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Spatial interferences in mental arithmetic:
Evidence from the motion–arithmetic
compatibility effect
Michael Wiemersa, Harold Bekkeringa & Oliver Lindemannab
a Donders Institute for Brain, Cognition and Behaviour, Radboud University
Nijmegen, Nijmegen, The Netherlands
b Division of Cognitive Science, University of Potsdam, Potsdam, Germany
Accepted author version posted online: 31 Jan 2014.Published online: 05
Mar 2014.
To cite this article: Michael Wiemers, Harold Bekkering & Oliver Lindemann (2014) Spatial interferences
in mental arithmetic: Evidence from the motion–arithmetic compatibility effect, The Quarterly Journal of
Experimental Psychology, 67:8, 1557-1570, DOI: 10.1080/17470218.2014.889180
To link to this article: http://dx.doi.org/10.1080/17470218.2014.889180
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Spatial interferences in mental arithmetic: Evidence from
the motion–arithmetic compatibility effect
Michael Wiemers
1
, Harold Bekkering
1
, and Oliver Lindemann
1,2
1
Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen, Nijmegen,
The Netherlands
2
Division of Cognitive Science, University of Potsdam, Potsdam, Germany
Recent research on spatial number representations suggests that the number space is not necessarily
horizontally organized and might also be affected by acquired associations between magnitude and
sensory experiences in vertical space. Evidence for this claim is, however, controversial. The present
study now aims to compare vertical and horizontal spatial associations in mental arithmetic. In
Experiment 1, participants solved addition and subtraction problems and indicated the result verbally
while moving their outstretched right arm continuously left-, right-, up-, or downwards. The analysis of
the problem-solving performances revealed a motion–arithmetic compatibility effect for spatial actions
along both the horizontal and the vertical axes. Performances in additions was impaired while making
downward compared to upward movements as well as when moving left compared to right and vice
versa in subtractions. In Experiment 2, instead of being instructed to perform active body movements,
participants calculated while the problems moved in one of the four relative directions on the screen. For
visual motions, only the motion–arithmetic compatibility effect for the vertical dimension could be
replicated. Taken together, our findings provide first evidence for an impact of spatial processing on
mental arithmetic. Moreover, the stronger effect of the vertical dimension supports the idea that
mental calculations operate on representations of numerical magnitude that are grounded in a vertically
organized mental number space.
Keywords: Mental arithmetic; Numerical cognition; Spatial–numerical associations; Embodied
cognition.
Classical research on mathematical cognition has
emphasized the importance of spatial codes
for the representation of numerical magnitude
(see e.g., Fias & Fischer, 2005; Hubbard, Piazza,
Pinel, & Dehaene, 2005). Very recently, the idea
of spatial–numerical associations has been utilized
to formulate theories of mental arithmetic, which
basically hold that adding or subtracting numbers
involves a cognitive process that can be described
as right- or leftward attentional shifts on a spatial
representation of numerical magnitude (Knops,
Viarouge, & Dehaene, 2009; McCrink, Dehaene,
& Dehaene-Lambertz, 2007; Pinhas & Fischer,
2008).
Studies on spatial associations in number
processing have traditionally focused on numerical
judgements and interference effects between
numerical information and horizontal space. The
most prominent example of this coupling is the
effect of spatial–numerical association of response
codes (SNARC; Dehaene, Bossini, & Giraux,
1993), which reflects the tendency to react faster
Correspondence should be addressed to: Michael Wiemers, Donders Institute for Brain, Cognition and Behaviour, P.O. Box 9104,
6500 HE Nijmegen, The Netherlands. E-mail: m.wiemers@donders.ru.nl
© 2014 The Experimental Psychology Society 1557
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Vol. 67, No. 8, 1557–1570, http://dx.doi.org/10.1080/17470218.2014.889180
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with right-side responses to large numbers and
with left-side responses to small numbers. This
number–response compatibility effect indicates
that numerical magnitudes are associated with pos-
itions along the horizontal axis with small numbers
on the left and large number on the right (so-called
mental number line hypothesis; Dehaene et al.,
1993; Fias & Fischer, 2005).
Interestingly, research has shown that the inter-
action between number and space also extends to
the vertical dimension (Holmes & Lourenco,
2012; Ito & Hatta, 2004; Schwarz & Keus, 2004;
Shaki & Fischer, 2012). Schwarz and Keus (2004)
were amongst the first to report a vertical number–
response compatibility effect and demonstrated
that upward saccades were initiated faster in
response to large numbers, while downwardsaccades
were initiated faster in response to small numbers.
However, studies on number representation that
compared vertical and horizontal spatial–numerical
couplings provide so far no consistent evidence on
whether the vertical or horizontal organization is
predominant (Holmes & Lourenco, 2012; Sell &
Kaschak, 2012; Shaki & Fischer, 2012; Winter &
Matlock, 2013). One the one hand, Shaki and
Fischer (2012) reported that Hebrew-speaking
participants who do not show a SNARC effect
with horizontally aligned responses due to incon-
sistent processing habits for words and numbers
exhibit a vertical SNARC effect. This finding
suggests a more robust and culturally independent
mapping between vertical space and magnitude.
