Evidence from task-irrelevant fact retrieval

Q233232—QJEP(A)08802 / Sep 20, 04 (Mon)/ [?? pages – 1 Tables – 2 Figures – 0 Footnotes – 3 Appendices] . Centre single caption • cf. [no comma] • Shortcut keys • UK Spelling. [RJ] THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2004, 57A (8), 1485–1511

Capacity and contextual constraints on product activation: Evidence from task-irrelevant fact retrieval Elena Rusconi University of Padua, Padua, Italy

Giovanni Galfano University of Trento, Rovereto, Italy

Viviana Speriani Ospedale S. Camillo, Venezia, Italy

Carlo Umiltà University of Padua, Padua, Italy Three experiments tested the limiting conditions of multiplication facts retrieval in a numbermatching task (LeFevre, Bisanz, & Mrkonjic, 1988). By presenting two digits as cue and by requiring participants to decide whether a subsequent numerical target had been present in the pair, we found interference when the target coincided with the product of the cue digits. This was evidence for obligatory activation of multiplication facts. Also, we showed that multiplication facts retrieval occurred even in the absence of any arithmetic context (i.e., a multiplication sign between the cue digits) and did not require processing resources (i.e., the process met the capacity criterion of automaticity; Jonides, 1981), whereas manipulation of the spatial relation between the two operands (cue digits) negatively affected retrieval. The present work appears to be unique in the context of previous similar studies on mental calculation, which invariably adopted an arithmetic task as the primary demand. We identify this difference as the reason for the failure of all previous studies in revealing independence of multiplication facts from attentional resources. Furthermore, we suggest the application of a contextual definition of automaticity to this kind of retrieval, given the fact that it might depend both on association strength and on contextual setting variables.

Correspondence should be addressed to Carlo Umiltà, Dipartimento di Psicologia Generale, Università di Padova, via Venezia, 8, 35131, Padova, Italy. Email: [email protected] Preparation of this manuscript was supported in part by grants from the European Commission (RTN grant HPRN-CT-2000-00076), MURST, and University of Padua to Carlo Umiltà. We would like to thank Mark Ashcraft, André Vandierendonck, and an anonymous reviewer for their helpful suggestions on earlier drafts of the manuscript. Ó 2004 The Experimental Psychology Society http://www.tandf.co.uk/journals/pp/02724987.html DOI:10.1080/02724980343000873

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Adults are thought to possess well-developed mental representations of simple (i.e., with two single-digit operands) number facts that allow them to retrieve, rather than to calculate, the result of simple arithmetic problems (see, e.g., Ashcraft, 1992; Campbell & Oliphant, 1992). For instance, Ashcraft (1987) proposed that number facts are represented in a network that is similar to the networks for word representations, where the nodes are numbers instead of words, and the links between them represent arithmetic relationships. In general, a network is a graph consisting of nodes that represent physical or conceptual objects and arcs that describe the relationship to other concepts or units. The information is stored in a network by interconnecting nodes with labelled arcs. Upon presentation of an arithmetic problem, activation in the network would automatically spread from activated operands to linked nodes, such as the sum. Individuals who are skilled at arithmetic are assumed to have automatic access to stored arithmetic facts via activation in their network of arithmetic knowledge. Processing strategies other than retrieval are prominent only in Siegler’s model (Shrager & Siegler, 1998; Siegler, 1988), which is also the only explicitly developmental model. (Note that nonretrieval strategies are more prominent in children than in adults.) The use of strategies, however, is not incompatible with network retrieval models. Some studies (e.g., LeFevre, Bisanz, & Mrkonjic, 1988; LeFevre & Kulak, 1994; Thibodeau, LeFevre, & Bisanz, 1996; Zbrodoff & Logan, 1986) addressed the issue of whether activation of arithmetic facts is indeed automatic, in the sense that activation of related nodes occurs without intention upon presentation of the appropriate stimuli. For example, in the Zbrodoff and Logan study, participants were asked to verify simple arithmetic equations of the form a + b = c and a × b = c. Half of the equations were false, and, more importantly, half of these false equations contained a result that would have been correct if the participant had been asked to perform a different operation. For instance, the authors found that more time was required to reject 4 × 3 = 7 than 4 × 3 = 6. In the former case, the multiplier, multiplicand, and false product were associated through knowledge about addition. Obligatory activation of the addition fact presumably interfered with retrieval of the relevant multiplication fact (also see Winkelman & Schmidt, 1974). Similarly, it took longer to reject 3 + 4 = 12 than 3 + 4 = 8. These findings, however, provide only limited support for obligatory activation of arithmetic facts because participants had been instructed to perform an arithmetic operation, although a different one. Therefore, the interference effect may reflect processing that is obligatory only if the participant is performing arithmetic with both kinds of operation. Indeed, Zbrodoff and Logan found a reduction or even disappearance of associative confusion when their participants were asked to verify just one kind of operation in a block. LeFevre et al. (1988) provided more convincing evidence that the activation of simple arithmetic facts is obligatory because they showed that it occurs even when mental arithmetic is completely irrelevant to the task at hand. In their study, pairs of numbers were presented (e.g., 5 + 1). After a variable delay, the pair of numbers disappeared, and a target number appeared. If the target was one of the previously presented numbers (e.g., 5), then the participant responded “yes” by pressing a response key. If the target was not one of the previously presented numbers (e.g., 3), then the participant responded “no” by pressing another key. It was found that a “no” item was rejected more slowly when the target was the sum of the presented pair (e.g., 5 + 1 and 6) than when the target was arithmetically unrelated (e.g., 5 + 1 and 3). That showed that activation of the sum occurs even when mental arithmetic is completely irrelevant to the task at hand and interferes with performance in a number-

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matching task. This interference effect was present at brief delays between cue and target (60 and 120 ms) but not at longer delays (180, 240, and 480 ms), indicating either that the automatic activation of arithmetic facts is short-lived or that its effects can be inhibited at longer intervals. Very importantly, obligatory activation did not depend on the presence of an addition sign between the initial pair of numbers, nor was it restricted to presentation in an arabic format. That is, interference occurred even when the addition sign was omitted (e.g., 5 1 and 6), or the target was a word (six). Thibodeau et al. (1996) tried to demonstrate the interference effect of obligatory activation of arithmetic facts in a number matching task by using multiplication facts. However, the results were not fully convincing because the condition without the multiplication sign was missing. Thus, the interference effect for multiplication is in need of further investigation. In a recent study, Galfano, Rusconi, and Umiltà (2003) addressed this question and demonstrated the presence of an obligatory activation of multiplication facts even in the absence of the multiplication sign in a number-matching task. In particular, they explored the possibility of obtaining an automatic activation of the nodes (i.e., the mental representations corresponding to numbers) that precede or follow, in the multiplication table, the product node (i.e., the mental representation of the product number). Their findings suggest that multiplication facts are stored in a highly related network in which activation spreads automatically from the product node to adjacent nodes (i.e., to the preceding and subsequent multiples of both the first and the second operand of the initial pair). The notion of obligatory activation is a central component of theories about automatic processes (e.g., Kahneman & Chajczyk, 1983; Raaijmakers & Shiffrin, 1981). However, obligatory activation is not the only criterion of automaticity. In particular, Jonides (1981) proposed three criteria of automaticity. The first criterion is capacity (or load insensitivity). If a process, like accessing arithmetic facts, is automatic, it should not require processing resources. Therefore, it should be less impaired than a controlled process by a concomitant, resourceconsuming task. The second criterion is resistance to suppression. An automatic process should be more difficult to suppress than a controlled process. The third criterion is expectancy. An automatic process should be triggered by presentation of the relevant information, regardless of the participant’s expectations about the task to be performed. Often, the last two criteria are subsumed under the single criterion of intentionality. Previous studies (e.g., LeFevre et al., 1988, as regards expectation; Zbrodoff & Logan, 1986, as regards resistance to suppression) tested two criteria only. To our knowledge, the criterion of capacity has yet to be tested in the context of task-irrelevant activation of arithmetic facts. It has been tested, instead, in the context of explicit recall (in production tasks) or recognition (in verification tasks) of arithmetic facts by adopting dual-task paradigms (e.g., De Rammelaere, Stuyven, & Vandierendonck, 1999, 2001; Lee & Kang, 2002; Lemaire, Abdi, & Fayol, 1996). The underlying rationale is very simple: If production and verification of an arithmetic fact were limited by the amount of available attentional resources, then requiring a participant to perform a secondary resource-demanding task during retrieval or verification should cause performance impairment with respect to baseline. It is likely that the greater the extent to which a process is limited by available resources, the more susceptible it should be to disruption by a resource-demanding task. Another issue that is presently debated in the literature, and that will be addressed in the present study, is the exact role played by verbal processes in elementary arithmetic. There can

