W3L2 - Behavioral and Neural Evidence for Numbers Flashcards
1.) Numbers are important and why we want to study them 2.) Number processing is distinct from language 3..) Performance on enumeration and number comparison tasks points to multiple number mechanisms in the brain 4.) Non-symbolic comparison processes are related to arithmetic 5.) The representation of numbers on a mental number line may explain some basic number phenomena 6.) Abstract coding of number in the brain allows basic calculation
Why are numbers important to study?
- ) Numbers are central to who we are (process 16,000 numbers a day)
- ) Basis of civilization
- ) Technological advancement depends on number
- ) Number deficits affect people’s opportunities in society
Central ; Civilization; Technology; Opportunities
What do we mean by numbers? Some questions…
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?
What are evidences suggesting numbers are different from language. Broad Evidences
- ) Neuropsychological (Dyslexia vs Dyscalculia)
- ) Animals
- ) Preverbal Children
- ) Cultures with limited language for numbers
How do we describe numbers and language
Numbers: Foundation of Knowledge
Language: Basis of communication
Animal Studies. Why are numbers important for animals
Crucial for survival
What are the 6 animals discussed in W3L2
- ) Clever Hans (Horse)
- ) Jakob (Raven)
- ) Alex (Parrot)
- ) Desert Ants
- ) Lions
- ) Chimps (Ayumi)
Animals in Numbers: Clever Hans Horse
Initially thought to be able to do math by hoof stamping (including square roots). But later found he was reading owner’s face
Animals in Numbers: Jakob Raven
- ) Could select a pot with a specific number of dots on the lid (1-7)
- ) Trained to open boxes and eat the seeds contained in them until a precise number of seeds had been eaten
Animals in Numbers: Alex Parrot
- ) Numerical ability in context of langauge
- ) 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.
Animals in Numbers: Desert Ants
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
Animals in Numbers: Lions
Lions decide to attack based on the ratio of their numbers compared to the number of voices in an “intruder” pride
Animals in Numbers: Chimps Ayumi
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)
Preverbal Children. What did they show. 2 Studies
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
Preverbal Children - Concept of number changes with time. Siegler & Booth (2004)
Move from logarithmic to linear representation
May reflect formal education about the number line
Preverbal Children - How did Siegler & Booth (2004) show concept of number changes with time
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.
Preverbal Chldren - Siegler & Booth (2004): Concept of number changes with time. Results?
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.
Preverbal Children - Understanding of Arithmetic (Karen, 1992)
Babies understood 1 + 1 = 2
Stared longer when 3 dolls appeared instead of 2
Cultures with limited language for numbers: Munduruku (Pica, 2004). Overview
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)
Cultures with limited language for numbers: Munduruku (Pica, 2004). Sense of Verbal Range
Similar to children. Logarithmic. Suggest role of schooling/language in Log > Linear.
Cultures with limited language for numbers: Aboriginals. Overview (Butterworth & Reeve)
- 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…
What are the core aspects of numerical competence
(1) Identify (2) Order (3) Compare
Numerical quantities
What are two tasks examining basic cognitive processes of number
- ) Enumeration: Verbalize precise number
2. ) Number comparison: Magnitude Comparison (non-symbolic)
Object Enumeration. What are the steps
- Judge the number of a bunch of spots.
- Accuracy and RT
Steps:
- ) Symbolic encoding of visual information
- ) Accessing that symbolic representation for combining into a total or sum
Object Enumeration. What are the results about 2 distinct enumeration systems?
- ) “Sibsitising” < 4ish
- Rapid & accurate enumeration - ) “Approximate Number System” >4ish
- Slow & approximate enumeration
RT curve shows elbow at the 4ish number
Number comparison: What are the types
Symbolic (8+3)
Non-Symbolic (Dots)
Cross-Modal (8 + Dots)
Number comparison: Does it require a verbal response
Not necessarily
Number comparison: What are the steps
- Comparing the magnitude of two regions of visual information
- Accuracy and reaction time (RT)
- ) Symbolic encoding of visual information
- ) Symbolic mapping into numerical information
- ) Accessing that symbolic representation for combining into a total or sum
Number comparison: What are the results. Symbolic Vs Non-Symbolic
Symbolic: Number distance effect
Non-Symbolic: Weber’s Ratio
Number comparison: What are the results. Symbolic Elaboration
Number Distance Effect:
Close in numerical distance = Slower RT
- Suggests that neural mechanisms are ordered in a functional way (mental number line)
Number comparison: What are the results. Non-Symbolic Elaboration
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
What do Weber’s Ratio on a log scale look like
Linear.
Width are the same. Just like the log number spacing for children & aboriginal cultures on the number line task
Is Weber Ratio related to arithmetic competence
- ) Correlated with school arithmetic competence
2. ) Children with dyscalculia are less accurate in comparison of two sets of dots compared with age-matched controls
Weber’s Ratio and Dot Comparison Task. Halberda, Mazzocco & Feigenson (2008)
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)
How well does the Weber fraction predict arithmetic competence?
- 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
Cross model number comparison: Is it something about dots rather than math? (Barth, Kanwisher & Spelke (2003)). What did they do?
Compared unimodal and cross modal sets of 10, 20 & 30 tones and spots. Asked to to give the total number
Cross model number comparison: Is it something about dots rather than math? (Barth, Kanwisher & Spelke (2003)). Results?
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
Enumeration vs. number comparison. Do they involve same mechanism?
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
