Different brain mechanisms mediate two strategies in arithmetic: evidence from Event-Related brain Potentials

Transcription

1 Neuropsychologia 41 (2003) Different brain mechanisms mediate two strategies in arithmetic: evidence from Event-Related brain Potentials Radouane El Yagoubi a,, Patrick Lemaire b, Mireille Besson c a Université de Provence & CNRS, case 66, 3 Place Victor Hugo, Marseille Cedex 3, France b Université de Provence, CNRS, & Institut Universitaire de France, 3 Place Victor Hugo, Marseille Cedex 3, France c Institut de Neurosciences Physiologiques et Cognitives (INPC), CNRS, 31 ch. Joseph Aiguier, Marseille Cedex 20, France Received 15 February 2002; accepted 5 September 2002 Abstract Participants were asked to verify if complex additions were smaller than 100 or not. Two hundred and forty arithmetic problems were presented, with half the problems being small-split problems (i.e. proposed sums were 2 or 5% away from 100) and half being large-split problems (i.e. proposed sums were 10 or 15% away from 100). Behavioral and ERPs data indicate that participants may use two different strategies to verify complex inequalities, a whole-calculation strategy for small-split problems and an approximate-calculation strategy for large-split problems. The choice between these two strategies occured within 250 ms post-stimulus presentation, and strategy execution was lateralized. Implications for our understanding of the brain mechanisms underlying arithmetic problem solving are discussed Elsevier Science Ltd. All rights reserved. Keywords: Strategy execution; Split effects; Inequalities verification; ERP 1. Introduction One of the most fascinating findings in the area of arithmetic processing is that people use several strategies to solve problems as simple as 4 8 [15,19,34]. Arithmetic is not special in that respect since people use several strategies in a wide variety of cognitive domains such as serial recall, question answering, sentence verification, reading, and naïve physics [20,23,33,36]. Yet, rarely as in arithmetic have strategic aspects underlying people s performance been documented with such a great level of details. The present study was conducted to further investigate how people use two specific arithmetic strategies (i.e. wholeand approximate-calculation strategies) when they solve two-digit problems like Event-related brain potentials (ERP) and behavioral data were analyzed to investigate whether each strategy could be distinguished on the basis of brain potentials, mean latencies, and percent errors. One of the main advantages of the ERP method is to allow us to precisely determine when the computations involved in the two conditions of a problem solving task are sufficiently different for the ERPs elicited in these two conditions to start Corresponding author. Tel.: ; fax: address: radouane@up.univ-mrs.fr (R. El Yagoubi). to diverge. Above and beyond the arithmetic domain, the present data were collected to shed light on the dynamics underlying strategic choice processes. Previous studies in arithmetic have reported a number of important findings regarding which strategies are used, how often they are used, how they are executed, how they are selected as a function of a variety of task parameters (e.g. types of problem and task), and how these strategic dimensions change with age and/or learning [7,15,22,23,35,38]. For example, people are slower to say that = 13 is false than = 19, presumably because they use a fast plausibility-checking strategy on the latter and a whole-calculation strategy on the former [2,12,29,38]. In the whole-calculation strategy, people encode the problem, compute the solution and retrieve the correct answer from long-term memory, compare the retrieved and proposed answers, make a true/false decision, and then respond. In the fast plausibility-checking strategy, people dispense with the retrieval and comparison processes, as the proposed answer is too far away from the correct answer. Thus, they encode the problem, decide whether it is close or far from the proposed solution, find an estimate (such as too small or too large ), and then respond. Multiple strategy use is not restricted to arithmetic verification tasks. It has been found in production tasks, such as when people are asked to find the result of problems like 8 7or [1,15,36,38] /02/$ see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S (02)00180-X

