Numerical Cognition Flashcards
what are the non-symbolic number representations?
doesn’t need language , tracking numerosity in a non symbolic way
analog magnitude system
object individuation system
analog magnitude system
yields noisy representations of approximate numbers that capture the inter-relation between different enough numerosities, e.g., 10 and 20.
- coarse and imprecise , does not require counting
object individuation system
tracks small numbers (up to 4) and precise representation of the numerosity of small sets without counting e.g., can quickly count 2 from 4.
what do these non-symbolic number representations provide evidence of?
- the human mind having access to distinct non-symbolic systems for representing numerosity, which is able to track numbers without even having to think of or know number words
- human language not necessary for this system to emerge as it shows in animals (evolutionary concept )
Methods to study how infants represent numerical info
Looking time methods (violation of expectation + preferential looking )
Manual search
Choice
Manual search
You have babies search for objects that you hide in front of them
Choice
- When babies are old enough where they’re able to crawl
- Experimental design where you set the things in front of them, letting the baby crawl to it
- Observing the choices baby makes
- More direct than just making inferences from looking behaviour
what did spelke and xu (2002) find after testing visual modality of 6m infants?
dishabituated when presented with a different number of dots. regardless of the no of dots presented to them in habituation
evidence of extracting no of dots presented to them and come up with a numerical representation via analog magnitude system
Limitation of spelke and xu study on visual modality of 6m infants
What if what the babies are sensitive to is not the numerosity but the visual property of stimuli they’re seeing
what did izard (2009) find about newborns and the AMS?
newborns can match numerical arrays across visual and auditory modalities (sets of beeps vs visual blocks)
looked longer at matching visual arrays
they have access to the analog magnitude system at birth to represent abstract properties of the world, displaying sensitivity to abstract aspects of numerocity
where is there evidence of developmental progression in the precision of AMS?
analog system is subject to a ratio limit in newborns – can differentiate large numerosities only if they are sufficiently different
- at birth, babies can discriminate ratios 1:3 but not 1:2
At 6 months, they can now do 1:2 but not 2:3
development of AMS
the precision of the system improves over the first year of life and is a part of core number knowledge
OIS: what did wynn (1992) find about 5m infants?
displayed surprise when the screen revealed the wrong number of puppets after occluder dropped (1 +1= 1)
- suggesting at around this age, babies are able to recruit their object individuation system to precisely represent the numerosity of the set of 2
- also reveals they were able to interpret events involving addition and subtraction
OIS: feigenson (2002) tested 10-12m infants and found…
they could track precise numbers and use this information to guide their choices (when crawling) , due to the development of their working memory
crawled to the bucket that had the most crackers ( 2 vs 3)
so mental representations inform choices
manual search task evidence to support OIS in 12-14m
- 2 balls in box, removed 1 in front of baby, baby then searched longer in this condition than when two got removed
when 12-14m infants still expected objects to be in the box and continued to look longer (feigenson and carey, 2003)
- so mental representations inform search behaviour
when did infants learn to extract and match numerosity across different modalities? (ages)
at 6-8m (starkey, 1990) via preferential looking tasks
looked longer to visual displays that matched the simultaneously presented drumbeats
they had cross-modal number representations for audition and vision
what is the OIS limited to tracking?
- tracking at maximum 3 objects in parallel (feigenson, 2002) and is operational during the first year of life.
- increasing number of crackers decreased chance level with behaviour behaving randomly even with big differences
object individuation system efficient with set sizes below 3
what are symbolic number representations?
humans developed number words and counting which allowed for precise representations of numerical information, even when recording very large numbers
Background observations of symbolic number representations
- begin to learn langauge in utero
- 6m can develop a receptive vocab, learning meanings of common words
when can infants recite the count list?
at 2y
when do infants work out how counting works?
at 4y
they struggle to figure out what number words mean, despite rapidly learning the meaning of object labels
dont understand how a sequence of words relate to the number of things
give-n task
tests developing understandings of number, via asking for N number of objects
What do toddlers understand about counting
- involves a stable order
- one to one
stable order
using the same labels in the same order, even if 2ys may consistently recite an incorrect count list (Gelman and Gallistel, 1978)
one-to-one
using one label per object, linking one object to a item.number word (Gelman and Meck, 1983)
what did lee and samecka (2010) find about the development of precise symbolic number representations ?
children learn number words in stages, which may span across several month
stages of number representation
- they learn the exact meaning of individual number words without knowing how counting encodes number (one-knowers – four-knowers
- make an inductive leap to understand the counting algorithm is governed by the cardinal principle and the successor function to reach the ‘cardinal principle knowers stage’
what does learning to count require?
understanding complex concepts and rules, slow process
cardinality
cardinality principle
successor function
cardinality
the number of elements in a set
cardinality principle
the number word applied to the final item in a set represents the number of elements in the set
successor function
tells us what the relations are between the numerals (e.g., if numeral N represents cardinality N, then the next numeral represents the cardinality N+1)
what did jara-ettinger (2017) find about the universality of counting ?
the stage-like process of discovering how counting works is universal, and independent of culture and mother tongue, although timing may vary across cultures
Western develops a bit earlier
Process of counting appears in languages without little forming schooling
examples of qualitative differences in how children learn number words of different sizes
1, 2, 3 – slow, stage-like, one number word at a time
4, 5, 6, 7 – fast, subsequent number words understood straight away
why is counting hard?
- number words work differently to other words, as they refer to sets rather than individuals/categories
- counting relies on an algorithm that children need to discover (cardinality principle + successor)
what does evidence suggest a relationship between?
early numerical skills at 6m and later formal maths learning at 3.5y (Starr, 2013), and greater learning of symbolic numbers in school
Emerging number systems and our mathematical knowledge
our emerging number systems are important building blocks of our formal mathematical knowledge that we develop throughout life
continuity in early number representation and learning
- children retain access to non-symbolic number systems
- there is a link between number skills and later maths learning
discontinuity in early number representation and learning
- discontinuity as non-symbolic and symbolic number systems represent different kinds of numerical information.
symbolic systems allow for precise unlimited representations of any number , non-symbolic systems have limitations