Also, Winter and Matlock (2013) provided
support in favour of a strong vertical organization
of number space by showing that the production
of random numbers is affected stronger by vertical
than by horizontal head movements. On the
other hand, a predominance of vertical number
associations has been challenged by Holmes and
Lourenco (2012). The authors tested SNARC-
like effects in a parity judgement task with different
spatial response button layouts and found that
when pitting both dimensions directly against
each other, only a horizontal but not a vertical
number–space mapping could be observed.
Although the empirical evidence for a predomi-
nantly vertical organization of number space is
ambiguous at this point, we argue that this idea is
in line with recent theoretical developments in psy-
chology of knowledge representations and the
general hypothesis that any conceptual represen-
tation is somehow grounded in sensorimotor
experiences (e.g., Barsalou, 2008; Lakoff, 1987).
Following such an embodied view on cognition,
Fischer and Brugger (2011) pointed out that
especially sensorimotor experience in vertical
space play a crucial role in the development of
abstract number concepts. They stated that due to
a strong correlation between vertical space and
numerical magnitude in the world—for example,
through piling objects when comparing the numer-
osity of two sets of objects and the fact that stacks
extend upwards as they grow—an association
between these two dimensions emerges from early
life on. However, in the action of piling objects,
numerical magnitude is confounded with stack
height. As a consequence, vertical space should
become associated with numerical magnitude and
magnitude at an abstract level. Therefore, it is
more accurate to speak of an association of vertical
space and magnitude. Crucially, no systematic
relationship between horizontal space and
numerical magnitude exists when arranging
objects horizontally. One might therefore assume
that mappings of numbers onto vertical space
provide the grounding for any type of spatial–
numerical mapping (henceforth referred to as
the “grounding hypothesis of spatial–numerical
associations”).
As mentioned above, the activation of spatial
codes is not restricted to the representation of
single numbers but also seems to play a crucial
role for mental arithmetic (McCrink et al.,
2007; Pinhas & Fischer, 2008). Until now,
studies of the interplay between mental arithmetic
and spatial processing have merely focused on the
horizontal dimension. McCrink et al. (2007) were
amongst the first to discuss the interaction
between spatial processing and mental arithmetic.
The authors reported that when adding or sub-
tracting the numerosities of two dot clouds, par-
ticipants tend to overestimate the result of
additions and underestimate the result of subtrac-
tions (see also Knops, Viarouge, et al., 2009).
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WIEMERS ET AL.
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This so-called operational momentum effect has
been interpreted as evidence for the idea that
addition and subtraction are accompanied by
attentional shifts to the right (addition) and left
(subtraction) on the mental number line, inducing
an over- or underestimation, respectively (Knops,
Viarouge, et al., 2009; McCrink et al, 2007).
Similar effects of the operation have also been
observed in symbolic arithmetic. For instance,
Pinhas and Fischer (2008), who instructed par-
ticipants to indicate the outcome of symbolic
additions and subtractions by pointing to the cor-
responding location on the horizontal number
line, found a systematic response bias towards
larger numbers (i.e., the right side) after adding
and towards smaller numbers (i.e., the left side)
after subtracting.
The notion that shifts on a spatial or spatial–
numerical continuum are causing the operational
momentum effect received so far only support
from two findings. First, Knops, Viarouge, et al.
(2009) reported a space-to-arithmetic mapping.
That is, participants preferred to choose the upper
right dot cloud for additions and the upper left
dot cloud for subtractions. Second, neuroimaging
research (Knops, Thirion, Hubbard, Michel, &
Dehaene, 2009) demonstrated that activity in the
posterior superior parietal cortex related to horizon-
tal eye movements was correlated with mental
additions and subtractions. This evidence is
however rather indirect, and a strong test of the
coupling of space and arithmetic, such as an effect
of spatial processes on the performances in arith-
metic tasks, is still missing.
Taken together, the grounding hypothesis of
spatial–numerical associations predicts predomi-
nant vertical organization of number space and
stronger interference effects for vertical rather
than horizontal spatial information in number pro-
cessing. Since research on number representations
provided only inconclusive evidence for this idea
and since vertical mappings have been neglected
in studies on mental arithmetic so far, the present
study aims to compare vertical and horizontal
spatial associations while solving addition and
subtraction problems.
We addressed these issues by employing a dual-
task-like paradigm, in which participants per-
formed slow continuous arm movements in the
horizontal (left- or rightward) or vertical dimen-
sion (up- or downwards) while solving addition
or subtraction problems. We expected an effect
of the compatibility between the performed move-
ment and the mental arithmetic task (motion–
arithmetic compatibility effect). That is, based on
studies on the operational momentum effect
(e.g., Knops, Viarouge, et al., 2009), a motion–
arithmetic compatibility effect should be reflected
in the horizontal dimension by better performance
for addition for a rightward movement than for a
leftward movement and better performance in sub-
traction for leftward than for rightward move-
ments. Importantly, the grounding hypothesis of
spatial–numerical associations predicts further-
more a stronger motion–arithmetic compatibility
effect along the vertical dimension, with facilitated
processing of additions while moving upward and
facilitated processing of subtractions for downward
movements.
