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.     

How counting represents number: what children must learn and when they learn it

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowers) to those who do not (subset-knowers), in order to better characterize the knowledge itself. New results are that (1) Many children answer the question "how many" with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.     

Cross-linguistic relations between quantifiers and numerals in language acquisition: Evidence from Japanese

A study of 104 Japanese-speaking 2- to 5-year-olds tested the relation between numeral and quantifier acquisition. A first study assessed Japanese children’s comprehension of quantifiers, numerals, and classifiers. Relative to English-speaking counterparts, Japanese children were delayed in numeral comprehension at 2 years of age but showed no difference at 3 and 4 years of age. Also, Japanese 2-year-olds had better comprehension of quantifiers, indicating that their delay was specific to numerals. A second study examined the speech of Japanese and English caregivers to explore the syntactic cues that might affect integer acquisition. Quantifiers and numerals occurred in similar syntactic positions and overlapped to a greater degree in English than in Japanese. Also, Japanese nouns were often dropped, and both quantifiers and numerals exhibited variable positions relative to the nouns they modified. We conclude that syntactic cues in English facilitate bootstrapping numeral meanings from quantifier meanings and that such cues are weaker in classifier languages such as Japanese.     

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.     

From grammatical number to exact numbers: early meanings of 'one', 'two', and 'three' in English, Russian, and Japanese

This study examined whether singular/plural marking in a language helps children learn the meanings of the words 'one,' 'two,' and 'three.' First, CHILDES data in English, Russian (which marks singular/plural), and Japanese (which does not) were compared for frequency, variability, and contexts of number-word use. Then young children in the USA, Russia, and Japan were tested on Counting and Give-N tasks. More English and Russian learners knew the meaning of each number word than Japanese learners, regardless of whether singular/plural cues appeared in the task itself (e.g., "Give two apples" vs. "Give two"). These results suggest that the learning of "one," "two" and "three" is supported by the conceptual framework of grammatical number, rather than that of integers.     

The logical syntax of number words: Theory, acquisition and processing

Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n = 32) and adult speakers of English (n = 32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language Acquisition, 11(4), 197-218]. In doing so, this work ties together research on the acquisition of the number vocabulary with a growing body of work on the development of quantification and sentence processing abilities in young children [Geurts, 2003; Lidz, J., Musolino, J. (2002). Children’s command of quantification. Cognition, 84, 113-154; Musolino, J., Lidz, J. (2003). The scope of isomorphism: Turning adults into children. Language Acquisition, 11(4), 277-291; Trueswell, J., Sekerina, I., Hilland, N., Logrip, M. (1999). The kindergarten-path effect: Studying on-line sentence processing in young children. Cognition, 73, 89-134; Noveck, I. (2001). When children are more logical than adults: Experimental investigations of scalar implicature. Cognition, 78, 165-188; Noveck, I., Guelminger, R., Georgieff, N., & Labruyere, N. (2007). What autism can tell us about every . . . not sentences. Journal of Semantics,24(1), 73-90. On a more general level, this work confirms the importance of integrating formal and developmental perspectives [Musolino, 2004], this time by highlighting the explanatory power of linguistically-based models of language acquisition and by showing that the complex structure postulated by linguists has important implications for developmental accounts of the number vocabulary.     

The relation between teachers' math talk and the acquisition of number sense within kindergarten classrooms

The aim of the present study was to investigate the relation between teachers' math talk and the acquisition of number sense within kindergarten classrooms. The mathematical language input provided by 35 kindergarten teachers was examined with 9 different input categories. The results of this study indicate that the role of each of these math talk categories is not as straightforward as was hypothesized. Although significant positive relations were found for math talk categories such as cardinality and conventional nominatives, the relations between the categories' calculation and number symbols and children's score on specific number sense tasks were negative. Moreover, a large diversity in math talk was negatively related to kindergartners' number sense acquisition. These results suggest that teachers should be careful and selective with the amount of math talk that they offer to young children.     

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.     

The development of language and abstract concepts: the case of natural number

What are the origins of abstract concepts such as "seven," and what role does language play in their development? These experiments probed the natural number words and concepts of 3-year-old children who can recite number words to ten but who can comprehend only one or two. Children correctly judged that a set labeled eight retains this label if it is unchanged, that it is not also four, and that eight is more than two. In contrast, children failed to judge that a set of 8 objects is better labeled by eight than by four, that eight is more than four, that eight continues to apply to a set whose members are rearranged, or that eight ceases to apply if the set is increased by 1, doubled, or halved. The latter errors contrast with children's correct application of words for the smallest numbers. These findings suggest that children interpret number words by relating them to 2 distinct preverbal systems that capture only limited numerical information. Children construct the system of abstract, natural number concepts from these foundations.     

Sensitivity to number: Reply to Gebuis and Gevers

Past research showing a bias towards the larger non-symbolic number by adults and children in line bisection tasks (de Hevia & Spelke, 2009) has been challenged by Gebuis and Gevers, suggesting that area subtended by the stimulus and not number is responsible for the biases. I review evidence supporting the idea that although sensitivity to number might be relatively affected by visual cues, number is a major, salient property of our environment. The influence of non-numerical cues might be seen as the concurrent processing of dimensions that entail information of magnitude, without implying that number is constructed out of those dimensions.     

Incomplete understanding of complex numbers Girolamo Cardano: a case study in the acquisition of mathematical concepts

In this paper, I present the case of the discovery of complex numbers by Girolamo Cardano. Cardano acquires the concepts of (specific) complex numbers, complex addition, and complex multiplication. His understanding of these concepts is incomplete. I show that his acquisition of these concepts cannot be explained on the basis of Christopher Peacocke’s Conceptual Role Theory of concept possession. I argue that Strong Conceptual Role Theories that are committed to specifying a set of transitions that is both necessary and sufficient for possession of mathematical concepts will always face counterexamples of the kind illustrated by Cardano. I close by suggesting that we should rely more heavily on resources of Anti-Individualism as a framework for understanding the acquisition and possession of concepts of abstract subject matters.     

Grey parrot number acquisition: The inference of cardinal value from ordinal position on the numeral list

A Grey parrot (Psittacus erithacus) had previously been taught to use English count words ("one" through "sih" [six]) to label sets of one to six individual items (Pepperberg, 1994). He had also been taught to use the same count words to label the Arabic numerals 1 through 6. Without training, he inferred the relationship between the Arabic numerals and the sets of objects (Pepperberg, 2006b). In the present study, he was then trained to label vocally the Arabic numerals 7 and 8 ("sih-none", "eight", respectively) and to order these Arabic numerals with respect to the numeral 6. He subsequently inferred the ordinality of 7 and 8 with respect to the smaller numerals and he inferred use of the appropriate label for the cardinal values of seven and eight items. These data suggest that he constructed the cardinal meanings of "seven" ("sih-none") and "eight" from his knowledge of the cardinal meanings of one through six, together with the place of "seven" ("sih-none") and "eight" in the ordered count list.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

Finding one's meaning: a test of the relation between quantifiers and integers in language development

We explored children’s early interpretation of numerals and linguistic number marking, in order to test the hypothesis (e.g., Carey (2004). Bootstrapping and the origin of concepts. Daedalus, 59-68) that children’s initial distinction between one and other numerals (i.e., two, three, etc.) is bootstrapped from a prior distinction between singular and plural nouns. Previous studies have presented evidence that in languages without singular-plural morphology, like Japanese and Chinese, children acquire the meaning of the word one later than in singular-plural languages like English and Russian. In two experiments, we sought to corroborate this relation between grammatical number and integer acquisition within English. We found a significant correlation between children’s comprehension of numerals and a large set of natural language quantifiers and determiners, even when controlling for effects due to age. However, we also found that 2-year-old children, who are just acquiring singular-plural morphology and the word one, fail to assign an exact interpretation to singular noun phrases (e.g., a banana), despite interpreting one as exact. For example, in a Truth-Value Judgment task, most children judged that a banana was consistent with a set of two objects, despite rejecting sets of two for the numeral one. Also, children who gave exactly one object for singular nouns did not have a better comprehension of numerals relative to children who did not give exactly one. Thus, we conclude that the correlation between quantifier comprehension and numeral comprehension in children of this age is not attributable to the singular-plural distinction facilitating the acquisition of the word one. We argue that quantifiers play a more general role in highlighting the semantic function of numerals, and that children distinguish between numerals and other quantifiers from the beginning, assigning exact interpretations only to numerals.     

