Barbara W Sarnecka | University of California, Irvine (original) (raw)

Papers by Barbara W Sarnecka

A large and growing proportion of American students are dual-language learners, and many live in ... more A large and growing proportion of American students are dual-language learners, and many live in or near poverty. In order to serve these students, educators must understand their needs. In this chapter we discuss the role of bilingualism in acquiring early number knowledge. Then we briefly review what is known about the ways in which young children represent number, including the approximate number system (ANS) and symbolic representations of exact numbers—both spoken and written. Next, we describe our own large study of early numeracy in preschool-age dual-language-learners (DLLs) from low-socioeconomic-status (SES) households, with comparison groups of high- and low-SES English monolingual preschoolers, as well as dual-language learners from high-SES households. The main conclusions we draw from this study are the following. (1) Educational delays in acquiring basic number skills are attributable to poverty, not to being a dual-language learner. (2) Pre-kindergarten programs such as Head Start seem to provide experience with counting and numbers that is crucial for low-SES children. (3) Given limited resources, it is reasonable to assess the early numeracy of low-income dual-language learners by testing only in their language of instruction, rather than in both of their languages. Testing only in the language of instruction is much less costly and yields very similar information. (4) The SES-related gap in math achievement is much more a gap in symbolic number knowledge (spoken and written numbers) than in ANS acuity. Thus, improving children’s symbolic number knowledge should be the focus of interventions seeking to narrow this gap.

Bookmarks Related papers MentionsView impact

Different kinds of knowledge are acquired in different ways. In this chapter, I consider four kin... more Different kinds of knowledge are acquired in different ways. In this chapter, I consider four kinds of knowledge: (1) Core knowledge, which is implicit, nonlinguistic and learned easily and automatically; (2) Explicit, linguistic knowledge that fits into an existing conceptual structure (i.e., a new word for an old concept); (3) Explicit, linguistic knowledge that does not fit into any prior conceptual structure (e.g., how to knit); and (4) Explicit, linguistic knowledge that is incompatible with some prior conceptual structure, making it very difficult to learn (e.g., knowing that the earth is a round ball floating in space.) It is this fourth type of learning that Carey (2009) calls ‘conceptual change.’ In this chapter I argue that mathematics includes all four types of knowledge, but that natural numbers are an example of the fourth and most interesting type—knowledge that is the result of conceptual change.

Bookmarks Related papers MentionsView impact

This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., th... more This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., three, seven, two hundred, etc.). I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey (2009). In this framework, the counting list (‘one,’ ‘two,’ ‘three,’ etc.) and the counting routine (i.e., reciting the list and pointing to objects, one at a time) form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers (e.g., There are 24 children in my class, so I need to bring 24 cupcakes for the party.) A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information (i.e., large, exact cardinal values) that they had no way of representing before.

Bookmarks Related papers MentionsView impact

A person’s belief about whether intelligence can change (called their implicit theory of intellig... more A person’s belief about whether intelligence can change (called their implicit theory of
intelligence) predicts something about that person’s thinking and behavior. People who
believe intelligence is fixed (called entity theorists) attribute failure to traits (i.e., “I failed
the test because I’m not smart.”) and tend to be less motivated in school; those who
believe intelligence is malleable (called incremental theorists) tend to attribute failure to
behavior (i.e., “I failed the test because I didn’t study.”) and are more motivated in school.
In previous studies, researchers have characterized participants as either entity or
incremental theorists based on their agreement or disagreement with three statements.
The present study further explored the theories-of-intelligence (TOI) construct in two
ways: first, we asked whether these theories are coherent, in the sense that they show
up not only in participants’ responses to the three standard assessment items, but on a
broad range of questions about intelligence and the brain. Second, we asked whether
these theories are discrete or continuous. In other words, we asked whether people
believe one thing or the other (i.e., that intelligence is malleable or fixed), or if there is
a continuous range of beliefs (i.e., people believe in malleability to a greater or lesser
degree). Study (1) asked participants a range of general questions about the malleability
of intelligence and the brain. Study (2) asked participants more specific questions about
the brains of a pair of identical twins who were separated at birth. Results showed
that TOI are coherent: participants’ responses to the three standard survey items are
correlated with their responses to questions about the brain. But the theories are not
discrete: although responses to the three standard survey items fell into a bimodal
distribution, responses to the broader range of questions fell into a normal distribution
suggesting the theories are continuous.

