Dyscalculia: Why do numbers make no sense to some people?

Dr. Anna J. Wilson Research Fellow Department of Psychology University of Auckland

My background • BSc, The University of Auckland (Psychology) – Exchange to University of California, Berkeley

• PhD, University of Oregon (Psychology) – Dissertation: Numerical & spatial cognition – Supporting area: Math learning disabilities

• Postdoctoral fellowship, INSERM U562, Paris – Development & testing of remediation software for dyscalculia (with Stanislas Dehaene)

• Research fellow, University of Auckland – Neural correlates of dyscalculia & relationship between dyscalculia & dyslexia (with Karen Waldie)

www.aboutdyscalculia.org • PhD, University of Oregon – Dissertation: Numerical & spatial cognition – Supporting area: Math learning disabilities

• Postdoctoral fellowship, INSERM U562, Paris – Development & testing of remediation software for dyscalculia (with Stanislas Dehaene)

• Research fellow, University of Auckland – Neural correlates of dyscalculia & relationship between dyscalculia & dyslexia (with Karen Waldie)

www.aboutdyscalculia.org

Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

Developmental dyscalculia • Severe difficulty in mathematics presumed to be due to a specific impairment in brain function • Also called “Mathematics Disorder” (DSM-IV), or “mathematical learning disabilities” • Prevalence: around 6% (same as dyslexia!) • Has genetic component (runs in families) • Understudied compared to dyslexia Kosc, 1974; Shalev & Gross-Tsur, 2001; Geary, 1993, 2004; Badian, 1983; Lewis, Hitch, & Walker, 1994

Surface symptoms Delay in acquisition of:

3x2= 3x3= 3x4=

– Counting – Addition strategies (counting on vs. counting all) – Memorization of number facts (e.g. times tables) Geary (1993, 2004) - review 2+5=?

Difficulties with story problems? (e.g. Nancy Jordan’s research) - esp. when dyslexia present!

Counting all “1..2... 1...2...3...4..5.. 1...2...3...4...5...6...7” Counting on “2...3...4...5...6...7” Counting on (max) “5...6...7”

Core cognitive symptoms • Difficulty representing quantity (“number sense”). – Slow to compare numbers

(Llanderl et al., 2004)

– Slow to enumerate 1-3 objects (“subitizing”) (Reeve et al., in press)

• Number symbols processed less automatically – Number stroop task (Rouselle & Nöel, 2007; Rubinsten & Henik 2005)

7

9 7

• Mental number line slow to develop

9

Some examples…

Llanderl, Bevan & Butterworth, 2004. Note that the impairment is in response time as well as accuracy

7

Llanderl, Bevan & Butterworth, 2004.

9

Mental number line development 0

"Put a mark where 64 goes"

Siegler & Booth, 2004

100

Individual differences on this task correlate with maths achievement scores.

Mental number line in dyscalculia "Put a mark where 64 goes"

0

a) Number placement at T1: 0-100 number line

100

b) Number placement at T1: 0-1000 number line

100

1000

Mean estimated magnitude

Mean estimated magnitude

900

80

60

40

20

Dyscalculic Control

0 0

20

40

60

Actual magnitude

80

100

800 700 600 500 400 300 200

Dyscalculic

100

Control

0 0

100 200 300 400 500 600 700 800 900 1000

Actual magnitude

Wilson, Krinzinger, Nuerk, Dehaene & Willmes, in prep

Likely other symptoms Difficulty with: • Subtraction • Using finger counting (slow, inaccurate, trouble recognising finger configurations) • Decomposing numbers (e.g. recognizing that 10 is made up of 4 and 6) • Understanding place value • Learning/understanding multi-step calculation procedures and problem solving Anxiety about or negative attitude towards maths

Consequences in adults • Blocked from certain professions (lower salary) • Difficulty managing money • Difficulty understanding statistics/numbers (influence on decision making) • Low self-esteem, anxiety, avoidance

“I have always had difficulty with simple addition and subtraction since young, always still have to ‘count on my fingers quickly’ e.g. 5+7 without anyone knowing. Sometimes I feel very embarrassed! Especially under pressure I just panic.”