2 856 R. El Yagoubi et al. / Neuropsychologia 41 (2003) Most previous studies looking at multiple strategy use in arithmetic are limited in at least two ways. First, very few studies have looked at strategies people use in complex arithmetic problem solving (see in [21,36], for exceptions). Most studies have investigated strategy use in simple arithmetic problems and this may differ for more complex problems. Second, none of the previous studies have determined when, in the time course of arithmetic information processing, people choose among alternative strategies. The present study was designed to address these two questions by studying how and when people use whole- and approximate-calculation strategies while verifying inequalities involving two-digit operands. To achieve this goal, we collected behavioral and ERP data in a new arithmetic task, namely the inequality verification task. In the present version of the inequality verification task, people were presented problems with two-digit addends (e.g ) and had to decide whether the sum was smaller than 100. The basic manipulation concerned correct sums. Half the problems had correct sums that were close to 100 (by 2 or 5%, e.g ), whereas, the other half had correct sums that were far from 100 (by 10 or 15%, e.g ). Such split manipulations have been used in previous research on simple equation verification tasks in which people have to say whether equations like 8+4 = 13 or 8+4 = 19 are true or false [2,12,29,38]. Split effects have never been tested in simple or complex inequalities verification tasks. This study provided the opportunity to test the generality of split effects across different types of problems and tasks. Even more than in simple problem verification tasks, people were expected to use two strategies to verify inequalities in the present experiment. They were expected to use a whole-calculation strategy on small-split problems (i.e. problems with correct sums that were close to 100) and an approximate-calculation strategy on large-split problems (i.e. problems with correct sums that were far from 100). Using different strategies to accomplish this inequality verification task leads to the following predictions. First, people should have better behavioral performance (i.e. faster latency and higher accuracy) on large-split problems than on small-split problems, thus replicating standard split effects. Second, temporal analyses of brain electrical activity, through the ERPs, should indicate when, in the course of problem solving, the ERPs associated with the resolution of small- and large-split problems start to diverge. By inference, this will provide important information regarding the potential use of different strategies for solving the two types of problems. Previous theoretical and experimental studies suggest that the critical neural structures subserving whole- and approximate-calculation strategies are different. In a study in which people were presented with simple arithmetic problems like 4 + 5, and then with two candidate responses like nine or seven, and were asked to either choose the exact answer or select the most plausible answer, Dehaene et al. [11] found that exact- and approximate-calculations involved distinct neural networks. Approximate-calculation was found to recruit bilateral areas of the parietal lobule, the cerebellum, the precentral, and dorsolateral prefrontal cortex, whereas, exact calculation involved the left inferior prefrontal cortex [8 10,29 31,37]. More recently, Stanescu-Cosson et al. [37], measured cerebral activity with functional MRI and event related potentials while healthy volunteers performed exact- and approximate-calculation tasks. The fmri results showed bilateral intraparietal, precentral, dorsolateral and superior prefrontal regions activation during approximation, while the left inferior prefrontal cortex and the bilateral angular regions were more activated during exact calculation. The ERPs results showed that the effect of task was significant as early as ms following problem presentation. The results were also in-line with the implication of two cerebral networks for number processing. To summarize, as the ERP method offers a high temporal resolution in the range of milliseconds and, since ERPs precisely reflect the temporal sequence of perceptive and cognitive computations, we hoped to determine when computations (strategies) start to diverge, a piece of information that cannot be obtained with reaction times (RTs) alone. Moreover, analyses of the scalp distribution of the variations in brain electrical activity associated with the two types of problems may help determine whether distinct cerebral networks underlie whole- and approximate-calculation strategies in arithmetic inequality verification tasks. If indeed different lateralized brain structures are involved in both the cases, as reported in previous brain-imaging experiments, we may expect inter-hemispheric differences. 2. Method 2.1. Participants Fifteen adults (six males and nine females; with a mean age of 23 years and 10 months; ranges = 19, 4 32, 7) were tested individually. All were right-handed, neurologically normal and had normal or corrected-to-normal vision. All participants were paid for their participation. Due to technical problems in ERP recordings, data from three participants were excluded from the analyses Stimuli The stimuli were 240 arithmetic problems (i.e. additions), presented in a standard form (i.e. a + b) with the operands a and b being two digit numbers. Numbers were displayed at the center of a computer screen (SVGA color computer screen placed 60 cm in front of participants) and participants were asked to decide whether the result was smaller than 100. Results were smaller than 100 (e.g ) for half of the additions and larger than 100 (e.g ) for the other half. Inequations were constructed so as to have two experimental conditions, depending upon the size of

3 R. El Yagoubi et al. / Neuropsychologia 41 (2003) the splits between 100 and correct sums: (a) for small-split problems, correct sums were ±2 or ±5% away from 100 (e.g ; ), and (b) for large-split problems, correct sums were ±10 or ±15% away from 100 (e.g and ). Based on previous findings in arithmetic, problems were selected according to several constraints in order to avoid a number of potential confounds. First, the side of the larger operand was controlled by having half the problems with their first operand being the largest (e.g ). Second, no operand had a unit digit equal to 0 or 5 [4,20,22]. Third, no problems had operands with the same unit digit (e.g ). Finally, we presented an equal number of problems with two even operands (e.g ), with two odd operands (e.g ), and with one even operand (e.g ; [22]) Procedure Participants were comfortably seated in a Faraday box and were instructed to solve the problems mentally as quickly and accurately as possible. They were asked to press a YES button if the sum was smaller than 100, and the NO button if the sum was larger than 100. Response hands were counterbalanced across participants. The set of 240 problems was divided into four blocks of 60 problems each, with an equal number of correct and incorrect solutions within each block. The order of blocks was counterbalanced across participants, and problems were randomized for each participants within each block. Each block of trials lasted approximately 7 min and short rest periods were provided between blocks. To familiarize participants with the task, the experiment started with a practice session including 16 problems with a similar structure, but different from the experimental problems. The sequence of events within a typical trial was as follows. A warning-fixation stimulus was displayed in the center of the screen for 300 ms, followed by the first operand, displayed for 500 ms. Then, the second operand replaced the first operand on the screen and remained on the screen until the participant responded. A clock began timing when the second operand appeared and stopped when the participant pressed one of the two response keys. Participants were given 2200 ms from second operand onset to provide their answer. The intertrial interval (ITI), lasting for 2000 ms, followed participant s response. During the ITI, four Xs appeared at the center of the screen to inform participants that they could blink and move their eyes. Participants were asked to refrain from moving (except for the button pressing response) or blinking during the critical phase of EEG recording. During the ITI, participants heard an acoustic feedback when they responded incorrectly Data acquisition and analysis EEG was recorded for 3200 ms, starting 200 ms before the onset of the warning signal (baseline) until a row of XXXX appeared on the screen (ITI), from 28 scalp electrodes mounted on an elastic cap and located at standard left and right hemisphere positions over frontal, central, parietal, occipital, and temporal areas (international 10/20 systems sites Fz, Cz, Pz, Oz, Fp1, Fp2, F3, F4, C3, C4, P3, P4, O1, O2, F7, F8, T3, T4, T5, T6, Fc5, Fc1, Fc2, Fc6, Cp5, Cp1, Cp2, Cp6). These recording sites plus an electrode placed on the right-mastoid were referenced to the left mastoid electrod. The data were re-referenced off-line to the algebraic average of the left and right mastoids. Impedances of these electrodes never exceeded 3 k. The horizontal electro-oculogram (EOG) was recorded from a bipolar montage with electrodes placed 1 cm to the left and right of the external canthi; the vertical EOG was recorded from an electrode beneath the right eye, referenced to the left mastoid, to detect blinks and vertical eye movements. Trials containing ocular artifacts, movement artifacts, or amplifier saturation were excluded from the averaged ERP waveforms. The EEG and EOG were amplified by an SA Instrumentation amplifier with a band pass of Hz, and were digitized at 250 Hz by a PC-compatible microcomputer (Compaq Prosignia 486). ERP data were analyzed by computing the mean amplitude in selected latency windows relative to a 200 ms baseline. Analyses of variance (ANOVAs) were used for all statistical tests. In this report, unless otherwise noted, differences are significant to at least P < Topographic maps were computed using ICA ( scott/ica.html) [25]. 