Facts, rules, and strategies in single-digit multiplication: evidence from event-related brain potentials

Cognitive Brain Research 20 (2004) 183 – 193 www.elsevier.com/locate/cogbrainres

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Facts, rules, and strategies in single-digit multiplication: evidence from event-related brain potentials Kerstin Jost *, Ulrike Beinhoff, Erwin Hennighausen, Frank Ro¨sler Experimental and Biological Psychology, Department of Psychology, Philipps-University of Marburg, Gutenbergstrasse 18, Marburg D-35032, Germany Accepted 27 February 2004 Available online 14 April 2004

Abstract It has been hypothesized that zero vs. nonzero operands in single-digit multiplication problems invoke distinct solution strategies. We studied such problems in an implicit production task with event-related brain potentials (ERPs) recorded from 61 scalp positions in 18 participants. The topography of a slow negative wave, which accompanied the implicit production of the multiplication result, varied with problem type. In comparison to small problems, larger problems evoked a stronger negativity over fronto-central and right temporal sites, and zero problems evoked a left anterior negativity. These topographic differences indicate not only that zero and small nonzero problems are solved by means of distinct strategies—most likely rule application vs. fact retrieval—but also that larger, less practiced problems invoke other processes than pure fact retrieval. Moreover, ERPs showed a positive deflection around 450 ms with a centro-parietal topography (P300), whose amplitude reflected differences in anticipated problem difficulty. D 2004 Elsevier B.V. All rights reserved. Theme: Neural basis of behavior Topic: Cognition Keywords: Mental calculation; Multiplication facts; Problem size effect; Rule application; Slow waves; P300

1. Introduction It is widely accepted that single-digit multiplication problems are stored as facts in long-term memory, representing the associative structure of operands, problems, and solutions (for review, see Refs. [3,4]). LeFevre et al. showed that fact retrieval can account for 80% of the solutions of simple addition and multiplication problems, while other strategies (e.g., decomposition into simpler memorized facts) are applied only if immediate retrieval fails [28,29]. However, there are some other exceptions. Multiplications involving zero must not necessarily be solved by fact retrieval because the corresponding result can be generated by applying a rule (i.e., n  0 = n). This is easily demonstrated by any multiplication with a zero * Corresponding author. Tel.: +49-6421-2822160; fax: +49-64212828948. E-mail addresses: [email protected] (K. Jost), [email protected] (F. Ro¨sler). URL: http://www.staff.uni-marburg.de/~cablab/. 0926-6410/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cogbrainres.2004.02.005

operand, which can be solved immediately irrespective of its size (e.g., 758  0). Further evidence for a distinction between ‘‘rule-based’’ and ‘‘retrieval-based’’ multiplications has been provided by double dissociations observed in brain-damaged patients with calculation impairments [36,44]. Sokol et al. described a patient whose performance on fact-retrieval problems was not uniform. Error rates and the time course of improvement varied for problems without zero. In contrast, all multiplication problems with zero were equally impaired at the beginning but improved suddenly at the same moment and to the same extent. This suggests that an impairment of a general rule affected the performance on all problems with zero. Although there is little doubt that some multiplication problems can be solved by applying the zero rule, the empirical evidence for how multiplications with a zero operand are actually solved is sparse and somewhat contradictory. Stazyk et al. [46] (see also Ref. [35]), for example, found that verification times of n  0 problems were unusually long and error-prone compared to problems without zero, although the zero problems were uniformly rated as

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less difficult than nonzero problems. Thus, the authors concluded that zero problems are solved by applying the appropriate rule, rather than by directly retrieving the result. Rule knowledge might be less well established and therefore its activation might take longer than fact retrieval. However, in production tasks, response latencies are usually found to be faster for zero than for nonzero problems [22,28,32]. This discrepancy may be due to processing differences between production and verification. While production problems require full retrieval of the result, verification problems can be solved without producing the solution [49]. Instead, some sort of familiarity or plausibility judgment seems often used to decide whether the equation is valid or not (see also Ref. [9]). It could be that these processes caused the disadvantage of the n  0 problems in the Stazyk et al. study. However, even if only production tasks are considered, response times (RTs) are inconclusive with respect to the solution strategy invoked by zero problems. LeFevre et al. [28], for example, collected introspective reports in a production task and found that rule application and fact retrieval both resulted in fast and correct responses to zero problems. However, it is not only claimed that distinct solution strategies are possibly used for zero vs. nonzero problems, or for zero problems if they are embedded in different tasks. Recent studies provided evidence that single-digit multiplications without zero are also not necessarily always solved by one and the same strategy (Refs. [28,43]; for a comparable finding with addition problems, see Ref. [29]). In these studies, participants solved simple arithmetic problems in a reaction time task and gave immediate verbal reports on the type of strategy they had used for producing the result. In contrast to existing theories, which state that direct memory retrieval is the one and only procedure to access the result of single-digit problems (e.g., Ref. [7]; for review, see Refs. [3,4]), participants reported substantial use of nonretrieval strategies, such as repeated addition, number series generation, and retrieval of derived facts. For example, the problem 8  7 can be solved by retrieving 7  7 and then adding seven to the intermediate result, ‘‘49’’. Typically, these backup strategies involve multiple steps, including retrieval of well-known facts and applications of computational principles, presumably the reason why most of them are more time-consuming than simple, immediate fact retrieval [28]. LeFevre et al. found that procedures other than direct retrieval are more often used if larger operands (e.g., 78), compared to smaller operands (e.g., 23), are involved ([28,43]; for comparable findings with addition and division problems, see Refs. [27,29]). Furthermore, the authors could show that the well-established problemsize effect (i.e., an increase of RT with increasing size of the operands; e.g., see Refs. [8,32,46]) can be reduced dramatically if problems that invoke some nonretrieval strategy are eliminated. This finding contradicts theories

