in 0.8-3.2s) or varied number and held time constant (4-16 flashes in 1.6s). Adult humans classified novel stimuli as many/long as monotonic functions of both number and duration, consistent with simultaneous, nonverbal, analog encoding. Numerical sensitivity, however, was finer than temporal sensitivity, suggesting differential salience of time and number. These results support the notion that adults simultaneously represent the number and duration of stimuli but suggest a possible asymmetry in their representations.

Nonverbal estimation during numerosity judgements by adult humans

On an automated task, humans selected the larger of two sets of items, each created through the one-by-one addition of items. Participants repeated the alphabet out loud during trials so that they could not count the items. This manipulation disrupted counting without producing major effects on other cognitive capacities such as memory or attention, and performance of this experimental group was poorer than that of participants who counted the items. In Experiment 2, the size of individual items was varied, and performance remained stable when the larger numerical set contained a smaller total amount than the smaller numerical set (i.e., participants used numerical rather than nonnumerical quantity cues in making judgements). In Experiment 3, reports of the number of items in a single set showed scalar variability as accuracy decreased, and variability in responses increased with increases in true set size. These data indicate a mechanism for the approximate representation of numerosity in adult humans that might be shared with nonhuman animals.

The influence of an external symbol system on number parity representation, or what's odd about 6?

The aim of this study was to investigate whether language-specific properties influence mental number processing. German Sign Language (DGS) numbers differ from those in spoken German not only in terms of modality but also in their basic language structure. A group of 20 congenitally deaf German signers participated in a number parity (odd/even) judgment task with DGS and printed German number words. The results indicated that two-handed DGS number signs are processed in a decomposed way. This language-specific effect also generalized to another linguistic number notation, German number words, but not to Arabic digit notation. These differences are discussed with respect to two possible routes to number parity.

Algebra in a man with severe aphasia

We report a dissociation between higher order mathematical ability and language in the case of a man (SO) with severe aphasia. Despite severely impaired abilities in the language domain and difficulties with processing both phonological and orthographic number words, he was able to judge the equivalence of and to transform and simplify mathematical expressions in algebraic notation. SO was sensitive to structure-dependent properties of algebraic expressions and displayed considerable capacity to retrieve algebraic facts, rules and principles, and to apply them to novel problems. He demonstrated similar capacity in solving expressions containing either solely numeric or abstract algebraic symbols (e.g., 8-(3-5)+3 versus b-(a-c)+a). The results show the retention of elementary algebra despite severe aphasia and provide evidence for the preservation of symbolic capacity in one modality and hence against the notion of aphasia as asymbolia.

One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles

Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.

An evolutionary perspective on learning disability in mathematics

A distinction between potentially evolved, or biologically-primary forms of cognition, and the culturally-specific, or biologically-secondary forms of cognition that are built from primary systems is used to explore mathematical learning disability (MLD). Using this model, MLD could result from deficits in the brain and cognitive systems that support biologically-primary mathematical competencies, or from the brain and cognitive systems that support the modification of primary systems for the creation of secondary knowledge and secondary cognitive competencies. The former include visuospatial long-term and working memory and the intraparietal sulcus, whereas the latter include the central executive component of working memory and the anterior cingulate cortex and lateral prefrontal cortex. Different forms of MLD are discussed as related to each of the cognitive and brain systems.

Core systems in human cognition

Research on human infants, adult nonhuman primates, and children and adults in diverse cultures provides converging evidence for four systems at the foundations of human knowledge. These systems are domain specific and serve to represent both entities in the perceptible world (inanimate manipulable objects and animate agents) and entities that are more abstract (numbers and geometrical forms). Human cognition may be based, as well, on a fifth system for representing social partners and for categorizing the social world into groups. Research on infants and children may contribute both to understanding of these systems and to attempts to overcome misconceptions that they may foster.

