AN EVENT-RELATED BRAIN POTENTIAL STUDY OF THE ARITH METIC

SPLIT EFFECT

M. Isabel Núñez-Peña1,2 and Carles Escera2

1 Department of Behavioral Sciences Methods, Faculty of Psychology, University of

Barcelona, Spain 2 Cognitive Neuroscience Research Group, Department of Psychiatry and Clinical

Psychobiology, Faculty of Psychology, University of Barcelona, Spain

Number of text pages: 24

Number of Figures: 3

Number of Tables: 1

Corresponding author:

Dr. María Isabel Núñez-Peña

Department of Behavioral Sciences Methods

Faculty of Psychology

University of Barcelona

Passeig Vall d’Hebron, 171

08035 Barcelona (SPAIN)

Tel. +34 93 312 58 53

Fax. +34 93 402 13 59

E-mail: inunez@ub.edu

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Abstract

This study explores the split effect on arithmetical sequence processing using event-related

brain potentials (ERPs). This effect has been reported in arithmetic problem verification

tasks and refers to an increase in reaction time when an incorrect solution close to the

correct one is presented. The use of different strategies has been suggested to account for

this effect: a whole-calculation strategy for close incorrect solutions (small-split problems),

and a plausibility-checking strategy for far or obviously incorrect solutions (large-split

problems). In the present study, participants were asked to verify whether the last item of a

numerical sequence was correct or not, according to the rule established by the preceding

numbers. Subjects were presented with sequences of numbers, and the distance between the

proposed and the correct ending was manipulated by presenting the correct ending, a small-

split ending or a large-split ending. To avoid problems that violated the parity rule only

even numbers were presented. Two ERP components were evident whenever the

arithmetical sequence was broken: an early negativity peaking at about 270 ms and a

subsequent late positivity component (LPC) peaking between 550 and 650 ms. The early

negativity was larger at posterior sites. The late positivity had a centro-parietal scalp

distribution, and its amplitude was sensitive to the numerical distance from the correct

number: the greater the distance, the larger the positivity. The present results suggest that

the centro-parietal LPC can be taken as an index of the split effect and may broaden our

knowledge about arithmetical processing strategies.

Key Words: arithmetic; split effect; odd-even effect; sequence processing

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INTRODUCTION

Number processing and mathematical calculation is a field of research with a growing

interest in cognitive neuroscience (for a recent review, see Campbell, 2005). Several

phenomena have been described concerning the processing of numerical information, but

one of the most robust phenomena is the effect of the manipulation of the numerical

distance between the proposed and the correct solutions in a problem verification task. This

is usually known as the split effect (Ashcraft and Battaglia, 1978; Ashcraft and Stazyk,

1981) and it is the core of this paper.

The split effect appears in verification tasks of simple mental calculation problems

where incorrect solutions close or far from the correct solution are presented. Put in a

nutshell, this effect consists of a slower and less accurate response when the proposed

solution is a number close to the correct solution. The suggested explanation for the split

effect is that two different strategies to solve arithmetical problems are used (Duverne and

Lemaire, 2005; El Yagoubi, et al., 2003; 2005). When the proposed solution is close to the

correct solution, subjects may use an exhaustive verification strategy — the whole-

calculation strategy — to achieve the exact calculation. However, when the proposed

solution is a long way from the correct one, subjects may not need to obtain the exact

calculation and may give their answer by using an easier and faster estimation strategy —

the plausibility-checking strategy. Thus, one can speculate that when an incorrect solution

near the correct one is presented in an arithmetical task (from now on, small-split solution),

subjects need to calculate the problem to decide about the correctness of the proposed

solution. By contrast, when an incorrect solution that is a long way from the correct one is

presented (from now on, far-split solution), subjects may not need to calculate the problem

to give their answer of incorrectness.

Event-related potential (ERP) studies of brain activity have reported a late positive slow

wave at parietal sites functionally related to mental arithmetical calculation (Iguchi and

Hashimoto, 2000; Núñez-Peña et al., 2005, 2006; Pauli et al., 1994, 1996). This component

can be useful to shed new light on the split effect and to find out whether this effect reflects

the type of strategy used to solve arithmetical problems. If subjects need to calculate the

problem when a small-split solution is presented in a verification task, the late positive slow

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wave is expected to be elicited, because exact calculation is necessary. However, this late

positive slow wave should not be elicited by large-split solutions, because the problem can

be solved by a plausibility-checking strategy.

Previous studies with ERPs have reported a number of findings regarding the effects of

violations of arithmetical rules in verification tasks (Niedeggen and Rösler, 1999; Núñez-

Peña and Honrubia-Serrano, 2004; Szűcs and Csépe, 2004, 2005). Niedeggen and Rösler

(1999) presented simple multiplication problems and asked participants to judge their

correctness. Incorrect solutions elicited a negativity peaking at about 400 ms followed by a

posteriorly distributed late positive component (LPC). In the incorrect solutions, the

distance from the correct number was manipulated and the results showed that the

amplitude of the late positivity increased monotonically with the size of the numerical

distance between the incorrect and correct solutions. These authors related this positive

deflection to the P3b component, which is assumed to indicate the amount of surprise or

context updating (Donchin and Coles, 1988). Similar results to those of Niedeggen and

Rösler (1999) have been reported by Szűcs and Csépe (2004, 2005) when participants

verified simple additions. These authors also reported a late positive component elicited by

incorrect solutions. Although they manipulated the numerical distance from the correct

solutions, no modulation of the amplitude of the late positive component was found. Szűcs

and Csépe (2004, 2005) also reported a negative component in response to incorrect

arithmetical results, but in this case it was earlier (at about 270 ms post-stimulus) than that

reported by Nieddeggen and Rösler (1999).

In a recent study (Núñez-Peña and Honrubia-Serrano, 2004), we have reported a similar

positive component using numerical sequences. Sequences of seven numbers were

presented, and participants were asked to judge the correctness of the last number in the

sequence. For the incorrect endings, the numerical distance was manipulated in such a way

that the target could deviate by ±1 or +25 from the correct ending. Incorrect numbers

elicited an early negativity, peaking at about 270 ms, followed by a late positive

component. The late positivity had a latency between 500 and 600 ms post-stimulus and a

centro-parietal scalp distribution. Moreover, its amplitude was sensitive to numerical

distance, being largest when the number presented was furthest from the correct one. On

the basis of these results, we suggested that this positivity was functionally similar to the

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syntactic P600 component (Frisch et al., 2002; Kaan et al., 2000; Kaan and Swaab, 2003a,

2003b; Osterhout et al., 1994), and so similar neurophysiological processes may be

required for processing numerical sequences and linguistic syntactic structures.

Summarising, in all previous studies a pattern of a negative component followed by a

LPC has been described whenever an incorrect solution to an arithmetical problem is

presented. The modulation of the positive component as a function of the distance between

correct and incorrect solutions has been explained in different ways, but it may also be

explained in terms of the type of arithmetic problem solving strategy that subjects used to

verify the problem. It has been suggested elsewhere (Iguchi and Hashimoto, 2000; Núñez-

Peña et al., 2005, 2006; Pauli, et al., 1994, 1996) that a parietally distributed positive slow

wave can be taken as a direct electrophysiological correlate of the exact calculation.

Therefore, the positive slow wave elicited by small-split problems might be due to the use

of a whole-calculation strategy, whereas the sharp late positive wave elicited by large-split

problems might result by the use of a plausibility-checking strategy. Although the late

positive slow wave reported in these studies (Niedeggen and Rösler, 1999; Núñez-Peña and

Honrubia-Serrano, 2004) could be taken as an index of the use of an exact calculation

strategy to verify small-split problems, this conclusion could be invalidated by the fact the

split effect might have been disturbed in these studies by another effect that is well

established in the literature: the odd-even effect. The odd-even effect consists of the faster

rejection in verification tasks of an incorrect proposed result if its parity differs from the

parity of the expected correct result. This effect has been described both in additions — i.e.,

4 + 6 = 13 is rejected faster than 4 + 6 = 12 — (Kruenger and Hallford, 1984; Vandorpe et

al., 2005) and multiplications — i.e., 4 x 6 = 25 is rejected faster than 4 x 6 = 26 —

(Krueger, 1986; Lemaire and Fayol, 1995; Masse and Lemaire, 2001). The explanation of

this effect is that subjects use a parity rule such as “a sum must be odd if an odd number of

its addends are odd, otherwise it must be even” and “a product must be even if either one or

both of its multiplier is even, otherwise it must be odd” (Vandorpe et al., 2005). The fact is

that if a small-split solution that violates the parity rule is presented in a verification task,

participants may not need to calculate the problem to give their answer of incorrectness

(i.e., 4 + 6 = 11), because they can solve the problem by means of the parity-checking

strategy. As a consequence, a whole-calculation strategy would not be used to achieve the

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exact calculation although the distance between the correct and the proposed solution was

small. As we have previously mentioned, some studies that have investigated the split

effect on the ERPs while subjects verify arithmetic problems have not accounted for the

possibility that the split effect could have been disturbed by the odd-even effect. Therefore,

the ERP pattern observed in small-split problems would not be attributed to the fact that

subjects used an exhaustive calculation strategy in these studies.

The present study

The purpose of the present study was to investigate the split effect on arithmetical sequence

processing with ERPs. To prevent subjects from using the parity-checking strategy — in

other words, to isolate the split effect from the odd-even effect — only even numbers were

used. Participants were presented with sequences of five even Arabic numbers that could be

finished by a number that completed the sequence correctly (e.g. 6-10-14-18-22) or by a

number that violated the sequence (e.g. 6-10-14-18-24). The size of the violations varied:

the incorrect number could deviate by ±2 (from now on, this will be referred to as the

small-split ending) or +26 from the correct ending (referred to as the large-split ending).

Based on the literature reviewed above, our first prediction is that the distance between

the proposed and the exact solution will influence subjects’ performance. Specifically,

small-split endings are expected to elicit less accurate responses than both correct and

large-split endings, replicating the standard split effect. Second, we hypothesized

differences in the ERPs depending on the size of the split. A sharp late positive component

is expected to be elicited by large-split problems and a late positive slow wave is expected

to be elicited by small-split endings. The amplitude modulation of this LPC might suggest

that different strategies are used to solve small- and large-split problems: a late positive

slow wave has been related to mental arithmetic calculation and calculation strategies are

considered to be necessary to verify small-split problems. Finally, a negative component

preceding the positive component is expected to be elicited by both small- and large-split

endings, as reported in previous studies.

METHODS

Participants

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Sixteen healthy volunteers were tested in this study (ten women; age 20-38 years,

mean = 25.06, standard deviation = 5.52; fifteen right-handed). All were university students

and had normal or corrected-to-normal visual acuity. Because of a large number of

artifacts, the data from three participants were excluded from the ERP data analysis.

Thirteen subjects (nine women; age 20-38 years, mean = 24.30, standard deviation = 4.89;

twelve right-handed) took part in ERP experiment. Subjects had no history of neurological

or psychiatric disorders, and gave written informed consent to participate after the nature of

the study had been explained to them.

Stimuli

The stimuli were sequences of five even Arabic numbers, which were constructed in the

following way: even numbers were selected to begin sequences, and the same constant

quantity1 was consecutively added. Thus, forty-eight sequences were generated. The

sequences ended in a number that was: a) a number that continued the sequence according

to the same rule — correct ending; b) an incorrect number that deviated by +26 from the

correct one — large-split ending; and c) an incorrect number that deviated by ±2 from the

correct one — small-split ending. For example, in the sequence 6-10-14-18, the correct

ending was 22, the large-split ending was 48, and the small-split ending was 24.

The experiment was controlled by the STIM 2.0 program (NeuroScan Inc., Herndon,

Va). Numbers were presented in white against a black background, and subtended a visual

angle of 1.76º vertically and 1.10º (for one digit stimuli) or 2.42º (for two digits stimuli)

horizontally.

Procedure

Each experimental session lasted approximately 2 hr and 30 min, including preparation.

Participants were seated in an electrically-shielded, sound-attenuating room at a distance of

130 cm from the screen, whose center was at eye level. The session began with a training

block consisting of the presentation of some trials similar to those used in the recording

session. To ensure that subjects understood the task to be performed, the training period

finished when one of the following learning criteria was reached: a) the participant had

answered correctly the first 10 consecutive trials; or b) the participant correctly answered

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90% of trials. Feedback was given whenever a participant’s response was wrong. There

was a 30 sec break after every 20 trials.

When the training period was over, the recording period started. Each trial consisted of a

sequence of five Arabic numbers that were successively presented on the screen. Numbers

remained on the screen for 1000 ms and the inter-stimulus interval (ISI) was 1500 ms. After

the presentation of the last number of the sequence, an asterisk was shown for 500 ms. The

inter-trial interval was 1500 ms. Subjects were asked to judge whether the last number in

the sequence was correct or incorrect by pressing the corresponding STIMPAD keys, with

the same finger of each hand (responding fingers were counterbalanced across subjects).

They were required to wait until an asterisk cue before responding. The asterisk also

informed the subjects that the sequence had finished. Subjects were instructed to relax and

to keep their eyes on the screen. They were also advised to blink during the presence of the

asterisks or the break messages in order to reduce the probability of eye movements in the

critical epochs. A message indicating a 30 s break appeared on the screen after 12 trials and

there was a 5 min break in the middle of the recording session. In contrast with the training

session, during the recording session no feedback was given in the case of incorrect

responses.

All participants were tested on 432 trials, 144 for each type of stimulus. Twelve blocks

of 36 trials were presented to every participant. A block included twelve trials of each type

of stimulus (correct, small-split and large-split ending) presented pseudorandomly so that

no more than three correct or three incorrect trials could appear consecutively.

Electrophysiological recording

EEG was recorded with the SynAmps/SCAN 3.0 hardware and software (NeuroScan,

Inc., Herndon, Va) from 31 tin electrodes mounted in a commercial electro-cap (Electro-

Cap International, Eaton, OH). Nineteen electrodes were positioned according to the 10-20

International System: three electrodes were placed over midline sites at Fz, Cz, and Pz

locations, along with 8 lateral pairs of electrodes over standard sites on frontal (FP1/FP2,

F7/F8, F3/F4), central (C3/C4), temporal (T3/T4, T5/T6), parietal (P3/P4), and occipital

(O1/O2) positions. Two electrodes were placed at Fpz and Oz, and ten electrodes were

placed halfway between the following additional locations: fronto-central (FC1/FC2),

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fronto-temporal (FT3/FT4), centro-parietal (CP1/CP2), temporo-parietal (TP3/TP4), and

mastoids (M1/M2). The common reference electrode for EEG and EOG measurements was

placed on the tip of the nose. EEG channels were continuously digitized at a rate of 500 Hz

by a SynAmpTM amplifier (5083 model, NeuroScan, Inc., Herndon, Va). A band pass filter

was set from 0.05 to 30 Hz, and electrode impedance was always kept below 5 kΩ. For

monitoring eye movement and blinks, Fp1, Fp2, Fpz (for the vertical EOG), and an

electrode placed at the external canthus of the right eye (for the horizontal EOG) were used.

Data Analysis

Behavioral Data

The percentage of correct responses was analyzed with a repeated-measures ANOVA,

taking ending type (correct, small-split and large-split) as within-subjects factor (the

Greenhouse-Geisser correction for sphericity departures was applied when appropriate).

The F value, the uncorrected degrees of freedom, the probability level following correction,

and the ε value are reported. Whenever a main effect reached significance, pairwise

comparisons were conducted using t tests, and the Bonferroni adjustment was used to

control for the increase in type I error. Tests of simple effects were calculated in the

presence of a significant interaction. In this report, differences were considered significant

at p < .05 for overall ANOVA.

EEG analysis

Analysis of the electrophysiological responses was carried out on the fifth number of the

sequence. Firstly, epochs for every subject in each type of ending were averaged relative to

a pre-stimulus baseline that was made up of the 100 ms of activity preceding the epoch of

interest. Secondly, trials with artifacts (voltage exceeded ±50 µV in any channel) and those

with response errors were excluded from the ERP average. The mean number of epochs

included in each ERP average varied between 105 and 114 for the various types of stimuli

used, and the minimum number of epochs included in any individual average was 49.

Finally, ERPs were quantified as mean amplitude measures in the 250-300 and 550-650 ms

latency windows following the onset of the fifth number of the sequence, which was the

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stimulus of interest. The first window covers the early negativity effect and the second

window the late positivity effect.

A 3 x 3 x 5 repeated-measures ANOVA was performed on the ERP amplitudes at 15

electrodes (F7, F3, Fz, F4, F8, T3, C3, Cz, C4, T4, T5, P3, Pz, P4, T6) taking as factors,

ending type (correct, small-split and large-split), frontality (frontal, central, and parietal)

and laterality (five levels from left to right – L1, L2, L3, L4, and L5). ERP scalp

distribution analysis was performed on normalized amplitudes according to the across-

subject average vector length (McCarthy and Wood, 1985; Picton et al, 2000). Statistical

analyses were performed in the same way described to behavioral data. Topographic maps

were plotted using the EEProbe 3.1 program (ANT Software BV, Enschede, The

Netherlands).

RESULTS

Behavioral data

The main purposes of the correct/incorrect judgment task were to ensure that

participants were paying attention to the sequences during the measurement of ERPs, and to

record accuracy data, which would allow us to establish to what extent split effect was

present in our data. Reaction times were not recorded in this experiment because subjects

were asked to respond only after an asterisk mark appeared on the screen, in order to avoid

a contamination of the ERP by response-related activity.

ANOVA results showed a significant effect of ending type, F(2,30) = 16.82, p < .001,

ε = .615. Paired contrasts were carried out to determine more specifically which ending

types differed from one another. Small-split endings were responded less accurately than

correct endings, F(1,15) = 12.03, p = .003, and than large-split endings, F(1,15) = 24.90,

p < .001. There were no significant differences in performance accuracy between correct

and large-split endings. Means and standard errors (in parentheses) of percentage of correct

response were 92.4 (.01), 95.3 (.02) and 82 (.04) for correct, large-split and small-split

endings, respectively.

Event-Related Brain Potentials

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The grand average ERPs for the three ending types are shown in Figure 1. Visual

inspection of these ERPs reveals different waveforms for the three ending types. When a

large-split ending was presented, the presence of an early negativity, peaking between 250

and 300 ms, was noted, followed by a sharp positive wave, peaking between 500 and 600

ms. This positive shift increased from frontal to posterior sites with an equal distribution

over the two cerebral hemispheres. The effect was largest over centro-parietal sites (voltage

maps indicating the distribution of the effects are shown in Figure 2). A similar ERP

pattern can be seen in small-split endings, but, in this case, the early negativity was

followed by a late positive slow wave, smaller than that elicited by large-split endings, and

present only at posterior sites (difference waves obtained after subtracting small-split and

correct endings, and large-split and correct endings are plot in Figure 3). Again, inspection

of the grand-averages and voltage maps showed no hemispheric asymmetries in the size of

this late positivity. Finally, correct endings elicited a 250 ms negative peak of smaller

amplitude than that elicited by incorrect endings, followed by a sharp positive wave

peaking around 300 ms.

Early negativity effect

The ANOVA performed on the 250-300 ms window with normalized data showed

statistically significant effects for Ending x Laterality [F(4,48) = 5.78, p = 0.002, ε = 0.39],

and for Ending x Frontality x Laterality, [F(16,192) = 14.23, p < 0.001, ε = 0.33]. In order

to analyze the higher order interaction (Ending x Frontality x Laterality), separate

ANOVAs were carried out for the five levels of laterality. Results showed that the Ending

effect was statistically significant for all the five levels of laterality (F values ranged from

17.67 to 38.92; all p-values < .001), but the Ending x Frontality interaction was only

significant at the left laterality L1 [F(4,48) = 5.92, p < 0.001, ε = 0.51].

A more detailed analysis of the ending effect was performed by means of paired

comparisons conducted for the three types of ending at frontal, central and parietal sites,

and at all levels of laterality. Table 1 shows the analysis at frontal, central and parietal sites.

It can be seen that a larger negativity was elicited whenever an incorrect ending was

presented. Both incorrect endings — small- and large-split endings— were different from

the correct endings (p-values < .001); however, no reliable differences were found between

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the two incorrect endings. Furthermore, the mean differences in Table 1 show that the

differences in voltage increase from frontal to parietal sites. The results for paired contrasts

for all levels of laterality showed that again, there was a widespread negativity effect: the

difference between correct and both incorrect endings reached significance at all five levels

of laterality (all of the F values ranged from -1.97 to -3.27; all p-values < .001).

Because the Ending x Frontality interaction reached statistical significance at the left

laterality L1, ANOVAs taking Ending as within-subjects factor were conducted at F7, T3

and T5, as those were the electrodes in laterality L1. The results showed that the ending

effect was only present at T3 [F(2,24) = 16.77, p < .001, ε = .863], and T5

[F(2,24) = 23.79, p < .001, ε = .851]. Ending type had no effect on ERP amplitude at F7 (p-

value > .05)

Overall, an early negativity effect was elicited by the incorrect endings and this was

unaffected by the distance between incorrect and correct solution. This negativity effect

was larger over central and posterior sites.

Late positivity effect

In order to study the late positivity effect, the normalized mean amplitude in the 550-650

ms latency window was analyzed using ANOVAs. The interaction Ending x Frontality was

statistically significant [F(4,48) = 5.69, p = 0.009, ε = 0.52]. Because we were particularly

interested in the differences in scalp distribution between the three types of endings,

pairwise comparisons were conducted for the three conditions at frontal, central and

parietal sites.

Table 1 shows the pairwise contrasts for the three types of endings at frontal, central and

parietal sites. Results showed significant differences between large-split and correct

endings, and between large- and small-split endings at all levels of frontality (p-

values < .001 at frontal, central, and parietal sites). It can be seen that large-split endings

elicited more positive voltages than both the correct and the small-split endings. Although

this late positivity effect was widespread, the ERP differences between large-split and

correct endings were larger over parietal sites (5.9 µV) than over central (4.8 µV) and

frontal sites (3.4 µV). These differences can be seen in the voltage map in Figure 2B (right

column). The results showed that the small-split endings elicited a larger positivity than the

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correct endings only over parietal sites (p = .01). This posteriorly distributed effect for the

small-split endings can also be observed in the voltage map in Figure 2B (left column). In

summary, the late positivity effect was more pronounced at posterior than at other sites, as

shown in the statistical analysis and in the voltage maps. This effect was present both in

large-split endings and in small-split endings over posterior sites, although it was largest for

large-split endings.

To sum up, the results indicate that incorrect endings elicit a late positivity effect. This

positive shift increased from frontal to parietal sites, and was larger over centro-parietal

sites. No hemispheric asymmetries in the size of this component were observed. Moreover,

the amplitude of this positive component was modulated by the size of the split, being

larger in the large-split endings than in the small-split endings. The split effect was present

at centro-parietal sites.

DISCUSSION

The experiment reported here used ERPs to examine the split effect on arithmetical

sequence processing. Three predictions were made. First, we expected that the size of the

split would influence participants’ accuracy, causing them to respond less accurately to

small-split endings than to correct and large-split endings. Second, the size of the split also

would influence the amplitude of a posteriorly distributed late positive component: a late

positive slow wave was expected to be elicited by small-split endings and a larger and

sharper positive wave was expected to be elicited by large-split endings. Third, we

predicted that a negative component preceding this late positive component would be

elicited by both incorrect endings.

The first prediction addressed the question of how the distance between the proposed

and the correct solution to an arithmetical problem would influence participants’ accuracy.

Previous studies have shown that when subjects are presented with an incorrect solution to

an arithmetical problem, they perform slower and less accurately if the number presented is

close to the correct solution than if the number is very different (Ashcraft and Battaglia,

1978; Ashcraft and Stazyk, 1981). In the present experiment, the split effect was indeed

found on accuracy as suggested by previous research. Small-split endings elicited less

accurate responses than correct and large-split endings. Nonetheless, the difference in

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accuracy is rather unusual (small-split endings yielding an accuracy rate of more than 10%

below those of correct and large-split endings — namely 82% relative to 92.4 and 95.3%,

repectively). Generally, in healthy participants differences between small- and large-split

trials are most obvious on response latencies and not so much on accuracies. One possible

interpretation of the present result might be that participants tend to double check the small-

split endings, and upon being confronted with a limitation of response latency (a relatively

short inter-trial-interval of 1500 ms), the likelihood of errors could increase dramatically.

This explanation is plausible considering the high working memory demand upon solving

these problems: it has to be reminded that one trial consisted of five stimuli that have to be

held in working memory in order to solve the task successfully.

The second prediction deals with the question of whether there is a neurophysiological

evidence for the split effect. The present results show that arithmetical incorrectness has an

impact on processing and induces a clear neurophysiological response in the form of a LPC

effect. As predicted, an amplitude modulation of a LPC was associated to the split-effect.

Small-split endings elicited a late positive slow wave that was largest over parietal sites,

distributed equally over both hemispheres. This positive shift might suggest that the exact

calculation is needed to decide on the incorrectness of these problems. Previous results

(Iguchi and Hashimoto, 2000: Núñez-Peña et al., 2005, 2006; Pauli et al., 1994, 1996) have

shown that a late positive slow wave may be functionally related to mental arithmetical

calculation. This late positive slow wave has been suggested to constitute a brain signature

of calculation, so the fact that in the present experiment small-split problems elicited a late

positive slow wave might support the idea that subjects used a whole-calculation strategy to

solve these problems.