EXPERIMENT 1
Participants were instructed to perform constant-
speed arm movements with the fully extended
right arm, starting pointing straight forward and
moving left-, right-, up-, or downwards. Arm
movements had to be initiated at the presentation
of an arithmetic (addition or subtraction) problem
on the screen and had to be continued until an
answer was given verbally. Assuming that the
spatial coding of numerical magnitudes is function-
ally involved in mental arithmetic, it is expected
that the execution of arm movements in space has
a systematic impact on solving simple additions
and subtraction problems. In particular, we expect
that the processing of arithmetic problems is
more efficient while performing a compatible
action than while performing an incompatible
action (motion–arithmetic compatibility effect).
Moreover, we expect the motion–arithmetic
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SPATIAL INTERFERENCE IN MENTAL ARITHMETIC
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compatibility effect for vertical arm movements to
be stronger than that for horizontal arm
movements.
METHOD
Participants
A total of 38 students from the Radboud University
Nijmegen (11 males, mean age of 20.49 years, 20
native Dutch and 18 native German speakers)
with normal or corrected-to-normal vision, naive
to the purpose of the study, took part in the study
in exchange for 15 euros or credit points.
Experimental set-up and apparatus
Participants were standing at approximately 3 m
distance from a 17′′ CRT monitor holding the
Wii®-controller (Nintendo) in their right hand,
which was used to record the hand position with
a sampling rate of 100 Hz. To detect the voice
onset, a microphone was placed in front of the
head of the subject. The set-up consisted of two
interconnected computers, one for the recording
of the Wii controller position data and one for pre-
sentations of the stimuli and the voice onset detec-
tion. The experiment was controlled using the
Python software package Expyriment (Krause &
Lindemann, 2013). The experimenter registered
the verbally given answers and the correctness of
the executed movements.
Stimuli
All problems consisted of two operands and one
operator (plus or minus) and were presented cen-
trally in black Arial font on a light grey background
(see Table 1). At a viewing distance of 3 metres, the
problems were spanning approximately 4° (hori-
zontal) ×2° (vertical) in visual angle. Subtraction
and addition problems were matched for overall
problem size (i.e., average of all numbers in the
correct equation) by inverting addition into sub-
traction problems (e.g., 34 +58 =92, 92 −58 =
34). The arithmetic problems in the experimental
block consisted of 15 small-size addition and sub-
traction problems (problem size ranging from 22
to 34, mean result: 13.2 for additions and 6.73
for subtractions) and 15 large-size addition and
Table 1. Arithmetic problems used for the experimental phase
Problem number Operand 1 Operation Operand 2 Result
19+817
27+916
39+615
45+914
59+413
63+912
78+715
86+814
98+513
10 3 +811
11 7 +613
12 5 +712
13 7 +411
14 6 +511
15 9 +211
16 27 +36 63
17 39 +25 64
18 29 +38 67
19 27 +44 71
20 49 +23 72
21 28 +53 81
22 37 +45 82
23 44 +39 83
24 56 +28 84
25 47 +38 85
26 57 +29 86
27 25 +66 91
28 38 +54 92
29 26 +67 93
30 36 +59 95
31 17 −89
32 16 −97
33 15 −69
34 14 −95
35 13 −49
36 12 −93
37 15 −78
38 14 −86
39 13 −58
40 11 −83
41 13 −67
42 12 −75
43 11 −47
44 11 −56
45 11 −29
46 63 −36 27
47 64 −25 39
48 67 −38 29
49 71 −44 27
50 72 −23 49
(Continued overleaf)
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subtraction problems (problem size ranging 124 to
190, mean result: 80.60 for additions and 37.67 for
subtractions). A different set of arithmetic pro-
blems was used in the training phase.
A black square frame of 2° ×2° in visual angle
was used as a starting position of each arm move-
ment. Block arrows pointing to the left, to the
right, up, or down served as movement cues. A
black diamond (3° by 3° visual angle) indicated
that no movement had to be executed.
Procedure
The experiment started with two practice blocks.
First, participants were required to verbally report
the correct result of visually presented arithmetic
problems. The second practice block was identical
to the subsequent experimental block (see below).
The experiment started when participants were
able to perform both tasks correctly simultaneously.
In the experimental block, every trial started
with a movement instruction, indicated by an
arrow pointing in one of the four relative directions
or a black filled diamond symbol indicating that no
movement had to be performed in this trial.
Participants had to bring the arm to a centre pos-
ition by moving a red dot into a centrally presented
square frame (2° ×2°of visual angle). After a
random interval of 1000–1500 ms, the arithmetic
problem was presented. With problem onset, par-
ticipants were required to start the instructed arm
movement and to continue with a steady speed of
approximately 10° angle/s until an answer was
given.
If the arm movement did not start within 1000
ms, an error signal was presented, and the trial was
interrupted. Participants were instructed to report
the result of the presented problem as soon as poss-
ible by first saying “is”followed by the result (in
their native language). The experimenter moni-
tored the movements and the verbal response and
entered both into the computer. If a movement
was stopped before an answer was given or the
answer was not given in the instructed manner,
the trial was scored as erroneous. After every trial,
feedback messages were presented saying “Correct
Movement”or “Correct Answer”with green font
colour or “Wrong Movement”or “Wrong
Answer”in red font colour. Afterwards, the exper-
imenter started the next trial.
Design
The experimental block comprised 30 addition and
30 subtraction problems each presented once in
each of the five movement conditions (left, right,
up, down, no movement), resulting in 300 trials
in total. All trials were presented in a randomized
order. The experiment lasted approximately
90 min.
