Simple additions: Dissociation between retrieval and counting with electrophysiological indexes

International Journal of Psychophysiology 149 (2020) 48–59

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International Journal of Psychophysiology journal homepage: www.elsevier.com/locate/ijpsycho

Simple additions: Dissociation between retrieval and counting with electrophysiological indexes Gloria Tejero, Pedro Macizo

T

⁎

University of Granada, Mind, Brain and Behaviour Research Centre (CIMCYC), Spain

ARTICLE INFO

ABSTRACT

Keywords: Simple arithmetic Retrieval from memory Counting Event-related potentials Associative memory Late positivity N400

There is current debate about the way adult individuals solve simple additions composed of one-digit operands. There are two opposing views. The first view assumes that people retrieve the result of additions from memory, whilst the second view states that individuals use automatized counting procedures. Our study aimed to dissociate between these two hypotheses. To this end, we analysed the type of problem effect when participants resolved simple additions by comparing additions with operands between 1 and 4 and control additions with at least one operand larger than 4. Brain-waves activity of a group of 30 adult individuals were recorded with 64 scalp electrodes mounted on an elastic cap, referenced against an electrode between Cz and CPz and re-referenced to an average reference offline. We considered two electrophysiological indexes, event-related potentials, ERPs, time-locked to the addition problems to distinguish between retrieval from memory and the use of procedures: A late positivity component (LP, 500–650 time window) over posterior regions associated to memory retrieval difficulty with higher LP positivity when participants resolve difficult vs. easy additions, and a negative component (N400, 250–450 ms time window) over fronto-central regions related to the use memory retrieval vs. procedures with more pronounced N400 amplitudes when the difficulty in the retrieval of semantic information increased. LP modulations were observed depending on the type of problem over posterior regions, P3 and Pz electrodes, whilst the N400 component was not affected. This pattern of results suggests that adult individuals use retrieval from memory to solve simple additions.

1. Introduction In cognitive arithmetic, it has been accepted for decades that people solve simple arithmetic problems (e.g., additions and multiplications with one-digit operands) through automatic retrieval of arithmetic facts from memory (e.g., network interference model, Campbell, 1987; Campbell et al., 2016; Chen and Campbell, 2016). Thus, in order to solve a simple math problem (e.g., 4 + 2, 3 × 2), an individual would retrieve the solution (e.g., 6) from semantic memory. According to this perspective, it is assumed that people have stored arithmetic facts (the operands, the operation sign, and the result) in long-term memory due to the frequent practice they have in everyday life with diverse arithmetic operations such as additions and multiplications. Evidence in support of the retrieval of arithmetic facts comes from studies in which participants have to explicitly perform arithmetic operations (e.g., oral report of the results of multiplication problems) and addition verification tasks in which participants decide whether the proposed result is correct or not (e.g., Chen and Campbell, 2015; Lemaire et al., 1991; Megías and Macizo, 2016; see Hinault and

Lemaire, 2016, for a recent review of strategies used in cognitive arithmetic and EEG evidence). However, perhaps the most direct evidence in favor of the automatic retrieval from memory comes from studies in which participants do not have to explicitly perform the arithmetic problem. For example, behavioural (Rusconi et al., 2006) and electrophysiological data (Galfano et al., 2009) obtained with the number-matching paradigm are unequivocally interpreted as reflecting task-irrelevant activation of arithmetic facts. In this paradigm, pairs of cue numbers are presented (e.g., 3 and 2) followed by a target number and participants have to indicate if the target is one of the cues numbers. In this task, the performance is poorer when the target number is the result of multiplying the cue numbers (e.g., 6) than when it is not (e.g., 8). This effect would imply that the presentation of the cue numbers produced the automatic activation of the multiplication result which made difficult to reject the target number when it coincided with the multiplication result. Thus, this interference effect would support the retrieval of multiplication facts from memory. With regards to the resolution of simple additions, there is also abundant evidence in favor of the retrieval from memory account (e.g.,

⁎ Corresponding author at: Departamento de Psicología Experimental, Facultad de Psicología, Universidad de Granada, Campus de Cartuja, s/n., 18071 Granada, Spain. E-mail address: [email protected] (P. Macizo).

https://doi.org/10.1016/j.ijpsycho.2020.01.001 Received 17 May 2019; Received in revised form 29 November 2019; Accepted 3 January 2020 Available online 10 January 2020 0167-8760/ © 2020 Elsevier B.V. All rights reserved.