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be little doubt that verbal processes are involved, for example, in understanding the operands of arithmetic problems or in producing a response. The debated issue concerns the format in which arithmetic facts are stored in long-term memory. Dehaene and his colleagues (e.g., Dehaene, 1992; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999) have proposed that arithmetic facts that have been learned by rote at school, especially the addition and multiplication tables, are stored in long-term memory in a verbal format. Also, they would be retrieved as automatic verbal associations. In contrast, most subtraction and division problems, which are not commonly learned by rote (e.g., Dehaene & Cohen, 1997), would be solved through mental manipulations of the quantities represented by the operands. However some simple subtraction and division problems would be memorized in a verbal format like multiplication problems. And, like multiplication and addition problems, their results would be retrieved as automatic verbal associations. Moreover, in a recent study, Lee and Kang (2002) have suggested that arithmetic knowledge might be related in a specific manner to workingmemory subsystems. In their study, simultaneous articulatory suppression significantly delayed the performance in solving multiplication but not subtraction problems. Holding an image in mind had the opposite effect. The purpose of Experiment 1 was twofold. First, we tried to replicate, by using the product and in the absence of a multiplication sign, the interference effect that Galfano et al. (2003) found with the nodes adjacent to it, therefore complementing the Thibodeau et al. (1996) study. Second, we tried to determine whether this effect, if present, satisfies the capacity criterion of automaticity and whether it is reduced, or even eliminated, when participants have to perform a resource-demanding secondary task. We addressed the question of the existence of an obligatory verbal access to arithmetic knowledge in Experiment 2, in which both a number-matching and a verification task have been employed, respectively, with multiplication and with multiplication and addition facts. Three secondary tasks have been adopted: Two of them are supposed to tap on attentional resources (backward subtraction and random spatial tapping) and one on the phonological loop (articulatory suppression). The presence of two primary tasks (a direct and an indirect arithmetical task), allowed us to distinguish between two kinds of associative effects, both attributed to retrieval of arithmetic stored facts, and to test the efficacy of our secondary-task manipulation. With Experiment 3 we attempted to qualify the concept of automatic retrieval by manipulating the stimulus arrangement, in order to investigate the effects of an unusual visuo-spatial presentation on the activation of arithmetic facts.

EXPERIMENT 1 Our first experiment was divided into two conditions. We asked participants to perform a number-matching task only in one condition, and a number-matching task with a concomitant random spatial tapping (Experiment 1A) or backward subtraction (Experiment 1B) in the other. The order of the two conditions was counterbalanced between participants. In the case of the number-matching task only, we expected to find a significant slowing of reaction times (interference) when the target corresponded to the product of the cue digits rather than being unrelated to them. Given the absence of any arithmetical hint, this would be a stronger


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evidence for automaticity of product activation when arithmetic is irrelevant or even interfering for the task at hand, as Thibodeau et al. (1996) interposed a multiplication sign between their cue digits. In the case of combined random tapping, or backward counting, and the number-matching task, if the task-irrelevant activation of multiplication facts satisfies the capacity criterion of automaticity, it should not be affected by any one of the concomitant tasks. Backward subtraction and random spatial tapping task, indeed, have classically been assumed to heavily tap on attentional resources (e.g., Logie, Gilhooly, & Wynn, 1994). In the number-matching task, product interference emerges from a task-irrelevant cognitive operation, therefore, under dual-task condition we might expect a reduction, or even a total disappearance of interference, if interference in a single-task condition depended on the availability of spared resources. Lavie (1995; Lavie & Cox, 1997) has shown that the locus of the selective attention filter in the spatial domain can be changed as a function of the perceptual/cognitive load of the task at hand. She found that interference effects in the flanker paradigm could be eliminated by increasing task difficulty and thus force participants to adopt an attentional set completely focused on the target-defining attributes. By applying this notion to a numbermatching paradigm, one may argue that the interference effect results from the task not imposing a heavy perceptual/cognitive load on the participants’ attentional state. Therefore, assuming a nearly total engagement of processing resources because of the execution of a concomitant task, the interference effect should be eliminated. If, instead, performance in the number-matching task is impaired by a concomitant backward subtraction task but not by a random spatial tapping, the disruption would be unambiguously attributed to a structural competition, as backward subtraction implies numbers exactly like product retrieval. A selective interference by random spatial tapping, on the contrary, would probably reveal an involvement of the spatial sketch-pad in multiplication facts retrieval.

Method Participants A total of 23 undergraduates from the University of Padua (most of them enrolled in humanistic faculties) participated in the experiment as volunteers. Their age ranged from 22 to 36 years, with a mean of 25.3 years. All had normal or corrected-to-normal vision and were naive to the purpose of the experiment.

Apparatus, stimuli, and procedure Experiments 1A and 1B. Primary task. All the participants performed the primary task in two sessions: one in the single-task condition and the other in the dual-task condition. The order of the conditions was counterbalanced between participants. A secondary random spatial tapping task was assigned to one group of participants (Experiment 1A), and a secondary backward subtraction task was assigned to the other group (Experiment 1B). The basic experimental paradigm was that of the number matching for multiplication facts. Each trial included an initial cue (e.g., 9 8) and a subsequent target (e.g., 75). Ties (e.g., 4 4) were excluded, because they appear to have an easier access to the memory store than do other problems (e.g., Graham & Campbell, 1992). There were six types of stimuli; three belonging to the nonmatching category and three belonging to the matching category. Nonmatching/product stimuli consisted of targets that were correct products of the cue (e.g., 3 8 and 24). Nonmatching/unrelated stimuli had the same cues as the product stimuli but the target was

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unrelated to either digit in the cue (e.g., 3 8 and 49). For both product and unrelated stimuli, the target did not match or include either number of the digit cue. Nonmatching/filler stimuli had a double-digit number in the cue and a single-digit number in the target, so that participants saw nonmatching trials that included double-digit cues and also single-digit targets that required a no response (e.g., 48 7 and 9). Matching/target-balancing stimuli had the same targets as those used in the nonmatching product trials, so that participants saw cases in which the double-digit number in the cue matched the target (e.g., 7 53 and 53). In matching/cue-balancing stimuli there were the same cues of nonmatching/product stimuli. This condition was included so that participants saw matching trials that consisted of two single-digit numbers in the cue (4 3 and 3). Matching/filler stimuli had a double-digit number in the cue (e.g., 97 3 and 3), so that participants saw trials in which the single digit in the cue matched the target (the opposite happening for target-balancing stimuli). The experimental list contained 10 stimuli for each type (see Appendix A), presented at three different stimulus onset asynchronies (SOAs). In total, there were 180 stimuli for each block. The experimental session was divided into two blocks (single-task and dual-task conditions), differing, apart from the requirement of performing the secondary task, only for the randomized order of stimuli. None of the number cues or targets was 0 or 1, in order not to elicit back-up strategies instead of direct access to a result (e.g., Baroody, 1983). Combinations of cues and targets that may evoke activation on the basis of some relation among the items other than multiplication (e.g., addition or counting: 5 7 and 12 or 3 4 and 5) were discarded. Participants were required to respond no if the one- or two-digit target did not match any of the cue numbers (e.g., 6 9 and 54 or 6 75 and 8) and to respond yes if the target matched either number in the cue (e.g., 6 9 and 9 or 2 89 and 89). An IBMcompatible 486 computer connected to a 15-inch colour VGA monitor controlled timing of events and generated stimuli. Display background was black, and stimuli appeared in white. Each trial began with a 100-ms 500-Hz tone as warning signal. At the same time, the fixation point (a hash sign) appeared for 400 ms. Then the cue appeared at the centre of the screen and was visible for 60 ms. It was then replaced by a 40-ms backward mask consisting of seven hash signs. Following the mask, a blank screen was presented for a variable time interval (20 ms, 170 ms, or 300 ms), therefore producing three different SOAs (120 ms, 270 ms, and 400 ms). Each SOA occurred with the same frequency. The target was then presented at the centre of the screen (see Figure 1). Participants were required to decide whether the

Figure 1. Experimental procedure for the number-matching task in Experiments 1 and 2.


