Embodied numerosity: implicit hand-based representations influence symbolic number processing across cultures.

Abstract

In recent years, a strong functional relationship between finger counting and number processing has been suggested. Developmental studies have shown specific effects of the structure of the individual finger counting system on arithmetic abilities. Moreover, the orientation of the mental quantity representation ("number line") seems to be influenced by finger counting habits. However, it is unclear whether the structure of finger counting systems still influences symbolic number processing in educated adults. In the present transcultural study, we pursued this question by examining finger-based sub-base-five effects in an Arabic number comparison task with three different groups of participants (German deaf signers, German and Chinese hearing adults). We observed sub-base-five effects in all groups, but particularly so for both German groups who use an explicit sub-base-five system in their finger counting habits. It is concluded that bodily experiences--namely finger counting--influence the structure of the abstract mental number representations even in adults. Thus, the present findings support the general idea that even seemingly abstract cognition may at least partially be rooted in our bodily experiences.

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Embodied numerosity: Implicit hand-based representations influence
symbolic number processing across cultures
Frank Domahsa,b,c,*, Korbinian Moellerd, Stefan Hubere, Klaus Willmesa,c,
Hans-Christoph Nuerkd
aLehr- und Forschungsgebiete Neuropsychologie und Neurolinguistik, Universitätsklinikum der RWTH Aachen, Pauwelsstr. 30, Aachen, Germany
bAbteilung Klinische Linguistik, Philipps-Universität Marburg, Germany
cInterdisziplinäres Zentrum für Klinische Forschung IZKF ‘BIOMAT.’, Universitätsklinikum der RWTH Aachen University, Aachen, Germany
dInstitute of Psychology, Eberhard Karls University Tuebingen, Germany
eDepartment of Psychology, Paris-Lodron University Salzburg, Austria
a r t i c l ei n f o
Article history:
Received 21 January 2009
Revised 21 March 2010
Accepted 4 May 2010
Keywords:
Magnitude comparison
Deaf signers
German
Chinese
Finger counting
Representational effects
Decade break
Five break
Embodied cognition
Base-system
a b s t r a c t
In recent years, a strong functional relationship between finger counting and number pro-
cessing has been suggested. Developmental studies have shown specific effects of the
structure of the individual finger counting system on arithmetic abilities. Moreover, the
orientation of the mental quantity representation (‘‘number line”) seems to be influenced
by finger counting habits. However, it is unclear whether the structure of finger counting
systems still influences symbolic number processing in educated adults.
In the present transcultural study, we pursued this question by examining finger-based
sub-base-five effects in an Arabic number comparison task with three different groups of
participants (German deaf signers, German and Chinese hearing adults). We observed
sub-base-five effects in all groups, but particularly so for both German groups who use
an explicit sub-base-five system in their finger counting habits. It is concluded that bodily
experiences – namely finger counting – influence the structure of the abstract mental num-
ber representations even in adults. Thus, the present findings support the general idea that
even seemingly abstract cognition may at least partially be rooted in our bodily
experiences.
? 2010 Elsevier B.V. All rights reserved.
1. Introduction
Across different cultures and ages, numbers are and
have been represented using body parts, invariantly
including hands and fingers (Butterworth, 1999). Only in
phylogenetically very recent times symbolic numerical
representations like the so called Arabic number system
have become more and more ubiquitous. However, even
massive use of these more abstract external number repre-
sentations may not lead to the development of fully ab-
stract mental number representations. Rather, we will
argue that mental representations of numerosity may to
a certain degree still remain embodied, i.e. they may be
strongly related to physical quantity representations –
namely finger counting. While the major importance of fin-
ger-based representations for numerical development is
nowadays largely accepted, their functional role in edu-
cated adults is still under debate: Are finger-based numer-
ical representations just an important – yet transitory –
step towards the development of abstract numerical repre-
sentations?Ordofinger-based
influence every-day number processing in numerate
adults? In the following sections, we will first review
developmental aspects of finger counting habits and then
turn to an evaluation of recent evidence on the role of
representationsstill
0010-0277/$ - see front matter ? 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.cognition.2010.05.007
* Corresponding author at: Lehr- und Forschungsgebiete Neuropsy-
chologie und Neurolinguistik, Universitätsklinikum der RWTH Aachen,
Pauwelsstr. 30, Aachen, Germany
E-mail address: domahs@neuropsych.rwth-aachen.de (F. Domahs).
Cognition 116 (2010) 251–266
Contents lists available at ScienceDirect
Cognition
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finger-based numerical representations in adults. After-
wards we will discuss potential functional relationships
between finger counting and numerical cognition and
present the rationale of our study.
1.1. Finger counting in children
The significant role of finger-based number representa-
tions in children is supported by several lines of evidence.
For instance, it has been shown that finger counting may
precede verbal counting in some children’s development
(Brissiaud, 1992; Descoeudres, 1921). Furthermore, even
children born with congenital absence of both forearms
and hands use phantom hands to count and to calculate
(Poeck, 1964). Moreover, finger gnosia has been found to
be a significant predictor of numerical abilities in children
(Fayol, Barrouillet, & Marinthe, 1998; Noël, 2005). It has
even been suggested that training of finger gnosia may
generalize to untrained numerical performance (Gracia-
Bafalluy & Noël, 2008; but see Fischer, 2010).
These studies were important in that they have estab-
lished a general functional link between numerical and fin-
ger representations in children. Specific developmental
stages in finger-based number representations have been
described by several authors (e.g. Baroody, 1987; Fuson &
Kwon, 1992; Svenson & Sjöberg, 1983). Crucially, there is
converging evidence that quantity-based finger counting1
systems typically exhibit a sub-base-five2similar to the ori-
ginal Roman number system (for a taxonomy of number sys-
tems with different base and sub-base systems see Zhang &
Norman, 1995). That is, finger patterns for numbers >5 inva-
riantly include a full hand representation (Marton & Neu-
man, 1990). For instance, in all such systems seven is
shown as h5 + 2i (five fingers of one hand and two of the
other) and never as h4 + 3i (for an example, see the German
finger counting system depicted in Fig. 1). Note that this fea-
ture of quantity-based finger counting systems does not ap-
ply to (partly) symbolic finger counting as in the Chinese
system (see Fig. 1).
Importantly, finger-based representations do not only
influence numerical processing in children when overt fin-
ger counting is actually used. Rather, it has been demon-
strated that early external finger-based representations
become internalized during primary school. Domahs,
Krinzinger, and Willmes (2008) observed above chance
proportions of split-five errors in complex addition and
subtraction problems (i.e. errors deviating by exactly ±5
from the correct result, e.g. 18 – 7 = 6). Note that these
split-five errors occurred in a number range up to 20 – that
is with numbers which cannot be represented by fingers in
a transparent one-to-one relationship between number of
fingers and numerosity to be represented any more. Inter-
estingly, during a specific phase of development, such
split-five errors occurred also in attempts to solve simple
problems via the direct retrieval of arithmetic facts from
memory. This can be taken as evidence that – at least at
a certain stage of numerical development – children build
up mental representations which inherit the sub-base-five
property of their hands.
Insum,thereisnowsoundevidencethatfingerrepresen-
tationsplayamajorroleduringnumericaldevelopmentand
that the specific structure of finger representations – in
particular,theirsub-base-five structure–influencesmental
arithmetic in children. However, it remains still unknown,
whether these findings are restrained to a specific develop-
mentalstageorwhethertracesoffingercountingcanstillbe
observed in numerate adults.
1.2. Finger counting in adults
To our knowledge, specific sub-base-five effects have
not been observed in adults so far. However, there is
behavioral and neurocognitive evidence for a general
(cor-)relation between hand and number representations.
In behavioral studies with adults, a common denominator
between fingers and numbers is their spatial representa-
tion. Conson, Mazzarella, and Trojano (2009) reported
SNARC-like effects (SNARC = spatial-numerical association
of response codes, Dehaene, Bossini, & Giraux, 1993) –
i.e. an association between small numbers and left hand
– when participants had to judge the laterality of hand
stimuli consistent with an egocentric perspective, and a re-
verse effect (i.e. an association between small numbers and
right hand) in judging hand stimuli coded from an allocen-
tric perspective. They concluded that the basic left-to-right
arrangement of the mental number line is defined with re-
spect to the body-centered egocentric reference frame.
Brozzoli and colleagues (2008) found that with their hand
resting palm down, participants performed better when
reporting tactile stimuli delivered to the little finger after
presentation of number 5 than number 1. Interestingly,
this pattern reversed when the hand was in a palm-up pos-
ture, showing that space- and body-based representation
of numbers may interact. Fischer (2008) observed that
the association of numbers with space (i.e. the SNARC-ef-
fect) is correlated with the hand with which people start
to count (see also di Luca, Granà, Semenza, Seron, & Pesen-
ti, 2006). Participants who started counting with their left
hand and finished counting with their right hand showed a
stronger left-to-right orientation of their mental number
line (as reflected by their SNARC-effect) than participants
who started counting with their right hand. Interestingly,
Fischer and Lindemann (2009)3presented data suggesting
that counting habits differ substantially between cultures.
Western cultures predominantly start counting with the left
hand while people from middle-eastern countries start
1Strictly speaking, finger counting and the use of finger patterns are
different stages of development (e.g. Fuson & Kwon, 1992). However, we
will employ the term finger counting in a broader sense, including the use of
finger patterns.
2If there is a number b such that numbers are represented in terms of
sums of multiples (up to b ? 1) of powers of b, then that number system is
said to have base b. If a system seems to have base s up to the number b
which is the real base of the system, it is said to have sub-base s. For
example, using this terminology of base and sub-base, Babylonian numer-
als form a base 60-system with a sub-base of 10 – seemingly base-10 up to
60, where the true base becomes evident (Sizer, 2004; Zhang & Norman,
1995).
3The authors thank Oliver Lindemann and Martin Fischer for making
part of their yet unpublished data available to us.
252
F. Domahs et al./Cognition 116 (2010) 251–266

