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-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons

Evolutionary groundingDeep and systematic parallels imply number sense is evolutionarily grounded:

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)

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Evolutionary groundingNumber size effect:

The ability to detect the difference between 2 numbers decreases with the size of the numbers, (6,8 easier than 78,80)

Rat

Dehaene et al. (1998)

Detection follows Weber’s Law - number is represented like a physical phenomenon

4 8 12 16

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8 12 16 25

10 15 20 25(Target)% r

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Human

1-9 10-19 20-99

10 20 50 (Target)% r

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Human,

damaged

Left Posterior

Development of Number SenseInfants 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 1LOOK at FIGURE 1

Interplay between ancient and culturally acquired number ability

Approximate ancient:

Bilateral intraparietal sulci

- language independent (no cost of switching

- 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)

Left (x=-44)

z=0

=approximate= exact

LOOK at FIGURE 2LOOK 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

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)

Learning maths: Language joins two initial systems?

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 trainingDelazer 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 trainingDelazer et al (2003)

By Natalie M. Zahr, Ph.D., and Edith V. Sullivan, Ph.D. [Public domain], via Wikimedia Commons

Where activity is reduced…..DLPFC

Reduction in working memory demands and numeric processing

DLPFC = Dorsolateral prefrontal cortex

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 multi-perspective approaches

social biological

experiential

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 dyslexiaSome 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 birth-weight 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 3,4LOOK at FIGURES 3,4

Butterworth (2011) LOOK at FIGURE 5LOOK at FIGURE 5

Mathematics

Behaviour(learning)

Mind (cognitive)

Brain(Biological)

Number symbols

Math fact retrieval

Numerosity representation, manipulation

Spatial abilities

Concepts, principles, procedures

Fusiform gyrus

Intraparietal sulcus

Angular gyrus

Prefrontal cortex

Simple number tasks

EDUCATIONAL CONTEXT

Exercises on manipulation of numbers

Exposure to digits and facts

Experiences of reasoning about numbers

Practice with numerosities

genetics

Lobes: Occipital Parietal frontal

“Rescue Calcularis” (Kucian et al. 2012)

Improved number line and maths - dyscalculics and controls.

Reduced frontoparietal activity (esp. dyscalculia)

Impact on education?

Brain-basis or brain outcome?

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?

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?

Neuroconstructivist CaveatsRecalling 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

* 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.

Neuroconstructivist Caveats

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)