Consequences in adults “I struggled through school with maths to the point the teachers gave up on me. I can only count on my fingers or with a calculator. I can't count out change, no matter how small and often get flustered with any tasks involving numbers. Despite trying hard I could never remember my 'times tables'. Division etc just bewildered me totally. English was one of my best subjects at school.” “I have no trouble whatsoever reading or writing, understanding literary concepts and theories etc., but spend an hour sitting in the bank trying to work out how much money is in my cheque account! Last year I returned to University, attempting to avoid any papers containing mathematics, but hidden in nearly everything are formulas and calculations.”

Co-morbid difficulties Both verbal and non-verbal: • Dyslexia (50%) • ADHD (30%) • Dyspraxia • Spatial difficulties

Why is there such a high association between these disorders?? What is the implication for remediation?

The big questions What causes dyscalculia? How can it be indentified early? Can it be “prevented”? What is the best type of remediation? Why does it co-occur so often with other learning disabilities? Are there subtypes; if so do they need different remediation approaches? In order to answer these we need to know about how maths works in the brain.

Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

Numerical cognition • Study of representation of number in the brain • Methods: Animals, infants, cross-cultural linguistics, brain imaging, cognitive psychology • Good introductory books:

Stanislas Dehaene

Mathematics is componential • Non-verbal – number, approximation, comparison

• Verbal – number facts (multiplication, addition)

• Logical – problem solving, higher maths

• Spatial – geometry? Number line?

Non-verbal bases of number • Number is not “constructed” or dependent on logic/language as Piaget thought • Animals can add, subtract, compare quantities! • As can pre-verbal human infants... Platt & Johnson (1971). Rats taught to press button a certain number of times for reward. Mode at the right value, but responses approximate.

Approximate number: Demonstration

Which side has more dots?

+

+

+

Ratio = 0.5

Ratio = 0.79

Faster, more accurate

Slower, less accurate

Approximate number Ability to discriminate depends on ratio of the two numbers. This "distance effect" is found in animals, and human adults and children.

e.g. see Brannon (2003) for review

+

12

24

+

19

24

+

12

24

24

19

Ratio = 0.5

Ratio = 0.79

Dots: faster, more accurate Digits: the same!!

Dots: slower, less accurate Digits: the same!!

Approximate number Ability to discriminate depends on ratio of the two numbers. This "distance effect" is found in animals, and human adults and children.

e.g. see Brannon (2003) for review

Approximate arithmetic Barth (2005) – Five year old children

Approximate number What have we learned about it so far? • • • • •

Non-verbal Non-symbolic Present in animals / human infants Still accessed in skilled adults Used for representation and operations

Next: Has a specific brain basis

Number sense “ Number sense ” is a short-hand term for our ability to quickly understand, approximate, and manipulate numerical quantities. (Dehaene, 2001) Caution: term used in different ways in education (vs. in numerical cognition) see Berch, 2005 review

Number sense in adults Using number sense activates the intraparietal sulcus (IPS): (This same area is involved in thinking about space.) Axial slice

Left hemisphere x = - 48

z = 44

Right hemisphere z = 49

x = 39

50 %

HIPS

22 % Dehaene, Piazza, Pinel, & Cohen (2003)

Tasks that activate this region: • Comparison of numbers • Subtraction • Approximation • Estimation e.g. comparison • Non-symbolic tasks Automatically activated by viewing numbers

Neurons & approximate number The “distance effect shows up in responses of neurons Areas that respond to a change of numerosity

Nieder & Miller, 2004

% of activation to a number of dots after a adaptation period

% d’activation

0.4

0.4

0.2

0.2

Weber fraction

0

0

-0.2

-0.2

-0.4

0.5

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Ratio / target

-0.4

Weber fraction

0.5

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Ratio / target

Piazza, 2004

Number sense in children Neural correlates the same as in adults. Non symbolic tasks

Cantlon, Brannon, Carter & Pelphrey 2006 fMRI in 4 year olds

Mathematics is componential • Non-verbal

Intraparietal sulcus (IPS) – number, approximation, comparison

• Verbal

Perisylvian language network – number facts (multiplication, addition)

• Logical

Frontal lobes?

