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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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Krajcsi et al. The Source of the Symbolic Numerical Effects

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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Krajcsi et al. The Source of the Symbolic Numerical Effects

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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Krajcsi et al. The Source of the Symbolic Numerical Effects

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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Krajcsi et al. The Source of the Symbolic Numerical Effects

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

e−1

2x−(r−1)

w√1+r22

√−2πw√1+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(x1−1×x2−1). 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 =value−1×10. 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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Krajcsi et al. The Source of the Symbolic Numerical Effects

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.

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Krajcsi et al. The Source of the Symbolic Numerical Effects

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