W3L2 - Behavioral and Neural Evidence for Numbers

Question 1

Why are numbers important to study?

Answer 1

1. ) Numbers are central to who we are (process 16,000 numbers a day)
2. ) Basis of civilization
3. ) Technological advancement depends on number
4. ) Number deficits affect people’s opportunities in society

Central ; Civilization; Technology; Opportunities

Question 2

What do we mean by numbers? Some questions…

Answer 2

Q1: Exact representation / Approximate Numbers?
Q2: Enumeration (Precise Number) / Comparison (More or Less)
Q3: Symbolic (1,2,3,4) / Non-symbolic (Number of cats)
Q4: Ability to do math

Are there any specialized brain regions?

Question 3

What are evidences suggesting numbers are different from language. Broad Evidences

Answer 3

1. ) Neuropsychological (Dyslexia vs Dyscalculia)
2. ) Animals
3. ) Preverbal Children
4. ) Cultures with limited language for numbers

Question 4

How do we describe numbers and language

Answer 4

Numbers: Foundation of Knowledge
Language: Basis of communication

Question 5

Animal Studies. Why are numbers important for animals

Answer 5

Crucial for survival

Question 6

What are the 6 animals discussed in W3L2

Answer 6

1. ) Clever Hans (Horse)
2. ) Jakob (Raven)
3. ) Alex (Parrot)
4. ) Desert Ants
5. ) Lions
6. ) Chimps (Ayumi)

Question 7

Animals in Numbers: Clever Hans Horse

Answer 7

Initially thought to be able to do math by hoof stamping (including square roots). But later found he was reading owner’s face

Question 8

Animals in Numbers: Jakob Raven

Answer 8

1. ) Could select a pot with a specific number of dots on the lid (1-7)
2. ) Trained to open boxes and eat the seeds contained in them until a precise number of seeds had been eaten

Question 9

Animals in Numbers: Alex Parrot

Answer 9

1. ) Numerical ability in context of langauge
2. ) Could squawk “1-6” how many specified colour blocks there are

3,) Colour of the larger or smaller of two objects, suggesting he had sense of greater or less than as well.

Question 10

Animals in Numbers: Desert Ants

Answer 10

Judge distance by counting steps

• If their legs are clipped they undershoot the journey
• If they are given stilts, the go too far when returning from foraging

Question 11

Animals in Numbers: Lions

Answer 11

Lions decide to attack based on the ratio of their numbers compared to the number of voices in an “intruder” pride

Question 12

Animals in Numbers: Chimps Ayumi

Answer 12

1.) Number tasks that exceed the ability of humans (Reducing time of visible numbers, Ayumi no effect but kids performance drop significantly)

(Superior performance on this task may reflect additional processes. VSTM required exceeds the 4-5 items limit of humans)

Question 13

Preverbal Children. What did they show. 2 Studies

Answer 13

Preferential looking task on dot displays that differ (Ability do discriminate)

• Babies looked longer at displays that had change in number 2-2 (1.9s) vs 2-3 (2.5s)
• 6mo babies can distinguish between 8 & 16 and 16 & 32

Question 14

Preverbal Children - Concept of number changes with time. Siegler & Booth (2004)

Answer 14

Move from logarithmic to linear representation

May reflect formal education about the number line

Question 15

Preverbal Children - How did Siegler & Booth (2004) show concept of number changes with time

Answer 15

Number Line Task: children are asked to place a vertical line at the location where the above number would appear between the given number range.

Question 16

Preverbal Chldren - Siegler & Booth (2004): Concept of number changes with time. Results?

Answer 16

Children often overcompensate for the position of the given number (Change is nonlinear. Develops from non-linear to linear)

• The early representation of symbolic number represents ratio difference rather than linear separation.

Question 17

Preverbal Children - Understanding of Arithmetic (Karen, 1992)

Answer 17

Babies understood 1 + 1 = 2

Stared longer when 3 dolls appeared instead of 2

Question 18

Cultures with limited language for numbers: Munduruku (Pica, 2004). Overview

Answer 18

There are tribes in the Amazon whose only number words are ‘one’, ‘two’ and ‘many’.

The Munduruku have words that go up to five. Beyond 5 those are only reflecting the approximate number. (smaller = precise; larger = imprecise)

Question 19

Cultures with limited language for numbers: Munduruku (Pica, 2004). Sense of Verbal Range

Answer 19

Similar to children. Logarithmic. Suggest role of schooling/language in Log > Linear.