International Journal of Psychophysiology 149 (2020) 48–59

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interference effects with additions in the number-matching paradigm described above, Lefevre et al., 1988; associative confusion effect in addition verification tasks, Lemaire et al., 1991; Megías and Macizo, 2016; retrieval practice effects with simple addition, Campbell and Thompson, 2012). However, in recent years, an alternative explanation proposed 35 years ago (Baroody, 1983) has revived to explain the resolution of simple sums. This perspective argues that adults solve addition problems with the use of a fast counting procedure carried out with the involvement of working memory (e.g., Uittenhove et al., 2016). The resolution of an addition would imply rapid counting of both operands from the higher operand of an addition to the result (e.g., in 4 + 2 =, four steps of counting for the first operand and then two steps of counting for the second operand to reach the solution, i.e., 6). This view is based on the way children develop arithmetic skills at school. It is widely accepted that children learn to resolve simple addition problems by using counting strategies and, during typical childhood development, there is a shift from the use of counting strategies to the use of retrieval from memory to solve simple arithmetic problems (Geary et al., 1991; Geary et al., 2004; Lemaire and Siegler, 1995; Siegler and Shrager, 1984). The view of counting procedures in adulthood implies that educated adults would continue using these counting procedures in a faster, unconscious, and automatic manner. An important source of evidence supporting the counting account comes from behavioural studies in which reaction times (RT) and accuracy are measured when individuals resolve simple additions (Barrouillet and Thevenot, 2013; Uittenhove et al., 2016). Overall, in these studies, a problem size effect was found during the resolution of very small addition problems: simple additions with small operands between 1 and 4 (additions 1–4, for short). Additions 1–4 with smaller operands (e.g., 4 + 2) were solved much faster and more accurately than additions 1–4 with larger operands (e.g., 4 + 3). However, the problem size effect was not observed when participants resolved medium small problems (problems with sums from 7 to 10 with at least one operand larger than 4). At first sight, this pattern of results with additions 1–4 could not be explained in terms of the retrieval from memory account. According to this view, all arithmetic facts are represented in memory so the time needed to retrieve them would be similar regardless of the size of the problem. On the other hand, the problem size effect can readily be explained by the counting view: the larger the size of the operands, the higher the number of counting steps required to reach the solution and thus more time is needed to resolve the problem. For example, in the additions 4 + 2 and 4 + 3, besides the four counting steps for the first operand, the resolution of the addition 4 + 2 implies two counting units for the second operand whilst the addition 4 + 3 implies one more counting step for the second operand and so the latter problem would require additional processing time. Thus, according to this perspective, the presence of the problem size effect with additions 1–4 would indicate that these problems were resolved by an automated counting procedure whilst the absence of the problem size effect with medium small problems would indicate that these problems were resolved by memory retrieval (Uittenhove et al., 2016, p. 298). Nevertheless, the retrieval from memory account could also explain the problem size effect found with additions 1–4. This account assumes that arithmetic fact retrieval involves accessing the approximate magnitude of the addition result (Campbell and Graham, 1985). Thus, if we consider that magnitude access is more efficient with small numbers than with large numbers (Ashcraft, 1992), the resolution of addition problems would also depend on the magnitude of the operands. Moreover, according to the network interference model (e.g., Campbell, 1995), a problem size effect on RTs would arise in simple fact retrieval because interference from competing arithmetic facts increases with problem size (see Chen and Campbell, 2018, for a detailed explanation). Therefore, in behavioural studies, the same pattern of results (longer RTs and lower accuracy when problem size increases) can be accounted by both the retrieval from memory view and the explanation based on counting procedures. The current report made use of electrophysiological indexes to dissociate between these two explanations.

1.1. Electrophysiology of simple arithmetic There are several ERP components associated to numerical cognition and arithmetic processing. The N100 potential is sensitive to variations in non-symbolic magnitudes (Hyde and Spelke, 2009), and P100 modulations have been linked to the implicit estimation of ordinal information (Rubinsten et al., 2013). When participants are presented an addition with a result (e.g., 4 + 2 = 9) and they verify the correctness of the solution, large vs. small distances between the proposed result and the correct result are related to more negative N2b amplitude and more positive P3b amplitude (Avancini et al., 2015, for a review). We did not consider these components here because the underlying cognitive processes they index did not apply in the current report (e.g., we did not evaluate non-symbolic quantities or magnitude estimation, and we did not use an addition verification task). Instead, we focused on two components of special relevance in our study. Firstly, a late positivity component (LP) has been observed in relation to the magnitude of the addition operands (Núñez-Peña et al., 2005; Núñez-Peña et al., 2011; Van Beek et al., 2014). The LP component consists of a large positivity over posterior regions (parietal distribution) on a late time window (between 300 ms to 800–1000 ms). The LP is sensitive to the size of the addition problems, with larger LP amplitudes for large addition problems (additions with results ≥10) than for small addition problems (additions with results < 10) (Van Beek et al., 2014). This LP component has been related to the difficulty in retrieving arithmetic facts from associative memory (e.g., Núñez-Peña et al., 2005; Pauli et al., 1994; Pauli et al., 1996). The most accepted explanation for the problem size effect is that it is due to the fact that small problems have greater associative strength in long-term memory than large problems because small problems are solved more often than large problems (Núñez-Peña, 2008; Zbrodoff and Logan, 2005). Therefore, a modulation of the LP component during the resolution of additions would indicate that participants used associative memory retrieval to resolve these problems. We also considered a second component, the N400, whose amplitude is sensitive to the processing of semantic information in language (Kutas and Hillyard, 1980) and arithmetic processing (El Yagoubi et al., 2003; Luo et al., 2009). The N400 allows the dissociation between complex arithmetic problems (e.g., additions with operands of more than one digit) depending on the difficulty in semantic processing. For example, Luo et al. compared two ways of solving sums, one involving the use of rounding shortcuts and the other that did not involve shortcuts but the whole computation of the problem. A greater N400 was observed over anterior locations (e.g., frontal regions) in the nonshortcut problems compared to the resolution of the problem with shortcuts. The authors interpreted this N400 modulation as an index of the greater difficulty in semantic processing associated to the resolution of additions without shortcuts compared to the use of shortcuts. El Yagoubi et al. observed a greater N400 amplitude when verifying small split additions versus large split additions (additions whose correct result was close or distant to a proposed result, respectively). According to the authors, this N400 modulation was due to the fact that the small split additions were solved by an exact calculation which implied greater difficulty in semantic processing than the large split additions whose resolution involved the use of an approximation procedure. In the field of simple arithmetic (e.g., problems with one-digit operands) it is widely assumed that the resolution of some problems (e.g., multiplications) necessarily implies semantic processing (i.e., the retrieval of arithmetic facts from memory, Campbell, 1995). Moreover, the N400 has been shown to be an index of the retrieval of arithmetic facts with onedigit multiplication problems (e.g., Niedeggen and Rösler, 1999; PrietoCorona et al., 2010). In verification of multiplications tasks, the N400 is inversely proportional to the degree of relationship between the proposed result and the correct result (e.g., Niedeggen and Rösler, 1999). The N400 would allow the dissociation between the use of memory retrieval and the use of procedural strategies to verify simple arithmetic problems. In particular, the retrieval from memory perspective would imply a reduced 49

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difficulty in semantic processing of the stimuli than the automatic counting strategy because the use of procedures does not involve the retrieval of arithmetic facts stored in memory. Previous studies have observed N400 modulations depending on the difficulty of semantic processing of the stimuli. For example, the processing of semantically rich words (open-class words or content words such as nouns, verbs or adjectives) produces smaller N400 amplitudes compared to the analysis of words whose semantic processing is difficult (closed-class words or function words with very limited semantic information such as conjunctions, determinants or prepositions) (Neville et al., 1992; Nobre and McCarthy, 1995; Pulvermüller et al., 1995). In our study, we considered this dissociation to evaluate the use of memory retrieval vs. counting procedures. Specifically, if medium small problems are resolved by retrieval and additions 1–4 by procedures (Barrouillet and Thevenot, 2013; Uittenhove et al., 2016), N400 amplitudes would be smaller for the first type of additions relative to additions 1–4. On the contrary, if all simple additions were resolved by memory retrieval (e.g., Campbell, 1987), there would be no differences in the N400 amplitude between these types of additions because all of them would imply access to semantic information (the retrieval of arithmetic facts). However, from the retrieval from memory view, LP modulations would be predicted with a larger positivity for medium small additions (control additions) than for small additions (additions 1–4). In this study, we address these predictions.

