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The Source of the Symbolic Numerical Distance and Size Effects

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Human number understanding is thought to rely on the analog number system (ANS), working according to Weber’s law. We propose an alternative account, suggesting that symbolic mathematical knowledge is based on a discrete semantic system (DSS), a representation that stores values in a semantic network, similar to the mental lexicon or to a conceptual network. Here, focusing on the phenomena of numerical distance and size effects in comparison tasks, first we discuss how a DSS model could explain these numerical effects. Second, we demonstrate that the DSS model can give quantitatively as appropriate a description of the effects as the ANS model. Finally, we show that symbolic numerical size effect is mainly influenced by the frequency of the symbols, and not by the ratios of their values. This last result suggests that numerical distance and size effects cannot be caused by the ANS, while the DSS model might be the alternative approach that can explain the frequency-based size effect.
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fpsyg-07-01795 November 17, 2016 Time: 15:41 # 1
ORIGINAL RESEARCH
published: 21 November 2016
doi: 10.3389/fpsyg.2016.01795
Edited by:
Roberta Sellaro,
Leiden University, Netherlands
Reviewed by:
Thomas J. Faulkenberry,
Tarleton State University, USA
Naama Katzin,
Ben-Gurion University of the Negev,
Israel
*Correspondence:
Attila Krajcsi
krajcsi@gmail.com
Specialty section:
This article was submitted to
Cognition,
a section of the journal
Frontiers in Psychology
Received: 18 July 2016
Accepted: 31 October 2016
Published: 21 November 2016
Citation:
Krajcsi A, Lengyel G and
Kojouharova P (2016) The Source
of the Symbolic Numerical Distance
and Size Effects.
Front. Psychol. 7:1795.
doi: 10.3389/fpsyg.2016.01795
The Source of the Symbolic
Numerical Distance and Size Effects
Attila Krajcsi1*, Gábor Lengyel2and Petia Kojouharova1,3
1Department of Cognitive Psychology, Institute of Psychology, Eötvös Loránd University, Budapest, Hungary, 2Department
of Cognitive Science, Central European University, Budapest, Hungary, 3Doctoral School of Psychology, Eötvös Loránd
University, Budapest, Hungary
Human number understanding is thought to rely on the analog number system (ANS),
working according to Weber’s law. We propose an alternative account, suggesting that
symbolic mathematical knowledge is based on a discrete semantic system (DSS), a
representation that stores values in a semantic network, similar to the mental lexicon or
to a conceptual network. Here, focusing on the phenomena of numerical distance and
size effects in comparison tasks, first we discuss how a DSS model could explain these
numerical effects. Second, we demonstrate that the DSS model can give quantitatively
as appropriate a description of the effects as the ANS model. Finally, we show that
symbolic numerical size effect is mainly influenced by the frequency of the symbols, and
not by the ratios of their values. This last result suggests that numerical distance and
size effects cannot be caused by the ANS, while the DSS model might be the alternative
approach that can explain the frequency-based size effect.
Keywords: numerical cognition, numerical distance effect, numerical size effect, analog number system, discrete
semantic system
AN ALTERNATIVE TO THE ANALOG NUMBER SYSTEM
According to the current models understanding numbers is supported by an evolutionary ancient
representation shared by many species (Dehaene et al., 1998;Gallistel and Gelman, 2000;Hauser
and Spelke, 2004), the analog number system (ANS). One defining feature of the ANS is that it
works similarly to some perceptual representations in which the ratio of the stimuli’s intensity
determines the performance (Weber’s law) (Moyer and Landauer, 1967;Walsh, 2003;Cantlon et al.,
2009). Two critical phenomena supporting the ratio based performance are the distance and the
size effects: when two numbers are compared, the comparison is slower and more error prone when
the distance between the two values is smaller (distance effect) or when the two numbers are larger
(size effect), (Moyer and Landauer, 1967) (Figures 1 and 2). Thus, in the literature, the numerical
distance and size effects are considered to be the sign of an analog noisy numerical processing
system working according to Weber’s law. The distance and the size effects are observable both in
non-symbolic and symbolic number processing, reflecting that the same type of system processes
numerical information, independent of the number notations (Dehaene, 1992;Eger et al., 2003).
However, the distance and size effects in symbolic comparison can also be explained
by a different representation. Quite intuitively, one might think that symbolic and abstract
mathematical concepts, like numbers could be handled by a discrete semantic system (DSS), similar
to conceptual networks or to the mental lexicon, i.e., representations that process symbolic and
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FIGURE 1 | The sources of the distance and size effects according to the two models.
FIGURE 2 | (A) Reaction time (RT) function for the ANS model (based on Crossman, 1955;Moyer and Landauer, 1967) (left) and a hypothetical RT function for the
DSS model where the reaction time is proportional to a combination of the specific forms of the distance and the frequencies of the numbers (right). Notations: large:
larger number; distance: distance between the two numbers; x1and x2: the two numbers; a, a1, a2and b are free parameters. (B) The prediction of the models on a
full stimulus space in a number comparison task of numbers between 1 and 9. Numbers 1 and 2 are the two values to be compared. Green denotes fast responses,
red denotes slow responses (note that numerically the ANS function increases, and the DSS function decreases toward the high ratio, but the direction is irrelevant in
the linear fit below). The distance effect can be seen as the gradual change when getting farther from the top–left bottom–right diagonal, and the size effect is seen
as the gradual change from top–left to bottom–right. In the figures the parameters aand a2are set to 1, a1is 0.4, and parameter bis set to 0.
abstract concepts. In this DSS model, numbers are stored in
a network of nodes, and the strength of their connections is
proportional to the strength of their semantic relations. We
propose that this DSS account could be responsible for symbolic
number processing; whereas non-symbolic number processing
is still supported by the ANS (see some additional details about
the relation of the two models below). The main aim of the
present study is to investigate the feasibility of the DSS model as
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Krajcsi et al. The Source of the Symbolic Numerical Effects
a comprehensive explanation of the symbolic numerical effects,
and to contrast it with the ANS model.
DSS Explanation for the Distance and
Size Effects
How can a DSS explain the symbolic numerical distance and
size effects? (1) Regarding the distance effect, the strength of the
connections between the nodes can produce an effect which is
proportional to their strength, and since in a network storing
numbers the strength of the connections is proportional to
the numerical values and numerical distance, this system could
produce a numerical distance effect. In fact, a similar semantic
distance effect was shown in a picture naming task (Vigliocco
et al., 2002): Naming time slowed down when the picture of
the previous trial was semantically related to the present picture,
and a small semantic distance between the previous and the
actual word caused stronger effect than a large semantic distance,
similar to the numerical distance effect1. This semantic distance
effect cannot be the result of a continuous representation similar
to the ANS, because the stimuli were categorical (e.g., finger,
car, smile, etc.)2. Thus, a discrete representation potentially can
produce a numerical distance effect. Several mechanisms can
be imagined how a numerical distance effect is generated. One
can imagine that the semantic distance information, that can
be revealed in a semantic priming, could generate a distance
effect. Alternatively, it is possible that the strength of the
association between the numbers and the large–small categories
create the numerical distance effect (Verguts and Fias, 2004;
Verguts et al., 2005). Here, we do not want to specify the
exact mechanism behind the numerical distance effect, but only
propose that several possible mechanisms are already available in
the literature. (2) Turning to the size effect, this effect also could
be generated by a DSS. It is known that smaller numbers are more
frequent than larger numbers, and the frequency of a number
is proportional to the power of its value (Dehaene and Mehler,
1992). Since the numbers observed more frequently could be
processed faster, the size effect could result from this frequency
pattern3. Thus, the DSS model can also explain the appearance of
distance and the size effects (Figure 1).
1Comparison distance effect (e.g., which of two numbers is larger) and
priming distance effect (whether previous stimulus influences the actual stimulus
processing based on the distance of the two stimuli) are known to be two different
mechanisms (Verguts et al., 2005;Reynvoet et al., 2009). While we want to find
a DSS explanation for the comparison distance effect, the cited semantic distance
effect is more similar to a priming distance effect. Importantly, we are not stating
that these two effects are the same, but we suggest that a distance-based effect is
possible in a DSS, independent of the exact mechanism behind that effect.
2A similar proposal is that the numerical distance effect might emerge from
the order property of numbers, and a distance effect can be observed not
only in numbers, but also in non-numerical orders, e.g., days or letters (Potts,
1972;Verguts and Van Opstal, 2014). However, (a) it might be possible that
in those examples the non-numerical orders are transformed to the numerical
representation, which is not possible for the categorical words in the cited picture
naming task (Vigliocco et al., 2002), and (b) the DSS model has less strict
constrains, i.e., no order structure is presupposed, but a more general series of
associations is sufficient to explain the distance effect.
