Weber-Fletcher law of perceptual discrimination
It is easier to choose if the difference between to be compared quantities is larger. Animals can decide where there is more food. Chimpanzees selected the larger of two piles of chocolate bit. Adults humans decide whether they saw 12 dots without counting (they were shown very fast). Performance for both gets worse the more similar the quantities get to each other.
The number sense intuition
Dehaene (1997) The number sense intuition for magnitude may be evolutionarily grounded. It is a biologically primitive ability. Animals have to assess quantities often in the wild. Animals don’t know the exact numbers, it is a perceptually based assessment of magnitude.
Number discrimination in 10 month old infants
Starkey et al (1980) Assessed number discrimination in 10 month old infants using a habituation task. skills in babies. Showed showed dots of one number and then a different one, do they dishabituate, can they tell the difference. Looking time recovers in such a situations which shows they can tell the difference between the two conditions.
Number discrimination: changing length and density
Starkey et al (1980) They were worried the infants in the original study were just dishabituating to the perceptual change. Used a shorter length of the same number of dots. This is a way to control for perceptual display. If they dishabituate to the same number squashed then it’s not about the number, it’s just about how they are presented. Infants can discriminate between these two categories. Considered as evidence that infants can discriminate numbers. However, infants looking time did not recover for bigger numbers. So infants discriminate small numbers but not large quantities.
Number discrimination: visual properties co-changing with number
Clearfield and Mix et al (1999) When changing the number on number discrimination tests, the visual properties may also be co-changing with the number e.g. the size of the dots might be changing.
Researchers try to control for visual parameters. In each of the displays, we can’t control everything that changes with the changing of the numbers. We can’t control every perceptual change no matter what
e.g. If we change from presenting 2 to 3, we are changing the density of them, we are changing the overall circumference of them, we are changing the areas of them. Mathematically, it is impossible to keep every perceptual property the same so we test keeping one perceptual property each time e.g. first changing the number and keeping the perceptual property and secondly changing the perceptual property and keeping the number. They found recovery of interest when the perception changed and not the number but not when the number changed and the perception stayed the same. Therefore, the evidence for babies being able to count is fairly controversial.
Ancient number representation
Elementary sense (intuition) of magnitude
Evolutionarily grounded
Biologically primitive ability (precursor)
Approximate
Non symbolic quantities (animals, babies, adults)
Physical phenomena in general
Cultural number representation
Symbol use (e.g. 1, 2, …)
Language of maths
Symbol manipulation
Makes higher maths possible
Symbolic numbers
e.g. comparing 2 to 3 or 2 to 7
Concrete operations we test in children
- Class induction
- Conservation
- Transivity
Number conservation/invariance: Classical results
Piaget. Children asked to select the row that has more marbles. Both rows have the same number of marbles but one is more spread out than the other so the line of marbles looks longer. About changing the superficial quantity of representations. He found the vast majority of 2-4 year olds choose the row that appears longer. Children showed a failure of number conservation so they think the concept of number changes with superficial changes. Before the age of 7, number would not be conserved.
Number conservation/invariance: child friendly version
Mehler and Bever (1968) children asked to select the row they wanted of two rows of m and m’s. One row had more but they were closer together and one row had less but were more spread out so the line looked longer. Seeing if they can understand for themselves that they should pick the bigger quantity. They found results that were inconsistent with Piaget’s original findings. Most children picked the line with more m and m’s, not the one that just looked longer.
Math from about 2/3 years of age
Spontaneous focusing on numerosity (SFON). Some children and their parents talk about numbers, pay attention to numerical information, others do not e.g. paying attention to how many of something children have/see
Math in kindergarten
Counting is initially learnt as verbal automatism. Children often count correctly in terms of verbal symbols but jump over objects being counted, skip some objects being counted, do not even look at the objects being counted and continue counting after there are no more objects to count.
5 basic principles of learning how to count
Gelman and Gallistel (1978)
Stable order - number labels come in order
One to one correspondence - each object is counted only once
Cardinal principle of last number - the last number represents numerosity/quantity of a set, children are called ‘cardinality knowers’ if they understand this
Abstraction - specific properties of to be counted objects are irrelevant so you can count anything
Order irrelevance - you can count things in any order, the important bit is to count them only once
Principles for advanced counters
Cardinality conserves - changing irrelevant properties after counting will not change the count
Grouping - grouping is possible e.g. pairs can be counted as one unit
Successor function - n+1 counting can go on forever
Domain specific cognition for mathematical development
Understanding the concept of number
Counting
Order relations
Intuition for magnitude (checking feasibility)
Understanding of symbols
Understanding of referents of symbols (abstract number)
Operand precedence
Domain general cognition for mathematical development
Planning and strategy
Attentional focusing on items
Memory for items
memory for order of items
Attentional focusing on internal representations in memory
Memory for partial results
Mental representations for numbers
Magnitude
Ordering
Word symbols
Ordering symbols
Ordered symbols and magnitudes
The mental number line
Dehaene et al (1993) The mental number line is a cultural construct. Spatial-numerical association of response codes (SNARC effect).
Left to right writing systems:
Within a number range, small numbers associated with the left side of space and large numbers are associated with the right side of space.
Right to left writing systems:
Reversed associations
This helps to make sense of magnitude relations
The development of left/right spatial associations
Szucs et al (2012) 65 grade 1, 2 and 3 British children. Parity judgements using a few numbers. Kids have to decide which numbers are odd or even. In one case they have to press left for odd and right for even and vice versa, counter balanced. They are better at responding when the bigger number is on the right. Automatic association of space/magnitude. The task does not require analysis of magnitude. Their findings may be because children are used to seeing bigger numbers on the right.
Maths in 9-10 year olds
Szucs et al (2014) Nearly all domain general cognitive skills correlate with mathematical achievement scores. It seems hard to find something that does not correlate with maths development.
Mathematical learning difficulties (MLD) and developmental dyscalculia (DD)
May be related to maths specific and domain general cognitive problems. One of which is visuo-spatial working memory that can be a big deal for maths development.
Is maths a special subject?
Szucs et al (2014) Maths is complex, precise, step by step and abstract. Lots of skills and representations to use to reach a result. You have to use them in concert and one after the other, coordinate a plan. Usually there is one good outcome, one correct answer. It require abstraction, a new language, everyday language is not enough. Maths seems to be really more demanding than other subjects for many pupils and adults. It has high demands and are particularly anxiety inducing.
Maths anxiety
Emotional factors and maths. Mathematics anxiety is a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in ordinary life and academic situations. Maths anxiety is the fear of learning and doing maths. It ranges from feeling mild tension to experiencing string fear of mathematics and can be experienced in school or in everyday life. People who are relatively good in math can also have high maths anxiety, it is not restricted to poor maths achievers.
Short term consequences:
Leads to worries that can occupy students thoughts during test situations, negatively affecting maths scores
Medium term consequences:
Students with high math anxiety may avoid elective math classes even if they are good at math so the maths achievement level of these students may not reach their full potential
Long term consequences:
Students and adults may avoid math related careers altogether. Math anxious adults may experience lesser quality of life e.g. failing to reflect properly on mortgage payments and gambling risks.
Gender and maths anxiety
Szucs et al (2018) Females tend to have more maths anxiety than males. It can lead to a relative lack of women in math heavy careers e.g. STEM. Maths anxiety is often much higher in girls than in boys in highly gender equal countries, gender equality paradox. Most studies in most countries found that girls and boys perform equally well in maths in school. However, maths anxiety is much higher in girls than in boys even in highly gender equal countries. This discrepancy in self reported math anxiety between girls and boys can be due to several reasons:
- There are often gender stereotypes about boys being better in maths and science than girls. Many parents, teachers and students use these stereotypes. Children can learn these stereotypes very early in life even before entering school. They have a fear of confirming negative stereotypes.
- Girls often report lower self confidence in maths than boys even if they perform equally well. Negative stereotypes about girls and maths can lead to high maths anxiety and low confidence in maths.
- Girls may be generally more anxious than boys. For example, often girls also have higher general anxiety and test anxiety than boys. Overall higher anxiety levels in girls may predispose them to higher level of maths anxiety as well.
- Girls may be more willing to admit their anxiety in many cultures than boys. Boys are often taught to be tough and not admit their anxieties.
- Girls may be able to recognise their anxiety better than boys of the same age. Girls are often emotionally more mature than boys at the same age, having better meta cognitive awareness.
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Theories of development34
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Prenatal development12
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Developmental genetics17
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Language development11
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Fundamentals of attachment theory9
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Observational Methods2
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Developmental Neuroscience Methods10
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Global developmental neuroscience17
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Early pathways to neurodiversity33
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Measuring development10
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Social perception and cognitive neuroscience15
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Cognition in infancy12
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Cognition in childhood21
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Reading27
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Dyslexia25
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Early memory development36
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Memory development beyond preschool20
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Mathematics development28