(e.g., Ref. [7]; for review, see Ref. [3]) that attribute the problem size effect to differences in the accessibility of results (e.g., due to differences in the frequency of problem usage) [1,2]. All in all, there is good reason to assume that singledigit multiplication problems are not only solved by pure fact retrieval, but that other strategies are used as well. To date, however, the empirical evidence of which problem is solved with which strategy is inconclusive. As outlined, findings with response times are inconsistent and, to some degree, task-dependent. Self-reports, on the other hand, are very suggestive, but their validity is questionable. Kirk and Ashcraft [22], for example, demonstrated that strategy reports for simple addition and multiplication problems are partially dependent on instructions. Moreover, self-reports can change the behavior of some individuals (see Ref. [43]). Brain imaging studies (e.g., Refs. [10,25]) also addressed the question of whether arithmetic problems invoke distinct solution processes, but so far, single-digit multiplication problems involving zero and nonzero, or small and large operands, were not analyzed systematically with this approach. To get more insight into whether multiplications with small and large or with zero operands invoke different solution strategies, we recorded event-related brain potentials (ERPs). ERPs are on-line measures of brain activity, which reflect timing, extent, and quality of processing mechanisms before an overt response is observed (e.g., see Ref. [40]). Previous studies demonstrated that ERPs could be successfully used to delineate the associative network structure of arithmetic knowledge [20,33,34], distinct processes during mental calculation [21], or strategy variations in verifying complex inequalities [14]. In the present study, the production of a multiplication result was embedded in a chain calculation problem—a sequence of simple calculations had to be performed and a final result, presented at the very end of the sequence, had to be verified (e.g., 3  5 + 2 = 17). The advantage of such a procedure is twofold. First, participants must implicitly produce the interim result of the multiplication problem before they can go on with the next calculation step. This prevents, as can happen in one-step verification problems (e.g., 3  5 = 16), that a decision is based on some sort of plausibility judgment without invoking production of the result (see Refs. [9,49]). Second, the multiplication result has to be produced implicitly (i.e., without any overt response) and this prevents movement artifacts in the electroencephalogram (EEG) recording, as they are most likely in one-step production tasks with an immediate verbal response. Three types of multiplications—problems with small or large nonzero operands and problems with one zero operand—were compared with a control condition not involving any type of mental calculation proper. In this condition, the operands of the first problem had to be stored only and one of the operands had to be used for another addition or

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subtraction.1 This provides an ‘‘active’’ baseline because the operands must be perceived and stored in each multiplication condition as well. Thus, computing the differences between multiplication conditions and the storage condition should reveal the net effects due to different solution strategies. We analyzed two ERP phenomena: the P300 component and a subsequent slow negative wave evoked in the production phase. P300 is an endogenous positive component occurring approximately 300 – 600 ms after stimulus onset that reflects higher-level cognitive processes such as event categorization [13,19]. Among others, the amplitude of P300 reflects variations in anticipated task difficulty: stimuli which notify a subjectively more demanding task elicit larger amplitudes [5,48]. Slow negative waves accompany almost any cognitive processing with two interesting features: their topographic maximum is determined by the modality and the type of task; for instance, linguistic tasks evoke a left anterior maximum and spatial tasks elicit a parietal maximum, and their amplitude reflects processing effort, that is, larger negativities are evoked by more difficult tasks) [38]. There is also strong evidence that slow negative waves reflect a relative increase of excitatory postsynaptic potentials within cortical cell assemblies [6]. Thus, slow negative waves indicate to what extent identical or distinct neural cell assemblies are recruited during a particular processing episode. Differences between calculation strategies should become manifest in the ERPs as follows: if the three conditions—problems with zero, and small and large problems without zero—are inherently categorized as varying in subjective difficulty, then an amplitude modulation of the P300 should be observed. More specifically, all multiplication conditions should evoke larger amplitude than the storage condition and the amplitude should be largest for the nonzero problems with large operands. 1 As shown by Ref. [26], arithmetic facts may be activated automatically by a set of two digits, even if participants are not explicitly asked to perform a calculation. In Ref. [26], subjects were shown pairs of numbers (e.g., 3 + 2) that were replaced, after a variable SOA, by a target number. Subjects were required to decide whether the target number was one of the numbers in the initial pair, referred to as number-matching task. LeFevre and Kulak found that subjects were slower to respond ‘‘no’’ when the target was the sum of the pair (e.g., 3 + 2 = 5) than when it was unrelated to the pair (e.g., 4). Thus, our storage condition could have caused a similar automatic activation of multiplication results, although the two digits were not shown together with a multiplication sign in this condition. However, even if such an effect was present in our control condition, it was of minor importance because the interference effect observed in Ref. [26] only occurred at very short SOAs (i.e., 40 and 60 ms), indicating that automatically activated results are very short-lived. Since the implicit production of the multiplication results took much longer than 60 ms in our study (see response times of Experiment I) and the main ERP differences between multiplication and storage also appeared much later in time (not earlier than 300 ms after problem presentation), it seems very unlikely that such an automatic spread of activation effect will have contaminated the contrasts of the multiplication problems with the control (storage) condition.

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Distinct solution strategies—rule application for zero, fact retrieval for nonzero small problems, and backup strategies for nonzero large problems, respectively—should become apparent in distinct topographies of the negative slow waves accompanying production proper. We expected one topographic difference between rule-based and retrieval-based problems and another one between retrieval-based problems involving small and large operands. We ran two studies: a behavioral study (Experiment I) and an ERP study (Experiment II). In the behavioral study, the timing of the event sequences was partially controlled by the participants: they saw the first part of the problem sequence (e.g., 34) and indicated with a finger movement when they had implicitly generated the solution to the multiplication problem and felt ready for the next operation (e.g., + 2). This allowed for an estimation of the implicit production times of the different problem types. We expected that responses are shorter for zero than for nonzero problems, and that they show the typical problem-size effect for nonzero problems (i.e., longer RTs for larger problems than for smaller problems). The estimate of the longest implicit production time was then used in the ERP study to specify a constant interval between the multiplication and the next operation, which was long enough to avoid any interference between the two operations.

2. Materials and methods 2.1. Participants Students of the University of Marburg were recruited for the behavioral and the ERP study. All were right-handed, native speakers of German, had normal or corrected-tonormal vision, and were naive with respect to the purpose of the study. They received either course credit points or a monetary compensation. Thirteen students participated in the behavioral study. One participant had to be excluded because of too many errors. The final sample comprised data from 12 participants (nine women) with a mean age of 25 years (range 20 – 34 years). Twenty-one students took part in the ERP study, none of whom had participated in the behavioral study. Three participants had to be excluded because of too many EEG recording artifacts. The final sample comprised data from nine women and nine men. The mean age was 23 years (range 18– 27 years). 2.2. Materials Operands ranging from 2 to 9 (ties were excluded; e.g., 3  3) were combined to form 56 multiplication problems without zero. Two levels of problem size were defined, each containing 28 problems: Problems with both operands V 5

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(products ranging from 6 to 20) were classified as small; problems with both operands >5 (products ranging from 42 to 72) were classified as large (for a similar classification, see Refs. [21,49]). Mixed problems with one operand V5 and one operand >5 (e.g., 38 or 56) were defined as small if they contained either a 2 or a 5. Thus, all problems with 2 and 5 were defined as small. This classification is based on the observation that in several response time studies, problems with 2 were found to be solved faster than other problems and that problems with 5 show a latency advantage relative to other problems of the same magnitude (e.g., Refs. [8,27]). In a previous study, this classification had provided a reliable problem-size effect, in both RTs and ERPs (Ref. [20]; for a similar classification, see Ref. [9]).2 Another 16 problems were constructed from the same digits by setting one operand to zero (n  0 or 0  n). All operand combinations were also used for the storage condition. Each problem was repeated seven times in the multiplication and in the storage condition. In total, there were 1008 trials. The additional, subsequent operation was either an addition or a subtraction by 1, 2, or 3. Incorrect solutions in the verification task ( p = 0.50) were realized by adding or subtracting 1, 2, or 3 from the correct result. 2.3. Design and procedure Participants sat in a recording chamber in front of a computer screen located at eye level at a distance of 70 cm. Numeric characters (vertical size 0.9 cm) were presented in black within a white rectangle in the center of the screen on a light-gray background. In the behavioral study, a trial started with a rectangle in the center of the screen in which the first operand appeared on its left side 1000 ms later. The next frame was presented another 500 ms later, and added the second operand on the right side of the rectangle and the operation sign in the center. Storage was indicated by ‘‘!’’ and multiplication by ‘‘  ’’. After another 500 ms, both digits disappeared and the operation sign remained visible during the ‘‘production phase.’’ The participants had to indicate with a finger response when they had generated the result and felt ready for the next operation. With this procedure, solution times were measured and an appropriate interval could be defined for the ERP study. The stimulus sequence was continued by the presentation of the addition2