A Magnitude Code Common to Numerosities and Number Symbols in Human Intraparietal Cortex

Activation of the horizontal segment of the intraparietal sulcus (hIPS) has been observed in various number-processing tasks, whether numbers were conveyed by symbolic numerals (digits, number words) or by nonsymbolic displays (dot patterns). This suggests an abstract coding of numerical magnitude. Here, we critically tested this hypothesis using fMRI adaptation to demonstrate notation-independent coding of numerical quantity in the hIPS. Once subjects were adapted either to dot patterns or to Arabic digits, activation in the hIPS and in frontal regions recovered in a distance-dependent fashion whenever a new number was presented, irrespective of notation changes. This remained unchanged when analyzing the hIPS peaks from an independent localizer scan of mental calculation. These results suggest an abstract coding of approximate number common to dots, digits, and number words. They support the idea that symbols acquire meaning by linking neural populations coding symbol shapes to those holding nonsymbolic representations of quantities.

Core knowledge

Human cognition is founded, in part, on four systems for representing objects, actions, number, and space. It may be based, as well, on a fifth system for representing social partners. Each system has deep roots in human phylogeny and ontogeny, and it guides and shapes the mental lives of adults. Converging research on human infants, non-human primates, children and adults in diverse cultures can aid both understanding of these systems and attempts to overcome their limits.

Cognitive cladistics and cultural override in Hominid spatial cognition

Current approaches to human cognition often take a strong nativist stance based on Western adult performance, backed up where possible by neonate and infant research and almost never by comparative research across the Hominidae. Recent research suggests considerable cross-cultural differences in cognitive strategies, including relational thinking, a domain where infant research is impossible because of lack of cognitive maturation. Here, we apply the same paradigm across children and adults of different cultures and across all nonhuman great ape genera. We find that both child and adult spatial cognition systematically varies with language and culture but that, nevertheless, there is a clear inherited bias for one spatial strategy in the great apes. It is reasonable to conclude, we argue, that language and culture mask the native tendencies in our species. This cladistic approach suggests that the correct perspective on human cognition is neither nativist uniformitarian nor "blank slate" but recognizes the powerful impact that language and culture can have on our shared primate cognitive biases.

The role of the human ventral premotor cortex in counting successive stimuli

Adult humans have the ability to count large numbers of successive stimuli exactly. What brain areas underlie this uniquely human process? To identify the candidate brain areas, we first used functional magnetic resonance imaging, and found that the upper part of the left ventral premotor cortex was preferentially activated during counting of successive sensory stimuli presented 10-22 times, while the area was not activated during small number counting up to 4. We then used transcranial magnetic stimulation to assess the necessity of this area, and found that stimulation of this area preferentially disrupted subjects' exact large number enumeration. Stimulation to the area affected neither subjects' number word perception nor their ability to perform a non-numerical sequential letter task. While further investigation is necessary to determine the precise role of the left ventral premotor cortex, the results suggest that the area is indispensably involved for large number counting of successive stimuli, at least for the types of tasks in this study.

Neural correlates of arithmetic and language comprehension: a common substrate?

There is debate as to the relationship between mathematical ability and language. Some research has suggested that common processes underlie arithmetic and grammar while other research has suggested that these are distinct processes. The current study aimed to address this issue in a large group of 68 left hemisphere stroke patients who were all tested on analogous arithmetic and language comprehension measures. The behavioral data revealed a significant correlation between performance on the comprehension and arithmetic measures, although a subset of patients showed a dissociation in performance on the two tasks. To determine the brain regions critical for performance on each measure, patients' lesions were analyzed using Voxel-based Lesion Symptom Mapping. Arithmetic was associated with a small number of foci, with the most significant region located in the left inferior parietal lobule (Brodmann areas 39 and 40). Comprehension was associated with a larger number of brain regions, most extensively in the left middle and superior temporal gyri. There was also overlap between the arithmetic and comprehension maps in a number of regions, such as the inferior frontal gyrus. Our findings suggest that arithmetic and language comprehension are mediated by partially overlapping brain networks. These findings are discussed in light of previous work on the neural basis of arithmetic ability and its relationship to language.

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered. The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated. In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Weber's Law signature, and was independent of conventional number knowledge.

Temporal and Spatial Enumeration Processes in the Primate Parietal Cortex

Humans and animals can nonverbally enumerate visual items across time in a sequence or rapidly estimate the set size of spatial dot patterns at a single glance. We found that temporal and spatial enumeration processes engaged different populations of neurons in the intraparietal sulcus of behaving monkeys. Once the enumeration process was completed, however, another neuronal population represented the cardinality of a set irrespective of whether it had been cued in a spatial layout or across time. These data suggest distinct neural processing stages for different numerical formats, but also a final convergence of the segregated information to form most abstract quantity representations.

