Mathematical development

Question 1

What are the benefits of mathematical abilities?

Answer 1

Independence
Organisational skills
Predict financial succes
Predict educational success
Financial skills

Question 2

What are the different things done with human numerical knowledge?

Answer 2

1. Algebra, trigonometry and calculus.
2. Estimate
   Tell small numbers apart fast,
   accurately and spontaneously.
   Count

Question 3

Why are numbers a psychological phenomenon?

Answer 3

Different property, not like colour, shape, size.
Attribute of sets
Relationship with language in humans.

Question 4

What do babies know?

Answer 4

Empirical evidence shows rudimentary representations in nonverbal human babies.

Question 5

Explain an experiment on a baby’s rudimentary knowledge of numbers.

Answer 5

Wynn (1992) showed that 4-month-old babies can add and subtract small numbers of objects.

Infants look longer at impossible (e.g. 1 + 1 = 1) than at possible outcomes (1 + 1 = 2) in ‘addition’ and ‘subtraction’ events.

Question 6

What is the ratio model for?

Answer 6

In order to understand infant numerical knowledge.

Question 7

What is the ratio model?

Answer 7

• A mechanism for approximate representation of numerical magnitudes.
• Acculator Model
• Numbers are represented as a function of the ratio between numerosities, not their absolute value - 1:2 than 2:3, for example.
• Prediction of the model: small and large numerosities represented equally.
• Response time and accuracy are related to the numerical distance between choices and the size of the two numbers being compared.

Question 8

What is the accumulator model?

Answer 8

Explains that magnitudes represent numerosities as energy pulses in an accumulator

Question 9

Define accumulator.

Answer 9

A receptacle of energy which stores these pulses as each entity is ‘counted’.

Question 10

What is the individuation model?

Answer 10

The visual system encodes numbers of entities via a tracking system.
Parallel Individuation model: individuals in small sets of objects represented with a tracking system with an index that sticks to each object as they move in space and time.
Representations of individuals stored in STM.
Limits on the number of indexes in the process of individuation determine the limits on how many individuals can be represented.
Reaction time effects: discrimination of 1 - 4 entities same; slope increases after 4; effects of processing.

Question 11

Explain the habitation study.

Answer 11

7 month-old infants set up expectations for the correct outcome of 1+ 1 only when they see the first object in full view, not when 2 objects disappear being occluder.
This is not predicted by the ratio model. But supports the individuation model.

Question 12

Explain babies understanding of larger numbers.

Answer 12

In numerical discrimination tasks, babies are found to discriminate numbers:
6 months old infants
Can discriminate 8 and 16 or 16 and 32 dots (ratio 1: 2).
Fail to discriminate 8 and 12 dots (ratio 2 : 3) (Xue and Spelk 2000; Xu, Spelke and Goddard, 2005).
9 months old infants:
Can discriminate 8 and 12 dots (ratio 2:3)
Fail to discrimate 10 and 12 dots (ratio 5) (Xu and Arriage 2007)

Question 13

What do children need to know about numbers?

Answer 13

Counting
Estimation
Number Line
How to read digits
How to write digits
Number magnitudes: which is more?
Relation between quantities and digits
Arithmetic operations
Problem solving.

Question 14

What age does counting take place?

Answer 14

Age 2 to 6 and follows 5 principles.

Question 15

What are the five principles of counting?

Answer 15

1. Stable Order
2. One-to-one Correspondence
3. Cardinality
   These 3 principles are considered by Gelman and Gallistel to be the ‘how to count’ principles as they specify the ways in which children must execute a count.
   The remaining 2 are ‘what to count’ principles, as they define what can actually be counted:
4. Abstraction
5. Order Irrelevance

Question 16

Explain stable order.

Answer 16

Number obey a particular order.

Question 17

Explain one-to-one correspondence.

Answer 17

Numbers are assigned to one object at a time.

Question 18

Explain cardinality.

Answer 18

The last number in the count refers to the total number of items in the count.

Question 19

Explain abstraction.

Answer 19

One can count anything regardless of what it is.

Question 20

Explain order irrelevance.

Answer 20

The order in which items are counted does not matter.

Question 21

What does a child start to learn verbal count?

Answer 21

At approximately 18 - 24 months, children start the counting routine: one, two, three
From the age of 2 - 2 1/2 years, children do know that counting words refers to distinct, unique numerosities; they just don’t know which number goes with which numerosity.

Question 22

Summarise Wynn (1990) and Le Corre and Carey (2007, 2010) study.

Answer 22

Children’s understanding of counting
Cardinality Principle test
‘How many’ and ‘Give a number’ task - give puppet
Strong within-child consistency, children learn the cardinal word principle at roughly 3 1/2 years of age
Children learn the meanings of smaller number words before larger number words within counting range 1 - 4.
Cardinal principle learned by 3 1/2, then induction over all number words within counting range.

Question 23

Summarise the development of counting.

Answer 23

Babies’ understanding of number go beyond perceiving more from less
The representations are accurate results of “problems” (e.g, Wynne 1992; Xu and Spelke, 2000) both in the small and large number systems
Two models seem to work in concert in the representation of small number and large number.
When children start counting, there is the translation of nonverbal number representations into linguistic representations.