3. Behavioral results Preliminary analyses revealed similar outcomes for problems smaller and larger than 100. Therefore, subsequent analyses collapsed over this factor. RTs for correct responses and percentages of errors were analyzed using an ANOVA with type of problems (small- and large-splits) as a within-subject factor. RTs shorter than 300 ms and longer than 2200 ms were considered as errors (0.9% for both anticipations and non-responses), and were excluded from analyses. Results showed a main effect of split, with longer RTs on small-split problems than on large-split problems (872 ms versus 687 ms; F(1, 11) = 67.63, MSe = ), yielding a 185-ms split effect. Moreover, more errors were made on small-split than on large-split problems (6.84% versus 2.15%; F(1, 11) = 50.85, MSe = 132), ruling out potential speed-accuracy trade-offs. 4. ERP results Grand mean ERPs (averaged across 12 participants) are compared for small- and large-split problems at midline (Fig. 1), and at lateral electrodes (Fig. 2). ERPs for correct responses only are presented for the entire recording

4 858 R. El Yagoubi et al. / Neuropsychologia 41 (2003) Table 1 Summary of results of the statistical analyses in the different latency ranges for each factor Midline Lateral A B AB A C AC D AD CD ACD ms ms ms ms ms ms + + Note: (+) significant effect (P <0.05); ( ) non-significant effect. A, split; B, electrode; C, hemisphere; D, localisation; ms, warning fixation stimulus; ms, first operand display; second operand shown 800 ms after warning signal. Fig. 1. Grand average ERPs, recorded from 12 participants at the midline electrodes, (Fz, frontal; Cz, central; Pz, parietal; and Oz, occipital). ERPs are time-locked to the onset of the sequence. The vertical bars indicate the onset of the warning signal, first and second operands. ERPs are compared in two experimental conditions: small- and large-splits. In this and subsequent figures, amplitude ( v) is represented on the ordinate, with negative voltage up, and time (ms) on the abscissa. period (3200 ms). Lateral and midline electrodes were analyzed separately. ANOVAs for midline electrodes involved two within-subjects factors, two splits (small, large) and four electrodes (Fz,Cz,Pz,Oz). ANOVAs for lateral electrodes was conducted using 18 electrodes, and involved four within-subjects factors, two splits by two hemispheres (right and left) by three regions (frontal, central and parietal) by three electrodes (for each region). These analyses were performed in successive latency bands (as shown in Table 1 for a summary of results). From 0 to 300 and 300 to 800 ms: As shown in Fig. 1, an N1 P2 complex is associated with the presentation of the warning stimulus and the first operand. Moreover, as expected, the ERPs elicited in the two experimental conditions (small- and large-splits) perfectly overlap. Accordingly, analyses in these latency bands showed no significant main effect of split, either at midline or at lateral electrodes. Thus, this split-factor had no effect on the ERPs to the warning fixation and the first operand. From 800 to 1050 ms: (i.e ms latency band after second operand onset). As shown in Fig. 1, an N1 P2 complex is associated with the presentation of the second operand. Again, the amplitude of the N1 P2 components do not differ as a function of type of problems. Consequently, analyses in this latency range, corresponding to the first 250 ms after the presentation of the second operand, showed no main effect of split, either at midline or at lateral electrodes. From 1050 to 1400 ms: (i.e ms latency band after second operand onset). The main effect of split was significant for both midline (F(1, 11) = 6.89, MSe = 48.92) and lateral electrodes (F(1, 11) = 4.91, MSe = ). ERPs to small-split problems generated less positivity compared to large-split problems. Moreover, at lateral sites, the main effect of split interacted with the hemisphere factor (split hemisphere: F(1, 11) = 4.98, MSe = 12.75). This interaction resulted from the ERP difference between small- and large-split problems being larger over the left than over the right hemisphere (as shown in Figs. 3 and 4) for all regions considered for analysis (frontal, central, and parietal). From 1400 to 1700 ms: ( ms latency band after second operand onset). Split effects were again significant in this latency band with small-split problems associated with less positivity than large-split problems at both midline (F(1, 11) = 26.32, MSe = ) and lateral electrodes

5 R. El Yagoubi et al. / Neuropsychologia 41 (2003) Fig. 2. Overlapped are the grand average ERPs (12 participants) to small- and large-split recorded from 22 scalp sites. Recordings from parietal sites (P3 and P4) are enlarged at the bottom of the figure.

6 860 R. El Yagoubi et al. / Neuropsychologia 41 (2003) Fig. 3. Mean amplitude ( v) of the ERPs in the ms latency range. Split effects (i.e. the difference between small- and large-split problems) are plotted as a function of hemispheres. (F(1, 11) = 15.04, P<0.01, MSe = ). However, this split effect did not interact with the effect of hemisphere. From 1700 to 3200 ms: There was no difference between conditions, as ERP traces were coming back together. In summary, the main electrophysiological findings of interest started in the ms band (i.e. 250 ms after the second operand was displayed on the computer screen): small-split problems were associated with less positive components than large-split problems. Moreover, the ERP split effects were larger over the left than the right hemisphere in this latency band. 5. Discussion The behavioral data showed clear split effects both on RTs (185 ms split effects) and on error rates (i.e. poorer performance on small-split problems than on large-split problems), that did not result from speed-accuracy trade-off. It is important to note that these findings are in-line with those previously reported in simple arithmetic equation verification tasks [2,12,38]. Thus, with rather large numbers (e.g ) and a complex arithmetic inequation verification tasks, we were able to demonstrate that such differences in split (2 5% versus 10 15%) are large enough to trigger differences in behavioural and ERP measures that may reflect the use of different strategies: a whole-calculation strategy to verify small-split inequations and an approximate-calculation strategy to verify large-split problems. In the whole-calculation strategy, people encode the problem, engage an initial strategy selection phase, choose the whole-calculation strategy, execute it to find the sums of decade and unit digits successively, compare the solution with 100, make a smaller/larger decision, and give their answer. In the approximate-calculation strategy, people encode the problem, engage an initial strategy selection phase, and choose the approximate-calculation strategy. However, in contrast to the whole-calculation strategy, people do not need to calculate the exact solution, because a quick estimate of the solution is sufficient to make a smaller/larger decision. While such interpretations may account for the results, an alternative interpretation should be considered. Participants may choose to always use the wholecalculation strategy. In this case, the difference between small- and large-split problems would be explained by an additional stage of double-checking for small-split problems, because risks of errors increase with this type of problems. Double checking would allow participants to make fewer errors, but would increase RTs, thus resulting in speed-accuracy trade-offs that were, however, not observed in the present results. Further, it predicts that the ERPs differences between small- and large-split problems, would be likely to develop later than observed in the present data. Overall the ERPs results from this experiment are consistent with the behavioral data and add new important information: (a) ERPs differences between small- and large-split problems were found to start as early as 250 ms after the presentation of the second operand; (b) ERPs to small-split problems were associated with a more negative-going component than ERPs to large-split problems, from 250 to 900 ms after second operand onset; and (c) the mean amplitude differences between small- and large-split problems were larger over the left than the right hemisphere. It is important to note that ERP differences between smalland large-split problems were not significant before the second operand was presented, showing that participants did not anticipate the type of problem on the basis of the first operand. This is consistent with the hypothesis that people did not choose strategies before the experiment. Rather, they selected strategies on a problem-by-problem basis. Both the bar of fixation and the first operand are associated with the Fig. 4. Topographic maps computed as an integration of mean amplitude values at specific points in time in the difference waves (large-split minus small-split). These maps illustrate the time course of the positivity for the split effect with a left-lateralized distribution.

7 R. El Yagoubi et al. / Neuropsychologia 41 (2003) occurrence of an N1 P2 complex, reflecting the cortical activation bound to visual encoding. This encoding lasts around 200 ms, as shown in previous research [6,26]. According to the process model outlined previously, the encoding stage is followed by the selection and execution of the strategies necessary to perform the task. Interestingly, it is clear from Figs. 1 and 2 that the ERPs to small and large-split problems start to diverge as early as 250 ms after second operand onset. Therefore, as mentioned in the introduction, one can make the inference that, this is the upper-time limit from which different processes are engaged to solve small- and large-split problems. Based upon behavioral results, authors have argued that different strategies (whole-calculation versus approximation strategies) are called into play to solve small- and large-split problems. The differences seen in the ERPs to solve small- and large-split problems are consistent with the interpretation that different strategies are selected and used. However, the present results do not rule out the possibility that both types of strategies run in parallel, and that one strategy (approximation) terminates faster than the other (whole-calculation). Moreover, it was also possible that similar approximation strategies are used in both the cases, but that their relative frequency of use differ for small- and large-split problems. Moreover, the ERPs associated with the resolution of small-split problems were less positive going in the ms after second operand onset than the ERPs associated with large-split problems. Previous ERP research has shown that positive components, that develop in this latency band, often belong to the family of P300 components [13]. Interestingly, P300 amplitude has been shown to vary as a function of memory load with smaller P300s associated with increased memory load [14,28]. In so far as small-split problems require specific computations to be solved, memory load would be increased compared to large-split problems that make less processing demands. Note, however, that ERP differences between small- and large-split problems, that we interpret as a reduction of P300 amplitude with processing demands when participants use the whole-calculation strategy, could also be interpreted as an increased negativity. ERP research on language processing has demonstrated that a negative component (N400) within the ms latency band has been considered as the signature of semantic processing [3,16,17,18]. Thus, one may argue that the greater neural activity occurring in response to small-split problems observed here suggests that the whole-calculation strategy involves more semantic information in memory [27] than the approximate-calculation strategy. This interpretation is in-line with results obtained from other brain-imaging methods, such as fmri showing activation of language areas in solving arithmetic problems with exact-calculation strategy [30,37]. However, the scalp distribution of the split effect (as shown in Fig. 4) is not typical of either the P300 (parietal) or the N400 components (centro-parietal, slightly larger over the right than the left hemisphere in the visual modality). Interestingly, analyses of the scalp distribution of the ERP differences between small- and large-split problems revealed that split effects were larger over the left than right hemisphere, in the ms latency band following the presentation of the second operand. Thus, the two hemispheres do not seem to equally contribute to the resolution of small- and large-split problems, and the left hemisphere may be more involved when problem solving requires a whole-calculation strategy. This interpretation is in-line with Dehaene and Cohen [9,10] neuroanatomical model of number processing according to which the exact calculation would involve language areas, as well as subcortical structures, mainly in the left hemisphere. In contrast, verifying large-split problems (i.e. execution of the approximate-calculation strategy) would activate the cerebral pathways in a bilateral way. Previous results in the literature have indeed pointed to an absence of strong hemispheric differences in the numerical comparison tasks, may be because analogical magnitude representation is computed and represented in both the left and right hemispheres [5,31,32]. In sum, the time-course of the processes involved in the present arithmetic inequality verification task can be as follows: (a) visual encoding of the second operand and strategy selection within the first 250 ms; (b) strategy execution that involves different processes to solve small- and large-split problems, as reflected by the ERP difference between ms after second operand onset; and (c) the decision and motor response processes which again differ for small- and large-split problems. It is important to note that these different processing stages are not necessarily serially organized. They may partly overlap and some may run in parallel, as proposed by Logan and Gordon [24], Näätänen [26] and Reder and Ritter [33]. In conclusion, the results of the present experiment seem to indicate that participants used two different strategies to verify complex inequalities, a whole-calculation strategy for small-split problems and an approximate-calculation strategy for large-split problems. The choice between these two strategies was made within 250 ms post-stimulus presentation, and the strategy execution was lateralized. Although the ERPs provide evidence that different processes were called into play to solve small- and large-split problems, it is not yet possible to specify in which aspects the nature of these processes differ. These data contribute to the larger goal of arithmetic research, that is building up a complete functional neuro-anatomical model of arithmetic problem verification tasks. Much work remains to be done to achieve this end. To mention just two of them, future research may be directed at determining which brain areas are involved in using each strategy, or where is the initial strategy selection phase localized? These questions can be addressed by using fmri. Moreover, it seems particularly interesting to investigate individual differences (e.g. age- or skill-related differences) in neuropsychological and behavioral changes in the selection and execution of strategies.

8 862 R. El Yagoubi et al. / Neuropsychologia 41 (2003) Acknowledgements This research was supported by the CNRS, a Cognitique grant from the French Ministère de la Recherche to Patrick Lemaire and by a grant from the International Fundation for Music Research (RPA 194) to Mireille Besson. We thank Abdelrhani Benraïss, Daniele Schön, and Cyril Magne for their valuable help. References [1] Allen PA, Ashcraft MH, Weber TA. On mental multiplication and age. Psychology and Aging 1992;7: [2] Ashcraft MH, Battaglia J. Evidence for retrieval and decision processes in mental addition. Journal of Experimental Psychology, Human Learning and Memory 1978;4: [3] Besson M, Magne C, Regnault P. Le traitement du langage. L imagerie fonctionnelle électrique (EEG) et magnétique (MEG): ses applications en sciences cognitives. Hermes, Sciences Cognitives, in press. [4] Campbell JID. Architectures for numerical cognition. Cognition 1994;53:1 44. [5] Cohen L, Dehaene S. Cerebral networks for number processing: evidence from a case of posterior callosal lesion. Neurocase 1996;2: [6] Coles MGH, Rugg MD. Event-related brain potentials: an introduction. In: Rugg MD, Coles MGH, editors. Electrophysiology of minds: ERPs and cognition. Oxford University Press, 1995; p [7] Cooney JB, Swanson H, Ladd SF. Acquisition of mental multiplication skill: evidence for the transition between counting and retrieval strategies. Cognition and Instruction 1988;5: [8] Dehaene S. The organization of brain activations in number comparison: event-related potentials and the additive-factors methods. Journal of Cognition and Neuroscience 1996;8: [9] Dehaene S, Cohen L. Two mental calculation systems: a case study of severe acalculia with preserved approximation. Neuropsychologia 1991;29: [10] Dehaene S, Cohen L. Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex 1997;33: [11] Dehaene S, Spelke E, Pinel P, Stanescu R, Tviskin S. Sources of mathematical thinking: behavioral and brain-imaging evidence. Science 1999;284: [12] De Rammelaere S, Stuyven E, Vandierendonck A. Verifying simple arithmetic sums and products: are the phonological loop and the central executive involved? Memory and Cognition 2001;29: [13] Donchin E. Surprise!...Surprise? Psychophysiology 1981;18: [14] Donchin E, Coles MGH. Is the P300 component a manifestation of context-updating? Behavioural and Brain Sciences 1988;11: [15] Geary DC, Whiley JG. Cognitive addition: strategy choices and speed-of-processing differences in young and elderly adults. Psychology and Aging 1991;6: [16] Kounios J, Holcomb PJ. Structure and process in semantic memory: evidence from event-related brain potentials and reaction times. Journal of Experimental Psychology: General 1992;121: [17] Kutas M, Hillyard SA. Reading senseless sentences: brain potentials reflect semantic incongruity. Science 1980;207: [18] Kutas M, Hillyard SA. Brain potentials during reading reflect word expectancy and semantic association. Nature 1984;307: [19] LeFevre J, Bisanz J, Daley KE, Buffone L, Sadesky GS. Multiple routes to solution of single-digit multiplication problems. Journal of Experimental Psychology: General 1996;125: [20] Lemaire P, Fayol M. When plausibility judgments superside fact retrieval: the example of the odd/even effects on product verification. Memory and Cognition 1995;23: [21] Lemaire P, Lecacheur M, Farioli F. Children s strategy use in computational estimation. Canadian Journal of Experimental Psychology 2000;54: [22] Lemaire P, Reder L. What affects strategy selection in arithmetic? An example of parity and five effects on product verification. Memory and Cognition 1999;22: [23] Lemaire P, Siegler RS. Four aspects of strategic change: contributions to children s learning of multiplication. Journal of Experimental Psychology: General 1995;124: [24] Logan GD, Gordon RD. Executive control of visual attention in dual-task situations. Psychological Review 2001;108: [25] Makeig S, Westerfield M. Jung TP. ICA Toolbox for Psychophysiological Research. Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, [26] Näätänen R. Implications of the ERPs data for psychological theories of attention. Biological Psychology 1988;26: [27] Niedeggen M, Rösler F. N400 effects reflect activation spread during retrieval of arithmetic facts. Psychological Science 1999;10: [28] Pelosi L, Hayward M, Blumhardt LD. Which event-related potentials reflect memory processing in a digit-probe identification task? Cognitive Brain Research 1998;6: [29] Pesenti M, Thioux M, Seron X, De Volder A. Neuroanatomical substrates of arabic number processing, numerical comparison, and simple addition: a PET study. Journal of Cognitive Neuroscience 2000;12: [30] Pesenti M, Zago L, Crivello F, Mellet E, Samson D, Duroux B, Seron X, et al. Mental calculation in a prodigy is sustained by right prefrontal and medial-temporal areas. Nature Neuroscience 2001;4: [31] Pinel P, Dehaene S, Rivière D, LeBihan D. Modulation of parietal activation by semantic distance in a number comparison task. NeuroImage 2001;14: [32] Ratinckx E, Brysbaert M, Reynvoet B. Bilateral field interactions and hemispheric asymmetry in number comparison. Neuropsychologia 2001;39: [33] Reder LM, Ritter FE. What determines initial feeling of knowing? Familiarity with question terms, not with the answer. Journal of Experimental Psychology: Learning, Memory, and Cognition 1992;18: [34] Rickard TC. Bending the power law: a CMPL theory of strategy shifts and the automatization of cognitive skills. Journal of Experimental Psychology: General 1997;126: [35] Siegler RS. Strategy choice procedures and the development of multiplication skills. Journal of Experimental Psychology: General 1988;117: [36] Siegler RS, Lemaire P. Older and younger adults strategy choices in multiplication: testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General 1997;126: [37] Stanescu-Cosson R, Pinel P, Van de Moortele PF, Le Bihan D, Cohen L, Dehaene S. Understanding dissociations in dyscalculia: a brain-imaging study of the impact of number size on the cerebral networks for exact and approximate-calculation. Brain 2000;123: [38] Zbrodoff NJ, Logan GD. On the relation between production and verification tasks in the psychology of simple arithmetic. Journal of Experimental Psychology: Learning, Memory, and Cognition 1990;16:83 97.