Lecture 12: Non-Symbolic Number Processing Flashcards
Learning objectives
Lecture overview:
Part 1: What is mathematical cognition?
Part 2: Non-symbolic numerical processing
Part 3: Small exact system
Part 4: Approximate Number System
- - - - - - - - - - - - - - - -
Learning objectives:
- Explain what is mathematical cognition and why it is a relevant subject of study.
- Describe the characteristics of the system involved in the exact perception of small numerosities.
- Describe the characteristics of the system involved in the approximate perception of large numerosities and the methodological issues related to assessing it.
- Discuss how the two above-mentioned numerical systems develop and relate to mathematical achievement.
Part 1: What is mathematical cognition?
What is maths cogniton?
Seeks to understand how individuals understand mathematical ideas, how people learn numbers and mathematical procedures and problems
Mathematical cognition develops from infancy to adulthood, there are individual differences (explanatory factors) and factors of mathematical disability (low maths skills – dyscalculia).
Mathematical cognition as a subject of study is a multidisciplinary field as it overlaps with
- psychology (in terms of cognitive systems)
- neuroscience (brain areas and processes)
- education (in terms of instruction and intervention of learning).
Cog psych starts with a behav, and we measure accuracy e.g. reaction times –> extract a model/ theory/ knowledge to be combined e.g. one thing influences the other –> then connect this to a brain area (neuroscience) and see how the environment affects this (learning of the skill).
Part 1: What is mathematical cognition?
Why study mathematical cognition?
Window into complex thinking, describes how cognitive system processes numerical information and mathematical ideas. Also, low numeracy is a problem.
> 24% of adults have a numeracy level below that one needed to function in everyday life
20% of adults cannot perform two-steps calculation, understand decimal, fractions and proportions
Low numeracy in childhood has a negative impact on future quality of life.
- Career path and wealth (lots of jobs are cognitively demanding e.g. chemical engineering, physics, medicine, computer science etc)
- Health
Part 1: What is mathematical cognition?
2 kinds of numerical processing…
Non-symbolic numbers are representing numerical information represented in a physical, analogue way e.g. dots
Symbolic numbers are represented in a symbolic way visually or verbally e.g. words or symbols - three/ 3 etc
Part 2: Non-symbolic numerical processing
Two systems for language independent numerical processing…
- One system is involved in small exact numerical perception (object tracking system, OTS; parallel individuation, PI)
- One system is involved in large approximate numerical perception (approximate number system, ANS)
Part 3: Small exact system
Small exact numerical perception
What is subitising?
Subitising: fast and accurate enumeration of small (3-4 items) numerical quantities
(from the Latin word subitus = sudden)
Subitising is the mechanism that allows the very fast perception of small numerical quantities (up to about 3) without having to count. When there are more elements (dots) the number must be perceived via counting, which takes more time.
Part 3: Small exact system
Subitising system development
Study (1990s):
- showed kids small numerical sets in pairs and asked them which set showed a larger number of elements (e.g. “Which box has more?”)
Found:
- kids were good at this when the numerical sets were small e.g. 1 vs 2, 2 vs 3, but with larger sets e.g. 3 vs 4, 4 vs 5, their accuracy went down.
Means:
- preschool children can enumerate 1-3 items quickly, using subitising
- when older, around 7 years, they are quite good at subitising up to about 3-4.
- individual differences remain, some people are better than others
Part 3: Small exact system
Where does the limited capacity of subitising come from?
Visuo-Spatial working memory explanation…
Asked people to perform an enumeration task where subitising was assessed, and a dual task* (visuo spatial working memory task).
*means you are asking ppts to complete two tasks at the same time
The addition of a dual task means that if performance in the second task goes down, the tasks probably share some kind of cognitive mechanism during processing, cognitive resources being limited, the performance worsens.
The subitising limit lessened in the dual task study, meaning visuo-spatial working memory is probably implicated in the process of subitising –> number of elements able to be stored in the VSWM effects ability to subitise
It has been found that children with developmental dyscalculia have steeper subitising slopes –> suggests ability to perceive small numerical quantities is important to build symbolic numerical skills & mathematical knowledge.
Part 3: Small exact system
Example of limited subitising capacity in real life
Roman numerals
instead of using IIII for for, we use IV, instead of IIIII it’s V etc etc, numerals can be percieved quicker
Part 3: Small exact system
Summary of the small exact system
> System dedicated to the perception of small numerosities (~3-4 items) –> Subitizing.
It shows a rapid development: from 1-3 items in preschool children to 3-4 items from childhood to adulthood (but there is variability!).
This system is most likely related to visuospatial working memory.
Some evidence of reduced subitizing in children with math disability.
Part 4: Approximate Number System
Large approximate numerical perception
What is the ratio effect?
Trying to perceive large numerical values with no time for counting leads to estimation, which is the representation of large numerical quantities.
Ratio effect: the ability to discriminate between large numerical quantities varies as a function of the ratio of the numerical sets
Close to 1: hard to discriminate
Far from 1: easy to discriminate.
Part 4: Approximate Number System
Ratio effect examples
Comparison - 12 vs 16
Ratio - 16:12, 4:3= 1.33
Distance - 16-12= 4
Comparison - 32 vs 36
Ratio - 36:32, 9:8= 1.13
Distance - 4
As the ratio has been moved closer to 1, despite the difference remaining the same it has become harder to perceive the difference between the numerical values.
Part 4: Approximate Number System
ANS acuity: Gaussian curves
Each numerosity is represented as a Gaussian curve of activation on an internal continuum, the mental number line.
Narrow curves of activation = less overlap = better performance in numerosity comparison task
Wide curves of activation = more overlap = worse performance in numerosity comparison task
e. g. 4 vs 8 –> Ratio 2 –> no overlap –> high accuracy
e. g. 9 vs 8 –> Ratio 1.13 –> huge overlap –> low accuracy
Part 4: Approximate Number System
Development of the ANS
Development of the ANS: Infants
6 month olds are able to perceive numerical differences with ratios of 2, but not 1.5.
10 month olds are able to perceive numerical differences with ratios of 1.5 but not 1.2
This illustrates that this is a skill that develops with age… BUT…
The skill to discriminate between large numerical quantities rapidly develops from childhood to adulthood, then experiences a slow decline
Part 4: Approximate Number System
ANS… universal skill?
The approximate number system is shared with other species, meaning, for example, monkeys are able to discriminate between large numerical quantities.
Why is this a universal skill?
Seeking larger quantities of food?
- There was a study done on lions, if they perceive that another group is approaching (based on the roars) if they work out that they are more numerous (more of them) than the group that’s approaching then they will continue to approach and fight them, if the group approaching is bigger, then they will leave –> Enemies
