1. Number Development Gaia Scerif Room 426, Ext. 67926 gs@psychology.nottingham.ac.uk Office Hours: Mon 2-4

2. Learning Objectives 1.Methods available to investigate infants’ knowledge 2.Necessary requirements for a mathematical system 3.Do infants represent an abstract number concept? 4.Can infants perform basic operations on number? 5.Linguistic and environmental influences on later number development

3. Methods: Habituation Changes in looking time as a measure of habituation to repeated presentations and dishabituation to novel stimuli Changes in looking time as a measure of habituation to repeated presentations and dishabituation to novel stimuli

4. Methods: Habituation Changes in looking time as a measure of habituation to repeated presentations and dishabituation to novel stimuli Changes in looking time as a measure of habituation to repeated presentations and dishabituation to novel stimuli Stimulus presented until infant’s attention wanes (looking time recorded) Stimulus presented until infant’s attention wanes (looking time recorded) Stimulus is presented again until looking time reaches criterion, e.g., half of looking time during first presentation ( = “habituation”) Stimulus is presented again until looking time reaches criterion, e.g., half of looking time during first presentation ( = “habituation”) Novel stimulus is presented: increased looking compared to last habituation trial ( = “ dishabituation”)? Novel stimulus is presented: increased looking compared to last habituation trial ( = “ dishabituation”)?

5. Methods: Habituation Changes in looking time as a measure of habituation to repeated presentations and dishabituation to novel stimuli Changes in looking time as a measure of habituation to repeated presentations and dishabituation to novel stimuli

6. Methods: Familiarization Relies on changes in looking time as measures of understanding (// habituation procedure) Relies on changes in looking time as measures of understanding (// habituation procedure) Identical object is presented to the left and right of fixation, for a set number of trials Identical object is presented to the left and right of fixation, for a set number of trials After a fixed number of trials, the familiar object is presented either to the right or left of fixation, paired with a novel stimulus After a fixed number of trials, the familiar object is presented either to the right or left of fixation, paired with a novel stimulus Longer looking to the novel stimulus? Longer looking to the novel stimulus? Unlike habituation, the overall time of presentation of stimuli is controlled by the experimenter (e.g., 12 trials), rather than by the infant Unlike habituation, the overall time of presentation of stimuli is controlled by the experimenter (e.g., 12 trials), rather than by the infant

7. Methods: Violation of Expectancy Also labelled “surprise test procedure” (Lee, 2000). Also labelled “surprise test procedure” (Lee, 2000). Relies on changes in looking time as measures of understanding (// habituation procedure) Relies on changes in looking time as measures of understanding (// habituation procedure) Objects are presented to the infant Objects are presented to the infant Scene is hidden by a screen, while an action is performed (e.g., adding or removing objects from behind the screen) Scene is hidden by a screen, while an action is performed (e.g., adding or removing objects from behind the screen) Screen is removed to reveal a display either consistent or inconsistent with the action Screen is removed to reveal a display either consistent or inconsistent with the action Longer looking to inconsistent test displays? Longer looking to inconsistent test displays?

8. Number: Mathematical Systems Represent numbers Represent numbers In terms of a set of mathematical entities In terms of a set of mathematical entities Small and large natural numbers Small and large natural numbers Operate on these representations Operate on these representations Using mathematical functions Using mathematical functions Addition Addition Subtraction Subtraction Multiplication, etc. Multiplication, etc. Requirements:

9. Early competency in number 1.Represent numbers In terms of a set of mathematical entities In terms of a set of mathematical entities 2.Operate on these representations Using mathematical functions Using mathematical functions If infants can be shown to both: Innate number concepts?

10. 1. Representing Number Piaget: it is only in the concrete and formal operational stages that children understand “Number” (Piaget, 1952) Piaget: it is only in the concrete and formal operational stages that children understand “Number” (Piaget, 1952) Challenges: children show understanding of the counting sequence before 6 years of age (Gelman & Gallistel, 1978) Challenges: children show understanding of the counting sequence before 6 years of age (Gelman & Gallistel, 1978)

11. 1. Representing Number One-week-old infants: habituated to 3-dot displays, dishabituate to 2-dot displays (Antell & Keating, 1983) One-week-old infants: habituated to 3-dot displays, dishabituate to 2-dot displays (Antell & Keating, 1983) Older infants: habituated to 2- object visual displays, dishabituate to 3-beat sounds (Starkey et al. 1980, 1983, 1990) Older infants: habituated to 2- object visual displays, dishabituate to 3-beat sounds (Starkey et al. 1980, 1983, 1990) Evidence of recognition of numerical sequences across modalities Evidence of recognition of numerical sequences across modalities