type of problem would modulate the LP with greater amplitude for control additions compared to additions 1–4. Furthermore, no differences in the N400 should be found between additions 1–4 and control additions since they would all involve the retrieval of semantic information (arithmetic facts) from memory. On the contrary, if very small problems are resolved by procedures and small medium problems are resolved by memory retrieval (Uittenhove et al., 2016); the N400 would be sensitive to the use of these two ways of solving the problems with larger N400 amplitude for problems resolved by counting procedures (additions 1–4) because the difficulty in semantic processing would be greater in this case than with the use of retrieval from memory (control additions). 2. Method 2.1. Participants Thirty students from the University of Granada took part in the study, 20 women and 10 men. The sample of participants was randomly selected. The average age of participants was 21.67 years old (SD = 2.34, from 18 to 26 years old). Handedness was determined through self-report, 27 participants were right-handed and 3 participants were left-handed. The ethnic composition of the group of participants included 29 Caucasian participants and 1 Asian participant. The required sample size was determined using the G*Power program 3.1.9.4 (Faul et al., 2007). It was calculated that for the analyses conducted in the study (Problem size × ROI and Type of problem × ROI, see below), 2 × 9 repeated measures analyses of variance (MANOVA) to achieve 95% statistical power with α = 0.05 and an a Cohen's effect size of 0.50, assuming a zero correlation among repeated measurements, the total sample size needed was N = 20. Thus, the sample used in this study was sufficient to capture the problem size and type of problem effects evaluated in the experiment. All the participants gave informed consent before conducting the experiment, and their participation was rewarded with academic credits. The participants had normal or corrected-to-normal visual acuity. None had reported any history of neurological or psychiatric disorders that could modulate the recording of the electroencephalogram (EEG). Furthermore, the participants did not report present or past numerical processing difficulties (e.g., dyscalculia). All procedures performed in this study were approved by the Ethical Committee on Human Research at the University of Granada (Spain) (Number issued by the Ethical Committee: 86/CEIH/2015), associated to the Grant PSI2016-75250-P awarded to Pedro Macizo by the Spanish Ministry of Economy, Industry and Competitiveness, and in accordance with the 1964 Helsinki declaration and its later amendments.

1.2. The current study In the current study, we made use of electrophysiological measures with the goal of dissociating between two accounts that explains the way adult individuals resolve simple additions by retrieving the result from memory or by using an automated counting procedure. To do so, we recorded electrical brain activity whilst a group of people performed simple additions. In our study, we used an arithmetic production task in which participants received additions and they had to orally report the solution. This task was selected for two reasons, firstly, because it is a naturalistic task that would reflect the usual cognitive processes that people carry out when resolving additions in their daily lives and, secondly, because this task has been used in studies in favor of the two views about the resolution of addition problem, the retrieval from memory account (Campbell and Thompson, 2012) and the rapid counting procedures (Barrouillet and Thevenot, 2013; Uittenhove et al., 2016). To distinguish between retrieval from memory and the use of procedures, we considered two electrophysiological components: the LP was taken as an index of the involvement of associative memory; in particular, this component is sensitive to the difficulty in retrieving arithmetic facts from memory; whilst we employed the N400 as a marker of the difficulty in semantic processing when resolving simple additions. To evaluate this issue, we considered two variables in our study, the problem size and the type of problem depending on the result of the problem and the operands of the addition, respectively. First of all, we focused on the problem size effect. We wanted to confirm in our study that the LP is sensitive to the difficulty of the retrieval process. To this end, we explored the problem size effect by comparing: (1) addition whose result was larger than 10 vs. (2) additions with results smaller than 10. We expected to observe higher positivity on the LP when participants resolved larger additions, which would confirm that the LP measures the difficulty associated to the retrieval of arithmetic facts in memory (e.g., Núñez-Peña et al., 2005; Van Beek et al., 2014). Once corroborated in our study that the LP component was sensitive to the retrieval of arithmetic facts, we explored the type of problem effect. To this end, we compared: (1) additions 1–4 defined by Uittenhove et al. (2016) as those that imply automated counting procedures, and (2) control additions composed of other one-digit simple additions with operands larger than 1–4, which do not involve the use of procedures but instead require retrieval from memory (Barrouillet and Thevenot, 2013; Uittenhove et al., 2016). The predictions were as follows: in the case that participants resolve all simple one-digit additions by retrieving their answer from memory, the

2.2. Design and materials In the current study, the participants were asked to solve simple additions whilst the EEG was recorded continuously. The tasks comprised 5 blocks of additions. In each block, 100 possible additions of one-digit operands from 0 to 9 were randomly presented. Thus, each participant resolved a total of 500 simple additions. In our study, two within-participant variables were considered: The problem size and the type of problem. To evaluate the problem size effect, we compared additions whose results were 10 or larger than 10 (38 additions) with those whose results were smaller than 10 (18 additions). To determine the type of problem effect, we conducted the same comparison described by Uittenhove et al. (2016). There were 1–4 additions defined as those whose operands were between 1 and 4 when tie problems were excluded (10 addition with equal operands, e.g., 5 + 5). The control additions were defined as those with results from 7 to 17 that did not contain 1 as operand and with at least one operand larger than 4 (50 addition problems). None of the conditions described above for the problem size variable and the type of problem variable included tie problems, because it is widely assumed that both involve direct access to associative 50

