Memory & Cognition 2000, 28 (3), 358-365

The odd-even effect in multiplication: Parity rule or familiarity with even numbers?

AUETTELOCHYandXA~ERSERON Universite Catholique de Louvain, Louvain-La-Neuve, Belgium

MARGARETE DELAZER UniversiUitklinikjilr Neurologie, Innsbruck, Austria and BRIAN BUTTERWORTH University College ofLondon, London, England

This study questions the evidence that a parity rule is used during the verification of multiplication. Previous studies reported that products are rejected faster when they violate the expected parity, which was attributed to the use of a rule (Krueger, 1986; Lemaire & Fayol, 1995). This experiment tested an alternative explanation of this effect: the familiarity hypothesis. Fifty subjects participated in a verifi­ cation task with contrasting types of problems (even X even, odd X odd, mixed). Some aspects of our results constitute evidence against the use of the parity rule: False even answers were rejected slowly, even when the two operands were odd. Wesuggest that the odd-even effect in verification of multipli­ cation could not be due to the use of the parity rule, but rather to a familiarity with even numbers (three quarters of products are indeed even).

Besides the issues concerning the representation for­ feet has been observed for both addition (Krueger & mat in which parity information is coded and the way this Hallford, 1984) and multiplication (Krueger, 1986), and information is retrieved (Campbell & Clark, 1992; Clark it has been attributed to the use ofa rule. Inproduct ver­ & Campbell, 1991; Dehaene, Bossini, & Giraux, 1993; ification, subjects likewise use the rule "the true product McCloskey, Aliminosa, & Sokol, 1991; McCloskey, Cara­ must be even ifany multiplier is even; otherwise it must mazza, & Basili, 1985), the role of parity in arithmetic be odd" (Krueger, 1986). For Krueger, evenness is more has also been studied (Krueger, 1986; Krueger & Hall­ salient than oddness since detecting a single even multi­ ford, 1984; Lemaire & Fayol, 1995; Lemaire & Reder, plier indicates that the product must be even, whereas all 1999). Infact, several studies have proposed that subjects multipliers must be odd for the result to be odd. This au­ have access and sometimes use parity information with­ thor conceived the parity rule as a strategy that permits out explicit awareness when doing mental calculation. For subjects to bypass a normal two-step verification process instance, it seems that in some circumstances, when sub­ in which the subjects first retrieve the result in memory, jects are verifying simple arithmetic operations (e.g., 2 and then compare it with the proposed one. These two pro­ X 3 = 6 or 2 + 3 = 5), they do not retrieve the result but cesses, the parity rule checking one and the verification use a shorthand strategy for checking for the plausibility one (retrieval plus comparison), would be initiated in a ofthe proposed result. Among the cues used for check­ parallel-race model. ing that plausibility, its parity has proven to be relevant. This parallel-race interpretation was supported by the It has been found that adults reject a false result more results ofLemaire and Fayol (1995). These authors spec­ rapidly when its parity is incongruent with the parity of ulated that ifa parity rule checking process is going on in the correct answer (e.g., 2 X 3 = 7) than when it is con­ parallel with the retrieval ofthe correct answer, it should gruent with it (e.g., 2 X 3 = 8). This incongruence ef- be more efficient early in the verification process and for difficult problems-that is, as long as the correct answer is not completely retrieved from memory. That is exactly odd~ This work was supported by the Belgian National Scientific Re­ what they found: In a product verification task, the search Funds (FNRS) and by the Belgian Federal government (I.U.A.P. even effect was significant only for difficult problems on temporal control of dynamic tasks situations and the nature of and for the shortest delay between the presentation ofthe knowledge representation). We thank Marie-Pascale Noel and Agnesa operands and the onset ofthe proposed answer. Pillon for their helpful comments on this work. Correspondence should be addressed to A. Lochy, Universite Catholique de Louvain, Faculte de Lemaire and Siegler (1995) found that second-grade Psychologie, 10 place Cardinal Mercier, 1348 Louvain-La-Neuve, Bel­ children committed errors in multiplication that reflected gium (e-mail: [email protected]). the odd-even status ofthe target response. However, the