12. 1. Representing Number 6-month-old infants: habituated two groups to sequence (2 or 3 actions), dishabituate to 3 or 2 actions (Wynn, 2000). Taken as evidence of: 6-month-old infants: habituated two groups to sequence (2 or 3 actions), dishabituate to 3 or 2 actions (Wynn, 2000). Taken as evidence of: Spatial-temporal integration of number Spatial-temporal integration of number When motion is continuous, infants can parse into discrete number of segments When motion is continuous, infants can parse into discrete number of segments

13. Representing Number: Limits? Is infant number “Abstract Number”? Results with small numbers may depend on a system of object-files, rather than abstract number understanding (Carey, 2001): Results with small numbers may depend on a system of object-files, rather than abstract number understanding (Carey, 2001): Abilities and limits may depend on visual capacity to individuate and track small numbers of items rather than number itself (Trick & Pylyshyn, 1994; Uller et al., 1999) Abilities and limits may depend on visual capacity to individuate and track small numbers of items rather than number itself (Trick & Pylyshyn, 1994; Uller et al., 1999) They may also depend on attentional abilities (Simon et al., 1995, 1997) They may also depend on attentional abilities (Simon et al., 1995, 1997) Discrimination along continuous dimensions other than number could explain early number (Clearfield & Mix, 1999; Feigenson et al. 2002) Discrimination along continuous dimensions other than number could explain early number (Clearfield & Mix, 1999; Feigenson et al. 2002) E.g., Contour subtended, Area subtended E.g., Contour subtended, Area subtended

14. Approximate and Exact Number Approximate number: Approximate number: = Magnitude estimations Comparisons exhibit: Distance effect: Distance effect: RT: 3 v 6 < 3 v 4 Size effect: Size effect: RT: 3 v 4 < 8 v 9 Scalar variability (Moyer & Landauer, 1967): Scalar variability (Moyer & Landauer, 1967): Variability depends on size of number Exact Number: Exact Estimation Language dependent? Bilingual errors in approximate vs. exact estimations tasks: discussed in more detail later (Spelke & Tsivkin, 2001) Varieties of numerical abilities (Dehaene, 1993)

15. Evidence for infants estimating number using an approximate system: Evidence for infants estimating number using an approximate system: For small numbers (Wynn, 2000 vs. Carey, 2001) For small numbers (Wynn, 2000 vs. Carey, 2001) For large numbers (Xu & Spelke, 2000; Xu, 2003) For large numbers (Xu & Spelke, 2000; Xu, 2003) Approximate and Exact Number

16. Evidence for infants estimating number using an approximate system: Evidence for infants estimating number using an approximate system: For small numbers (Wynn, 2000 vs. Carey, 2001) For small numbers (Wynn, 2000 vs. Carey, 2001) For large numbers (Xu & Spelke, 2000; Xu, 2003) For large numbers (Xu & Spelke, 2000; Xu, 2003)

17. 1. Representing Number Summary of the evidence: Initial evidence suggested abstract representations of small numbers Initial evidence suggested abstract representations of small numbers Independent of modality of input Independent of modality of input Independent of perceptual properties of specific array Independent of perceptual properties of specific array This interpretation is highly debated. Issues and alternatives: This interpretation is highly debated. Issues and alternatives: Number needs to be teased apart from its continuous dimensions (area, contour, etc.) Number needs to be teased apart from its continuous dimensions (area, contour, etc.) Distinction between small numbers and large number approximate estimation? Distinction between small numbers and large number approximate estimation?

18. 2. Operating on Number 5-month-old infants: Addition and subtraction events (Wynn, 1992, in Lee, 2000) 5-month-old infants: Addition and subtraction events (Wynn, 1992, in Lee, 2000) Groups: Groups: 1 + 1 = 2 vs. 1 or 3? 1 + 1 = 2 vs. 1 or 3? 2 – 1 = 1 vs. 2? 2 – 1 = 1 vs. 2?

19. 2. Operating on Number 5-month-old infants: Addition and subtraction events (Wynn, 1992, in Lee, 2000) 5-month-old infants: Addition and subtraction events (Wynn, 1992, in Lee, 2000) Two groups: Two groups: 1 + 1 = 2 vs. 1 or 2 vs. 3? 1 + 1 = 2 vs. 1 or 2 vs. 3? 2 - 1 = 1 vs. 2? 2 - 1 = 1 vs. 2?