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counting with their right hand and this difference in finger
counting may correspond to between-culture difference in
spatial-numerical associations (Shaki & Fischer, 2008). Rela-
tions between number magnitude and spatial attributes of
hand movements beyond counting habits (e.g. grip aperture)
have been demonstrated by Andres and colleagues (Andres,
Davare, Pesenti, Olivier, & Seron, 2004; Andres, Ostry, Nicol,
& Paus, 2008; Badets, Andres, Di Luca, & Pesenti, 2007).
The idea that neurocognitive relations between finger
and number representations also exist in adults is not
new. In 1930, Gerstmann (see also Gerstmann, 1940) de-
scribed a neuropsychological syndrome in which acalculia
and finger agnosia were two of the core symptoms (besides
impaired left–right-orientation and agraphia). Although,
this syndrome was controversially debated since its first
report, subsequently many patients have been described
with combined impairments of finger and number repre-
sentations. What is more, the relation between finger and
number has been corroborated by modern experimental
neurocognitive methods. For instance, Rusconi and col-
leagues applied rTMS over the left angular gyrus to disrupt
both finger gnosia and number processing (Rusconi, Walsh,
& Butterworth, 2005). However, not only common impair-
ments due to real or virtual lesions have been reported. Re-
cently, Sato, Cattaneo, Rizzolatti, and Gallese (2007) have
investigated the excitability of hand muscles during
numerical judgments. In a parity judgment task they ob-
served an increase of motor evoked potentials for right
hand muscles to be modulated by number magnitude
(see Andres, Seron, & Olivier, 2007, for similar findings in
a counting task). In a similar vein, fMRI studies on numer-
ical cognition have repeatedly suggested relations between
finger and number magnitude activations. Göbel, Johan-
sen-Berg, Behrens, and Rushworth (2004) found more acti-
vation for one-digit as compared to two-digit numbers in
the anterior IPS (see also Wood, Nuerk, & Willmes, 2006,
Fig. 1. Finger counting systems as used in German, DGS, and Chinese depicted from the signer’s perspective. Note that, although the actual hand with which
counting is started may vary (Fischer, 2008), the system itself remains principally unaffected. In Chinese, there are variants, especially for the numbers 7–
10. Crucially, however, the symbolic character (sometimes related to written Chinese numeral characters) is preserved irrespective of the variant used.
F. Domahs et al./Cognition 116 (2010) 251–266
253

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for similar effects for tens and units of two-digit numbers),
an area which has repeatedly been shown to be associated
with finger movements (Manthey, Schubotz, & von Cra-
mon, 2003). Similarly, Thompson, Abbott, Wheaton, Syn-
geniotis, and Puce (2004) found the left IPS to be more
activated when participants had to extract number magni-
tude from finger movements as compared to lip move-
ments. Kaufmann et al. (2008) also observed activation in
the post-central gyrus for a non-symbolic numerical judg-
ment task with finger stimuli. Interestingly, activation in
the pre- and post-central gyrus for non-symbolic numeri-
cal processing was significantly more pronounced in chil-
dren thaninadults, possibly
activation of finger-based representations in numerical
tasks may be more important for children than for adults.
Moreover, in a single-digit addition task with adult partic-
ipants, Venkatraman, Ansari, and Chee (2005) observed
precentral activation in an area commonly associated with
finger counting. Taken together, these studies clearly sug-
gest that there is some relationship between finger and
number representations even in adults, although, it may
be less pronounced than in children.
indicatingthat the
1.3. A functional relation between finger and number
representations?
The fact that there are associations between finger and
number representations does not necessarily imply that
finger representations play a functionally important role
in adult number processing. As reviewed above, there is re-
cent evidence for an influence of the direction of finger
counting on the direction of the mental number represen-
tation (Fischer, 2008). However, to the best of our knowl-
edge there is no study which shows that the specific
structure of the finger counting system influences the
way numbers are processed. The only exception so far is
an investigation with German deaf signers by Iversen,
Nuerk, Jager, and Willmes (2006). These authors investi-
gated the question whether the sub-base-five system of
GermanSign Language(Deutsche
DGS)4influences performance in a parity judgment task.
To pursue this question, they used the so-called MARC-effect
Gebärdensprache,
(Nuerk, Iversen, & Willmes, 2004), which describes the find-
ing that even numbers are faster responded to by the right
hand and odd numbers by the left hand. The important point
about the sub-base-five finger counting system in DGS (see
Fig. 1) is that the parity of numbers >5 is no longer straight-
forwardly assigned. So, the number 6, being even with re-
spect to the base-10 system, is odd with respect to the
base-five system, as 6 is h5 + 1i (which could be written as
[1–1], when the base of the leftmost digit is five). In line
with this rationale, Iversen et al. (2006) observed that the
MARC-effect reversed for the numbers 6–9 (as compared
to the numbers 1–5) and thus followed the parity of the
sub-base-five system when DGS number signs were pre-
sented to deaf signers in a parity judgment task. Most
importantly for the present study, Iversen et al. (2006) also
presented German written number words (e.g. neun ‘nine’)
to the deaf signers which produced the same pattern of re-
sults as DGS number signs. For the motivation of the current
study this finding is crucial because German (as well as Eng-
lish) written number words exhibit no external5sub-base-
five system. Therefore, when the MARC-effect for written
number words is influenced by a sub-base-five, this can only
be due to a mental representation of the sub-base-five sys-
tem. These findings indicate that, in principle, the hand-
based sub-base-five structure can influence adult number
processing – at least for the deaf population – and that the
influence of the sub-base-five structure is not constrained
to external sub-base-five representations (i.e. signed num-
bers) but can also be attributed to internal sub-base-five
number representations. However, so far this has only been
shown for deaf signers whose native language is sign lan-
guage and whose linguistic number system exhibits an
inherent sub-base-five structure. Therefore, it remains cur-
rently unknown, whether the sub-base-five structure of fin-
ger counting can also influence number processing in
hearing adults.
1.4. The present study
In the current study, we aimed at investigating whether
the sub-base-five system associated with finger counting
influenced performance of hearing educated adults in a
standard numerical task with symbolic stimuli – Arabic
number magnitude comparison. We hypothesized that
the influence of the sub-base-five system may vary as a
function of the structure and salience of the specific finger
counting system in use.
Therefore, we chose a transcultural approach with three
different cultural groups: (i) we tested deaf German sign-
ers who have previously shown sub-base-five effects in ba-
sic numerical cognition (Iversen et al., 2006). As DGS
4Functionally, number signs in sign language correspond to the spoken
number words of hearing people (Klann, Kastrau, Kemeny, & Huber, 2001).
In German Sign Language (DGS), the taxonomy of the number word system
exhibits a base-10/sub-base-five structure (see Fig. 1). The base-10
structure corresponds roughly to that of the Arabic number system (with
the exception of teens). In this system, the number 7 is related to the
number 17: The base-10 is added to 7 to symbolize the number 17 as
h10 + 7i. Similarly, a sign indicating the decade is subsequently added to the
sign indicating the unit in two-digit DGS numerals.
The sub-base-five system in DGS is similar to the Roman numeral
system. For instance, on the dominant hand the number seven is signed in
the very same way as the number two. The only difference between these
two numbers is the power indicated on the non-dominant hand. When all
five fingers are stretched on the non-dominant hand, this implies that the
number seven is signed as h5 + 2i, otherwise the number two is signed as
h0 + 2i. Similarly, in Roman numerals the number II is identical to the
number VII except that the sub-base V is appended in front of the number
II. In the Arabic number system (as in most verbal number systems
including the English, German, and Chinese), there is no sub-base-five, as
the external representation of the number 7 is in no way related to that of
the number 2.
5In the theory of distributed representations (Zhang & Norman, 1995),
external representations are the representations in the environment, as
physical symbols or objects (e.g. written symbols, beads of abacuses) and
external rules, constraints, or relations embedded in physical configura-
tions (e.g. spatial relations of written digits). The information in external
representations can be picked up by perceptual processes. In contrast,
internal representations are the sensations in the mind, as propositions,
productions, schemas, mental images, neural networks, or other forms. The
information in internal representations has to be retrieved from memory by
cognitive processes.
254
F. Domahs et al./Cognition 116 (2010) 251–266