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target matched one of the digits in the cue and to press the appropriate key (e.g., 1 for yes, and 3 for no) on the numerical key-pad on the computer keyboard as fast as possible. Reaction time and accuracy were recorded, and, at the end of each trial, participants were given a visual feedback concerning their performance. They sat at an approximate distance of 50 cm from the screen. Each single digit measured 50 mm in height and 30 mm in width, and the two digits in the cue were always separated by three spaces. Note that there was no multiplication sign between the two digit cues. Half of the participants were instructed to respond yes by pressing a key with their right index finger and no by pressing another key with their right middle finger; the other half received opposite instructions. Note that the presence of the interference effect was indexed by significantly slower reaction times for nonmatching/product trials than for nonmatching/unrelated trials. We hypothesized that participants, upon presentation of the cue (e.g., 9 3) activate a set of possible targets that includes the actual number presented (e.g., 9, 3), as well as associated numbers (e.g., 27). Therefore, they should be slower to respond no when the target was the product (i.e., 9 3 and 27) than when the target was unrelated to the pair (e.g., 9 3 and 28). For what concerns the effect of SOA, we might expect the same pattern of results as that found by LeFevre and her colleagues (LeFevre et al., 1988; LeFevre & Kulak, 1994)—that is, interference with the shorter SOA (120 ms), which declines and/or disappears when the SOA becomes longer (270 ms and 400 ms). Alternatively, based on previous data from our own laboratory (Galfano et al., 2003), we might expect a significant interference effect also at the longer SOAs. Experiment 1A. Secondary task: Random spatial tapping. In the concomitant task, 12 participants had to press with their left index finger six different keys in a random sequence, with a constant 1-s rhythm, paced by a metronome that continued throughout the entire dual-task block. The keys belonged to the numerical key-pad of a keyboard positioned on a table to the participant’s left and connected to an I-Mac processor. They were the 1, 2, 3, 4, 5, and 6 keys, isolated from the neighbour keys by a cardboard screen and covered with white labels. After practising for about 3 minutes, participants were instructed to maintain their gaze fixed to the centre of the PC screen, in order to begin performing their primary task. The putatively random sequence of tapped keys was recorded in a SimpleText file on the I-Mac. During the whole session, the experimenter was present for assuring that participants adhered to the instructions (constant rhythm, random generation, and gaze fixed to the centre of the screen). Experiment 1B. Secondary task: Backward subtraction. In the concomitant task, 11 participants had to count backward by three, starting from a three-digit number provided by the experimenter, and with a constant 1.8-s rhythm paced by a metronome, which continued throughout the entire dual-task block. The articulation of a three-digit number requires time: This is the reason why we chose a slower pace with respect to the random spatial tapping. Using two-digit numbers instead would allow a faster pace. However, as the primary task lasted for a few minutes, this would have required frequent interruptions of the task, as the participant would reach one-digit numbers in about one minute from backward subtraction. Experimental instructions explicitly required participants to pronounce aloud the result of every subtraction, so that their performance was under control. During each dual-task block, which lasted about 9 minutes, the experimenter provided a total of three starting numbers (one every 3 minutes, alternating odd and even numbers), in order to maintain constant the articulation time for each number word. After practising for about 3 minutes, participants were instructed to maintain their gaze fixed to the centre of the screen, in order to begin performing their primary task, while the experimenter pronounced aloud the first three-digit number. Performance on the subtraction task was manually recorded by the experimenter, who also reminded the participant to be accurate, if more than three consecutive results were wrong. A response was not considered wrong if it was three steps away from a previous wrong answer (e.g., in the response sequence “100, 97, 93, 90, 87 . . . ” there is only one error).

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Results Experiment 1A Performance in the number-matching task (single- and dual-task conditions). All analyses reported here and in the corresponding sections of Experiments 2 and 3 refer to nonmatching/product and nonmatching/unrelated trials—that is, to the crucial manipulation of our experimental design. For the first group of participants, mean correct reaction times (RTs) were submitted to an analysis of variance (ANOVA) with a between-participants factor, order of conditions (single task/dual task vs. dual task/single task), and three within-participants factors, condition (single task vs. dual task), SOA (120 ms vs. 270 ms vs. 400 ms) and target type (product vs. unrelated). It revealed significant main effects of condition, F(1, 10) = 20.03, p < .001, SOA, F(2, 20) = 18.87, p < .0001, and target type, F(1, 10) = 10.55, p = .009. Mean RT in the singletask condition (number matching task: M = 692 ms, SE = 38) was 181 ms faster than mean RT in the dual-task condition (number matching + random tapping: M = 873 ms, SE = 70). RT decreased as SOA increased (120-ms SOA: M = 862 ms, SE = 56; 270-ms SOA: M = 750 ms, SE = 52; 400-ms SOA: M = 737 ms, SE = 42). That presumably reflects the well-known temporal warning effect (see, e.g., Niemi & Näätänen, 1981) and was not considered further in the present and in the subsequent discussions. Responses to product targets required, on average, 814 ms (SE = 55) to be executed, whereas responding to unrelated targets required 752 ms (SE = 45; 62 ms of interference effect). The crucial Condition × Target Type interaction was far from significance (F < 1), which does not allow one to accept the hypothesis of a difference between the interference effect found in the single-task condition (67 ms) and that found in the dual-task condition (57 ms). The mean error rate was about 2%. A 2 (order of blocks) × 2 (condition) × 3 (SOA) × 2 (target type) ANOVA on the number of errors revealed only a significant main effect of condition, F(1, 10) = 9.35, p < .01, indicating a lower accuracy in the dual-task condition than in the single-task condition, and of target type, F(1, 10) = 5.00, p < .05, indicating lower accuracy on product trials than on unrelated trials. The Condition × Target Type interaction was not significant (F < 1).

Performance in the secondary task (random spatial tapping). We assumed that the two conditions differed for the amount of resources available to execute the primary task. However, the nonsignificant difference in the interference effect between the single-task condition and the dual-task condition might be due to the fact that participants did not comply with the instructions concerning the secondary task. Participants’ performance in randomization tasks is commonly assessed by measures of stereotypy (e.g., frequency of adjacent items in an ordinal sequence or preferential selection of particular pairs), and usage of the response alternatives (e.g., whether each item in the response set is equally likely to be selected). In the present dual-task conditions, the mean redundancy (R) score (see Towse & Neil, 1998) was 1.45% (range: 0.19–3.22%), the mean Coupon score was 10.13 (range: 8.63–12.50), the mean Guttman’s Null Score Quotient (NSQ) was 3.09% (range: 0.00–8.57%) and the mean Turning Point Index (TPI) was 89.29% (range: 63.80–104.13%).


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Experiment 1B Performance in the number-matching task (single- and dual-task conditions). A 2 (order of conditions) × 2 (condition) × 3 (SOA) × 2 (target type) ANOVA on RTs revealed significant main effects of condition, F(1, 9) = 8.47, p = .02, SOA, F(2, 18) = 33.24, p < .0001, and target type, F(1, 9) = 5.57, p = .04. The mean latency of response for the single-task condition (number-matching task: M = 660 ms, SE = 40) was about 206 ms faster than the mean latency for the dual-task condition (number matching + backward subtraction: M = 866 ms, SE = 73). RT decreased as SOA increased (120-ms SOA: M = 839 ms, SE = 59; 270-ms SOA: M = 762 ms, SE = 55; 400-ms SOA: M = 688 ms, SE = 44). Responses to product targets required on average 789 ms (SE = 57) to be executed, while responding to unrelated targets required 737 ms (SE = 47; 52 ms of interference effect). The crucial Condition × Target Type interaction was far from significance (F < 1), which does not allow one to accept the hypothesis of a difference between the mean interference effect found in the single-task condition (65 ms) and that found in the dual-task condition (40 ms). A further ANOVA with two between-participants factors, group (tapping vs. subtraction) and order of execution (single/dual vs. dual/single), and three within-participant factors, condition (single- vs. dual-task condition), SOA (120, 270, and 400 ms) and target type (product vs. unrelated), was executed, with RT data from both the first and the second group, to assess the hypothesis of a quantitative difference between the two dual-task manipulations. In particular, the 40-ms interference effect found with the concomitant backward-subtraction task might be significantly lower than the 57-ms effect found with random spatial tapping. The ANOVA revealed significant main effects of condition, F(1, 19) = 23.66, p < .0001, SOA, F(2, 38) = 46.08, p < .0001, and target type, F(1, 19) = 15.42, p = .0009, but no main effect of the two between-participants factors (Fs < 1). No two-way or three-way interactions reached significance, including the crucial Group × Target Type, and Group × Condition × Target Type interactions (Fs < 1). The mean error rate was 4%, and a 2 (order of conditions) × 2 (condition) × 3 (SOA) × 2 (target type) ANOVA on the number of errors revealed only a significant main effect of condition, F(1, 9) = 25.06, p < .001, showing a lower accuracy in the dual-task condition than in the single-task condition, and of target type, F(1, 9) = 12.18, p = .007, showing lower accuracy with product trials than unrelated trials. The Condition × Target Type interaction was not significant (F < 1). Performance on the secondary task (backward subtraction). The mean percentage of errors was 4%, identical to the mean percentage of errors for the primary task. During task performance, the experimenter noted on preprinted tables calculations and errors produced by the participant. The participant showing the best performance was accurate on 99% of counts, the participant showing the worst performance was accurate on 87% of counts. A qualitative classification of errors indicated a substantial homogeneity, with a predominance of subtractions by 1, 2, or 4 and hundred or decade substitutions.

