Conceptual Change and Bootstrapping REESE 2011
See also BBS A precis.
Innate Primitives Learning mechanisms Adult conceptual repertoire
CS1 Learning mechanisms CS2
Three Theses Innate Primitives: Core Cognition Change: Discontinuities Learning Mechanism: Quinian Bootstrapping
Three Theses CS1: Rich initial systems of representation Change: Discontinuities Learning Mechanism: Quinian Bootstrapping
Discontinuity Conceptual Change Increases in Expressive Power
Case Study Natural Number Peano s Axioms: order, successor function Frege: 1-1 correspondence, 2 nd order logic
Innate Representations with Numerical Content Three Systems of Core Cognition 1) Analog magnitude representations of number. Dehaene s number sense. 2) Parallel representation of small sets of individuals. 3) Natural language quantification. Setbased quantification that underlies linguistic quantifiers.
Core System 1: Analog Magnitude Representations Representations of approximate cardinal values of large sets of individuals (at least to the 100s)
Analog Magnitude Models Number represented by a quantity linearly or logarithmically related to the cardinal value of the set. Weber s law. One: Two: Three: Seven: Eight:
mathematics achievement Four findings in human adults & children 1. ANS compromised in children and adults with dyscalculia. 2. 2. Middle-school students who have gotten better grades at math have sharper nonsymbolic number representations. Non-symbolic numerical abilities at 14 years of age are specifically associated with symbolic math achievement back to third grade (~age 8). finer number discrimination coarser (Halberda, Mazzoco & Feigenson, 2008)
"Look, here come some blue dots!" "Look, here come some blue dots!" B Four findings in human adults & children "Now they're being covered up!" "Now they're being covered up!" "And here come some red dots!" "Here come some more blue dots. Now they're ALL back there!" "Are there more blue dots, or more red dots?" 3. Children who are better at nonsymbolic arithmetic go on to greater Preschool achievement addition in task: first-year more blue school dots or mathematics. more red dots?................................ "And here come some red dots!" "Are there more blue dots, or more red dots?" +math & literacy achievement test 100 scores 90 C 100% 95% 90% 85% 80% Sum larger (with literacy Sum smaller score controlle 40 answer = "blue" answer = "red" 75% Nonsymbolic addition predicts achievement in kindergarten math. 70% 65% (Gilmore, McCarthy & Spelke) 80 70 60 50 40 50 60 70 80 90 100 Non-Symbolic Addition Score (%) r =.587, p<.001 r =.530, p=.001)
Four findings in human adults & children 4. Before children learn symbolic arithmetic, they draw on the approximate number system to solve symbolic number problems. ratio dependence addition = comparison subtraction < addition also: numbers on line (Gilmore, MCarthy & Spelke,)
Back to infants... number? addition? 100 90 80 70 60 50 40 40 50 60 70 80 90 100 Non-Symbolic Addition Score (%) 4 + 5 > 6 6 + 5 < 17 Infants & animals including birds, rodents, and monkeys have a cognitive system with genuine numerical content: a foundation for human learning & performance of formal mathematics.
Characteristics Evolutionary and ontogenetic continuity. Amodal Analog format Numerical computations carried out over these representations: --addition, subtraction, numerical comparison, ratio calculation, division by 2, multiplication by 2 (at least).
Core System 2 Parallel individuation, attention/short term memory signature, limited to 3 or 4 items attended to in parallel Not a dedicated number representational system. Implicitly represents the number of individuals in small sets
Evidence for Parallel Individuation The choice task (Feigenson, Carey & Hauser, 2002)
Infant Ordinal Choice Percent Correct 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 10 mos 1vs2 12 mos 1vs2 10 mos 2vs3 12 mos 2vs3 10 mos 3vs4 12 mos 3vs4 3vs6 2vs4 1vs4
Parallel Individuation Models 1 box 2 boxes 3 boxes --one symbol for each individual --no symbols for integers (or any other quantifiers), just symbols for individuals
Not a dedicated number representation system Computations defined over these representations include: --sum total spatial extent, comparisons of total volume, area (as in cracker choice task) --1-1 correspondence, establish numerical equality/inequality; numerical more/less (as in spandex slit manual search task) --chunking/set binding (can do two sets, each subject to limit of 3 e.g., cracker choice)
Properties Evolutionary and ontogenetic continuity Amodal Probably iconic format Numerical computations carried out over these representations: comparison of sets on the basis of 1-1 correspondence to establish numerical equivalence, computatations of numerical identity, updating models of sets of objects as individuals are added or subtracted.
Representations of Sets Attentional mechanisms pick out sets of individuals AM only summary symbol for approximate cardinal values of sets PI working memory model of set, one symbol for each individual, no symbols for quantifiers or cardinal values.
{ } {a,b,c} {a,b,d} {b,c,d} {a,c,d}... {a,b} {a,c} a b {a,d} {b,c} {b,d} {c,d} c d... = At...
