The Mathematical Brain
Number sense is important to most
species, for food and defence.
E.g. Lionesses decide to attack intruders
by comparing the number of unfamiliar
roaring individuals and the number of their
own pride present (McComb et al. 1994)
Number sense has arisen through
evolution - it is not a “modern” human
faculty
By Falense (Own work) [GFDL
(http://www.gnu.org/copyleft/fdl.html) or CC-BYSA-3.0 (http://creativecommons.org/licenses/bysa/3.0/)], via Wikimedia Commons
Evolutionary grounding
Distance effect: ability to
discriminate between two
numbers improves with
distance between them (16,60
easier than 60,66)
(also true for humans with arabic
symbols => automatic conversion to
analogue form)
error rates
Deep and systematic parallels
imply number sense is
evolutionarily grounded:
The ability to detect the
difference between 2
numbers decreases with
the size of the numbers,
(6,8 easier than 78,80)
Dehaene et al. (1998)
% responses
Detection follows
Weber’s Law - number is
represented like a
physical phenomenon
% responses
Number size effect:
% responses
Evolutionary grounding
1-9
4
8
12 16
Rat
5
8
10
15
12 16
20 (Target)
25
Human
10
10-19
15
20-99
20
25(Target)
Human,
damaged
10
20
Left Posterior
50 (Target)
Development of Number Sense
Infants lose interest if they cannot discriminate the arrival of
a new stimulus within a certain time period.
Discriminable ratio develops during childhood.
At 6 months, children can discriminate between large
numbers (e.g. 16:32) if the ratio is between 1:2 and about
2:3 (Xu and Spelke, 2005).
- large approximate numerosity
Exact number awareness (up to 3-4) is also present in
infants and even in other primates (Hauser et al., 1996)
- small exact numerosity (object-file system)
Neural correlates of intuitive
number sense
Eger (2003) presented
a sequence of stimulus
items randomly
interleaved across
modalities and
categories - asked P’s
to spot the target
LOOK at FIGURE 1
Interplay between ancient and
culturally acquired number ability
Approximate ancient:
Bilateral intraparietal sulci
- language independent (no cost
of switching
Left
(x=-44)
- transfers well to novel facts
Exact acquired:
left frontal and angular gyri(BA39)
(language/word association areas)
Transfers poorly to diff. Language
or novel facts
figures interpreted from Dehaene et al. (1999)
z=0
=approximate
= exact
LOOK at FIGURE 2
Intraparietal Sulcus
Sebastian023 [CC-BY-SA-3.0
(http://creativecommons.org/licenses/by-sa/3.0)], via
Wikimedia Commons
Learning maths: Language
joins two initial systems?
New maths concepts (working towards large exact
calculations) may be constructed by bringing together
two initial systems (large approximate/small exact)
Credited to Spelke and Carey by Mark Johnson (2005)
*
*
*
*
*
*
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*
* * *
*
*
*
But…object file system limited (~3)
and specific to object (Huang 2010)
• 3 dogs is [DOG DOG DOG] and doesn’t mean
you can count 3 cats
• Moving from 3 to 4 appears to require a more
generalising, but approximate sense of number,
so that 4 dogs is [DOG x 4] and its mastery
means you can count 4 cats as well
• Possibly a shift from object file system to a
more generalisable system using approximate
numerical magnitude (IPS) systems
• Finally, a shift to exact formal abstract system
Learning maths: Language
joins two initial systems?
E.g. when learning counting words and routine:
Step 1. Children initially map only “one” to the object-file
system, can identify elements amongst arrays 1 and 4, not 2
and 4.
Step 2. Children use systems together to identify 2 and 3
(i.e. there’s a detectable ratio difference between 1 and 2, 2
and 3), all other numbers being mapped to “some”
Step 3. Children surmise that each word in the count
sequence corresponds to an additional element in the array.
(they never learn to count 4 before 3)
Neuroimaging studies can also show the time course of
learning in terms of brain activity, indicating how the use of
the brain changes as a learning task proceeds……..
Delazer (2003) Learning complex arithmetic – an fMRI
study, Cognitive Brain Research, 18, 76-88.
Training of adults on a set of 18
complex multiplication problems –
comparing brain activity when
multiplying before and after training.
Effects of multiplication training
Delazer et al (2003)
Where activity is
increased…..higher
levels of automatic
processing
By Polygon data were generated by Database Center for Life Science(DBCLS)[2].
