Thinking about Influences on
Number Skills and Simple Arithmetic
for Children Aged 4-8
Dr Silke Göbel & Dr Anuradha J. Bakshi
INTRODUCTIONS
YORK, England
Frankfurt, Germany
FRANKFURT, Germany
MUMBAI, India
BANGALORE, India
٣
23 + 65
百
427
不可思议
Typical development
From Butterworth (2010)
Overview
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•
•
•
•
•
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Preverbal number skills
Counting
Number Symbols
Arithmetic
Working memory
Cross-cultural differences
Struggling with math
PREVERBAL NUMBER SKILLS
from Dehaene (1999)
Subitizing
Counting
Subitizing
from Dehaene (1999)
Human Infants
Human Infants
Xu & Spelke (2000)
• 6-month-old
infants
discriminate 8
from 16
6-month-old Infant Numerical Discrimination
Ratios they could do
Ratios they could not do
4 vs 8
4 vs. 6
8 vs 16
8 vs. 12
16 vs 32
16 vs. 24
1: 2 YES
2:3 NO
From Brannon (2005)
Development of Number Sense Acuity
Halberda &
Feigenson (2008)
• 3, 4, 5, 6 year olds
& adults
Development of number sense acuity
Halberda &
Feigenson (2008)
• 3, 4,5, 6 year olds
& adults
Weber Fraction
• Equal to the difference between the two
numbers divided by the smaller number
• The closer the number is to zero the better
the number discrimination
• Example:
– ratio 7:8
– Weber fraction (8-7/7) = 0.14
Predictors of math performance:
Weber fraction
Halberda et al. (2008)
COUNTING
Group Activity
USE YOUR FINGERS!
HOW DO YOU COUNT?
Finger Counting
• Mundurucu finger counting
WHICH HAND FIRST?
from Lindemann et al. (2011)
A Small Survey of Counting Up To 10 Using
Fingers/Hands in India (n=54)
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•
•
•
Children, young adults & older adults
Both female and male
Grown up in villages and cities
Grown up in different parts of India
(e.g., Delhi, Chandigarh, W. Bengal,
Rajasthan, Gujarat, Mumbai, Chennai)
• From different communities
WHICH HAND FIRST?
• Handedness?
• Orthography?
• Yet, fewer count starting with the left hand; most
start with their right hand. Right-starters
outnumber the left-starters (40 out of 54).
• ―Finger counting habits are related to eye
scanning habits in visual perception outside of
reading‖ (Lindemann et al., 2011, p. 574).
WHICH FINGER FIRST?
• Thumb?
• Index finger?
• Little finger?
• Smallest finger heuristic more representative of
the Indian participants (39 out of 54 participants).
SOME REPEATED TYPES OF COUNTING UP
TO 10 USING FINGERS/HANDS
SOME REPEATED TYPES OF COUNTING UP
TO 10 USING FINGERS/HANDS
SOME REPEATED TYPES OF COUNTING UP
TO 10 USING FINGERS/HANDS
SOME REPEATED TYPES OF COUNTING UP
TO 10 USING FINGERS/HANDS
 One-third of the
participants counted
using phalanges (n=
18).
 Eight participants
counted using 4
subdivisions/finger.
Counting
• Children typically learn to count before they go to school
• Gelman & Gallistel’s how to count principles:
– 1-1 principle (count each thing once)
– Stable order principle (use count words in fixed order)
– Cardinality principle (last count word represents number of things
counted)
– Abstraction principle (any collection of objects can be counted)
– Order irrelevance principle (order of counting objects doesn’t affect
outcome)
• Counting skills form the basis of early arithmetic skills
Counting principles
Counting direction
Counting direction
Percentage of adults
100
left starting
right starting
80
60
40
20
0
Left to right
Right to left
Mixed
Illiterate
Reading Experience
from Shaki, Fischer & Göbel (submitted)
Counting direction
Reading Direction
100
Left to right
Mixed
Right to left
left starting
right starting
Percentage of children
80
60
40
20
0
pre-school school
pre-school school
pre-school school
from Shaki, Fischer & Göbel (submitted)
COUNTING DIRECTION: INDIA
NUMBER SYMBOLS
LEFT OR RIGHT?
WHICH NUMBER IS LARGER?
2
8
7
1
9
3
8
9
When is it easier to identify
the larger number?