A common representational system governed by Weber's law: nonverbal numerical similarity judgments in 6-year-olds and rhesus macaques

This study compared nonverbal numerical processing in 6-year-olds with that in nonhuman animals using a numerical bisection task. In the study, 16 children were trained on a delayed match-to-sample paradigm to match exemplars of two anchor numerosities. Children were then required to indicate whether a sample intermediate to the anchor values was closer to the small anchor value or the large anchor value. For two sets of anchor values with the same ratio, the probability of choosing the larger anchor value increased systematically with sample number, and the psychometric functions superimposed when plotted on a logarithmic scale. The psychometric functions produced by the children also superimposed with the psychometric functions produced by rhesus monkeys in an analogous previous experiment. These examples of superimposition demonstrate that nonverbal number representations, even in children who have acquired the verbal counting system, are modulated by Weber's law.

Shared system for ordering small and large numbers in monkeys and humans

There is increasing evidence that animals share with adult humans and perhaps human infants a system for representing objective number as psychological magnitudes that are an analogue of the quantities they represent. Here we show that rhesus monkeys can extend a numerical rule learned with the values 1 through 9 to the values 10, 15, 20, and 30, which suggests that there is no upper limit on a monkey's numerical capacity. Instead, throughout the numerical range tested, both accuracy and latency in ordering two numerical values were systematically controlled by the ratio of the values compared. In a second experiment, we directly compared humans' and monkeys' performance in the same ordinal comparison task. The qualitative and quantitative similarity in their performance provides the strongest evidence to date of a single nonverbal, evolutionarily primitive mechanism for representing and comparing numerical values.

An open trial assessment of "The Number Race", an adaptive computer game for remediation of dyscalculia

Background

In a companion article [1], we described the development and evaluation of software designed to remediate dyscalculia. This software is based on the hypothesis that dyscalculia is due to a "core deficit" in number sense or in its access via symbolic information. Here we review the evidence for this hypothesis, and present results from an initial open-trial test of the software in a sample of nine 7–9 year old children with mathematical difficulties.

Methods

Children completed adaptive training on numerical comparison for half an hour a day, four days a week over a period of five-weeks. They were tested before and after intervention on their performance in core numerical tasks: counting, transcoding, base-10 comprehension, enumeration, addition, subtraction, and symbolic and non-symbolic numerical comparison.

Results

Children showed specific increases in performance on core number sense tasks. Speed of subitizing and numerical comparison increased by several hundred msec. Subtraction accuracy increased by an average of 23%. Performance on addition and base-10 comprehension tasks did not improve over the period of the study.

Conclusion

Initial open-trial testing showed promising results, and suggested that the software was successful in increasing number sense over the short period of the study. However these results need to be followed up with larger, controlled studies. The issues of transfer to higher-level tasks, and of the best developmental time window for intervention also need to be addressed.

Principles underlying the design of "The Number Race", an adaptive computer game for remediation of dyscalculia

Background

Adaptive game software has been successful in remediation of dyslexia. Here we describe the cognitive and algorithmic principles underlying the development of similar software for dyscalculia. Our software is based on current understanding of the cerebral representation of number and the hypotheses that dyscalculia is due to a "core deficit" in number sense or in the link between number sense and symbolic number representations.

Methods

"The Number Race" software trains children on an entertaining numerical comparison task, by presenting problems adapted to the performance level of the individual child. We report full mathematical specifications of the algorithm used, which relies on an internal model of the child's knowledge in a multidimensional "learning space" consisting of three difficulty dimensions: numerical distance, response deadline, and conceptual complexity (from non-symbolic numerosity processing to increasingly complex symbolic operations).

Results

The performance of the software was evaluated both by mathematical simulations and by five weeks of use by nine children with mathematical learning difficulties. The results indicate that the software adapts well to varying levels of initial knowledge and learning speeds. Feedback from children, parents and teachers was positive. A companion article [1] describes the evolution of number sense and arithmetic scores before and after training.

Conclusion

The software, open-source and freely available online, is designed for learning disabled children aged 5–8, and may also be useful for general instruction of normal preschool children. The learning algorithm reported is highly general, and may be applied in other domains.