Discussion The present experiment confirms and extends the Thibodeau et al. (1996) study regarding the obligatoriness of product retrieval. In a number-matching task we found significantly slower

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RTs to nonmatching/product than to unrelated targets, even in the absence of any arithmetical cue (e.g., a multiplication sign between the cue digits). We further assumed that, if the interference effect was still present in spite of the need to perform a resource-consuming concomitant task, that would fulfil the capacity criterion of automaticity. Thus, there would be evidence that the activation of multiplication facts is both obligatory and resource independent. That in turn would be evidence that the activation of multiplication facts is automatic, according to the criteria proposed by Jonides (1981). Our results provide clear evidence in favour of a truly automatic activation for multiplication facts. The interference effect is significant and present in both the single-task and the dual-task conditions, and its time course does not differ between the two conditions. The lack of interaction between SOA and target type is consistent with other data from our laboratory on neighbour nodes (Galfano et al., 2003), but contradicts other studies that employed the same number-matching task (e.g., LeFevre et al., 1988; Thibodeau et al., 1996). At the moment, we are not able to identify the reason for this difference. An analysis of the studies on multiplication facts points to the selection of stimuli as a potentially critical factor. The rapid decrease (if not the disappearance or even the inversion) of the interference effect in the Thibodeau et al. study might be due to the presence of a higher percentage of product targets (20% vs. 15% in our experiments) and the inclusion of 5-related facts, which have an easier access than other multiplication facts (Campbell & Graham, 1985). The relative accessibility and frequency of critical targets may have produced an earlier activation peak and earlier activation decay. Further experimental work is necessary to test the plausibility of this provisional explanation. The interference found in the dual-task conditions was smaller than that found in the single task, even if the Condition × Target Type interaction was far from significance. Assuming that our experiment lacked statistical power and that this difference is real, it might be attributed to the slower RTs in the dual-task conditions. It has been proposed (e.g., De Jong, Liang, & Lauber, 1994) that, when an automatically activated representation is not task relevant it decays rapidly, and its effect on RTs varies as a function of the relative response speed. That is, with longer RTs the probability to detect the effects of automatic activation decreases. In the present experiment, the temporal interval between cue onset and selection of the response was substantially longer in dual-task conditions, which may explain the apparent decrease in magnitude of the interference effect. However, this interpretation was not sustained by a significant interaction between target type and SOA. As at the shortest SOA RTs were significantly longer than at the longest SOA, we should have registered a larger interference effect at the longest SOA. At any rate, it is clear that the activation of the product occurs here and even under very unfavourable conditions. However, we have to be cautious in concluding that processing resources play no role in the retrieval of arithmetic knowledge. In fact, retrieval could be only partially automatic. Rather surprisingly, we did find a significant worsening of performance on the primary task with a concomitant backward subtraction but no significant reduction of the interference effect. Given the fact that both number matching and the activation of arithmetic knowledge require processing of number representations, we must consider at least two possibilities, besides the one that invokes lack of statistical power. The first is that the backward subtraction task actually did not require processing resources. We show in Experiment 2 that this was not


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the case. The second possible explanation appeals to cognitive and neuropsychological models of number cognition. According to the triple-code model (Dehaene, 1992; Dehaene & Cohen, 1995), not all types of arithmetic operation are handled by the same functional and neural system. Familiar multiplication problems are often learned by rote and would be solved through the retrieval of automatic verbal associations. In contrast, other problems, particularly subtraction problems, would not be systematically memorized and would require the manipulation of the quantities represented by the operands instead. Neuropsychological evidence points to the existence of two dissociable types of arithmetic routines verbal and semantic (e.g., Dehaene & Cohen, 1997). Our data fit well with the functional separability of different arithmetic operations, as the retrieval of multiplication facts was not impaired by the execution of a concomitant subtraction task. According to Dehaene (1992), a prerequisite for the retrieval of multiplication facts is that arabic operands be first encoded in a verbal format. One wonders, of course, how our participants could efficiently retrieve the product of two digits flashed on a screen, if an interfering task was loading their phonological loop. If Dehaene’s hypothesis were correct, results should have revealed effects of a structural verbal competition on the spontaneous activation of multiplication facts. That did not happen. However, we cannot define our backward subtraction task as a proper concomitant verbal task, given the slow rate imposed by the choice of three-digit numbers. The hypothesis of a verbal structural interference was addressed in Experiment 2.

EXPERIMENT 2 In Experiment 2, all participants took part in eight sessions, equally distributed between two days on consecutive weeks (e.g., if a participant performed the first four sessions on Thursday, he or she completed the remaining four on the next Thursday). Number matching was performed as the primary task in four out of the eight sessions, and verification of simple arithmetical problems was performed as the primary task in the other four sessions. Except for the two single-task conditions (number matching or verification only), a concomitant task was performed during each session: backward subtraction, random spatial tapping, or articulatory suppression. The number-matching task has already been described for Experiment 1, and its basic characteristics remained the same in the present experiment. However, stricter criteria were adopted in the selection of the stimulus set, as described in the following section. As in the previous experiment, we were interested in analysing the interference effect under different conditions—that is, during the execution of tasks that required attentional resources (spatial random tapping and backward subtraction) and during a task selectively interfering with verbal processing (articulatory suppression). As regards the effects of secondary tasks requiring attentional resources, we expected to replicate the findings we observed in the previous experiment. In particular, we expected to find a significant interference effect (related trials slower than unrelated trials) both in the single-task condition and in the conditions with concomitant spatial random tapping and backward subtraction. The condition with concomitant articulatory suppression was introduced in order to test the hypothesis of obligatory access to the verbal code for the retrieval of multiplication facts (e.g., Dehaene & Cohen, 1997). Were this hypothesis correct, a concomitant task interfering with the articulatory loop

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should disrupt the retrieval process, which in turn would result in a significant reduction, or disappearance of the interference effect. If, on the other hand, interference persisted even with concomitant articulatory suppression, then this would imply that multiplication facts can be accessed without necessarily employing a verbal code. In addition to number matching, we introduced as a primary task the arithmetic verification task that Zbrodoff and Logan (1986) adopted to test the intentionality criterion in simple mental arithmetic. In the mixed block condition of their Experiments 1 and 2, participants verified simple multiplication and addition problems; half of the equations were true and half were false. Half of the false equations, defined as associative lures, would have been true, had participants performed the alternative arithmetic operation (e.g., 4 + 7 = 28; 4 × 7 = 11), and the remaining false equations, called nonassociative lures, would not have been true under any conventional arithmetic operation (e.g., 4 + 7 = 26; 4 × 7 = 13). Nonassociative lures were responded to faster and more accurately than associative lures, showing the presence of an associative confusion effect (i.e., a cost in RT and error rate to associative vs. nonassociative lures). By manipulating the relevance of the irrelevant operation, the authors demonstrated that this effect is only partially automatic. Indeed, intending to perform just one operation (pure block condition) reduced its magnitude. We were not interested in the effect of our dualtask conditions on the verification task per se. Rather, we were interested in the associative confusion effect: Being only partially automatic (Zbrodoff & Logan, 1986), the associative confusion effect should disappear or significantly decrease when participants are performing a concomitant random spatial tapping or a concomitant backward subtraction. Moreover, if arithmetic facts were obligatorily retrieved by accessing their verbal code, a structural competition effect should emerge with concomitant articulatory suppression, thus eliminating the associative confusion deriving from product associative lures (i.e., during the verification of addition problems). We expected a different pattern as regards the influence of the concomitant resourcedemanding tasks on the associative confusion effect (in the verification task) and on the interference effect (in number-matching task). In particular, we predicted that the interference effect would remain significant at least in three out of the four conditions in which number matching was performed—that is, in the single-task condition and even in the presence of resource-consuming concomitant tasks (backward subtraction and spatial random tapping) as in Experiment 1. By contrast, we expected the associative confusion effect to be present in the single-task condition but not in the conditions with concomitant backward subtraction and random spatial tapping. According to the triple-code model (e.g., Dehaene & Cohen, 1995), neither the interference effect in the number-matching task nor the associative confusion effect in the verification task should be reliable in the conditions with concomitant articulatory suppression. To summarize, in designing Experiment 2, we introduced three modifications. First, we added another primary task. This was aimed at showing that the dual-task manipulations used in Experiment 1 selectively influence an associative effect (observed under conditions of task-relevant retrieval), leaving the interference effect (observed under conditions of taskirrelevant retrieval) relatively unchanged. In doing so, we empirically tested the arguments of task ineffectiveness and/or lack of power that might be invoked as alternative accounts for the results of Experiment 1. Second, we adopted, as an additional concomitant task, the classical articulatory suppression task at 1-s rate (e.g., Logie et al., 1994) to test the