Bookmarks Related papers MentionsView impact

Oxford Handbooks Online, 2014

Bookmarks Related papers MentionsView impact

British Journal of Developmental Psychology, 2014

Although everyone perceives approximate numerosities, some people make more accurate estimates th... more Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word 'six' (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu-Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact-number knowledge may not understand the phrase 'more dots' to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact-number knowledge in normally developing children disappears.

Bookmarks Related papers MentionsView impact

Developmental Science, 2014

Does speaking more than one language help a child perform better on certain types of cognitive ta... more Does speaking more than one language help a child perform better on certain types of cognitive tasks? One possibility is that bilingualism confers either specific or general cognitive advantages on tasks that require selective attention to one dimension over another (e.g. Bialystok, 2001; Hilchey & Klein, 2011). Other studies have looked for such an advantage but found none (e.g. Morton & Harper, 2007; Paap & Greenberg, 2013). The present study compared monolingual and bilingual children's performance on a numerical discrimination task, which required children to ignore area and attend to number. Ninety-two children, ages 3 to 6 years, were asked which of two arrays of dots had ‘more dots’. Half of the trials were congruent, where the numerically greater array was also larger in total area, and half were incongruent, where the numerically greater array was smaller in total area. All children performed better on congruent than on incongruent trials. Older children were more successful than younger children at ignoring area in favor of number. Bilingual children did not perform differently from monolingual children either in number discrimination itself (i.e. identifying which array had more dots) or at selectively attending to number. The present study thus finds no evidence of a bilingual advantage on this task for children of this age.

Bookmarks Related papers MentionsView impact

Developmental Psychology, 2014

Children’s understanding of numbers is often assessed using a number-line task, where the child i... more Children’s understanding of numbers is often assessed using a number-line task, where the child is shown a line labeled with 0 at one end and a higher number (e.g., 100) at the other end. The child is then asked where on the line some intermediate number (e.g., 70) should go. Performance on this task changes predictably during childhood, and this has often been interpreted as evidence of a change in the child’s psychological representation of integer quantities. The present article presents theoretical and empirical evidence that the change in number-line performance actually reflects the development of measurement skills used in the task. We compare 2 versions of the number-line task: the bounded version used in the literature and a new, unbounded version. Results indicate that it is only children’s performance on the bounded task (which requires subtraction or division) that changes markedly with age. In contrast, children’s performance on the unbounded task (which requires only addition) remains fairly constant as they get older. Thus, developmental changes in performance on the traditional bounded number-line task likely reflect the growth of task-specific measurement skills rather than changes in the child’s understanding of numerical quantities.

Bookmarks Related papers MentionsView impact

Frontiers in psychology, 2014

This mini-review focuses on the question of how the grammatical number system of a child’s langua... more This mini-review focuses on the question of how the grammatical number system of a child’s language may help the child learn the meanings of cardinal number words (e.g., “one” and “two”). Evidence from young children learning English, Russian, Japanese, Mandarin, Slovenian, or Saudi Arabic suggests that trajectories of number-word learning differ for children learning different languages. Children learning English, which distinguishes between singular and plural, seem to learn the meaning of the cardinal number “one” earlier than children learning Japanese or Mandarin, which have very little singular/plural marking. Similarly, children whose languages have a singular/dual/plural system (Slovenian and Saudi Arabic) learn the meaning of “two” earlier than English-speaking children. This relation between grammatical and cardinal number may shed light on how humans acquire cardinal-number concepts. There is an ongoing debate about whether mental symbols for small cardinalities (concepts for “oneness,” “twoness,” etc.) are innate or learned. Although an effect of grammatical number on number-word learning does not rule out nativist accounts, it seems more consistent with constructivist accounts, which portray the number-learning process as one that requires significant conceptual change.

Bookmarks Related papers MentionsView impact

Cognitive Science, 2013

Understanding what numbers are means knowing several things. It means knowing how counting relate... more Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.

Bookmarks Related papers MentionsView impact

Cognition, 2013

The present study asks when young children understand that number words quantify over sets of dis... more The present study asks when young children understand that number words quantify over sets of discrete individuals. For this study, two- to four-year-old children were asked to extend the number word five or six either to a cup containing discrete objects (e.g., blocks) or to a cup containing a continuous substance (e.g., water). In Experiment 1, only children who knew the exact meanings of the words one, two and three extended higher number words (five or six) to sets of discrete objects. In Experiment 2, children who only knew the exact meaning of one extended higher number words to discrete objects under the right conditions (i.e., when the problem was first presented with the number words one and two). These results show that children have some understanding that number words pertain to discrete quantification from very early on, but that this knowledge becomes more robust as children learn the exact, cardinal meanings of individual number words.

Bookmarks Related papers MentionsView impact

Evolution and Human Behavior, 2013

Cooperation between nonrelatives is common in humans. Reciprocal altruism is a plausible evolutio... more Cooperation between nonrelatives is common in humans. Reciprocal altruism is a plausible evolutionary mechanism for cooperation within unrelated pairs, as selection may favor individuals who selectively cooperate with those who have cooperated with them in the past. Reciprocity is often observed in humans, but there is only limited evidence of reciprocal altruism in other primate species, raising questions about the origins of human reciprocity. Here, we explore how reciprocity develops in a sample of American children ranging from 3 to 7.5 years of age, and also compare children's behavior to that of chimpanzees in prior studies to gain insight into the phylogeny of human reciprocity. Children show a marked tendency to respond contingently to both prosocial and selfish acts, patterns that have not been seen among chimpanzees in prior studies. Our results show that reciprocity increases markedly with age in this population of children, and by about 5.5 years of age children consistently match the previous behavior of their partners.

Bookmarks Related papers MentionsView impact

Child Development, 2012

How is number-concept acquisition related to overall language development? Experiments 1 and 2 me... more How is number-concept acquisition related to overall language development? Experiments 1 and 2 measured number-word knowledge and general vocabulary in a total of 59 children, ages 30–60 months. A strong correlation was found between number-word knowledge and vocabulary, independent of the child’s age, contrary to previous results (D. Ansari et al., 2003). This result calls into question arguments that (a) the number-concept creation process is scaffolded mainly by visuo-spatial development and (b) that language only becomes integrated after the concepts are created (D. Ansari et al., 2003). Instead, this may suggest that having a larger nominal vocabulary helps children learn number words. Experiment 3 shows that the differences with previous results are likely due to changes in how the data were analyzed.

Bookmarks Related papers MentionsView impact

Behavior Research Methods, 2012

Number-knower-levels indicate what children know about counting and the cardinal meaning of numbe... more Number-knower-levels indicate what children know about counting and the cardinal meaning of number words, which is an important developmental variable. The Give-N methodology, which is used to diagnose knower-level, has been highly refined – in contrast, the field’s analysis of Give-N data remains somewhat crude. Here we work with a model by Lee and Sarnecka (2010), which is a generative model of how children perform in Give-N, allowing more-principled inference of knower-level. We present a close approximation of the model’s inference that can be computed by Microsoft Excel, as well as a worked implementation and instructions for its usage. This should give developmental researchers access to sharper inference about young children’s number-word knowledge.

Bookmarks Related papers MentionsView impact

Advances in Child Development and Behavior, 2012

Bookmarks Related papers MentionsView impact

Cognition, 2011

Lee and Sarnecka (2010) developed a Bayesian model of young children’s behavior on the Give-N tes... more Lee and Sarnecka (2010) developed a Bayesian model of young children’s behavior on the Give-N test of number knowledge. This paper presents two new extensions of the model, and applies the model to new data. In the first extension, the model is used to evaluate competing theories about the conceptual knowledge underlying children’s behavior. One, the knower-levels theory, is basically a “stage” theory involving real conceptual change. The other, the approximate-meanings theory, assumes that the child’s conceptual knowledge is relatively constant, although performance improves over time. In the second extension, the model is used to ask whether the same latent psychological variable (a child’s number-knower level) can simultaneously account for behavior on two tasks (the Give-N task and the Fast-Cards task) with different performance demands. Together, these two demonstrations show the potential of the Bayesian modeling approach to improve our understanding of the development of human cognition.