– problem solving, higher maths

• Spatial – geometry? Number line?

Parietal lobes?

Superior Frontal Gyrus

Intraparietal Supramarginal sulcus gyrus Angular gyrus

Middle Frontal Gyrus Inferior Frontal Gyrus

Middle Temporal Gyrus Inferior temporal Gyrus

Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

Causes of dyscalculia "Core deficit" hypothesis: hypothesis: Deficit in number sense (Butterworth, 1999; Gersten & Chard, 1999; Wilson & Dehaene, 2007)

left hemisphere

quantity

right hemisphere

quantity

verbal "six"

visual 6

visual 6

Dehaene, S. (1992). Cognition, 44, 1-42. Dehaene, S., & Cohen, L. (1995). Mathematical Cognition, 1, 83-120.

Brain bases of dyscalculia

Dyscalculic children - less grey matter in IPS (Rotzer et al., 2008)

Dyscalculic adults born pre-term – less gray matter in IPS (Isaacs, Edmonds & Lucas, 2001) Superimposed images of sulci Controls

Turner subjects

Dyscalculic children – less activation in IPS during magnitude tasks (Kucian et al., 2006) Molko, Cachia and Riviere (2004) Turners subjects structural and functional alternations in IPS.

Causes of dyscalculia "Access" hypothesis : Deficit in link between number sense and symbols (Rouselle & Nöel, 2007)

left hemisphere

quantity

verbal "six"



visual 6

"Core deficit" hypothesis: hypothesis: Deficit in number sense (Butterworth, 1999; Gersten & Chard, 1999; Wilson & Dehaene, 2007)

right hemisphere

quantity


visual 6

Dehaene, S. (1992). Cognition, 44, 1-42. Dehaene, S., & Cohen, L. (1995). Mathematical Cognition, 1, 83-120.

One subtype proposal • Number sense / number sense access – Everything affected except counting, fact retrieval – May have difficulty with non-symbolic tasks

• Verbal – Difficulty with counting, fact retrieval, word problems – Associated with dyslexia?

• Executive – Difficulty with fact retrieval, use of strategy/procedure – Associated with ADHD??

• Spatial – Difficulty with subitizing, apprehension of non-symbolic quantity… mental number line? Wilson & Dehaene (2007)

Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

The big questions What causes dyscalculia? How can it be indentified early? Can it be “prevented”? What is the best type of remediation? Why does it co-occur so often with other learning disabilities? Are there subtypes; if so do they need different remediation approaches?

Auckland comorbidity project Postdoctoral fellowship with Karen Waldie, funded by Univ. of Auckland. Phase I: 80 adults with dyscalculia, dyslexia, both or neither (20 per group) • Cognitive testing (symptoms, subtypes) • Brain imaging (fMRI; neural markers) Phase II (if funded!): similar study in children

An aside... Many people mistakenly think that “if it’s in the brain it can’t be changed” Nothing could be more wrong! • The brain is the basis of all learning • Brain function and even structure is highly “plastic”, especially at a young age • The mild impairments associated with learning disabilities are nothing like the brain damage caused by stroke/lesion

An example from reading... Fast ForWord (Merzenich et al., 2006; Tallal et al., 2006; Scientific Learning Corporation) • Adaptive game software for reading • Specific Language Impairment, Dyslexia • Intensive individual intervention (1h/day for 12 wks)

Temple et al. 2003

Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

Identification Test for: • Mathematics level (standardised test) – e.g. PAT, Woodcock Johnson, WRAT, KeyMath

• Profile of performance in different components • IQ (rule out general difficulties) • Dyslexia, ADHD, spatial difficulties, dyspraxia if suspected Important to rule out: educational experiences, motivation

Profiling tests Ideally: Measurements of response time as well as accuracy. Separate breakdowns for different operations and components • KeyMath (5-22 yrs) • TEMA-3 (3-8 yrs) • CMAT (7-19 yrs) • Diagnostic mathematics profiles (AUS) • Booker Profiles? (AUS)

Dyscalculia Screener (nferNelson) Brian Butterworth, University College London www.mathematicalbrain.com Computerised, for use in schools – Number stroop – Subitizing / Counting – Mental arithmetic Administration time: 30 minutes Advantages: Precise measures including reaction time, standardised, fast Disadvantages: Assumes dyscalculia caused by core deficit in number sense

Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

Individual remediation • • • • • • •

Focus on understanding (esp. quantity) Drilling of facts only useful up to a point Use concrete materials Start at an easy level (success important!) Provide lots of practice Reduce need for memorisation Ask a lot of questions to get the child engaged and thinking • Make learning active and fun

How to help in the classroom • Give children their own set of work, at their level • Allow extra time • Use written and verbal instructions and questions • Extra scaffolding, especially for multi-step procedures • Reduce opportunity for comparison with peers

What about subtypes? In the absence of a verdict from research a good way to approach subtypes is by using a componential analysis to plan remediation. e.g. If child is good at multiplication but has trouble with number sense, focus on number sense! If child has dyslexia and trouble with word problems, focus on reading/interpreting. Note that this necessitates a componential assessment

Remediation workbooks

Dyscalculia Guidance by Brian Butterworth & Dorian Yeo. (2004). The Dyscalculia Toolkit: Supporting Learning Difficulties in Maths by Ronit Bird (2007). Dyscalculia: Action Plans for Successful Learning in Mathematics by Glynis Hannell. (2005). Dyslexia, Dyspraxia and Mathematics by Dorian Yeo. (2003). Mathematics for dyslexics including dyscalculia by Steve Chinn and Richard Ashcroft. (2007, 3rd Edn). The Trouble with Maths: A Practical Guide to Helping Learners with Numeracy Difficulties by Steve Chinn. (2004).

Software

Bubble Reef by ICDC

Number Shark by White Space

To Market, To Market by Learning in Motion

The Number Race by myself and Stan Dehaene

Knowsley Woods by ICDC

The Number Race http://www.unicog.org/main/pages.php?page=NumberRace Adaptive game to remediate/teach early number sense • Non-profit model ("open source" = free to obtain, copy, distribute, modify) • Programmed by myself Wilson et al. 2007a,b

Languages:

Research based instructional principles • Enhance number sense – intensive number comparison (e.g. largest of 3, 9?) – speed deadline – link between number and space

• Cement non-symbolic symbolic links – repeated association of non-symbolic & symbolic numbers – encourage increasing reliance on symbols

• Conceptualise and automatise arithmetic – concrete representations of operations – speed deadline

• Maximize motivation – positive reinforcement – difficulty adaptation – entertaining format (game!)

Adaptive algorithm • AI (artificial intelligence) module builds a model of children's "knowledge space" • Presents problems on borders of knowledge (not too easy, not too hard) • Tries to push these borders • General: can be used for any task if learning dimensions known 1

notation

0.8 0.6 0.4 0.2 0 1

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distance

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speed

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Talk outline • • • • • • •

What is dyscalculia? Numerical cognition Causes of dyscalculia Auckland comorbidity project Identification Remediation The future

Future goals Identify children as young as possible and provide preventative intervention Why? • Brain more "plastic" at young age • Maths is highly cumulative • Avoid negative experiences associated with failure • Quicker and cheaper

Early identification • Currently can do behavioural screening, either – Non-symbolic in infancy – Symbolic in kindergarten

• Can predict future performance from behavioural measures in kindergarten (e.g. Mazzocco & Thompson, 2005)

• But still have many false positives • More work needed to develop measures

Future goals Identify genetic markers so we know which kids are at risk. Need to: find genes Need dyscalculic families! Identify neural markers. Need to: develop inexpensive and quick brain imaging techniques.

A copy of this presentation ...Can be found on: My website: www.aboutdyscalculia.org (look under “news”) Contact information: [email protected]