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hypothesis of an obligatory translation into a verbal code for retrieving the product of two single-digit numbers (e.g., Dehaene & Cohen, 1997). Third, we opted for a within-participants design, which should enhance sensitivity to experimental manipulations and statistical power.

Method Participants A total of 17 undergraduates from the University of Padua (most of them enrolled in humanistic faculties) participated in the experiment as paid volunteers. Their age ranged from 22 to 29 years, with a mean of 24.5 years. All had normal or corrected-to-normal vision and were naive to the purpose of the experiment. None had participated in the previous experiment.

Apparatus, stimuli, and procedure Primary tasks. Number matching task. All the participants performed number matching as primary task in four sessions: one in the single-task condition and the others in each of the three dual-task conditions. The order of conditions was roughly counterbalanced between participants, and, for each participant, it was the same as the order of conditions in the verification task. The basic experimental paradigm was the same as that used in the previous experiment. A total of 11 stimuli for each of the six target conditions were created, and additional criteria were used to select the stimulus set, besides those used in Experiment 1. First, we selected only unrelated targets having the same parity as the product targets (i.e., an odd product was coupled with an odd unrelated target, and an even product was coupled with an even target), thus increasing their semantic similarity. Second, the average magnitude of unrelated and product targets was nearly the same—that is, five of the unrelated targets were obtained by adding two units to the product and six of them by subtracting two units. This would eliminate the possibility that product targets were rejected more slowly because they were closer to the cue digits (also see Galfano et al., 2003). Furthermore, in the present experiment, if the product contained one of the single digits presented in the cue, its corresponding two-digit unrelated target also presented a partial matching. That was allowed in order to have a fair number of stimuli that met the new selection criteria. Apparatus and procedure were the same as those in Experiment 1. The only exception was the presence of only one SOA (120 ms) instead of three, as in the previous experiment SOA did not interact with any of the other factors in the experimental design. The total number of trials was 132, resulting from two repetitions of the entire set of 66 stimuli reported in Appendix B. Verification task. The stimuli were simple equations of the form a + b = c and a × b = c and are reported in Appendix B. The same pairs of digits that had been presented as critical cues in the matching task were used as pairs of operands in the verification task. Half of the trials were true equations, and half were false equations. Addition and multiplication problems were equally distributed in the two conditions. For addition problems, 11 true equations were formed such that c was the sum of a and b (e.g., 4 + 7 = 11), 11 associative lures were formed such that c was the product of a and b (e.g., 4 + 7 = 28), and 11 nonassociative lures were formed such that c was neither the sum nor the product of a and b (e.g., 4 + 7 = 26). Similarly, for the multiplication problems, 11 true equations, 11 associative lures (e.g., 4 × 7 = 11), and 11 nonassociative lures (e.g., 4 × 7 = 13) were created. In both the addition and multiplication problems, the proposed results for the nonassociative lures were chosen to match the distance between correct results and associative lures. Average distance between lures and sums for addition problems was + 28.09 in the associative condition and + 27.90 in the nonassociative condition; average distance

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between lures and products for multiplication problems was –28.09 in the associative condition and – 28.27 in the nonassociative condition. The stimuli appeared centred on the screen located in front of the participants. Each single digit measured 50 mm in height and 30 mm in width. The operands and the operator were always separated by a single space. On each trial, participants first saw a fixation hash sign lasting for 500 ms, serving as a warning signal. This was accompanied by a 100-ms, 500-Hz tone. After fixation offset, the operation appeared and remained visible for 2,500 ms or until a response was made, whichever came first. The allowed response keys were the same as those for the number-matching task. A total of 8 participants responded “false” by pressing the “1” key and “true” by pressing the “3” key; the others did the opposite. In each of the four sessions with verification task as primary task, the participants received a block of 172 trials. Within each block, 22 true equations were repeated four times, and 44 lures were repeated two times each. The order of trials was randomized independently during each block. Participants were told that they had to verify simple arithmetic equations by pressing one of two response keys and were asked to be as quick and accurate as possible. Secondary tasks. Random spatial tapping. In order to render the random spatial tapping more demanding than in Experiment 1A, participants had to press with their left index finger nine different keys in a random sequence, with a constant 1-s rhythm, paced by a metronome that continued through the entire dual-task block. As in Experiment 1A, the keys belonged to the numerical key-pad of a different keyboard from that used for performing the primary task. They were the 1, 2, 3, 4, 5, 6, 7, 8, and 9 keys, isolated from the neighbour keys by a cardboard screen and covered with white labels. The procedure was identical to that described for Experiment 1A. Backward subtraction.

Apparatus, stimuli, and procedure were the same as those in Experiment 1.

Articulatory suppression. Our participants were asked to repeat aloud the word bah at a constant rhythm of 1 s, paced by a metronome, while performing the primary task. The experimenter was always present to monitor the correct execution of the task.

Results Performance in the primary tasks Number matching. Mean correct RTs were submitted to a two-way repeated measures ANOVA with condition (single task vs. concomitant tapping vs. concomitant subtraction vs. concomitant suppression) and target type (product vs. unrelated) as within-participant factors. It revealed significant main effects of condition, F(3, 48) = 12.92, p < .0001 (mean RTs of 619 ms for the single-task condition, 667 ms for the condition with concomitant tapping, 751 ms for the condition with concomitant backward subtraction, and 618 ms for the condition with articulatory suppression), and target type, F(1, 16) = 8.74, p = .009 (mean RTs of 678 ms for the product and 650 ms for unrelated targets). The Condition × Target Type interaction was not significant (F < 1). However, in order to test our specific hypotheses more directly, we performed separate planned comparisons for target type in each condition. In the single-task condition there was a significant interference effect of 18 ms: product, M = 628 ms, SE = 20; unrelated, M = 610 ms, SE = 19; t(16) = 1.83, p = .04, one-tailed. A significant 25-ms interference effect was present in the condition with concomitant random spatial tapping (product, M = 679 ms, SE = 21; unrelated, M = 654 ms, SE = 21), t(16) = 1.87, p = .04, one-tailed, and in


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the condition with concomitant articulatory suppression (product, M = 631 ms, SE = 30; unrelated, M = 606 ms, SE = 30), t(16) = 2.68, p < .01, one-tailed. In the condition with concomitant backward subtraction the interference effect showed only a trend towards significance (product, M = 772 ms, SE = 33; unrelated, M = 729 ms, SE = 31), t(16) = 1.35, p = .07, one-tailed) in spite of being larger (43 ms) than in the other conditions (see Table 1). The mean error rate was about 25%. A 4 (condition) × 2 (target type) ANOVA on the number of errors confirmed the analysis on latencies, revealing a significant main effect of condition, F(3, 48) = 27.48, p < .001 (2.76 errors in the single-task condition, 5.11 errors in the condition with concomitant tapping, 7.58 in the condition with concomitant backward subtraction, and 2.32 in the condition with articulatory suppression), and target type, F(1, 16) = 75.35, p < .001, due to participants making more errors to product targets (M = 5.76, SE = 0.48) than to unrelated targets (M = 3.13, SE = 0.42). The two-way interaction was also significant, F(3, 48) = 5.13, p = .004. In the single-task condition, although participants made more errors to product targets (M = 3.25, SE = 0.58) than to unrelated targets (M = 2.62, SE = 0.49), the interference effect was not significant (p = .135, one-tailed). It was significant, instead, in all the dual-task conditions—that is, random tapping, t(16) = 1.95, p = .03 (onetailed; product, M = 5.70, SE = 0.70; unrelated, M = 4.52, SE = 0.64), backward subtraction, t(16) = 2.51 p < .01, (one-tailed; product, M = 9.64, SE = 0.72; unrelated, M = 7.58, SE = 0.92), and articulatory suppression, t(16) = 3.11, p = .004 (one-tailed; product, M = 5.26, SE = 1.07; unrelated, M = 4.13, SE = 1.01). In summary, the analysis of errors confirmed and extended the evidence found with RTs. The interference effect was present and significant in every condition with either RTs or accuracy. In the single-task condition, interference was reliable with RTs but not with accuracy, in the random spatial tapping and articulatory suppression it appeared both with RTs and accuracy, and in backward subtraction the interference was significant with accuracy only. The reason why interference was significant with only one dependent variable in the single-task condition and in the condition with concomitant backward subtraction is not clear. It is not clear either why errors were much more frequent in Experiment 2 (mean error rate 25%) than in Experiment 1 (mean error rate 3%). Perhaps, for unknown reasons, participants in Experiment 2 were more concerned with speed TABLE 1 Mean reaction timesa, standard errors, and associative effects in Experiment 2 Single task —————— M SE Associative lure Nonassociative lure Associative confusion Product Unrelated Interference a