Question 20

Cultures with limited language for numbers: Aboriginals. Overview (Butterworth & Reeve)

Answer 20

• Anindilyakwa have one, two, three (which sometimes means four) and many
• Warlpiri have one, two and many

Children no words for the numbers four, five, six and seven, yet were perfectly able to hold those amounts in their heads

> > > Words not necessary to understand exactness…

Question 21

What are the core aspects of numerical competence

Answer 21

(1) Identify (2) Order (3) Compare

Numerical quantities

Question 22

What are two tasks examining basic cognitive processes of number

Answer 22

1. ) Enumeration: Verbalize precise number

2. ) Number comparison: Magnitude Comparison (non-symbolic)

Question 23

Object Enumeration. What are the steps

Answer 23

• Judge the number of a bunch of spots.
• Accuracy and RT

Steps:

1. ) Symbolic encoding of visual information
2. ) Accessing that symbolic representation for combining into a total or sum

Question 24

Object Enumeration. What are the results about 2 distinct enumeration systems?

Answer 24

1. ) “Sibsitising” < 4ish
   - Rapid & accurate enumeration
2. ) “Approximate Number System” >4ish
   - Slow & approximate enumeration

RT curve shows elbow at the 4ish number

Question 25

Number comparison: What are the types

Answer 25

Symbolic (8+3) Non-Symbolic (Dots) Cross-Modal (8 + Dots)

Question 26

Number comparison: Does it require a verbal response

Answer 26

Not necessarily

Question 27

Number comparison: What are the steps

Answer 27

- Comparing the magnitude of two regions of visual information - Accuracy and reaction time (RT) 1. ) Symbolic encoding of visual information 2. ) Symbolic mapping into numerical information 3. ) Accessing that symbolic representation for combining into a total or sum

Question 28

Number comparison: What are the results. Symbolic Vs Non-Symbolic

Answer 28

Symbolic: Number distance effect | Non-Symbolic: Weber's Ratio

Question 29

Number comparison: What are the results. Symbolic Elaboration

Answer 29

Number Distance Effect: Close in numerical distance = Slower RT - Suggests that neural mechanisms are ordered in a functional way (mental number line)

Question 30

Number comparison: What are the results. Non-Symbolic Elaboration

Answer 30

Weber's Ratio: Errors depend on the ratio of the magnitudes (Non-Linear) - Smaller Weber ratio > Higher sensitivity to ratio differences - Difficult measure ratio in subitising (<4) range

Question 31

What do Weber's Ratio on a log scale look like

Answer 31

Linear. Width are the same. Just like the log number spacing for children & aboriginal cultures on the number line task

Question 32

Is Weber Ratio related to arithmetic competence

Answer 32

1. ) Correlated with school arithmetic competence | 2. ) Children with dyscalculia are less accurate in comparison of two sets of dots compared with age-matched controls

Question 33

Weber's Ratio and Dot Comparison Task. Halberda, Mazzocco & Feigenson (2008)

Answer 33

Weber ratio each individual past scores on standardized maths achievement tests Compared with past maths performance. Predictive of overall ability do maths. (Less Ratio = Greater ability)

Question 34

How well does the Weber fraction predict arithmetic competence?

Answer 34

- Large individual differences in the data - Complicated relationship o Even easy items may require formal calculation skills o Studies of over-practiced indices of maths competence (adding two single digits) result in ceiling performance for youngest children. o Influence of experience would blur the real relationship

Question 35

Cross model number comparison: Is it something about dots rather than math? (Barth, Kanwisher & Spelke (2003)). What did they do?

Answer 35

Compared unimodal and cross modal sets of 10, 20 & 30 tones and spots. Asked to to give the total number

Question 36

Cross model number comparison: Is it something about dots rather than math? (Barth, Kanwisher & Spelke (2003)). Results?

Answer 36

Cross modal number as accurate as unimodal. - Little/ No accuracy cost for comparing numerosities across stimulus format or modality - Judgments of approximate numerosity are likely to be based on abstract representations of number - Number comparison is a general brain mechanism, not a visual

Question 37

Enumeration vs. number comparison. Do they involve same mechanism?

Answer 37

Enumeration requires counting - Eye movements / Shifts of attention - Increases in RT MAY BE DUE TO these factors Comparison can be rapid - Preclude eye movement / attention shift - BUT, the subitising limit is close to the point where the weber ratio makes discrimination difficult