Fast Comprehension of Geometrical Rules in Human Adults and Preschoolers

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Fast Comprehension of Geometrical Rules in Human Adults and Preschoolers

Marie Amalric, Lìpíng Wáng, Pierre Pica, Santiago Figueira, Mariano Sigman, Stanislas Dehaene

To cite this version:

Marie Amalric, Lìpíng Wáng, Pierre Pica, Santiago Figueira, Mariano Sigman, et al.. Fast Compre-hension of Geometrical Rules in Human Adults and Preschoolers. 28th Annual Convention of the Association for Psychological Science, May 2016, Chicago, United States. �halshs-02434355�

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Marie Amalric

1,2,*

, Liping Wang

3 , Pierre Pica

4 , Santiago Figueira

5 , Mariano Sigman

6 , Stanislas Dehaene

1,7

1 _{Cognitive Neuroimaging Unit, CEA DSV/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France}

2_{Sorbonne Universités, UPMC Univ Paris 06, IFD, 4 place Jussieu, Paris, France}

3 _{Institute of Neuroscience, Shanghai Institutes of Biological Science, Chinese Academy of Science, Shanghai, China} 4_{UMR 7023 Structures Formelles du Langage CNRS, Université Paris 8, France}

5 _{Department of Computer Science, FCEN, University of Buenos Aires, Pabellon I, Ciudad Universitaria, Buenos Aires, Argentina.}

6_{Laboratory of Integrative Neuroscience, Physics Department, Faculty of Exact and Natural Sciences, University of Buenos Aires, Argentina} 7_{Collège de France, Paris, France}

Fast comprehension of embedded geometrical primitives

and rules in human adults and preschoolers

 Studies of sequence learning have outlined one possible mechanism by which complex mental representations are constructed out of simpler primitives: the human ability to

extract complex nested structures from sequential inputs.

 Experiments in infants, preschoolers, and adults without

access to education have demonstrated the existence of innate “core knowledge” for space, endowing humans with spontaneous intuitions of geometry.

 The question therefore arises whether a capacity for the internal representation and manipulation of nested

sequences also underlies the acquisition of mathematics.

 We propose to formalize the human sensitivity to mathematical rules through a “language of thought” that allows the formation of complex representations from a small repertoire of primitives.

d’

CONCLUSIONS

 Simple rotations and axial symmetries

were all detected and quickly used by

human adults and 5-years-old children. Point symmetry was more challenging for French preschoolers or Munduruku adults than for French adults.

 Human subjects were able to detect most of the embedded expressions we

used to define our visuospatial

sequences such as simple repetition, concatenation, and some repetition with variation.

 The analysis of error patterns provided

direct evidence for hierarchical

embedding: superficial rules were

acquired more quickly and induced fewer errors than deeper rules.

 With age, a geometrical language

endowed with nested rules seems to

arise even in the absence of formal schooling, as Munduruku adults who

lacked school-based education,

performed better than 5-years-old kids.  In children, the failure with complex

sequences could arise from limitations in working memory and not necessarily to a lack of understanding nested structures.

 The theoretical complexity of a

sequence was an excellent predictor of its mean error rate, and we confirm that

minimal description length is a

reasonable approach of adult sequence learning capacity.

 Additional primitives, both geometrical and non-geometrical still need to be added to our model to complete its description of “core geometry”.

RESULTS

Subjects saw the first locations of a given sequence and had to point to the next ones. If they were mistaken, the sequence restarted.

Participants:

23 French adults and 47 5-years-old children, and 14 Munduruku teenagers and adults.

Repeat +1 01234567 K=5 Repeat +2 02460246 K=5 [+0, [+1]7_] _{[+0, [+2]}7_] Alternate 0213243546 K=8 2arcs 01237654 K=8 [+0, +2]8_{+1}] _{[[+0, [+1]}3_]2_<V>] 2squares 02467135 K=8 4segments 01726354 K=7 4diagonals 04152637 K=7 2points 02020202 K=7 [[+0, [+2]3_]2 _<−1>] _{[[+0, H]}4_{−1}] _{[[+0, P]}4_{+1}] _{[[+0, +2]}4_{+0}] 2rectangles 05416327 K = 10 2crosses 04512673 K=7 Irregular 04715236 K = 16 4points 02360236 K = 11 [[[+0, −3]2_{P}]2_<−2>] _{[[+0, P ]}4_{−3}] _{[ +0,P,B,+2,P,B,+1,H ]} _{[[+0,+2,+1,+3]}2 _<+0>] +1 or H +2 -1 or A -2 0 B V P

Knowledge of geometrical primitive rules

French adults: French children: Munduruku adults:

