Lecture 4 - Space and Number

Lecture 4 - Space and Number

1. Historical research on space, number and maths

   • Piaget - Conservation tasks. Striking failures of reasoning about physical properties until the "concrete operational" stage around 6-7years.
   • The three mountains task. Piaget and Inhelder 1956 - Do children understand how things look from another viewpoint? Preschool children (≤ 4 yrs) 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” Consistently correct solutions not until 10-12 years
   • Piaget’s classic tasks show that explicit reasoning about formal properties (e.g. of space, number, area, volume) takes a long time to develop. 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
     • When and how do the basic abilities develop? Are some of them innate? When and how do the formal abilities develop? Building on or replacing basic ones?

2. Development of spatial representations

   • Recent and current research, using simple direct responses rather than explicit reasoning or problem-solving. Many tasks are directly comparable with those used with animals. From birth - spatial orienting (egocentric), in first year - from egocentrism to spatial updating, at 18-24 months - use of spatial updating and landmarks, 5 years + - flexible coding using indirect landmarks.
   • Neonatal spatial orienting (egocentric)
     • Newborns can roughly localise visual, auditory and tactile stimuli in space Early spatial coding is largely egocentric – only relative to own body
   • 1st year - egocentrism to spatial updating
     • Acredolo, 1978; Acredolo, 1980. 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 to go beyond egocentrism
     • 1. Spatial updating = keep track of locations as you move. 2. Landmark use = code where a target is relative to landmarks. More sophisticated kind of spatial coding, can allow flexible navigation and recall from new viewpoints. Water maze - Morris 1981.
       • 18+ months: spatial updating and landmark use. Sandbox task Huttenlocher et al (1994); Newcombe et al (1998). See toy hidden. Either have to retrieve from same side (can solve ego centrically simply by encoding relative to body) OR opposite side (must update body encoding to take own movements into account, with or without extra landmarks in the room visible). Same side : above chance at 16 months. Opposite side: also above chance at 16 months. Adding extra landmarks improved performance only from 22 months.
       • 18+ months: landmark use – room geometry. “Disorientation” tasks, Cheng (1986) – rats; Hermer & Spelke (1994) - humans. Another way to dissociate updating from landmark use is to disorient participants. Forces them to rely on landmarks – so a strict test for landmark use. Hermer & Spelke (1994): 18-24 months use room “geometry” as a landmark, but ignore other cues – like adult rats (Cheng, 1986). Spelke: a dedicated, innate “geometric module” for processing room shape in humans, as in rats? Part of Spelke’s “core knowledge” proposal – see later. And more on modularity in “disorders” lecture. The task is solved at 4 years, Spelke and colleagues attribute this to development of language abilities. “Core” (innate) geometric understanding combined with linguistic coding for left/right/next to, etc. Many criticisms of this account (see 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 –does this have any relation to later understanding of formal geometry, and if so what is the relation between the two?
   • 5 years - flexible landmark use
     • Nardini et al (2006). 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
   • What develops?
     • Spelke: nativist approach. “core knowledge” of basic spatial concepts supplemented by education and language (see e.g. Spelke & Kinzler, 2007) Newcombe: empiricist or “neoconstructivist” approach. Increasingly sophisticated spatial coding schemes are constructed from experience (see e.g. Newcombe, 2011)
     • Insights from animal models and neuroscience- Dissociable neural bases for different kinds of spatial coding – e.g. “place” vs “response” learning in rats (similar dissociation to Acredolo orienting studies); “place” learning and flexible landmark use associated with the hippocampus - in animals, human patients and healthy humans. To what extent does development of cognitive abilities depend on brain development, e.g. of the hippocampus? Some work in this direction, e.g. Sluzenski et al, 2004
       • Prior data have revealed striking contrasts between 18 and 24-month-old children in place learning, an ability knownto dependon the hippo campus(Newcombe, Huttenlocher, Drummey, & Wiley, 1998). The current research examined the development of three other basic abilities of mature spatial competence: the representation of multiple locations, the learning of relations among objects, and the recall of a single location after a substantial filled delay. Results indicated a transition from 18 to 24 months in all three abilities. This evidence supports a general transition in spatial representation that occurs towards the end of infancy. Existing neurobehavioral data suggest that a corresponding change in hippocampal functioning underlies this development.

3. Space - formal abilities

   • Maps
     • Map use becomes increasingly sophisticated and flexible in childhood, presumably partly with education and instruction (Liben & Downs, 1993) Visualisations / diagrams: direct training on formal spatial tasks has been shown to be beneficial, e.g. for engineering students (Sorby & Bartmaans, 1996, 2000) --> improved spatial reasoning and better achievement on courses
       • Liben and Downs 1993 - Drawing from cognitive-developmental and cartographic theory, the authors tested children's understanding of person–map–space relations. After introducing maps of the classroom, an adult moved to different positions in the room and pointed straight ahead. Five- to 12-yr-old children (N = 259) placed colored arrow stickers on the map to show the person's location and heading, once when the map was aligned with the room and once when rotated 180°. Performance was better on the aligned condition and when headings were parallel to the nearby wall; most younger children failed to understand point of view; and sex differences favored boys. Data from 168 children given additional spatial tasks were consistent with the hypothesized importance of projective spatial concepts for mapping. Data suggest that it is geometric (spatial) rather than representational (symbolic) space–map correspondences that pose particular difficulty for older children.
     • A map (or photograph or model) symbolises the real environment - related to development of symbolic thought. Can find a toy in real room based on location in a model room at 3 years (deLoache et al, 1987) --> shows symbolic understanding, though nature of spatial coding unclear as all locations were unique (e.g. there is only one chair). More detailed tests of spatial coding – Spelke and colleagues. --> e.g. Winkler-Rhoades et al, 2013: Put Kermit in the place shown on the map: better than chance (49% correct vs 33% expected by chance) at 2.5 years -> Early developing abilities to use maps as symbols, and relate spatial relations in maps to spatial relations in real layouts
   • Geometry
     • Formal geometry and number are abstract concepts. They relate to real objects, but idealised geometric objects do not exist in the real world (e.g. points so small they have no extent, lines so thin they have no width). Euclid (~300BC) deduced the principles of what is now called Euclidean geometry from a small set of axioms. Kant (18th Century) argued that the human mind has Euclidean intuitions a priori (independently of experience). To what extend is geometric knowledge either “innate” or dependent on experience? A great question for developmental and cross-cultural psychology: studies with young children studies with remote groups who have had no formal education in geometry
       • Geometry in US adults, US children, and Mundurucu* adults (*Amazonian culture without instruction in geometry). Studies by Izard, Pica, Spelke and Deheane Review: Izard et al (2011) In Brannon & Dehaene (Eds.), Attention and Performance Vol. 24. Which is the odd one out? Testing different geometric properties, 45 in total. Correlations: Mundurucu and US children find similar items easy vs difficult. Also Mundurucu and US adults, and US children and US adults. Interpretation: although education increases % correct, perception of similar geometric features independent of education Dehaene et al (2006) Izard & Spelke (2009)
       • “Universal”* perception of geometric features (Dehaene et al 2006) * in children and adults across two very different cultures in any case --> Universal understanding of geometric concepts that go beyond perceptual experience? Izard et al (2011). Reasoning about points and lines on perfectly flat and perfectly spherical worlds - Mundurucu, US and French children and adults – Evidence for universal and early developing understanding of geometric concepts But some errors persist even in geometrically educated adults (who are not mathematicians) --> a role for education in refining these.
   • Summary
     • Maps: Early developing abilities to use maps as symbols, and relate spatial relations in maps to spatial relations in real layouts (starting in 2nd year). Map use improves in childhood, and better use of diagrams/visualisations can be trained in students
     • Geometry: There may be a continuum between infants’ early sensitivity to geometry for reorientation and later geometric abilities. See Spelke (2011), in Brannon & Dehaene (Eds.), Attention and Performance Vol. 24. Similar perception of geometry, and reasoning about geometric problems, in Western-educated children and adults, and in indigenous groups. Are geometric concepts innate? Difficult to answer, but evidence that not much experience is needed (as young children have them), and that education contributes relatively little (as similar concepts across cultures)

4. Number: basic ablities

   • Keeping track of how many of something there are is a key perceptual and cognitive ability. Small numbers – keep track of nearby objects of interest. Large numbers – judge which of two sets is more numerous. Infants have both basic abilities early on.
     • Small number tracking in infants, Wynn 1992. "Addition and subtraction by human infants." Showed an object being placed, screen goes up and then other object added. Then either a possible outcome, revealing 2 objects, or impossible outcome, revealing 1 object.5-month olds look longer at the impossible event --> keep track of how many there are, and understand the effect of adding or subtracting 1. Likewise, untrained birds and primates have been found to represent the exact numerosity of small sets of objects (e.g. Hauser et al, 1996). Exact number representations are limited to about 3-4. May be related to the “object file” system that evolved to allow us to keep track of 3-4 moving objects (Carey, 2001)
     • Approximate numerosity of large sets in infants. Xu and Spelke 2000 - 6 month olds discriminate 8 vs 16 dots but not 8 vs 12. Therefore, 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
     • Two basic number systems. Infants (and many nonhuman animals) can keep track of exact numbers up to about 3-4 and discriminate larger numbers with >1.5 ratios. Later, humans uniquely also learn to deal with exact large numbers (e.g. 752 + 191 = 943) - abstract concepts with no immediate basis in perception. How do they do this?
       • Exact large-number numerosity - An important foundation for exact numerosity is learning to count. Gelman and colleagues: counting builds on previous nonverbal knowledge and serves as the basis for future numerical understanding (Gallistel & Gelman, 2005). A number of skills needed, including knowing the relative values of the number words, e.g. that 5 > 2 ; that the last counting word also represents the total number; that objects can be counted in any order. 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) (Siegler & Robinson, 1982). 4-5 years, count to 20, appreciating most of these principles
         • Acquiring the exact number system for large numbers seems to depend on education: Young children and adults from indigenous groups without formal education represent large numbers on a non-linear (log) scale
         • Number line tasks: show where on the line this number falls. Children: Presented with numbers. Cross-cultural: with the Mundurucu, like the geometry studies. Presented with dots or tones.
         • Approximate large number system is non-linear – can discriminate the same ratio irrespective of set size (Weber’s law – as with other perceptual properties) Acquiring a linear number line, and appreciating that adding 1 to 100 has the same effect as adding 1 to 1, seems to require formal education. Relationship between the two systems? Accuracy of children’s approximate number system is correlated with their maths ability with exact numbers (Halberda et al, 2008) – suggests some continuity between the two systems
       • Exact large-number numerosity - An important foundation for exact numerosity is learning to count. Gelman and colleagues: counting builds on previous nonverbal knowledge and serves as the basis for future numerical understanding (Gallistel & Gelman, 2005). A number of skills needed, including knowing the relative values of the number words, e.g. that 5 > 2 ; that the last counting word also represents the total number; that objects can be counted in any order. 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) (Siegler & Robinson, 1982). 4-5 years, count to 20, appreciating most of these principles
   • Summary: Basic- Two early-developing systems, also shared with nonhuman 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)