Generating a lexicon without a language model: Do words for number count?

Response to commentaries on What Babies Know

Twenty-five commentaries raise questions concerning the origins of knowledge, the interplay of iconic and propositional representations in mental life, the architecture of numerical and social cognition, the sources of uniquely human cognitive capacities, and the borders among core knowledge, perception, and thought. They also propose new methods, drawn from the vibrant, interdisciplinary cognitive sciences, for addressing these questions and deepening understanding of infant minds.

Number Stroop Effects in Arabic Digits and ASL Number Signs: The Impact of Age and Setting of Language Acquisition

Multiple studies have reported mathematics underachievement for students who are deaf, but the onset, scope, and causes of this phenomenon remain understudied. Early language deprivation might be one factor influencing the acquisition of numbers. In this study, we investigated a basic and fundamental mathematical skill, automatic magnitude processing, in two formats (Arabic digits and American Sign Language number signs) and the influence of age of first language exposure on both formats by using two versions of the Number Stroop Test. We compared the performance of individuals born deaf who experienced early language deprivation to that of individuals born deaf who experienced sign language in early life and hearing second language learners of ASL. In both formats of magnitude representation, late first language learners demonstrated overall slower reaction times. They were also less accurate on incongruent trials but performed no differently from early signers and second language learners on other trials. When magnitude was represented by Arabic digits, late first language learners exhibited robust Number Stroop Effects, suggesting automatic magnitude processing, but they also demonstrated a large speed difference between size and number judgments not observed in the other groups. In a task with ASL number signs, the Number Stroop Effect was not found in any group, suggesting that magnitude representation might be format-specific, in line with the results from several other languages. Late first language learners also demonstrate unusual patterns of slower reaction time for neutral rather than incongruent stimuli. Together, the results show that early language deprivation affects the ability to automatically judge quantities expressed both linguistically and by Arabic digits, but that it can be acquired later in life when language is available. Contrary to previous studies that find differences in speed of number processing between deaf and hearing participants, we find that when language is acquired early in life, deaf signers perform identically to hearing participants.

Emergent Morphology in Child Homesign: Evidence from Number Language

Human languages, signed and spoken, can be characterized by the structural patterns they use to associate communicative forms with meanings. One such pattern is paradigmatic morphology, where complex words are built from the systematic use and re-use of sub-lexical units. Here, we provide evidence of emergent paradigmatic morphology akin to number inflection in a communication system developed without input from a conventional language, homesign. We study the communication systems of four deaf child homesigners (mean age 8;02). Although these idiosyncratic systems vary from one another, we nevertheless find that all four children use handshape and movement devices productively to express cardinal and non-cardinal number information, and that their number expressions are consistent in both form and meaning. Our study shows, for the first time, that all four homesigners not only incorporate number devices into representational devices used as predicates , but also into gestures functioning as nominals, including deictic gestures. In other words, the homesigners express number by systematically combining and re-combining additive markers for number (qua inflectional morphemes) with representational and deictic gestures (qua bases). The creation of new, complex forms with predictable meanings across gesture types and linguistic functions constitutes evidence for an inflectional morphological paradigm in homesign and expands our understanding of the structural patterns of language that are, and are not, dependent on linguistic input.

Unraveling the contribution of left-right language on spatial perspective taking

We examine whether acquiring left/right language affects children's ability to take a non-egocentric left-right perspective. In Experiment 1, we tested 10-13 year-old Tseltal (Mayan) and Spanish-speaking children from the same community on a task that required they retrieve a coin they previously seen hidden in one of four boxes to the left/right/front/back of a toy sheep after the entire array was rotated out of view. Their performance on the left/right boxes correlated positively with their comprehension and use of left-right language. In Experiment 2, we found that training Tseltal-speaking children to apply left-right lexical labels to represent the location of the coin improved performance, but improvement was more robust among a second group of children trained to use gestures instead.

Language-specific numerical estimation in bilingual children

We tested 5- to 7-year-old bilingual learners of French and English (N = 91) to investigate how language-specific knowledge of verbal numerals affects numerical estimation. Participants made verbal estimates for rapidly presented random dot arrays in each of their two languages. Estimation accuracy differed across children's two languages, an effect that remained when controlling for children's familiarity with number words across their two languages. In addition, children's estimates were equivalently well ordered in their two languages, suggesting that differences in accuracy were due to how children represented the relative distance between number words in each language. Overall, these results suggest that bilingual children have different mappings between their verbal and nonverbal counting systems across their two languages and that those differences in mappings are likely driven by an asymmetry in their knowledge of the structure of the count list across their languages. Implications for bilingual math education are discussed.