In other studies, multiplication problems were categorized with respect to the size of the product irrespective of the size of the operands, that is, all products <25 were classified as small, all >25. However, there is some evidence that all multiples of 2 and all multiples of 5 show a response time advantage even if the second operand is >5. Therefore, we used the outlined classification procedure because we think that it separates easy and more difficult problems better. Following the recommendation of one reviewer, we reanalyzed the response times and the ERPs by using the more traditional classification, with the cutoff set at 25. Surprisingly, this reclassification provided almost identical results as found with our classification.

al operation for 600 ms. In the storage condition, the position of the third operand indicated which of the two stored digits should be used. Presentation on the left side of the rectangle indicated to add/subtract on basis of the first, left-sided operand, and presentation on the right side indicated to use the second, right-sided operand. The additional operation was followed by a blank screen of 200 ms, and the final result was presented until a response was given, or for a maximum of 1100 ms. The intertrial interval varied randomly between 2 and 3 s. Participants were instructed either to multiply or to store the first two operands and to indicate when result production was finished. This and the verification response had to be given by a brief upward finger movement, which interrupted a light gate. Allocation of the index fingers to the two response alternatives in the verification task was varied systematically across participants. Delayed and incorrect responses were fed back and excluded from further analysis. Stimulus presentation in the ERP study was identical with one exception. Participants had a fixed interval to produce the result of the multiplication task or to store the two operands: the operation sign was presented for 2800 ms (see Fig. 1), which is equivalent to the maximum RT observed in the behavioral study. The following additional operation and the verification task were as described for the behavioral study. In both studies, trials were presented in 28 blocks (with 36 trials each), with multiplication and storage trials appearing equally often within one block. A new randomized trial

Fig. 1. Event sequence of a trial, grand average ERPs, and measurement epochs. A trial started with a rectangle in which the first operand appeared on its left side followed by the second operand on the right and the operation sign in the center. From this point on, participants had 3300 ms to produce the result of the multiplication problem, or to store the two operands. After the production phase, an additional operation was presented and followed by the final result, which had to be verified. Shading shows where the positivity and the slow negative wave were measured (i.e., relative to the presentation of the second operand, between 300 and 600 ms and between 900 and 3100 ms, respectively). Negativity is up in this figure and in the following figure.

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sequence was created for each participant with the restriction that no operand combination was repeated within a block. Participants received some practice trials in advance in order to familiarize themselves with the task. An obligatory break of at least 20 s separated the blocks, which could be prolonged by moving both fingers out of the light gate. Participants were advised to use the short breaks for recreation. A longer break was inserted after approximately half of the blocks. In total, the experiments took about 2.5 h (plus about 1.5 h of electrode montage in the ERP study). 2.4. EEG recording, artifact handling, and signal extraction The EEG was recorded with Ag/AgCl electrodes from 61 locations and referenced to the nose tip. Additional electrodes were attached at the outer canthi of both eyes and the suborbital and supraorbital ridges of the left eye for horizontal and vertical EOG monitoring. The left mastoid served as ground. Impedances were always kept below 7 kV. Bandpass of the amplifier system (SYNAMPS; NeuroScan) ranged from DC to 40 Hz and the signals were digitized with 200 Hz. A DC reset was initiated automatically prior to the beginning of each experimental block. Trials containing eye blinks were detected by wavelet analysis and corrected using a linear interpolation algorithm. Drift artifacts were corrected by a regression method [17]. ERPs were extracted by averaging trials separately for subjects, electrodes, and experimental conditions, and were based on a minimum of 90 trials each. 2.5. Dependent variables and statistical analysis 2.5.1. Behavioral data (Experiment I) Mean response times were computed for each participant and experimental condition. Trials with response times shorter than 200 ms and larger than 10000 ms were excluded. This procedure eliminated less than 1% of the trials. 2.5.2. EEG data (Experiment II) Two intervals were defined to analyze the positive deflection and the slow negative wave in the production phase (see Fig. 1). P300 was measured between 300 and 600 ms by computing the average amplitudes of 10 consecutive time windows of 30 ms width. The slow negative wave was measured between 900 and 3100 ms by means of 11 average amplitudes, each covering 200 ms. The measures were referenced to a baseline of 100 ms preceding the onset of the first operand (see Fig. 1). To avoid an inflation of the degrees of freedom in the analysis of interactions with factor electrode, we selected 17 standard electrodes: frontal (FPz, F7, F3, Fz, F4, F8), central (C3, Cz, C4), temporal (T3, T5, T4, T6), parietal (P3, Pz, P4), and occipital (Oz). Moreover, the analysis of P300 effects was restricted to Pz and surrounding electrodes (P3, P4, C3, Cz, C4) because the P300 is known to have a

