1. No category

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
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•
•
•
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•
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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
1
2
Ratio / target
-0.4
Weber
fraction
0.5
1
2
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
0.8
0.6
0.4
0.2
distance
0
0
0.2
0.4
speed
0.6
0.8
1
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).