Page 5

constitutes their native language6and finger-based number
signs are their native number words, the sub-base-five sys-
tem should be highly salient in their culture and sub-base-
five effects should most likely be observed, (ii) we assessed
hearing German adults to evaluate whether there is a spe-
cific influence of their finger counting system on numerical
cognition even when there is no explicit finger representa-
tion involved in the task. Hearing Germans have a very sim-
ilar (sub-base-five) finger counting system as deaf German
signers (see Fig. 1). However, it should be less salient be-
cause finger signs do not constitute part of their native lin-
guistic system. Thus, sub-base-five effects may also be
observed, but they may be less pronounced than for deaf
German signers and (iii) we tested Chinese hearing adults.
The Chinese finger counting system is different from those
of the two German groups. In particular, this system exhibits
no direct sub-base-five structure (see Fig. 1), because the
hand symbols for the numbers >5 continue to comprise
one hand only. However, although there is no explicit sub-
base-five system, the five still plays some role for the Chi-
nese finger counting system: whereas the numbers 1–5
are transparently represented in one-to-one correspondence
as finger quantities, only the numbers 6–10 are coded sym-
bolically. In consequence, there is a conceptual change
which occurs just after the number 5 when quantities can
no longer be represented transparently by the fingers of
one single hand. Thus, although the Chinese finger counting
system exhibits no actual sub-base-five, the five may still
play some role in structuring the system. Therefore, sub-
base-five effects may be completely absent or should at least
be less pronounced in Chinese as compared to German par-
ticipants (both deaf and hearing).
In this study, we chose to investigate numerical magni-
tude which can be regarded as core numerical representa-
tion. We examined numerical magnitude representation
using a very simple numerical task: number comparison.
In this task, participants had to single out either the smal-
ler or the larger of two presented numbers which always
had a distance of 2 (e.g. 4_6, 5_7, and so on). It is well
known that in the number comparison task, there is a mag-
nitude effect, meaning that participants become slower as
the magnitude of the problem increases (e.g. Brysbaert,
2005; Restle, 1970), although the exact nature of this in-
crease in reaction time (linear vs. logarithmic) is still under
debate (Dehaene, 2007; Moeller, Pixner, Kaufmann, &
Nuerk, 2009; Opfer & Siegler, 2007; Siegler, Thompson, &
Opfer, 2009). In any way, this magnitude effect must be
partialled out when examining other effects. We therefore
fitted magnitude both logarithmically and linearly to the
reaction time (RT) data of our number comparison items
and then looked at the residuals. In this way, the objective
of our study comes down to the question whether these
residuals are systematically influenced by the sub-base-
five structure or whether they just vary in an unsystematic
fashion.
What may be the specific nature of the hypothesized
sub-base-five effects? Two types of possible sub-base-five
influences can be distinguished which we termed ‘compar-
ison hypothesis’ and ‘generation hypothesis’, respectively.
The comparison hypothesis (see Fig. 2) assumes that some
hand-based representations can be compared faster than
others. In particular, this should apply to pairs in which
quantities are represented on a different number of hands,
respectively. In fact, similarity ratings have shown that
mental number representations based on a different num-
ber of hands are more distinct than representations using
the same number of hands – even when Arabic digits are
used as input (Shepard, Kilpatric, & Cunningham, 1975).
In our task this should affect the items 4_6 and 5_7,7be-
cause these items are the only ones in single-digit number
comparisons which use different numbers of hands in Ger-
man (but not in Chinese) finger representations. In both
pairs, one number is represented by one hand while the
other number is represented by two hands. Therefore, it is
not necessary to decode the quantity of individual fingers in-
volved; rather the larger number can be singled out by
checking which number is represented by two hands instead
of just one. A comparison strategy of this sort has been re-
ported by Girelli (1998, p. 97) for Erica, a four-and-a-half
year old girl. When she was to compare 4 and 9 she stated:
‘‘It is definitely the 9! For nine you need two hands.” All
other one-digit number pairs cannot be solved via that sim-
ple strategy because the number of hands involved is always
identical for both numbers (i.e. two times one hand or two
times two hands, respectively). Therefore, the comparison
hypothesis would predict particularly fast RT for the items
4_6 and 5_7 as compared to the other one-digit number
pairs. Note that in principle similar effects may be expected
for all number pairs crossing five and ten breaks (e.g. 9_11,
10_12, 14_16, 15_17 and so on). However, these effects
may fade out as the importance of finger-based representa-
tions decreases and the influence of other factors increases
(see Section 4).
The generation hypothesis (see Fig. 2) would predict the
opposite pattern of performance. In line with neurocogni-
tive studies mentioned above, the generation hypothesis
states that when comparing Arabic numbers, hand-based
representations are also activated. These representations
may be based on motor imagery, visualization, and/or
any other modality. In German (but not Chinese) finger-
based representations for numbers >5 may be more com-
plex to generate than those for numbers 65, because two
hands (rather than just one) have to be activated. In fact,
although a review of the literature on motor control re-
veals a complex and heterogeneous picture, there is evi-
dence for a disadvantage for bimanual as compared to
unimanual movements in behavioral (Aglioti, Berlucchi,
Pallini, Rossi, & Tassinari, 1993; Anson & Bird, 1993; Garry
& Franks, 2000; Sabate, Gonzalez, & Rodriguez, 2004) as
well as in neurophysiological (Donchin et al., 2001;
Goerres, Samuel, Jenkins, & Brooks, 1998; Jancke et al.,
6Strictly speaking, this may only be true for congenitally deaf signers.
However, making sure that sign language has been acquired early, the
probability that knowledge of other languages (spoken languages in
particular) interferes is low. Note that in our sample, mean age of
acquisition of DGS was 3.1 years (see Section 2).
7Here and in the remainder of this paper number pairs are only
indicated in the order stating the smaller number first (e.g. 4_6). Never-
theless, this notation is meant to include the opposite order of the pair, too
(e.g. 6_4, see Section 2).
F. Domahs et al./Cognition 116 (2010) 251–266
255