Copyright 2000 Psychonomic Society, Inc. 358 EVENNESS FAMILIARITY EFFECT 359

pattern oferrors first tended to correspond to the addition that noting evenness as a feature invoking the parity rule pattern ofthe rule, and changed over the school year' to­ is twice as likely when there are two even operands; how­ ward the multiplication pattern. The parity effect thus ever, no explanation was proposed for the reverse trend seems to develop with practice and with the constitution noted for two-odd problems. of an integrated network of multiplication facts, consis­ Our proposition for an evenness familiarity effect may tent with the adaptive strategy choice model (ASCM; Sieg­ explain the pattern ofresults obtained in Krueger's (1986; ler & Lemaire, 1997; Siegler & Shrager, 1994). Krueger & Hallford, 1984) studies. Since three quarters Furthermore, Lemaire and Reder (1999) found that of the products proposed as incongruent answers are the use of the parity rule depended on stimuli and con­ odd, they may be rejected faster because of their lower textual characteristics. They observed that when subjects likelihood of being the results of multiplication. More­ were verifying complex problems (49 X 8 = 394), the over, some characteristics ofthe stimuli used in these stud­ incongruence effect was greater when both operands were ies might have reinforced the evenness familiarity effect even and when the efficiency ofthat strategy was high­ by favoring even numbers as plausible. Indeed, the prob­ that is, when the proportion ofproblems violating the rule ability ofbeing correct was much higher for even (36.11% was high. in Experiment 1) than for odd (13.8%) results, which were The results of these studies revealed a parity incon­ then probably more easilyjudgedas false (a ratio of1:2.6).2 gruence effect in product verification. However, they do In summary, the source ofthe so-called incongruence not compel us to interpret the effect in terms ofa parity effect observed in verification tasks is ambiguous: It could rule. In this paper, we will propose an alternative inter­ be due either to the application of a parity rule, which pretation: We will argue that this effect could arise from predicts the parity of the product given the parity of the the distributional properties of the products. Within the operands, or to the natural asymmetry existing in the multiplication tables, even numbers are three times more odd-even distribution ofthe products in simple multipli­ likely to be the result ofmultiplication than odd numbers, cation tables. Moreover, this effect might also arise, in since three quarters ofsimple multiplication give an even part, from the imbalance in the stimulus material presented result. On this basis, even numbers might seem more to the subjects (a higher proportion ofeven answers were plausible than odd ones in product verification tasks. In correct), since this kind ofsituational variation has been other words, the incongruence effect might be, in fact, an shown to influence performance (Lemaire & Reder, evenness familiarity effect. 1999; Reder, 1988). An "evenness familiarity effect" may indeed be con­ The aim ofthis study was to question further the exis­ fused with a "parity incongruence effect" since for at tence ofa parity rule- and to contrast this hypothesis with least 75% of the products both effects would result in a the evenness familiarity hypothesis by focusing more shortening ofthe reaction time (RT) when the subject is closely on the different types ofproblems and each oftheir verifying a false incongruent answer. Indeed, for the latency patterns. We wanted to use a verification task like even X even (E X E), even X odd (E X 0), and odd X that ofthe aforementioned authors while controlling the even (0 X E) types ofproblems, an incorrect odd answer potential biases that might explain the facilitation effects will be more quickly rejected than an even one, either by found on incongruent splits. the application ofa parity rule checking strategy (because Simple arithmetic problems were presented for verifi­ of the violation of the odd-even congruence with the cation. False results were either congruent or incongru­ rule) or by a familiarity judgment because an odd answer ent with the parity of the correct answers. In the design looks less familiar than an even one. The crucial type of of this task, we carefully equated the proportion of odd product that may allow one to disentangle the two inter­ and even products in our list by adding filler items in pretations is the odd X odd product (0 X 0). For that type order to avoid any response bias due to the unequal dis­ of problem only do the two interpretations diverge for tribution ofodd and even products and their correctness. the expected performance: Ifthe parity congruence rule In order to obtain evidence allowing us to disentangle is used, presenting an incongruent even product would the "parity rule hypothesis" and the "evenness familiar­ also result in a shortening ofRT since it violates the ex­ ity hypothesis," we considered the results for three types pected parity. On the contrary, ifthe verification process ofproblems: E X E,O X 0, and mixed (E X 0 and 0 X is influenced by evenness familiarity, an even incorrect E) problems. The two hypotheses predict different ef­ product would be less easy to reject because even results fects on these types ofproblems: Ifsubjects use a rule of are the more likely products in multiplication. the type "the true product will be even ifany multiplier The studies reviewed above did not assess the incon­ is even; otherwise it must be odd (or, schematically: E X gruence effect for each ofthe four product types, and thus E = E; E X 0 = E; 0 X E = E; 0 X 0 = 0)," then an they do not allow us to choose between the two interpre­ incongruence effect (i.e., shorter RTs for incongruent tations. An exception is the Lemaire and Reder (1999) answers) should be observed for each type of problem. study, which compared types of problems and found a However, if evenness is more salient than oddness (as greater parity effect when both operands were even than suggested by Krueger, 1986, p. 142), this incongruence when one operand was even, and a reverse, but not sig­ effect ought to be the largest when both multipliers are nificant, effect on 0 X 0 problems. The authors proposed even, smaller when one is even and the other odd, and 360 LOCHY, SERON, DELAZER, AND BUTTERWORTH

smallest when both are odd. If, on the contrary, a famil­ These 12 problems were presented eight times each, for a total of iarity effect of the type "an even answer is more plausi­ 96 regular trials. In halfofthe eight presentations, the proposed an­ ble as a result of a multiplication and an odd answer is swer was the correct product of the two operands, and in the other more unfamiliar and less plausible," two specific pre­ half, problems were presented with a false answer. The distance be­ tween the correct and the false proposed answer (i.e., splits) was dictions can be made. First, the so-called incongruence ± I, 2, 3, or 4 (e.g., 9 X 7 = 62, 61, 66, 59). Incongruent answers effect should be observed only for E X E and mixed were at splits of± I or 3, and congruent ones were at splits of±2 or problems because, in these cases only, the false incon­ 4 from the correct answers (Table I). The direction of these splits gruent answer is odd (i.e., looks less familiar). In the was controlled to avoid potential biases: The sum of positive and case of0 X 0 problems, by contrast, this "incongruence negative splits was equal to zero in each split condition; six false an­ effect" should be reversed; that is, the facilitation effects swers were larger and six were smaller than the correct answer. The problems were of three different types consisting of four should be observed on congruent false answers since equations each (halfofthem were complementary: a X b = c, b X they are odd and look less familiar. For example, and a = c): even X even (E X E), odd X odd (0 X 0), and mixed (E contrary to the use ofa rule, 7 X 9 = 61 would be rejected X 0 and 0 X E). This factor should allow us to determine whether more quickly than 7 X 9 = 62. This would clearly mean the mathematical rule (N X even = even; odd X odd = odd) or that, even in the case of0 X 0 problems, subjects expect evenness familiarity [P(even product) > P(odd product)] was re­ even answers, which would then be rejected more slowly. sponsible for the incongruence effect. Second, subjects should take longer to reject even answers, False answers were chosen in order to avoid potential confusion. First, all the false answers were not plausible multiplication results, which should lead to higher error rates (i.e., to be erro­ because table-related products might increase response latencies neously accepted more often) than odd ones, whatever (Ashcraft, 1992; