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The development of number concepts: Discrete quantification and numerosity

Dissertation
Author: Emily Beth Slusser
Abstract:
This line of research examines how children develop natural-number concepts through the use of emergent language and innate representational resources. Specifically, how children come to understand that number words (1) refer to numerosity, as opposed to some other property of individuals or sets, and (2) quantify over discrete entities. Some researchers contend that children understand these semantic restrictions from very early on and that this knowledge, in fact, guides children's learning of the first few numberwords. Conversely, it has been argued that children extrapolate these restrictions from the first few number-word meanings. Results from the present studies support this latter view by providing evidence that number concepts are constructed over time, via a predictable developmental trajectory. Study 1 demonstrates that only children who understand the cardinal principle of counting (those who are able to use counting to determine the number of items in a set) reliably extend number words to numerosity. Study 2 shows that children must learn the cardinal meanings of three or more number words in order to apply higher, unknown number words to discrete entities. Each of these two studies provides evidence that children do not connect number words to specific discrete quantities upon first encounter with the number-word list. Rather, they make the connection around the same time they figure out the cardinal principle. Study 3 addresses how children make this connection by exploring the proposal that children use the pairing of number words with count nouns as an indication that objects are preferred referents of number words. The results of this last study indicate that linguistic context facilitates learning the semantic functions of number words well before a child learns the specific meaning of each of these words. Additionally, this study provides supporting evidence for the claim that Mandarin noun classifiers provide information analogous to count mass distinctions in English. Implications for theories of number-concept construction will be discussed in relation to the findings of each of these studies.

TABLE OF CONTENTS

Page

LIST OF FIGURES...........................................................................................................vi LIST OF TABLES............................................................................................................vii ACKNOWLEDGMENTS...............................................................................................viii CURRICULUM VITAE....................................................................................................ix ABSTRACT OF THE DISSERTATION.........................................................................xii CHAPTER 1: General Introduction....................................................................................1 1.1

Motivation........................................................................................................................1

1.2

Background......................................................................................................................2

1.2.1

Core Cognition...........................................................................................................2

1.2.2

Language....................................................................................................................4

1.2.3

Constructionism and Bootstrapping...........................................................................7

1.3

Questions..........................................................................................................................8

1.3.1

When Do Children Understand Number Words Refer to Numerosity?.....................8

1.3.2

When Do Children Understand Number Words Refer to Discrete Quantification?10

1.3.3

Do Children Use Linguistic Context to Learn Number Word Semantics?..............12

CHAPTER 2: Connecting Number Words to Cardinality and Numerosity.....................14 2.1

Introduction....................................................................................................................14

2.2

Experiment 1A...............................................................................................................17

2.2.1

Method.....................................................................................................................17

2.2.1.1

Participants......................................................................................................17

2.2.1.2

Match-to-Sample Task....................................................................................18

2.2.1.3

Give-N Task....................................................................................................22

2.2.2

Results......................................................................................................................24

2.2.2.1

Give-N Task....................................................................................................24

2.2.2.2

Match-to-Sample Task: Number Word Trials.................................................25

2.2.2.3

Match-to-Sample Task: Color/Mood Trials....................................................26

2.3

Experiment 1B...............................................................................................................28

2.3.1

Method.....................................................................................................................28

iv 2.3.1.1

Participants......................................................................................................28

2.3.1.2

Match-to-Sample Task....................................................................................28

2.3.1.3

Give-N Task....................................................................................................31

2.3.2

Results......................................................................................................................31

2.3.2.1

Give-N Task....................................................................................................31

2.3.2.2

Match-to-Sample Task....................................................................................31

2.4

Discussion......................................................................................................................39

CHAPTER 3: Connecting Number Words to Discrete Quantification............................41 3.1

Introduction....................................................................................................................41

3.2

Experiment 2A...............................................................................................................42

3.2.1

Method.....................................................................................................................42

3.2.1.1

Participants......................................................................................................42

3.2.1.2

Cups Task........................................................................................................42

3.2.1.3

Give-N Task....................................................................................................45

3.2.2

Results......................................................................................................................45

3.2.2.1

Give-N Task....................................................................................................45

3.2.2.2

Cups Task Number Word Trials......................................................................46

3.2.2.3

Cups Task Quantifier Trials............................................................................47

3.3

Experiment 2B...............................................................................................................49

3.3.1

Method.....................................................................................................................51

3.3.1.1

Participants......................................................................................................51

3.3.1.2

Cups Task........................................................................................................51

3.3.1.3

Give-N Task....................................................................................................53

3.3.2

Results......................................................................................................................53

3.3.2.1

Give-N Task....................................................................................................53

3.3.2.2

Cups Task........................................................................................................54

3.3

Discussion......................................................................................................................58

CHAPTER 4: Using Linguistic Context to Connect Number Words to Discrete Quantification...................................................................................................................60 4.1

Introduction....................................................................................................................60

4.2

Experiment 3A...............................................................................................................61

4.2.1

Method.....................................................................................................................61

4.2.1.1

Participants......................................................................................................61

v 4.2.1.2

Procedure.........................................................................................................61

4.2.2

Results......................................................................................................................62

4.3

Experiment 3B...............................................................................................................64

4.3.1

Method.....................................................................................................................64

4.3.1.1

Participants......................................................................................................64

4.3.1.2

Cups Task........................................................................................................65

4.3.1.3

Give-N Task....................................................................................................67

4.3.2

Results......................................................................................................................68

4.3.2.1

Give-N Task....................................................................................................68

4.3.2.2

Cups Task........................................................................................................69

4.4

Experiment 3C...............................................................................................................71

4.4.1

Method.....................................................................................................................72

4.4.1.1

Participants......................................................................................................72

4.4.1.2

Cups Task........................................................................................................72

4.4.1.3

Give-N Task....................................................................................................74

4.4.2

Results......................................................................................................................74

4.4.2.1

Give-N Task....................................................................................................74

4.4.2.2

Cups Task........................................................................................................74

4.5

Experiment 3D...............................................................................................................76

4.5.1

Method.....................................................................................................................77

4.5.1.1

Participants......................................................................................................77

4.5.1.2

Cups Task........................................................................................................77

4.5.1.3

Give-N Task....................................................................................................78

4.5.2

Results......................................................................................................................78

4.5.2.1

Give-N Task....................................................................................................78

4.5.2.2

Cups Task........................................................................................................78

4.6

Discussion......................................................................................................................80

CHAPTER 5: General Discussion & Conclusions...........................................................83 5.1

Summary of Results.......................................................................................................83

5.2

Discussion of Issues.......................................................................................................84

5.3

Overall Conclusions.......................................................................................................87

REFERENCES.................................................................................................................89

vi

LIST OF FIGURES

Page

Figure 1. Experiment 1A: Examples of Color Stimuli.....................................................18 Figure 2. Experiment 1A: Examples of Size Stimuli........................................................19 Figure 3. Experiment 1A: Examples of Number Trial.....................................................20 Figure 4. Experiment 1A: Examples of Color Trial.........................................................21 Figure 5. Experiment 1A and 1B: Age Ranges of Participants........................................25 Figure 6. Experiment 1A: Results for Number, Color, & Mood......................................27 Figure 7. Experiment 1A and 1B: Comparison of Stimuli...............................................29 Figure 8. Experiment 1B: Results for Number, Color, & Mood......................................32 Figure 9. Experiment 1B: Example of Trial Type G........................................................33 Figure 10. Experiment 1B: Results for Trial Type G.......................................................34 Figure 11. Experiment 1B: Example of Trial Type F.......................................................35 Figure 12. Experiment 1B: Results for Trial Type F........................................................35 Figure 13. Experiments 1A & 1B: Comparison of Results..............................................37 Figure 14. Experiment 2A: Design...................................................................................45 Figure 15. Experiment 2A: Age Range of Participants....................................................46 Figure 16. Experiment 2A: Responses to “Which cup has five [six]?”............................47 Figure 17. Experiment 2A: Responses to “Which cup has more [a lot]?”.......................48 Figure 18. Experiment 2B: Design...................................................................................53 Figure 19. Experiment 2B: Age Range of Child Participants...........................................54 Figure 20. Experiment 2B: Results for Number Word Trials...........................................55 Figure 21. Experiment 2B: Results for Color Word Trials...............................................57 Figure 22. Experiment 3B, 3C, & 3D: Design..................................................................67 Figure 23. Experiments 3B, 3C, & 3D: Age Range of Child Participants.......................69 Figure 24. Experiment 3B: Results...................................................................................70 Figure 25. Experiment 3C: Results...................................................................................75 Figure 26. Experiment 3D: Results...................................................................................79

vii

LIST OF TABLES

Page

Table 1. Experiment 1A: Trial Types...............................................................................22 Table 2. Experiment 1A & 1B: Trial Types.....................................................................30 Table 3. Experiment 1A & 1B: Mean Percentage of Correct Responses.........................38 Table 4. Experiment 3A: Counts of Number Word Utterances........................................63 Table 5. Experiment 3A: Counts of Cardinal Number Word Utterances.........................63

viii

ACKNOWLEDGMENTS

I would like to extend my thanks to the children, families, and staff of the preschools where this research was conducted. Thanks also to Francesca DelGobbo, Meghan Goldman, Geoff Iverson, Michael Lee, Penelope Maddy, James Negen, Lisa Pearl, Barbara Sarnecka, and Ted Wright for their helpful comments on earlier versions of the manuscript. Finally, I would like to offer my thanks to lab managers Helen Braithwaite and Emily Carrigan and to research assistants Elaine Cheong, Pierina Cheung, Stephanie Cline, Annie Ditta, Meghan Li, Roxanna Salim, Jill Sorathia, Rochelle Telebrico, and Farah Touiller. This research was supported by NIH R03HD054654 granted to the dissertation chair, Dr. Barbara Sarnecka. Preliminary analyses of a subset of these data were presented at the International Conference on Infant Studies in March, 2010 and the biennial meetings of the Society for Research in Child Development in March, 2007 and April, 2009.

ix

CURRICULUM VITAE

EDUCATION University of California, Irvine (2010), Ph.D. Psychology. University of California, Irvine (2008), M.A. Psychology. University of California, Irvine (2003), B.A. Psychology (with minor in Education). PUBLICATIONS Slusser, E. & Sarnecka, B. W. (under revision). Connecting numbers to discrete quantification: A step in the child’s construction of integer concepts. Slusser, E. & Sarnecka B. W. (submitted). Another picture of eight turtles: A child’s understanding of cardinality and numerosity. Slusser, E., Sarnecka, B. W., & Cheung, P. (in preparation). Using linguistic context to connect number words with discontinuous quantification: Converging evidence from English and Mandarin. CONFERENCE PRESENTATIONS Slusser, E. & Sarnecka, B. W. (2010). Children’s use of morpho-syntactic information to connect number words to discrete quantification. Paper presented at the International Conference on Infant Studies. Baltimore, MD. Slusser, E. (2009). Distinguishing number words from quantifiers and adjectives. Paper presented at the Symposium on Cognitive and Language Development. Los Angeles, CA. Slusser, E. & Sarnecka, B. W. (2009). Children’s partial understanding of number words. Poster presented at the Society for Research in Child Development. Denver, CO. Slusser, E. (2008). Understanding that number words refer to the number of discrete items in a set. Paper presented at the Symposium on Cognitive and Language Development. Irvine, CA. Slusser, E. & Sarnecka, B. W. (2007). Are number words exclusively reserved for number? Paper presented at the Symposium on Cognitive and Language Development. Los Angeles, CA. Slusser, E. & Sarnecka, B. W. (2007). When do young children connect number words to discrete quantification? Poster presented at the Society for Research in Child Development. Boston, MA. Slusser, E. (2006). When do young children understand that number refers to discrete quantification? Paper presented at the Symposium on Cognitive and Language Development. Irvine, CA.

x Slusser, E. (2003). Early childhood gender differences along various dimensions of social behavior. Paper presented at the UC, Irvine Undergraduate Research Symposium. Irvine, CA. Slusser, E. (2003). Early childhood gender differences along various dimensions of social behavior. Paper presented at the Stanford Undergraduate Psychology Conference. Palo Alto, CA. Slusser, E. (2003). Early childhood gender differences along various dimensions of social behavior. Paper presented at the University of Washington Undergraduate Research Symposium. Seattle, WA. Slusser, E. (2003). Early childhood gender differences along various dimensions of social behavior. Paper presented at the Southern California Conference of Undergraduate Research. Pasadena, CA. RESEARCH EXPERIENCE 2005-2010: Lab Member and Manager: UC, Irvine Cognitive Development Lab. Designing and implementing a research paradigm investigating how young children connect language to number concepts. 2004-2005: Co-Investigator: proposed development and data collection. “Measuring Effects of ‘Jumpstart’ Intervention on Child’s English Language, Literacy Skills”. 2001-2003: Lead Investigator: Summer Undergraduate Research Fellowship sponsored by UC Irvine. “Early Childhood Gender Differences along Various Dimensions of Social Behavior”. 2001-2003: Research Assistant: Child Development Center. “Methyl-Phenidate Efficacy and Safety in Preschoolers with ADHD” 2002-2003: Research Assistant: UC Irvine Consortium for Integrated Health Studies. “Monitoring Adolescent Stress and Health”. 2002-2003: Research Assistant: UC Irvine Consortium for Integrated Health Studies. “PalmTop Partners”. TEACHING EXPERIENCE Fall 2006-Spring 2010: Teaching Assistant for UC, Irvine. Courses assisted: Cognitive Development (Psych 141D), Developmental Psychology (Psych 120D), Human Memory (Psych 140M), Intro to Psychology (Psych 7A), Intro to Syntax (Ling 20), and Intro to Linguistics (Ling 3). Fall 2005 & Winter 2006: Instructor for UC, Irvine. Courses instructed: Jumpstart Pre-K Ed (Psych 141A / Educ 141J) RELEVANT WORK EXPERIENCE 8/2003-12/2005: Site Manager for Jumpstart at UC, Irvine. Initiated Jumpstart program at UC Irvine in Fall 2003; hired and supervised over 150 undergraduate students; customized and facilitated over 60 hours of training on early childhood education;

xi initiated and supported assessments on language/literacy and social development of preschool children. HONORS AND AWARDS UC Irvine Social Sciences Fellowship Recipient Order of Merit, Phi Beta Kappa, Cum Laude Graduate UC Irvine Undergraduate Research Fellowship Recipient

xii

ABSTRACT OF THE DISSERTATION

The Development of Number Concepts: Discrete Quantification and Numerosity By Emily Beth Slusser Doctor of Philosophy in Psychology University of California, Irvine, 2010 Professor Barbara W. Sarnecka, Chair

This line of research examines how children develop natural-number concepts through the use of emergent language and innate representational resources. Specifically, how children come to understand that number words 1) refer to numerosity, as opposed to some other property of individuals or sets, and 2) quantify over discrete entities. Some researchers contend that children understand these semantic restrictions from very early on and that this knowledge, in fact, guides children’s learning of the first few number- words. Conversely, it has been argued that children extrapolate these restrictions from the first few number-word meanings. Results from the present studies support this latter view by providing evidence that number concepts are constructed over time, via a predictable developmental trajectory. Study 1 demonstrates that only children who understand the cardinal principle of counting (those who are able to use counting to determine the number of items in a set) reliably extend number words to numerosity. Study 2 shows that children must learn the cardinal meanings of three or more number words in order to apply higher, unknown

xiii number words to discrete entities. Each of these two studies provides evidence that children do not connect number words to specific discrete quantities upon first encounter with the number-word list. Rather, they make the connection around the same time they figure out the cardinal principle. Study 3 addresses how children make this connection by exploring the proposal that children use the pairing of number words with count nouns as an indication that objects are preferred referents of number words. The results of this last study indicate that linguistic context facilitates learning the semantic functions of number words well before a child learns the specific meaning of each of these words. Additionally, this study provides supporting evidence for the claim that Mandarin noun classifiers provide information analogous to count mass distinctions in English. Implications for theories of number-concept construction will be discussed in relation to the findings of each of these studies.

1

CHAPTER 1: General Introduction

1.1 Motivation Acquiring the precise meanings of number words is a challenging task. Although ostensibly a task of word learning, number word acquisition cannot be accomplished through a basic one-to-one mapping of word to concept. While the ability to perceive and detect numerosity is available from infancy and a preliminary list of number words is acquired only two years later, it takes another two years to map number words to even the smallest numerosities. Hence, one of the key questions in the field of number word development: What accounts for this sustained period between the acquisition of the word and meaning? Researchers have proposed that children acquire natural-number concepts in a piecemeal fashion, via a process characterized as conceptual-bootstrapping (see Carey, 2009). This perspective predicts that there is a point at which children secure an understanding of the semantic properties inherent to number words, so as to map them to specific numerosities. For instance, children must understand that number words refer to properties of sets of discrete individuals. They must also understand that number words refer specifically to numerosity rather than, for example, summed spatial extent (i.e. quantity). It is possible that the time it takes to learn the semantic role of number words may, at least partially, account for the delay. The motivation for this research is to examine when and how these inductions are made.

2 1.2 Background 1.2.1 Core Cognition The following proposal will assume that several domains of core cognition 1 serve as a foundation for the construction of natural-number concepts (see Carey, 2009). Although the precise characterization of core cognition has many philosophical and theoretical implications for any study of human thought, this paper will restrict the definition to: Conceptual knowledge emergent in early development (infancy) that, due to its central and integrated nature, is fundamentally different from representations provided through low-level sensation and perception. Cognition of this sort has been identified for several domains including, but not restricted to: Causality, agency, navigation, objects, and number. What follows is a discussion of those domains particularly important for aspects of number word learning explored in this paper. Objects. There is extensive evidence supporting the notion of core object cognition. Infants use information inherent only to rigid, cohesive objects (i.e. spatio- temporal continuity) to infer their existence as individualized objects (Kellman & Spelke, 1983; Spelke, Kestenbaum, Simons, & Wein, 1995). Infants maintain specific expectations for these objects, such as cohesion (Spelke & Van de Walle, 1993), solidity (Baillargeon, Needham, & DeVos, 1992; Spelke, Breinlinger, Macomber, & Jacobson, 1992), and common fate (Baillargeon, 2004). Likewise, when shown examples of solid objects and continuous (non-solid and non-rigid) substances, infants as young as 8

1 The reality of innate representations of this sort is debatable. The scope of this paper does not permit an argument for its existence but will rely on evidence that conceptual knowledge is, in fact, present in infancy. Whether it is acquired through domain-general learning devices or innate domain-specific mechanisms is irrelevant for the purposes of this paper.

3 months old demonstrate an understanding of object permanence and numerical identity for the discrete objects only (Huntley-Fenner, Carey, & Solimando, 2003). More recently, evidence has emerged suggesting that infants hold expectations for continuous substances that are distinctly different than those that they hold for objects (Hespos, Ferry, & Rips, 2009). For example, infants reason that substances can be penetrated and can change form (so as to fit different containers) whereas solid objects cannot. Expectations for objects and substances are thus mutually exclusive. These studies provide compelling evidence that this conceptual distinction is available from infancy, long before children acquire language reflecting the distinction. In fact, studies that pit children’s reasoning about ontological categories against various language cues clearly reinforce this point (Soja, 1992; Soja, Carey, & Spelke, 1991). Quantity and Numerosity. Core cognition of quantity and numerosity has been well investigated. There is clear evidence that infants can make distinctions between two sets according to continuous quantity. That is to say, infants respond to changes in total spatial extent (such as total contour length) when shown, for example, an array of two large squares with a total contour length of 16 cm and an array of two small squares with a total contour length of 24 cm (Clearfield & Mix, 1999; 2001). Infants also demonstrate sensitivity to changes in total volume, such as when making distinctions between glasses containing differing amounts of liquids (Gao, Levine, & Huttenlocher, 2000). Importantly, infants also show sensitivity to cardinal values of sets. That is to say, they are able to discriminate among differences in numerosity, when continuous quantity is not a factor (Brannon, 2002; Carey & Xu, 2001; Feigenson & Carey, 2003; 2005; Hauser & Carey, 2003; Lipton & Spelke, 2003; McCrink & Wynn, 2004; Wood &

4 Spelke, 2005; Xu, 2003; Xu & Spelke, 2000). Studies have shown that infants as young as six months are able to discriminate among approximate magnitudes of large numerosities (Brannon, 2002; Lipton & Spelke, 2003; McCrink & Wynn, 2004; Wood & Spelke, 2005, Xu & Spelke, 2000). They are also able to make more precise discriminations among arrays of one, two, or three objects (Carey & Xu, 2001; Feigenson & Carey, 2003; 2005; Hauser & Carey, 2003; Xu, 2003). These studies extensively control for continuous properties (such as summed spatial extent) showing that these discriminations are based on number alone. 1.2.2 Language Morphology. The ability to represent the categorical distinctions between objects and substances discussed above (§1.2.1) is manifested linguistically only shortly after children begin learning language. By the time they turn two, children accurately pluralize count nouns and use appropriate morphology to denote the difference between mass and count nouns (Barner, Chow, & Yang, 2009; Barner, Thalwitz, Wood, Yang, & Carey, 2007; Brown, 1973; Cazden, 1968; Clark & Nikitina, 2009; Gordon, 1988; Kouider, Halberda, Wood, & Carey, 2006; Mervis & Johnson, 1991; Soja, Carey, & Spelke, 1991). That is to say, they accurately use plural marking when referencing sets of discrete individuals and do not use this marking when referencing a mass of continuous substance. Children can also use morphological distinctions between objects and substances so as to extend novel words to substances that comprise objects, such as material or color, accordingly (Markman & Wachtel, 1988). Semantics. Children learn the meaning of approximate quantifiers (such as “more” and “a lot”) by about two years old (Dale & Fenson, 1996; Le Corre & Carey,

5 2007). Although they lack knowledge of the precise meanings of number words at this time, they do recognize that number words form a special class of words. Experiments with 2 to 5 year old children show that very few children (e.g. 0 out of 500 children 3 to 5 years old and 3 out of 40 children 2 year olds (Fuson, 1988, pg. 35)) mistakenly used anything other than number words when asked to recite the number word sequence or count an array of objects (Fuson, 1988; R. Gelman & Gallistel, 1978). Similarly, when given a letter, numeral, or Chinese character and asked “What number is this?” children as young as 2 years old (1;10 to 2;0) reliably offered only number words, showing that they understand that “number words” form a subordinate class (Schatz, 2005; Shatz & Backsheider, 2001). Being that children learn to recite the number sequence during the same time, it is likely that this experience helps children identify that number words belong to a certain class of words. Some maintain that experience with the counting routine facilitates children’s acquisition of the precise meanings of number words (e.g. Fuson, 1988; Wynn, 1990). However, the prolonged period of time necessary to learn that number words represent exact, cardinal numbers (e.g., Briars & Siegler, 1984; Frye, Braisby, Lowe, Maroudas, & Nicholls, 1989; Fuson, 1988; Le Corre, Van de Walle, Brannon, & Carey, 2006; Sarnecka & Carey, 2008; Wagner & Walters, 1982) calls this proposal into question. In fact, a robust pattern of number word acquisition has been shown to take place over a period of many months (often up to two years). During this time, children work out the cardinal meanings of the first few number words, one at a time and in order (Le Corre et al., 2006; Sarnecka & Lee, 2009; Wynn, 1990; 1992). In doing so, they go

6 through a series of number-knower levels, which are found not only for child speakers of English, but also of Japanese (Barner, Libenson, Cheung, & Takasaki, 2009; Sarnecka, Kamenskaya, Yamana, Ogura, & Yudovina, 2007), Mandarin Chinese (Li, Le Corre, Shui, Jia, & Carey, 2003), and Russian (Sarnecka et al., 2007). The pattern shows up clearly on the Give-N task (Frye et al., 1989; Le Corre et al., 2006; Sarnecka & Lee, 2009; Wynn, 1990; 1992), in which children are asked to produce a set of a particular number (e.g., ‘Please give three bananas to the puppet’). At the earliest level, the child’s responses are unrelated to the number requested indicating that the child has not yet learned the exact meanings of any of the number words. For example, children at this level (which we will call the pre-number-knower level) often give just one object, no matter what number word was requested. (Another common response is to give all the objects in the bowl, no matter what number word was requested.) At the next level, children give one object when asked for “one”, and two or more objects when asked for any other number word. We can call these children “one”- knowers, because they seem to have learned that “one” means one and that other number words have some cardinality greater than one. The “one”-knower level is followed by the “two”-knower level, then the “three”-knower level, and sometimes the “four”-knower level. Children in these levels can be called subset-knowers, as they have learned the meanings of “one,” “two,” “three” and/or “four” but have not yet figured out the cardinal principle of counting (Le Corre et al., 2006). The cardinal principle of counting (R. Gelman & Gallistel, 1978) is the rule that maps the number-word list to natural numbers by making the cardinal meaning of any number word dependent on that word’s position in the list. For example, the twenty-

7 seventh word in any number-word list (in any language) must mean 27. In practical terms the cardinal principle also means that the last word spoken in counting expresses the cardinality of the set that was counted. Children who understand the cardinal principle of counting can be called cardinal-principle-knowers. Cardinal-principle-knowers effectively know the meanings of all the words in their list, as a result of knowing the cardinal principle. Subset-knowers perform very differently from cardinal-principle-knowers on many tasks: Cardinal-principle-knowers use counting to generate sets of a particular number, to check the number of items in a set, and to fix their answers when they make mistakes. Subset-knowers know how to count, but they do not use counting to generate, check or fix sets. They seem not to understand how counting reveals cardinality (Frye et al., 1989; Fuson, 1988; 1992; Le Corre et al., 2006; Sarnecka & Carey, 2008). Studies have shown that cardinal-principle-knowers apply number words to collections of objects as well as a variety of other types of entities, so long as they can be perceptually individuated. For example, children will apply number words to series of events (Wagner & Carey, 2003), pieces of whole objects (Shipley & Shepperson, 1990), or sets of holes (Giralt & Bloom, 2000). Nevertheless, results from each of these studies suggest that, when given the option to choose between objects and another type of individuated entity, children prefer to apply number words to objects. 1.2.3 Constructionism and Bootstrapping The cognitive and linguistic abilities above lay the groundwork to ask how humans make inferences from innate (or at least early developed) cognitive resources to create new conceptual representations, thus forming more sophisticated abstract thought

8 processes. Specifically, how are natural-number concepts constructed from primitive resources that do not have numerical content sufficient to represent all positive integers (Carey & Sarnecka, 2005)? Some posit that conceptual-role bootstrapping (or Quinean bootstrapping) accurately describes this process (Carey, 2009; Quine, 1960). This hypothesis proposes that children acquire the natural-number concepts by first learning a set of symbols (in this case, the number words) and then gradually imbue the words with meaning, piece by piece. In doing so, they are actually constructing the concepts that the words will stand for. Although each piece of meaning must build on what was there before, changes accumulate in such a way that the final knowledge state is truly, deeply different than initial knowledge state. Thus, the output (i.e. the numeral list representation of natural number) has more power than its antecedents. In the Quinean bootstrapping account, children’s construction of the concepts 5 and 6 is identical to their construction of meanings for the words “five” and “six.” Thus, the question of interest becomes whether we can identify a point where children figure out that “five” and “six” denote specific quantities of discrete individuals. 1.3 Questions 1.3.1 When Do Children Understand Number Words Refer to Numerosity? Part of knowing the meaning of any number word is knowing that it denotes a specific numerosity. Imagine that you (the reader) are shown a picture of some happy, green turtles, and are told, “This is a picture of hachi turtles. Show me another picture of hachi turtles.” Then you are shown two other pictures: one has sad, orange turtles—about as many as in the original picture. The other has happy, green turtles—about twice as

9 many as in the original picture. Which do you choose? If you guess that hachi is an adjective (perhaps meaning green or happy) you choose the second option. But if you know that hachi is a number word, you choose the first. And this you can do without knowing exactly what hachi means 2 because you know that number words are about numerosity. When do children know this? One possibility is that children connect number words with numerosity from the time they figure out what “one” means (Sarnecka & S. Gelman, 2004; Wynn, 1992), or even earlier (R. Gelman & Gallistel, 1978). If so, then they could use numerosity as the basis for extending a number word from one set to another, even if they don’t know the exact meaning of that particular number word. They can then use this knowledge as a guide when learning of the first few number-word meanings. An alternative possibility is that early on, children do not see number words as connected to numerosity. In this case, children would need to extrapolate this knowledge from the first few number-word meanings (e.g. Condry & Spelke, 2008). This projection falls in line with the conceptual-role bootstrapping hypothesis described above (§1.2.3) in that children necessarily construct their concept of numerosity gradually. As of yet, there is empirical evidence supporting both of these possibilities. Some researchers have argued that children understand all number words as picking out numerosities (and even understand that each number word picks out a specific numerosity) much earlier than constructivist accounts would predict (R. Gelman, 1977; R. Gelman & Gallistel, 1978). Two studies using the knower-levels framework (Sarnecka

2 Hachi means eight, in Japanese.

10 & S. Gelman, 2004; Wynn, 1992) concluded that by the time children are “one”-knowers (long before they figure out the cardinal principle), they already see higher number words as referring to numerosity. In fact, these studies make an even stronger claim—that children understand each number word to pick out a specific numerosity, even though the children don’t yet know the numerosity associated with each word. More recently, however, researchers present evidence that children do not see higher number words as picking out specific numerosities until the children figure out the cardinal principle of counting (Condry & Spelke, 2008). Chapter 2 will elaborate on studies behind these claims as well as present two new experiments designed to reconcile these discrepancies by drawing a distinction between quantity (e.g. summed spatial extent) and numerosity. 1.3.2 When Do Children Understand Number Words Refer to Discrete Quantification? Another piece of knowledge integral to natural-number concepts is the knowledge that number is a property of sets of discrete individuals, and not of continuous masses. To illustrate, the following dialogue ensued after James, a 2½-year-old, turned off the faucet in his bathtub. Mother: Oh, you turned off the faucet because you noticed that it was full. James: Yes, we needed enoughs of water. We needed one, two, three, four, five, six, seven, eight, nine, ten of water. (Sarnecki, 2007)

This exchange is interesting both for what James appears to know and for what he does not know about number words. Although he seemed to understand that number words quantify, he did not restrict their quantification to discrete entities. Instead, he applied them to water. Again, this intermediate state reflected by James’ partial

11 knowledge of number concepts is predicted by conceptual-role bootstrapping. That is to say, it demonstrates that children may acquire an understanding of the broad semantics of number words as they acquire the specific meaning of number words. Chapter 3 will investigate this further by addressing the question: When do children learn that number words form a semantic class whose referents are restricted to discrete individuals? This entails that they refer to sets of things that are individually separate and perceptually distinct from one another, rather than non-individuated continuous substances such as water 3 . As discussed above (§1.2.1), children bring a robust conceptual distinction between objects and substances into this process of number word learning. Since they also recognize number words as belonging to a subordinate class (§1.2.2), it is possible that upon encountering a number word children identify it as a member of this class and appropriately extend it only to discrete individuals. There are no cognitive, conceptual, or linguistic limitations that would impede this task so this connection could conceivably be made immediately. However, in line with the above anecdote, it is possible that children attain this knowledge at a later stage of number word learning, perhaps not until they induce the cardinality principle. Chapter 3 will present two experiments designed to investigate when children make this connection relative to their understanding specific meanings of number words.

3 Adults, of course, do use number words to quantify over continuous substances, but only by adding an intermediate unit to make the substances discrete (e.g., five drops/cups/buckets of water). Children, however, do not typically use discretizing measurement terms until around 8 years of age (Fuson, 1988; Huntley-Fenner, 2001).

Full document contains 109 pages
Abstract: This line of research examines how children develop natural-number concepts through the use of emergent language and innate representational resources. Specifically, how children come to understand that number words (1) refer to numerosity, as opposed to some other property of individuals or sets, and (2) quantify over discrete entities. Some researchers contend that children understand these semantic restrictions from very early on and that this knowledge, in fact, guides children's learning of the first few numberwords. Conversely, it has been argued that children extrapolate these restrictions from the first few number-word meanings. Results from the present studies support this latter view by providing evidence that number concepts are constructed over time, via a predictable developmental trajectory. Study 1 demonstrates that only children who understand the cardinal principle of counting (those who are able to use counting to determine the number of items in a set) reliably extend number words to numerosity. Study 2 shows that children must learn the cardinal meanings of three or more number words in order to apply higher, unknown number words to discrete entities. Each of these two studies provides evidence that children do not connect number words to specific discrete quantities upon first encounter with the number-word list. Rather, they make the connection around the same time they figure out the cardinal principle. Study 3 addresses how children make this connection by exploring the proposal that children use the pairing of number words with count nouns as an indication that objects are preferred referents of number words. The results of this last study indicate that linguistic context facilitates learning the semantic functions of number words well before a child learns the specific meaning of each of these words. Additionally, this study provides supporting evidence for the claim that Mandarin noun classifiers provide information analogous to count mass distinctions in English. Implications for theories of number-concept construction will be discussed in relation to the findings of each of these studies.