Ontogenetic Origins of Human Integer Representations

Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.

Number gestures predict learning of number words

When asked to explain their solutions to a problem, children often gesture and, at times, these gestures convey information that is different from the information conveyed in speech. Children who produce these gesture-speech "mismatches" on a particular task have been found to profit from instruction on that task. We have recently found that some children produce gesture-speech mismatches when identifying numbers at the cusp of their knowledge, for example, a child incorrectly labels a set of two objects with the word "three" and simultaneously holds up two fingers. These mismatches differ from previously studied mismatches (where the information conveyed in gesture has the potential to be integrated with the information conveyed in speech) in that the gestured response contradicts the spoken response. Here, we ask whether these contradictory number mismatches predict which learners will profit from number-word instruction. We used the Give-a-Number task to measure number knowledge in 47 children (Mage  = 4.1 years, SD = 0.58), and used the What's on this Card task to assess whether children produced gesture-speech mismatches above their knower level. Children who were early in their number learning trajectories ("one-knowers" and "two-knowers") were then randomly assigned, within knower level, to one of two training conditions: a Counting condition in which children practiced counting objects; or an Enriched Number Talk condition containing counting, labeling set sizes, spatial alignment of neighboring sets, and comparison of these sets. Controlling for counting ability, we found that children were more likely to learn the meaning of new number words in the Enriched Number Talk condition than in the Counting condition, but only if they had produced gesture-speech mismatches at pretest. The findings suggest that numerical gesture-speech mismatches are a reliable signal that a child is ready to profit from rich number instruction and provide evidence, for the first time, that cardinal number gestures have a role to play in number-learning.

Language, gesture, and judgment: Children's paths to abstract geometry

As infants, children are sensitive to geometry when recognizing objects or navigating through rooms; however, explicit knowledge of geometry develops slowly and may be unstable even in adults. How can geometric concepts be both so accessible and so elusive? To examine how implicit and explicit geometric concepts develop, the current study assessed, in 132 children (3-8 years old) while they played a simple geometric judgment task, three distinctive channels: children's choices during the game as well as the language and gestures they used to justify and accompany their choices. Results showed that, for certain geometric properties, children chose the correct card even if they could not express with words (or gestures) why they had made this choice. Furthermore, other geometric concepts were expressed and supported by gestures prior to their articulation in either choices or speech. These findings reveal that gestures and behavioral choices may reflect implicit knowledge and serve as a foundation for the development of geometric reasoning. Altogether, our results suggest that language alone might not be enough for expressing and organizing geometric concepts and that children pursue multiple paths to overcome its limitations, a finding with potential implications for primary education in mathematics.

Working Memory for Linguistic and Non-linguistic Manual Gestures: Evidence, Theory, and Application

Linguistic manual gestures are the basis of sign languages used by deaf individuals. Working memory and language processing are intimately connected and thus when language is gesture-based, it is important to understand related working memory mechanisms. This article reviews work on working memory for linguistic and non-linguistic manual gestures and discusses theoretical and applied implications. Empirical evidence shows that there are effects of load and stimulus degradation on working memory for manual gestures. These effects are similar to those found for working memory for speech-based language. Further, there are effects of pre-existing linguistic representation that are partially similar across language modalities. But above all, deaf signers score higher than hearing non-signers on an n-back task with sign-based stimuli, irrespective of their semantic and phonological content, but not with non-linguistic manual actions. This pattern may be partially explained by recent findings relating to cross-modal plasticity in deaf individuals. It suggests that in linguistic gesture-based working memory, semantic aspects may outweigh phonological aspects when processing takes place under challenging conditions. The close association between working memory and language development should be taken into account in understanding and alleviating the challenges faced by deaf children growing up with cochlear implants as well as other clinical populations.

