Sources of mathematical thinking: behavioral and brain-imaging evidence

Does the human capacity for mathematical intuition depend on linguistic competence or on visuo-spatial representations? A series of behavioral and brain-imaging experiments provides evidence for both sources. Exact arithmetic is acquired in a language-specific format, transfers poorly to a different language or to novel facts, and recruits networks involved in word-association processes. In contrast, approximate arithmetic shows language independence, relies on a sense of numerical magnitudes, and recruits bilateral areas of the parietal lobes involved in visuo-spatial processing. Mathematical intuition may emerge from the interplay of these brain systems.

Modularity and development: the case of spatial reorientation

In a series of experiments, young children who were disoriented in a novel environment reoriented themselves in accord with the large-scale shape of the environment but not in accord with nongeometric properties of the environment such as the color of a wall, the patterning on a box, or the categorical identity of an object. Because children's failure to reorient by nongeometric information cannot be attributed to limits on their ability to detect, remember, or use that information for other purposes, this failure suggests that children's reorientation, at least in relatively novel environments, depends on a mechanism that is informationally encapsulated and task-specific: two hallmarks of modular cognitive processes. Parallel studies with rats suggest that children share this mechanism with at least some adult nonhuman mammals. In contrast, our own studies of human adults, who readily solved our tasks by conjoining nongeometric and geometric information, indicated that the most striking limitations of this mechanism are overcome during human development. These findings support broader proposals concerning the domain specificity of humans' core cognitive abilities, the conservation of cognitive abilities across related species and over the course of human development, and the developmental processes by which core abilities are extended to permit more flexible, uniquely human kinds of problem solving.

Human spatial representation: insights from animals

HUMAN NAVIGATION IS SPECIAL: we use geographic maps to capture a world far beyond our unaided locomotion. In consequence, human navigation is widely thought to depend on internalized versions of these maps - enduring, geocentric 'cognitive maps' capturing diverse information about the environment. Contrary to this view, we argue that human navigation is best studied in relation to research on navigating animals as humble as ants. This research provides evidence that animals, including humans, navigate primarily by representations that are momentary rather than enduring, egocentric rather than geocentric, and limited in the environmental information that they capture. Uniquely human forms of navigation build on these representations.

Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.

The construction of large number representations in adults

What is the nature of our mental representation of quantity? We find that human adults show no performance cost of comparing numerosities across vs. within visual and auditory stimulus sets, or across vs. within simultaneous and sequential sets. In addition, reaction time and performance in such tasks are determined by the ratio of the numerosities to be compared; absolute set size has no effect. These findings suggest that modality-specific stimulus properties undergo a non-iterative transformation into representations of quantity that are independent of the modality or format of the stimulus.

Non-symbolic arithmetic in adults and young children

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.

The double-edged sword of pedagogy: Instruction limits spontaneous exploration and discovery

Motivated by computational analyses, we look at how teaching affects exploration and discovery. In Experiment 1, we investigated children's exploratory play after an adult pedagogically demonstrated a function of a toy, after an interrupted pedagogical demonstration, after a naïve adult demonstrated the function, and at baseline. Preschoolers in the pedagogical condition focused almost exclusively on the target function; by contrast, children in the other conditions explored broadly. In Experiment 2, we show that children restrict their exploration both after direct instruction to themselves and after overhearing direct instruction given to another child; they do not show this constraint after observing direct instruction given to an adult or after observing a non-pedagogical intentional action. We discuss these findings as the result of rational inductive biases. In pedagogical contexts, a teacher's failure to provide evidence for additional functions provides evidence for their absence; such contexts generalize from child to child (because children are likely to have comparable states of knowledge) but not from adult to child. Thus, pedagogy promotes efficient learning but at a cost: children are less likely to perform potentially irrelevant actions but also less likely to discover novel information.

Initial knowledge: six suggestions

Although debates continue, studies of cognition in infancy suggest that knowledge begins to emerge early in life and constitutes part of humans' innate endowment. Early-developing knowledge appears to be both domain-specific and task-specific, it appears to capture fundamental constraints on ecologically important classes of entities in the child's environment, and it appears to remain central to the common-sense knowledge systems of adults.

Infants' discrimination of number vs. continuous extent

Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate with these properties. Experiment 1 extended the standard habituation/dishabituation paradigm to a 1 vs 2 comparison with three-dimensional objects and confirmed that when number and total front surface area are confounded, infants discriminate the arrays. Experiment 2 revealed that infants dishabituated to a change in front surface area but not to a change in number when the two variables were pitted against each other. Experiments 3 through 5 revealed no sensitivity to number when front surface area was controlled, and Experiments 6 and 7 extended this pattern of findings to the Wynn (1992) transformation task. Infants' lack of a response to number, combined with their demonstrated sensitivity to one or more dimensions of continuous extent, supports the hypothesis that the representations subserving object-based attention, rather than those subserving enumeration, underlie performance in the above tasks.

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.

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.