Data analysis
The response time (RT) was defined as the time
between problem presentation and onset of the
vocal response.
For the analysis, all trials with anticipation
responses (RTs ,400 ms), missing responses (no
responses and RTs .10,000 ms), trials in which
triggering of the voice key was not due to the
verbal response of the participant, and trials with
incorrectly performed arm movements (e.g., move-
ments in the wrong direction, movements that
stopped before verbal response, very slowly per-
formed movements) were excluded (7.8% of the
trials).
To control for a possible trade-off between
speed and accuracy of responding or different strat-
egies of the participant—that is, an emphasis on
speed or accuracy depending on the experimental
condition—we analysed efficiency scores.
Efficiency scores integrate the two performance
Table 1. Continued.
Problem number Operand 1 Operation Operand 2 Result
51 81 −53 28
52 82 −45 37
53 83 −39 44
54 84 −28 56
55 85 −38 47
56 86 −29 57
57 91 −66 25
58 92 −54 38
59 93 −67 26
60 95 −59 36
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SPATIAL INTERFERENCE IN MENTAL ARITHMETIC
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indices speed and accuracy of responding by divid-
ing the accuracy rate by response time in seconds.
Motion–arithmetic compatibility was defined as
follows. In addition trials, right- and upward
motions were defined as motion–arithmetic com-
patible. Left- and downward movements were con-
sidered as incompatible. For subtractions,
compatible and incompatible motions were in the
opposite direction to that for additions.
Results
The average error rate was 3.23% for problems
with small size and 17.67% for large-size problems,
t(41) =−10.02, p,.001.
Efficiency scores (ES) were submitted to a four-
way repeated measures analysis of variance
(ANOVA) with the within-subject factors problem
size (small, large), operation (addition, subtraction),
spatial dimension (horizontal, vertical), and motion–
arithmetic compatibility (compatible, incompatible).
The analysis revealed two main effects of the factors
problem size, F(1, 37) =344.20, MSE =0.10,
p,.001, η
2
=.90, and operation, F(1, 37) =
129.59, MSE =0.02, p,.001, η
2
=.78. That is,
performance was better for additions (0.575 effect
size, ES) than for subtraction problems (0.453 ES)
and better for small-size problems (0.749 ES) than
for large-size problems (0.279 ES). Both effects
reflect two well-established findings in the field of
mental arithmetic. Moreover, there was a significant
interaction between the factors operation and
problem size, F(1, 37) =37.03, MSE =0.01,
p,.001, η
2
=.50, reflecting that the difference in
performance between addition and subtraction
trials was more pronounced for small- (0.176 ES)
than for large-size problems (0.072 ES).
Importantly, there was a significant effect of
motion–arithmetic compatibility, F(1, 37) =8.43,
MSE =0.03, p,.01, η
2
=.19, suggesting that
problem-solving performance was better when
adding and moving up or to the right or subtracting
and moving down or to the left (motion–arithmetic
compatible; 0.521 ES) than when adding and
moving down or left or subtracting and moving up
or right (motion–arithmetic incompatible; 0.507
ES). Interestingly, there was no interaction
between the factors motion–arithmetic compatibility
and spatial dimension, F(1, 37) =1.14, MSE =
0.003, p=.29, η
2
=.03, suggesting that the
motion–arithmetic compatibility effects for the hori-
zontal and vertical dimensions did not differ from
each other. However, post hoc ttests revealed that
only for the vertical dimension was the interaction
between movement direction and arithmetic
operation significant, t(37) =2.00, p,.05. The
motion–arithmetic compatibility effect for the hori-
zontal dimension did not reach significance, t(37) =
1.10, p=.14 (see Figure 1). This pattern suggests a
predominant role of vertical space for numerical pro-
cessing in the context of mental arithmetic.
In order to specify whether the vertical motion–
arithmetic compatibility effect reflects an interfer-
ence and/or facilitation effect, the compatible and
incompatible conditions were compared with the
no-movement condition. The ttests indicated sig-
nificantly lower performance in both the motion–
arithmetic compatible (0.523 ES) and incompatible
conditions (0.505 ES) than in the no-movement
condition (0.534 ES), t(37) =2.81, p,.05, and
t(37) =3.57,p,.01. This pattern suggests inter-
ference of motion, which is incompatible with the
arithmetic operation, as well as greater overall cog-
nitive demands for the movement than for the no-
movement conditions.
Since the average result size in additions was
larger than that in subtractions (13.2 for additions
and 6.73 for subtractions in small-size problems
and 80.60 for additions and 37.67 for subtractions
in large-size problems), it might be objected that
Figure 1.Efficiency scores (accuracy rate/response time, acc/RT, in s)
as a function of the factors spatial dimension and motion–arithmetic
compatibility in Experiment 1. The asterisk indicates a significant
difference between conditions, p,.05.