Functional imaging of numerical processing in adults and 4-y-old children

Adult humans, infants, pre-school children, and non-human animals appear to share a system of approximate numerical processing for non-symbolic stimuli such as arrays of dots or sequences of tones. Behavioral studies of adult humans implicate a link between these non-symbolic numerical abilities and symbolic numerical processing (e.g., similar distance effects in accuracy and reaction-time for arrays of dots and Arabic numerals). However, neuroimaging studies have remained inconclusive on the neural basis of this link. The intraparietal sulcus (IPS) is known to respond selectively to symbolic numerical stimuli such as Arabic numerals. Recent studies, however, have arrived at conflicting conclusions regarding the role of the IPS in processing non-symbolic, numerosity arrays in adulthood, and very little is known about the brain basis of numerical processing early in development. Addressing the question of whether there is an early-developing neural basis for abstract numerical processing is essential for understanding the cognitive origins of our uniquely human capacity for math and science. Using functional magnetic resonance imaging (fMRI) at 4-Tesla and an event-related fMRI adaptation paradigm, we found that adults showed a greater IPS response to visual arrays that deviated from standard stimuli in their number of elements, than to stimuli that deviated in local element shape. These results support previous claims that there is a neurophysiological link between non-symbolic and symbolic numerical processing in adulthood. In parallel, we tested 4-y-old children with the same fMRI adaptation paradigm as adults to determine whether the neural locus of non-symbolic numerical activity in adults shows continuity in function over development. We found that the IPS responded to numerical deviants similarly in 4-y-old children and adults. To our knowledge, this is the first evidence that the neural locus of adult numerical cognition takes form early in development, prior to sophisticated symbolic numerical experience. More broadly, this is also, to our knowledge, the first cognitive fMRI study to test healthy children as young as 4 y, providing new insights into the neurophysiology of human cognitive development.

This functional imaging study provides evidence for a neurobiological link between early non-symbolic numerical abilities of 4 year-old children and the more symbolic numerical processing of adults.

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.

Linguistic Relativity: Does Language Help or Hinder Perception?

A recent study has explored the interplay between language and perception through the brain's visual pathways. The results suggest an influence of linguistic categories on the speed of colour discrimination.

The representation of numerical magnitude

The combined efforts of many fields are advancing our understanding of how number is represented. Researchers studying numerical reasoning in adult humans, developing humans and non-human animals are using a suite of behavioral and neurobiological methods to uncover similarities and differences in how each population enumerates and compares quantities to identify the neural substrates of numerical cognition. An important picture emerging from this research is that adult humans share with non-human animals a system for representing number as language-independent mental magnitudes and that this system emerges early in development.

Weber's Law influences numerical representations in rhesus macaques (Macaca mulatta)

We present the results of two experiments that probe the ability of rhesus macaques to match visual arrays based on number. Three monkeys were first trained on a delayed match-to-sample paradigm (DMTS) to match stimuli on the basis of number and ignore continuous dimensions such as element size, cumulative surface area, and density. Monkeys were then tested in a numerical bisection experiment that required them to indicate whether a sample numerosity was closer to a small or large anchor value. Results indicated that, for two sets of anchor values with the same ratio, the probability of choosing the larger anchor value systematically increased with the sample number and the psychometric functions superimposed. A second experiment employed a numerical DMTS task in which the choice values contained an exact numerical match to the sample and a distracter that varied in number. Both accuracy and reaction time were modulated by the ratio between the correct numerical match and the distracter, as predicted by Weber's Law.

Sex differences in intrinsic aptitude for mathematics and science?: a critical review

This article considers 3 claims that cognitive sex differences account for the differential representation of men and women in high-level careers in mathematics and science: (a) males are more focused on objects from the beginning of life and therefore are predisposed to better learning about mechanical systems; (b) males have a profile of spatial and numerical abilities producing greater aptitude for mathematics; and (c) males are more variable in their cognitive abilities and therefore predominate at the upper reaches of mathematical talent. Research on cognitive development in human infants, preschool children, and students at all levels fails to support these claims. Instead, it provides evidence that mathematical and scientific reasoning develop from a set of biologically based cognitive capacities that males and females share. These capacities lead men and women to develop equal talent for mathematics and science.