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memory (Campbell et al., 2016). Furthermore, n + 0/0 + n problems were not used in any experimental condition because it is widely accepted that they are resolved by procedures (Chen and Campbell, 2015). Additions n + 1 and 1 + n were used in the 1–4 additions condition but not in any other condition. The complete list of additions used in each experimental condition is reported as Online Supplementary Material 1. 2.3. Procedure The experiment was designed and controlled by E-prime experimental software (Schneider et al., 2002). The stimuli were always presented in the centre of the screen in black colour (Arial font, 30 point size) on a white background. Participants were tested individually and they were seated at approximately 60 cm from the computer screen. The procedure used for presenting the addition problems was similar to that utilized by Van Beek et al. (2014). Each trial began with a fixation point (####) in the centre of the screen for 1000 ms followed by a simple addition (e.g., 3 + 4) for another 1000 ms. Following this, the screen became white for 500 ms and then the sign “?” appeared on the screen. At that time, the participants had to say aloud the result of the addition (Fig. 1). Hence, a delayed version of the addition task was used in which oral responses were produced after 1500 ms of presenting the arithmetic problem. This delayed answer procedure was used to avoid the influence of motor and articulatory movement related to the production of oral responses on the late ERP time window considered for analyses (500–650 ms, see below). In fact, the average RT required by participants from different cultures to name one-digit additions with small results is about 500 ms (Campbell and Xue, 2001). Data points were considered incorrect responses if the participants produced an incorrect answer to the addition or if they did not give an oral response (1.47% of data). Only trials with correct responses were included in the EEG analyses. Incorrect responses were submitted to the behavioural analyses (accuracy measures).

Fig. 2. Scalp distribution of electrodes. Boxes indicate the electrodes considered for statistical analyses.

The recording was filtered to delete additions in which participants gave an incorrect response and additions in which the EEG was contaminated by eye movements or amplifier saturations. In addition, blinks were corrected as follows: Visual inspection of the activity in the electrodes placed above and below the left eye was carried out for each participant separately in order to determine the voltage range associated to blinks. A voltage threshold was then individualized for each participant to capture as blink artefacts those epochs exceeding the voltage criterion (the mean voltage threshold across participants was 150 μV). The blinks were then averaged for each participant separately using an average of 357 blinks (with a minimum of 37 blinks) for each participant and later corrected with linear regression in the time domain (Neuroscan Scan 4.5 software, El Paso, TX). Individual epochs were conducted for each experimental condition beginning with a 100 ms pre-stimulus baseline. Average ERP waveforms were time-locked to the presentation of the addition problem. Of the total of 500 additions presented to each participant, the maximum number of observations for each level of the variables considered in our study was as follows: Problem size factor: 190 trials for additions larger than 10, 90 trials for additions smaller than 10. Type of problem factor: 60 trials for additions 1–4, and 250 trials for control additions. After eliminating trials with incorrect responses and trials in which the EEG was contaminated by eye movements or amplifier saturation, the average number (and minimum-maximum number in parentheses) of observations entering each grand average per participant and condition was as follows: Problem size factor: 184 (175–190) trials for additions larger than 10, 89 (88–90) trials with additions smaller than 10. Type of problem factor: 60 (58–60) trials for additions 1–4, and 244 (235–250) trials for control additions. The statistical analyses were carried out on the mean amplitude in four temporal windows according to previous electrophysiological studies on simple arithmetic (Núñez-Peña et al., 2005, 2011; Van Beek et al., 2014): 100–150 ms, 150–250 ms, 250–450 ms and 500–650 ms. The last two temporal windows were particularly important for the purpose of our research. The 250–450 ms time window associated to semantic processing during the resolution of additions (N400 component), and the 500–650 ms time window selected to evaluate the LP component related to the difficulty in retrieving arithmetic facts. On each window, two statistical analyses were conducted, one to evaluate the problem size effect and another to determine the type of problem effect. In both analyses, we also included the region of interest (ROIs) in order to evaluate the topographic distribution of the effects investigated in our work. According to previous studies, the LP component associated to the problem size and the retrieval of information in

2.4. Electrophysiological recording The EEG was recorded from 64 scalp electrodes (see Fig. 2) mounted on an elastic cap in accord with the international 10–20 system (Jasper, 1958). Following this, the EEG was amplified with a band pass of 0.01–100 Hz with Neuroscan Synamps 2 (El Paso, TX), and digitalized at a sampling rate of 500 Hz. To control for vertical and horizontal eye movements, two additional pairs of electrodes were used: a) two electrodes placed one above and the other below the left eye on the outer canthi, allowing blink artefact to be corrected; b) two electrodes placed on the external canthi, with one electrode on the left and another on the right side, allowing eye movements to be rejected. During recording, impedances were kept below 5 kΩ. The EEG was initially recorded against an electrode placed on the midline of the cap (between Cz and CPz) and rereferenced off-line against a common average reference. The software NeuroScan 4.5 (El Paso, TX) was used to record and analyse the EEG.

Fig. 1. Procedure used to present addition problems. 51

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3.2.1. Problem size The ROI effect was significant in the 100–150 ms time window, F(8, 232) = 21.944, p < .001, ε = 0.237, ηp2 = 0.431; the 150–250 ms time window, F(8, 232) = 9.694, p < .001, ε = 0.199, ηp2 = 0.251; and the 250–450 ms time window, F(8, 232) = 39.081, p < .001, ε = 0.364, ηp2 = 0.574. However, in these time windows, there were no significant effects of problem size or the problem size × ROI interactions, all ps > 0.05 (see Table 1 for detailed analyses). Thus, in the 250–450 time window, the problem size effect was not significant in any of the electrodes considered in the ROI variable (all ps > 0.05).

associative memory is located at centro-posterior regions (Núñez-Peña et al., 2005). In order to establish a direct comparison between our data and those reported in previous studies (e.g., problem size effect, Núñez-Peña et al., 2005, Van Beek et al., 2014), central electrodes (C3, Cz C4) and parietal electrodes (P3, Pz, P4) were selected. On the other hand, the N400 component in arithmetic is located over anterior regions (e.g., Luo et al., 2009). For this reason, we include frontal electrodes (F3, Fz, F4) within the ROI variable. All of these electrodes have been considered in a large number of studies on cognitive arithmetic (e.g., Núñez-Peña et al., 2005, 2011; Van Beek et al., 2014; Xiang et al., 2016). In our study, the number of recording electrodes was higher than the number of electrodes included in the analysis, an approximation similar to that used in previous research (Blackford et al., 2012; Kuipers and Thierry, 2015; Kuperberg et al., 2018; Weber et al., 2004). Luck (2005, p. 109) indicates that even when multiple electrode sites are recorded, it is usually best not to include measurements from electrode sites spanning the entire scalp because sites where an ERP component is not present might add noise to the analyses. Thus, in our study, we selected exactly the 9 electrodes described by Luck (2005) that has been used for statistical analyses in previous studies (Blackford et al., 2012; Kuipers and Thierry, 2015; Kuperberg et al., 2018; Weber et al., 2004). According to these authors, this subarray of 9 electrodes represent a good balance between simplicity of design and a topographic distribution that follows the International standard 10–20 System location and maintain a symmetrical distribution in the lateral and anteroposterior axis. To evaluate the problem size effect, a 2 × 9 within-participants factorial design was used. Repeated-measure analyses of variance (ANOVAs), were conducted with the problem size factor (additions with results larger than 10, additions with results smaller than 10) and the ROIs (9 electrodes) as within-participants variables. To evaluate the type of problem effect, ANOVAs were conducted with the type of problem factor (additions 1–4, control additions with operands larger than 1–4) and ROIs (9 electrodes) as within-participants variables. The ROI was introduced in these analysis as a single within-participants factor in order to make our analyses comparable with those previously conducted in EEG studies on the problem size effect (e.g., in Núñez-Peña et al., 2005, the location variable entered the analysis as a single variable with 7 electrodes). The Greenhouse-Geisser correction (Greenhouse and Geisser, 1959) for nonsphericity of variance was used for all F-ratios with more than one degree of freedom in the denominator; reported here are the original df, the corrected probability level, and the ε correction factor.

Table 1 Problem size effect. ERP analyses. F

p

ε

ηp2

100–150 ms Problem size ROI Problem size × ROI

1.446 21.944 2.214

0.239 0.001⁎ 0.084

1 0.237 0.420

0.047 0.431 0.071

150–250 ms Problem size ROI Problem size × ROI

0.063 9.694 1.903

0.803 0.001⁎ 0.126

1 0.199 0.425

0.002 0.251 0.062

250–450 ms Problem size ROI Problem size × ROI

0.392 39.081 0.281

0.536 0.001⁎ 0.862

1 0.364 0.425

0.013 0.574 0.010

500–650 ms Problem size ROI Problem size × ROI

5.148 21.121 1.013

0.031⁎ 0.001⁎ 0.380

1 0.427 0.305

0.151 0.421 0.034

Note. Summary of statistical analyses conducted to evaluate the problem size effect (additions with results larger vs. smaller than 10). ⁎ p < .05.

In the last time window (500–650 ms), the problem size effect was significant, F(1, 29) = 5.148, p = .031, ε = 1, ηp2 = 0.151. In particular, more positive amplitudes were found when the addition result was larger than 10 relative to additions with results smaller than 10. The ROI effect was significant, F(8, 232) = 21.121, p < .001, ε = 0.427, ηp2 = 0.421, but the problem size × ROI failed to reach significance, F(8,232) = 1.013, p = .380, ε = 0.305, ηp2 = 0.034. However, we analysed the problem size effect in each electrode separately because we had a priori hypothesis about the topographic distribution of the LP effect (posterior regions, Núñez-Peña et al., 2005, 2011; Van Beek et al., 2014). The effect was significant only in the Pz electrode, t(29) = −3.576, p < 0.001, d = 0.703; but was not significant at any other location, all ps > 0.05 (see Fig. 3). In order to confirm that the pattern of results did not depend on differences in the number of trials with additions larger than 10 and additions smaller than 10, we conducted further analyses introducing as covariant the difference in number of trials per participant. The results of these analyses showed the same pattern of results as that reported above. In the 100–150 ms, 150–250 ms and 250–450 ms time windows, the problem size effect was not significant nor was the Problem size × ROI interaction (all ps > 0.05). However, in the 500–650 ms time window, the problem size effect was significant, F(1, 28) = 6.44, p = .02, with more positive amplitude for large additions than for small additions.

3. Results Firstly, we report the results obtained in the analyses conducted with behavioural measures (i.e., accuracy measures) followed by the results found with electrophysiological measures. 3.1. Behavioural measures 3.1.1. Problem size The problem size effect was significant, t(29) = 6.682, p < .001, d = 1.220. The percentage of errors was higher in additions with a result larger than 10 (M = 2.347%, SD = 4.073), relative to additions with a result smaller than 10 (M = 0.518%, SD = 0.681). 3.1.2. Type of problem The problem type effect was significant, t(29) = 7.936, p < .001, d = 1.449. The percentage of errors was higher in control additions (M = 2.413%, SD = 4.038) than in additions 1–4 (M = 0.278%, SD = 0.461).

3.2.2. Type of problem In the early 100–150 ms time window, the main effect of ROI was significant, F(8, 232) = 19.490, p < .001, ε = 0.228, ηp2 = 0.402, and the type of problem × ROI interaction was also significant, F(8, 232) = 2.698, p = .036, ε = 0.483, ηp2 = 0.085. Detailed analyses revealed a significant effect of type of problem at the F3 electrode, t (30) = 2.497, p = .018, d = 0.456; whilst the effect was not significant in any other electrode (all ps > 0.05). In the next two time windows, the only significant effect was the

3.2. Event-related potentials As with the behavioural data, we report the analysis of the problem size followed by analysis of the type of problem. In these analyses, we report the effects for the four time windows described in the method section. 52

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Fig. 3. Problem-size effect: Grand average ERPs for addition problems with results larger than 10 and addition problems with results smaller than 10. The box indicates the time window and ROI in which the problem size effect was significant at Pz (larger than 10, SD = 0.55; smaller than 10, SD = 0.51).

ROI effect in the 150–250 ms time window, F(8, 232) = 10.525, p < .001, ε = 0.204, ηp2 = 0.266; and the 250–450 ms time window, F (8, 232) = 32.171, p < .001, ε = 0.349, ηp2 = 0.526. The type of problem and the interaction between this variable and ROI were not significant, all ps > 0.05 (see Table 2 for detailed analysis).

The results obtained in the analyses conducted in the last time window (500–650 ms) are shown in Fig. 4. The main effect of the type of problem was not significant, F(1, 29) = 1.120, p = .299, ε = 1, ηp2 = 0.037. The main effect of ROI was significant, F(8, 232) = 21.332, p < .001, ε = 0.395, ηp2 = 0.424. Importantly, the type of problem × ROI interaction was also significant, F(8, 232) = 3.402, p < .001, ε = 0.464, ηp2 = 0.105. Separate analyses of the type of problem effect were conducted for each location. The type of problem effect was significant at two posterior electrodes: P3, t (29) = 2.431, p = .021, d = 0.444, and Pz, t(29) = 3.848, p < .001, d = 0.703. The type of problem effect was not significant over any other location, all ps > 0.05.

Table 2 Type of problem effect. ERP analyses. F

p

ε

ηp2

100–150 ms Type of problem ROI Type of problem × ROI

0.013 19.490 2.698

0.909 0.001⁎ 0.036⁎

1 0.228 0.483

0.000 0.402 0.085

150–250 ms Type of problem ROI Type of problem × ROI

0.233 10.525 1.523

0.633 0.001⁎ 0.196

1 0.204 0.535

0.008 0.266 0.050

250–450 ms Type of problem ROI Type of problem × ROI

1.588 32.171 1.606

0.218 0.001⁎ 0.179

1 0.349 0.491

0.052 0.526 0.052

500–650 ms Type of problem ROI Type of problem × ROI

1.120 21.332 3.402

0.299 0.001⁎ 0.001⁎

1 0.395 0.464

0.037 0.424 0.105

3.2.3. Very small addition problems In the analyses reported above, the comparison between additions 1–4 vs. one-digit control additions with operands larger than 1–4 did not modulate mean amplitude in the N400 time window whilst control additions showed large LP amplitudes over posterior regions. Uittenhove et al. (2016, p. 295) conducted additional analyses with very small problems in which they compared additions with operands 1–4 vs. control additions with additions 7–10 that did not involve 1 with at least one operand larger than 4. The analyses conducted with exactly the same additions; type of problem: additions 1–4 vs (60 additions, 12 additions repeated 5 times each) vs. control additions (90 additions, 18 additions repeated 5 times each) showed the same pattern of result than that described above. In the N400 time window, the type of problem was not significant, F < 1, nor this variable interacted with ROI, F(8, 232) = 1.407, p = .245, ε = 0.397. However, in the LP time

Note. Summary of statistical analyses conducted to evaluate the type of problem effect (additions 1–4 vs. control additions). ⁎ p < .05. 53

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Fig. 4. Type of problem effect: Grand average ERPs for additions 1–4 and control additions. The boxes indicate the time window and ROIs in which the type of problem was significant at P3 (small additions 1–4, SD = 0.58; control additions, SD = 0.46) and Pz (small additions 1–4, SD = 0.56; control additions, SD = 0.48).

axis × Laterality axis three-way interaction was marginal, F(4, 116) = 2.39, p = .06, ε = 0.85. The Problem size × Anterior-posterior axis interaction was significant over parietal regions, F(2, 58) = 6.85, p = .004, ε = 0.86, but not at frontal and central regions (all ps > 0.05). Regarding the type of problem, the 2 × 3 × 3 ANOVA conducted in the 250–450 ms time window with Type of problem (1–4 additions, control additions) × Anterior-posterior axis (frontal, central, parietal) × Lateral axis (left, midline, right) showed no significant effect of the type of problem and this factor did not interact with any other (all ps > 0.05). In the 500–650 time window, the Type of problem × Anterior-posterior axis interaction was significant, F(2, 58) = 5.99, p = .01, ε = 0.66. The type of problem effect was significant over parietal regions, F(1, 29) = 15.09, p < .001, but not at frontal and central regions (all ps > 0.05). Thus, the results of these analyses confirm and strengthen those reported in the main text. The problem size and type of problem effects were not significant in the N400 time window. However, both effects were significant in the LP time window and were located over posterior regions. The topographic distribution of the effects explored in the study, the problem size and the type of problem, were analysed further with the aim of controlling the false discovery rate. The two effects were considered in each topographic location (ROIs, 9 locations) in the time windows of interest associated to the N400 component (250–450 ms) and the LP component (500–650 ms) and false discovery rate (FDR) corrections were computed (Benjamini and Hochberg, 1995). Following the Benjamini-Hochberg procedure, the p values associated to each comparison were ranked from lowest to highest where the smallest p value had a rank of i = 1 and the largest a value of i = 9 (9 scalp locations). Afterwards, the Benjamini-Hochberg critical value, (i/m)Q, was computed, where i was the rank, m the total number of comparisons, and Q the false discovery rate. The Q value was set at 0.10

window, the type of problem × ROI interaction was significant, F(8, 232) = 4.898, p < .001, ε = 0.554. Similarly to that reported in text, the effect was significant over posterior regions (P3 electrode, t (30) = 4.589, p < .001). Furthermore, when new analyses were conducted with an equal number of observations for each experimental condition: 1–4 additions (12 additions repeated 5 times each) and 12 medium small problems (12 additions repeated 5 times randomly selected from the set of control additions); the posterior effect in the LP time window remained: P3 electrode, t(30) = 2.304, p = .03. The analyses conducted with the same number of items in the N400 time window did not show significant effect of type of problem, F(2, 29) = 2.45, p = .12, or Type of problem × ROI interaction, F < 1. 3.2.4. Topographic distribution of problem size and type of problem effects The topographic distribution was considered in our study as a single ROI factor (9 locations). This was done based on previous studies on the problem size effect with addition problems (i.e., Núñez-Peña et al., 2005). However, we acknowledge that ROI has a very large topographic distribution, which could camouflage significant main effects or interactions. In order to examine this confounding factor, we performed further analyses to explore the problem size and type of problem effects on the critical time windows, the N400 time window (250–450 ms) and the LP time window (500–650 ms) considering scalp locations with a smaller topographic distribution. The 2 × 3 × 3 ANOVA conducted in the 250–450 ms time window with Problem size (small additions, large additions), Anterior-posterior axis (frontal, central, parietal) and Lateral axis (left, midline, right), did not show significant the problem size effect, nor the interaction between problem size and any other variable (all Fs < 1). In the 500–650 time window, the Problem size × Anterior-posterior 54

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Fig. 7 shows the results found in these analyses (a detailed table of Wilcoxon t-test values and permutation p-values approximated by Monte Carlo estimate is provided in Online Supplementary Material 2). Regarding the LP component, the pattern of results was similar to that found with parametric tests but it delimited the time windows in which the effect occurred. To be more specific, the problem size effect was significant in the 498–542 ms and the 566–650 ms LP time windows, whilst the type of problem effect in the LP time window appeared earlier (468–496 ms and the 520–650 ms time windows).

following McDonald's recommendations (2014, p. 260, Q value between 0.10 and 0.20). According to the Benjamini-Hochberg procedure, the largest p value that has p < (i/m)Q is significant, and all of the p values smaller than it are also significant. Concerning the problem size in the N400 time window, none of the comparisons were significant. However, in the LP time window, the type of problem effect was significant at the Pz electrode, p = .001 < (i/m)Q = 0.011, no other comparison was significant (see Table 3). Regarding the type of problem, none of the comparisons were significant at the N400 time window. However, the type of problem modulated the LP amplitude at the Pz electrode, p = .001 < (i/m)Q = 0.005, and the P3 electrode, p = .021 < (i/m)Q = 0.097. The type of problem effect was not significant at any other location. Thus, the pattern of results remained after controlling for Type I error. On the other hand, the topographic maps associated to the problemsize effect (Fig. 5) and the type of problem effect (Fig. 6), in the time windows of 250–450 ms and 500–650 ms revealed the typical distribution of the N400 component over anterior regions and the LP component over posterior regions.

4. Discussion There is currently no consensus regarding the way in which individuals resolve simple addition problems composed of one-digit operands. Two accounts are proposed, the retrieval explanation (Campbell and Graham, 1985; Campbell et al., 2016) and the procedural view (Barrouillet and Thevenot, 2013; Uittenhove et al., 2016). In our study, we aimed at dissociating between these two views through the use of electrophysiological indexes. In particular, we propose that depending on the theoretical perspective (retrieval from memory vs. counting procedures) ERP modulations would be found in the N400 or the LP component. Specifically, if additions 1–4 involve counting and control additions memory retrieval (Barrouillet and Thevenot, 2013; Uittenhove et al., 2016), N400 modulations would be expected with smaller amplitudes for control additions vs. additions 1–4 because the semantic processing would be easier with medium small additions (retrieval of arithmetic facts) than with very small additions (counting procedures). Conversely, if all simple additions involve semantic access by retrieving arithmetic facts (Campbell, 1987), no N400 but LP modulations would be obtained. Thus, the type of problem would modulate the LP component with greater amplitude for control additions compared to additions 1–4, because retrieval from memory is difficult with medium small problems (control additions) than with small problems (additions 1–4). In our study, we evaluated two factors, the problem size and the type of problem effect. Regarding the first factor, the results revealed that the amplitude of the LP component was modulated by the size of the addition result. In particular, LP amplitudes were more positive when the problem size was larger as opposed to when the size of the problem was smaller than 10. However, the type of problem did not modulate the ERP amplitude in the N400 time window. This electrophysiological pattern of results replicates previous findings suggesting that the LP component is sensitive to the difficulty of the problem (Núñez-Peña et al., 2005, 2011; Van Beek et al., 2014). The problem size effect was located over posterior regions (Pz electrode). This topographic distribution of the LP component is consistent with previous studies regarding the difficulty of arithmetic problems (Núñez-Peña

3.2.5. Nonparametric permutation testing A Bottom-up approximation was also used to examine in depth the time course of the problem size and type of problem effects found in the study over posterior regions (i.e., Pz electrode). To this end, we compared the averaged ERP waveforms of the problem size (additions larger than 10, additions smaller than 10) and the type of problem (additions 1–4, control additions) at a particular electrode (Pz) that was sensitive to electrophysiological differences between conditions previously examined with parametric tests. Differences between conditions were evaluated over time (100–650 ms time window, time-locked to the addition problem onset) by performing a separate test every 2 ms. Type I error rate due to the large number of statistical comparisons (276 contrasts for each factor, problem size and type of problem) was controlled for with nonparametric cluster-based permutation tests (Maris and Oostenveld, 2007). This permutation testing procedure began the same way as the usual parametric tests by computing statistic of the observed data for the problem size and the type of problem effects (the nonparametric Wilcoxon t-test was used). All cluster-level statistics, defined as the sum of t values within each cluster, were evaluated under the permutation distribution of the maximum (minimum) cluster level statistic. This permutation distribution was approximated by drawing 9999 random permutations of the observed data. The obtained p values represented the probability under the null hypothesis (no difference between the two conditions of the problem size and the type of problem factors) of observing a maximum or minimum cluster-level statistic that was larger or smaller (respectively) than the observed cluster-level statistics. Table 3 Problem size and type of problem effects in N400 and LP time windows. Electrode

Problem size

Type of problem

N400 (250–450 ms)

F3 Fz F4 C3 Cz C4 P3 Pz P4

LP (500-650 ms)

N400 (250–450 ms)

LP (500–650 ms)

p value

(i/m)Q

p value

(i/m)Q

p value

(i/m)Q

p value

(i/m)Q

0.482 0.702 0.804 0.669 0.318 0.794 0.977 0.499 0.449

0.905 0.905 0.905 0.905 0.905 0.905 0.977 0.905 0.905

0.589 0.297 0.244 0.868 0.247 0.770 0.227 0.001⁎ 0.982

0.884 0.535 0.535 0.976 0.535 0.976 0.535 0.011 0.982

0.042 0.087 0.262 0.370 0.148 0.753 0.580 0.615 0.452

0.374 0.391 0.589 0.665 0.444 0.753 0.692 0.692 0.678

0.080 0.632 0.355 0.405 0.329 0.613 0.021⁎ 0.001⁎ 0.081

0.181 0.632 0.521 0.521 0.521 0.632 0.097 0.005 0.181

Note. Problem size effect (additions larger than 10, additions smaller than 10) and Type of problem effect (additions 1–4, control additions) at each scalp location (electrode) in the N400 time window (250–450 ms) and the LP time window (500–650 ms). (i/m)Q = critical Benjamini-Hochberg critical p value. ⁎ p significant comparisons after controlling for false discovery rate. 55

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Fig. 5. Topographic maps of the Problem size effect obtained when participants resolved additions whose results were larger than 10 or smaller than 10 at the N400 time-window (250–450 ms) and the LP time-window (500–650 ms).

et al., 2005, 2011). The reference-dependent surface potentials explored in our study did not allow direct inference about the underlying generator sources, however, the topographic distribution of the effects over posterior regions are in line with previous studies showing the role of the parietal cortex during arithmetic processing (e.g., Menon et al., 2000; Zago and Tzourio-Mazoyer, 2002). The LP modulations found in the current study might be considered an example of the difficulty associated to the retrieval of information stored in associative memory, regardless of whether this information involves arithmetic knowledge (Núñez-Peña et al., 2005, 2011) or other types of contents such as new associations created when individuals study pairs of unrelated words (Donaldson and Rugg, 1999). However, more important for the purpose of the current study was the pattern of results obtained when the type of problem was examined. If very small additions (additions 1–4) are resolved by counting procedures and simple additions with larger operands (control additions) by memory retrieval, we expected to find differences due to the type of problem. Specifically, memory retrieval would involve an easy access to semantic information than the use of automated procedural strategies, so the N400 amplitude would be more pronounced in additions 1–4 than in control additions. However, the mean amplitudes in the N400 time window were similar in these two types of additions. No difference between additions was found even at frontal region where N400 modulations are observed depending on the way arithmetic problems are resolved (e.g., exact calculation vs. rounding, Luo et al., 2009). It

could be argued that the absence of N400 modulations regarding the type of problem was a consequence of a reduced statistical power. However, a priori analyses for the experimental design and sample size of participants in our study yielded a statistical power > 95%. Importantly, the results of the study did show differences associated to the type of problem in the LP time window. Problems with large results (control additions and additions with results larger than 10) showed larger LP amplitudes over posterior locations relative to additions with small results (additions 1–4 and additions with results smaller than 10). As in other studies, these effects were found over parietal region related to the retrieval of contents from associative memory (Donaldson and Rugg, 1999; Núñez-Peña et al., 2005; Van Beek et al., 2014). It could be argued that the type of problem effect (1–4 additions, control additions) was irremediably the same effect as the problem size (small additions, large additions). In fact, the additions 1–4 condition (12 additions) included 50% of items of the small additions condition and 0% of items of the large additions condition. However, the similarity between the levels of the type of problem and the problem size variables cannot explain the pattern of results. On the one hand, the critical condition (additions 1–4) included problems that were used only in this condition (1 + n and n + 1 addition problems) and were not part of the problem size factor. On the other hand, differences between the problem size and type of problem effects were confirmed when we compare (a) the type of problem effect reported in our results with (b) the additional analyses reported in the results section with a

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Fig. 6. Topographic maps of the Type of problem effect obtained when participants resolved additions with operands between 1 and 4 (Additions 1–4) and additions with operands between 7 and 17 (Control Additions) at the N400 time-window (250–450 ms) and the LP time-window (500–650 ms).

subset of items used by Uittenhove et al. (2016). In the first analysis (a), the control condition was composed of 24% small additions and 76% large additions (greater percentage of large additions than of small additions in this condition). In contrast, in the second analysis (b) the control additions condition was composed of 67% small additions and 33% large additions (lower percentage of large additions than of small additions in this condition). Although the weight of the large/small additions percentages in the control additions condition was reversed in these two analyses (a) vs. (b), the pattern of results was similar in both. Uittenhove et al. (2016) proposed that “very small problems are solved by a rapid sequential procedure whereas medium small and ties problems are solved through retrieval” (p. 298). The data found in our study are difficult to accommodate within this proposal. On the one hand, regarding the N400 component, the idea of an easy semantic access to arithmetic facts for medium small problems (control condition) but a reduced semantic processing through counting procedures for very small problems (additions 1–4) would predict small N400 negativity with medium small problems relative to very small problems (e.g., as observed during the processing of stimuli that varies in semantic content, e.g., open-class words vs. closed-class words, Neville et al., 1992; Nobre and McCarthy, 1995; Pulvermüller et al., 1995). On the other hand, in our study, we found greater LP positivity in control additions than in additions 1–4. In our opinion, these LP modulations are difficult to be accounted for in terms of differences in the underlying cognitive mechanism (counting for very small problems, retrieval for medium small problems). On the contrary,

variations in LP amplitude depending on the type of problem would be easily explained if we assume that the main difference between these two types of sums is their problem size (larger for medium small problems than for very small problems). In fact, late positive modulations have been recurrently interpreted as reflecting the difficulty of accessing arithmetic facts as the size of arithmetic problems increases in multiplications (Pauli et al., 1994, 1996) and additions (Núñez-Peña et al., 2005). Therefore, in our view, the data found in our study are difficult to reconcile with the proposal of two different cognitive mechanisms, retrieval and counting, for two subsets of simple sums, medium small and very small, respectively. On the contrary, our results are in line with the idea that all simple additions are resolved by retrieving the result from memory. 4.1. Conclusions The most accepted view in cognitive arithmetic assumes that educated adults solve simple additions through memory retrieval. Accepting the existence of alternative cognitive mechanisms underlying the resolution of simple additions in adulthood (e.g., a counting procedure) would have profound implications for the teaching of simple arithmetic in childhood. The electrophysiological results obtained in our study suggest that adults solve simple additions by retrieving the result from memory, according to the well stablished and commonly accepted view.

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Fig. 7. Nonparametric permutation testing of ERP waveforms at a particular electrode (Pz) as a function of the Problem size (upper graph) and the Type of Problem (lower graph). Time sample specific contrasts were performed every 2 ms (100–650 ms time window time-locked to stimulus onset). Vertical lines represent the contrasts for which the statistic value exceed the critical value that corresponded to an alpha-level of 0.05.

Acknowledgments

References

This work was supported by the Spanish Ministry of Economy and Competitiveness (research project PSI2016-75250-P).

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