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2000; Nair, Purcott, Fuchs, Steinberg, & Kelso, 2003) mea-
sures. Thus, according to the generation hypothesis pairs
including numbers, which have to be coded by two hands
(4_6 and 5_7 for one digit; 6_8 and 7_9 for both digits of
the pair), should be responded to slower than pairs, where
both digits can be coded by only one hand (1_3, 2_4, 3_5).
The magnitude effect can be partialled out.
Note that all three participant groups (hearing Ger-
mans, deaf German signers, and Chinese) may be expected
to show consistent effects related to the perception of vi-
sual properties of the stimuli used, i.e. external representa-
tional effects in the sense of Zhang and colleagues (Zhang &
Norman, 1995; Zhang & Wang, 2005). Specifically, pairs
including both a one-digit and a two-digit Arabic number
(i.e. 8_10 and 9_11) should be decided on faster than ex-
pected on the basis of their magnitude alone, as it is suffi-
cient to decode the number of Arabic digits involved (i.e. a
decade break effect, see Fig. 2).
In sum, this rationale leads to three hypotheses: (i) if
the finger counting sub-base-five system does not influ-
ence Arabic number comparison (null hypothesis), there
should be no systematic variation due to the five break,
(ii) as stated by the comparison hypothesis there should
be an influence of the sub-base-five system (‘five break ef-
fect’), facilitating RT at the break when one number is >5
and the other 65, (iii) the generation hypothesis predicts
the opposite, i.e. an inhibitory kind of influence of the
sub-base-five system for numbers >5, as their finger-based
representations are more complex to generate. Moreover,
the facilitation predicted by the comparison hypothesis
should be restricted to the pairs 4_6 and 5_7, whereas
the inhibitory effect predicted by the generation hypothe-
sis should affect all pairs with at least one number >5
(including 6_8 and 7_9).
Finally, if there is a sub-base-five effect of any kind, it
should be influenced by the salience or structure of the fin-
500
520
540
560
580
600
620
640
123456789 10 11 12 13 14 15 16 17 18
3456789 10 11 12 13 14 15 16 17 18 19 20
Fictitious RT in ms
Number Pair
Comparison Hypothesis
German
DGS
Chinese
500
520
540
560
580
600
620
640
12345678 9 10 11 12 13 14 15 16 17 18
3456789 10 11 12 13 14 15 16 17 18 19 20
Fictitious RT in ms
Number Pair
Generation Hypothesis
German
DGS
Chinese
Fig. 2. RT-effects predicted by the comparison and the generation hypothesis and their expected cultural modulation. Both graphs include the assumption
of a magnitude effect (instantiated as overall logarithmic increase of RT) as well as a decade break effect (local decrease of RT for number pairs 8_10 and
9_11, due to features of the external stimulus [i.e. different number of digits involved in Arabic notation]) for all three groups of participants. In addition to
this, the comparison hypothesis expects a local RT-decrease for number pairs 4_6 and 5_7 due to the different number of hands involved in the assumed
internal representation of German participant groups. The generation hypothesis predicts a local increase of RT for number pairs 4_6, 5_7, 6_8, and 7_9, due
to an increase of complexity for the mental generation of every additional hand involved in both German participant groups (see Section 1 for further
details). Both effects may be more pronounced for DGS users as compared to hearing German participants, because number signs are assumed to be more
salient for the former as compared to the latter. Note that although only number pairs in ascending order are indicated (e.g. 4_6), this is also meant to
include the respective number pairs in descending order of presentation (e.g. 6_4).
256
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Page 7

ger counting system used. Both comparison and generation
effects should be least pronounced for Chinese participants
whose finger counting system exhibits no direct sub-base-
five system and most pronounced for DGS signers who use
the sub-base-five finger counting system as their native
language number word system.
2. Method
2.1. Participants
Participants from three different cultural backgrounds
(24 hearing Germans, 25 deaf Germans, and 27 hearing
Chinese) were recruited. Hearing Germans (mean age:
27.2 years, mean years of formal education: 15.3 years,
23 right handers, 12 women) were native speakers of Ger-
man. All deaf German participants (mean age: 29.6 years,
mean years of formal education: 14.2 years, 24 right hand-
ers, 13 women) were fluent users of DGS (mean age of
acquisition: 3.1 years). Most of them (15/25) currently
used DGS, an autonomous sign language, as their primary
language. The others mainly used LBG (Lautsprachbeglei-
tende Gebärden), a sign system literally translating from
and into German.8In DGS as well as in LBG numbers are
signed using both hands (see Fig. 1). Note that although
LBG number signs used by deaf participants are identical
to the finger counting system used by hearing Germans, it
is used much more frequently by the former as compared
to the latter, as it replaces the verbal counting system of deaf
users. All Chinese participants (mean age: 24.1 years, mean
years of formal education: 17.8 years, 26 right handers, 18
women) were recruited in Germany, where they were en-
rolled as university students. All of them were native speak-
ers of Chinese, having spent at least 19 years (mean
22.0 years) in China before they came to Germany. All par-
ticipants had normal or corrected-to-normal vision and re-
ported no mathematical deficits.
All participants were asked about their counting habits,
confirming that they used finger counting systems consis-
tent with our hypotheses. With respect to the Chinese
group, this means in particular, that all participants used
finger counting without any explicit sub-base-five system,
using one hand for numbers <10 and two hands for num-
bers P10. Moreover, all Chinese participants showed a
break between transparent (numbers 6 5) and symbolic
(numbers > 5) finger counting. We also made sure that all
hearing German and DGS participants used completely
transparent finger counting systems in the number range
up to 10, meaning that they used one hand for numbers
65 and two hands for numbers >5.
Unfortunately, we did not systematically ask about the
participants’ starting hand. Recent evidence shows that
finger counting habits may vary substantially both within
and between cultures (Fischer & Lindemann, 2009). Fur-
thermore, the hand on which finger counting is initiated
may influence the SNARC-effect (Fischer, 2008). Yet, at
the time the present experiment was conducted we were
not aware of this relationship. Note, however, that in the
present study – thus complementing the studies of Fischer
and colleagues – we were examining the influence of finger
counting habits on the structure of the mental magnitude
representation rather than its spatial orientation. In con-
trast to the within-culture variation concerning the start-
ing hand there was no variation within-culture with
respect to the above-described use of a sub-base-five sys-
tem in our sample.
2.2. Stimuli
All number pairs with a distance of 2 within the number
range from 1 to 20 were used in both orders (i.e. from 1_3
to 18_20 and from 3_1 to 20_18, respectively; see7). We
did not restrict ourselves to the number range from 1 to
10, which had typically been used in previous studies, to
avoid the five break – presumably present in the perma-
nent mental representation – to be at the same time the
midpoint of transient, task specific representations causing
laterality effects (Cohen Kadosh, 2008b; Tzelgov, Meyer, &
Henik, 1992). Nevertheless, for various reasons our analy-
ses of potential five break effects were restricted to
single-digit number pairs (for more details see Section 4).
Stimuli were presented as Arabic numbers in Arial 60 pt
centered in black on a white computer screen with both
numbers in the same line but separated by seven blanks.
2.3. Procedure
Participants were instructed to answer as fast and accu-
rately as possible. Half of the participants started with the
instruction to press a button on the side of the smaller
number, the other half started with the instruction to press
a button on the side of the larger number. Response keys
were the ‘s’ and the ‘l’ key on a standard keyboard and thus
were aligned with the presentational layout of the stimuli
on the screen. After half of the experiment, response
assignments were changed. Under each response assign-
ment, every number pair was presented six times per or-
der. For example, the pair 4_6 was presented six times
with one response assignment and six times with the other
one and the same was true for the pair 6_4. Thus, the pre-
sentation of 36 number pairs ? 6 repetitions ? 2 response
assignments led to a total of 432 experimental trials per
participant. Experimental trials were presented in six
blocks per response assignment, each block including all
36 number pairs in randomized order. Each response
assignment was preceded by an additional training block
of all 36 number pairs. Training results were not included
in the analyses.
Each trial started with a blank screen (500 ms), fol-
lowed by a centered fixation cross (200 ms) and another
blank screen (200 ms). Afterwards, the stimulus pair was
presented for a maximum of 2000 ms or until a response
was given. Trials were initiated in a self-paced manner,
i.e. participants pressed the space-bar on the keyboard to
proceed to the next trial.
8Although, not all deaf participants used DGS as their primary language
at the time of this investigation, we will use the abbreviation DGS to refer
to the whole group of deaf participants in this study.
F. Domahs et al./Cognition 116 (2010) 251–266
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2.4. Analyses
Only trials followed by a correct response were incorpo-
rated in the reaction time (RT) analyses. First, a univariate
ANOVA (ANalysis Of VAriance) with the between-subject
factor participant group (German vs. DGS vs. Chinese)
was conducted to reveal possible intergroup differences
in overall RT and error rates (please note that error rates
were arcsine transformed prior to the analysis to approxi-
mate normal distribution). Additionally, individual mean
RT was correlated with individual average error rate to
appraise the presence of a speed–accuracy-trade-off.
Furthermore, to investigate SNARC-like congruity effects
of hand-to-response assignment with numerical magni-
tude, a 2 ? 2 ? 3 ANOVA with the within-subject factors
comparison condition (select smaller vs. select larger)
and response side (left vs. right) and the between-subject
factor participant group (German vs. DGS vs. Chinese)
was conducted on response latencies.
Subsequently, possible decade as well as five break ef-
fects were computed as follows for each participant indi-
vidually:first,thelogarithm
response latencies per experimental number pair (both or-
ders collapsed; e.g. 3_5 and 5_3) was calculated. Then, a
logarithmic function of the type y = a ? ln(x) + b was fitted
to these individual data with x representing the mean of
the to be compared interval (e.g. four for the pair 3_5).
The fitting procedure was designed to minimize the root
mean squared error of approximation (RMSEA) and to
maximize R2as measures of goodness of fit. Logarithmic
fitting was chosen first because of evidence for a logarith-
mically compressed quantity representation (see Sec-
tion1).Afterwards, the
logarithms of the observed response latencies and the val-
ues predicted by the fitting function (i.e. the residuals)
were computed by subtracting the predicted values from
the actual logarithm of the RT. Finally, these residuals were
standardized to a mean of 0 and a standard deviation (SD)
of 1 to allow for reliable evaluation of differences of the
residuals between the three participant groups.
The resulting residuals should be free from influences of
overall magnitude as these influences are captured by the
logarithmic fitting. This assumption was confirmed as a
correlation analysis showed no reliable interrelation of
either magnitude (i.e. the mean of the two to-be-compared
numbers) or the ratio of the two-be-compared numbers
with the computed residuals for any of the three partici-
pant groups (all r < .11, all p > .68). This also held for the
correlations of the absolute size of the residuals with either
the magnitude or the ratio of the two numbers (all r < .22,
all p > .38). Finally, a missing relation between the absolute
residuals and the variance in RT of each item (which is sup-
posed to increase with increasing magnitude) again cor-
roborated the assumption of the residuals being not
determined by magnitude or related properties (all
r < .23, all p > .35). These analyses are an important prere-
quisite for the analyses below, because they show that
any effect observed for the residuals cannot be explained
by any confound with magnitude, because a whole variety
of different magnitude measures were not correlated with
the size of the residuals.
(ln)oftheaveraged
differencesbetween the
Another confound might be caused by possible cultural
differences in mathematical competence favouring East
Asian participants (e.g. Campbell & Xue, 2001; Geary,
Bow-Thomas, Liu, & Siegler, 1996; Miller, Kelly, & Zhou,
2005; Miura, Okamoto, Kim, Steere, & Fayol, 1993). To
evaluate such influences it was of interest whether there
were differences in the influence of magnitude on perfor-
mance between the three participant groups. Therefore,
we inspected whether the slope of the logarithmic fitting
function or the obtained R2differed reliably. Two ANOVAs
showed that neither the slope of the fitting function
[F(2, 73) < 1] nor the resulting R2[F(2, 73) < 1] were modu-
lated by participant group.
Note that the results were identical when linear fitting
of magnitude wasapplied
F(2, 73) < 1]. Moreover, a direct comparison revealed that
logarithmic and linear fittings accounted for a comparable
amount of variance over all participant groups (R2
R2
between the logarithmic and the linear fitting may be dri-
ven by the exceptionally fast responses to pairs 8_10 and
9_11 (see Figs. 3 and 4), because only in those trials a visual
shortcut based on the number of Arabic digits involved can
be used. As the linear runs below the logarithmic fitting
function at this part of the distribution, this may have re-
sulted in an irregularly good fit for these two data points
which in turn may have determined overall R2. To account
for this possible artifact we reran the analyses for logarith-
mic and linear fitting but excluded the two pairs 8_10 and
9_11. Again, there were no differences between the partic-
ipant groups for either slope [logarithmic: F(2, 73) < 1; lin-
ear:
F(2, 73) < 1]orthe
F(2, 73) < 1; linear: F(2, 73) < 1]. However, logarithmic fit-
ting outperformed linear fitting now [R2
F(1, 75) = 80.09, p < .001], suggesting that the comparable
performance of both fittings may have indeed been a
methodological artifact due to the exceptionally fast RT
for pairs 8_10 and 9_11.
In sum, based on these findings we are confident that
analyses resting on the residuals are valid for the evalua-
tion of our hypotheses concerning differences in finger
counting habits between participant groups.
The decade break effect was assessed by testing
whether the mean of the individual residuals was reliably
smaller than zero within each group of participants indi-
cating actual RTs to be faster than those predicted by the
fitting procedure. For the decade break effect the pooled
difference between the observed and the predicted laten-
cies of the two relevant number pairs (8_10 and 9_11)
was evaluated. Finally, the size of the decade break effect
was compared between the three participant groups
(German vs. DGS vs. Chinese) using a one-way ANOVA to
check whether this effect differed between the participant
groups.
For the five break effect analyses were twofold: (i) in
line with the procedure for the decade break effect, the
validity of the comparison hypothesis was assessed by
evaluating whether the average residual of the two rele-
vant single-digit number pairs (4_6 and 5_7) was reliably
smaller than zero indicating specific facilitation of re-
sponses. Subsequently, the size of the five break effect
[slope:
F(2, 73) < 1;
R2:
log: .44 vs.
lin: .44; F(1, 75) < 1). The fact that there was no difference
resulting
R2
[logarithmic:
log: .56 vs. R2
lin: .49;
258
F. Domahs et al./Cognition 116 (2010) 251–266

Page 9

was contrasted between the participant groups by a one-
way ANOVA and (ii) evaluating the adequacy of the gener-
ation hypothesis comprised two steps. In a first step, we
examined within each participant group whether the aver-
age residuals for the number pairs involving at least one
single-digit number >5 (4_6, 5_7, 6_8, and 7_9) were sig-
nificantly larger than zero as this would index actual RTs
for these number pairs to be slower than those predicted
by the fitting procedure. In a second step, we investigated
for each group, whether the residuals for the number pairs
specified above were larger than those for number pairs
exclusively including numbers 65 (1_3, 2_4, and 3_5).
German-speaking
6.20
6.30
6.40
6.50
1
3
2
4
3
5
4
6
5
7
6
8
7
9
89 10
12
11
13
12
14
13
15
14
16
15
17
16
18
17
19
18
20 1011
Number pair
ln(RT)
ln(RT)
Fitting
y = 0.085 * ln(x) + 6.253
R2 = 0.59
RMSEA = 0.04
Chinese-speaking
6.10
6.20
6.30
6.40
1
3
2
4
3
5
4
6
5
7
6
8
7
9
89 10
12
11
13
12
14
13
15
14
16
15
17
16
18
17
19
18
20 1011
Number pair
ln(RT)
y = 0.094 * ln(x) + 6.125
R2 = 0.63
RMSEA = 0.05
German Sign Language (DGS)
6.20
6.30
6.40
6.50
1
3
2
4
3
5
4
6
5
7
6
8
7
9
8910
12
11
13
12
14
13
15
14
16
15
17
16
18
17
19
18
20 10 11
Number pair
ln(RT)
ln(RT)
Fitting
ln(RT)
Fitting
y = 0.085 * ln(x) + 6.185
R2 = 0.65
RMSEA = 0.04
Fig. 3. Mean ln (RT) curves for all three groups of participants (actual data) and their fitting curves.
F. Domahs et al./Cognition 116 (2010) 251–266
259