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centro-parietal maximum. This was also confirmed by a first inspection of the data. ERP effects were tested with repeated-measures ANOVAs with factors measurement epoch (10 and 11 levels), electrode (6 and 17 levels), and problem type (4 levels). The four levels of problem type were given by zero problems, small and large problems without zero, and the storage condition. Storage with small, large, and zero operands was combined because a first analysis had not revealed any differences between them, neither for the P300 nor for the slow negative wave. A superordinate ANOVA comprising all three factors revealed a significant triple interaction, which justified more detailed analyses. To avoid an increase in the likelihood of type I errors, all further tests were guided by the following hierarchical constraints: ‘Local’ ANOVAs for one measurement epoch at one electrode were computed only if the particular electrode had shown a significant Epoch  Problem Type interaction and the particular measurement epoch had shown a significant Electrode  Problem Type interaction in superordinate ANOVAs. Thus, the number of local tests was restricted in advance. Amplitude differences were considered as reliable only if at least two consecutive time windows had signaled significance ( p < 0.05) for the particular effect. Fstatistics were corrected according to the formulas of Huynh and Feldt [18]. The uncorrected degrees of freedom, the corrected p value, and the respective epsilon values are reported. To compare the topographies of slow negativity, ANOVAs were also run with z-standardized scores such that mean and variance across electrodes were equalized for experimental conditions and subjects. The standardization permits evaluation of pure topographic differences between conditions that are not caused by amplitude or variance differences (see Refs. [30,47]).

3. Results 3.1. Experiment I: behavioral data Response times showed a reliable problem-size effect. The implicit production took longer for larger (mean 1344 ms; S.D. F708 ms) than for smaller problems (841F277 ms; F(1,11)=11.55, p=0.0059). Moreover, multiplications with zero (690F185 ms) were solved faster than both small [ F( 1,11 )= 16. 01 , p =0 .00 21 ] a nd large pro bl e m s [F(1,11)=14.20, p=0.0032]. The average time for the storage condition amounted to 744(F199) ms with no differences between zero, small, or large digits. 3.2. Experiment II: ERPs Grand average ERPs in the production phase (starting with the onset of the second operand and the operation sign) are presented in Fig. 2. As can be seen, both ERP phenom-

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Fig. 2. Grand average ERPs for the four problem types in the production phase for a selected set of frontal (F7, Fz, F8), central and parietal (Cz, Pz), and temporal electrodes (T3, T4, T5, T6). The figure shows the amplitude modulation of the centro-parietal positivity (maximum at Pz) between 300 and 600 ms and of the subsequent slow negativity. The second operand and the operation sign were presented at time 0 s. ERPs were referenced to a 100-ms-long baseline (between 600 and 500 ms).

ena of interest, the P300 and the subsequent slow wave, were affected by the experimental conditions. 3.2.1. P300 All four conditions evoked a phasic positivity with a clear centro-parietal maximum around 450 ms (see Fig. 2). Latency and topography qualify this component as P300 [19]. The amplitude of the positive peak measured at Pz was systematically affected between 390 and 600 ms by factor problem type [minF(3,51) = 5.82, p = 0.0076, e = 0.6356; maxF(3,51) = 20.25, p < 0.0001, e = 0.7439]. As is obvious from Fig. 2, the amplitude is smallest for storage, intermediate for multiplication problems with zero, and largest for multiplication problems without zero. Post-hoc tests revealed significant amplitude differences between storage and multiplications with zero between 450 and 600 ms [minF(1,17) = 4.71, p = 0.0444; maxF(1,17) = 8.78, p = 0.0087]. The amplitude was also significantly smaller for multiplications with zero than for multiplications without zero between 390 and 570 ms [minF(1,17) = 8.09, p = 0.0112; maxF(1,17) = 46.90, p < 0.0001]. Amplitude differences are not restricted to electrode Pz. Surrounding electrodes (i.e., P3, P4, Cz, and C4; see Cz in Fig. 2) also show reliable differences between storage and multiplication with zero between 450 and 570 ms [minF(1, 17) = 4.53, p = 0.0483; maxF(1,17) = 8.70, p = 0.0090] and between multiplication with and without zero between 390 and 480 [minF(1,17) = 4.51, p = 0.0486; maxF(1,17) = 20.90, p = 0.0003]. Problem size did not affect the amplitude of the positivity.