Spatial knowledge and geometric representation in a child blind from birth

A series of experiments demonstrated that a congenitally blind 2 1/2-year-old child-as well as sighted but blindfolded children and adults-can determine the appropriate path between two objects after traveling to each of those objects from a third object. This task requires that the child detect the distances and the angular relationship of the familiar paths and that she derive therefrom the angle of the new path. Our research indicates that the locomotion of the young blind child is guided by knowledge of the Euclidean properties of a spatial layout and by principles for making inferences based on those properties.

Beyond core knowledge: Natural geometry

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems.

Exact Equality and Successor Function: Two Key Concepts on the Path towards understanding Exact Numbers

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of exact numbers: the fact that all numbers can be generated by a successor function, and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Mundurucu (an Amazonian language), and young western children (3-4 years old) understand these fundamental properties of numbers.

When is four far more than three? Children's generalization of newly acquired number words

What is the relationship between children's first number words and number concepts? We used training tasks to explore children's interpretation of number words as they acquired the words' meanings. Children who had mastered the meanings of only the first two or three number words were systematically provided with varied input on the next word-to-quantity mapping, and their extension of the newly trained word was assessed across a variety of test items. Children who had already mastered number words to three generalized training on four to new objects and nouns, such that their representation of the newly learned number was approximate. In contrast, children who had mastered only one and two learned to apply three reliably within a single count-noun context (e.g., three dogs), but did not generalize training to new objects labeled with different nouns (e.g., three cows). Both findings suggest that children fail to map newly learned words in their counting routine to the fully abstract concepts of natural numbers.

Reaching and grasping a moving object in 6-, 8-, and 10-month-old infants: laterality and performance

The goal of this study was to investigate some of the visuo-motor factors underlying an infant's developing ability to grasp a laterally-moving object. In particular, hand preference, midline crossing, and visual-field asymmetry were investigated by comparing performance as a function of the object's direction of motion. We presented 6-, 8-, and 10-month-old infants with a graspable object, moving in a circular trajectory in the horizontal plane. Six-month-old infants reached for the object with the ipsilateral hand and grasped it with the contralateral hand. Eight-month-old infants showed a strong right-hand bias for both reaching and grasping. Ten-month-old infants showed a greater diversity of strategy use including bimanual and successful ipsilateral grasping following ipsilateral reaching in both directions of motion. Thus, motor constraints due to spatial compatibility, hand preference and bimanual coordination (but not midline crossing) must be taken into account to understand age differences in grasping a moving object.

Children's multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with nonhuman primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports children's predictions about the outcomes of halving and perhaps also doubling transformations. A total of 138 kindergartners and first graders were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number.

Will any doll do? 12-month-olds’ reasoning about goal objects

Infants as young as 5 months of age view familiar actions such as reaching as goal-directed (Woodward, 1998), but how do they construe the goal of an actor's reach? Six experiments investigated whether 12-month-old infants represent reaching actions as directed to a particular individual object, to a narrowly defined object category (e.g., an orange dump truck), or to a more broadly defined object category (e.g., any truck, vehicle, artifact, or inanimate object). The experiments provide evidence that infants are predisposed to represent reaching actions as directed to categories of objects at least as broad as the basic level, both when the objects represent artifacts (trucks) and when they represent people (dolls). Infants do not use either narrower category information or spatiotemporal information to specify goal objects. Because spatiotemporal information is central to infants' representations of inanimate object motions and interactions, the findings are discussed in relation to the development of object knowledge and action representations.

What exactly do numbers mean?

Number words are generally used to refer to the exact cardinal value of a set, but cognitive scientists disagree about their meanings. Although most psychological analyses presuppose that numbers have exact semantics (two means EXACTLY TWO), many linguistic accounts propose that numbers have lower-bounded semantics (AT LEAST TWO), and that speakers restrict their reference through a pragmatic inference (scalar implicature). We address this debate through studies of children who are in the process of acquiring the meanings of numbers. Adults and 2- and 3-year-olds were tested in a novel paradigm that teases apart semantic and pragmatic aspects of interpretation (the covered box task). Our findings establish that when scalar implicatures are cancelled in the critical trials of this task, both adults and children consistently give exact interpretations for number words. These results, in concert with recent work on real-time processing, provide the first unambiguous evidence that number words have exact semantics.

Early concepts of intimacy: Young humans use saliva sharing to infer close relationships

Across human societies, people form "thick" relationships characterized by strong attachments, obligations, and mutual responsiveness. People in thick relationships share food utensils, kiss, or engage in other distinctive interactions that involve sharing saliva. We found that children, toddlers, and infants infer that dyads who share saliva (as opposed to other positive social interactions) have a distinct relationship. Children expect saliva sharing to happen in nuclear families. Toddlers and infants expect that people who share saliva will respond to one another in distress. Parents confirm that saliva sharing is a valid cue of relationship thickness in their children's social environments. The ability to use distinctive interactions to infer categories of relationships thus emerges early in life, without explicit teaching; this enables young humans to rapidly identify close relationships, both within and beyond families.