It's in the sample: the effects of sample size and sample diversity on the breadth of inductive generalization

Developmental studies have provided mixed evidence with regard to the question of whether children consider sample size and sample diversity in their inductive generalizations. Results from four experiments with 105 undergraduates, 105 school-age children (M = 7.2 years), and 105 preschoolers (M = 4.9 years) showed that preschoolers made a higher rate of projections from large samples than from small samples when samples were diverse (Experiments 1 and 3) but not when samples were homogeneous (Experiment 4) and not when the task required a choice between two samples (Experiment 2). Furthermore, when a property occurred in large and diverse samples, preschoolers exhibited a broad pattern of projection, generalizing the property to items from categories not represented in the evidence. In contrast, adults followed a normative pattern of induction and never attributed properties to items from categories not represented in the evidence. School-age children showed a mixed pattern of results.     

Children's evaluation of the certainty of another person's inductive inferences and guesses

In three studies, 5–10-year-old children and an adult comparison group judged another's certainty in making inductive inferences and guesses. Participants observed a puppet make strong inductions, weak inductions, and guesses. Participants either had no information about the correctness of the puppet's conclusion, knew that the puppet was correct, or knew that the puppet was incorrect. Children of all ages (but not adults) rated the puppet as more certain about statements the child knew to be correct than statements the child knew to be incorrect. When assessing another's certainty, children have difficulty inhibiting their own knowledge and focusing on the other's perspective. Children were more likely to differentiate between inductions and guesses when the puppet made an Incorrect Statement, but even the oldest children did not differentiate consistently. The distinction between induction and guessing appears to be only acquired gradually but is important as a contributor to more advanced forms of reasoning and epistemological understanding.     

Developing access to number magnitude: a study of the SNARC effect in 7- to 9-year-olds

The SNARC (spatial-numerical association of response codes) effect refers to the finding that small numbers facilitate left responses, whereas larger numbers facilitate right responses. The development of this spatial association was studied in 7-, 8-, and 9-year-olds, as well as in adults, using a task where number magnitude was essential to perform the task and another task where number magnitude was irrelevant. When number magnitude was essential, a SNARC effect was found in all age groups. But when number magnitude was irrelevant, a SNARC effect was found only in 9-year-olds and adults. These results are taken to suggest that (a) 7-year-olds represent number magnitudes in a way similar to that of adults and that (b) when perceiving Arabic numerals, children have developed automatic access to magnitude information by around 9 years of age.     

Tracking irregular morphophonological dependencies in natural language: evidence from the acquisition of subject-verb agreement in French

This study examines French-learning infants’ sensitivity to grammatical non-adjacent dependencies involving subject-verb agreement (e.g., le/les garçons lit/lisent ‘the boy(s) read(s)’) where number is audible on both the determiner of the subject DP and the agreeing verb, and the dependency is spanning across two syntactic phrases. A further particularity of this subsystem of French subject-verb agreement is that number marking on the verb is phonologically highly irregular. Despite the challenge, the HPP results for 24- and 18-month-olds demonstrate knowledge of both number dependencies: between the singular determiner le and the non-adjacent singular verbal forms and between the plural determiner les and the non-adjacent plural verbal forms. A control experiment suggests that the infants are responding to known verb forms, not phonological regularities. Given the paucity of such forms in the adult input documented through a corpus study, these results are interpreted as evidence that 18-month-olds have the ability to extract complex patterns across a range of morphophonologically inconsistent and infrequent items in natural language.     

The evolution of frequency distributions: Relating regularization to inductive biases through iterated learning

The regularization of linguistic structures by learners has played a key role in arguments for strong innate constraints on language acquisition, and has important implications for language evolution. However, relating the inductive biases of learners to regularization behavior in laboratory tasks can be challenging without a formal model. In this paper we explore how regular linguistic structures can emerge from language evolution by iterated learning, in which one person’s linguistic output is used to generate the linguistic input provided to the next person. We use a model of iterated learning with Bayesian agents to show that this process can result in regularization when learners have the appropriate inductive biases. We then present three experiments demonstrating that simulating the process of language evolution in the laboratory can reveal biases towards regularization that might not otherwise be obvious, allowing weak biases to have strong effects. The results of these experiments suggest that people tend to regularize inconsistent word-meaning mappings, and that even a weak bias towards regularization can allow regular languages to be produced via language evolution by iterated learning.