3.2.2. Slow negative wave The positivity was followed by a sustained negativity with an initial phasic maximum at frontal and temporal electrodes and a subsequent gradual increase at central to occipital sites. Problem type affected the amplitude and topography of this negativity significantly. The negativity was most pronounced for large problems, least pronounced for storage, and intermediate for small and zero problems. This is substantiated by a significant main effect of problem type between 900 and 2700 ms [minF(3,51) = 3.11, p = 0.0343, e = 1; maxF(3,51) = 7.64, p = 0.0004, e = 0.9080]. Pairwise posthoc tests for the mentioned time epoch revealed that the average amplitudes for large problems differ significantly from that of small problems (between 1100 and 2700 ms), zero problems (between 1100 and 1500 ms), and storage [between 900 and 2700 ms; minF(1,17) = 4.91, p = 0.0407; maxF(1,17) = 12.77, p = 0.0023, for the mentioned contrasts]. The difference between storage and multiplications with zero proved to be reliable between 1100 and 2300 ms [minF(1,17) = 3.21, p = 0.0911; maxF(1,17) = 8.24, p = 0.0106]. Beside these main effects, factor problem type interacted with factor electrode for the entire measurement epoch [minF(48,816) = 2.87, p = 0.0006, e = 0.2851; maxF(48,816) = 4.89, p < 0.0001, e = 0.2730], indicating distinct scalp distributions of the negativity for the various problem types. To investigate topographical differences, we first computed the net effect of each multiplication type by subtracting the storage (control) condition. As will be shown, the

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three types of multiplication problems evoked the relative negativity with distinct maxima. To delineate which electrodes are relevant for the topographic and the amplitude differences, we also computed the difference scores between the multiplication conditions. Fig. 3 shows the topographies of these difference waves for two consecutive time windows of 400 ms length between 1100 and 1900 ms. This interval was chosen because differences between the problem types are maximal and stable throughout this interval (see below;

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note that the ANOVAs were computed for 200 ms time windows). Furthermore, ANOVAs showed reliable interactions of problem type and electrode, both for the contrasts of each multiplication type vs. storage and for the contrasts of zero vs. small problems and large vs. small problems. These effects provide the basis for subsequent ‘local’ contrasts and detailed analyses of the topographic differences. Relative to the storage condition, small multiplications without zero evoked the negativity with an occipital max-

Fig. 3. Topographies. Interpolated maps for two consecutive time windows of 400 ms length between 1100 and 1900 ms where the effect sizes were largest. Net effects (multiplication minus storage) were computed for the three types of multiplication problems. Dark shading indicates a relatively more negative potential in comparison to the storage condition. Topographies of the difference waves of zero minus small problems and large minus small problems (problem-size effect) are presented in the second and fourth rows. Notice the different scales, which were adjusted to compensate for differences in effect size.

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imum extending to left temporal and central sites (see Fig. 3, middle row). The difference was significant (with p < 0.05) at C3, P3, T3, T5, Oz, and T6 for all 200 ms time epochs between 1100 and 1900 ms (except T6 where the effect was significant only until 1500 ms). Multiplications with zero (upper row) evoked a clear left anterior negativity between 1100 and 1500 ms, differing significantly from storage at left frontal and central electrodes (F7, F3, C3). A second maximum appeared between Oz and T6 (all contrasts p < 0.05). In the following time window, the relative negativity is again significant at the mentioned electrodes, but also at left temporal (T3, T5) and parietal positions (P3). Amplitude differences between large multiplications without zero and storage (see lower row) existed at almost all electrodes (F7, F3, Fz, T3, C3, T4, T5, P3, P4, T6, Oz) with p < 0.05 for all measurement epochs between 1100 and 1900 ms, but clear maxima were present at frontal (Fz), occipital (Oz), and temporal (T5, T6) electrodes. To objectify the topographic differences between the three multiplication conditions, pairwise ANOVAs with zstandardized difference scores were run for the consecutive time windows of 200 ms between 1100 and 1900 ms. The topography for large multiplication problems proved to be significantly distinct from those of either small multiplications and multiplications with zero throughout the whole epoch [comparison large vs. small: minF(16,272) = 2.07, p = 0.0304, e = 0.6152; maxF(16,272) = 2.35, p = 0.0180, e = 0.5410; comparison large vs. zero: minF(16,272) = 2.58, p = 0.0220, e = 0.3820; maxF(16,272) = 3.11, p = 0.0069, e = 0.3902]. Small multiplications differed from zero problems only marginally between 1100 and 1500 ms but significantly between 1500 and 1900 ms [minF(16,272) = 1.78, p = 0.0948, e = 0.4464; maxF(16,272) = 2.80, p = 0.0100, e = 0.4370]. To delineate which electrodes are responsible for the topographic and amplitude differences, we also computed the difference scores between multiplication conditions. The differences between zero and small multiplications and between large and small problems are plotted as topographic maps in the second and fourth rows of Fig. 3. Zero problems evoked a relatively stronger negativity than small multiplications at left anterior to left central sites. Towards left temporal and occipital sites, a polarity shift occurred where small multiplications were more negative than zero problems especially in the time window between 1100 and 1500 ms (see also Fig. 2). However, only the differences at left frontal and central electrodes (F3, Fz, C3) proved to be reliable between 1500 and 1900 ms (with p < 0.05). Thus, the topographic differences are attributable to the left anterior negativity for zero problems that were not observed for small problems. The slow wave was more negative for large problems than for small problems at almost all electrodes ( p < 0.05), varying between 0.5 and 2.0 AV (with nearly identical topography in the two time windows), but most pronounced

