EEG-based prediction of cognitive workload induced by arithmetic: a step towards online adaptation in numerical learning

Abstract

Numeracy is a key competency for living in our modern knowledge society. Therefore, it is essential to support numerical learning from basic to more advanced competency levels. From educational psychology it is known that learning is most effective when the respective content is neither too easy nor too demanding in relation to learners’ prerequisites. However, so far it is difficult to assess individual’s cognitive workload independently from performance to adapt learning environments accordingly. In the present study, we aim at identifying learners’ cognitive workload induced by addition tasks of varying difficulty using electroencephalography (EEG). To this end, a classifier using specific features in the EEG-signal is trained to differentiate between different levels of task difficulty significantly above chance level and with high consistency over all participants. Importantly, our model even allows for the prediction of cognitive demands induced by the addition tasks in a cross-participant approach. Closer inspection of the crucial EEG features indicates that oscillations in the theta and alpha band recorded from parietal electrodes are most reflective of current task difficulty. In summary, we are able to differentiate cognitive workload of participants independently from performance based on data of only a small number of electrodes. This suggests that a reduced EEG-setup combined with cross-participant classification may be a feasible approach to assess learners’ cognitive workload.

Full text

1 3
DOI 10.1007/s11858-015-0754-8
ZDM Mathematics Education
ORIGINAL ARTICLE
EEG‑based prediction of cognitive workload induced
by arithmetic: a step towards online adaptation in numerical
learning
Martin Spüler
1
· Carina Walter
1
· Wolfgang Rosenstiel
1
· Peter Gerjets
2,3
·
Korbinian Moeller
2,3
· Elise Klein
2,4
Accepted: 29 December 2015
© FIZ Karlsruhe 2016
summary, we are able to differentiate cognitive workload
of participants independently from performance based on
data of only a small number of electrodes. This suggests
that a reduced EEG-setup combined with cross-participant
classification may be a feasible approach to assess learners’
cognitive workload.
Keywords Electroencephalography (EEG) · Brain-
computer interface · Numerical cognition · Arithmetic ·
Cognitive workload
1 Introduction
Numeracy is a key competency for life at the beginning of
the 21st century (e.g., Askew 2015; Geiger, Goos, & For-
gasz 2015). Managing every-day life is associated substan-
tially with the ability to appropriately deal with numbers. In
almost all everyday activities such as buying a cup of coffee,
telling the time or boarding the correct bus we are exposed
to numbers—not to mention all the mathematic algorithms
for operating the latest technological advances such as smart
phones, etc. As a consequence, numeracy is seen as the ulti-
mate goal of mathematics education with a broad consen-
sus that it is essential to assist numeracy learning from basic
to more advanced competency levels. However, the precise
nature and implications for teaching numeracy in primary/
elementary and secondary school are still under debate (e.g.,
Calder 2015; Geiger, Goos, & Forgasz 2015).
This is of specific scientific but also general interest
because deficits in numeracy entail both considerable per-
sonal handicaps and massive socio-economic costs (e.g.,
Butterworth et al. 2011; Gross et al. 2009). A relevant
percentage of the population suffers from insufficient
numeracy skills with prevalence rates of developmental
Abstract Numeracy is a key competency for living in
our modern knowledge society. Therefore, it is essential to
support numerical learning from basic to more advanced
competency levels. From educational psychology it is
known that learning is most effective when the respective
content is neither too easy nor too demanding in relation
to learners’ prerequisites. However, so far it is difficult to
assess individual’s cognitive workload independently from
performance to adapt learning environments accordingly.
In the present study, we aim at identifying learners’ cog-
nitive workload induced by addition tasks of varying dif-
ficulty using electroencephalography (EEG). To this end,
a classifier using specific features in the EEG-signal is
trained to differentiate between different levels of task dif-
ficulty significantly above chance level and with high con-
sistency over all participants. Importantly, our model even
allows for the prediction of cognitive demands induced by
the addition tasks in a cross-participant approach. Closer
inspection of the crucial EEG features indicates that oscil-
lations in the theta and alpha band recorded from parietal
electrodes are most reflective of current task difficulty. In
M. Spüler and C. Walter contributed equally.
* Korbinian Moeller
k.moeller@iwm-tuebingen.de
1
Department of Computer Engineering, Eberhard-Karls
University Tuebingen, Tübingen, Germany
2
Leibniz-Institut für Wissensmedien, Tübingen, Germany
3
Department of Psychology, Eberhard-Karls University
Tuebingen, Tübingen, Germany
4
Department of Neurology, Section Neuropsychology,
University Hospital, RWTH Aachen University, Aachen,
Germany

M. Spüler et al.
1 3
mathematics learning difficulties between 3 and 6 % (e.g.,
Kucian & von Aster 2015). Moreover, acquired numerical
deficits constitute a frequent (up to 50 % of patients with
left- and 30 % with right-hemispheric lesions) and incapac-
itating disorder after brain damage and degenerative brain
conditions (e.g., Willmes & Klein 2014).
Against this background, it is important to note that there
is an ongoing debate in mathematics education research
on how to keep learners within their individual zone of
proximal development during learning (e.g., Calder 2015;
Askew 2015). Importantly, learning outcomes seem most
promising when training programs and learning environ-
ments are tailored to the individual’s specific needs (e.g.,
Gerjets & Hesse 2004; Richards et al. 2007; for numerical
interventions see Dowker 2009; Karagiannakis & Coore-
man 2014); this means that training content should nei-
ther be too easy nor too demanding in relation to learners’
individual prerequisites. Thus, participants should be kept
in an optimal range of cognitive workload to ensure learn-
ing progress (Gerjets, Scheiter, & Cierniak 2009; Sweller
et al. 1998). Therefore, it would be desirable to adapt the
difficulty of the training content to individual competencies
(i.e., strengths and weaknesses) of the learner.
A currently popular approach to ensure learning within
the zone of proximal development is scaffolding (Moschk-
ovich 2015; Wischgoll, Pauli, & Reuscher 2015; Makar,
Bakker, & Ben-Zvi 2015). Scaffolding can rely on tempo-
rary, adaptive support provided by a more knowledgeable
other to assist a student to solve a problem (e.g., Smit &
Van Eerde 2013). Scaffolding can also be used in whole-
class settings (Smit, Van Eerde, & Bakker 2013) or in
long-term supporting systems such as curricula (Cazden
2001). Other approaches are, for instance, student-centered
learning (Calder 2015) or computer-supported learning
(Kirschner & Gerjets 2006). The latter, computer-supported
learning environments, seems specifically suited for imple-
menting adaptivity because it is, in principle, easy to imple-
ment algorithms that allow for adaptation, for instance, the
order of presentation of specific learning contents based
on learners’ interaction behavior. This allows for a person-
alization of the learning environment to the specific needs
of a learner to keep him/her within the zone of proximal
development, which is assumed to ensure efficient learn-
ing progress. However, so far, adaptive computer-supported
learning environments primarily rely on learners’ explicit
interaction behavior for adaptation (e.g., correctness of
responses, reaction times, etc., cf. Käser et al. 2013b) and
not on direct evidence for learners’ current cognitive state.
In numerical cognition, researchers have already devel-
oped and evaluated the first computer-supported and adap-
tive training environments (e.g., the Number Race by Wil-
son et al. 2006, see Li & Ma 2010 for a meta-analyses on
effects of computer technology on mathematics learning).
One of the currently most sophisticated environments is the
“Calcularis” training environment for children with difficul-
ties in learning mathematics (Käser et al. 2012, 2013a, b).
Calcularis combines training basic numerical competencies
and their interrelations with arithmetic operations. To this
end, it is composed of different games following a hierar-
chical structure from number understanding in general to
arithmetic operations to arithmetic word problems. Each
content domain builds up on knowledge and competencies
addressed in previous levels of the program. Therefore, it
is essential to correctly assess children’s current numeri-
cal competencies as well as their development to present
them with training content of the adequate difficulty level.
In Calcularis this is achieved by means of a user model
which considers the accuracy of the learner’s interactions
with the program to estimate and monitor the actual compe-
tency level of the learner (see Käser et al. 2012, 2013b for a
detailed description of the procedure). While an evaluation
of the program indicates that this procedure works very well
(e.g., Käser et al. 2013a), it should be noted that the adap-
tation procedure depends exclusively on logging behavioral
input data (e.g., correctness of the response). Importantly,
these are rather indirect and distal measures that are not
very specific with respect to the inference of the cognitive
processes required to perform the task at hand. For instance,
more errors in a row may not only be caused by the diffi-
culty of the task itself but also by task-unspecific processes
(e.g., lapses of attention, fatigue, or disengagement).
In response, there are now attempts to use so-called
brain-computer interfaces for designing adaptive computer-
supported learning environments. Brain-computer interfaces
are thought to allow for a more direct and implicit monitor-
ing of learners’ states like cognitive workload by means of
measuring specific neural correlates of these states (Gerjets
et al. 2014; Zander & Kothe 2011). One possibility for such
an interface is to evaluate the signature of learners’ electroen-
cephalogram (EEG), i.e., the specific patterns of electric sig-
nals produced by the brain. EEG is an electrophysiological
method for monitoring the electrical activity of the brain. It
is typically noninvasive, with electrodes placed on the scalp
of participants. EEG measures the brain’s spontaneous as
well as task induced electrical activity over time using mul-
tiple electrodes placed across the scalp. Thereby, it measures
voltage fluctuations resulting from ionic current flow within
the neurons of the brain (see Niedermeyer & da Silva 2004
for an overview). Current brain-computer interfaces mostly
focus on spectral aspects of EEG, that is, the type of neu-
ral oscillations (popularly called “brain waves”) that can
be observed in EEG signals. The critical methodological
point here is to guarantee an effective adaptation of learn-
ing contents based on EEG signals by developing methods
for real-time assessments of learners’ cognitive state (e.g.,
from bored to overwhelmed). Beyond using EEG, functional

EEG-based prediction of cognitive workload induced by arithmetic: a step towards online…
1 3
magnetic resonance imaging (fMRI) has also been pro-
posed as a basis for real-time brain-computer interfaces that
may eventually allow for more specific adaptation of learn-
ing environments to cognitive workload as induced by, for
instance, arithmetic tasks (for reviews see McFarland et al.
2011 for brain-computer interfaces using EEG; Sulzer et al.
2013 for brain-computer interfaces using fMRI).
In the present study, we investigate whether it is possible
to predict cognitive workload induced by the difficulty of
arithmetic problems based on participants’ EEG signatures
while solving the task. In particular, we are interested in
whether it is possible to classify different levels of work-
load induced by arithmetic tasks of low, medium or high
task difficulty based on EEG signatures. Moreover, we test
how well this classification works across participants. Addi-
tionally, we evaluate whether the crucial EEG signatures
used for classification are in line with the literature on EEG
correlates of numerical cognition. This reflects the first step
in developing a learning environment for numerical com-
petencies, which is adapted to the individual learners’ com-
petency levels in real time by means of potentially domain-
specific EEG correlates. Before reporting the details of the
empirical study, we will therefore provide a brief overview
of known EEG correlates of numerical cognition.
Generally, numerical cognition is associated with a
fronto-parietal brain network implementing domain-spe-
cific numerical processes. These processes are subserved
by cortex areas around the intraparietal sulcus, and are
supported by less number-specific processes associated
with (pre)frontal cortices (e.g., working memory, etc.;
Dehaene et al. 2003; Klein et al. 2013, 2016). Interestingly,
this is also reflected in the EEG correlates of numerical
cognition. In the first EEG study on numerical cognition,
Dehaene (1996) observed reliable modulations in event-
related potentials at parietal electrodes associated with the
specific processing of number magnitude. In particular, he
found that the numerical distance effect (i.e., slower deci-
sions in singling out the larger of two numbers when these
are numerically close, e.g., 4 and 5 vs. 1 and 9, Moyer &
Landauer 1967) is reflected in participants’ EEG signa-
tures. Meanwhile, several studies substantiated that in
particular, number magnitude processing modulates event-
related potentials at parietal electrodes (e.g., Jost et al.
2004; Galfano et al. 2009; Hsu & Szucs 2012; Libertus,
Woldorff, & Brannon 2007; Turconi et al. 2004). In addi-
tion to event-related potentials analyses investigating the
time-course of brain activity, there is also growing interest
in evaluating oscillatory EEG activity induced by number
processing (e.g., Grabner & De Smedt 2011, 2012; Har-
mony et al. 1999; Micheloyannis et al. 2005; Moeller et al.
2010). Physiologically, these oscillations are related to
the coupling and uncoupling of functional networks in the
brain. Thereby, they provide incremental insights into how
task-related neuronal networks are formed and interact with
each other (Neuper & Pfurtscheller 2001; Klimesch et al.
2005). Moreover, in contrast to event-related potentials
whose analysis requires averaging over at least 50 trials,
reliable oscillatory measures of induced EEG activity can
be obtained based on only a few (or even single) trials (e.g.,
Grabner & De Smedt 2011). Therefore, these measures are
promising candidates for the development of electrophysi-
ological markers of task difficulty that might eventually
allow for real-time adaptations in learning environments
(Gerjets et al. 2014). Studies considering oscillatory EEG
data provided convincing evidence suggesting differen-
tial functional significance of various frequency bands. In
numerical cognition—similar to other tasks inducing cog-
nitive workload—it was observed that cognitive demands
imposed by an arithmetic task result in task-related band
power increases in the theta band and decreases in the alpha
band (Harmony et al. 1999). Furthermore, the coincidence
of parietal theta band increase and alpha band decrease is
in line with a strong involvement of specific fronto-parietal
networks related to number processing (see also Micheloy-
annis et al. 2005; Moeller et al. 2010). Therefore, EEG data
and in particular the evaluation of the theta and alpha band
power may provide evidence for the specific neural activa-
tion of the numerical fronto-parietal network.
Against this background, the present study follows a
three-step procedure: In the first step, we explore whether
the task difficulty of arithmetic addition problems is reliably
reflected in the individual EEG signatures of the participants.
As postulated, for instance, by cognitive load theory (e.g.,
Gerjets et al. 2009; Sweller et al. 1998), increasing task diffi-
culty should increase cognitive workload, since by definition,
an increase of task difficulty demands additional cognitive
processing resources that might be provided by domain-gen-
eral (e.g., frontal) structures as well as by domain-specific
(e.g., parietal) structures. Therefore, we hypothesize that an
increase of cognitive workload should result in an increase in
the theta band EEG power, and a decrease in the alpha band
EEG power (Pesonen 2007). In the second step, we train a
regression model to predict item difficulty and thereby dif-
ferentiate the presented arithmetic problems into three cat-
egories ranging from ‘low’ to ‘medium’ to ‘high’ difficulty.
In the final third step, we explore whether it is possible to
reduce the number of electrodes needed for classification
towards a more practical application.
2 Methods
2.1 Participants
Ten students (6 female and 4 male; range 17–32 years,
M = 24.9 years, SD = 5.3 years) participated voluntarily

M. Spüler et al.
1 3
in this study and received monetary compensation for par-
ticipation. All participants reported to have normal or cor-
rected to normal vision and no mathematical problems.
Participants were chosen randomly. Nevertheless, all par-
ticipants were university students (with different fields of
study) and can thus be considered as having a high educa-
tional background.
2.2 Stimuli, task, and procedure
Each participant had to solve 240 addition problems with
an increasing level of difficulty while his/her EEG was
measured. These math problems required the addition of
two numbers ranging from two single-digit to two four-
digit numbers. Depending on the constituting digits, the
addition problems required either no, one, or more carry
over procedures. In total, 43 % of the 240 trials required a
carry over.
Problems are presented in six blocks of increasing level
of difficulty. The first block consists of 90 easy problems
with single-digit addends and no carry over (e.g., 2 + 4).
The following 5 blocks increases in difficulty and consist
of 30 problems each.
The difficulty of the addition problems is defined in
terms of their so-called Q-value. The Q-value was sug-
gested by Thomas (1963) to reflect the information content
of an arithmetic task. The main parameters for problem dif-
ficulty in addition are the size of the summands involved
(i.e., problem size, see Stanescu-Cosson et al. 2000) and
whether or not a carry over operation is needed (see Kong
et al. 2005). Importantly, the Q-value takes into account
both problem size and the need for a carry over operation.
Therefore, the Q-value is a more comprehensive measure
of task difficulty as compared to using only one parameter
such as problem size. Moreover, the Q-value also consid-
ers additional aspects, which should affect task difficulty
(e.g., the reduced difficulty of specific problems such as
1000 + 1000). Therefore, we are confident that the Q-value
is an adequate measure of task difficulty.
Basically, the Q-value considers the magnitudes of the
single digits and whether a carry over is needed or not.
Given an arithmetic problem x + y, with x
i
reflecting digit
i of x (x
1
reflects the units, x
2
reflects the tens of the first
summand, etc.), and y
i
denoting digit i of y (y
1
reflects
the units, y
2
reflects the tens of the second summand, etc.)
Q(x + y) is computed as follows:
For single-digit problems that do not require a carry over:
Considering the problem 2 + 4, this would equal:
Q
(
x + y
)
= log

x
1
+ y
1
+

x
1
+ y
1

.
Q
(
2
+
4
)
=
log
[
2
+
4
+
(
2
+
4
)
]
=
log
(
2
+
4
+
6
)
=
log
(
12
)
=
1.07.
For single-digit problems that require a carry over:
Given the problem 7 + 9, this would mean:
For multi-digit problems an additional parameter c = 1
indicates the carry over from one position to the next:
For the example problem 27 + 49, the Q-value equals:
Q-Values for addition problems used in this study ranged
from Q = 0.6 (easy single-digit problems) to Q = 7.2 (dif-
ficult 4-digit problems).
Each trial consists of 3 phases. The problem was pre-
sented for 5 s. During this period participants had to cal-
culate the result which they had to type within a 3.5 s long
response period. This was followed by an inter-trial interval
of 1.5 s, resulting in a total length of about 40 min for the
experiment.
To avoid perceptual-motor confounds the time windows
used for analyzing EEG-data should not contain motor
events. As typing in the answer leads to motor artifacts, the
calculation phase was used for EEG analysis (Fig. 1).
2.3 EEG recording
A set of 28 active electrodes (actiCap, BrainProducts
GmbH), attached to the scalp, placed according to the
extended international electrode 10–20 placement system
(FPz, Afz, F3, Fz, F4, F8, FT7, FC3, FCz, FC4, FT8, T7,
C3, Cz, C4, T8, CPz, P7, P3, Pz, P4, P8, PO7, POz, PO8,
O1, Oz, O2), was used to record EEG signals. Three addi-
tional electrodes were used to record an electrooculogram
(EOG); two of them were placed horizontally at the outer
Q
(
x + y
)
= log

x
1
+ y
1
+

x
1
+ y
1

+ 10 +

x
1
+ y
1
− 10

.
Q
(
7
+
9
)
=
log
(
7
+
9
+
16
+
10
+
6
)
=
log
(
48
)
=
1.68.
Q

x
1
x
2
+ y
1
y
2

= Q

x
1
+ y
1

+ Q

x
2
+ y
2
+ c

= log

x
1
+ y
1
+

x
1
+ y
1

+ 10 +

x
1
+ y
1
− 10

+ log

x
2
+ y
2
+

x
2
+ y
2

+ c
2
+

x
2
+ y
2
+ c
2

.
Q
(
27
+
49
)
=
Q
(
7
+
9
)
+
Q
(
2
+
4
+
1
)
=
Q
(
7
+
9
)
+
log
(
2
+
4
+
6
+
1
+
7
)
=
2.98.
Fig. 1 Schematic illustration of the time course of the experiment.
The grey area indicates the considered activation interval (IA),
whereas the white area represents response and inter-stimulus inter-
val (ISI). At the beginning and the end of the experiment, a fixation
cross was displayed

EEG-based prediction of cognitive workload induced by arithmetic: a step towards online…
1 3
canthus of the left and the right eye, respectively, to measure
horizontal eye movements and one was placed in the mid-
dle of the forehead between the eyes to measure vertical eye
movements. Ground and reference electrodes were placed
on the left and right mastoids. EOG- and EEG-signals were
amplified by two 16-channel biosignal amplifier systems
(g.USBamp, g.tec) and sampled at a rate of 512 Hz. EEG
data were high-pass filtered at 0.5 Hz and low-pass filtered
at 60 Hz during the recording. Furthermore, a notch-filter
was applied at 50 Hz to filter out power line noise.
2.4 Data processing and analysis
To remove the influence of eye movements, we applied an
EOG-based regression method (Schlögl 2007). For data
analysis, we estimated the power spectrum during the cal-
culation phase of each trial. We used Burg’s maximum
entropy method (Cover and Thomas 2006) with a model
order of 32 to estimate the power spectrum from 1 to 40 Hz
in 1 Hz bins.
Before using a linear ridge regression
1
to predict task
difficulty, we calculated the squared correlation coefficient
(R
2
) between the power at each frequency bin (for each
electrode) and Q-values of the corresponding trials. This
analysis serves as an estimate at which electrodes and fre-
quencies the EEG signal is associated strongest with task
difficulty. In turn, this allows for the identification of the
EEG features that are most important to predict the cogni-
tive workload of a learner.
2.5 Predicting task difficulty
The prediction of task difficulty and the evaluation of this
prediction involve three steps:
In the first step, we applied the previously calculated
power spectrum as features to predict task difficulty as
indicated by Q, using a regression approach. For the regres-
sion, Q was considered a continuous variable with Q-values
separated into bins of 0.1. With a range from Q = 0.6 to
Q = 7.2, this summed up to 64 Q-bins. To correct for inter-
participant variability in the participant’s baseline of EEG
power, the first 30 trials (easy trials, with Q < 1) were used
for normalization. The mean and standard deviation for
1
When aiming at identifying features from EEG data that are mean-
ingful to predict cognitive workload it is important to note that there
a lot of possible features available when using EEG (e.g., different
electrodes, different band frequencies, etc.). Therefore, it is necessary
to choose an analysing approach, which is able to reduce features
systematically to identify those which are relevant for prediction. To
this end, the method of choice for analysing our EEG data was ridge
regression (Hoerl 1962), which allows for such a way of data reduc-
tion.
each frequency bin at each electrode was calculated using
the first 30 trials and the remaining 210 trials were scaled
according to these means and standard deviations. The 30
trials used for normalization were not used any further in
the prediction process (neither for training nor for testing
the model).
Based on the normalized data of 9 participants we
trained a linear ridge regression model with the regulari-
zation parameter λ = 10
3
. This regularization parameter
was determined by cross-validation on the training data.
In particular, we performed a nested cross-validation to
determine the best regularization parameter for the cross-
subject regression. In this process, only the training set of
the 9 subjects was used for parameter estimation (nine-
folds, training on 8 subjects, testing on 1). Regularization
parameters were tested in the range from 10
−5
to 10
5
in
exponential steps. Using this procedure, we found λ = 10
3
to result in the best classification performance on average.
Consequently, we skipped the optimization step in further
analysis and used a fixed value of 10
3
. To ensure that the
results of the within-subject and cross-subject analyses are
comparable, we used the same parameter setting in both
analyses (see below Sect. 3.5).
In the next step, we used only every second frequency
bin (1, 3, 5 Hz, …) to train the model in order to reduce the
number of features. When not specified otherwise, we used
the 17 inner electrodes (F3, Fz, F4, FC3, FCz, FC4, C3, Cz,
C4, CPz, P3, Pz, P4, POz, O1, Oz, O2), resulting in a total
of 340 features for prediction purposes.
The regression model was trained using the data of 9
participants (1890 samples in total) to predict the difficulty
level (Q-value) of each of the problems for the remaining
participant. To smooth the prediction result, we calculated
the moving average with a window length of 7 trials as the
final output. The moving average leads to a more robust
prediction (see Walter et al. 2014), but costs the system to
react slower to sudden changes in workload, which is feasi-
ble since it is not recommendable to adapt an online learn-
ing environment every 8.5 s (=single trial duration).
In the second step, we used a leave-one-participant-out
cross-validation to evaluate prediction performance. There-
fore, the data of 9 participants were used for training and
then tested on the one remaining participant. This was
repeated 10 times so that the data of each participant were
used once for testing. As criterion for performance evalua-
tion of the presented prediction method we used the corre-
lation coefficient (CC) to reflect the statistical relationship
between the actual and the predicted Q-values, as well as
the root mean squared error (RMSE) to examine the differ-
ence between the actual and the predicted Q-values (Spüler
et al. 2015).
We assumed that a precise prediction of task difficulty
will not be necessary in an adaptive learning environment

M. Spüler et al.
1 3
as it is more important to identify when the task gets too
easy or too difficult in order to keep the learner in his/her
optimal range of cognitive workload. Therefore, in the third
step, we used the output of the regression model to predict
the current workload state of each learner and differentiated
three difficulty levels, which were based on learners’ per-
formance levels (cf. Fig. 2). For Q < 2 the trial was con-
sidered easy, for Q > 4 it was considered too difficult, and
the trial was given a medium difficulty level when Q was
between 2 and 4. To estimate the validity of our classifica-
tion procedure, we applied this classification system to the
actual Q and the predicted Q and evaluated the classifica-
tion accuracy (CA), which was given as the percentage of
trials correctly classified.
Finally, we are interested in how much the prediction of
task difficulty depends on the number of electrodes upon
the prediction is based. Therefore, we repeated the analyses
described above using different subsets of electrodes.
3 Results
3.1 Behavioral results and arithmetic performance
Overall, 64.75 % of the trials were solved correctly. A
further analysis of the performance data shows that the
Q-value is a suitable measure for task difficulty since it
correlates well [r(64) = −0.945, p < 0.001] with the per-
centage of correctly solved trials in the 64 Q-bins (see
Fig. 2).
Although there are inter-individual differences between
participants, trials with Q < 2 were solved correctly in
nearly all cases, while trials with a difficulty of Q ~4 were
solved correctly in about 30 % of the trials only. None of
the participants was able to solve trials with a Q > 6.
3.2 EEG analyses
Considering R
2
values for the association between the
Q-value, the power at each electrode, and frequency bin
showed that task difficulty is reflected in the EEG power
spectrum (see Fig. 3). In the delta frequency band, there is
a small difficulty-related effect over the central electrodes,
while the effect in the theta, alpha, and beta frequency band
is located at the parieto-occipital electrodes. This effect is
strongest for the alpha band. While the lower beta band
(13–24 Hz) still shows some effects related to task dif-
ficulty, they cannot be observed in the upper beta band
(25–40 Hz).
3.3 Difficulty prediction
The result of the regression based difficulty prediction for
each of the 10 participants is shown in Fig. 4. The average
correlation coefficient between the actual and the predicted
Q-value for all participants is CC = 0.74 with an average
RMSE of 1.73.
Fig. 2 Percentage of correctly solved trials depending on the diffi-
culty level of each trial measured in terms of Q. Each participant is
shown by the thin colored lines and the average of all participants is
shown by the bold black line
Fig. 3 Topographic display of R
2
values averaged over all participants, showing the influence of the Q-value for each electrode in different fre-
quency bands. R
2
values are color coded with increasing red reflecting higher R
2
values

EEG-based prediction of cognitive workload induced by arithmetic: a step towards online…
1 3
While the increase in Q was generally well predicted
by the model for Q < 4, the model did not work well with
Q > 4, where it either reached a plateau or even predicted a
decline in task difficulty for about half of the participants.
Due to this finding we tested two different regression
models: one trained on all data, and one trained on all tri-
als with Q < 6. This cut-off was chosen because hardly any
problem with Q ≥ 6 was solved correctly by any of the par-
ticipants. While the detailed results for the model trained
on all data are shown in Fig. 4, a comparison of the average
prediction performance metrics for both models is shown
in Table 1. The results regarding CC and RMSE when all
data were used for training did not differ significantly from
the results when only Q < 6 data were used for training.
However, average classification accuracy was signifi-
cantly higher [t(9) = 2.46, p < 0.05] when only the Q < 6
data were used for training, reaching an average classifi-
cation accuracy of 56 % (33 % represents chance level).
It should also be noted that in general the classification
worked well for the detection of easy and medium dif-
ficulty levels with classification accuracy of 68 and 84 %,
respectively. Yet, it performed below chance level with a
classification accuracy of only 16 % for difficult problems.
Training the model on trials with Q < 6 and reducing the
test data to trials with Q < 6 resulted in significantly better CC
[t(9) = −3.32, p < 0.01] and RMSE [t(9) = 8.48, p < 0.001],
while the classification results did not differ significantly.
3.4 Reduction of electrodes
Since the application of an EEG-based adaptive learning
environment would benefit from a reduced number of elec-
trodes, we also evaluated how the use of different subsets
of electrodes would affect prediction performance. As it
can be seen in Table 2, there is no significant difference in
prediction performance (all t < 1.1, all p > 0.1) when either
all electrodes were used for prediction, only the 17 inner
electrodes were used, or a selection of central or parieto-
occipital electrodes were used. When only one single elec-
trode was used to produce a difficulty prediction, POz was
the electrode that showed the best results. While its classifi-
cation accuracy is significantly lower than the 4 previously
Fig. 4 Difficulty prediction using linear ridge regression on features
from 17 EEG channels (blue line) and actual difficulty level (red
line) for all participants. The correlation coefficient (CC) between the
actual and predicted Q and the root mean squared error (RMSE) are
shown at the top half of each plot. At the bottom half of each plot,
trials solved (TS) are shown with 1 indicating a correctly solved and
0 reflecting an incorrectly solved problem. Each light blue cross rep-
resents one addition problem, while the black line is a smoothed ver-
sion of the trials solved vs. trials plot, depicting the smoothed per-
centage of correctly solved problems over time

M. Spüler et al.
1 3
mentioned electrode setups (all t > 2.5, all p < 0.05), there
is no significant difference in terms of CC and RMSE.
3.5 Predictions within vs. across participants
The above results reflect a cross-participant prediction, in
which the performance of one participant was accounted
for by a model trained on the data of the other nine par-
ticipants. For reasons of comprehensiveness, we also per-
formed a within-participant prediction, in which the model
was trained and tested on the data of the same participant
using a 10-fold cross-validation. For this we only consid-
ered the data from all inner electrodes and Q < 6. Other-
wise, the procedure was identical to the methods and
parameters as in the cross-subject prediction.
While the cross-participant prediction resulted in an
average CC of 0.82 and an RMSE of 1.34 (as reported
above), the within-participant prediction achieved an aver-
age CC of 0.90 and an RMSE of 0.95, which is signifi-
cantly better for CC [t(9) = −3.28, p < 0.01] and RMSE
[t(9) = −7.04, p < 0.001]. As discussed in Walter et al.
(2014), the benefit of reduced calibration and training effort
in cross-participant prediction seems to outweigh the per-
formance increase of the within-participant prediction.
4 Discussion
The present study aims at identifying math-related cogni-
tive workload imposed onto learners by using oscillatory
EEG data following a three-step procedure. In the first
step, we aimed at predicting cognitive workload induced by
solving addition problems from the EEG signatures of par-
ticipants. In the second step, we trained a classifier to dif-
ferentiate the presented arithmetic problems into three dif-
ficulty categories (i.e., easy, medium, and difficult). In the
third step, we were interested in the number-specificity of
the considered EEG features and whether it is possible to
reduce the number of electrodes needed for classification.
Our data are meaningful in all three respects: First, we
were successful in predicting the workload induced by the
difficulty of arithmetic problems from participants EEG
signature. Second, classification accuracy was significantly
better than chance level even when classification was done
across participants. Finally, we observed that it is well
possible to achieve reliable classification accuracy with a
reduced set of electrodes or even a single electrode. The
most relevant electrodes and oscillatory EEG features for
classification are well in line with the literature on EEG
signatures in numerical cognition. In the following, these
points will be discussed in greater detail.
With respect to the prediction of task difficulty by EEG
signatures we find that the Q-value is indeed a good indi-
cator for task difficulty as reflected by the reliable corre-
lation between Q and calculation accuracy. Furthermore,
the present data corroborate our hypothesis that it should
be possible to predict task difficulty and thus induce cogni-
tive workload by means of participants’ EEG signatures. A
ridge regression analysis clearly indicated that the Q-value
of an arithmetic problem can be predicted significantly well
Table 1 Prediction
performance with different data
(either all, or only trials with
Q < 6) for training and testing
averaged over all participants
For performance prediction we used the correlation coefficient (CC), root mean squared error (RMSE) and
the Classification Accuracy (CA) for the three difficulties (easy, medium, and difficult) as well as the aver-
age of these three difficulty levels
Train Test CC RMSE CA
Easy
(%) CA
Normal
(%) CA
Hard
(%) CA (%)
All All 0.745 1.726 44.8 74.2 30.4 49.8
Q < 6 All 0.764 1.778 67.7 84.2 16.0 56.0
Q < 6 Q < 6 0.820 1.343 67.7 84.2 15.3 55.8
Table 2 Average prediction
performance of all participants
for different electrode subsets
Each row provides a description of the subset, including the number of electrodes used, the approximate
location of the electrodes, as well the position of all electrodes in the subset according to the 10–20 system.
As for performance metric, the table shows the correlation coefficient (CC), the root mean squared error
(RMSE) and the average classification accuracy (CA)
Number Description Positions CC RMSE CA (%)
28 All See Sect. 2 0.807 1.292 55.6
17 Inner See Sect. 2 0.820 1.343 55.8
7 Central Fz, FCz, Cz, CPz, Pz, POz, Oz 0.839 1.352 54.7
9 Frontocentral F3, Fz, F4, FC3, FCz, FC4, C3, Cz, C4 0.725 1.465 45.5
7 Parietooccipital P3, Pz, P4, POz, O3, Oz, O4 0.831 1.383 54.1
1 Best electrode POz 0.788 1.479 50.5

EEG-based prediction of cognitive workload induced by arithmetic: a step towards online…
1 3
by a selection of EEG features with contributions from
parietal electrodes in the alpha and theta band being the
most predictive.
In the next step, we observe that in all cases, the model,
which was trained on the EEG data of nine participants, is
capable of differentiating between easy, medium, and dif-
ficult arithmetic problems based on the EEG data of the
remaining participant with a classification accuracy sig-
nificantly above chance. As expected, this capability is
even more pronounced for within-participant classification.
However, it needs to be noted that the classifier works best
for values of Q smaller than 4. As can be taken from Fig. 4,
the model is not able to reliably categorize problems with
a Q value larger than 4. The phenomenon of decline in Q
prediction for the latter problems together with the fact
that our results become better when considering only with
a Q-value smaller than 6 indicate non-linear effects in Q.
Yet, considering how the difficult problems (with Q > 6)
are constructed, this does not seem surprising. The respec-
tive problems mainly require the addition of two 4-digit
numbers with the need for at least one carry over (e.g.,
4326 + 5618). Given the limited time of 5 s for calcula-
tion, participants might have experienced cognitive over-
load when solving these problems. As a consequence, the
observed decline in classification performance for these
items may be due to the fact that participants may no longer
have tried to solve the respective problems but may have
become disengaged simply because the tasks were too dif-
ficult. As this would change the associated EEG signature
considerably and thus disrupt the systematic increase in
workload, it might well account for the drop of prediction
performance for these very difficult problems. However,
this needs to remain speculative, as it is difficult to detect
with the current study design not considering any subjec-
tive measures of, for instance, disengagement. Neverthe-
less, for future learning environments it might be desirable
to specifically consider EEG correlates of cognitive over-
load and/or disengagement to increase performance of the
prediction model.
A finding of high relevance for future practical applica-
tions is that the model still performed above chance level
when the number of electrodes were reduced to subsets of
the original set of electrodes or even when only one elec-
trode (POz) was used. The fact that it is possible to reduce
the number of electrodes needed for the model to produce a
classification accuracy above chance level is highly impor-
tant for the prospects of future practical applications of EEG-
based adaptive systems, because it largely increases the usa-
bility of these systems. In terms of adapting the content in
a computer-supported learning environment based on EEG
signatures, such a system would rely much more directly on
cognitive processes as compared to existing programs such
as Calcularis (Käser et al. 2012, 2013a), which only consider
behavioral input given by users (for a more comprehensive
discussion of this aspect see Gerjets et al. 2014).
Furthermore, it is worth to mention that the current find-
ings are in line with previous literature on both the neu-
ral correlates of numerical cognition in general as well as
EEG correlates of numerical cognition in particular. In the
present study, EEG band power signatures from parietal
electrodes (P3, P4, and Pz) turn out to be the most predic-
tive of task difficulty and thus important for classification.
This replicates earlier results from studies evaluating ERP
(Dehaene 1996; Galfano et al. 2009; Hsu & Szucs 2012;
Libertus, Woldorff, & Brannon 2007; Turconi et al. 2004)
as well as oscillatory EEG effects induced by numerical
tasks (e.g., Grabner & De Smedt 2011, 2012; Harmony
et al. 1999; Micheloyannis et al. 2005; Moeller et al. 2010).
We did not use ERP analyses, because they usually require
a large number of trials for results to be meaningful and are
therefore not suitable for algorithms used for online adapta-
tion. Instead, we consider oscillatory EEG data and observe
that parietal electrodes are most relevant for classification.
Moreover, we observe that modulations in the theta and
alpha band are associated most closely with changes in
cognitive workload states induced by arithmetic difficulty,
which is in line with previous studies. While electrodes P3
and P4 are actually situated on the scalp over the posterior
part of the intraparietal sulcus (e.g., Okamoto et al. 2004) it
should be noted that localization has to be interpreted very
cautiously for EEG data. In fact, while the good fit with the
literature on the neural correlated numerical cognition (e.g.,
Arsalidou & Taylor 2011 for a review) is tempting, this
does not prove a connection between the anatomical struc-
ture and the EEG data measured at these electrodes.
At last, some limitations to this study need to be dis-
cussed. The study uses a block design in which task diffi-
culty increases steadily from block to block. This might be
considered a confounding factor as non-stationary effects
(like fatigue) may have influenced the EEG data. One might
thus suspect that prediction performance of our classifica-
tion is based on such effects rather than problem difficulty.
However, there are several reasons why this seems unlikely.
For instance, the assumption of the continuous increase of
task difficulty over blocks cannot explain the discontinuous
drop in classification performance for Q > 6. We choose
the respective block design on purpose to realize a steady
increase of task difficulty, which is in line with the order of
items in other speed performance tests such as intelligence
tests. Comparable to these tests, we aim to reduce factors
such as the lack of motivation and/or participants’ frustra-
tion, which might be caused by having to start with a block
of very difficult (e.g., four-digit + four-digit number) addi-
tion questions. Furthermore, the aim of this project is to
pave the way for the later development of a realistic adap-
tive learning environment, in which task difficulty usually