3Frequency is essential in other numerical tasks to produce size effect (Zbrodoff
and Logan, 2005), and the role of frequency in size effect was also proposed in
other alternative models of number comparison (Verguts and Van Opstal, 2014).
DSS Explanation for Other Numerical
Effects
Whereas in the present work we focus on the DSS explanation of
the distance and size effects, the DSS explanation can be readily
extended to other effects, too, and it can be a comprehensive
model of symbolic number processing. The following details
can demonstrate that despite its radical difference from the
ANS model, DSS might be a viable option to explain symbolic
numerical phenomena. Many of these explanations have already
been proposed in the literature, although these explanations
usually focused on single specific phenomena, and they did not
offer a comprehensive model.
Several interference effects can be explained in the DSS
framework. For example, the SNARC effect (interference between
numerical value and response location in a task) was originally
interpreted as the interference of the ANS’s spatial property and
the response locations (Dehaene et al., 1993), however, it is also
possible that the effect is the result of the interference of the
left-right and large-small nodes in a semantic network similar
to the DSS (Proctor and Cho, 2006;Leth-Steensen et al., 2011;
Patro et al., 2014; Krajcsi et al., unpublished). Similarly, while
the size congruency effect (Stroop-like interference between the
numerical value and the physical size of symbols; Henik and
Tzelgov, 1982) can be thought of as an interference between the
ANS and a representationally similar analog size representation,
it can also be thought of as an interference between the many-few
and the physically large-physically small nodes.
While there are many empirical and theoretical works in the
literature that support the ANS model, in fact there are only a
handful effects that are cited to support the ANS model, and we
propose that most of these effects (in fact to our knowledge all
of them at the moment) can also be explained by the DSS. While
mostly it would not be too difficult to find DSS explanations for
different phenomena, in the present work we only focus on the
numerical distance and size effects in comparison tasks.
Different Representations for Symbolic
and Non-symbolic Numbers
As it was mentioned above, the DSS model can only account
for symbolic number processing. Clearly, there are cases when
the DSS cannot handle numerical information, for example,
when the symbolic mental tools are not available, like in the
case of infants (Feigenson et al., 2004), animals (Hauser and
Spelke, 2004), or adults living in a culture without number words
(Gordon, 2004;Pica et al., 2004), therefore, the ANS seems to
be a sensible model to explain these non-symbolic phenomena.
It also seems reasonable that because of their representational
structure, the two systems could be specialized for different forms
of numbers: The DSS could be responsible for the precise and
symbolic numbers, while the ANS could process the approximate
non-symbolic stimuli.
This idea of different representations for symbolic and
non-symbolic numbers is supported by the increasing number
of findings in the literature, suggesting that symbolic and
non-symbolic number processing is supported by different
representations. For example, it has been shown that
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performance of the symbolic and non-symbolic number
comparison tasks do not correlate in children (Holloway and
Ansari, 2009;Sasanguie et al., 2014), and in an fMRI study the
size of the symbolic and non-symbolic number activations did
not correlate (Lyons et al., 2015). As another example, whereas
former studies found common brain areas activated by both
symbolic and non-symbolic stimuli (Eger et al., 2003;Piazza
et al., 2004), later works with more sensitive methods found
only notation-dependent activations (Damarla and Just, 2013;
Bulthé et al., 2014, 2015). According to an extensive meta-
analysis, although it was repeatedly found that simple number
comparison task (the supposed sensitivity of the ANS) correlates
with mathematical achievement, it seems that non-symbolic
comparison correlates much less with math achievement,
than symbolic comparison (Schneider et al., 2016). In another
example, Noël and Rousselle (2011) found that whereas older
than 9- or 10-year-old children with developmental dyscalculia
(DD) perform worse in both symbolic and non-symbolic tasks
than the typically developing children; younger children with
DD perform worse than control children only in the symbolic
tasks, but not in the non-symbolic tasks. The authors concluded
that the deficit in DD can be explained in the terms of two
different representations: The deficit is more strongly related to
the symbolic number processing, and the impaired non-symbolic
performance is only the consequence of the symbolic processing
problems. See a more extensive review of similar findings in
Leibovich and Ansari (2016). All of these findings are in line with
the present proposal, suggesting that symbolic and non-symbolic
numbers are processed by different systems.
Related Models for Symbolic Number
Processing
There are former models in the literature that are potential
alternatives to the ANS model, and some of those models can
be fitted into a DSS framework, or they could be considered as
implementations of more specific aspects of the DSS account.
Verguts et al. (2005) and Verguts and Van Opstal (2014)
proposed a connectionist model describing several phenomena
of number processing and more generally several phenomena of
ordinal information processing. According to their simulations
and experiments, this model offers a superior description of
number naming, parity judgment and number comparison than
the ANS model, and their model can also explain non-numerical
order processing phenomena. Their model includes a hidden
layer representing the values of the numbers in a place-code with
a fixed width of noise. This means that the nodes of the hidden
layer represent numbers on a linear scale, and a number most
strongly activates the node mainly representing that number, but
additional activation also can be found in the neighboring nodes.
The distance these additional activations can reach to do not
depend on the source number, i.e., the noise has a fixed width.
Although the authors suggest that this model implements an
analog representation, it contradicts the ANS model, because on
a linear inner scale the size of the noise is not proportional to the
size of the number, and relatedly it could not generate ratio-based
performance. In line with this representational issue, the model
in itself cannot produce a size effect, and an uneven frequency
of numbers should be introduced to generate the numerical size
effect (Verguts and Fias, 2004;Verguts et al., 2005), questioning
whether this model can be seen as an ANS-like model. However,
we propose here that the model can be interpreted as a discrete
symbolic representation: Activation in the neighboring nodes is
not the noise of that representation but it is a spreading activation
in the hidden layer. With this alternative interpretation the model
can be seen as a specific implementation of the discrete symbolic
system when stimuli are arranged as an ordered list. Note that
in their model the comparison distance effect is not explained by
the spreading activation, but by the connection weights between
the value nodes and the response nodes (Verguts et al., 2005;
Verguts and Van Opstal, 2014). This model as a potential DSS
implementation can give a more precise description for a whole
range of phenomena, the ANS model could not account for,
thus, strengthening the DSS explanation of symbolic number
processing.
Tracking a different line, Henik and Tzelgov (1982)
investigated automatic processing of numbers with the size
congruency effect (interference between the physical size and
numerical value properties of the stimuli). Based on their results
they suggested that some basic elements (primitives) are stored
in the long term memory, e.g., integers from 1 to 9 and the
number 0 (Pinhas and Tzelgov, 2012), while other numbers
are not stored as basic elements, e.g., negative numbers and
ratios (Kallai and Tzelgov, 2009;Tzelgov et al., 2009). The basic
elements or primitives can be considered as the nodes of the DSS:
These basic elements could be the values that are stored in the
nodes of the network, while other numbers are the combination
of the primitives, somewhat similar to the relation of words
and sentences. Also, the size congruency effect can be used as a
method to find whether a number is stored as a unit in the DSS.
Possible Quantitative Descriptions of
Symbolic Comparison Performance in
the DSS Model
While the DSS model can explain why the numerical distance
and size effects appear in a comparison task, the ANS model
not only suggests that there should be numerical distance and
size effects, but it offers a quantitative description for the
performance. For example, Moyer and Landauer (1967) proposed
that the reaction time of a comparison task is proportional to
the following function: K×log (large_number/(large_number –
small_number)). (See Dehaene, 2007 for a more detailed
description of the ANS predictions for behavioral numerical
decisions.)
One of the next challenges for the DSS model is to find a
quantitative description similar to the ANS model. As in the
ANS model where the details of the model were borrowed
from psychophysics models, we borrow the details of the DSS
model from psycholinguistics and semantic network models.
Unfortunately, whereas in many cases the psychophysics models
offer quantitative descriptions of the performance (Dehaene,
2007;Kingdom and Prins, 2010), the bases of the DSS model
do not have consensual quantitative descriptions. Additionally,
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our description does not build upon a detailed working model
with specific mechanisms (e.g., as it was mentioned, there could
be different candidates that could generate the distance effect),
but a functional description of these potential effects are given
here. Thus, our quantitative proposal is unavoidably speculative,
although there are some constrains we can build upon. First,
one term of this quantitative description should depend on the
distance between the two values. Second, another term should
depend on the frequencies of the values, where the frequency of
the number is the power of that number (Dehaene and Mehler,
1992). Current theoretical considerations do not specify what
distance and size functions should be used, how the frequency
of the two numbers should be combined, and how exactly the
two terms create performance, thus these details are unavoidably
speculative at the moment, and future work can refine the
versions offered here. However, based on these few starting
points, a number of alternative versions of the DSS model can be
created, and many of them display a qualitatively similar pattern
of number comparison performance. One simple example is
displayed on Figure 2, where, as the mathematically simplest
version, the distance effect is a linear function, the frequencies
of the numbers are summed up, and the distance and size
components are added up. This DSS-motivated function creates
a qualitatively very similar pattern to the function of the ANS
model: Looking at the patterns, the two models are rather similar,
also reflected in the high correlation between the two models
(r= −0.89). Thus, one can create a hypothetical quantitative
description based on the DSS account that seemingly can explain
the comparison performance in a similar way as the ANS model4.