at the frontal (Fz) and right temporal (T6) electrodes— electrodes that already showed the maximum of the net effect for large multiplications.

4. Discussion The objective of the present study was to determine by means of event-related brain potentials whether different solution strategies are used to produce the results of singledigit multiplication problems with small, large, or zero operands. The experimental conditions affected significantly both ERP phenomena of interest, the P300 component peaking around 450 ms after problem presentation and the subsequent slow negative wave. Amplitude of the P300 was smallest for storage, intermediate for zero multiplications, and largest for nonzero multiplications, irrespective of their size. The peak latency of this effect was much shorter than the average solution times estimated in the behavioral study (the fastest responses were found for zero multiplications with 690 ms). Therefore, it can be concluded that the P300 reflects a processing difference that precedes the actual production of a solution. This finding is compatible with the idea that different problems are immediately categorized according to their expected difficulty and that these differences are reflected by the P300 amplitude (see also Refs. [5,48]).3 In this sense, the amplitude modulation can be interpreted as a measure of cognitive preparation in terms of adjusting attentional and processing resources (see Ref. [48]). However, the observed amplitudes do not fully reflect the task difficulty as may be inferred from the response times or the later negativity. RTs differed for small and large nonzero multiplications (841 vs. 1344 ms) while the P300 amplitude was the same for these two conditions. Moreover, the observation of some kind of precategorization also fits well with the model of arithmetic performance in children of Siegler and Shipley [42] (see also Ref. [41]). This model contains a decision component that selects a solution strategy prior to problem solution. The P300 effects in the present study suggest that a similar decision stage is passed in adults as well. The subsequent slow negative wave whose timing coincides with the production phase proper was most pronounced for large multiplications, intermediate for small and zero problems, and smallest for the storage condition. This suggests that the different problems recruit neural resources to a different extent.

3

It is unlikely that event probability—in general a major factor for P300 amplitude variations—was the main determinant for the observed amplitude variations: zero multiplications had the smallest probability ( p = 0.111) while small and large multiplications without zero had each a larger probability of 0.194, or, if both were combined to one category, of 0.389. Nevertheless, the amplitude of P300 was smallest for the zero problems with the lowest probability.

K. Jost et al. / Cognitive Brain Research 20 (2004) 183–193

The most important finding, however, is that the negativity revealed clear topographic differences between the problem types. These indicate an activation of distinct generator ensembles [47] and, therefore, support the hypothesis that single-digit multiplication problems are solved by means of different solution strategies. 4.1. Problem size Large problems in contrast to small problems evoked a larger negative slow wave, suggesting greater mental load for large problems (cf. Refs. [6,38]). This result agrees with the problem-size effect observed in the behavioral study. There the RT difference between small and large problems amounted to 500 ms—an effect that is substantial and larger than in other studies (cf. Refs. [8,9]). More important is the topographic difference between small and large problems. Small multiplications (compared to storage) evoked the negativity with an occipital maximum extending to left temporal sites. In contrast, larger problems evoked a more substantial negativity over centralfrontal areas (Fz) and a larger amplitude over right temporal areas (T6). Such clearly distinct topographies are difficult to reconcile with pure differences in processing difficulty. In that case, one and the same generator configuration should have been affected and one and the same surface topography should have shown an amplitude variation only. The distinct topographies are more compatible with the hypothesis that large one-digit multiplications invoke other processes than pure fact retrieval. Of course, by means of ERPs, we cannot decide which strategies our participants used to solve such problems. ‘‘Derived facts’’ is the most reported strategy to solve problems with larger operands (see Refs. [28,43]). Thus, it is likely that participants in the present study sometimes decomposed large problems into a series of smaller, easily accessible facts. However, P300 amplitude was the same for small and large problems, indicating that small and large problems were initially classified as equally demanding and that the same amount of processing resources has been allocated to the two problem types. Thus, the first categorization of the problems seems not to fully anticipate the actual processing demands. From this, it can be also concluded that use of strategies other than fact retrieval might not be the default option for large problems, but rather that such strategies are activated only when fact retrieval fails (cf. Refs. [41,42]). Although the topography of scalp-recorded slow waves can give only a rough estimate of the location of underlying generators, the location of the most prominent topographic differences between small and large problems is nevertheless suggestive. The amplitude difference over occipital to right temporal electrodes indicates stronger activation of the underlying areas. This agrees with previous findings [20,21], which related ERP effects measured at right inferior parietal to temporal areas to number comparison and magnitude estimation processes [11,12]. Accordingly, it might

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be concluded that magnitude estimation processes are triggered when the results of larger single-digit multiplication problems are retrieved. The prominent frontal activation observed in this study agrees with similar findings of other studies [24,45] and indicates that strategies other than pure fact retrieval seem to be involved when larger operands are multiplied. Gruber et al. [15] reported that more complex problems involving calculation rules and decomposition strategies showed increased activity over frontal areas. They related these differences to linguistic and working memory functions. Likewise, the frontal negativity in the present study could indicate that larger problems impose more load on working memory and executive functions if they are decomposed into a chain of smaller and easily accessible facts [12]. 4.2. Facts vs. rules In the pilot study, zero problems were solved much faster (690 ms) than all nonzero problems (841 and 1344 ms for small and large problems), a finding that has also been observed in other studies [22,28,32]. Moreover, the amplitude of the P300 also suggests that the zero problems are categorized as potentially less demanding than nonzero multiplications. The negativity evoked by zero problems shows its most prominent amplitude over left fronto-central sites, and it deviates in this aspect from the topography of small and large nonzero problems. The fact that zero problems evoke a much larger amplitude over left anterior electrodes than small nonzero problems (Fig. 3, second row) strongly suggests that they are solved by a strategy that is clearly different from fact retrieval. Most likely, the correct results are generated by applying the zero rule. It is intriguing that this negativity, which most likely accompanies a rule-based strategy in mental calculation, has a similar topography as a conceptually related phenomenon observed with linguistic material, that is, the left anterior negativity (LAN). LANs with a maximum at left frontal to central sites have been observed in two situations: first in sentences with a rule violation (e.g., incorrect morphosyntax or incorrect usage of an auxiliary; e.g., see Refs. [16,37]) and, second, in grammatically and semantically correct sentences with less used, and therefore less easily parsed, grammatical constructions (see Refs. [23,39]). For example, Ro¨sler et al. [39] presented German sentences in which the order of sentence elements (subject, indirect object, and direct object) was systematically varied. Note that the German language—due to direct case marking—allows constructing grammatically correct sentences of different complexity by changing the word order. In this study, determiners, which function as case markers in German, evoked a left anterior negativity whenever they indicated that a noun phrase sequence would not continue in its canonical (expected) word order. Ro¨sler et al. interpreted the LAN as indicating that a deviation from the canonical

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word order triggers an additional rule-based parsing step (e.g., that the next noun phrase is not immediately assigned to the syntactic role of the actor). The fact that a left anterior negativity is evoked with similar topography by both linguistic and arithmetic material whenever a rule application is most likely invoked suggests some conceptual parallelism between linguistic and arithmetic processing (see Refs. [20,34] for a comparison of arithmetic and semantic N400 effects). Possibly similar mechanisms and overlapping neural structures are involved if linguistic and arithmetic representations are modified in working memory by rulebased operations. In other words, the common feature between the arithmetic and the linguistic LAN is seen in the fact that in both cases, default strategies—as extrapolation of overlearned grammatical structures or direct access to multiplication facts—fail and have to be replaced by a distinct rule-based processing step, which imposes extra load on executive and/or working memory functions. These could be reflected by the LAN.

[6] [7]

[8] [9] [10]

[11]

[12] [13] [14]

[15]

5. Conclusion [16]

By means of ERPs, we could provide evidence that strategies other than pure fact retrieval are used to solve single-digit multiplication problems. Our data sustain the distinction between rule-based and fact-retrieval problems and they support the notion that problem size determines the solution process, too [28]. These functional dissociations observed here with ERPs are congruent with neuropsychological findings showing differential impairments of acalculia patients, in particular, a dissociation between facts and rules, on one hand [44], and problems with small and large operands, on the other hand [31].

[17]

[18]

[19] [20]

[21]

Acknowledgements [22]

This research was supported by grants Ro529/10 and Ro529/17-1 from the German Research Foundation (DFG) assigned to Frank Ro¨sler.

[23]

[24]

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