As for large-split endings, the largest and sharpest LPC was elicited. This result is

consistent with previous studies. First, Niedeggen and Rösler (1999) and our own group

(Núñez-Peña and Honrubia-Serrano, 2004) reported a LPC modulation in arithmetic

problem verification tasks depending on the numerical distance between the proposed and

the presented solution: the greater the distance, the larger the amplitude. Second, El

Yagoubi et al. (2003, 2005) reported a modulation on a LPC in an inequality verification

task that was interpreted as an index of the different strategies that subjects used to verify

complex inequalities: a whole-calculation strategy for small distance problems and a

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plausibility-checking strategy for large distance problems. Although there are some

differences in the scalp distribution of the LPC reported by El Yagoubi et al. — larger at

frontal than at parietal sites and left-lateralized — and the LPC reported in the present study

— parietally distributed and without hemispheric asymmetries —, this discrepancy may be

explained. First, neuropsychological and neurophysiological studies have indicated that

both the prefrontal and the parietal cortices are involved in arithmetical tasks (see for

example, Menon et al., 2000, Stanescu-Cosson et al., 2000, or Zago and Tzourio-Mazoyer,

2002). The involvement of the parietal cortex in number processing has been reported in

data from brain-lesioned patients, and by using positron emission tomography (PET) and

functional magnetic resonance imaging (fMRI) (for a review see Dehaene et al., 2003); it

has been proposed that the parietal lobe contributes to the representation of numerical

quantity (Dehaene and Cohen, 1995). Second, the experimental tasks were different. In the

present study, a problem verification task was used, consisting of verifying if the last

number of a sequence was correct or not. By contrast, El Yagoubi et al. used an inequality

verification task that consisted of deciding if the addition of two numbers was smaller or

larger than 100.

It is tempting to speculate that the LPC elicited by arithmetical violations in a

verification task reflects the application of arithmetical rules in working memory, especially

because parietal regions are believed to be involved in arithmetical processing. However, a

more accurate characterization of the LPC reported in the present experiment will have to

await future investigation, because there are arguments that suggest that these findings are

not limited to this particular cognitive domain. First, we have suggested elsewhere (Núñez-

Peña and Honrubia-Serrano, 2004) that the modulation of the amplitude of this late positive

component might be a response to the difficulty of integrating the incoming element to the

previous structure, because some studies have shown a comparable LPC amplitude

modulation in grammatical (Frisch et al., 2002; Kaan et al., 2000; Kaan and Swaab, 2003a,

2003b; Osterhout et al., 1994) and musical structure processing (Patel, 2003; Patel et al.,

1998). The more difficult it is to integrate an element into the previous structure, the larger

the amplitude of this component. Consequently, this ERP component might be a brain

response related to the ability to repair structural incongruities in sequences governed by

rules. Second, the LPC reported in the present experiment is also similar to the well-known

15

P3b, whose amplitude has been suggested to vary as a function of memory load, with

smaller amplitudes associated with increased memory load. When a small-split ending is

presented, a whole-calculation strategy would be expected to involve a greater memory

load than a plausibility-checking strategy. Summarizing, the question as to whether the

LPC reported in the present experiment is a specific computational index or a reflection of

a more general mechanism is unresolved. To clarify this question an additional experiment

should be performed in order to assess the distinctiveness of brain responses to anomalies

that involve calculation and to anomalies that do not involve calculation. Although there is

no firm evidence for the arithmetical specificity of the LPC effect reported in the present

experiment, we considered the LPC can be a useful dependent variable in mathematical

cognition research, because it is an ERP effect triggered by operations at some level of

human calculation processing .

Finally, our last prediction concerned the early negativity elicited by incorrect endings.

As predicted, in the present study, an early negativity —with latency at about 270 ms post-

target and maximum amplitude over parietal sites — was observed. This component was

elicited whenever an incorrect number was presented and was not modulated by the

distance between the proposed and the exact solution. As differences between ERPs can be

used to monitor the onset and time course of electrophysiological effects attributable to

variations of the independent variables, the present result suggests that subjects detect the

incorrect solution as early as 270 ms after the proposed solution onset. When the last item

in the sequence is presented in a problem verification task, participants might perhaps

already have an estimation of the solution. The discrepancy between the estimation and the

proposed number might be the origin of this early negative component. When this

discrepancy is produced, then subjects might select the optimal strategy to solve the

problem. If the proposed number differs a lot from the estimation, then enough information

is available to make a decision and produce a response: in other words a plausibility-

checking strategy might be used. However, if both numbers are very close, then more

information is needed to solve the problem, and its correct solution is completely

calculated: the whole calculation strategy might be used.