M. Spüler et al.
1 3
increases steadily. Nevertheless, it is true that the chosen
procedure may have reduced unsystematic error variance.
Therefore, it is highly desirable for future studies to evalu-
ate how our prediction model performs in a scenario where
trials of differing difficulty are presented in randomized
order to generalize the validity of the approach.
Moreover, we need to take into considerations that EEG
oscillations in the theta and alpha band have also been
associated with domain-general processes such as working
memory (for an overview see Gerjets et al. 2014; Klimesch
1999). An increase in working memory demands is found
to lead to increases in theta band oscillations over frontal-
midline electrodes (e.g., Gevins et al. 1997; Sauseng et al.
2010) as well as decreases in alpha band oscillations over
parieto-occipital electrodes (e.g. Gevins et al. 1997; Krause
et al. 2010; Scharinger, Kammerer, & Gerjets 2015a; Schar-
inger, Soutschek, Schubert, & Gerjets 2015b). Therefore,
future studies are needed to further disentangle correlates
of domain-general and domain-specific numerical processes
in the EEG signatures, in order to clarify the issue to what
extent the features identified to predict task difficulty in this
study are specific to the domain of number processing.
Finally, one might suspect that the classifier simply
predicted qualitatively different solution strategies such
as arithmetic fact retrieval and magnitude manipulations
instead of the differences in cognitive workload induced by
task difficulty. Following this rationale one would assume a
discontinuity at the point where participants need to switch
from one strategy to another. However, when considering
Fig. 4 (red lines) it becomes obvious that this is not the
case. Instead, discontinuities are caused by the progression
from single- to two- to three- and to four-digit numbers
(reflected by steep increases in the Q-value). Additionally,
our classifier successfully differentiated three different lev-
els of task difficulty whereas it is most common to differ-
entiate only two solution strategies (i.e., fact retrieval and
magnitude manipulations). In sum, this argues against the
idea that classification results simply reflect qualitatively
different solution strategies.
In this context, it should be noted that there is increasing
evidence suggesting that this does not seem to be a question
of whether it is magnitude manipulation or fact retrieval.
Rather, it is suggested that—depending on task difficulty—
it is a question of how much magnitude manipulation vs.
how much fact retrieval (e.g., Klein et al. 2014). On the one
hand, even for very easy tasks (e.g., additions with sum-
mands <5) there is evidence for the consideration of the
summands magnitude (e.g., Thevenot et al. 2007). On the
other hand, arithmetic fact retrieval is claimed to be used
when, for instance, calculating the sums of units, tens, hun-
dreds, etc. in multi-digit addition (e.g., 214 + 672 = 2 + 6
at the hundreds, 1 + 7 at the tens, and 4 +2 at the ones
positions (e.g., Dehaene & Cohen 1995).
In conclusion, there are only few studies on the EEG-cor-
relates of number processing. However, the existing litera-
ture is highly consistent in revealing significant modulation
of EEG signatures by number processing at electrodes placed
above the parietal cortex (e.g., Dehaene1996; Libertus,
Woldorff, & Brannon 2007; Turconi et al. 2004). In the cur-
rent study, we did not aim to replicate these findings but show
that, based on this knowledge, EEG signatures from these
electrodes are meaningful to reliably predict difficulty of an
addition task. In the near future, this might be used to adapt
learning environments to keep task difficulty within the zone
of proximal development to corroborate individualized learn-
ing. Moreover, our results indicate that classification perfor-
mance in the present study is unlikely to capitalize on chance
or unspecific cognitive processes. Instead, the EEG band
power signatures upon which classification is based seem to
reflect cognitive workload states induced by number-specific
processes (cf. Harmony et al. 1999; Grabner & DeSmedt
2012). Against this background, the implementation of an
EEG-based online adaptation for a computer-supported learn-
ing environment for numerical competencies seems possible:
(i) We observe significant classification accuracy with only
a small number of electrodes and (ii) the prediction model
was trained in a cross-participant approach in which no EEG
data of the actual user was needed for training—meeting two
important criteria for practical application.
Acknowledgments The authors would like to thank Philipp Wolter
for programming the experimental setup and collecting the EEG data.
The current research is supported by the Baden-Württemberg Stiftung
(GRUENS), the German Research Foundation (DFG; SP 1533/2-1)
and the Leibniz ScienceCampus Tübingen “Informational Environ-
ments.” Carina Walter is a doctoral student of the LEAD Graduate
School [GSC1028], her work is funded by the Excellence Initiative
of the German federal and state governments. Elise Klein is supported
by the Leibniz-Competition Fund (SAW-2014-IWM-4) as well as by
a Margarete-von-Wrangell Fellowship (European Social Fund and
the Ministry of Science, Research and the Arts Baden-Württemberg).
Korbinian Moeller, Wolfgang Rosenstiel and Peter Gerjets are princi-
ple investigators of the LEAD Graduate school at the Eberhard Karls
University Tuebingen funded by the Excellence Initiative of the Ger-
man federal government.
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