In the first section, so far we have introduced the DSS model,
an alternative to the ANS explanation of number processing,
where the basic building blocks of the representation are nodes
with appropriate connections. We have reasoned that the DSS
framework can be a comprehensive explanation of symbolic
number processing. While focusing on the comparison distance
and size effects, we have demonstrated that the DSS model is
capable of giving as appropriate a description of the comparison
performance as the ANS model. In the following parts we turn
to empirical tests. First, we investigate which model describes
better an Indo-Arabic comparison task. Then, we investigate a
very specific aspect of number comparison where the two models
have clearly different predictions: Whether the size effect depends
on the frequency of the numbers (predicted by the DSS model) or
on the ratio of the numbers (predicted by the ANS model).
EXPERIMENT 1 – GOODNESS OF THE
TWO QUANTITATIVE DESCRIPTION OF
THE MODELS IN INDO-ARABIC
COMPARISON
After creating a quantitative description for the DSS model, we
can contrast the two models, testing which model (Figure 2) fits
4After creating additional versions of the DSS quantitative prediction with
considering the constrains described here, we found qualitatively similar patterns.
See another example in the Methods section of Experiment 1.
better the empirical data in an Indo-Arabic number comparison
task. Although the two models strongly correlate, and the
differences between them are subtle, still, there are differences
between them, and it is possible that those differences are
detectable in a simple comparison task, supposing that the noise
is relatively low.
Methods
Participants
Twenty university students participated in the study. Pilot
studies with Indo-Arabic and new symbols (see also the second
experiment) aiming to refine the applied paradigms revealed that
the main effects to be observed can be detected reliably with
a sample size of around 20. After excluding two participants
because of a higher than 5% error rate, the sample included
18 participants (15 females, mean age 21.5 years, standard
deviation 2.8 years). All studies reported here were carried out
in accordance with the recommendations of the Department of
Cognitive Psychology ethics committee with written informed
consent from all subjects. All subjects gave written informed
consent in accordance with the Declaration of Helsinki.
Stimuli and Procedure
The participants compared Indo-Arabic number pairs. In a
trial two numbers between 1 and 9 were shown until response
and the participants chose the larger one. All possible number
pairs including numbers between 1 and 9 were shown 10
times, excluding ties, resulting in 720 trials. Presentation of the
stimuli and measurement of the responses were managed by the
PsychoPy software (Peirce, 2007).
Analysis Methods
In the analysis, we contrasted the two models with analyzing
the reaction times, the error rates, and the diffusion analysis
drift rates. (1) Reaction time analysis was used, because response
latency may be a more sensitive measurement than the error
rate, and the results are comparable with many former results,
including the seminal Moyer and Landauer (1967) paper.
However, there is no strong consensus which function could
describe the ANS model (see the applied version below). (2) Error
rate analysis was chosen, because the function describing error
rate performance is well established (Dehaene, 2007;Kingdom
and Prins, 2010), even if the measurement is not as sensitive as
the reaction time data. (3) Finally, drift rate was applied, because
diffusion analysis is thought to be more sensitive than the error
rate or the reaction time, although its parameter recover methods
could be debated. In the recent decades, the diffusion model
and related models became increasingly popular to describe
simple decision processes (Smith and Ratcliff, 2004;Ratcliff and
McKoon, 2008). In the diffusion model, decision is based on
a gradual accumulation of evidence offered by perceptual and
other systems. Decision is made when an appropriate amount
of evidence is accumulated. Reaction time and error rates partly
depend on the quality of the information (termed the drift
rate) upon which the evidence is built. Importantly for our
analysis, observed reaction time and error rate parameters can
be used to recover the drift rates (Ratcliff and Tuerlinckx, 2002;
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Wagenmakers et al., 2007). Drift rates can be more informative
than the error rate or reaction time in them, because drift
rates reveal the sensitivity of the background mechanisms more
directly (Wagenmakers et al., 2007).
Because different versions of the ANS models and the DSS
models can be proposed, multiple versions of the models were
tested, when it was necessary. For the ANS model the following
functions were used in the analysis. (1) Regarding the reaction
time analysis, although there are several considerations how to
describe the reaction time function of continuous perceptual
comparisons (Crossman, 1955;Welford, 1960;Dehaene, 2007),
it is not straightforward which version should be applied to
describe the ANS model (Kingdom and Prins, 2010). First, we
used the version used by Moyer and Landauer (1967), displayed
in Figure 2. Second, we applied the RT α1/(log(large/small))
function suggested by Crossman (1955), which function he finds
to be more superior compared to the previous function. (2) For
the error rate analysis we used the ANS model described in
Dehaene (2007, equation 10), which supposes a linear scaling in
the ANS,
pcorrect(n1,n2)=
+∞
w
o
e1
2x(r1)
w1+r22
2πw1+r2dx
where n1and n2are the two numbers to be compared, ris the
ratio of the larger and the smaller number, and wis the Weber
ratio. (3) Regarding the drift rates, in the ANS model the stored
values to be compared can be conceived as two random Gaussian
variables, and the difficulty of the comparison might depend on
the overlap of the two random variables: Larger overlap leads to
worse performance (see the detailed mathematical description
in Dehaene, 2007). It is supposed that in a comparison task
the drift rate depends purely on the overlap of the two random
variables (Palmer et al., 2005;Dehaene, 2007). According to the
current theories, drift_rate =k×task_difficulty, (Palmer et al.,
2005;Dehaene, 2007), or it could also include a power term
as a generalization, drift_rate =k×task_difficultyβ, although
the exponent is often close to 1, thus the first, proportional
model approximates the second, power model. Task difficulty
is measured as stimulus strength, which is calculated with the
distance/large_number function as suggested by Palmer et al.
(2005) for psychophysics comparison. Because in an analog
representation as the task becomes more difficult (i.e., the two
stimuli become indistinguishable) the drift rate tends to zero,
in the linear fit this means that the intercept is forced to be
zero. To summarize, the drift_rate =k×distance/large_number
function was used in the drift rate analysis fit for the ANS
model.
For the DSS model, two versions were used in the analysis.
First, the simple linear version was applied, as described in
Figure 2. Additionally, a logarithmic version of the DSS model
was also used, in which the logarithm of the two terms are used,
i.e., RT αlog(distance) +log(x11×x21). This logarithmic
version seems reasonable, because strictly speaking the distance
effect cannot be linear, since that would result in negative reaction
time or error performance for sufficiently large distances (even
if the linear version could be an appropriate approximation).
Additionally, the logarithmic distance effect is partly confirmed
by the second experiment and by the inspection of the residuals
(results not presented here).
Detecting the Distance and Size Effects
The present analysis is not relevant in contrasting the ANS
and DSS models, but in the second and third experiments the
existence of the numerical distance and size effects was tested,
and the same analysis was run in the present experiment, to be
able to use these results as a point of reference. The slopes of the
specific effects were tested (1) with multiple linear regressions,
and (2) with simple linear regressions.
Methods for multiple linear regression
Average error rates and median reaction times of the correct
responses were calculated for each number pair for each
participant. Error rates and reaction times were fitted with two
regressors for all participants: (a) distance effect (the absolute
difference of the two values), (b) size effect (the sum of the two
values). See the values of the regressors for the whole stimulus
space on Figure 3. (The end effect regressor is used only in the
second and third experiments.) This analysis gives a more stable
result compared to the more commonly applied simple linear
regression analysis (see below). The weights of the regressors
were calculated for each participant in both error rates and
reaction times, and all regressors’ values were tested against
zero.
Methods for simple linear regression
To test our data with a more commonly applied simple linear
regression, all multiple linear regression analyses were retested.
For the distance effect the trials were grouped according to
distance (absolute difference between the two numbers) for all
participants. For the size effect the trials were grouped according
to the sum of the two numbers, excluding trials with distance
larger than 3. The latter was necessary, because otherwise the
specific shape of the stimulus space and the distance effect might
cause an artifact size effect: Cells from the middle part of the size
range include more large-distance cells than cells from the end
part of the size range do. Linear slope was fitted both on the
error rates and on the reaction times for both the distance and
size effects for all participants, then the slopes were tested against
zero. Because the simple linear regression analysis gave the very
same pattern as the multiple linear regression for all experiments
of the present work, the results of this analysis are not presented
here.
Results and Discussion
Fitting the Functions of the ANS and the DSS Models
to the Reaction Times
For the reaction time analysis median reaction time of the
correct responses for each number pair and for each participant
was calculated. The mean of the participants data for all
number pairs (Figure 4) were fit linearly with the least
square method. Four models were fit to the group mean:
The Moyer and Landauer version of the ANS function,
the Crossman version of the ANS function, the linear DSS
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FIGURE 3 | Values of the three regressors applied in the multiple linear regression in the whole stimulus space.
FIGURE 4 | Error rates (left), response times in ms (middle) and drift rates (right) in the Indo-Arabic digits number comparison for the whole stimulus
space. Green denotes fast and error-free responses, red denotes slow and erroneous responses. Results show distance and size effects.
function, and the logarithm DSS function (see Methods for their
descriptions).
For the Moyer and Landauer version the data showed a
quite appropriate fit, with R2=0.884, AIC =613.8, while the
Crossman version of the ANS function fit was somewhat worse,
although similar, with R2=0.769 and AIC =663.5. Regarding
the DSS models, the fit for the linear version was R2=0.808,
AIC =652.4, and the fit for the logarithm version was R2=0.893,
and AIC =610.3.
Overall, fitting the functions of the four versions of the two
models resulted in similar AICs within the same range, therefore
no clear preference for any model can be pronounced. It seems
that either the appropriate function is not precise enough to
have a higher fit (which could be true for either the ANS or the
DSS model), and/or with the current noise of the data the subtle
differences between the models cannot be investigated. Thus,
reaction time analysis with the current functions and the available
signal-to-noise ratio could not be decisive in contrasting the ANS
and DSS model.
Fitting the Functions of the Models to the Error Rates
For the error rate analysis, the mean error rate for each number
pair and for each participant was calculated, then the average
of the participants was computed (Figure 4). To test the ANS
model, first, we looked for the Weber ratio that gives the same
mean error rate for the stimulus space used here (all possible
number pairs for numbers between 1 and 9, ties excluded) as it
was measured in our data (2.5%). The found 0.11 Weber ratio
was used to generate the predictions of the ANS model for all
cells of the stimulus space (see Methods for the function), and the
model was linearly fit to the error rate data with the least square
method. The goodness of fit was R2=0.625, AIC = −371. In
testing the DSS model, the goodness of fit for the linear version
was R2=0.505, AIC = −341, and the logarithmic DSS model
gave a goodness of fit of R2=0.667, AIC = −377.
Like in the case of the reaction time, the goodness of fit of the
ANS and the DSS models are indistinguishable in the error rates
data. This again shows that with the signal-to-noise ratio of the
present data, the two models are indistinguishable, or the DSS
model is not precise enough to show a higher fit.
Fitting the Functions of the Models to the Drift Rates
To recover the drift rates for all number pairs in the two
notations, the EZ diffusion model was applied, which can be
used when the number of trials per cells is relatively small
(Wagenmakers et al., 2007). For edge correction we used the
half trial solution (see the exact details about edge correction in
Wagenmakers et al., 2007). The scaling within-trials variability
of drift rate was set to 0.1 in line with the tradition of the
diffusion analysis literature. Drift rates for each number pair and
participant were calculated in both notations. The mean drift
rates of the participants for the full stimulus space are displayed
in Figure 4.
According to the goodness of fit of the models, the ANS model
is worse (AIC = −140.1) than the DSS model (AIC = −332.4
and AIC = −348.1 for the linear and logarithmic DSS model
versions, respectively). (Because in a linear fit with zero intercept,
the R2is much higher than in a linear fit with non-zero
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intercept (as a consequence of some of the mathematical
properties of R2), and because the ANS model uses 0 intercept,
but the DSS model does not, the R2values are not reported
here.)
Looking at the drift rates of the comparison task (Figure 4)
might reveal why the ANS model is worse than the DSS model:
While the ANS model predicts that the drift rate tends to
zero as the stimuli become indistinguishable (e.g., 8 vs. 9), the
recovered drift rates are in fact much larger, tending to the 0.2
values. This problem is analogous to a conceptual problem: How
is it possible that an imprecise representation solves a precise
comparison task? In other words, if the Weber fraction of the
ANS is around 0.11, how is it possible that small ratio number
pairs, e.g., 8 vs. 9, can still be differentiated with relatively high
precision.
Thus, in the diffusion model analysis the DSS model seems
to offer a better prediction than the ANS model, however, it
is important to note that (a) the EZ diffusion model analysis
and more generally any diffusion models have some constrains
(Wagenmakers et al., 2007), and consequently, it is possible that
in this case the recovered parameters are not entirely reliable, and
(b) task difficulty can be defined in different ways (Palmer et al.,
2005;Dehaene, 2007), and it might be debated which definition
is appropriate. Thus, while the present diffusion model analysis
reveals the advantage of the DSS model over the ANS model, the
uncertainties of the methods might question how reliable these
results are. (The methods and the models are investigated in more
details in Krajcsi et al., unpublished).
Presence of the Distance and the Size Effects
According to the multiple linear regression analysis, both
the distance and the size effects were present both in the
error rates and in the reaction times, 95% CI for the slope
was [1.16%, 0.65%], t(17) = −7.42, p<0.001 for the
distance effect in error rates, and CI of [23.6 ms, 15.5 ms],
t(17) =10.1, p<0.001 in reaction times, CI with [0.3%, 0.59%],
t(17) =6.57, p<0.001 for the size effect in error rates, and
CI with [4.8 ms, 9.1 ms], t(17) =6.78, p<0.001 in reaction
times.
Summary
First, we found that reaction time and error rate patterns in Indo-
Arabic number comparison (Figure 4) could not be decisive in
contrasting the ANS and the DSS models. Even if the two models
correlate, the correlation is not perfect, and there was a chance
that the present test could have decided. Still, with the present
models and/or signal-to-noise ratio, the test was not decisive.
On the positive side, this means that the DSS model is a viable
alternative to the ANS model, because the goodness of fit of the
DSS model is in the same range as the goodness of fit of the ANS
model. Second, we found that in a diffusion model analysis the
drift rate pattern is more in line with the DSS model than with
the ANS model, although the uncertainties about the method
may question the reliability of these results. Overall, while the
performance in the Indo-Arabic comparison task suggests that
the DSS model is a viable model, this paradigm could not decide
firmly which model is preferred. Thus, in the next experiment a
new approach is utilized in which we investigate the role of the
frequency in the size effect.
EXPERIMENT 2 – ROLE OF THE
FREQUENCY IN THE SIZE EFFECT
In a different approach, we tested whether the distance and
the size effects are strongly related as suggested by the ratio-
based ANS model, or whether the two effects can dissociate. In
the present experiment we investigated whether size effects can
dissociate from distance effect if the frequency of the symbols
is manipulated. (See another type of test for the dissociation of
the two effects in Krajcsi, 2016) To manipulate the frequency of
the symbols, it might be more appropriate to use new symbols,
instead of the well-known Indo-Arabic symbols, because the
frequency of the already known symbols might be well established
and learned.
Thus, to investigate the role of the frequency in the size
effect, participants learned new number symbols in a simple
number comparison task, and the frequency of the symbols was
manipulated in the experiment. According to the DSS model,
the size effect could be changed as a function of the symbol
frequencies (Figure 1), if the reaction time depends on the
frequency of the symbol, and not the frequency of the concept.
For example, if the distribution of the frequencies is uniform,
then according to the DSS model, the size effect should vanish.
In contrast, according to the ANS model, even with uniform
frequency distribution the size effect should be visible, because
the size effect is rooted in the ratio of the two values, independent
of the frequency (Figure 5). It is important to stress that although
according to the ANS model it might be possible that the
frequency of the symbols have an effect on the performance, the
effect should be relatively weak: Although in the ANS model the
role of the frequency is not discussed, it states that the largest
part of the performance variance should be explained by the
ratio (Moyer and Landauer, 1967;Dehaene, 2007), which means
that any other factors could have only a minor effect on the
performance.
Methods
Participants learned new symbols (Figure 6) for the numbers
between 1 and 9 to compare, while the frequency of all symbols
was manipulated in two conditions.
It is possible that the new symbols are not connected to the
numerical values they represent, and they may be processed only
as a non-numerical ordered series. This could cause a problem,
because the ANS could not process this non-numerical order5.
To ensure that the new symbols were connected to the numerical
values they represent, at the end of the experiment we used
a priming task to measure the priming distance effect (PDE)
between the newly learned symbols and familiar Indo-Arabic
digits (Figure 6). In a PDE the reaction time to the target is faster
5Note, however, that several works suggest that order processing and quantity
processing rely on the same mechanisms (Leth-Steensen and Marley, 2000;
Marshuetz et al., 2006;Verguts and Van Opstal, 2014), thus, ANS should be
activated even when the new symbols are non-numerical orders.
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FIGURE 5 | Prediction of the two models for the symbol frequency manipulation in Experiment 2. Bar charts show the frequency of the stimuli used in the
uniform distribution condition and in the Indo-Arabic-like distribution condition. (In the Indo-Arabic-like distribution the resulting performance is computed as
0.4 ×Distance +Frequency.)
when the numerical distance between the prime and the target
is smaller, reflecting a semantic relation between the prime and
the target (Koechlin et al., 1999;Reynvoet and Brysbaert, 1999;
Reynvoet et al., 2009).
Participants
Eighteen university students participated in the uniform
frequency distribution condition. After excluding 2 of them
because the error rate did not fall below 5% even after the
5th block, and excluding 2 further participants showing higher
than 5% error rates in the main comparison task, the data of
14 participants was included (11 females, mean age 20.6 years,
standard deviation 2.1 years). Fifteen university students
participated in the Indo-Arabic-like frequency distribution
condition. After excluding two participants because their error
rates were higher than 5% either in the main comparison or in
the priming comparison task, the data of 13 participants was
analyzed (13 females, mean age 24.3 years, standard deviation
6.9 years).
Stimuli and Procedure
The participants first learned new symbols for the numbers
between 1 and 9. Then, in a comparison task they decided
which number is larger in a simultaneously presented symbol
pair. Finally, in a priming comparison task the participants
decided whether one-digit numbers are smaller or larger than 5
(Figure 6).
New symbols were introduced to represent values between
1 and 9. The new symbols were chosen from writing
systems that were mostly unknown to the participants,
and the characters had similar vertical and horizontal size.
The symbols were randomly assigned to values for all
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FIGURE 6 | Tasks in the new symbol experiments.
participants, i.e., the same symbol could mean a different
value for different participants, from the following characters:
.
To ensure that the participants have learned the symbols, in
the symbol learning phase, the symbols were practiced until a
threshold hit rate was reached. In a trial of the new symbol
learning phase a new symbol and an Indo-Arabic digit were
shown simultaneously, and the participant decided whether
the two symbols denote the same value. The stimuli were
visible until response. After the response, auditory feedback
was given. In a block all symbols were presented 10 times (90
trials in a block) in a randomized order. In half of the trials
the symbols denoted the same values. The symbol learning
phase ended if the error rate in a finished block was smaller
than 5%, or the participant could not reach that level in five
blocks.
In the main comparison task, the same procedure was
used as in the first experiment, but here the numbers were
denoted with the new symbols. In the uniform frequency
distribution condition the number of the presentation of a
digit were the same as in the first experiment (all possible
number pairs were shown 10 times). In the Indo-Arabic-
like frequency distribution condition the frequencies of the
specific values followed the frequencies of the numbers in
everyday life (Dehaene and Mehler, 1992), specified with the
following formula: frequencyvalue =value1×10. This formula
generated the following frequencies (value:frequency): 1:10, 2:5,
3:4, 4:3, 5:2, 6:2, 7:2, 8:2, 9:1 (Figure 5). The 2-permutations of
these numbers excluding ties were presented, resulting in 794
trials.
In the priming comparison task in odd (prime) trials a new
symbol was visible, and the participant decided whether it was
smaller or larger than 5. Two hundred ms after the response
in an even (target) trial an Indo-Arabic digit was shown, and
the participant decided whether it was smaller or larger than
5. Two thousand ms after the response a new odd (prime)
trial was shown. The stimuli were visible until response. The
instruction included the value of 5 in both notations: For even
trials Indo-Arabic notation (“5”), for odd trials the new notation
(e.g., “ ”) was used. All possible new symbols were presented
with all possible Indo-Arabic digits three times, resulting in 192
trials.
Results and Discussion
To summarize the main results, in the uniform distribution
comparison task the distance effect was present, but the size
effect was not (Figure 7A). This result is in line with the DSS
model, but not with the ANS model. On the other hand, in
the Indo-Arabic-like, biased frequency comparison task both
the distance and the size effects were visible (Figure 7B) in a
similar pattern as observable in Indo-Arabic number comparison
(Figure 4), suggesting that it is the frequency manipulation that
is responsible for the size effect.
Distance and Size Effects in the Uniform Frequency
Distribution
The same analysis methods were applied as in the first experiment
with two exceptions. Descriptive data clearly shows an end
effect (Leth-Steensen and Marley, 2000). Thus, an end effect
regressor was also included in the multiple linear regressions
(Figure 3) with a value of 1 if any of the presented numbers
were 9, 0.5 if any of the numbers were 8 or 1, and 0
otherwise. These values were specified with first calculating
the average reaction time for all presented numbers, then the
distance effect (distance from 5) of the middle number range (i.e.,
without end effect) was extrapolated, and finally, the deviation
from this extrapolation at the end of the number range was
estimated.
In the multiple linear regression the slope of the distance
effect deviated from zero, 95% CI was [1.04%, 0.48%],
t(13) = −5.84, p<0.001 for error rates, and CI was [73.6 ms,
26.1 ms], t(13) = −4.53, p=0.001 for reaction time. On the
other hand, the slope of the size effect did not differ from zero, CI
with [0.15%, 0.06%], t(13) = −0.933, p=0.368 for error rates,
and CI with [26.6 ms, 13.9 ms], t(13) = −0.679, p=0.509 for
reaction time. The end effect was present for the reaction time, CI
of [430.6 ms, 147.6 ms], t(13) = −4.41, p=0.001, and more
unstably for the error rates, CI with [0.23%, 2%], t(13) =1.71,
p=0.111.
These results also demonstrated an end effect (the most
extreme values in the set are easier to respond than other
values) (Leth-Steensen and Marley, 2000), however, while the
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FIGURE 7 | Error rates (left) and response times in ms (right) in the new symbol number comparison for the whole stimulus space. Green denotes fast
and error-free responses, red denotes slow and erroneous responses. (A) Equal frequencies condition, showing distance and end effects. (B) Biased frequencies
condition, showing distance, size and end effects.
end effect can be in line with the DSS model (Leth-Steensen and
Marley, 2000), it is also possible that the effect is irrelevant in
the description of the representation processing the numerical
values (Balakrishnan and Ashby, 1991;Piazza et al., 2003),
consequently, the presence of this effect is not decisive in the
present question.
Distance and Size Effects in the Indo-Arabic-Like
Frequency Distribution
The slope of the distance effect differed from zero in both the
error rates, CI with [1.56%, 0.5%], t(12) = −4.25, p=0.001,
and in reaction times, [55.7 ms, 28.9 ms], t(12) = −6.87,
p<0.001. The non-zero slope of the size effect was also
observable, [0.20%, 0.68%], t(12) =3.99, p=0.002 for the
error rate, and CI with [28.4 ms, 50.2 ms], t(12) =7.85,
p<0.001 for the reaction time. Additionally, the end effect
was observable in the reaction times, CI with [622.5 ms,
294.9 ms], t(12) = −6.1, p<0.001, but not in the error rates,
CI with [2.76%, 0.7%], t(12) = −1.3, p=0.217.
We tested directly whether the size effects of the two
frequency conditions differed. The size effect slopes between
the uniform frequency distribution and the Indo-Arabic-like
frequency distribution conditions differed significantly in both
the error rates, U=13, p<0.001, and in the reaction times,
U=15, p<0.001.
Priming Distance Effect
In this analysis the error rates and median reaction
times of the correct responses of the target Indo-Arabic
numbers were analyzed as a function of the prime (new
symbols) – target (Indo-Arabic digit) distance (Figure 8).
Only the trials in which the response was the same for
the prime and distance (i.e., both numbers were smaller
than 5, or both numbers were larger than 5) were analyzed
(Koechlin et al., 1999;Reynvoet and Brysbaert, 1999;
Reynvoet et al., 2009). Linear slope was calculated for the
PDE.
In the uniform frequency distribution the data of one
participant was not recorded due to technical problems. Because
in the symbol learning task participants practiced the new
symbol – Indo-Arabic pairs, the zero distance pairs could
have this extra practice gain, and not depend purely on the
semantic priming effect. Thus, the 0 distance pairs were not
included in the analysis. While the descriptive data showed
increasing priming effect with smaller distance (Figure 8), the
effect was not significant: In the uniform frequency condition
CI is [1.62%, 2.69%], t(12) =0.54, p=0.599 for the error
rate, and CI is [1.4 ms, 39.7 ms], t(12) =2.03, p=0.065 for
the reaction time, and in the Indo-Arabic frequency condition
CI is [0.13%, 1.63%], t(12) =1.85, p=0.089 for the error
rates, and CI is [8.9 ms, 27.2 ms], t(12) =1.11, p=0.290
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FIGURE 8 | Prime distance effect (PDE) measured in error rates (bars) and reaction time (lines), in equal frequency condition (left) and in biased
frequency condition (right) in Experiment 2. Error bars represent 95% confidence interval.
for the reaction time. The lack of significance could mean the
lack of PDE, or it could reflect the lack of statistical power, or
both. Looking at the gradual increase of error rate and reaction
time as the function of priming distance (Figure 8) and the
biased CIs, it seems more probable that the PDE could be
statistically significant with larger statistical power. To extend
the reasoning that the lack of the significance might be the
result of insufficient statistical power, we also analyzed three
unpublished similar experiments conducted in our laboratory,
where in the same design new symbols were learned with the
same stimuli and procedure as in the present works (in the
third unpublished experiment the learning and the comparison
were repeated for 5 days). In those experiments the PDE
was measured with similar sample sizes as in the experiments
presented here. We found that in all cases the confidence
interval was biased to the direction the PDE predicts, although
mostly it was only close to be significant. In the first two
unpublished experiments 95% CI is [6.2 ms, 17.7 ms], N=12,
p=0.312, and [29.1 ms, 75.1 ms], N=10, p<0.001. In
the third unpublished experiment the PDE was measured for
5 consecutive days which is especially informative about the
consistency and fluctuation of the PDE in a relatively small
sample: 95% CI [10.25 ms, 34.54 ms], N=13, p=0.002,
[2.09 ms, 28.21 ms], p=0.085, [1.05 ms, 14.39 ms],
p=0.084, [9.34 ms, 8.13 ms], p=0.882, [0.49 ms, 13.24 ms],
p=0.066, for the five days, respectively. A meta-analysis on
the five available experiments (second and third experiments
of the present paper and three unpublished experiments; only
day 1 was used from the last unpublished experiment; meta-
analysis of means in original units with random effect) revealed
95% CI [6.7 ms, 34.3 ms], p=0.004 (Cumming, 2013). The
analysis also confirms that the effect size would require much
larger sample to have a significant result reliably in a single
experiment: The estimated effect size could be as small as
d=0.3 (with around 25 ms standard deviation), which would
require a magnitude of 100 participants to reach 95% statistical
power (Faul et al., 2007). Taken together, based on (a) the
expected gradual pattern of the PDE (Figure 8), and (b) the
consistently biased CIs across experiments, (c) confirmed with
the meta-analysis, it is most reasonable to conclude that the
PDE is present, even if our usual sample size around 15 does
not guarantee the preferred 95% statistical power for a single
experiment.
Effect of the Frequency
To further demonstrate the effect of the frequency (because it
cannot be observed readily on Figure 7), the mean reaction
time was calculated for all cells that include a specific value
in both conditions (right of Figure 9). The reaction time
changes in line with the frequencies of the values: The more
frequent a number is in one condition compared to the other
condition (left of Figure 9), the faster it is to process (right
of Figure 9). In other words, the differences of the two
conditions for the values in the reaction time data are inversely
proportional to the differences of the two conditions for the
values in the frequency. Note that the reaction time data do
not include purely the frequency effect, because (a) middle
values are gradually slower to process because of the interaction
of the distance effect and the shape of the stimulus space,
and (b) end values are faster to process because of the end
effect.
Summary
In the second experiment the numerical distance and size
effects dissociated. More specifically, the numerical size effect
was missing when the frequency distribution was uniform,
and the size effect was present with the biased frequencies
of symbols, suggesting that size effect was guided by the
frequencies of the symbols. These results cannot be explained
by the ANS model, whereas they can be in line with the DSS
model. We highlight again that according to the ANS model
although the frequency might slightly modulate the performance,
it cannot change a large proportion of the variance in the
performance. However, the present result reveal that largest part
of the variance of the size effect is directed by the frequency,
while the ratio has no observable effect (as revealed by the
statistical lack of the size effect), contradicting the ANS model
prediction.
Results also show that the new numbers semantically primed
the Indo-Arabic digits as revealed by the PDE, demonstrating that
the new symbols were connected to the values they represent.
Thus, the lack of the size effect in the second experiment cannot
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FIGURE 9 | Frequencies of the specific values (left) and response latencies for those values (right) in Experiment 2.
be the result of potentially non-numeric new symbols which
could not be processed by the ANS.
EXPERIMENT 3 – ROLE OF THE
SEMANTIC CONGRUENCY EFFECT IN
THE SIZE EFFECT
As another potential confound, it is possible that in the
second experiment there was a size effect in the uniform
distribution condition, however, a semantic congruency effect
(SCE) extinguished it. According to the SCE, large numbers are
responded to faster than small numbers when the task is to
choose the larger number, resulting in a reversed size-like effect,
and small numbers are faster to decide on when the smaller
number should be chosen, resulting in a regular size-like effect
(Leth-Steensen and Marley, 2000). If the SCE was present in the
second experiment, this anti-size effect could have extinguished a
potentially existing size effect. To test this possibility, the uniform
frequency condition of the second experiment was rerun, but this
time the participants had to choose the smaller number. If the
SCE was present in the second experiment as a reversed size-
like effect, then it should be observed in the present experiment
as a regular size-like effect, increasing the size effect. However,
the size effect was not present in this control experiment,
demonstrating that the SCE did not mask a potentially existing
size effect.
Methods
The methods of the second experiment was applied, however,
participants had to choose the smaller number, not the larger, in
the comparison task. The priming comparison task was not run.
Eighteen university students participated in the study. Two
participants were excluded, because their error rates were higher
than 5% after the 5th learning block, and two participants were
excluded because they had higher than 5% error rate in the
comparison task. The data of 14 participants were analyzed, 10
females, with mean age of 25.4 years, and standard deviation
of 6.9.
Results
Distance and Size Effects
In the multiple linear regression analysis the distance effect
was present in both the error rate and the reaction time, CI
[1.54%, 0.48%], t(13) = −4.13, p=0.001, and CI [77.0 ms,
39.6 ms], t(13) =6.73, p<0.001, respectively. More critically,
the size effect was not observable neither in the error rate
nor in the reaction time, CI [0.18%, 0.15%], t(13) = −0.184,
p=0.857, and CI [25.0 ms, 26.2 ms], t(13) =0.0482,
p=0.962, respectively. Comparing the slopes of the uniform
frequency condition of the second and the present experiments,
the slopes of the size effects did not differ significantly, neither
in error rate nor in reaction time, t(26) =0.33, p=0.744,
and U=91.5, p=0.783, respectively. Thus, choosing the
smaller number did not change the size effect, consequently, the
SCE did not influence essentially the size effect in the second
experiment.
GENERAL DISCUSSION
We introduced the DSS model as a comprehensive alternative
account to the ANS model to explain symbolic number
processing. First, we have shown that the DSS model can explain
many symbolic numerical effects, and we demonstrated that
the DSS model could give a similar quantitative prediction for
symbolic number comparison performance as the ANS model.
Second, we tried to contrast the two models in Indo-Arabic
comparison task. However, because of the relatively high noise
and the uncertainties of the diffusion analysis method, it was
not possible to find a straightforward preference for any models.
On the other hand this result could show that the DSS model
prediction empirically fits the Indo-Arabic number comparison
as good as the ANS model prediction. Finally, the second and
third experiments revealed that in new symbol comparison tasks
the numerical size effect is the consequence of the frequency
manipulations of the symbols, as proposed by the DSS model,
and not the consequence of the ratios of the values, as predicted
by the ANS model. These data also show that the numerical
distance and size effects are not straightforward signs of the
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ANS, because an alternative mechanism could produce them as
well.
While the second and the third experiments utilized new
symbols, it is possible to extend our conclusion about other
symbolic number comparisons, for example, the Indo-Arabic
number comparison. Because all known numerical effects that
were observable in the new symbol comparison show the very
same pattern as in Indo-Arabic comparison (i.e., distance effect,
size effect and PDE), it is parsimonious to suppose that the same
mechanisms work behind new symbol comparison and Indo-
Arabic comparison, and our findings can also be generalized to
the Indo-Arabic and other symbolic number processing, unless
additional data show the opposite.
We argue that the ANS model is not in line with our results.
While one can try to modify the ANS model to align with the
present result, ratio-based performance is a defining feature of
the ANS, and changing that feature leads not only to a modified
ANS model, but to a completely new model. Additionally, adding
frequency effect to the ANS model cannot modify it to explain the
frequency-based size effect, because the ANS critically suggests
that the performance should mainly be driven by the ratio, which
ratio effect in fact was statistically invisible in the second and third
experiments.
We argue that the ANS model is not in line with the
present results, and the DSS can be an appropriate alternative.
However, one might question how strongly our results support
a DSS model. Obviously, one can only tell if a model is
in line with the empirical results, and whether the model is
coherent. We argue that the DSS is in line with the present and
previous results (e.g., it can explain the independent distance
and size effects, why symbolic and non-symbolic comparisons
are relatively independent, or how arbitrarily precise comparison
can be made), and it is a coherent model. Additionally, based
on current cognitive models, it is reasonable to suppose that
abstract symbolic operations are processed by a system that
is otherwise known to be used for other symbolic operations,
such as the mental lexicon or a conceptual network. On the
other hand, no one can exclude that an alternative, third
model could account for these results, and not the ANS or
the DSS models. Further research can tell whether the DSS
framework is an appropriate explanation for the symbolic
number processing or another alternative should be found.
Furthermore, it is possible that it is not a single representation
that is responsible for the discussed effects, but cooperation of
several representations is required, and although the ANS cannot
explain the distance effect in comparison task, still there could
be other symbolic numerical phenomena that could be rooted
in the ANS. Additional works can find out whether such a
partial role can be attributed to the ANS in symbolic number
processing.
The DSS model in its current form relies on models
about mental lexicon or conceptual networks. These starting
points could offer many properties of the models, while at
the same time, many other details are seemingly missing,
e.g., the exact quantitative description of the comparison
performance. While these shortcomings might make the
impression that the DSS model is less detailed than the ANS
model, these differences are the consequence of changing
the base of the explanations. While the ANS model is a
low-level perceptual model in its nature, the DSS model
is more like a linguistic or conceptual network model.
Models describing higher level functions are usually less
quantitative than models describing lower level functions, partly
because of methodological reasons, and from this viewpoint
it seems reasonable that a DSS model is less quantitative
than an ANS model. However, from a different—and more
relevant—viewpoint, the DSS model is as efficient as the
ANS model, because seemingly all relevant symbolic numerical
effects and phenomena can also be explained in the DSS
model, and a few examples can already be found where
the DSS model can give a better explanation than the ANS
model.
The ANS model is a widely accepted and deeply grounded
explanation for number processing. However, despite the
huge amount of papers discussing and supporting the
ANS view, they are relying on surprisingly few effects and
findings that demonstrate an ANS activation. In fact, the
few phenomena can also be explained in the alternative
DSS model as well. Additionally and more importantly, an
increasing number of findings are not in line with the ANS
model. For example, symbolic and non-symbolic performance
seems to be independent on many behavioral (Holloway
and Ansari, 2009;Sasanguie et al., 2014;Schneider et al.,
2016) and neural level (Damarla and Just, 2013;Bulthé
et al., 2014, 2015;Lyons et al., 2015). In a correlational
study it has been shown that distance and size effects
dissociate in Indo-Arabic comparison task (Krajcsi, 2016).
Some results show that the numerical representation is not
analog: Functional activation in the brain while processing
symbolic numbers seems to be discrete (Lyons et al., 2015),
and symbolic numbers can also interfere with the discrete
yes-no responses (Landy et al., 2008). The present finding
showing the frequency dependence of the size effect also
extends the list of results contradicting the ANS model. Future
research can tell whether the ANS can be reformulated to
account for these findings, or an alternative model, such
as the DSS, can characterize symbolic number processing
better.
AUTHOR CONTRIBUTIONS
All authors listed, have made substantial, direct and
intellectual contribution to the work, and approved it for
publication.
ACKNOWLEDGMENTS
We thank Réka Eszter Börcsök, Dávid Czikora, József Fiser,
Ildikó Király, and Krisztina Peres for their comments
on an earlier version of the manuscript. The study was
supported by the Institute of Psychology, Eötvös Loránd
University.
Frontiers in Psychology | www.frontiersin.org 14 November 2016 | Volume 7 | Article 1795
fpsyg-07-01795 November 17, 2016 Time: 15:41 # 15
Krajcsi et al. The Source of the Symbolic Numerical Effects
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... Among the most frequently cited evidence to support the notion that symbols are fundamentally linked to quantities is the finding that human adults produce a 'distance effect' when making comparative judgments of both symbolic and nonsymbolic numerical magnitudes (e.g., Dehaene et al., 1998;Holloway & Ansari, 2008Krajcsi et al., 2016;Moyer & Landauer, 1967;Pavese & Umiltà, 1998;van Opstal & Verguts, 2011). The distance effect is the highly replicable finding that humans are faster and more accurate at judging which of two numerical magnitudes is numerically greater when those magnitudes are numerically far apart, rather than close together (Moyer & Landauer, 1967). ...
... The distance effect is the highly replicable finding that humans are faster and more accurate at judging which of two numerical magnitudes is numerically greater when those magnitudes are numerically far apart, rather than close together (Moyer & Landauer, 1967). There have been many reports of similar distance effects during the processing of symbols and quantities that have been replicated across many studies (Buckley & Gillman, 1974;Holloway & Ansari, 2008;Krajcsi et al., 2016;Moyer & Landauer, 1967) and have been taken as evidence that symbols and quantities are represented using a shared analogue magnitude system (Dehaene, 2007;Dehaene et al., 1998). Numerical distance has been shown to influence the processing of numerical magnitudes when the symbol or quantity is the relevant dimension (Buckley & Gillman, 1974;Holloway & Ansari, 2009;Moyer & Landauer, 1967) and the irrelevant dimension (Henik & Tzelgov, 1982;Pavese & Umiltà, 1998, 1999. ...
... Notably, there are other explanations for the differences between the processing of symbolic and nonsymbolic numerical magnitudes. Converging recent behavioural data has indicated that the similar behavioural effects observed in different formats of numerical magnitudes (i.e., symbolic and nonsymbolic) do not correlate with each other (Holloway & Ansari, 2009;Krajcsi et al., 2016;Lyons et al., 2015), and may, in fact, be supported by two similar, but distinct representational systems. Indeed, while nonsymbolic numerical magnitudes are likely processed using an evolutionarily ancient analogue magnitude system, where the ratio of the stimuli's intensity affects performance (Weber's law) (Moyer & Landauer, 1967) the processing of symbols is likely supported by a different more exact system. ...
Article
Full-text available
Are number symbols (e.g., 3) and numerically equivalent quantities (e.g., •••) processed similarly or distinctly? If symbols and quantities are processed similarly then processing one format should activate the processing of the other. To experimentally probe this prediction, we assessed the processing of symbols and quantities using a Stroop-like paradigm. Participants (NStudy1 = 80, NStudy2 = 63) compared adjacent arrays of symbols (e.g., 4444 vs 333) and were instructed to indicate the side containing either the greater quantity of symbols (nonsymbolic task) or the numerically larger symbol (symbolic task). The tasks included congruent trials, where the greater symbol and quantity appeared on the same side (e.g. 333 vs. 4444), incongruent trials, where the greater symbol and quantity appeared on opposite sides (e.g. 3333 vs. 444), and neutral trials, where the irrelevant dimension was the same across both sides (e.g. 3333 vs. 333 for nonsymbolic; 333 vs. 444 for symbolic). The numerical distance between stimuli was systematically varied, and quantities in the subitizing and counting range were analyzed together and independently. Participants were more efficient comparing symbols and ignoring quantities, than comparing quantities and ignoring symbols. Similarly, while both symbols and quantities influenced each other as the irrelevant dimension, symbols influenced the processing of quantities more than quantities influenced the processing of symbols, especially for quantities in the counting rage. Additionally, symbols were less influenced by numerical distance than quantities, when acting as the relevant and irrelevant dimension. These findings suggest that symbols are processed differently and more automatically than quantities.
... While the ANS model is an elegant and parsimonious solution offering explanations for a series of phenomena, in the present section we outline an alternative explanation that supposes an entirely different architecture behind the same symbolic number processing phenomena (Krajcsi et al., 2016). ...
... Based on the idea of this potential alternative generator, our research group proposed a comprehensive alternative account for symbolic number processing that tries to account for all the D i s t a n c e S i z e phenomena the ANS accounts for (Krajcsi et al., 2016). Importantly, this alternative explanation deals only with symbolic numbers but not with nonsymbolic numerosities. ...
... Adding up the two effects results in a very similar pattern seen in the ANS model (see the distance and size effect components and their sum in Figure 4, and contrast this DSS description of the effect with the ANS description displayed in the left panel of Figure 2). (See more details in Krajcsi et al., 2016 about how hypothetical quantitative descriptions of the DSS could be formed.) The similarity of the DSS prediction and the ANS prediction can be captured in several ways. ...
Chapter
According to the dominant view in the literature, several numerical cognition phenomena are explained coherently and parsimoniously by the Approximate Number System (ANS) model, which supposes the existence of an evolutionarily old, simple representation behind many numerical tasks. We offer an alternative account that proposes that only nonsymbolic numbers are processed by the ANS, while symbolic numbers, which are more essential to human mathematical capabilities, are processed by the Discrete Semantic System (DSS). In the DSS, symbolic numbers are stored in a network of nodes, similar to conceptual or linguistic networks. The benefit of the DSS model and the benefit of the more general hybrid ANS–DSS framework are demonstrated using the crucial example of the distance and size effects of comparison tasks.
... Amongst the most frequently cited evidence to support the notion that symbols are fundamentally linked to quantities is the finding that human adults produce a 'distance effect' when making comparative judgements of both symbolic and nonsymbolic numerical magnitudes (e.g., Dehaene, Dehaene-Lambertz, & Cohen, 1998;Holloway & Ansari, 2008Krajcsi, Lengyel, & Kojouharova, 2016;Moyer & Landauer, 1967;Pavese & Umiltà, 1998;van Opstal & Verguts, 2011). The distance effect is the highly replicable finding that humans are faster and more accurate at judging which of two numerical magnitudes is numerically greater when those magnitudes are numerically close together, rather than far apart (Moyer & Landauer, 1967). ...
... There have been many reports of similar distance effects during the processing of symbols and quantities that have been replicated across many studies (Buckley & Gillman, 1974;Holloway & Ansari, 2008;Holloway, Price, & Ansari, 2010;Krajcsi, Lengyel, & Kojouharova, 2016;Moyer & Landauer, 1967) and taken as evidence that symbols and quantities are represented using a shared analogue magnitude system (Dehaene, 2007;Dehaene et al., 1998). Numerical distance has been shown to influence the processing of numerical magnitudes when the symbol or quantities is the relevant dimension (Buckley & Gillman, 1974;Holloway & Ansari, 2009;Moyer & Landauer, 1967) and the irrelevant dimension Pavese & Umiltà, 1998, 1999. ...
... Notably, there are other explanations for the differences between the processing of symbolic and nonsymbolic numerical magnitudes. Converging recent behavioural data has indicated that the similar behavioural effects observed in different formats of numerical magnitudes (i.e., symbolic and nonsymbolic) do not correlate with each other (Holloway & Ansari, 2009;Krajcsi et al., 2016;Lyons, Nuerk, & Ansari, 2015), and may, in fact, be supported by two similar, but distinct representational systems. Indeed, while nonsymbolic numerical magnitudes are likely processed using an evolutionarily ancient analogue magnitude system, where the ratio of the stimuli's intensity affects performance (Weber's law) (Moyer & Landauer, 1967) the processing of symbols is likely supported by a different more exact system. ...
Preprint
Are number symbols (e.g., 3) and numerically equivalent quantities (e.g., •••) processed similarly or distinctly? If symbols and quantities are processed similarly then processing one format should activate the processing of the other. To experimentally probe this prediction, we assessed the processing of symbols and quantities using a Stroop-like paradigm. Participants (NStudy1 = 80, NStudy2 = 63) compared adjacent arrays of symbols (e.g., 4444 vs 333) and were instructed to indicate the side containing either the greater quantity of symbols (nonsymbolic task) or the numerically larger symbol (symbolic task). The tasks included congruent trials, where the greater symbol and quantity appeared on the same side (e.g. 333 vs. 4444), incongruent trials, where the greater symbol and quantity appeared on opposite sides (e.g. 3333 vs. 444), and neutral trials, where the irrelevant dimension was the same across both sides (e.g. 3333 vs. 333 for nonsymbolic; 333 vs. 444 for symbolic). The numerical distance between stimuli was systematically varied, and quantities in the subitizing and counting range were analyzed together and independently. Participants were more efficient comparing symbols and ignoring quantities, than comparing quantities and ignoring symbols. Similarly, while both symbols and quantities influenced each other as the irrelevant dimension, symbols influenced the processing of quantities more than quantities influenced the processing of symbols, especially for quantities in the counting rage. Additionally, symbols were less influenced by numerical distance than quantities, when acting as the relevant and irrelevant dimension. These findings suggest that symbols are processed differently and more automatically than quantities.
... Importantly, the proposed neural coding scheme automatically entails that decisions based on these noisy representations will not be deterministic, but will probabilistically follow some distribution that originates in the specific properties of neural noise. This prediction has been confirmed many times, in the form of well-established psychophysical effects: (a) the distance effect (i.e., closer numbers are more difficult to compare than numbers that are further apart; e.g., the distance between 10 and 11 is more difficult to perceive than the distance between 10 and 20) (van Opstal and Verguts, 2011;Núñez-Peña and Suárez-Pellicioni, 2014;Dietrich et al., 2015); (b) the size effect (i.e., the same numeric difference between the two numbers is harder to perceive for larger numbers; e.g., it is easier to differentiate between 3 and 5 than to differentiate 123 and 125) Krajcsi et al., 2016); and, (c) Weber's law (the fundamental psychophysical principle of magnitude discriminability where our accuracy in comparing two numbers is determined by their difference-to-magnitude ratio) (Dehaene, 2011;Kacelnik and Brito e Abreu, 1998;Nieder, 2013). ...
... This is assumption is well-established, particularly in numerosity cognition Nieder and Merten, 2007;Miller, 2003, 2004;. In particular, larger overall magnitudes are have noisier mental representations than smaller magnitudes (Krajcsi et al., 2016;Pardo-Vazquez et al., 2019;, and noise proxies as a measure of the Weber fraction (Khaw et al., 2020). Related to magnitude coding, (4) assumes that the noise, δ, distorts the perception of the categories or sub-ranges. ...
... In this frame, if the exact distance between the two left-most digits is not computed, then it seems quite implausible that the ratio between both quantities might drive the congruity effect. Finally, we run additional multilevel regression analyses to separate the effects of congruity and ratio in the pooled data of Experiments 1 & 2 and in Experiment 3. The analyses are described in detail in Appendix 2 [Footnote 5: As pointed out by a reviewer, these analyses should be taken with caution due to two different factors: a) the high correlation between ratio and congruity; b) the indeterminacy of the models supporting ratio or ratio-like factors (e.g., see the Discrete Semantic System, Krajcsi et al., 2016) regarding the specific regressors to include in the analysis]. In Experiments 1 & 2, effects of fillers ratio and congruity were found whereas ratio was not significant. ...
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... Rather surprisingly, indeed, a ratio-dependency has been as well observed in symbolic number comparison tasks (i.e., either with number words or Arabic digits) (Moyer & Landauer, 1967;Gallistel & Gelman, 1992). Yet, this theoretical account has been challenged in the more recent years by an increasing body of evidence supporting two separate systems for the representation of symbolic and nonsymbolic numbers (Krajcsi, 2017;Krajcsi, Lengyel, & Kojouharova, 2016;Lyons, Ansari & Beilock, 2015;Marinova, Sasanguie & Reynvoet, 2020;Sasanguie et al., 2017). Some studies have shown, indeed, that the performance in number comparison tasks is fully ratiodependent only with non-symbolic numbers (e.g., Marinova et al., 2020). ...
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... It might be argued that familiarity with small numerals rather than the operationalization of the numerical system underpins the pattern of findings reported in this study. For instance, Verguts and van Opstal (2005) showed that the numerical size effect in symbolic comparison of small numbers is the result of the skewed frequency distribution of numbers, with smaller numbers being more frequent than larger numbers (see also, Krajcsi et al., 2016;Verguts & Van Opstal, 2014). Indeed, a frequency-based account would render similar predictions to those formulated in the current study, and the pattern of results would be hardly discernible. ...
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... It might be argued that familiarity with small numerals rather than the operationalization of the numerical system underpins the pattern of findings reported in this study. For instance, Verguts and van Opstal (2005) showed that the numerical size effect in symbolic comparison of small numbers is the result of the skewed frequency distribution of numbers, with smaller numbers being more frequent than larger numbers (see also, Krajcsi et al., 2016;Verguts & Van Opstal, 2014). Indeed, a frequency-based account would render similar predictions to those formulated in the current study, and the pattern of results would be hardly discernible. ...
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Recent years have witnessed an increase in research on how numeral ordering skills relate to children’s and adults’ mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers.
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