Bookmarks Related papers MentionsView impact

Journal of Experimental Child Psychology, 2011

An essential part of understanding number words (e.g., eight) is understanding that all number wo... more 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.

Bookmarks Related papers MentionsView impact

Cognitive Science, 2010

We develop and evaluate a model of behavior on the Give-N task, a commonly used measure of young... more We develop and evaluate a model of behavior on the Give-N task, a commonly used measure of
young children’s number knowledge. Our model uses the knower-level theory of how children represent
numbers. To produce behavior on the Give-N task, the model assumes that children start out with
a base rate that makes some answers more likely a priori than others but is updated on each experimental
trial in a way that depends on the interaction between the experimenter’s request and the
child’s knower level. We formalize this process as a generative graphical model, so that the parameters—
including the base rate distribution and each child’s knower level—can be inferred from data
using Bayesian methods. Using this approach, we evaluate the model on previously published data
from 82 children spanning the whole developmental range. The model provides an excellent fit to
these data, and the inferences about the base rate and knower levels are interpretable and insightful.
We discuss how our modeling approach can be extended to other developmental tasks and can be
used to help evaluate alternative theories of number representation against the knower-level theory.

Bookmarks Related papers MentionsView impact

Journal of Experimental Child Psychology, 2009

Researchers have long disagreed about whether number concepts are essentially continuous (unchang... more Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children’s incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the ‘discontinuity’ account of number development in general and the ‘knower-levels’ account in particular.

Bookmarks Related papers MentionsView impact

Cognition, 2008

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowe... more 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.

Bookmarks Related papers MentionsView impact

A large and growing proportion of American students are dual-language learners, and many live in ... more A large and growing proportion of American students are dual-language learners, and many live in or near poverty. In order to serve these students, educators must understand their needs. In this chapter we discuss the role of bilingualism in acquiring early number knowledge. Then we briefly review what is known about the ways in which young children represent number, including the approximate number system (ANS) and symbolic representations of exact numbers—both spoken and written. Next, we describe our own large study of early numeracy in preschool-age dual-language-learners (DLLs) from low-socioeconomic-status (SES) households, with comparison groups of high- and low-SES English monolingual preschoolers, as well as dual-language learners from high-SES households. The main conclusions we draw from this study are the following. (1) Educational delays in acquiring basic number skills are attributable to poverty, not to being a dual-language learner. (2) Pre-kindergarten programs such as Head Start seem to provide experience with counting and numbers that is crucial for low-SES children. (3) Given limited resources, it is reasonable to assess the early numeracy of low-income dual-language learners by testing only in their language of instruction, rather than in both of their languages. Testing only in the language of instruction is much less costly and yields very similar information. (4) The SES-related gap in math achievement is much more a gap in symbolic number knowledge (spoken and written numbers) than in ANS acuity. Thus, improving children’s symbolic number knowledge should be the focus of interventions seeking to narrow this gap.

Bookmarks Related papers MentionsView impact

Different kinds of knowledge are acquired in different ways. In this chapter, I consider four kin... more Different kinds of knowledge are acquired in different ways. In this chapter, I consider four kinds of knowledge: (1) Core knowledge, which is implicit, nonlinguistic and learned easily and automatically; (2) Explicit, linguistic knowledge that fits into an existing conceptual structure (i.e., a new word for an old concept); (3) Explicit, linguistic knowledge that does not fit into any prior conceptual structure (e.g., how to knit); and (4) Explicit, linguistic knowledge that is incompatible with some prior conceptual structure, making it very difficult to learn (e.g., knowing that the earth is a round ball floating in space.) It is this fourth type of learning that Carey (2009) calls ‘conceptual change.’ In this chapter I argue that mathematics includes all four types of knowledge, but that natural numbers are an example of the fourth and most interesting type—knowledge that is the result of conceptual change.