References Badian, N. A. (1983). Dyscalculia and nonverbal disorders of learning. In H. R. Myklebust (Ed.), Progress in Learning Disabilities (Vol. 5, pp. 235-264). New York: Stratton. Barth, H., La Mont, K., Lipton, J., & Spelke, E. S. (2005). Abstract number and arithmetic in preschool children. PNAS, 102(39), 14116-14121. Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339. Brannon, E. M. (2003). Number knows no bounds. Trends in Cognitive Sciences, 7(7), 279-281. Butterworth, B. (1999). The mathematical brain. London: Macmillan. Cantlon, J. F., Brannon, E. M., Carter, E. J., & Pelphrey, K. A. (2006). Functional imaging of numerical processing in adults and 4-y-old children. PLoS Biology, 4(5), e125. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1-2), 1-42. Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. Oxford: Oxford University Press. Dehaene, S. (2001). Précis of the number sense. Mind and Language, 16, 16-36. Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1(1), 83-120. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487-506. Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological and genetic components. Psychological Bulletin, 114(2), 345-362. Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37(1), 4-15. Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of special education, 33(1), 18. Isaacs, E. B., Edmonds, C. J., Lucas, A., & Gadian, D. G. (2001). Calculation difficulties in children of very low birthweight: A neural correlate. Brain, 124(9), 1701-1707. Kucian, K., Loenneker, T., Dietrich, T., Dosch, M., Martin, E., & von Aster, M. (2006). Impaired neural networks for approximate calculation in dyscalculic children: A functional mri study. Behavioral and Brain Functions, 2, 31. Kosc, L. (1974). Developmental Dyscalculia. Journal of Learning Disabilities, 7(3), 164-177. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: a study of 8-9-year-old students. Cognition, 93(2), 99-125. Lewis, C., Hitch, G. J., & Walker, P. (1994). The prevalence of specific arithmetic difficulties and specific reading difficulties in 9- to 10-year old boys and girls. Journal of Child Psychology and Psychiatry, 35(2), 283-292.

References cntd. Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability. Learning Disabilities Research and Practice, 20(3), 142-155. Merzenich, M. M., Jenkins, W. M., Johnston, P., Schreiner, C., Miller, S. L., & Tallal, P. (1996). Temporal processing deficits of language-learning impaired children ameliorated by training. Science, 271(5245), 77-81. Molko, N., Cachia, A., Riviere, D., Mangin, J. F., Bruandet, M., Le Bihan, D., et al. (2003). Functional and structural alterations of the intraparietal sulcus in a developmental dyscalculia of genetic origin. Neuron, 40(4), 847-858. Nieder, A., & Miller, E. K. (2004). A parieto-frontal network for visual numerical information in the monkey. PNAS, 101(19), 7457-7462. Piazza, M., Izard, V., Pinel, P., Le Bihan, D., & Dehaene, S. (2004). Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron, 44, 547-555. Rotzer, S., Kucian, K., Martin, E., Aster, M. v., Klaver, P., & Loenneker, T. (2008). Optimized voxel-based morphometry in children with developmental dyscalculia. NeuroImage, 39(1), 417-422. Rousselle, L., & Noel, M.-P. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing. Cognition, 102(3), 361-395. Rubinsten, O., & Henik, A. (2005). Automatic Activation of Internal Magnitudes: A Study of Developmental Dyscalculia. Neuropsychology, 19(5), 641. Shalev, R. S., & Gross-Tsur, V. (2001). Developmental dyscalculia. Pediatric Neurology, 24(5), 337-342. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428-444. Tallal, P., Miller, S. L., Bedi, G., Byma, G., Wang, X., Nagarajan, S. S., et al. (1996). Language comprehension in language-learning impaired children improved with acoustically modified speech. Science, 271(5245), 81-84. Temple, E., Deutsch, G. K., Poldrack, R. A., Miller, S. L., Tallal, P., Merzenich, M. M., et al. (2003). Neural deficits in children with dyslexia ameliorated by behavioral remediation: Evidence from functional mri. Proc Natl Acad Sci U S A, 100(5), 2860-2865. Wilson, A. J., & Dehaene, S. (2007). Number sense and developmental dyscalculia. In D. Coch, G. Dawson & K. Fischer (Eds.), Human behavior, learning and the developing brain: Atypical development. New York: Guilford Press. Wilson, A. J., Dehaene, S., Pinel, P., Revkin, S. K., Cohen, L., & Cohen, D. (2006a). Principles underlying the design of “the number race”, an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2(19). Wilson, A. J., Revkin, S. K., Cohen, D., Cohen, L., & Dehaene, S. (2006b). An open trial assessment of “the number race”, an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2(20).