b

Verification 904 39 892 38 12* Number matching 628 20 610 19 18*

In ms. Trend. * Significant effects (p < .05).

Random spatial tapping —————— M SE

Backward subtraction —————— M SE

Articulatory suppression —————— M SE

992 994 –2

33 39

1145 1152 –7

25 33

911 891 20*

35 35

679 654 25*

21 21

33 31

631 606 25*

30 30

772 729 b 43

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than with accuracy. That is supported by the observation that overall RTs were faster in Experiment 2 (M = 664 ms) than in Experiment 1 (M = 773 ms). Verification. Mean correct RTs for trials with “false” response were submitted to an ANOVA with condition (single task vs. concomitant tapping vs. concomitant subtraction vs. concomitant suppression), operation type (addition vs. multiplication) and lure type (associative vs. nonassociative) as within-participant factors. It revealed only a significant main effect of condition, F(3, 48) = 29.83, p < .001, with mean RTs of 898 ms for the single-task condition, 993 ms for the condition with concomitant tapping, 1,147 ms for the condition with concomitant backward subtraction, and 901 ms for the condition with concomitant articulatory suppression. The interactions involving condition and lure type were not significant: Condition × Lure Type, F(3, 48) = 1.596, p = .20; Condition × Operation × Lure Type, F(3, 48) = 1.03, p = .39. However, we performed a series of planned comparisons to test the presence of the associative confusion effect, which was expected to be significant at least in the single-task condition. Indeed, we found a 12-ms significant associative confusion effect in the single-task condition (associative lure, M = 903 ms, SE = 39; nonassociative lure, M = 891 ms, SE = 38), t(16) = 1.79, p = .04, one-tailed, and a 20-ms significant effect with concomitant articulatory suppression (associative lure, M = 911 ms, SE = 35; nonassociative lure, M = 891 ms, SE = 35), t(16) = 2.45, p < .01, one-tailed. Crucially, no effect was evident with concomitant spatial random tapping and concomitant backward subtraction (ts < 1). The mean error rate was about 8%. A 4 (condition) × 2 (operation) × 2 (lure type) ANOVA on the number of errors confirmed and extended the analysis on latencies, revealing a significant main effect of condition, F(3, 48) = 11.304, p < .001 (0.89 errors in the single-task condition,1.73 in the condition with concomitant tapping, 3.69 in the condition with concomitant backward subtraction, and 1.51 in the condition with articulatory suppression), and of lure type, F(1, 16) = 12.279, p = .003, showing a clear associative confusion effect (associative lure, M = 2.21, SE = 0.22; nonassociative lure, M = 1.70, SE = 0.20). Also, a trend towards significance emerged for the main effect of operation, F(1, 16) = 3.577, p = .07, due to participants’ tendency to make more errors for verifying multiplications (M = 2.13, SE = 0.23) than additions (M = 1.77, SE = 0.20). In addition, the ANOVA showed a significant Condition × Lure Type interaction, F(3, 48) = 3.325, p = .02, and a trend towards significance for the Lure Type × Operation interaction, F(1, 16) = 4.02, p = .06. As we did for RTs, we performed a series of planned comparisons to test the associative confusion effect for each condition. The associative confusion effect was reliable in the single-task condition, t(16) = 3.20, p = .001 (associative lure, M = 2.35, SE = 0.49; nonassociative lure, M = 1.23, SE = 0.44) and in the condition with concomitant articulatory suppression, t(16) = 4.36, p < .001 (associative lure, M = 4.23, SE = 0.95; nonassociative lure, M = 1.82, SE = 0.57). By contrast, no significant differences between associative and nonassociative lures emerged in the concomitant random spatial tapping and concomitant backward subtraction conditions (ts < 1). Performance in the secondary tasks Random spatial tapping. The performance was, on average, worse than that in Experiment 1 but did not systematically differ between sessions. In the session with number matching, the mean R score was 2.49% (range 0.42–8.47%), the mean Coupon score was 25.36


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(range 14.46–77.50), the mean NSQ was 15.51% (range 2.50–35.00%), and the mean TPI was 85.86% (range 56.09–106.48%). In the session with number matching, the mean R score was 2.57% (range 0.32–4.82%), the mean Coupon score was 24.05 (range 14.97–37.83), the mean NSQ was 17.43% (range 5.00–35.00%), and the mean TPI was 85.47% (range 67.37– 102.18%). Number matching and verification were thus performed under very similar dualtask conditions. Backward subtraction. The mean percentage of errors was 8.61% (range 0–15.15%) during the matching task and 6.53% (range 1.89–15.22%) during the verification task. Articulatory suppression. The participants easily complied with the experimental requests and showed no difficulties in maintaining a 1-s pace.

Discussion The main purpose of Experiment 2 was to test the replicability, under stricter conditions, of the results obtained in Experiment 1 and the validity of our dual-task manipulations. We opted for a repeated measures design and asked our participants to take part in eight experimental sessions, equally distributed between two days on consecutive weeks. On each day participants completed all the conditions (single task, concomitant tapping, concomitant subtraction, and concomitant articulatory suppression) with only one primary task (number matching or verification). Apart from the power issue, it might be the case that in Experiment 1 our secondary tasks insufficiently taxed processing resources. Therefore, we included a verification task, in which a different kind of arithmetic associative effect emerges. Zbrodoff and Logan (1986) demonstrated that the associative confusion effect is only partially automatic as, without intention, the activation of irrelevant facts cannot run on to completion. By comparing how a concomitant task affects interference in number matching and associative confusion in verification, it is possible to address both the power and the ineffectiveness arguments. Table 1 reports a summary of the obtained results. As concerns RT data in number matching, a significant interference effect was found in the single task, concomitant random spatial tapping, and articulatory suppression, whereas with a concomitant backward subtraction the effect showed only a trend towards significance. Nonetheless, significant interference emerged for the backward subtraction condition in accuracy data. As concerns verification, significant associative confusion appeared in both latency and accuracy data for the single task and concomitant articulatory suppression conditions only. Not even a trend toward significance was found with concomitant random spatial tapping or backward subtraction. The effects of our dual-task manipulations on performance in the two primary tasks were identical—that is, there was a significant slowing both with a concomitant random spatial tapping and with a concomitant backward subtraction, whereas comparable RTs were found in the single task and with concomitant articulatory suppression. In the verification task, this pattern is perfectly in line with the existing literature (e.g., De Rammelaere et al.,1999, 2001; Lemaire et al., 1996), in which a delay in responses to the primary task is assumed to index secondary task interference on arithmetic processes.

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In spite of this similarity in the effects of concomitant tasks on performance in the primary tasks, the associative confusion effect but not the interference effect was disrupted by the two resource-demanding secondary tasks. This confirms the study of Zbrodoff and Logan (1986) on associative confusion, which has been defined as a relatively automatic effect, and validates our secondary task manipulations. The persistence of the interference effect both in the backward subtraction and in the random tapping conditions cannot be attributed to low efficacy of our secondary tasks in requiring attentional resources, as the same secondary tasks were effective in disrupting associative confusion. The lack of effect of concomitant articulatory suppression both with primary tasks and with associative effects is especially interesting if considered in light of the distinction put forward by Dehaene and Cohen (1997), between rote verbal and quantitative knowledge of arithmetic. The phonological loop is not involved in recognizing false equations or in the pure retrieval of arithmetic facts. Together with the finding that the interference in number matching persisted during the execution of a backward subtraction task, our data point to a functional separability between operations that is not entirely accounted for by a distinction in processing modalities (i.e., verbal vs. semantic; see also Van Harskamp & Cipolotti, 2001, for neuropsychological data).

EXPERIMENT 3 What precisely does it mean to tag a process as automatic? Logan (1998) noted that many theories of automatization and skill acquisition, especially memory-based theories, assume that automatic performance depends on retrieval of past solutions from memory. But what exactly is learned during automatization? There is abundant evidence that spatial attention and focusing are necessary constraints for the conscious processing of visual stimuli (e.g., Turatto, Angrilli, Mazza, Umiltà, & Driver, 2002, concerning the phenomenon of change blindness), and that participants encode location even under incidental learning conditions (e.g., McCormack, 1982; Naveh-Benjamin, 1987, 1988). There is also evidence for obligatory encoding of location information in visual search tasks (e.g., Chun, 2000; Treisman, Vieira, & Hayes, 1992). As a consequence, if during a training period the crucial stimuli appear in a particular location or in a fixed spatial relation, the automatic skill should be disrupted when spatial arrangement is different in a transfer situation. Logan adopted a category search task, in which participants were required to search through one- or two-word displays for members of a target category. Targets, nontargets, and distractors appeared in consistent locations throughout training, and their locations were varied at transfer. Sensitivity to changes in location was assessed with implicit and explicit memory tests, and, given some cautions, the results revealed that participants encoded word locations during the automatization phase. We reasoned that the operands of a simple fact are held constant also during the formal visual learning phase of arithmetic (at least in Italy and in other European countries, the operands of a simple arithmetic fact are usually presented horizontally). If those attributes were encoded during automatization and retrieved to support automatic performance, then transfer performance should be worse when the attributes change. Transfer costs may arise because the changed attributes are less effective as retrieval cues, so that memory traces that supported automatic performance are no longer retrieved. In order to evaluate this hypothesis, we introduced a crucial manipulation in the experimental setting, by not reproducing the


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spatial relations between numbers, which are assumed to be canonical during the learning phase of basic arithmetic facts. The cue digits were presented vertically, one above and the other below fixation. We predicted reduction, or even disappearance of the interference effect, attributable to the automatic activation of the product.

Method Participants A total of 35 undergraduates from the University of Padua (most of them enrolled in humanistic faculties) participated in the experiment as volunteers. None of them had participated in the previous experiments. Their age ranged from 20 to 40 years, with a mean of 25.5 years, and their visual acuity was normal or corrected to normal.

Apparatus, stimuli, and procedure Apparatus and procedure were the same as those used for the single-task condition in Experiments 1 and 2. The main differences concerned the selection of stimuli (see Appendix C) and their spatial arrangement. The cue digits were presented along the vertical meridian, one above and the other below the centre of the screen, at the same distance (0.5 cm). The target number, instead, could appear 3.5 cm on the left or the right of the centre. Each digit measured 50 mm in height and 30 mm in width, and any time the target appeared in one of the two hemifields, a hash sign (if the target was a single digit) or two hash signs (if the target was a two-digit number) appeared on the opposite side. As in Experiment 1, six types of stimuli were used: Three required a negative response in the number-matching task—that is, their target was not present in the cue (nonmatching condition); the remaining three required a positive response—that is, their target was present in the cue (matching condition). In the nonmatching condition, two types of stimuli were critical: the product stimuli (in which the target always corresponded to the product of the cue digits) and the unrelated stimuli (in which the cue digits were identical to the product stimuli but the target did not share any arithmetic relation with them). Generally, the unrelated targets shared with the product targets the parity status, and half of the time they were greater and the other half smaller than the product targets. These characteristics allowed us to exclude that the interference effect, if any, would arise simply as a consequence of semantic distance (measured in magnitude and parity) between product and unrelated stimuli (for details about the use of an odd–even rule and plausibility judgements in verification tasks, see, e.g., Krueger, 1986; Lemaire & Fayol, 1995). We excluded from our set of cues 0 and 1, because they seem to trigger procedural rules instead of direct retrieval (Baroody, 1983; McCloskey, Aliminosa, & Sokol, 1991). All the arithmetic facts from 2 × 2 to 9 × 9 were represented within the nonmatching product stimuli, half of them appearing with the greater operand above and the other half with the greater operand below the centre of the screen. The entire set of multiplication facts was classified into four categories: 2-table (this has been considered apart, because the unrelated target stimuli did not have the same parity as that of product targets, otherwise they would be identical to the adjacent nodes of the product); ties (reaction time experiments requiring production or verification of arithmetic facts have shown that ties produce particularly fast and accurate performance; Campbell & Graham, 1985; Miller, Perlmutter, & Keating, 1984); 5-table (this table seems to be accessed differently from the other tables, at least in verification tasks; Masse & Lemaire, 2001); regular tables (all the other multiplication facts). If a partial matching (multiplication fact whose product shares a digit with one of the operands) was present, the corresponding neutral targets also presented the same characteristic, whenever possible. The 204 stimuli were each presented twice with an SOA of 270 ms between the cue and the target, which appeared once to the left side and once to the right side of fixation, in random order. Therefore, participants saw 408 trials in total. Half of the participants responded no with

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Figure 2. Experimental procedure for the number-matching task in Experiment 3.

the left key and the other half with the right key. Response keys were the same as those in Experiment 1 and were controlled by the index and the middle finger of the participant’s right hand. The cue remained on the PC screen for 120 ms and was then replaced by a 30-ms backward mask made of five couples of adjacent hash signs along the vertical meridian. Following the mask, a blank screen was presented, resulting in an SOA of 270 ms. The target was then presented on the left or on the right of the centre with identical probability for 150 ms (see Figure 2). We emphasized the need for the participant to keep the gaze on the centre of screen. The fact that the target display appeared for a very short time (150 ms) made eye movements useless. Participants were required to decide whether the target matched one of the digits in the cue and to press the appropriate key as fast as possible. RT and accuracy were recorded, and, at the end of each trial, participants were given visual feedback concerning their performance. Participants sat at an approximate distance of 50 cm from the screen. The whole session lasted about 20 min.

Results In the present design we had three within-participant factors: side of the target (left vs. right), multiplication (regular vs. ties vs. 5-table vs. 2-table), and target type (product vs. unrelated). A 2 × 4 × 2 ANOVA was performed on latency and accuracy data. It only revealed a significant interaction between multiplication and target type, F(3, 102) = 3.01, p = .03. Planned comparisons (t tests) revealed a significant interference effect for ties, t(34) = 1.77, p = .04, one-tailed (product, M = 759 ms, SE = 34; unrelated, M = 735 ms, SE = 29) and for 2-table, t(34) = 1.88, p = .03, one-tailed (product, M = 775 ms, SE = 36; unrelated, M = 728 ms, SE = 26) but not for regular, t(34) = –0.15, p = .43 (product, M = 757 ms, SE = 28; unrelated, M = 759 ms, SE = 29) and 5-table, t(34) = –1.25, p = .11 (product, M = 747 ms, SE = 25; unrelated, M = 763 ms, SE = 29). The mean global error rate was about 2%, and a 2 × 4 × 2 ANOVA performed on the number of errors in product and unrelated trials revealed only a significant main effect of side, F(1, 34) = 11.34, p < .001, indicating a lower accuracy when the target appeared on the right side. Also, the main effect of multiplication was significant, F(3, 102) = 11.62, p = .002, revealing that there was a gradual decrement in accuracy. Participants responded with higher


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accuracy to 2-table trials than to 5-table trials, which in turn were responded to with higher accuracy than ties and regular table trials. Neither the main effect of target type nor the interaction between multiplication and target type were significant. Instead, the interaction between visual field and target type showed a trend, F(1, 34) = 2.97, p = .09. As revealed by planned comparisons, there was interference effect on the left side, t(34) = 2.29, p < .01, onetailed, but not on the right side, t(34) = -0.56, p = .28, one-tailed.

Discussion The main effect of interference from multiplication facts in the number-matching task, which had persisted with concurrent articulatory suppression, random generation, and backward subtraction, was instead eliminated by simply modifying the spatial arrangement of the stimuli. However, this overall result is qualified by the fact that the interference effect disappeared for regular tables and 5-table only (note that the majority of items in Experiment 1 and the entire set of stimuli in Experiment 2 belonged to the category defined here as regular). In contrast, interference persisted and was quite important in magnitude with 2-table and ties. Even if it is possible that the 37-ms effect with 2-table was slightly overestimated because the unrelated targets had not the same parity status as the corresponding product targets, this selective suppression appears to confirm what studies on explicit arithmetic tasks inferred from latency and accuracy data: the presence of a greater associative strength between tie problems and their correct answer, and the fact that 2-table probably constitutes a class of very easy problems (Campbell & Graham, 1985). In Experiments 1 and 2, the task-irrelevant retrieval of the product resulted essentially unaffected by several concurrent task manipulations, which led us to conclude in favour of automaticity. Yet, by presenting a retrieval cue with unusual spatial characteristics, the same effect was eliminated (at least for regular tables), which leads us to be more cautious and to apply a contextual definition of automaticity, as suggested, for example, by Besner and collaborators (e.g., Besner & Stolz, 1999) on the grounds of verbal Stroop studies. Retrieving an arithmetic fact has been said to be automatic in the sense that adults cannot refrain from unconsciously retrieving the result of an arithmetic operation between two digits, despite its detrimental effects on the explicit task performance. Nevertheless, the unconscious activation of a mental process does not mean that this is inevitable: Besner, Stolz, and Boutilier (1997) argue that the “mental set determines the kind and extent of processing that is engaged . . . serving to bias processing at different levels in multilevel models” of mental processes (p. 221). We claim that environmental cues are able to control mental set by triggering processes, whose obligatoriness and autonomy depend on the degree of matching between the external and the stored template information.

GENERAL DISCUSSION Our first hypothesis concerned the presence of interference with multiplication facts in the absence of any arithmetic context. It is worth a reminder that Thibodeau et al. (1996) tried to extend the notion of obligatory activation to multiplication facts without eliminating the operation sign between the numbers in the cue. Arithmetic context provided by the operation symbol may thus have been instrumental in triggering the retrieval of the problem solution,

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which interfered with the number-matching task. In order to render the notion of obligatory activation more compelling, in the present study there was no multiplication sign between the numbers in the cue. In the single-task condition of Experiments 1 and 2, we observed a significant slowing of RTs when the target was the product of the cue digits. The interference effect with products, therefore, was replicated even though any influence of arithmetic context was eliminated. On the basis of the current results and of Galfano et al.’s (2003) findings, we can definitively rule out the alternative interpretation advanced for the interference effect with addition facts (LeFevre et al., 1988; LeFevre & Kulak, 1994), which attributes interference to the activation of an automatic counting procedure (e.g., Baroody, 1994). Dehaene’s triple-code model (e.g., Dehaene, 1992) assumes that multiplication facts are stored in a verbal word format, implying that a fact such as 7 × 3 = 21 cannot be accessed unless the problem is translated into a verbal code. This model predicts that articulatory suppression, which makes verbal coding almost impossible, should interfere with the activation of a stored problem. In other words, if multiplication facts are stored in long-term memory with a verbal format, then loading the phonological loop while a pair of digits is presented should lower the likelihood of product activation. In turn, the lack of product activation should abolish the interference effect in a number-matching task (LeFevre et al., 1988), in the presence of any concurrent task that loads the verbal representational system. In contrast, the results of Experiment 2 point to no effect of articulatory suppression on the access to multiplication facts, corroborating what other studies have already shown (see, e.g., De Rammelaere et al., 2001; Seitz & Schumann-Hengsteler, 2000) with explicit verification and production tasks. Neuroimaging studies (e.g., Pesenti, Thioux, Seron, & De Volder, 2000) also did not detect, during an arithmetic fact retrieval task, activation of any of those cerebral areas that are involved in verbal processing. A case study reported by Butterworth, Cipolotti, and Warrington (1996) showed the existence of intact arithmetic skills in the presence of an impaired articulatory loop. In summary, our results are congruent with a growing body of evidence that is at odds with the assumption of an obligatory verbal storage of multiplication facts. In the light of all these studies, Dehaene’s triple-code model clearly needs to be modified. An attenuated version, which assigns an important but not necessary role to verbal coding in the process of retrieval, would, however, be compatible with our data. The main target of our Experiment 2 was to test the capacity criterion for task-irrelevant product retrieval. If the interference effect, as found in the number-matching task, was limited by available attentional resources, then requiring our participants to perform secondary resource-demanding tasks should have caused the disappearance of interference. As a primary task, in addition to number matching, our participants were required to perform an explicit arithmetic (verification) task. The same secondary task manipulations were employed both with explicit and implicit arithmetic facts retrieval. Associative confusion in verification but not interference in number matching resulted, impaired by the concomitant execution of a random spatial tapping or a backward subtraction task; this confirms that task-irrelevant retrieval only is automatic according to the capacity criterion. This account of automaticity, however, offers no explanation for the decrease of interference under conditions of unusual spatial arrangement (see Experiment 3). According to a classical definition (e.g., Jonides, 1981), we are in the presence of an automatic process, and yet it could be simply eliminated with a change in the relative positions of environmental cues that are responsible to trigger retrieval. In fact, by simply modifying the spatial arrangement of the


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two cue digits we found an interesting pattern of interference effects. That is, the spreading of activation between related nodes in the multiplication network showed its characteristic automaticity (obligatory and unintentional) only in the case of the strongest associates (i.e., ties and 2-table). On the whole, the present results appear to be in line also with a study by Logan and Schulkind (2000) that demonstrates that people can retrieve information about one stimulus while they are processing another one. By using a psychological refractory period (PRP) paradigm, which allows one to control the presentation interval between targets relevant for two concomitant tasks, they asked whether a parallel retrieval of semantic information occurs. Their experiments showed that RT to the first stimulus was faster when it came from the same category as the second than when it came from a different category, but only when the same task requirements were applied to the two stimuli. This strict dependence of automaticity on task set led Logan and Schulkind to define the retrieval effect as conditionally automatic. Our work seems to suggest that this automaticity is not only limited by explicit task requirements but it is also modulated by contextual implicit variables, such as the typicality of a spatial arrangement.

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APPENDIX A Nonmatching/product and nonmatching/unrelated stimuli used for the numbermatching task in Experiment 1 Cue 2 8 2 4 3 9 4 7 6 9

Product 7 2 9 3 8 3 9 6 9 8

14 16 18 12 24 27 36 42 54 72

Unrelated 39 51 35 26 49 28 86 58 52 75

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APPENDIX B Nonmatching/product and nonmatching/unrelated stimuli used for the numbermatching task in Experiment 2 Cue 3 8 3 7 4 9 7 9 8 6 6

7 3 9 4 9 6 8 7 9 4 8

Product

Unrelated

21 24 27 28 36 54 56 63 72 24 48

19 26 25 26 38 52 54 65 74 26 46

Set of stimuli used for the verification task in Experiment 2 Multiplication problems Problem

Product

Associative lure

Nonassociative lure

21 24 27 28 36 54 56 63 72 24 48

10 11 12 11 13 15 15 16 17 10 14

8 7 16 13 15 13 13 20 13 14 10

Associative lure

Nonassociative lure

21 24 27 28 36 54 56 63 72 24 48

19 26 25 26 38 52 54 65 74 26 46

3×7 8×3 3×9 7×4 4×9 9×6 7×8 9×7 8×9 6×4 6×8 Addition Problems Problem 3+7 8+3 3+9 7+4 4+9 9+6 7+8 9+7 8+9 6+4 6+8

Sum 10 11 12 11 13 15 15 16 17 10 14


APPENDIX C Nonmatching/product and nonmatching/unrelated stimuli used for the number-matching task in Experiment 3 Cue 2 4 2 8 2 3 3 7 3 9 4 4 8 4 6 8 7 9 7 9 8 2 3 4 5 6 7 8 9 5 6 5 5 6

3 2 7 2 9 4 6 3 8 3 5 7 4 9 5 5 6 6 8 7 9 2 3 4 5 6 7 8 9 3 4 7 9 8

Product

Unrelated

Category

6 8 14 16 18 12 18 21 24 27 20 28 32 36 30 40 42 54 56 63 72 4 9 16 25 36 49 64 81 15 24 35 45 48

7 9 13 15 17 10 20 19 26 25 18 26 30 38 28 42 40 52 54 65 74 5 7 18 23 38 51 62 83 13 26 37 49 46

2-table 2-table 2-table 2-table 2-table Regular Regular Regular Regular Regular 5-table Regular Regular Regular 5-table 5-table Regular Regular Regular Regular Regular Tie Tie Tie Tie Tie Tie Tie Tie 5-table Regular 5-table 5-table Regular

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