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the interaction between movement direction and
arithmetic operation is driven by a difference in
result size and a sort of effect of spatial–numerical
compatibility (larger results might be associated
with right- and upward motions and smaller with
left- and downward motions). In order to rule out
that the difference in result size for additions and
subtractions is driving the compatibility effects,
we performed an additional three-way repeated
measures ANOVA with the within-subject
factors problem size (small, large), spatial dimen-
sion (horizontal, vertical), and spatial–numerical
compatibility (compatible, incompatible). The
analysis did not reveal a main effect of spatial–
numerical compatibility, F(1, 37) ,1, an inter-
action between spatial–numerical compatibility
and spatial dimension, F(1, 37) ,1, an interaction
between spatial–numerical compatibility and
problem size, F(1, 37) =1.14, MSE =0.002,
p=.29,η
2
=.03, or a three-way interaction
between spatial–numerical compatibility, spatial
dimension, and problem size, F(1, 37) ,1.
To further explore the effects from the efficiency
analyses, we performed repeated measures
ANOVAs for mean RT data and error rates with
the factors operation (Table 3) (addition, subtrac-
tion), spatial dimension (horizontal, vertical), and
motion–arithmetic compatibility (compatible,
incompatible). For the analysis of RT data, all
trials with a RT 2.5 standard deviations below or
above the grand mean were excluded. Similar to
the efficiency score data, the analyses revealed a sig-
nificant effect for motion–arithmetic compatibility,
F(1, 37) =8.36, MSE =0.43, p,01, η
2
=.18,
and no interaction between spatial dimension and
motion–arithmetic compatibility, F(1, 37) ,1.
The analyses of error rates did not show a signifi-
cant effect for motion–arithmetic compatibility or
an interaction between motion–arithmetic compat-
ibility and another factor.
Discussion
Experiment 1 provides evidence for an impact of
spatial processing on mental arithmetic by
showing a systematic effect of movement direction
on problem-solving performance of arithmetic
problems. That is, participants’arithmetic perform-
ance was impaired while moving the arm in an
incompatible direction with respect to the required
arithmetic operation (i.e., adding while moving left
or down, or subtracting while moving right or up)
as compared to the compatible movement con-
dition (adding while moving right or up, or sub-
tracting while moving left or down). The data
support the notion that spatial representations are
functionally involved in mental arithmetic (cf.
McCrink et al., 2007; Pinhas & Fischer, 2008).
Most importantly, the demonstration that arith-
metic operations are systematically affected by
movements along the vertical axis indicates that a
vertically organized number space is involved in
mental arithmetic (Fischer & Brugger, 2011).
It has to be emphasized that the grounding
hypothesis of spatial–numerical association predicts
a predominant role for the vertical associations.
While Experiment 1 revealed clear evidence for a
coupling of arithmetic operations and vertical
space, it provided only mixed support for a difference
in the motion–arithmetic compatibility effects along
the horizontal and vertical dimensions. The lack of
interaction between the factors motion–arithmetic
compatibility and spatial dimension in the
ANOVA suggests that compatibility effects are
not modulated by the spatial dimension.
The simple effects analysis, however, is in line
with the grounding hypothesis and supports our
assumption that motion–arithmetic compatibility
effects emerge predominantly for vertical move-
ments. The results of Experiment 1 are thus not
fully conclusive, and we interpret the outcome
therefore parsimoniously as a failure to find evi-
dence for a difference between horizontal and ver-
tical motion–arithmetic compatibility effects.
Another problem of Experiment 1 relates to the
physical effort associated with arm movements in
different directions. Due to the forces of gravity,
moving the arm upwards is more effortful than
moving it downwards. This difference in effort
for vertical arm movements has a potentially con-
founding effect, since the motion–arithmetic com-
patibility for the vertical dimension could also be
due to a “more-is-more”mapping (i.e., linking
physical effort with arithmetic operations, cf.
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Stavy & Tirosh, 2000) instead of a grounding in
terms of mental space. In contrast, horizontal
arm movements do not suffer from the same
confound.
We conducted a second experiment in order to
address the issue of an asymmetry in the required
energy for up- and downward arm movements
and to provide a more conclusive additional test
of a potential difference between the effect of hori-
zontal and vertical spatial representations on arith-
metic problem-solving performance.
EXPERIMENT 2
Experiment 2 further investigated the two motion–
arithmetic compatibility effects and aimed to rule
out the potential confound of a difference in effort
for the upward compared to the downward move-
ment condition being responsible for the compat-
ibility between vertical motion and arithmetic.
Whereas a difference in the effort for up- and
downward arm movements is clearly noticeable by
the participant, we believe that the difference in
the effort of up- and downward smooth pursuit
eye movements is only marginal and will not be
mentally represented. As a study by Collewijn,
Erkelens, and Steinman (1988) indicates, the brain
does not sufficiently take into account gravity and
the related difference in inertial forces for the plan-
ning of vertical eye movements. The authors
showed that upward saccades undershoot, while
downward saccades systematically overshoot relative
to target position. This finding supports our
assumption that the difference in the applied
energy for eye movements is small and negligible.
Experiment 2 also served to provide an
additional test of a difference between the impact
of horizontal and vertical spatial information on
arithmetic problem-solving performance, since the
results from Experiment 1 were inconclusive in
this regard.
In order to induce slow smooth pursuit eye
movements, we presented arithmetic problems
moving from screen centre in one of the four
relative directions with a rate of about 1.7° visual
angle/s.
Method
Participants
A total of 44 students of the Radboud University
Nijmegen (10 males, mean age of 21.77 years) par-
ticipated in the experiment in return for 10 euros or
course credits.