20. Operating on Number: Limits? Cohen & Marks (2002) Cohen & Marks (2002) Results could be explained by a combination of preference for: Results could be explained by a combination of preference for: familiar stimuli familiar stimuli displays containing large number of stimuli displays containing large number of stimuli Debate: Can infants add and subtract? Debate: Can infants add and subtract?Evaluate: Cohen & Marks (2002) Cohen & Marks (2002) Commentaries by Wynn, Carey, Mix Commentaries by Wynn, Carey, Mix Cohen’s response Cohen’s response

21. Summary of the evidence: Infants can detect differences in displays representing operations on numerical representations Infants can detect differences in displays representing operations on numerical representations The bases for these discriminations are highly controversial The bases for these discriminations are highly controversial Could perceptual (e.g., continuous extent) and mnemonic biases (e.g., familiarity) underlie development of numerical understanding? Could perceptual (e.g., continuous extent) and mnemonic biases (e.g., familiarity) underlie development of numerical understanding? 2. Operating on Number

22. Beyond Infancy: What develops? 1. Infants numerical knowledge is limited Small numbers: Exact number estimation (at least for small numbers), or perceptually and attentionally based object-files? Small numbers: Exact number estimation (at least for small numbers), or perceptually and attentionally based object-files? Large numbers: Approximate estimation? Large numbers: Approximate estimation? 2. The preverbal number systems we have discussed may form the bases for the acquisition of later referential number labels (Wynn, 1990, Gelman & Gallistel, 1992) 3. Later numerical knowledge is more likely to be linguistically and culturally-determined

23. 2. Beyond Infancy: What develops? What are the relationships between infant number and number in childhood? Relationship to early estimation is poorly understood (Huntley-Fenner & Cannon, 2000) Relationship to early estimation is poorly understood (Huntley-Fenner & Cannon, 2000) Understanding counting (KEY PAPER: Wynn, 1990): Understanding counting (KEY PAPER: Wynn, 1990): One-to-one correspondence and cardinality develop slowly One-to-one correspondence and cardinality develop slowly Understanding numerical equivalence (Mix, 1999): Understanding numerical equivalence (Mix, 1999): Gradual more sophisticated emergence of equivalence (initially greatly affected by perceptual appearance of stimuli) Gradual more sophisticated emergence of equivalence (initially greatly affected by perceptual appearance of stimuli)

24. Equivalence (Mix, 1999): Equivalence (Mix, 1999): Gradual more sophisticated emergence of equivalence (initially affected by perceptual appearance of stimuli) Gradual more sophisticated emergence of equivalence (initially affected by perceptual appearance of stimuli) 2. Beyond Infancy: What develops

25. 3. Beyond Infancy: Role of culture Linguistic factors: influence on later numerical knowledge? Language systems used to name numerical identities Language systems used to name numerical identities English vs. Chinese children learning 11-19 at differential rates (Miller et al., 2000) English vs. Chinese children learning 11-19 at differential rates (Miller et al., 2000) Bilingual errors in approximate vs. exact estimations tasks (Spelke & Tsivkin, 2001) Bilingual errors in approximate vs. exact estimations tasks (Spelke & Tsivkin, 2001)

26. 3. Beyond Infancy: Role of culture Cross-cultural factors: influence on later numerical knowledge? Nature of the number system learned Nature of the number system learned e.g., Babylonians, using base 60 vs 10 e.g., Babylonians, using base 60 vs 10 Type of instruction given Type of instruction given e.g., Children using the discovery method vs. didactic methods for simple operations (Siegler & Jenkins, 1989; Steel & Funnell, 2001) e.g., Children using the discovery method vs. didactic methods for simple operations (Siegler & Jenkins, 1989; Steel & Funnell, 2001)

27. Beyond Infancy: Summary Later exact numerical abilities are: Slow to develop, perhaps based on the early systems we discussed for infants Slow to develop, perhaps based on the early systems we discussed for infants Influenced by linguistic and environmental factors Influenced by linguistic and environmental factors

28. Summary and links to theory 1.It is highly debated whether infants represent an abstract number concept, and whether they can perform basic operations on number: 1.Nativist (e.g., Wynn, Spelke, Carey) vs. constructivist (e.g., Cohen) approaches 2.Domain-specific (e.g., Wynn, Spelke) vs. domain-general (e.g., Cohen, Simon – Carey?) approaches 2.Infant “number” may be based on a system of approximate numerosity. Exact number seems to be more dependent on linguistic and cultural influences

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