Core knowledge of geometry in an Amazonian indigene group

Does geometry constitute a core set of intuitions present in all humans, regardless of their language or schooling? We used two nonverbal tests to probe the conceptual primitives of geometry in the Mundurukú, an isolated Amazonian indigene group. Mundurukú children and adults spontaneously made use of basic geometric concepts such as points, lines, parallelism, or right angles to detect intruders in simple pictures, and they used distance, angle, and sense relationships in geometrical maps to locate hidden objects. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

Giving the boot to the bootstrap: how not to learn the natural numbers

According to one theory about how children learn the concept of natural numbers, they first determine that "one", "two", and "three" denote the size of sets containing the relevant number of items. They then make the following inductive inference (the Bootstrap): The next number word in the counting series denotes the size of the sets you get by adding one more object to the sets denoted by the previous number word. For example, if "three" refers to the size of sets containing three items, then "four" (the next word after "three") must refer to the size of sets containing three plus one items. We argue, however, that the Bootstrap cannot pick out the natural number sequence from other nonequivalent sequences and thus cannot convey to children the concept of the natural numbers. This is not just a result of the usual difficulties with induction but is specific to the Bootstrap. In order to work properly, the Bootstrap must somehow restrict the concept of "next number" in a way that conforms to the structure of the natural numbers. But with these restrictions, the Bootstrap is unnecessary.

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.

Preschool children's mathematical knowledge: The effect of teacher "math talk."

This study examined the relation between the amount of mathematical input in the speech of preschool or day-care teachers and the growth of children's conventional mathematical knowledge over the school year. Three main findings emerged. First, there were marked individual differences in children's conventional mathematical knowledge by 4 years of age that were associated with socioeconomic status. Second, there were dramatic differences in the amount of math-related talk teachers provided. Third, and most important, the amount of teachers' math-related talk was significantly related to the growth of preschoolers' conventional mathematical knowledge over the school year but was unrelated to their math knowledge at the start of the school year.

Semantic congruity affects numerical judgments similarly in monkeys and humans

Monkeys (Macaca mulatta) were trained to order visual arrays based on their number of elements and to conditionally choose the array with the larger or smaller number of elements dependent on a color cue. When the screen background was red, monkeys were reinforced for choosing the smaller numerical value first. When the screen background was blue, monkeys were reinforced for choosing the larger numerical value first. Monkeys showed a semantic congruity effect analogous to that reported for human comparison judgments. Specifically, decision time was systematically influenced by the semantic congruity between the cue ("choose smaller" or "choose larger") and the magnitude of the choice stimuli (small or large numbers of dots). This finding demonstrates a semantic congruity effect in a nonlinguistic animal and provides strong evidence for an evolutionarily primitive magnitude-comparison algorithm common to humans and monkeys.

Monkeys match the number of voices they hear to the number of faces they see

Convergent evidence demonstrates that adult humans possess numerical representations that are independent of language [1, 2, 3, 4, 5 and 6]. Human infants and nonhuman animals can also make purely numerical discriminations, implicating both developmental and evolutionary bases for adult humans' language-independent representations of number [7 and 8]. Recent evidence suggests that the nonverbal representations of number held by human adults are not constrained by the sensory modality in which they were perceived [9]. Previous studies, however, have yielded conflicting results concerning whether the number representations held by nonhuman animals and human infants are tied to the modality in which they were established [10, 11, 12, 13, 14 and 15]. Here, we report that untrained monkeys preferentially looked at a dynamic video display depicting the number of conspecifics that matched the number of vocalizations they heard. These findings suggest that number representations held by monkeys, like those held by adult humans, are unfettered by stimulus modality.

Abstract number and arithmetic in preschool children

Educated humans use language to express abstract number, applying the same number words to seven apples, whistles, or sins. Is language or education the source of numerical abstraction? Claims to the contrary must present evidence for numerical knowledge that applies to disparate entities, in people who have received no formal mathematics instruction and cannot express such knowledge in words. Here we show that preschool children can compare and add large sets of elements without counting, both within a single visual-spatial modality (arrays of dots) and across two modalities and formats (dot arrays and tone sequences). In two experiments, children viewed animations and either compared one visible array of dots to a second array or added two successive dot arrays and compared the sum to a third array. In further experiments, a dot array was replaced by a sequence of sounds, so that participants had to integrate quantity information presented aurally and visually. Children performed all tasks successfully, without resorting to guessing strategies or responding to continuous variables. Their accuracy varied with the ratio of the two quantities: a signature of large, approximate number representations in adult humans and animals. Addition was as accurate as comparison, even though children showed no relevant knowledge when presented with symbolic versions of the addition tasks. Abstract knowledge of number and addition therefore precedes, and may guide, language-based instruction in mathematics.