Symbolic Distance Effect
from Butterworth (1999)
Predictors of Math Performance:
Numerical Distance Effect
De Smedt et al. (2009):
N = 42 6-year-olds, tested again after 1 year
2
8
7
3
1
9
Was is it easy to identify the
larger number?
Was there any distraction?
Size Congruity Effect
• In the numerical
comparison task: size
congruency effect at
all ages
• In the physical
comparison task:
incongruency only
affected older children
and adults
• Automatization of
number processing
develops gradually
Butterworth (1999)
ARITHMETIC
2+ 5=?
Problem Size Effect - Addition
from Ashcraft (1995)
The Development of Arithmetic Skills
• At school children master single addition first, followed
by subtraction, multiplication and division (going on to
multi-digit numbers and problems associated with place
value)
Single digit addition may be solved in a
number of (increasingly sophisticated) ways
2 +5 = ?
Could you share what are some of the
ways in which a child may add this?
How do you add this?
– Count all strategy (2+3 –> 1, 2 …3, 4,
5)
– Count on strategy (2+3 –> 2 …3, 4, 5)
– Min strategy (2+3 –> 3 … 4, 5) – and
this involves the commutativity principle
(order or addends irrelevant)
– Finally with practice problems are solved
by direct retrieval from LTM (number
fact knowledge)
Strategies and Problem size
Siegler & Shrager (1984)
Addition in 4- and 5 year old children
• Use a variety of strategies:
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–
–
–
64% of the time: retrieval
15% counting on fingers
8% counting without finger use
13% using fingers without counting
Strategies and Problem size
Siegler (1987)
Addition in 6- and 8-year-old children
• Use a variety of strategies:
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–
–
–
–
35% of the time: retrieval
36% counting with minimum addend
8% counting with all digits
7% decomposition into simpler problems
14% guessed or failed to answer
WORKING MEMORY
Baddeley & Hitch’s Working Memory
Model (1974)
Articulatory
loop
Central
Executive
Visuo-spatial
Sketch-pad
B&H Working memory model
• The model distinguishes between two modality
specific stores (the phonological loop and the
visuo-spatial sketch pad) and an amodal
attentional control system (the Central
Executive)
• Deficits in any of these components might affect
the development of arithmetic skills
Predictors of math performance:
working memory
 In most studies (e.g. Passolunghi et al.,
2007) the central executive was a unique
predictor of later mathematics achievement
 Age-related differences with regard to the
contribution of the slave systems to
mathematics performance have been
suggested (De Smedt et al. ,2009)
CROSS-CULTURAL DIFFERENCES
Cross-cultural differences in
math performance
• Children from Japan, Korea and China outperform
American and European children on tests of math ability
• East Asian subjects produce smaller problem-size effects
than do North Americans (Campbell & Xue, 2001)
• The Hindu-Arabic base-10 system is the foundation of
modern arithmetic, and knowledge of base-10 is critical
to children’s math competence:
East Asian students tend to have a better grasp of the
base-10 system (Geary, 2006)
Cross-cultural differences in
math performance
What do you think are the possible reasons
explaining these differences?
Possible factors:
– Cultural influences
– Linguistic influences
Cultural Influences
• Influence of pre-school education
• Quantity and quality of math teaching (Perry et
al., 1993)
• Strategies used in problem solving problems
(Fuson & Kwon, 1999)
• Attitudes of teachers, parents and children
Group Activity
1. MATH TEACHING
2. ATTITUDES OF TEACHERS,
PARENTS &
CHILDREN
Linguistic Influences
• Number Words
Activity in small groups
NUMBER WORDS IN YOUR
LANGUAGE S???
Cultures differ in the number of
number words!
• Smallest number word in the
languages you know?
• Largest number word in the
languages you know?
English
Short Scale Long Scale
(US & Modern British) (Continental Europe
& Older British)
Sextillion
1024
1036
Octillion
1027
1048
Centillion
10303
10600
० शन्
ू य shoonya
१,०००
१,००,०००
१,००,००,०००
१,००,००,००,०००
१,००,००,००,००,०००
हजार
ऱाख
करोड़
अरब
खरब
hajaara
laakh
karod
arab
kharab
A few large numbers used in India
by about 5th century BCE (Ifrah)
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ahaha (अहाहा) —1070
ababa (अबाबा). —1077
atata (अटाटा) —1084
soganghika (सोगान्घीका) —1091
uppala (उप्पाऱा) —1098
matsya (मत्स्य) —10600infinities
dasavatara (दशावतारा) —1010000infinities
jyotiba (ज्योततबा) —1080000infinities
Cultures differ in the number of
number words!
• Two tribes in the Amazonian
area of Brazil have very small
lexica of number words:
– the Pirahã (Frank, Everett, Fedorenko,
Gibson, 2008; Gordon, 2004)
– the Mundurucu (Dehaene, Izard, Spelke,
& Pica, 2008; Pica et al., 2004).
Number Vocabulary Size &
Numerical Cognition
• The Pirahã have only three number
words:
– ―hói‖ (one)
– ―hoí‖ (two)
– ―aibaagi‖/―aibai‖ (many).
• When the Pirahã were tested in number
discrimination and reproduction tasks, their
performance with quantities larger than three
was poor.
Numeral
1
2
3
4
5
6
7
8
9
10
Modern
English
Old English/
Medieval English
(masculine;
feminine)
German Hindi
one
ān; forma
eins
ek
two
twēgen, twā, tū; ōðer
zwei
do
three
þrīe, þrēo; þridda
drei
teen
four
fēower; fēorða
vier
chaar
five
fīf; fīfta
fünf
panch
six
six; sixta
sechs
cheh
seven
seofon; seofoða
sieben
saat
eight
eahta; eahtoða
acht
aath
nine
nigon; nigoða
neun
non
ten
tīene; tēoða
zehn
das
Numeral
11
12
13
14
Modern
English
eleven
Old English/
Medieval English
(masculine; feminine)
endleofan; endleofta
German
Hindi
elf
gyaarah
twelve
twelf; twelfta
zwölf
baarah
thirteen
þrēotīene; þrēotēoða
dreizehn
terah
fourteen
fēowertīene; fēowertēoða vierzehn
choudah
15 fifteen
16 sixteen
17 seventeen
fīftīene; fīftēoða
fünfzehn
pandarah
sixtīene; sixtēoða
sechzehn
solah
seofontīene; seofontēoða siebzehn
satrah
18 eighteen
19 nineteen
20 twenty
eahtatīene; eahtatēoða
achtzehn
atharah
nigontīene; nigontēoða
neunzehn
unnees
twentig; twentigoða
zwanzig
bees
Numeral
Modern
English
20 twenty
21 twenty-one
Old English/
Medieval English
(masculine;
feminine)
twentig; twentigoða
German
Hindi
zwanzig
bees
ān and twentiġ (masculine einundzwanzig
ikkees
form)
22
23
24
39
40
43
55
56
twenty-two
twēgen and twentiġ
zweiundzwanzig
baiees
twenty-three
þrīe and twentiġ
dreiundzwanzig
thaiees
twenty-four
fēower and twentiġ
vierundzwanzig
chaubees
thirty-nine
nigon and þrītig
neununddreißig
unnchalis
forty
fēowertig
vierzig
chalis
forty-three
þrīe and fēowertig
dreiundvierzig
tetalis
fifty-five
fīf and fīftig
fünfundfünfzig
pachpann
fifty-six
six and fīftig
sechsundfünfzig
chhappan
Regularity of the number system
English
French
Japanese
One
Un/une
Ichi
Two
Deux
Ni
Three
Trois
San
Four
Quatre
Shi
Ten
Dix
Juu
Eleven
Onze
Juu-ichi
Twelve
Douze
Juu-ni
Twenty
Vingt
Ni-juu
Twenty-one
Vingt-et-un
Ni-juu-ichi
From Towse & Saxton (1998)
WHAT DIFFERENCES DO YOU OBSERVE IN
THE NUMBER WORDS ACROSS THE
DIFFERENT LANGUAGES?
• For example, in saying 43—what are the
differences across the languages?
• In your opinion, which way is a better system?
• Why?
Number Words
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
consistent
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
inconsistent
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
inconsistent
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
inconsistent
43
In English:
‘Forty-three’
In German:
‘Dreiundvierzig’
Number Words
• Consistent or inconsistent?
• The consistency or inconsistency of a numberword is a matter of perspective.
Number Words
• More errors in transcoding when inconsistent
• More mistakes in number comparison
• Decade distance (DD):
• Unit-decade compatibility
compatible
incompatible
32
37 62
57
small
large
32
32 87
47
Inversion for Austrians
From Pixner et al. (2011)
Summary of Linguistic influences
that may be relevant (cf. Dowker, 2005):
• Whether the language includes number words at
all and if there is an upper limit
• Whether there is a written number system
• The regularity of the spoken number system
• The degree and consistency of mapping
between the spoken and the written number
system
NUMBER GAMES?