Page 10

Following the generation hypothesis this contrast should
be significant for both German participant groups, as the
German gestures for numbers >5 involve both hands.
Finally, intergroup differences of the size of the five break
effect were evaluated in a one-way ANOVA as well.
3. Results
3.1. General findings
The ANOVAs revealed no reliable group differences for
either RT [mean RT (SD) in ms: German: 616 (104), DGS:
594 (99), Chinese: 568 (80); F(2, 73) = 1.67, p = .20] or error
rate [mean error rate (SD) in %: German: 3.6 (2.6), DGS: 4.2
(3.2), Chinese: 2.7 (1.7); F(2, 73) = 2.17, p = .12]. Moreover,
none of the subsequent Bonferroni corrected pair-wise
comparisons indicated a significant difference in any con-
trast (all p > .13). Finally, there was no indication of a
speed–accuracy trade-off in any of the three groups as re-
flected by the absence of a reliable negative correlation be-
tween RT and error rates in each group (all r > ?.30, all
p > .13).
3.2. SNARC-like effects
The ANOVA revealed a reliable interaction of response
side and comparison condition [F(1, 73) = 19.12, p < .001],
indicating that responses to the larger number were given
faster by the right hand than by the left hand (567 ms vs.
581 ms, respectively). Comparably, left hand responses to
the smaller number were faster than right hand responses
to the smaller number (609 ms vs. 616 ms). Additionally,
this two-way interaction was not further modulated by
participant group [F(2, 71) = 2.34, p = .11]. Finally, only
themaineffectof instruction
[F(2, 71) = 74.45, p = .001], with selecting the larger num-
ber being faster than selecting the smaller number
wassignificant
(574 ms vs. 612 ms). No further main effect or any further
interaction was reliable [all F < 2.49, all p > .12].
3.3. Decade break effect
The use of one-sample t-tests revealed that for all par-
ticipant groups the decade break effect was reliable [Ger-
man: t(23) = 16.97, p < .001, see Fig. 3 upper panel; DGS:
t(24) = 12.83, p < .001, see Fig. 3 intermediate panel; Chi-
nese: t(26) = 22.79, p < .001, see Fig. 3C lower panel]. Fur-
thermore, the ANOVA revealed that this effect was not
modulated by participant group [F(2, 73) < 1]. This indi-
cated that responses to trials involving one single-digit
and one two-digit Arabic number were equally facilitated
for all groups: the actual RTs were faster than those pre-
dicted by the fitting by about 1.5 SD [German: ?1.62 SD;
DGS: ?1.44 SD; Chinese: ?1.56 SD, see Fig. 4].
3.4. Five break effect
3.4.1. Comparison hypothesis
Contrary to this hypothesis, there was no facilitation for
the two relevant number pairs (4_6 and 5_7). Instead, one-
sample t-tests showed reliably increased RTs for all three
participant groups [German: t(23) = 7.47, p < .001; DGS:
t(24) = 4.64, p < .001; Chinese: t(26) = 2.65, p < .05, see
Fig. 3]. Moreover, the ANOVA revealed that this increase
was modulated by participant group [F(2, 73) = 3.55,
p < .05]. Subsequent pair-wise comparisons indicated that
the five effect did not differ between German-speaking
and DGS participants [+0.69 SD vs. +0.60 SD, respectively],
but was smallest in the Chinese participants [+0.29 SD, see
Fig. 4] with only the difference between German-speaking
and Chinese participants being significant at the p < .05 le-
vel (p = .15 for the difference between DGS and Chinese
participants).
Standardized Residuals
-2.00
-1.00
0.00
1.00
1
3
2
4
3
5
4
6
5
7
6
8
7
9
8910
12
11
13
12
14
13
15
14
16
15
17
16
18
17
19
18
201011
Number pair
z-Score
German-speaking
German Sign Language (DGS)
Chinese-speaking
Fig. 4. Standardized RT residuals of all three groups of participants.
260
F. Domahs et al./Cognition 116 (2010) 251–266

Page 11

3.4.2. Generation hypothesis
The t-tests revealed that the residuals for number pairs
involving at least one number >5 were reliably larger than
zero for the German-speaking and the DGS participants
only [German: t(23) = 9.55, p < .001; DGS: t(24) = 7.41,
p < .001, see Fig. 3]. For the Chinese participants the resid-
uals only tended to be marginally larger than zero
[t(26) = 1.77, p = .09, see Fig. 3]. Moreover, subsequent
comparisons showed that the residuals for the number
pairs including at least one number >5 were larger than
the residuals for the number pairs consisting of numbers
65 only for the German-speaking and the DGS participants
[German: t(23) = 6.14, p < .001; DGS: t(24) = 6.06, p < .001],
but not for the Chinese participants [t(26) = 0.41, p = .69].
The ANOVA substantiated these intergroup differences
for the five break effect (i.e. pooled residuals of trials with
at least one number >5 – pooled residuals of trials with
both numbers 65; [F(2, 73) 12.21, p < .001]). The size of
the five break effect did not differ between German-speak-
ing and DGS participants [+0.77 SD vs. +0.69 SD, respec-
tively]. However, the five break effect was significantly
smaller for the Chinese participants [+0.04 SD] as com-
pared to both other participant groups [at p < .001 see
Fig. 4].
Taken together, five break effects were more pro-
nounced in those participant groups in which finger count-
ing has a strong sub-base-five. Finally, it has to be noted
that five break effects were not driven by differences in
overall performance as would have been indicated by a
reliable correlation between the respective five break ef-
fect and overall RT. No reliable correlation between any
five break effect and overall RT was present for any partic-
ipant group (all r < .29, all p > .17).
4. Discussion
The results of the present study demonstrated that the
processing of Arabic number symbols is strongly influ-
enced by a sub-base-five system inherent in the finger
counting habits of both German groups of participants. In
a magnitude comparison task number pairs with one num-
ber larger than five were responded to slower than would
have been expected on the basis of their abstract magni-
tude alone. This influence of the finger-based sub-base-five
system was not only observed in DGS signers whose native
number words are hand signs. Rather, the present study
shows that a finger counting sub-base-five system may
also influence symbolic number processing in hearing
numerate adults in a functionally specific way. We suggest
that these data are evidence for effects of embodied numer-
osity indicating that mental finger representations of num-
erosity are neither restricted to deaf signers, nor to a
transient stage during numerical development. Instead,
the embodiment of numerosity representations persists
even in hearing numerate adults and exerts specific struc-
tural influences on mental number processing which are
functionally relevant in the most elementary numerical
tasks. Even simple magnitude processing with purely sym-
bolic input in adults is not exclusively abstract but influ-
enced by embodied counting experiences.
This view on numerical cognition is in line with recent
demonstrations of embodied cognition effects in other do-
mains such as language processing (e.g. Glenberg & Kas-
chak, 2002; Hauk, Johnsrude, & Pulvermüller, 2004;
Tettamanti et al., 2005; Zwaan & Taylor, 2006). While there
is converging evidence on effects of embodiment from
other domains in cognition, the embodied numerosity ac-
count is at least partially at odds with approaches suggest-
ing that the magnitude comparison task indexes an
abstract magnitude representation (e.g. Dehaene & Cohen,
1995). In our view, the present data cannot be explained by
a purely abstract magnitude representation, because there
is no reason to obtain sub-base-five effects in an abstract
magnitude representation. Rather, the current data corrob-
orate and extend recent theoretical accounts from Cohen
Kadosh and Walsh (2009): it is not only the notation or
the modality of the input format which is partially pre-
served in adult magnitude comparison (as comprehen-
sively argued by Cohen Kadosh and Walsh). Rather,
previous bodily experiences such as finger counting influ-
ence performance in a task with visual Arabic symbols in
which no finger counting images were used as input
modality or during the task. This suggests that magnitude
representation is not only abstract, not only modulated by
input notation or modality but also mediated by our bodily
experiences which we used when we learned about num-
bers decades ago. We will elaborate on this in more detail
below.
Importantly, sub-base-five influences have been dem-
onstrated to be culturally modulated. In DGS signers and
German hearing adults, who use an explicit sub-base-five
finger counting system, five break effects were larger than
in Chinese participants, who have no explicit sub-base-five
finger counting system. This finding is theoretically impor-
tant, because it implies that the five break effect found in
the present investigation is not due to some artifact of gen-
eral stimulus properties (such as visual similarity of Arabic
digits) but is modulated by cultural experience.9We sug-
gest that the cultural experience underlying the different
kinds of five break effects reflects the difference between
the respective finger counting systems concerning the pres-
ence or absence of a sub-base-five system. The importance
of structural properties of finger counting systems is in line
with recent evidence suggesting a major role of finger repre-
sentations for numerical cognition. Since Butterworth
(1999) strongly advocated the functional role of finger
counting for numerical cognition, there has been a growing
record of developmental, behavioral, and neurocognitive
evidence suggesting a close link between fingers and num-
ber. Our study adds to that evidence but at the same time
extends previous findings. There is not only a general rela-
tion between finger counting performance and arithmetic
ability (Fayol et al., 1998; Noël, 2005), not only a correlation
between spatial properties of finger counting and spatial
9Note that the observed cultural differences in the five break effect
cannot be due to the different number word systems because both
languages, German as well as Chinese, do not have any indication of a
sub-base-five in their number words. In particular, in German, where larger
sub-base-five effects are observed, there is no indication of any five break in
the number word system.
F. Domahs et al./Cognition 116 (2010) 251–266
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properties of the mental number line (Brozzoli et al., 2008;
di Luca et al., 2006; Fischer, 2008), not only a close vicinity
between brain areas activated by finger representations
and number processing (Andres et al., 2007; Rusconi et al.,
2005; Sato et al., 2007) – rather the specific structure of
the finger counting system determines the way number
magnitude is processed even in tasks in which only Arabic
numbers are presented. Similar influences of seemingly
irrelevant representations have already been demonstrated
in other cognitive domains. For instance, in visual word rec-
ognition it is now undisputed that even in adult silent read-
ing phonological representations
subcomponents are automatically accessed (e.g. van Orden,
1987). In a similar vein it may therefore not be too surpris-
ing that seemingly irrelevant mental representations – such
as those of finger counting – may exert some influence on
processing of Arabic numbers even in adults. However, the
specificity of this influence is as culture-specific (and poten-
tially also as individual; cf. Fischer, 2008) as the specificity of
finger counting habits themselves.
ofwordsandtheir
4.1. How the sub-base-five system specifically influences
Arabic number comparison
In Section 1, we have outlined three hypotheses about
possible ways in which the sub-base-five system may
influence Arabic number comparison: (i) Null hypothesis.
There is no influence of the sub-base-five system. (ii) Com-
parison hypothesis. Mental comparison of finger patterns
facilitates Arabic magnitude comparison if the numbers
of a given pair are represented on a different number of
hands. (iii) Generation hypothesis. Pairs in which at least
one number is represented on two hands take longer to
compare than pairs in which both numbers can be repre-
sented on only one hand.
The null hypothesis (i) can be rejected as we have ob-
served – albeit culturally modulated – influences of the
sub-base-five system in all three groups tested.
The comparison hypothesis (ii) predicted facilitating ef-
fects for the comparison of mental finger patterns for num-
ber pairs in which one number corresponds to a one-
handed sign and the other number corresponds to a two-
handed sign. When compared, such numbers do not need
to be encoded in detail (i.e. with respect to their number
or pattern of fingers) to determine which number is larger.
One only needs to compare the number of hands involved
(1 vs. 2). As the current data are in contrast with this pre-
diction, the comparison hypothesis can be rejected for all
three participant groups. Rather than observing facilitating
effects at the five break, we obtained interfering effects for
these pairs. Number pairs crossing the five break (i.e. 4_6
and 5_7) were responded to consistently slower than pre-
dicted by the fitted magnitude function. As the comparison
hypothesis suggests that items crossing the five break
should be particularly easy, it cannot account for the data.
Note that although we did not find evidence for the
comparison hypothesis with respect to mental finger-based
representations, a pattern consistent with a variant of the
comparison hypothesis was observed with respect to fea-
tures of the external representation: the largest deviations
of standardized residuals were observed for the number
pairs 8_10 and 9_11. These pairs were compared much fas-
ter than predicted by their magnitude (see Fig. 3). In our
view the reason for this facilitation is that in these two
number pairs a one-digit Arabic number was compared
with a two-digit Arabic number. This comparison can be
performed without encoding the specific Arabic symbols
involved. Instead, it can simply be solved by choosing the
number pair which has more digits (i.e. 2 vs. 1), when
the instruction is to choose the larger and vice versa in
the ‘‘choose the smaller” condition. The large negative
residuals suggest that such a visual shortcut strategy may
be employed when the external representation favors such
as strategy (which is 100% correct in the Arabic number
system). In sum, for the comparison hypothesis in Arabic
number comparison external and internal number repre-
sentations must be distinguished (Shepard et al., 1975;
Zhang & Norman, 1995; Zhang & Wang, 2005). For the
external Arabic base-10 number representation the data
are in line with this hypothesis. Yet, for the mental sub-
base-five finger patterns the comparison hypothesis is
not consistent with the data. Note, that this must not nec-
essarily mean that there is no facilitating effect when men-
tal comparison involves one vs. two hand-based mental
representations. This facilitation may, however, be over-
written by some stronger effect, which may be related to
the generation rather than to the comparison of internal
hand-based representations.
Finally, the generation hypothesis (iii) predicted that
performance for numbers >5 is slowed down, when mental
hand-based representations corresponding to those num-
bers are more complex to be generated than mental repre-
sentations corresponding to numbers 65. Therefore, this
hypothesis stated that for finger counting systems in
which numbers >5 are counted with both hands (German
and DGS), the Arabic number comparison for pairs includ-
ing such numbers (4_6, 5_7, 6_8, and 7_9) should be rela-
tively slower than for number pairs in which both numbers
can be represented with only one hand (1_3, 2_4, and 3_5).
The present data are most consistent with the generation
hypothesis. The residuals of item pairs with numbers >5
have been found to be larger than the residuals for pairs
with both numbers 65 for both DGS signers and hearing
German adults who use a transparent sub-base-five finger
counting system involving two hands for numbers >5. The
cultural modulation of the data, too, is consistent with the
generation hypothesis. In Chinese participants, who sign
all numbers <10 with the fingers of one single hand, the
generation hypothesis does not predict any difference for
numbers smaller or larger than 5. Indeed, when the resid-
uals of item pairs including a number >5 were compared
with those of number pairs with both numbers 65, there
was no significant difference between these item groups
for Chinese participants. In summary, the generation
hypothesis on the influence of a sub-base-five finger
counting system can explain the observed data pattern
best, both in general and with respect to the observed cul-
tural modulation. Moreover, it is in line with recent behav-
ioral and neurocognitive data about a close relationship
between finger and number representations (Andres
et al., 2007; Brozzoli et al., 2008; Fischer, 2008; Rusconi
et al., 2005; Sato et al., 2007). To conclude, our data suggest
262
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Page 13