Bookmarks Related papers MentionsView impact

This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., th... more This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., three, seven, two hundred, etc.). I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey (2009). In this framework, the counting list (‘one,’ ‘two,’ ‘three,’ etc.) and the counting routine (i.e., reciting the list and pointing to objects, one at a time) form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers (e.g., There are 24 children in my class, so I need to bring 24 cupcakes for the party.) A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information (i.e., large, exact cardinal values) that they had no way of representing before.

Bookmarks Related papers MentionsView impact

A person’s belief about whether intelligence can change (called their implicit theory of intellig... more A person’s belief about whether intelligence can change (called their implicit theory of
intelligence) predicts something about that person’s thinking and behavior. People who
believe intelligence is fixed (called entity theorists) attribute failure to traits (i.e., “I failed
the test because I’m not smart.”) and tend to be less motivated in school; those who
believe intelligence is malleable (called incremental theorists) tend to attribute failure to
behavior (i.e., “I failed the test because I didn’t study.”) and are more motivated in school.
In previous studies, researchers have characterized participants as either entity or
incremental theorists based on their agreement or disagreement with three statements.
The present study further explored the theories-of-intelligence (TOI) construct in two
ways: first, we asked whether these theories are coherent, in the sense that they show
up not only in participants’ responses to the three standard assessment items, but on a
broad range of questions about intelligence and the brain. Second, we asked whether
these theories are discrete or continuous. In other words, we asked whether people
believe one thing or the other (i.e., that intelligence is malleable or fixed), or if there is
a continuous range of beliefs (i.e., people believe in malleability to a greater or lesser
degree). Study (1) asked participants a range of general questions about the malleability
of intelligence and the brain. Study (2) asked participants more specific questions about
the brains of a pair of identical twins who were separated at birth. Results showed
that TOI are coherent: participants’ responses to the three standard survey items are
correlated with their responses to questions about the brain. But the theories are not
discrete: although responses to the three standard survey items fell into a bimodal
distribution, responses to the broader range of questions fell into a normal distribution
suggesting the theories are continuous.

Bookmarks Related papers MentionsView impact

Oxford Handbooks Online, 2014

Bookmarks Related papers MentionsView impact

British Journal of Developmental Psychology, 2014

Although everyone perceives approximate numerosities, some people make more accurate estimates th... more Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word 'six' (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu-Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact-number knowledge may not understand the phrase 'more dots' to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact-number knowledge in normally developing children disappears.

Bookmarks Related papers MentionsView impact

Developmental Science, 2014

Does speaking more than one language help a child perform better on certain types of cognitive ta... more Does speaking more than one language help a child perform better on certain types of cognitive tasks? One possibility is that bilingualism confers either specific or general cognitive advantages on tasks that require selective attention to one dimension over another (e.g. Bialystok, 2001; Hilchey & Klein, 2011). Other studies have looked for such an advantage but found none (e.g. Morton & Harper, 2007; Paap & Greenberg, 2013). The present study compared monolingual and bilingual children's performance on a numerical discrimination task, which required children to ignore area and attend to number. Ninety-two children, ages 3 to 6 years, were asked which of two arrays of dots had ‘more dots’. Half of the trials were congruent, where the numerically greater array was also larger in total area, and half were incongruent, where the numerically greater array was smaller in total area. All children performed better on congruent than on incongruent trials. Older children were more successful than younger children at ignoring area in favor of number. Bilingual children did not perform differently from monolingual children either in number discrimination itself (i.e. identifying which array had more dots) or at selectively attending to number. The present study thus finds no evidence of a bilingual advantage on this task for children of this age.

Bookmarks Related papers MentionsView impact

Developmental Psychology, 2014

Children’s understanding of numbers is often assessed using a number-line task, where the child i... more Children’s understanding of numbers is often assessed using a number-line task, where the child is shown a line labeled with 0 at one end and a higher number (e.g., 100) at the other end. The child is then asked where on the line some intermediate number (e.g., 70) should go. Performance on this task changes predictably during childhood, and this has often been interpreted as evidence of a change in the child’s psychological representation of integer quantities. The present article presents theoretical and empirical evidence that the change in number-line performance actually reflects the development of measurement skills used in the task. We compare 2 versions of the number-line task: the bounded version used in the literature and a new, unbounded version. Results indicate that it is only children’s performance on the bounded task (which requires subtraction or division) that changes markedly with age. In contrast, children’s performance on the unbounded task (which requires only addition) remains fairly constant as they get older. Thus, developmental changes in performance on the traditional bounded number-line task likely reflect the growth of task-specific measurement skills rather than changes in the child’s understanding of numerical quantities.

Bookmarks Related papers MentionsView impact

Frontiers in psychology, 2014

This mini-review focuses on the question of how the grammatical number system of a child’s langua... more This mini-review focuses on the question of how the grammatical number system of a child’s language may help the child learn the meanings of cardinal number words (e.g., “one” and “two”). Evidence from young children learning English, Russian, Japanese, Mandarin, Slovenian, or Saudi Arabic suggests that trajectories of number-word learning differ for children learning different languages. Children learning English, which distinguishes between singular and plural, seem to learn the meaning of the cardinal number “one” earlier than children learning Japanese or Mandarin, which have very little singular/plural marking. Similarly, children whose languages have a singular/dual/plural system (Slovenian and Saudi Arabic) learn the meaning of “two” earlier than English-speaking children. This relation between grammatical and cardinal number may shed light on how humans acquire cardinal-number concepts. There is an ongoing debate about whether mental symbols for small cardinalities (concepts for “oneness,” “twoness,” etc.) are innate or learned. Although an effect of grammatical number on number-word learning does not rule out nativist accounts, it seems more consistent with constructivist accounts, which portray the number-learning process as one that requires significant conceptual change.

Bookmarks Related papers MentionsView impact

Cognitive Science, 2013

Understanding what numbers are means knowing several things. It means knowing how counting relate... more Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.

Bookmarks Related papers MentionsView impact

Cognition, 2013

The present study asks when young children understand that number words quantify over sets of dis... more The present study asks when young children understand that number words quantify over sets of discrete individuals. For this study, two- to four-year-old children were asked to extend the number word five or six either to a cup containing discrete objects (e.g., blocks) or to a cup containing a continuous substance (e.g., water). In Experiment 1, only children who knew the exact meanings of the words one, two and three extended higher number words (five or six) to sets of discrete objects. In Experiment 2, children who only knew the exact meaning of one extended higher number words to discrete objects under the right conditions (i.e., when the problem was first presented with the number words one and two). These results show that children have some understanding that number words pertain to discrete quantification from very early on, but that this knowledge becomes more robust as children learn the exact, cardinal meanings of individual number words.

Bookmarks Related papers MentionsView impact

Evolution and Human Behavior, 2013

Cooperation between nonrelatives is common in humans. Reciprocal altruism is a plausible evolutio... more Cooperation between nonrelatives is common in humans. Reciprocal altruism is a plausible evolutionary mechanism for cooperation within unrelated pairs, as selection may favor individuals who selectively cooperate with those who have cooperated with them in the past. Reciprocity is often observed in humans, but there is only limited evidence of reciprocal altruism in other primate species, raising questions about the origins of human reciprocity. Here, we explore how reciprocity develops in a sample of American children ranging from 3 to 7.5 years of age, and also compare children's behavior to that of chimpanzees in prior studies to gain insight into the phylogeny of human reciprocity. Children show a marked tendency to respond contingently to both prosocial and selfish acts, patterns that have not been seen among chimpanzees in prior studies. Our results show that reciprocity increases markedly with age in this population of children, and by about 5.5 years of age children consistently match the previous behavior of their partners.