(Polygon data are from BodyParts3D[1]) [CC-BY-SA-2.1-jp
(http://creativecommons.org/licenses/by-sa/2.1/jp/deed.en)], via Wikimedia Commons
Effects of multiplication training
Delazer et al (2003)
Where activity is
reduced…..DLPFC
Reduction in working
memory demands
and numeric
processing
DLPFC =
Dorsolateral
prefrontal cortex
By Natalie M. Zahr, Ph.D., and Edith V. Sullivan, Ph.D.
[Public domain], via Wikimedia Commons
Difficulties with number
Social
- values and beliefs of society, family, peers
- personality and communication with teacher
Free-will (requires experiential investigation)
- motivation of pupils
- motivation of parents, teachers
Biological
- e.g. developmental dyscalculia
Education requires multiperspective approaches
experiential
social
biological
Number difficulties:
Interacting Issues
Evidence for non-biological influence:
Cross-cultural studies comparing abilities show strong
differences (Stevenson et al., 1993)
Evidence for biological influence:
Dyscalculia often comorbid with ADHD and dyslexia
Some cases associated with epilepsy, fragile X
chromosome, Turner Syndrome, Phenylketonuria,
Williams Syndrome
Core-deficit theory (or theories)
of Dyscalculia
• These propose that dyscalculia is a core deficit in an
inherited foundational capacity for numbers
• This may be detection of numerosity or, the ability to
code numerosity – e.g. the ability to categorise
examples of “twoness” together
Dyscalculia neural correlates?
Reduced gray matter
volume in parietal
lobes of low birthweight children with
number difficulties
Isaacs et al. (2001)
Diagram shows location of reduced gray matter
(-39,-49,45) interpreted from Isaacs et al (2001)
Also: Small region of reduced grey matter density in left IPS in
an adolescent dyscalculic (Rykhlevskaia et al., 2009)
LOOK at FIGURES 3,4
Butterworth (2011)
LOOK at FIGURE 5
Mathematics
Behaviour
(learning)
Mind
(cognitive)
EDUCATIONAL
CONTEXT
Exercises on
manipulation of
numbers
Simple
number
tasks
Number
symbols
Math fact
retrieval
Numerosity
representation,
manipulation
Lobes: Occipital
Brain
(Biological)
Fusiform
gyrus
Exposure to
Concepts,
digits and facts
principles,
procedures Experiences of
reasoning
Spatial
about numbers
abilities
Practice with
numerosities
Parietal
Intraparietal
sulcus
genetics
frontal
Angular
gyrus
Prefrontal
cortex
Impact on education?
“Rescue Calcularis” (Kucian et al. 2012)
Improved number line and maths - dyscalculics and controls.
Reduced frontoparietal activity (esp. dyscalculia)
Brain-basis or brain outcome?
We know that, even in the adult brain, education and
training can change brain structure
-is it just a result of low-birth weight, socio-economic
factors influencing other (e.g. educational) processes?
For argument sake, however, let us imagine that this
reduction in gray matter is also linked to a genetic
difference.
Does that make it a cause?
Neuroconstructivist Caveats
Recalling the neuroconstructivist approach:
* probabilistic epigenetics and Waddington, external
environmental effects, the importance of common stimuli
So, even with possible genes identified, non-genetic
environmental factors may still account for large amounts
of variance (e.g. more than half of the variance in dyslexia
data - DeFries et al., 1993)
* if neural pathway construction influenced by neural inputs
from other areas, then “atypicality” in one area will produce
“atypicality” elsewhere.
So, other atypical parts of the brain may be involved with
unusual parietal development
Neuroconstructivist Caveats
* damage to brain systems more devastating in
development terms than damage to cortical areas
It may be a “smaller” atypicality in a system of the brain
causing this - more difficult to detect - such as atypical
early thalamic input.
* different types of initial atypicality can result in same
outcomes – i.e. if the same brain system affected.
….and so this “smaller” atypicality may not be exclusive
to dyscalculia.
Many questions unanswered….
Are their genetic differences? What level of variance do
these account for in dyscalculia?
What gray matter differences are associated with
dyscalculia? Do these co-occur with other disorders?
When do these differences first emerge in development?
What sort of interventions influence mathematical and
neural outcomes (ability, activity, structure)?*
Beware of “cause”
Modern developmental cognitive neuroscientific
approaches emphasise that
Two of John Morton’s maxims when discussing
cause:
“there is no single cause of anything”
“nothing is determined”
(Morton, 2004).
“Cause is not an easy word. Its popular use
would be laughable if it was not so
dangerous, informing, as it does, government
policy on matters that affect us all.”
(Morton, 2004)