The Mathematical Brain

1. 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

2. Evolutionary grounding Deep 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) error rates

3. Evolutionary grounding Number size effect: The ability to detect the difference between 2 numbers decreases with the size of the numbers, (6,8 easier than 78,80) 4 8 12 16 Rat % responses 5 10 15 20 (Target) 8 12 16 25 Detection follows Weber’s Law - number is represented like a physical phenomenon Human % responses 10 15 20 25(Target) Human, damaged Left Posterior 1-9 10-19 20-99 % responses Dehaene et al. (1998) 10 20 50 (Target)

4. 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)

5. 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

6. Interplay between ancient and culturally acquired number ability Approximate ancient: Bilateral intraparietal sulci - language independent (no cost of switching - transfers well to novel facts Left (x=-44) Exact acquired: left frontal and angular gyri(BA39) (language/word association areas) Transfers poorly to diff. Language or novel facts z=0 =approximate = exact figures interpreted from Dehaene et al. (1999) LOOK at FIGURE 2

7. Intraparietal Sulcus Sebastian023 [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

8. 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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10. 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

11. 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)

12. 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.

13. 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

14. 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

15. 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

16. Education requires multi-perspective approaches experiential social biological

17. 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

18. 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

19. 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 3,4

20. Butterworth (2011) LOOK at FIGURE 5

21. Mathematics EDUCATIONAL CONTEXT Exercises on manipulation of numbers Exposure to digits and facts Experiences of reasoning about numbers Practice with numerosities Behaviour (learning) Simple number tasks Number symbols Math fact retrieval Concepts, principles, procedures Mind (cognitive) Numerosity representation, manipulation Spatial abilities Lobes: Occipital Parietal frontal Brain (Biological) Intraparietal sulcus Angular gyrus Prefrontal cortex Fusiform gyrus genetics

22. Impact on education? “Rescue Calcularis” (Kucian et al. 2012) Improved number line and maths - dyscalculics and controls. Reduced frontoparietal activity (esp. dyscalculia)

23. 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?

24. 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

25. 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.

26. 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)?*

27. 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).

28. “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)