Bookmarks Related papers MentionsView impact

Child Development, 2012

How is number-concept acquisition related to overall language development? Experiments 1 and 2 me... more How is number-concept acquisition related to overall language development? Experiments 1 and 2 measured number-word knowledge and general vocabulary in a total of 59 children, ages 30–60 months. A strong correlation was found between number-word knowledge and vocabulary, independent of the child’s age, contrary to previous results (D. Ansari et al., 2003). This result calls into question arguments that (a) the number-concept creation process is scaffolded mainly by visuo-spatial development and (b) that language only becomes integrated after the concepts are created (D. Ansari et al., 2003). Instead, this may suggest that having a larger nominal vocabulary helps children learn number words. Experiment 3 shows that the differences with previous results are likely due to changes in how the data were analyzed.

Bookmarks Related papers MentionsView impact

Behavior Research Methods, 2012

Number-knower-levels indicate what children know about counting and the cardinal meaning of numbe... more Number-knower-levels indicate what children know about counting and the cardinal meaning of number words, which is an important developmental variable. The Give-N methodology, which is used to diagnose knower-level, has been highly refined – in contrast, the field’s analysis of Give-N data remains somewhat crude. Here we work with a model by Lee and Sarnecka (2010), which is a generative model of how children perform in Give-N, allowing more-principled inference of knower-level. We present a close approximation of the model’s inference that can be computed by Microsoft Excel, as well as a worked implementation and instructions for its usage. This should give developmental researchers access to sharper inference about young children’s number-word knowledge.

Bookmarks Related papers MentionsView impact

Advances in Child Development and Behavior, 2012

Bookmarks Related papers MentionsView impact

Cognition, 2011

Lee and Sarnecka (2010) developed a Bayesian model of young children’s behavior on the Give-N tes... more Lee and Sarnecka (2010) developed a Bayesian model of young children’s behavior on the Give-N test of number knowledge. This paper presents two new extensions of the model, and applies the model to new data. In the first extension, the model is used to evaluate competing theories about the conceptual knowledge underlying children’s behavior. One, the knower-levels theory, is basically a “stage” theory involving real conceptual change. The other, the approximate-meanings theory, assumes that the child’s conceptual knowledge is relatively constant, although performance improves over time. In the second extension, the model is used to ask whether the same latent psychological variable (a child’s number-knower level) can simultaneously account for behavior on two tasks (the Give-N task and the Fast-Cards task) with different performance demands. Together, these two demonstrations show the potential of the Bayesian modeling approach to improve our understanding of the development of human cognition.

Bookmarks Related papers MentionsView impact

Journal of Experimental Child Psychology, 2011

An essential part of understanding number words (e.g., eight) is understanding that all number wo... more 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.

Bookmarks Related papers MentionsView impact

Cognitive Science, 2010

We develop and evaluate a model of behavior on the Give-N task, a commonly used measure of young... more We develop and evaluate a model of behavior on the Give-N task, a commonly used measure of
young children’s number knowledge. Our model uses the knower-level theory of how children represent
numbers. To produce behavior on the Give-N task, the model assumes that children start out with
a base rate that makes some answers more likely a priori than others but is updated on each experimental
trial in a way that depends on the interaction between the experimenter’s request and the
child’s knower level. We formalize this process as a generative graphical model, so that the parameters—
including the base rate distribution and each child’s knower level—can be inferred from data
using Bayesian methods. Using this approach, we evaluate the model on previously published data
from 82 children spanning the whole developmental range. The model provides an excellent fit to
these data, and the inferences about the base rate and knower levels are interpretable and insightful.
We discuss how our modeling approach can be extended to other developmental tasks and can be
used to help evaluate alternative theories of number representation against the knower-level theory.

Bookmarks Related papers MentionsView impact

Journal of Experimental Child Psychology, 2009

Researchers have long disagreed about whether number concepts are essentially continuous (unchang... more Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children’s incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the ‘discontinuity’ account of number development in general and the ‘knower-levels’ account in particular.

Bookmarks Related papers MentionsView impact

Cognition, 2008

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowe... more 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.

Bookmarks Related papers MentionsView impact