Space and Number
28 important questions on Space and Number
What does Piaget mean by egocentrism?
i.e. Children in the pre conceptual substage tend to view the world from their own perspective - p. 247 developmental textbook
What is meant by conservation and who was it proposed by?
What was the Three Mountains Task and what did it show?
- tested whether children understand how things look from another viewpoint
Results:
Preschool children (≤ 4 yrs - preoperational children) unable to choose pictures showing how the mountains would look from other points of view
= Tend to choose the picture of their own view: “egocentrism”
7-8 year olds realised that the mountain at the front of the view is at the back of the doll's view but still select the wrong viewpoint
Consistently correct solutions not until 9-10 years - concrete operations stage
(exp on p. 247 developmental textbook)
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What were some criticisms of Piaget's three-mountain task?
2. Reconstructing the display/ choosing the appropriate drawings may be beyond the ability of a young child
3. Choosing the correct perspective may be difficult to understand for young children
(info p. 248 developmental textbook)
What changes did Borke (1975) make to the 3 mountain task and what did he find?
- He asked children to rotate a scaled-down model of the display to present the appropriate viewpoint rather than reconstruct the display or choose from drawings
Results
- children as young as 3 were able to choose the right viewpoint from each of the 3 different positions
(p. 248 developmental textbook)
What changes did Hughes, 1975 make to the 3 mountain task? What did he find?
Asked the children where one doll could hide so the other doll couldn't see him
Results: children from 3.5 years to 5 years gave the right answers
= when the task is more comprehendible to children, they can perform better
(p. 248 developmental book)
What do Piaget's classic tasks show? (Conservation Task & 3 Mountains Task)
However, basic precursors of these abilities develop very early: studies show that infants have capabilities for representing space and number – as do nonhuman animals (Later-developing formal spatial and mathematical systems are uniquely human, and are also supported by language)
What did Piaget mean by "animistic thinking"?
Piaget used this to argue that egocentrism impacted preoperational thinking in lots of diff situations
e.g. the children may believe the wind and the river have feelings (they thought this because wind and rivers move) = egocentrism made it difficult for children to distinguish between the physical and psychological world i.e. children assume that activity is produced by something psychological
(info p. 248 developmental textbook)
What experiment challenges Piaget's view of "animistic thinking"?
Bullock, 1985 - also pointed out that Piaget's objects e.g. sun, moon, wind are often open to interpretation even for adults
(info p. 248 developmental textbook)
What are examples of simple and formal systems of space, number and maths?
Space - location coding
- navigation
Number and maths - small number tracking
- larger number discriminations
Formal:
Space - map reading
- geometry
Formal - exact numerosity
- arithmetic
What shows that newborns have some degree of spatial orientation?
Early spatial coding is largely egocentric – only relative to own body
What experiment shows when spatial representation changes from egocentrism to spatial updating?
Infant learns to orient to a window where an experimenter is playing “peekaboo” (a) whenever a buzzer sounds (infant’s RIGHT).
Infant is carried to the opposite side. The buzzer sounds. Look to the (a) correct window, (now infant’s LEFT) or (b) the incorrect (but egocentrically “correct”) window?
Ability to update position correctly when moved develops at around 1 year.
e.g. fail at 11 months but pass at 16 months in Acredolo, 1978
How do infants go beyond egocentrism?
2. Landmark use: code where a target is relative to landmarks (most sophisticated kind of spatial coding - can allow flexible navigation and recall from new viewpoints e.g. The Water Maze - when rats are put into a pool of milky water & have to find the hidden platform, Morris, 1981
What is a second way of testing spatial updating in infants?
"Disorientation" tasks, Cheng, 1986 - rats; Hermer & Spelke, 1994 - humans
Hermer & Spelke (1994) - the blue wall study: 18-24 months use room "geometry" as a landmark, but ignore other cues to find the toy - like adult rats (Cheng, 1986)
Spelke = proposed that there is an innate "geometric module" for processing room shape in humans
How do Spelke and colleagues explain infants solving spatial updating tasks?
- there are many criticisms of this account (Twyman & Newcombe, 2010 - Five Reasons to Doubt the Existence of a Geometric Module)
But overall, many studies show human infants and other species to be highly sensitive to room geometry (shape) (review, http://www.pigeon.psy.tufts.edu/asc/Cheng/ )
It seems in any case that humans have an early-developing capacity to perceive the shape (“geometry”) of 3D layouts
What experiment shows that 5+ year olds can demonstrate flexible landmark use?
3-6 year olds had to recall the location of a toy within an array surrounded by landmarks.
- Changed viewpoint produced by walking around: can use spatial updating = All ages (3+) can solve
- Changed viewpoint produced by the board being rotated: have to use the landmarks = Only 5 years+ can solve
According to Spelke & Newcombe, what develops in terms of spatial representatives?
Newcombe: empiricist or “neoconstructivist” approach. Increasingly sophisticated spatial coding schemes are constructed from experience (e.g. Newcombe, 2011)
What experiment demonstrates the geometry ability of adults, children and Mundurucu (amazon culture without instruction)?
Izard & Spelke (2009)(Review: Izard et al (2011) In Brannon & Dehaene (Eds.), Attention and Performance Vol. 24)
Test: participants had to say which shape was the odd one out out of a series of shapes?
Results: Mundurucu and US children find similar items easy vs difficult. Also, correlations between Mundurucu and US adults, and US children and US adults.
Interpretation: although education increases % correct, perception of similar geometric features is independent of education
What can be concluded on geometric formation?
- But some errors persist even in geometrically educated adults = a role for education in refining these
What basic number abilities do children have early on?
Large numbers – judge which of two sets is more numerous
What experiments assesses small number tracking in infants?
Task: look at slide 36
Results: 5-month olds look longer at the impossible event = shows they keep track of how many there are, and understand the effect of adding or subtracting 1
Exact number representations are limited to about 3-4
What experiment demonstrates the approximate numerosity of large sets in infants?
Can also be found in the auditory domain (Lipton & Spelke, 2003)
Disciminability depends on the ratio, not the difference in numbers. In general, infants can discriminate 2x ratio but not 1.5x
Birds, rodents, and primates can do this too (review, books by Dehaene, 1997; Gallistel, 1990). Dehaene (1997) The Number Sense; Gallistel (1990) The Organization of Learning
What are the 2 basic number systems in infants?
Infants (and many animals) can
1. keep track of exact numbers up to about 3-4
2. discriminate larger numbers with >1.5 ratios
What experiment demonstrates the exact large-number numerosity?
A number of skills needed, including knowing:
1. the relative values of the number words, e.g. that 5 > 2
2. that the last counting word also represents the total number
3. that objects can be counted in any order
Results
= at 3 years, can often count to 10, but without understanding most of the above
= At 4 years, can correctly answer “which is bigger” (e.g. 5 or 2)
Note: An important foundation for exact numerosity is learning to count
What does the exact number system for large numbers depend on?
Young children and adults from indigenous groups without formal education represent large numbers on a non-linear (log) scale (slide 41, 42 & 43 for experiment)
What experiment shows that counting ability and conservation of number are related to one another?
Children 4-6 years - Saxe distinguished children by their counting and number conservation abilities
Results:
- some children who had good counting skills were unable to conserve
- all the children who could conserve also had good counting skills
= counting ability is necessary but not a sufficient condition for the type of reasoning used in number conservation
(p. 312 developmental textbook)
What can be summarised from this lecture regarding numbers?
- Two early-developing systems, also shared with animals
---> exact number tracking up to 3-4
---> approximate large number discrimination
Formal
- Built up from basic via counting and language (Gelman)
- This depends on education (cross-cultural studies)
- But correlations suggest that the basic system continues to contribute to formal mathematical skill (Halberda et al)
What can be summarised from this lecture regarding space & number?
- Early-emerging, basic spatial and numerical abilities (animals also have this)
---> Evidence that some of these emerge very early is consistent with the idea that they are innate - though innateness is difficult to prove. (Present at birth? Independent of experience? Often difficult or impossible to test. )
- Later developing, uniquely human understanding of abstract concepts of geometry and number
---> These seem to depend on education and language to different degrees – more so for number, less so for geometry
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