that the mental generation rather than the mental compar-
ison of embodied representations modulates symbolic
magnitude comparison.
4.2. On the relationship between abstract magnitude
representation and embodied numerosity
We have entitled this study embodied numerosity as we
suggest that bodily experiences, namely activation of
hand-based representations, influence the way even edu-
cated hearing adults process numerosity. However, in mul-
tiple studiesand models
representation has been suggested (e.g. Dehaene & Cohen,
1995; Dehaene, Piazza, Pinel, & Cohen, 2003). How do
these two representations relate to each other? In our
view, the present data are no evidence against an impor-
tant role of an abstract number representation. It is crucial
to point out that we observed sub-base-five effects for the
residuals of a fitting function. In this fitting function, we
fitted the RTs of our items as a function of their magnitude,
which can be conceived as an index of an abstract numer-
osity representation. This means that all sub-base-five ef-
fects we have observed in the present study were only
observed after the effects of abstract magnitude were par-
tialled out. Only after these abstract magnitude effects
were considered, we observed sub-base-five effects which
we interpreted as traces of embodied numerosity repre-
sentations. To summarize, our study does show both: a
good prediction of the data by abstract number magnitude
and an explanation of the remaining residuals by sub-base-
five effects. Therefore, we suggest that at least two numer-
ical representations – an abstract number magnitude code
as well as an embodied finger counting representation –
exert their influence in a number comparison task. Thus,
although we do not deny the existence of an abstract rep-
resentation of numerical magnitude, our data add to rising
doubts that our numerosity representation is purely ab-
stract in nature (Campbell, 1994; Cohen Kadosh, 2008a;
Cohen Kadosh, Cohen, Kaas, Henik, & Goebel, 2007; Cohen
Kadosh & Walsh, 2009).
In principle, an abstract magnitude representation may
be indistinguishable from a transparent finger-based num-
erosity representation as in German, i.e. when one finger
representation is added for each increment in numerosity,
behavioral effects were identical with those based on ab-
stract linear magnitude coding. However, some aspects of
our results speak against such an interpretation: First, a
logarithmic fitting of the magnitude effect was superior
to a linear fitting, at least when items with external visual
shortcut strategies (number of digits comparison) were
eliminated. A finger-based account of the magnitude effect,
however, does not seem to predict a logarithmic magni-
tude function. Second, the slope of RT increase with
increasing magnitude did not differ between participant
groups, even though in Chinese there is no transparent
matching between number of fingers and quantity. Thus,
in the light of our data we would not argue that the mag-
nitude effect itself is a reflection of hand-based representa-
tions. Rather, it is its modulation at the five break that we
trace back to embodied numerosity representation.
anabstractmagnitude
4.3. Cultural modulation of sub-base-five effects
Above we have stated that the generation hypothesis
accounts best for the sub-base-five effects and their cul-
tural modulation, as German adults of both groups exhib-
ited larger sub-base-five effects than Chinese participants
who do not have an explicit sub-base-five finger counting
system. However, a closer inspection of the data revealed
that Chinese participants also exerted some, albeit small,
five break effects. In particular, the two number pairs
crossing the five break (i.e. 4_6 and 5_7) were relatively
slower than the other one-digit number pairs. How could
this finding be explained? Both the comparison and gener-
ation hypothesis cannot straightforwardly explain why
these number pairs are responded to more slowly than
the others, because all Chinese numbers 69 are signed
using one hand only. Therefore, the comparison of one-
handed vs. two-handed representations cannot be faster
nor can the generation of two-handed patterns be respon-
sible for any RT increase in these participants.
A possible hand-based explanation for five break ef-
fects in Chinese to be tested in the future may be a tran-
sition account referring to the transparency of the finger
counting system. In the Chinese system, there is a con-
ceptual break after the number 5: all numbers 65 are
transparently represented by the quantity of fingers
stretched out (see Fig. 1), while for the numbers 6–10
there is no transparent reflection of the quantity indi-
cated. Thus, the signs for those numbers are purely sym-
bolic (see Fig. 1). One may hypothesize that transparent,
quantity based and symbolic finger counts are repre-
sented in different ways. Comparison within one type of
representation may be easier than comparison across
the two different (i.e. transparent and symbolic) represen-
tations. Therefore, the number pairs 4_6 and 5_7 may be
compared unexpectedly slow, because one transparent,
quantity-based finger representation (4 or 5, respectively)
has to be compared with one symbolic finger representa-
tion (6 or 7, respectively). In all other pairs both numbers
have consistently either transparent finger quantity rep-
resentations (1_3, 2_4, and 3_5) or symbolic finger repre-
sentations (6_8, 7_9, and 8_10). This explanation makes
use of the fact that the transition from a transparent,
quantity based to a symbolic finger counting system falls
together with the five break. In consequence, the ob-
served five break effect in Chinese would have little to
do with the finger sub-base-five system but only with a
transition from a transparent, quantity based to a sym-
bolic finger representation which by accident occurs just
between the numbers 5 and 6 in Chinese. The way to test
this account is therefore straightforward: one would need
to study a finger counting system in which the transition
from a transparent, quantity based to a symbolic repre-
sentation occurs at a different point in the number se-
quence. For example, this transition occurs between 4
and5inKoreanSignLanguage
tube.com/watch?v=Us4BMLjer2U). In this case, the com-
parison of number pairs which cross that culture-
specific border (e.g. 3_5, 4_6 in Korean Sign Language)
should be relatively more difficult because they imply
comparisons across different types of representation.
(http://www.you-
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263