The early negativity elicited by incorrect solutions in the present experiment also

reminds the ‘selection negativity’, an event-related potential that has been reported in the

16

non-spatial attention literature. This component has been described as a negative shift,

maximal at occipital scalp sites, and which starts around 160-200 ms after stimulus onset.

The ‘selection negativity’ has been interpreted as a reflection of a selection process

(Talsma, Kok and Ridderinkhof, 2006), a post-selection process (Smid, Jakob and Heinze,

1999) or an early target/non-target classification (Talsma and Kok, 2001).

Although these arguments could explain the early negativity, previous studies have

reported the presence of a negative peak which precedes the LPC component for

arithmetical anomalies, with a different scalp distribution and latency. First, Niedeggen and

Rösler (1999), and Niedeggen, Rösler, and Jost (1999) found that incorrect solutions to

multiplication problems elicited a negativity peaking at about 400 ms with maximum

amplitude over centro-parietal brain areas. These authors related this component to the

semantic N400 effect (for review, see Kutas and Van Petten, 1994, and Kutas and

Federmeier, 2000). Second, Szűcs and Csépe (2004, 2005) reported an early negativity

peaking at about 270 ms elicited by incorrect solutions in addition problems. This

component was larger at fronto-central sites. Third, a similar fronto-central N270

component was reported by Wang, Kong, Tang, Zhuang and Li (2000), although these

authors used addition, subtraction, multiplication and division problems indistinctly.

Therefore, the early negativity elicited by incorrect solutions in the present experiment

remains an open question requiring further investigation.

The main conclusions of the present study are as follows. First, we demonstrated a

significant LPC effect in response to the size of the split in violations of arithmetical rules.

This LPC effect might indicate that two different strategies are used to verify arithmetical

problems: a whole-calculation strategy for small-split solutions and a plausibility-checking

strategy for large-split solutions. Second, in addition to the LPC effect, there was another

observation of interest: the presence of an early negativity, maximum over parietal sites, in

arithmetical violations. Although this finding raises the possibility that this negative

component is related to the discrepancy between the estimation of the following number

prepared by the subject and the presented number, further research needs to be carried out

in order to further our knowledge of the issue.

17

ACKNOWLEDGEMENTS

This research was supported by grants BS02003-02440 and SEJ2006-000496/PSIC from

the Spanish Ministry of Science and Technology and SGR2005-00953 from the Generalitat

de Catalunya. We thank two anonymous reviewers for their helpful comments on an earlier

draft of this paper.

18

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FOOTNOTES

(1) The following expression was used to generate the sequences, xi+1 = xi + c, where c is

a constant that could take the values -2, -4, -6, 2, 4, or 6.

23

Table 1. Pairwise contrasts for the three types of endings at frontal, central and parietal sites. Mean differences (in µV) between the three types of ending are shown for the 250-300 ms and the 550-650 ms windows.

250-300 550-650

Contrast Region Mean difference Mean difference

Large-split – Correct Frontal Central Parietal

-2.13** -2.84** -3.49**

3.42** 4.83** 5.86**

Small-split – Correct Frontal Central Parietal

-2.15** -2.84** -3.32**

.07 .68 1.75*

Large-split – Small-split Frontal Central Parietal

.01 .00 -.17

3.35** 4.16** 4.12**

* .01 ≥ p ≥ .001; ** p < .001

24

FIGURE CAPTIONS:

Figure 1: Averaged waveforms (n=13) elicited by the three types of endings at 16 scalp electrodes.

Figure 2: A) The spatial distribution of the early negativity effect (the voltage difference between 250 and 300 ms) over all electrodes at the scalp surface. Left: Voltage differences between small-split and correct endings. Right: Voltage differences between large-split and correct endings. B) The spatial distribution of the late positivity effect (the voltage difference between 550 and 650 ms) over all electrodes at the scalp surface. Left: Voltage differences between small-split and correct endings. Right: Voltage differences between large-split and correct endings. Figure 3: Difference waves (n=13) obtained after subtracting small-split and correct endings, and large-split and correct endings at 15 scalp electrodes.