Experimental set-up and apparatus
Participants were sitting at about 70 cm viewing
distance from a 17′′ CRT monitor. As in
Experiment 1, response onset was detected via
voice key. Responses were classified and scored by
the experimenter.
Stimuli
The same arithmetic problems as those in
Experiment 1 were presented. Problems were
moving after appearance at the screen centre to
the left, to the right, up, or down at approximately
1.7° visual angle/s. The problems were presented in
smaller size, spanning only 4° ×2° visual angle.
A black square (1° ×1° of visual angle) served as a
fixation point.
Procedure
Again, the experiment started with a practice block.
In the experimental block, each trial started with a
central presentation of the fixation point. After a
random interval of 1,000 to 1,500 ms, the addition
or subtraction problem was presented at the centre
of the screen and started moving in one of the four
directions or remained at screen centre. The par-
ticipants’task was to indicate the correct result as
fast as possible by first saying “is”followed by the
result. If an answer was not provided before
the arithmetic problem reached the border of the
screen, or within 6 s in the no-motion condition,
the trial was counted as erroneous. Messages
saying either “Correct Answer”in a green font or
“Wrong Answer”in a red font were presented at
the end of each trial as feedback.
Design and data analysis
The experimental design and data analysis were
identical to those in Experiment 1. The experiment
lasted about 60 min.
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Results
The average error rate was 3.22% for small-size
problems and 17.46% for large-size problems,
t(41) =−10.05, p,.001.
Efficiency scores were submitted to a repeated
measures ANOVA with the within-subject factors
problem size (small/large), operation (addition/sub-
traction), spatial dimension (horizontal/vertical),
and motion–arithmetic compatibility (compatible/
incompatible). As in the previous analyses, the
same pattern of effects for problem size, operation,
and the interaction between operation and problem
size was shown. The effect of motion–arithmetic
compatibility was not significant, F(1, 42) =2.63,
MSE =0.02, p=.11, η
2
=.06.
Most importantly, there was a significant
interaction between spatial dimension and
motion–arithmetic compatibility, F(1, 42) =4.50,
MSE =0.04, p,.05, η
2
=.10 (see Figure 2).
Post hoc ttests revealed that only vertical spatial
motion showed a significant motion–arithmetic
compatibility effect, with higher efficiency scores
for compatible (0.504 ES) than for incompatible
(0.493 ES) visual motion, t(42) =7.48, p,.01.
For the horizontal dimension, movement direction
did not modulate the performance in the
mental arithmetic task (0.498 ES vs. 0.500 ES),
t(42) ,1. Thus, arithmetic problem-solving per-
formance was better when adding and moving up
or subtracting and moving down (0.504 ES) than
when adding and moving down or subtracting
and moving up (0.493 ES). In order to further
specify the compatibility effect for the vertical
dimension, the compatible and incompatible con-
ditions were compared against the no-movement
condition. The ttests indicated significantly lower
performance in the motion–arithmetic incompati-
ble (0.493 ES) than in the no-movement condition
(0.510 ES), t(41) =3.70, p,.01. The motion–
arithmetic compatible condition did not differ
from the no-movement condition,t(41) =1.29,
p=.20, which suggests an interference of motion,
which is incompatible with the arithmetic
operation.
In order to rule out that differences in result size
for additions and subtractions are underlying the
motion–arithmetic compatibility effect, we per-
formed an additional ANOVA with the within-
subject factors problem size (small, large), spatial
dimension (horizontal, vertical), and spatial–
numerical compatibility (compatible, incompati-
ble). As in Experiment 1, the analysis revealed
neither a significant effect of spatial–numerical
compatibility nor a significant interaction between
spatial–numerical compatibility and any other
factor, all Fs,1.
As a supplement to the efficiency score analysis,
we also performed analyses for RT and error data
with the factors operation (addition/subtraction),
spatial dimension (horizontal/vertical), and motion–
arithmetic compatibility (compatible/incompatible)
(see Table 4).
For Experiment 2 the analyses of mean RTs
merely revealed a trend for an effect of motion–
arithmetic compatibility, F(1, 41) =1.06, MSE =
0.012, p=.06.
The error rate analyses showed a trend for an
interaction between spatial dimension and
motion–arithmetic compatibility, F(1, 41) =3.73,
MSE =226.19, p=.06, with a descriptive differ-
ence in the a motion–arithmetic compatibility for
the vertical dimension (9.87%, compatible vs.
11.07%, incompatible) and an inverted difference
for the horizontal dimension (10.99%, compatible
vs. 9.87%, incompatible). However, further ttests
failed to prove the significance of these differences
[t(41) =1.61, p=.06, and t(41) =1.56, p=.06,
for the vertical and horizontal spatial dimension,
respectively].
Figure 2.Efficiency scores (accuracy rate/response time, acc/RT, in s)
as a function of the factors spatial dimension and motion–arithmetic
compatibility in Experiment 2. The double asterisk indicates a
significant difference between conditions, p,.01.
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Discussion
Experiment 2 addressed the problem of a potential
confound of a difference of effort or required energy
for upward compared to downward movements,
underlying the compatibility between vertical
motion and arithmetic. In order to control for
this, we presented addition and subtraction pro-
blems moving from screen centre in one of the
four directions. The continuous motion induced
smooth pursuit eye movements, for which the
asymmetry in effort for up- and downward move-
ments is marginal or not even taken into account
by the brain (Collewijn et al., 1988). Importantly,
we found a compatibility between vertical motion
and mental arithmetic—that is, performance was
impaired for additions when moving downwards
(compared to upward movements) and for subtrac-
tions when moving upwards (compared to down-
ward movements). The replication of the vertical
motion–arithmetic compatibility effect in this
setting suggests that representations of space are
driving the compatibility effect in the vertical
dimension. Moreover, as in Experiment 1, the
motion–arithmetic compatibility for the horizontal
dimension was not significant, suggesting a stron-
ger coupling of vertical space and mental arithmetic
than the linkage with horizontal space. Together,
Experiment 2 supports our hypothesis of a stronger
coupling between mental arithmetic and the verti-
cal than with the horizontal dimension and sup-
ports the notion that mental addition and
Table 2. Mean efficiency scores in Experiment 2 as a function of the factors problem size, operation, spatial dimension, and motion–arithmetic
compatibility
Spatial dimension MAC
Problems
Small-size Large-size
Addition Subtraction Addition Subtraction
Horizontal dimension Compatible 0.8045 0.6469 0.2900 0.2524
Incompatible 0.8019 0.6512 0.2907 0.2550
Vertical dimension Compatible 0.8126 0.6602 0.2907 0.2469
Incompatible 0.7988 0.6419 0.2857 0.2436
Note: MAC =motion–arithmetic compatibility.
Table 3. Mean response time and error rates in Experiment 1 as a function of the factors problem size, operation, spatial dimension, and
motion–arithmetic compatibility
Dependent variable Spatial dimension MAC compatibility
Problems
Small-size Large-size
Addition Subtraction Addition Subtraction
Response time (s) Horizontal Compatible 1.235 1.602 3.095 3.483
Incompatible 1.282 1.581 3.125 3.652
Vertical Compatible 1.230 1.531 3.110 3.568
Incompatible 1.277 1.619 3.083 3.587
Error rate (%) Horizontal Compatible 2.76 3.67 8.43 13.91
Incompatible 2.90 3.10 7.87 12.65
Vertical Compatible 2.96 4.68 8.20 10.72
Incompatible 3.37 2.58 7.29 15.23
Note: MAC =motion–arithmetic compatibility.
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subtraction are grounded in vertical spatial–
numerical associations.
GENERAL DISCUSSION
Two experiments investigated the involvement of
spatial representations in mental arithmetic and
demonstrated for the first time an impact of
spatial information on the performances to solve
addition and subtraction problems. Interestingly,
the observed motion–arithmetic compatibility
effect is more reliable for the vertical than for the
horizontal dimension, supporting the idea that
the mapping between numbers and vertical space
has a predominant function in mental arithmetic.
If participants were required to move their arm
while solving arithmetic problems (Experiment
1), we observed impaired performance in addition
when moving left- or downwards than when
moving right- or upwards and vice versa in subtrac-
tion. Experiment 1 furthermore revealed a descrip-
tive difference between the compatibility effects
along the two dimensions, suggesting a stronger
association for the vertical dimension. However,
the interaction between the factors motion–arith-
metic compatibility and dimension did not reach
significance. The second experiment provided an
additional test of differences in the impact of verti-
cal and horizontal spatial information and
controlled for the potential confound of higher
effort associated with upward than with downward
arm movements as an alternative explanation for
the motion–arithmetic compatibility effect for the
vertical dimension. Experiment 2 revealed only a
vertical but not a horizontal motion–arithmetic
compatibility effect and this way confirmed our
hypothesis that the coupling of mental arithmetic
is more robust for the vertical dimension.
Together our data support the idea that spatial pro-
cesses are functionally linked to mental arithmetic
(Pinhas & Fischer, 2008) and indicate that both
horizontal and vertical spatial information are
involved in mental arithmetic. Importantly, the
greater robustness of the motion–arithmetic com-
patibility effect for the vertical than for the horizon-
tal dimension argues for a predominant role of the
vertical dimension in the coupling of space and
arithmetic operations.
The observed motion–arithmetic compatibility
effect is in line with the idea that mental arithmetic
is accompanied by attentional shifts in mental
number space (McCrink et al., 2007; Pinhas &
Fischer, 2008). The previous support for a spatial
origin of the operational momentum effect in
approximate arithmetic was exclusively based on
spatial response biases for symbolic and nonsym-
bolic arithmetic (Knops, Viarouge, et al., 2009;
Pinhas & Fischer, 2008) and the predictability of
addition and subtraction from neural correlates of
Table 4. Mean response time and error rates in Experiment 2 as a function of the factors problem size, operation, spatial dimension, and
motion–arithmetic compatibility
Dependent variable Spatial dimension MAC compatibility
Problems
Small-size problems Large-size problems
Addition Subtraction Addition Subtraction
Response time (s) Horizontal Compatible 1.300 1.650 3.106 3.518
Incompatible 1.330 1.628 3.168 3.497
Vertical Compatible 1.299 1.622 3.162 3.526
Incompatible 1.330 1.646 3.153 3.555
Error rate (%) Horizontal Compatible 3.23 4.28 16.50 19.95
Incompatible 2.23 3.64 13.64 19.98
Vertical Compatible 2.71 2.97 12.68 21.11
Incompatible 2.25 4.51 15.42 22.11
Note: MAC =motion–arithmetic compatibility.