Page 14

Regarding possible differences between hearing Ger-
mans and DGS users, we have expected sub-base-five ef-
fects to be more pronounced in the latter, given that for
deaf signers visual and verbal number systems fall to-
gether and number signs are thus much more salient than
for hearing participants. Clearly, this hypothesis was not
borne out by the present data. We can only provide tenta-
tive explanations for this finding. Given that finger count-
ing is so important during the development of numerical
cognition and is still used in certain contexts even by hear-
ing numerate adults (Butterworth, 1999), the lack of a sig-
nificant difference may just be due to the fact that saliency
has reached ceiling for hearing Germans as well as for DGS
users.
Some findings published very recently by Fischer and
colleagues (Fischer, 2008; Fischer & Lindemann, 2009) sug-
gest that there may be at least one further trace of finger
counting to be found in symbolic number processing: the
hand with which subjects start to count may modulate
their individual SNARC-effect. Specifically, the SNARC-ef-
fect may be enhanced in participants who start to count
on their left hand but it may be reduced in participants
starting on their right. However, there is large variability
both within and across cultures. For instance, about 60%
of Germans participating in an online survey stated that
they start with their left while the remaining 40% start
with their right (Fischer & Lindemann, 2009). Unfortu-
nately, we did not ask participants about their individual
starting hand. Therefore, although we could demonstrate
SNARC-like effects in general, we were not able to investi-
gate their modulation by individual counting habits in the
present data.
However, the studies of Fischer and colleagues (Fischer,
2008; Fischer & Lindemann, 2009) do in our view comple-
ment our study. Different aspects of mental number repre-
sentation seem to be influenced by finger counting: the
orientation of the mental number line (left–right) and the
structure of the magnitude representation (sub-base-five
system). Moreover, the studies of Fischer and colleagues
show that differences between and differences within-cul-
ture can influence individual numerical cognition. While
some embodied representations seem to be fairly consis-
tent within a culture (such as the use of sub-base-five sys-
tems in our study), others seem to vary even within a
culture (e.g. starting hand). Obviously, the demonstration
of specific impact of cultural and individual differences
limits the generality of findings observed in only one
culture.
4.4. What about larger numbers?
As outlinedinSection 2,we used the numberrange upto
20 to avoid potential confounds of five break and laterality
effects. Still, our analyses were restricted to the number
range up to 11, mainly focusing on one-digit numbers.
Although, the present data set seems to invite analyses of
larger numbers, too, any interpretation for two-digit num-
bers has to be treated with great caution. For example, in
German (in contrast to DGS or Chinese) there is no conven-
tionalfingercountingsystemfornumbers>10.Onecanonly
speculate,whetherhearingGermansrepeatthesystemused
for numbers 610 and keep some marker for the current
number range (e.g. teens) in mind. Furthermore, in DGS,
the signs for numbers >10 exhibit an additional movement
component (a rotation in the case of teens, a sequential
movementinthecaseofothertwo-digitnumbers).Itisvery
difficult to estimate the additional cost of this movement
component, if any, in number representation. Moreover, in
Chinesethesignfor10mayrequirethecoordinationofboth
hands (see Fig. 1). According to the generation hypothesis,
one may speculate that this particular representational
complexity has led to the pronounced peak observed for
the number pair 10_12. An alternative explanation for this
pronounced peak in the Chinese data may be that Chinese
(but not German) number words >10 are longer than those
610. For numbers >10, the Chinese finger counting system
adopts a place-value system, i.e. it depends on the hand a
specific number is signed with, whether it is the decade or
unitofatwo-digitnumber.Finally,thenumberwordsystem
inGerman(butmuchlesssoinChinese)isratherirregularin
theteensrangeinseveralrespects(e.g.intermsofnumberof
syllables and transparency). When a multimodal represen-
tationofquantity–includingaverbalrepresentation–isas-
sumed,thismayfurther
Notwithstanding all these potential pitfalls, visual inspec-
tion of the graphs in Figs. 3 and 4 points to the possibility
of sub-base-five effects in both German groups of partici-
pants and to the lack of such effects in the Chinese partici-
pants in the teens range, too. While the above mentioned
factors may – in general – complicate the interpretation of
the data found, we do not see how any of these points could
explaintheapparentspecificincreasearoundthenumber15
in German but not in Chinese. For example, the length of
numberwords(measuredinsyllables)doesnotdifferinthis
number range neither in German nor in Chinese. It remains
for further research to shed some light on the issue of sub-
base-five representations of larger numbers. However, we
wouldnotbetoosurprisedtofindevidenceforsub-base-five
structure in numerical cognition even beyond 10.
complicate thepicture.
5. Conclusions
This study set out to investigate the influences of a fin-
ger-based sub-base-five system on Arabic number process-
ing in adults. In three different populations (hearing
Germans, DGS users, and hearing Chinese), we observed
evidence for an influence of a sub-base-five system on Ara-
bic number comparison. Whereas such an influence has
previously been shown for children (Domahs et al., 2008)
and DGS signers (Iversen et al., 2006), this study is the first
to show sub-base-five influences in hearing educated
adults. We suggest that these sub-base-five effects are
due to the sub-base-five structure of the respective finger
counting systems and conclude that even in adult numer-
ical cognition finger counting is still influential for number
processing and possibly even for mental arithmetic. In this
respect, the present data support the general idea of
embodied numerosity: adult number representation –
according to this view – is not restricted to an abstract
magnitude representation or an exact number word sys-
tem. Rather, it is also shaped by bodily experience, that is
264
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Page 15

to say, by finger counting habits and structures (see also
Fischer, 2008, for number line orientation). Thus, the pres-
ent study suggests that finger counting habits are not only
important at a specific developmental stage, but even
influence the structure of adult number processing. These
findings from numerical cognition converge with results
from other cognitive domains such as language processing
and thus support the idea that the constitution of seem-
ingly abstract representations is at least partially rooted
in our bodily experiences.
Acknowledgements
We would like to thank Anna Hong Xian, Ege Karar, and
NinaKleiserfortheirhelpindatacollection,IsabelBihlmaier
for her assistance in the preparation of the manuscript as
well as Stefanie Keim for creating the line drawings. This
work was supported by funding from the Interdisciplinary
Centre for Clinical Research ‘BIOMAT.’ (project VVZ 51) of
theRWTHAachenUniversityHospitalsupportingF.Domahs
and K. Willmes. We also wish to thank two anonymous
reviewers for their helpful and constructive comments on
an earlier version of the manuscript.
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