MODEL FITS

0 % 50 % 100 % Error rate: Irregular baseline Regular sequence Clicks – Number of participants: Actual sequence 0 0.2 0.4 0.6 0.8 1 repeat 3 456 78910111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 alternate 34567 8910111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 4segments 3 456 78910111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 4diagonals 34567 8910111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 2arcs 3 4 5 6 7 8 9 10111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 2squares 3 4 5 6 7 8 910111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 2rectangles 3 4 5 6 7 8 9 10111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 2crosses 3 4 5 6 7 8 910111213141516 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 6 7 8 12 13 14 15 16 1 2 3 4 5 6 7 8

Learning of embedded rules

French children: Munduruku adults: repeat repeat+2 2points 2arcs 2squares 4points 4segments 4diagonals 2rectangles 2crosses irregular repeat alternate 2arcs 2squares 4segments 4diagonals 2rectangles 2crosses irregular 5 6 7 8 9 10 11 12 13 14 15 16 0 0.2 0.4 0.6 0.8 1 ρ = 0.75 5 6 7 8 9 10 11 12 13 14 15 16 0 0.2 0.4 0.6 0.8 1 5 6 7 8 9 10 11 12 13 14 15 16 0 0.2 0.4 0.6 0.8 1 Complexity 2crosses 4diagonals ρ = 0.52 ρ = 0.59 2crosses 4diagonals French adults: % errors

Kolmogorov complexity well predicts

subjects’ performance

Noisy adult model

Reduced instruction set (Recursion, P) Children data

Munduruku data

Reduced instruction set (P)

% correct Adult data

Model

Repeat 2arcs 2squares 4seg 4diag 2rect 2crosses irregular

Data point #

1. Dehaene S, Meyniel F, Wacongne C, Wang L, Pallier C. The Neural Representation of

Sequences: From Transition Probabilities to Algebraic Patterns and Linguistic Trees. Neuron. 2015;88: 2–19.

2. Saffran JR, Aslin RN, Newport EL. Statistical Learning by 8-Month-Old Infants. Science. 1996;274: 1926–1928.

3. Restle F. Theory of serial pattern learning: structural trees. Psychol Rev. 1970;77: 481 4. Martins MD, Laaha S, Freiberger EM, Choi S, Fitch WT. How children perceive fractals:

Hierarchical self-similarity and cognitive development. Cognition. 2014;133: 10–24.

5. Dehaene S, Izard V, Pica P, Spelke E. Core Knowledge of Geometry in an Amazonian Indigene Group. Science. 2006;311: 381–384.

6. Izard V, Pica P, Dehaene S, Hinchey D, Spelke E. Geometry as a Universal Mental Construction. Space, Time and Number in the Brain. Elsevier; 2011; pp. 319–332.

7. Fodor JA. The Language of Thought. Harvard University Press; 1975

8. Romano S, Sigman M, Figueira S. $LT2_C2_{$ : A language of thought with Turing-computable}

Kolmogorov complexity. Pap Phys. 2013;5

9. Mathy F, Feldman J. What’s magic about magic numbers? Chunking and data compression in short-term memory. Cognition. 2012;122: 346–362

10. Feldman J. The simplicity principle in human concept learning. Curr Dir Psychol Sci. 2003;12: 227–232.

REFERENCES

Repeat +1 Repeat +2 Alternate 2arcs

2squares 4segments 4diagonals 2points

2rectangles 2crosses irregular 4points

1 2 7 6 8 3 5 4 1 3 2 4 2 4 1 3 2 1 1 2 6 7 5 3 8 4 6 2 5 4 1 7 8 3 2 4 1 3 2 4 3 5 1 6 7 8 3 5 8 6 1 7 4 2 4 7 8 5 1 6 2 3 4 5 7 6 1 8 3 2 4 6 3 8 1 7 5 2 prf = [01] 01234567 … 01243567 6 … 15 prf = [012] σ Associated complexities (K) computed with noise σ

List of all possible sequences with their associated programs (P) Prefix K_max = 12 Choose P that minimizes K(P) • If K(P) ≤ Kmax: the next location is defined by P: • If K(P) > Kmax:

the next location is chosen randomly: prf = [prf, rand([1..8])] [+0, [+1]7_] … [+0,[[+1]²]²<A>,+2,[+1]²]  In comparison to irregular baseline, most of the regular sequences were well learnt.  Error pattern directly reflects

the hierarchical internal

representation of sequences.  Error rate at specific data

point indicates how well a given rule is understood: e.g. in “4segments”, even data points reveal that all axial symmetries are detected by all groups of participants.

Complexity threshold: