ORIGINAL PAPER

High-Resolution EEG Analysis of Power Spectral Density Mapsand Coherence Networks in a Proportional Reasoning Task

Giovanni Vecchiato • Ana Susac • Stavroula Margeti •

Fabrizio De Vico Fallani • Anton Giulio Maglione •

Selma Supek • Maja Planinic • Fabio Babiloni

Received: 7 July 2012 / Accepted: 18 September 2012

� Springer Science+Business Media New York 2012

Abstract Proportional reasoning is very important logi-

cal skill required in mathematics and science problem

solving as well as in everyday life decisions. However,

there is a lack of studies on neurophysiological correlates

of proportional reasoning. To explore the brain activity of

healthy adults while performing a balance scale task, we

used high-resolution EEG techniques and graph-theory

based connectivity analysis. After unskilled subjects

learned how to properly solve the task, their cortical power

spectral density (PSD) maps revealed an increased parietal

activity in the beta band. This indicated that subjects

started to perform calculations. In addition, the number of

inter-hemispheric connections decreased after learning,

implying a rearrangement of the brain activity. Repeated

performance of the task led to the PSD decrease in the beta

and gamma bands among parietal and frontal regions

along with a synchronization of lower frequencies. These

findings suggest that repetition led to a more automatic task

performance. Subjects were also divided in two groups

according to their scores on the test of logical thinking

(TOLT). Although no group differences in the accuracy

and reaction times were found, EEG data showed higher

activity in the beta and gamma bands for the group that

scored better on TOLT. Learning and repetition induced

changes in the pattern of functional connectivity were

evident for all frequency bands. Overall, the results indi-

cated that higher frequency oscillations in frontal and

parietal regions are particularly important for proportional

reasoning.

Keywords High-resolution EEG � Oscillations �Functional connectivity � Logical reasoning �Balance scale task � Mental arithmetic

Introduction

Proportional reasoning is an important logical skill

required in problem solving in mathematics and science,

but also in everyday life situations. One example is,

deciding which product is the better buy while shopping

(e.g., Capon and Kuhn 1979). Educational research found

that students’ proportional reasoning skill, together with

other logical abilities such as control of variables, combi-

national and probabilistic reasoning, can serve as a pre-

dictor of their academic performances in science courses

(e.g., Bird 2010; Coletta and Phillips 2005). Since pro-

portional reasoning is a capability that students will use in

their professional and daily life (e.g., Hoyles et al. 2001), it

is important to understand better its neurophysiological

correlates, to what extent such a capability/skill can be

improved, and to develop efficient teaching strategies.

G. Vecchiato � F. De Vico Fallani � F. Babiloni

Department of Physiology and Pharmacology,

University of Rome ‘‘Sapienza’’, Rome, Italy

G. Vecchiato � F. De Vico Fallani � F. Babiloni

IRCCS ‘‘Fondazione Santa Lucia’’, Rome, Italy

A. Susac (&) � S. Supek � M. Planinic

Department of Physics, Faculty of Science, University

of Zagreb, Bijenicka 32, 10000 Zagreb, Croatia

e-mail: ana@phy.hr

S. Margeti

Medical Division, Widen Laboratory, University of Crete,

Heraklion, Greece

A. G. Maglione

Department of Anatomy, Histology, Forensic Medicine and

Orthopedics, University of Rome ‘‘Sapienza’’, Rome, Italy

123

Brain Topogr

DOI 10.1007/s10548-012-0259-5

Different tasks have been employed to explore propor-

tional reasoning (for a review see, Tourniaire and Pulos

1985). The one that is most often used is the balance scale

task introduced by Inhelder and Piaget (1958). Siegler

(1976, 1981) applied a trial-by-trial strategy assessments,

and found a simple model of hierarchically related

sequence of four rules through which children of different

age progress on the task. More recently, an additional rule

and modification of the previous model have been intro-

duced (Jansen and van der Maas 2002). Measurement of

reaction times (RTs) provided the additional information

on the rule application (van der Maas and Jansen 2003).

According to the Piaget’s theory of cognitive develop-

ment, children start to reason proportionally at a formal

operational stage, the final stage of intellectual develop-

ment, which starts at the age of around 12 years and con-

tinues in the adulthood. Large literature base on students’

difficulties with proportional reasoning suggests that this

form of logical reasoning is difficult for learners of all ages

(e.g., Boyer et al. 2008; Karplus et al. 1983; Lawton 1993;

Mellar 1991; Noelting 1980; Tourniaire and Pulos 1985;

Vass et al. 2000). Namely, even many adults are not able to

reason proportionally (e.g., Capon and Kuhn 1979; Lamon

2005).

While strategies and problems with proportional rea-

soning in children and adults have been the topic of

numerous educational and developmental studies, its neural

basis and development over the life span are not yet fully

understood. In fact, to our knowledge, there is a complete

lack of functional neuroimaging studies on the proportional

reasoning. However, clinical neuropsychological tests have

shown that patients with heterogeneous brain injuries make

significantly more errors on proportional reasoning tasks

than individuals without brain damage (Allen et al. 2007a,

b; Donders 2001). A lack of difference on test perfor-

mances both between groups with right and left hemisphere

lesions (Allen et al. 2007b), and between groups with

anterior and posterior lesions (Donders 2001), suggest the

involvement of widespread brain networks in proportional

reasoning tasks.

Recently, a number of neuroimaging studies have been

conducted on various types of logical thinking, such as

deductive reasoning (for a review see, Goel 2007), ana-

logical reasoning (Bunge et al. 2005; Green et al. 2010; Qiu

et al. 2008) and probabilistic reasoning (Parsons and

Osherson 2001). Using mostly fMRI (functional magnetic

resonance imaging), neuroimaging studies have demon-

strated that different sorts of logical reasoning recruit dis-

tinct neural substrates. Extensive and distributed brain

networks are found to underlie logical reasoning, with a

consistent involvement of frontal areas (Goel 2007). Very

recent use of EEG (electroencephalography) with its mil-

lisecond temporal resolution offered a complementary

view on brain activity during logical reasoning (Bonnefond

and Van der Henst 2009; Luo et al. 2008; Medaglia et al.

2009; Prado et al. 2008; Qiu et al. 2008). Modulation of

ERP (event-related potential) components such as N2 and

P3b was reported for different logical reasoning tasks

(Bonnefond and Van der Henst 2009; Luo et al. 2008;

Prado et al. 2008).

In addition to logical thinking, proportional reasoning

also requires developed arithmetical skills. Many neuro-

imaging studies investigated neural basis of numerical

processing and its development (for a review see, Ansari

2008; Dehaene et al. 2004; Dehaene 2009; Zamarian et al.

2009). The parietal cortex is systematically activated in all

tasks related to numbers (e.g., Dehaene 2009). Anderson

et al. (2008) developed a model of brain activation during

complex cognitive task involving visual stimuli and man-

ual responses. It consists of six modules which are found to

correspond to particular brain regions found to be active in

fMRI studies. For example, an imaginary module, which

holds representation of a problem (e.g., representation of an

equation), is associated with posterior parietal cortex,

whereas a retrieval module, which is responsible for

retrieval and storage operations (e.g., algebraic rules and

arithmetic facts), is associated with frontal cortex.

Advances in brain imaging techniques allowed not only

analysis of electrical and hemodynamic responses in single

regions of interests but they also provided methods to

estimate the functional connectivity among several cerebral

areas, i.e. correlations between activities in different parts

of the human brain. A number of studies examined syn-

chronous brain activity in several cognitive tasks using

EEG (e.g., Andres and Gerloff 1999; Classen et al. 1998;

Ghilardi et al. 2000; Lachaux et al. 1999; Miltner et al.

1999). Many authors (Lachaux et al. 1999; Rappelsberger

et al. 1994; Singer 1999; Thompson and Varela 2001) have

shown that synchronous cerebral activity plays an impor-

tant role in higher cognitive functions, including associa-

tive memory, emotional tone, and motor planning

(Rodriguez et al. 1999). Such a synchronous cooperation

between two or more distant brain regions can be treated as

a complex network.

Recently, it was realized that the functional connectivity

networks estimated from brain-imaging data obtained by

EEG, MEG (magnetoencephalography) and fMRI can be

investigated using graph theory (Achard and Bullmore

2007; Bartolomei et al. 2006; Bassett et al. 2006; Eguiluz

et al. 2005; Micheloyannis et al. 2006; Salvador et al. 2005;

De Vico Fallani et al. 2008a; Stam 2004; Stam and

Reijneveld 2007). Since a graph is a mathematical repre-

sentation of a network that has been essentially reduced to

nodes and connections between them, the use of the graph-

theory approach is potentially relevant and useful

to quantify and describe the degree and modality of

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communication among different cerebral areas, as first

demonstrated on a set of anatomical brain networks

(Sporns 2002; Strogatz 2001).

The aim of this study was to investigate the brain

activity in healthy adults during the performance of a

proportional reasoning task, using high-resolution EEG

methods and parameters derived from the graph theory.

The high-resolution EEG technology has been developed

to enhance the spatial information content of the EEG

activity and was here employed in order to estimate the

cortical source signals (Babiloni et al. 2005; Bai et al.

2007; Dale et al. 2000; He et al. 1999; Nunez 1995).

We expected to find synchronization of neuronal oscil-

lations after subjects had learned to solve the balance scale

task, especially for higher frequencies which are usually

related to complex cognitive tasks. We also anticipated a

training effect, i.e. changes in the behavioral and EEG

responses due to repeated performance of the proportional

reasoning task. Repetition of the same task would lead to a

lower level of attention which would, consequently,

decrease synchronization of beta and gamma oscillations

and increase synchronization of alpha rhythm. We pre-

sumed that widespread cortical networks would be acti-

vated during the performance of a complex cognitive task,

such as the balance scale task. According to the Anderson’s

model, we expected activation of the prefrontal, parietal

and occipital cortex. As proportional reasoning cannot be

reduced only to the successful solving of the balance scale

task, we wanted to compare EEG responses of the subjects

who scored well and the subjects who scored poorly on the

TOLT behavioral test of proportional reasoning (Tobin and

Capie 1981).

Materials and Methods

Subjects

Twenty subjects (mean age 26.6 ± 5.2, 10 females), with

normal or corrected-to-normal vision, participated in the

study. Each participant gave an informed consent before

taking part in the experiment in agreement with the ethical

committee rules followed at the recording site (Fondazione

Santa Lucia, Rome, Italy).

Procedure

We measured the EEG activity of 20 healthy subjects while

they were solving the balance scale task. The subject sat in

a quiet, dimly lit room and viewed balance scale drawings

(Fig. 1) delivered on the computer screen by the Presen-

tation software (Neurobehavioral Systems Inc., Albany,

CA) subtending approximately 4.6 9 1.5� of visual angle.

On each side of the fulcrum there were four equidistant

pegs on which different numbers (maximum four) of

equally heavy weights were placed, all on a single peg. The

drawing was black, except for some grey pegs. The balance

task (BALANCE) was to predict the movement of the

scale, i.e. whether the scale would balance, or tip to the left

or to the right. The control task (CONTROL) was to

indicate on which side of the balance there were more grey

pegs. Subjects were instructed to press as correctly and as

quickly as possible one of the three arrow buttons of an

ordinary keyboard: left arrow button to indicate the scale

tip to the left (or more grey pegs on the left side), right

arrow button for the scale tip to the right (or more grey

pegs on the right side) and bottom arrow button for the

balanced scale (or the equal number of grey pegs on both

sides). They had 10 s to solve the task. During the first 5 s

the picture of the balance scale was shown while during the

remaining 5 s only a grey screen was visible. The trial

finished when the subject pressed a button or after 10 s.

The two tasks were alternated in short blocks with a

counterbalanced order across subjects; each short block

contained 12 trials. In each run, the subjects were presented

with 48 BALANCE trials and 48 CONTROL trials.

Twelve out of twenty subjects (NAIVE group) did not

know the torque rule needed to properly solve the balance

scale task (i.e. to compare the product of force, given by

the number of weights, and lever arm, which was the dis-

tance of weights from the fulcrum, for both the left and the

right side of the balance). After Run 1 they were told

the torque rule, and then participated in Runs 2 and 3.

Accordingly, NAIVE group consisted of 12 subjects who

participated in all three runs. The remaining 8 subjects who

knew the torque rule from the beginning participated only

in Runs 2 and 3.

After the EEG measurements the subjects were asked to

answer the first four questions from the test of logical

thinking (TOLT, Tobin and Capie 1981) in order to assess

their ability of proportional reasoning independently from

the balance scale task performance. Based on the TOLT

scores, the subjects were divided into PR (proportional

reasoning) and NPR (non proportional reasoning) groups.

The PR group consisted of 10 subjects who solved cor-

rectly all four questions from the TOLT. The remaining 10

Fig. 1 Example of a balance scale drawing used in the study

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subjects were part of the NPR group which just partially

overlapped with the NAIVE group; 6 subjects belonged to

both NPR and NAIVE group.

Behavioral Data Analysis

The accuracy (percentage of correct answers) and reaction

times (RTs) for different tasks, runs and subject groups

were evaluated using two-way repeated measures analyses

of variance (ANOVA). Where applicable, the results were

analyzed by using a two-tailed t test after ANOVA

(alpha = 0.05). Only data from the correct trials were used

in the analysis of RTs.

First, we wanted to assess differences in the accuracy

and RTs before and after the subjects knew the torque rule,

which was needed to properly solve the balance scale task.

Two repeated-measures ANOVAs were performed on

NAIVE group data using the factors Run (Run 1 vs. Run 2)

and Task (BALANCE vs. CONTROL), by employing

accuracy and RTs as dependent variables, respectively.

Second, training effect was evaluated for all subjects by

comparing their reaction times and accuracy in the two

runs when they knew the torque rule. As described above,

we performed two repeated-measures ANOVAs with the

factors Run (Run 2 vs. Run 3) and Task (BALANCE vs.

CONTROL) with accuracy and RTs as dependent vari-

ables, respectively.

Third, we compared behavioral performance (accuracy

and RTs) on balance scale task for the two groups of

subjects (PR and NPR), which differed in their perfor-

mance on the TOLT test. In this case we adopted a mixed

model ANOVA with the within-subjects factor Run (Run 2

vs. Run 3), and the between-subjects factor Group (PR vs.

NPR).

EEG Recordings and Pre-Processing

The brain activity was recorded with a 61-channel EEG

system (Brain Amp, Brainproducts GmbH, Germany) with

a sampling rate of 200 Hz. Electrode impedances were kept

below 10 kX. The raw EEG signals were band pass filtered

at 2–47 Hz and cleaned of ocular artifacts by employing

the independent component analysis (ICA): the compo-

nents due to eye blinks and ocular movements were

detected by eye inspection, and then removed from the

original signal. A manual procedure was also adopted to

reject trials presenting muscular and other kinds of arti-

facts. Recordings were initially extra-cerebrally referred

and then off-line converted to a common average refer-

ence. The recording sessions were segmented in order to

analyze the EEG activity elicited during the performance of

the balance scale task in both BALANCE and CONTROL

conditions. Each segment had different length, since it

started with the presentation of the balance stimulus and

ended up with the accomplishment of the task. Only arti-

facts-free trials with the correct behavioral response were

considered in the following analysis.

Estimation of Cortical Power Spectral Density

Cortical activity from the EEG scalp recordings was esti-

mated by employing the high-resolution EEG technologies

(Babiloni et al. 2000a, b; De Vico Fallani et al. 2007; Ding

et al. 2005; He et al. 1999; Nunez 1995) with the use of the

average head model from McGill University. The scalp,

skull and dura mater compartments were built by using

1,200 triangles for each structure, and the boundary ele-

ment model was then employed to solve the forward

electromagnetic problem. For each subject, the coordinates

of electrodes’ locations on the scalp surface were calcu-

lated by a non-linear minimization procedure (Astolfi et al.

2008). The cortical model consisted of 1,024 dipoles uni-

formly distributed on the cortical surface, and the estima-

tion of the current density strength for each dipole was

obtained by solving the electromagnetic linear inverse

problem according to the minimum norm solution as

described in the previous papers (Astolfi et al. 2007a, b;

Babiloni et al. 2005). Each dipole was modeled to be

perpendicular to the cortical surface.

The power spectral density (PSD) of the estimated

cortical signals was calculated using the Welch method

(Welch 1967), and then mapped onto the average cortex

model (MNI template, 1,024 cortical dipoles) as described

above. This procedure allowed us to obtain a measure of

PSD values for each estimated cortical location and for

each trial for all subject’s datasets, in the frequency range

of [1, 40] Hz with a resolution of 1 Hz. Obtained PSD data

were averaged across subject for each location.

We focused the present analysis on the canonical fre-

quency bands of interest of the EEG, i.e. theta [4, 7] Hz,

alpha [8, 12] Hz, beta [13, 24] Hz, gamma [25, 40] Hz. For

each comparison of interest, we employed the t test

(P \ 0.05) to contrast the cortical PSD for each cortical

dipole, and adopted the false discovery rate correction for

multiple comparisons (Vecchiato et al. 2010). When a

between group comparison was performed, we considered

the PSD elicited during the execution of CONTROL trials

as a baseline for the z score computation (Zar 2009), for

each dipole in the four frequency bands of interest. In this

case, the t test was performed between z score values.

Brain Network Analysis

The nodes of the functional networks were estimated

by considering 52 regions of interest (ROIs), defined

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according to the boundaries of the Broadmann areas (BAs).

In order to obtain a signal from each ROI we averaged the

time-series of the cortical sources within the corresponding

region. To perform the connectivity analysis, we employed

spectral coherence as an estimation of the level of syn-

chronicity between the pairs of ROIs (Andres and Gerloff

1999; Zouridakis et al. 2007a, b). This measure is a func-

tion operating in the frequency domain returning continu-

ous values ranging between 0 and 1. For each trial, spectral

coherence was calculated for all combinations of ROIs.

The resulting connectivity patterns were filtered (i.e. the

weakest links were removed) in order to generate networks

with the same number of unweighted connections. The

proper threshold to establish the number of valid connec-

tions was calculated by following the method explained in

details in our earlier study (De Vico Fallani et al. 2008b).

A graph as a representation of a network consists of a set

of vertices (nodes) and a set of edges (connections), indi-

cating the interaction between the vertices. The adjacency

matrix contains the information about the connectivity

structure of the graph. In this study we considered undi-

rected and unweighted graphs: if a connection between the

i-th and the j-th node exists, the related element of the

adjacency matrix is 1, 0 otherwise. In the present study, the

connectivity degree and the number of inter-hemispheric

connections were the two indices employed to describe the

network topology.

The degree of the graph’s node is given by its total

number of connections with other vertices. Its general

formulation for a given node i is: ki ¼P

j2N

aij,where aij is

the connection value between the i-th and j-th nodes and N

is the set of nodes. Higher degree values are associated

with higher centrality of a given node.

In the present work we define the number of inter-

hemispheric connections as the number of edges between

the two hemispheres. This index can be defined by ordering

the matrix, so that the first rows and columns correspond to

the ROIs belonging to the left hemisphere. Hence, the

following formula defines the number of inter-hemispheric

connections:

h ¼XN=2

i¼1

XN

j¼N=2þ1

aij þXN

i¼N=2þ1

XN=2

j¼1

aij

By definition, this index measures the communication

strength between the two hemispheres and its value ranges

between [0; N2

4].

For each comparison described in the above paragraph,

we employed the t test (P \ 0.05) to contrast the indices

calculated via graph analysis, and adopted the false dis-

covery rate correction for multiple comparisons, according

to the statistical design.

Just as in the case of statistical comparisons of PSDs, we

evaluated between group comparison of graph indices

related to the execution of CONTROL trials as a baseline

for the z score computation (Zar 2009), for each ROI in the

four frequency bands of interest. Accordingly, the t test

was performed between z score values.

Results

Behavioral Results

First, we found a learning effect on the behavioral

responses of NAIVE subjects (Table 1). ANOVA revealed

a significant main effect of both factors on accuracy, Run,

F(1,11) = 59.4; P \ 0.0001, and Task, F(1,11) = 31.0;

P = 0.0002, and a significant interaction Run 9 Task,

F(1,11) = 46.1; P \ 0.0001. In Run 1, subjects were less

accurate on the BALANCE task than on the CONTROL

task (P \ 0.0001), whereas there was no statistical differ-

ence between the tasks in Run 2. Subjects were more

accurate on the BALANCE task in Run 2 compared to Run

1 (P \ 0.0001), while no statistical difference was found

between the runs for the CONTROL task. We did not find

any statistically significant response pattern for NAIVE

subjects before they learned the torque rule.

Corresponding ANOVA on reaction times showed a

significant main effect of Run, F(1,11) = 21.0;

P = 0.0008, and Task, F(1,11) = 16.1; P = 0.002, and

a significant interaction Run 9 Task, F(1,11) = 16.3;

P = 0.002. RTs for the BALANCE and the CONTROL

tasks did not differ in Run 1. In Run 2, subjects had longer

RTs for the BALANCE task than for the CONTROL task

(P = 0.0004). RTs for the BALANCE task were longer in

Run 2 compared to Run 1 (P = 0.0005) and no statistical

difference between runs was found for the CONTROL

task.

Second, behavioral data of all subjects showed a training

effect in terms of an improvement of accuracy and a

decrease of reaction times during the experiment (Table 2).

Two-way repeated-measures ANOVA performed on the

accuracy data from all subjects showed main effect of Run,

F(1,19) = 9.4; P = 0.006, and no main effect of Task or

interaction. Subjects were more accurate in Run 3 com-

pared to Run2 (Table 2).

ANOVA performed on the RTs data revealed a signifi-

cant main effect of Run, F(1,19) = 11.6; P = 0.003, and

Task, F(1,19) = 27.4; P \ 0.0001, and a significant

interaction Run 9 Task, F(1,19) = 6.0; P = 0.02. Sub-

jects had longer RTs for the BALANCE task than for the

CONTROL task in Run 2 and in Run 3 (P \ 0.0001 and

P = 0.0007, respectively). RTs for the BALANCE task

were shorter in Run 3 than in Run 2 (P = 0.002) while no

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statistical difference between runs was found for the

CONTROL task.

Third, when accuracy and RTs for the BALANCE task

were compared for runs (Run 2 vs. Run 3), and groups (PR

vs. NPR), only main effect of the factor Run was found.

As previously shown, subjects were more accurate,

F(1,20) = 9.5; P = 0.006, and faster, F(1,20) = 12.4;

P = 0.002, on the BALANCE task in Run 3 than in Run 2.

No main effect of Group or interaction was found in the

behavioral data.

Comparisons of Cortical PSD Maps

The evoked cortical activations involved all frequency

bands for all subject. The first statistical comparison is

related to the NAIVE group; we compared PSD values

elicited during Runs 1 and 2 (Fig. 2). We did not find any

statistically different activation in the theta and alpha

bands, whereas we noticed significant increase of the PSD

values in the higher frequency ranges, beta and gamma,

during the performance of the BALANCE task in Run 2. In

particular, significantly different activity was found mostly

in the parietal areas for the beta band, while the areas

differentially activated in the gamma band were in the

frontal cortex. These activations were distributed mostly in

the left hemisphere, although some significant increase of

the PSD in the beta band was located also in the right

hemisphere.

The second contrast was performed by taking into

account the PSD activity of all subjects and comparing the

spectral power elicited in Runs 2 and 3 (Fig. 3). The

resulting statistical maps revealed a higher spectral power

during Run 2 which was spread among different frequency

bands and cortical regions. In Run 2, we found an increased

PSD in the prefrontal cortex for the alpha rhythm, whereas

in higher frequency range, such as beta and gamma, these

activations were also spread among other frontal and

parietal regions of the cortex. The cortical maps in Fig. 3

also show an increase of spectral activity in some brain

regions during Run 3. This increase was present in the low

frequency range of the EEG spectrum, i.e., theta and alpha

bands. In particular, statistical differences of the PSD were

evident in both hemispheres in frontal areas and parietal

regions for the theta and alpha bands, respectively.

In the third contrast (Fig. 4), we compared the z score

transformed PSD activity of our two experimental groups

of subjects, namely PR versus NPR, during Run 3. The

cortical maps showed larger activity for the PR group in the

beta and gamma bands which was bilaterally spread in

many cortical regions, mostly involving the right parietal

lobe and the prefrontal areas. Moreover, a significant

increase of activity for the PR group was found in the

superior parietal lobe in the theta band. Conversely, the

NPR group showed a larger activation in the frontal and

parietal regions, mostly in the left hemisphere, in the alpha

band, accompanied by some increase of the PSD in the

same areas in the theta band.

Table 1 Means and standard deviations for the accuracy and reaction

times for the NAIVE subjects before and after they learned the torque

rule

Run Accuracy (%) RT (s)

Task Task

BALANCE CONTROL BALANCE CONTROL

Run 1 43 ± 9 82 ± 14 2.16 ± 0.85 2.28 ± 0.54

Run 2 81 ± 20 85 ± 16 3.51 ± 0.73 2.55 ± 0.88

Table 2 Means and standard deviations for the accuracy and reac-

tion times for all subjects in Run 2 and Run 3

Run Accuracy (%) RT (s)

Task Task

BALANCE CONTROL BALANCE CONTROL

Run 2 86 ± 17 84 ± 15 3.71 ± 0.77 2.70 ± 0.82

Run 3 94 ± 11 89 ± 12 3.39 ± 0.81 2.61 ± 0.85

Fig. 2 t test maps of the cortical PSD values between Runs 1 and 2 for

the NAIVE group in the beta and gamma bands. The average head model

is shown from an above and the left frontal perspective (left and rightpanels, respectively), as the label A (anterior), P (posterior), L (left), R

(right) indicate in the first row. The same convention holds for the secondrow. Each row is related to a frequency band. Red color indicates cerebral

areas with increased spectral power in Run 1, while blue color shows

enhanced spectral power in Run 2 (P \ 0.05, false discovery rate

adjusted). Grey color is used to illustrate cortical areas with no significant

difference between the two runs (Color figure online)

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123

Brain Network Analysis

In the first contrast, the graph indices for the NAIVE group

in Runs 1 and 2 were compared. We found a significant

difference for the number of inter-hemispheric connections

in the theta band (t = 2.74, P = 0.02). As Fig. 5 shows,

the number of connections was larger in Run 1 than in

Run 2.

Moreover, in both theta and alpha bands the connec-

tivity degree values for the left parietal and occipital areas

as well as for the right frontal area, were lower in Run 2

(Fig. 6). In addition, we found a significant increase of the

connectivity degree during Run 2 in ROIs BA6 left, BA18

right in the theta band, and in ROIs BA21 right and BA40

right in the alpha band. In the gamma band, there was an

increase of the connectivity degree in the BA10 left in Run

2. There is no statistically significant difference in the beta

band. The related map is not shown.

The second contrast included all subjects in Runs 2 and

3. There was no statistical difference in terms of inter-

hemispheric connections. However, the connectivity degree

demonstrated significant differences in all frequency bands

(Fig. 7). In the theta and alpha bands, we mostly observed a

decrease of this index for the right temporal and parietal

ROIs in Run 3. On the contrary, in the beta band we found

an increase of the connectivity degree for BA40 right in Run

3. Similarly, in the gamma band an increase was evident in

the same run for the BA6 and BA44 right.

In the third contrast, we compared the graph indices for

the PR and NPR group in Run 3. There was no statistical

difference in the number of inter-hemispheric connections.

Nevertheless, the degree index differed across PR and NPR

groups (Fig. 8). In the theta band, we found a significant

increase of the connectivity degree for the PR group in the

right BA40, and an increase of the same parameter in the

left BA7 for the NPR group. In the alpha band, we

observed a bilateral increase of degree index in the BA42

for the NPR group. Finally, in the gamma band there was a

significant increase of the connectivity degree in the left

temporal areas BA21 and BA22 for the PR group. There is

no statistically significant difference in the beta band. The

related map is not shown.

Fig. 3 t test maps of the cortical PSD values between Runs 2 and 3

for all subjects in all frequency bands. Color and label convention as

in Fig. 2 is adopted (Color figure online)Fig. 4 t test maps of the cortical PSD values between the PR and

NPR group in Run 3 in all frequency bands. Color and label

convention as in Fig. 2 is adopted (Color figure online)

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123

Discussion

More than half of our participants (NAIVE group) did not

know how to properly solve the balance scale task. This

finding corroborates previous reports on adults’ difficulties

with proportional reasoning tasks (e.g., Capon and Kuhn

1979; Vass et al. 2000). Although our NAIVE subjects did

not know the torque rule, their accuracy for the BALANCE

task in Run 1 was above the chance level (i.e., 33 %). This

indicates that they probably took somehow into account

both the number of weights on each side of the scale and

their distance from the fulcrum as we did not have ‘‘easy’’

cases e.g., equal number of weights placed at different

distances from the fulcrum. However, their RTs did not

differ for the BALANCE and CONTROL tasks, suggesting

that they did not perform more complex calculation than

counting. As expected, after NAIVE subjects had learned

the torque rule, their accuracy greatly improved in Run 2,

and RTs increased as they calculated torque. The learning

of the torque rule affected neither accuracy nor RTs in the

CONTROL task.

EEG data allowed us to explore the neuronal correlates

of the observed behavioral effects. The comparison of the

cortical PSD in Runs 1 and 2 for the NAIVE group

revealed an increased activity in the parietal regions in the

beta band. Beta oscillations are often associated with active

concentration and cognitive processes (e.g., Ray and Cole

1985; Sheth et al. 2009). Increased beta activity in Run 2,

compared to Run 1, most likely reflects calculation of the

momentum in the BALANCE task. Many studies have

shown that mental calculation is mediated by a distributed

network within the parietal cortex (e.g., Zamarian et al.

Fig. 5 The number of the inter-hemispheric connections in Runs 1

and 2 for the NAIVE group in the theta band. The central red mark on

each box is the median, the edges of the box are the 25th and 75th

percentiles, the whiskers extend to the most extreme data points

without considering outliers, and outliers are plotted individually by

red crosses (Color figure online)

Fig. 6 t test maps of the cortical connectivity degree values between

Runs 1 and 2 for the NAIVE group in the theta, alpha and gamma

bands. The average head model is shown from an above perspective,

as the labels A (anterior), P (posterior), R (right), L (left) indicate for

the first band. The same convention is used for the following ones.

Red color indicates cerebral areas with increased degree values in

Run 1, while blue color shows enhanced degree in Run 2 (P \ 0.05,

false discovery rate adjusted). Grey color is used to illustrate cortical

areas with no significant difference between the two runs (Color

figure online)

Fig. 7 t test maps of the cortical connectivity degree values between

Runs 2 and 3 for all subjects in all frequency bands. Color and label

convention as in Fig. 6 is adopted (Color figure online)

Brain Topogr

123

2009), so our results are in line with the previous findings.

Small increase in the gamma activity in the left frontal

region might indicate retrieval of arithmetic facts and

algebraic rules, as suggested by Anderson et al. (2008).

Also, it is worth to note that the visible activations on the

corpus callosum are caused by the limit of the model

employed, which includes this cerebral region. In fact,

from the mathematical point of view, each cortical acti-

vation is a particular ‘‘projection’’ of the scalp signals on

the cortex. In such a case, the activation of the corpus

callosum would represent a cortical signal that would exist

if at that location there was the cortex. Consequently, this

kind of activation has no physical sense and this is why we

do not comment on that in the paper.

Moreover, the functional connectivity analysis per-

formed via graph theory revealed that the learning of how

to solve the balance scale task is correlated with a signif-

icant decrease of the number of inter-hemispheric con-

nections. This finding suggests a more focused brain

activity after the learning. The degree index also

demonstrated learning induced changes in the pattern of

functional connectivity. Besides changes in information

flow in the frontal and parietal areas, occipital areas also

modified their interactions with other brain regions. This

finding is also in agreement with a significant role of the

visual cortex in Anderson et al. (2008).

Since ancient time it is known that repetition plays an

important role in learning. Although it has been accepted

that learning of the multiplication tables, playing musical

instruments or sport performance strongly depend on rep-

etition, the same was, for many years, not considered

essential in learning of higher cognitive skills. Now it

becomes clear that repetition has an important part in

learning of any subject. Modern neurobiology has shown

that learning and memory are based on modifications of

synaptic strengths among neurons (Hebb 1949). Repeated

stimulation leads to an increase in synaptic strength

between neuronal cells and consequently to an improved

learning at the molecular level. The design of our study

allowed us to explore the effect of training, i.e., repeated

task performance on the behavioral results and the brain

activity.

Our behavioral data showed that repetition led to higher

accuracy and shorter RTs on the BALANCE task in Run 3

compared to Run 2. The training effect was less pro-

nounced for the CONTROL task. For example, RTs were

not statistically different between Runs 2 and 3 for all

subjects, nor between Runs 1 and 2 for the NAIVE sub-

jects. These results suggest that more complex tasks, such

as the proportional reasoning task, leave more space for an

improvement by training. According to the Anderson et al.

(2008), improvement could be made at different stages of

task solving. For example, training might advance per-

ceptual stage when important information is extracted from

the picture of a balance scale, or retrieval stage when

arithmetic facts are obtained.

The comparison of the PSD values between Runs 2 and

3 for all subjects showed larger activity among frontal and

parietal areas in the beta and gamma bands during the

BALANCE task performance during Run 2. This was

accompanied by a desynchronization of the alpha rhythm

in the parietal regions, which is often considered as an

index of attention (Klimesch 1999; Lachat et al. 2012).

Similarly, enhanced activity in the parietal and interhemi-

spheric areas was found in the theta band in Run 3 when

compared to Run 2. These results suggest that in the

beginning, subjects needed more neuronal resources to

solve the BALANCE task, as indicated by increased beta

and gamma oscillations in many brain regions. Repetition

of the same task resulted in decreased high-frequency

activity, i.e., the same task could be solved using less

resources. Corresponding effect of training was present in

the low-frequency bands. After repeating the same task,

Fig. 8 t test maps of the cortical connectivity degree values between

the PR and NPR group in Run 3 in the theta, alpha and gamma bands.

The labels A (anterior), P (posterior), L (left), R (right) indicate the

orientation of the cortical model for the three bands. Color convention

as in Fig. 6 is adopted (Color figure online)

Brain Topogr

123

subjects did not need to be so attentive as in the beginning;

increased alpha and theta oscillations were a sign of

decreased attention. Altogether, these results imply that

repetition leads to a more automatic performance of the

proportional reasoning task. Moreover, the observed cere-

bral activations are in agreement with the Anderson et al.

(2008) stating that the posterior parietal cortex is the base

of the representation of an equation, whereas the frontal

cortex is responsible for the algebraic rules and arithmetic

facts. However, the parietal cortex activation is also related

to arithmetical skills since it is often involved in tasks

concerning numbers. The bilateral cerebral activations

testify that higher order cognitive processes involved in

problem solving are not lateralized but generally distrib-

uted throughout the cortex (Allen et al. 2007a).

The functional connectivity analysis showed that right

temporal and frontal areas act as central hubs of commu-

nication between different brain regions for logical rea-

soning tasks, as already shown (Goel 2007). Repetition of

the same task led to an increase of the connectivity degree

for the higher frequencies, and a corresponding decrease

for the lower frequencies. Our results suggested that rep-

etition alters information flow which might be crucial in the

learning process.

The score on the balance scale task is not an absolute

indicator of person’s ability to reason proportionally. It is

possible that somebody who can solve the balance scale

task fails to solve other types of problem with proportional

reasoning, and vice versa, somebody who cannot solve the

balance scale task is able to reason proportionally in

another context. In our study, four subjects who were able

to solve the balance scale task without instructions could

not solve all four proportional reasoning questions from the

test of logical thinking. Conversely, six subjects could not

solve the balance scale task because they did not know the

torque rule, yet they solved all four TOLT questions.

When we divided all subjects in the PR and NPR

groups, according to their scores on the TOLT questions,

and compared their accuracies and RTs in Runs 2 and 3

(when all knew the torque rule), we did not find any sta-

tistically significant difference between the groups. How-

ever, when we compared their cortical spectral EEG

activity we found larger bilateral activity for the PR group

in the beta and gamma band. In the alpha band, the NPR

group showed a larger activation in frontal and parietal

regions, mostly in the left hemisphere. Although all sub-

jects knew how to solve the task in Run 3, our results

indicated different underlying brain activity. Subjects with

more developed proportional reasoning skill (PR group)

had more high frequency oscillations, which are usually

related with complex cognitive functions (e.g., Jensen et al.

2007; Ray and Cole 1985), and less alpha activity than

NPR group, which might indicate higher level of

task-related attention. Since theta power increases in a

large variety of tasks (e.g., Schacter 1977), it seems that

theta activity could in part reflect unspecific factors, such

as attentional demands, task difficulty and cognitive load.

The functional connectivity analysis has also revealed

differences in network patterns for the PR and the NPR

group in the theta band. In the gamma band, the PR group

had larger information flow in the left temporal areas than

the NPR group. These results provide initial support that

development of proportional reasoning might be related to

changes in the functional connectivity of the human brain.

Our results show that EEG data can reveal more

detailed information about complex cognitive processing

than only behavioral data. It is not easy to bridge the gap

between neuroscience and education (Ansari and Coch

2006). Namely, a straightforward transfer of the neuro-

science findings to the teaching practice is not possible.

However, some neuroscience evidence, like the results of

this study, could be relevant for education. Our results

support a well-known and obvious idea that learning

changes brain activity. We found changes in brain activity

associated with learning of how to solve the balance scale

task. We also showed functional measures of brain

changes related to training. Our results suggest that rep-

etition is important for the development of proportional

reasoning. In addition, the proportional way of thinking

becomes more automatic through repetition. However, it

does not mean that students should repeat one type of

problem many times. Proportional reasoning is a complex

ability. If somebody learns to solve certain type of a

problem with proportional reasoning, it does not neces-

sarily mean that they will be able to transfer that way of

thinking to a different problem. Overall, the results of the

present study demonstrate that learning and training rela-

ted to the balance scale task changed EEG responses in

various frequency ranges. The findings indicate that higher

frequency oscillations in frontal and parietal regions are

particularly important for proportional reasoning. The

repetition of the same task led to a more automatic brain

processing. Learning and training also led to the altered

brain functional connectivity. It is advisable to exercise

proportional reasoning in different contexts. That practice

could eventually lead to the adoption of that form of

logical thinking. This study suggests that high-resolution

EEG methods provide useful tools in the investigation

of complex cognitive processes such as proportional

reasoning.

Acknowledgments This study was supported by the ‘‘Fondazione

Santa Lucia’’, the European Union through the COST Actions

CONSCIOUSNESS (BM0605) and NEUROMATH (BM0601) and

the Croatian Ministry of Science, Education, and Sport (Grants

119-1081870-1252 and 119-0091361-1027). We thank Gianni Nicolai

and Marco Secci for their technical assistance.

Brain Topogr

123

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