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horizontal saccades (Knops, Thirion et al., 2009).
Motor response biases, however, do not provide a
strict test for the spatial processes and can be alter-
natively explained by the simple response heuristic
to point more toward smaller numbers after sub-
tracting and more towards large numbers after
additions in situations of uncertainty (see e.g.,
Knops, Zitzmann, & McCrink, 2013 for a discus-
sion of a similar alternative account). The present
study now sheds new light on the origin of the
operational momentum effect by showing that the
mere activation of spatial codes affects the overall
efficiency to solve arithmetic problems. The non-
spatial accounts of response heuristics predict a
systematic over- or underestimation or distortions
in pointing trajectories but do not account for the
influence reported here of spatial codes on arith-
metic performances. Therefore, the present
finding of an impact of spatial information on
arithmetic problem-solving performance also pro-
vides support for the previously made notions that
biases in nonsymbolic (McCrink et al., 2007) and
symbolic arithmetic (Pinhas & Fischer, 2008) orig-
inate from attentional shifts in mental numerical
space (Knops, Thirion, et al., 2009).
The present study substantially extends the lit-
erature on number processing by showing that
associations with vertical space are not restricted
to the processing of single numbers (Sell &
Kaschak, 2012; Shaki & Fischer, 2012; Winter &
Matlock, 2013) but are also present in mental arith-
metic. Importantly, the greater impact of vertical
spatial information on arithmetic provides direct
empirical support for the grounding hypothesis of
spatial–numerical associations (cf. Fischer &
Brugger, 2011; Lakoff, 1987). According to this
notion, sensory experiences with numbers will be
primarily associated with the vertical dimension
due to a correlation between vertical space and
magnitude in the world. Therefore, vertical space
becomes an important and maybe even the predo-
minant dimension in the organization of number
space. Following this line of reasoning, it has also
been argued that vertical spatial associations
provide the conceptual basis for the development
of any numerical knowledge (Fischer & Brugger,
2011). The presented motion arithmetic
compatibility effect and more robust effect of verti-
cal spatial information can therefore also be con-
ceived as support for the recently discussed
general hypothesis that sensorimotor experiences
shape the development of any kind of conceptual
knowledge (Barsalou, 2008).
One fundamental question concerning the
interference between spatial processing and
mental arithmetic is the direction of the influence.
Importantly, in Experiment 2 participants merely
had to follow the moving problem with their eyes.
We regard it as rather unlikely that mental arith-
metic has a strong impact on the control of eye
movements, since previous studies could show
that the control of such movements is highly auto-
matized and robust against cognitive interferences
(e.g., Kathmann, Hochrein, & Uwer, 1999;
Theeuwes, 2010). The motion–arithmetic compat-
ibility effect for visual motions therefore most likely
reflects an impact of spatial processing on mental
arithmetic. However, more research is needed to
determine the causality between space and
arithmetic.
Since problem size was controlled between
addition and subtraction trials by matching overall
numerical size (i.e., the sum of the operands and
the correct solution), it might be objected that
our findings are confounded by a difference in the
size of solutions for addition and subtraction
trials. If this was true, an interaction between
motion direction and problem size would be
expected. That is, independent of the arithmetic
operation performance should be enhanced for
large-solution trials when moving right- or
upwards compared to when moving left- or down-
wards. Similarly, small-solution problems should
be facilitated when moving left- or downwards
compared to when moving right- or upwards.
Additional analyses demonstrated that the data
provided no evidence for such an interaction
effect, which demonstrates that the effect of the
motion direction was driven by the compatibility
with the arithmetic operation.
As discussed above, vertical movements per
definition suffer from a confound of effort, since a
movement working against gravity requires more
energy than a movement going with the forces of
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gravity. This difference is, however, marginal for
smooth pursuit eye movements and not even fully
taken into account by the brain (see Collewijn
et al., 1988). We therefore argue that asymmetries
in effort do not account for the predominance of
the vertical mapping in the present study.
Nevertheless we acknowledge that in principle it
cannot be exclude that the stronger vertical
motion arithmetic compatibility effect is the result
of a “more-is-more”association (Stavy & Tirosh,
2000) between effort and numerical size, since a
difference in physically required energy for up and
downward eye movements does clearly exist.
The present finding of a motion–arithmetic
compatibility effect for horizontal and vertical
space might have implications for maths education
in elementary school or for remedial educational
programmes for dyscalculic individuals. As it has
already been shown that gestures can increase
mathematical learning (Goldin-Meadow, Cook,
& Mitchell, 2009), a systematic involvement of
both horizontal and vertical spatial mappings
might support the learning of numerical concepts
and arithmetic.
In conclusion, the present study shows that both
horizontal and vertical organizations of mental
numerical space are activated during mental arith-
metic and suggests a functional involvement of
both dimensions. Our finding of a more robust
effect of vertical space furthermore suggests a pre-
dominant role of the vertical organization of the
mental number space in mental calculations.
Original manuscript received 13 August 2013
Accepted revision received 19 December 2013
First published online 6 March 2014
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