Which number games or games involving
numbers did you play as a child?
Games Involving Numbers
Played by Children in India
Board Games
Board Games
Card Games
Traditional Indian Number
Games
: Pallanguli
Traditional Indian Number Games
7-Tiles (Pithoo; Saat Pathar; Lagori):
Other Games/Play Involving Numbers
•
•
•
•
Counting while skipping
Hopscotch (Stapu, Pandee)
Gittak (8 pebbles)
Maintaining a score in many outdoor
games such as badminton
STRUGGLING WITH MATH
Group Activity
WHAT ARE THE CHILDREN
STRUGGLING WITH ?
What is Mathematics Disorder?
Slow, limited or otherwise faulty development
of number processing, counting and/or
calculation in children who do not otherwise
give evidence of gross neurological or
psychiatric disability.
A case study
Claire
• 8 years old, second grade
• average cognitive skills, good motivation and attention
• struggles in math in school
• developmental lag in counting, understanding of place
value
• difficulties in addition and subtraction of one digit
numbers
– about 80% and 40 % accurate, using finger counting
An adult case
Mathematical Disorder
DSM-IV Diagnostic Criteria
Discrepancy definition:
―mathematical ability, as measured by individually
administered standardized tests is substantially
below that expected given the person’s
chronological age, measured intelligence and ageappropriate education‖
Prevalence of MD
• Lewis, Hitch & Walker
(1994)
– 9 and 10 year old
children (N = 1206)
– Criterion : Standard
score in math less than
85
– Normal nonverbal IQ
– Reading at or above
standard score of 90
• 1.3 % specific difficulty in
maths
• 2.3 % reading and maths
problems
•
Total of 3.6 %
mathematics disorder
Thus comorbidity between MD and RD is common.
MD/RD children tend to have more severe problems with Math.
Prevalence of MD
• Share, Moffitt & Silva (1988) in New Zealand:
11.2%
• Large US study by Baker & Cantwell (1985):
6%
• In Israel, Gross-Tsur, Manor & Shalev (1996):
6.5%
Mathematics Disorder
• What is there to explain?
–
–
–
–
Inaccurate calculations
Counting errors
Slow on number fact retrieval
Usage of immature strategies for calculation
(finger counting)
Cognitive Deficits in MD
• Four main ideas:
– Number (magnitude) representation problems:
• Basic difficulties in representing numbers
– Counting problems
– Number fact storage problems:
• Difficulties in learning and storing the solutions to
problems
– Attentional control and working memory problems
• Problems in executing the process required to solve a
problem
Cognitive Deficits in MD
• Four main ideas:
– Number (magnitude) representation problems:
• Basic difficulties in representing numbers
– Counting problems
Number Sense Deficit
– Number fact storage problems:
• Difficulties in learning and storing the solutions to
problems
– Attentional control and working memory problems
• Problems in executing the process required to solve a
problem
Cognitive Deficits in MD
• Four main ideas:
– Number (magnitude) representation problems:
• Basic difficulties in representing numbers
– Counting problems
Number Sense Deficit
– Number fact storage problems:
• Difficulties in learning and storing the solutions to
problems
– Attentional control and working memory problems
• Problems in executing the process required to solve a
problem
Working Memory Deficit
Other Influences: Math Anxiety
(Dowker, 2005; Geary, 2006)
• Rate your math anxiety on a 10-point
scale
• What explains your rating?
Other Influences:
Math Related Self-Conceptions of
Learners (de Corte & Verschaffel, 2006)
• Share some of the math-related selfconceptions you have?
• How do these impact math
performance?
―These beliefs exert a powerful influence on
students’ evaluation of their own ability, on their
willingness to engage in mathematical tasks, and
on their ultimate mathematical disposition‖ (in de
Corte & Verschaffel, 2006, p. 121)
Summary 1
 Preverbal infants can already possess an ability to
discriminate numerosities (ratio-dependent)
 This ability gets better with age
 Counting lays the foundations for arithmetic
Summary 2
 Arabic numerals have to be learned
―We learn number processing, and in particular
arithmetic and higher mathematical skills, through
formal instruction (Goebel, Shaki, & Fischer, 2011,
p. 544)‖.
 Research into longitudinal predictors
of math skills is in its infancy
 A comprehensive developmental
model of number processing
is missing