Transcending Core Cognition: The Descriptive Problem Object File Representations --No symbols for integers --Set size limit of 3 or 4 Analog Magnitude Representations --Cannot represent exactly 5, or 15, or 32 --Obscures successor relation Set Based Quantification --No symbols for cardinal values above 2 or 3. --Not embedded in arithmetical computations
The Numeral List Gelman and Gallistel s Counting Principles. Implements the successor function. Represents a finite subset of the positive integers. 7 is one more than 6, which is one more than 5, which is one more than 4 Child can think thoughts formulated over the concept seven.
Interim conclusion infants represent number yes, but not natural number. Infants, non-human animals cannot think thoughts formulated over the concept seven Nor can people in cultures/languages with no numeral list.
Peter Gordon: The Pirahã Hunter-gatherers Semi Nomadic Maici River (lowland Amazonia) Pop: about 160-200 Villages 10 to 20 people Monolingual in Pirahã Resist assimilation to Brazilian culture Limited trading (no money) No external representations (writing, art, toys )
Proportion Correct Brief Presentation (Subitizing) Brief Presentation 1 0.75 0.5 0.25 0 Targe t 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8-10 Proportion Correct One-to-One Line Match 1-1 Line M atch 1 0.75 0.5 0.25 0 Targe t
Line Copying
Proportion Correct. Line Draw Copy Line-draw Copy 1 0.75 0.5 0.25 0 Targe t 1 2 3 4 5 6 7
Coeff Var Mean SD Averaged Responses Across Tasks 9 2 8 7 6 1.5 5 4 1 Mean SD 3 2 0.5 1 0 1 2 3 4 5 6 7 8 9 0 Target 0.2 0.15 0.1 0.05 Coeff Var 0 1 2 3 4 5 6 7 8 9 Target
Discontinuity CS1-CS2 Difficult to learn Descriptive Within-child consistency Explanatory What learning process underlies CS1- CS2 transition?
Wynn s Difficulty of Learning Argument Give a number task No-number words. one- knowers two -knowers three -knowers ( four -knowers) CP-knowers
One, two, three, four and nothing more. Every child who knows the cardinal principles (i.e., how counting represents number) has assigned numerical meaning to one two three and four. No other numerals are assigned numerical meaning before the construction of the counting principles.
Conclusions Numerical meanings for one through four alone support the construction of the numeral list representation of the positive integers. These are not provided by AM system.
What ARE those meanings? Format? Process of assigning numerals to sets?
Format one {i} two {j k} three {l m n} four {o p q r} Enriched set-based quantification. Where i j k stand for numerically distinct individuals. Needn t be symbols for abstract individuals. E.g., two {left hand, right hand}
Process of assigning numerals to sets? Make a mental model in working memory of the set to be assigned a numerical value {dog dog}. Match this set, via 1-1 correspondence to stored models {j k}. Retrieve associated numeral: two. enriched parallel individuation
Explanatory Challenge: Quinian Bootstrapping Relations among symbols learned directly Symbols initially partially interpreted Symbols serve as placeholders Analogy, inductive leaps, inference to best explanation Combine and integrate separate representations from distinct systems
Bootstrapping the Integer List Representation of Integers How do children learn: The list itself? The meanings of each word? (that three has cardinal meaning three; that seven means seven)? How the list represents number (for any word X on the list whose cardinal meaning, n, is known, the next word on the list has a cardinal meaning n + 1).
Role of set-based quantification Details of the partial meanings children have for small numerals suggests that toddler s interpretation of one, two, ten is guided by quantifier representations. E.g., one- knowers: one functions like singular determiner a, two... ten function like plural, some.
A Bootstrapping Proposal Number words learned directly as quantifiers, not in the context of the counting routine. Supported by enriched parallel individuation and set-based quantification. One is learned just as the singular determiner a is. An explicit marker of sets containing one individual The plural marker -s is learned as an explicit marker of sets containing more than one individual.
continued Two, three, four are analyzed as quantifiers that mark sets containing more than one individual. Some children analyze two as a generalized plural quantifier, like some. Two is analyzed as a dual marker, referring to sets consisting of pairs of individuals. Three, four, contrast with two. Three is analyzed as a trial marker.
continued Meanwhile, the child has learned the counting routine. Child notices the identity of the first three words in the counting routine and the singular, dual, and trial markers one, two, three. Child notices analogy between two distinct follows relations next in the count list, and next in series of sets related by additional individual when sets compared on basis of 1-1 correspondence.
continued Induction If X is followed by Y in the counting sequence, adding an individual to an X collection results in what is called a Y collection. Adding an individual is equivalent to adding one, because one represents sets containing a single individual.
Giaquinto Observation Integration of integer list representation with analog magnitude representations important 101001 in base 2
What number is 101001? x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
41 What number is 101001?
Explanatory Challenge: Quinian Bootstrapping Relations among symbols learned directly Symbols initially partially interpreted Symbols serve as placeholders Analogy, inductive leaps, inference to best explanation Combine and integrate separate representations from distinct systems