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.

Our chimpanzee mind

Some might consider the title of this piece preposterous. Bishop Wilberforce would no doubt have shaken his fist at it, just as he disputed Huxley's championing of darwinian continuity. But the title of this essay is no more outrageous than one entitled 'The chimpanzee's bird brain', for there has been extensive evolutionary conservation of many neural and psychological functions across species. We share with chimpanzees some--but not all--mental functions, some of which are shared with other species as well. As the publication of the chimpanzee genome reveals, we also share a good deal of our DNA. Unfortunately, we are virtually in the dark when it comes to understanding how genes build minds. If comparative genomics is to enlighten our understanding of human origins, it must be accompanied by an equally rich description of animal psychology, both in terms of its underlying neural signatures and the evolutionary processes that led to convergence and divergence with other species.

Preschool children master the logic of number word meanings

Although children take over a year to learn the meanings of the first three number words, they eventually master the logic of counting and the meanings of all the words in their count list. Here, we ask whether children's knowledge applies to number words beyond those they have mastered: Does a child who can only count to 20 infer that number words above 'twenty' refer to exact cardinal values? Three experiments provide evidence for this understanding in preschool children. Before beginning formal education or gaining counting skill, children possess a productive symbolic system for representing number.

Making sense of number sense: implications for children with mathematical disabilities

Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of "number sense." I expand on their preliminary effort by examining in detail the diverse set of components purported to be encompassed by this construct. My analysis reveals some major differences between the ways in which number sense is defined in the mathematical cognition literature and its definition in the literature in mathematics education. I also present recent empirical evidence and theoretical perspectives bearing on the importance of measuring the speed of making magnitude comparisons. Finally, I discuss how differing conceptions of number sense inform the issue of whether and to what extent it may be teachable.

Number and language: how are they related?

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 numerical concepts. We challenge the 'bootstrapping' theory which claims that children move from using something like an object-file - 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.

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.

Counting on neurons: the neurobiology of numerical competence

Numbers are an integral part of our everyday life - we use them to quantify, rank and identify objects. The verbal number concept allows humans to develop superior mathematical and logic skills that define technologically advanced cultures. However, basic numerical competence is rooted in biological primitives that can be explored in animals, infants and human adults alike. We are now beginning to unravel its anatomical basis and neuronal mechanisms on many levels, down to its single neuron correlate. Neural representations of numerical information can engage extensive cerebral networks, but the posterior parietal cortex and the prefrontal cortex are the key structures in primates.

Psychological Universals: What Are They and How Can We Know?

Psychological universals, or core mental attributes shared by humans everywhere, are a foundational postulate of psychology, yet explicit analysis of how to identify such universals is lacking. This article offers a conceptual and methodological framework to guide the investigation of genuine universals through empirical analysis of psychological patterns across cultures. Issues of cross-cultural generalizability of psychological processes and 3 cross-cultural research strategies to probe universals are considered. Four distinct levels of hierarchically organized universals are possible: From strongest to weakest claims for universality, they are accessibility universals, functional universals, existential universals, and nonuniversals. Finally, universals are examined in relation to the questions of levels of analysis, evolutionary explanations of psychological processes, and management of cross-cultural relations.

Language and the origin of numerical concepts

Reports of research with the Pirahã and Mundurukú Amazonian Indians of Brazil lend themselves to discussions of the role of language in the origin of numerical concepts. The research findings indicate that, whether or not humans have an extensive counting list, they share with nonverbal animals a language-independent representation of number, with limited, scale-invariant precision. What causal role, then, does knowledge of the language of counting serve? We consider the strong Whorfian proposal, that of linguistic determinism; the weak Whorfian hypothesis, that language influences how we think; and that the "language of thought" maps to spoken language or symbol systems.