Language, procedures, and the non-perceptual origin of number word meanings

Perceptual representations of objects and approximate magnitudes are often invoked as building blocks that children combine to acquire the positive integers. Systems of numerical perception are either assumed to contain the logical foundations of arithmetic innately, or to supply the basis for their induction. I propose an alternative to this framework, and argue that the integers are not learned from perceptual systems, but arise to explain perception. Using cross-linguistic and developmental data, I show that small (~1-4) and large (~5+) numbers arise both historically and in individual children via distinct mechanisms, constituting independent learning problems, neither of which begins with perceptual building blocks. Children first learn small numbers using the same logic that supports other linguistic number marking (e.g. singular/plural). Years later, they infer the logic of counting from the relations between large number words and their roles in blind counting procedures, only incidentally associating number words with approximate magnitudes.

Gesture for Linguists: A Handy Primer

Humans communicate using language, but they also communicate using gesture - spontaneous movements of the hands and body that universally accompany speech. Gestures can be distinguished from other movements, segmented, and assigned meaning based on their forms and functions. Moreover, gestures systematically integrate with language at all levels of linguistic structure, as evidenced in both production and perception. Viewed typologically, gesture is universal, but nevertheless exhibits constrained variation across language communities (as does language itself ). Finally, gesture has rich cognitive dimensions in addition to its communicative dimensions. In overviewing these and other topics, we show that the study of language is incomplete without the study of its communicative partner, gesture.

Gesture as a window onto children's number knowledge

Before learning the cardinal principle (knowing that the last word reached when counting a set represents the size of the whole set), children do not use number words accurately to label most set sizes. However, it remains unclear whether this difficulty reflects a general inability to conceptualize and communicate about number, or a specific problem with number words. We hypothesized that children's gestures might reflect knowledge of number concepts that they cannot yet express in speech, particularly for numbers they do not use accurately in speech (numbers above their knower-level). Number gestures are iconic in the sense that they are item-based (i.e., each finger maps onto one item in a set) and therefore may be easier to map onto sets of objects than number words, whose forms do not map transparently onto the number of items in a set and, in this sense, are arbitrary. In addition, learners in transition with respect to a concept often produce gestures that convey different information than the accompanying speech. We examined the number words and gestures 3- to 5-year-olds used to label small set sizes exactly (1-4) and larger set sizes approximately (5-10). Children who had not yet learned the cardinal principle were more than twice as accurate when labeling sets of 2 and 3 items with gestures than with words, particularly if the values were above their knower-level. They were also better at approximating set sizes 5-10 with gestures than with words. Further, gesture was more accurate when it differed from the accompanying speech (i.e., a gesture-speech mismatch). These results show that children convey numerical information in gesture that they cannot yet convey in speech, and raise the possibility that number gestures play a functional role in children's development of number concepts.

From iconic handshapes to grammatical contrasts: longitudinal evidence from a child homesigner

Many sign languages display crosslinguistic consistencies in the use of two iconic aspects of handshape, handshape type and finger group complexity. Handshape type is used systematically in form-meaning pairings (morphology): Handling handshapes (Handling-HSs), representing how objects are handled, tend to be used to express events with an agent ("hand-as-hand" iconicity), and Object handshapes (Object-HSs), representing an object's size/shape, are used more often to express events without an agent ("hand-as-object" iconicity). Second, in the distribution of meaningless properties of form (morphophonology), Object-HSs display higher finger group complexity than Handling-HSs. Some adult homesigners, who have not acquired a signed or spoken language and instead use a self-generated gesture system, exhibit these two properties as well. This study illuminates the development over time of both phenomena for one child homesigner, "Julio," age 7;4 (years; months) to 12;8. We elicited descriptions of events with and without agents to determine whether morphophonology and morphosyntax can develop without linguistic input during childhood, and whether these structures develop together or independently. Within the time period studied: (1) Julio used handshape type differently in his responses to vignettes with and without an agent; however, he did not exhibit the same pattern that was found previously in signers, adult homesigners, or gesturers: while he was highly likely to use a Handling-HS for events with an agent (82%), he was less likely to use an Object-HS for non-agentive events (49%); i.e., his productions were heavily biased toward Handling-HSs; (2) Julio exhibited higher finger group complexity in Object- than in Handling-HSs, as in the sign language and adult homesigner groups previously studied; and (3) these two dimensions of language developed independently, with phonological structure showing a sign language-like pattern at an earlier age than morphosyntactic structure. We conclude that iconicity alone is not sufficient to explain the development of linguistic structure in homesign systems. Linguistic input is not required for some aspects of phonological structure to emerge in childhood, and while linguistic input is not required for morphology either, it takes time to emerge in homesign.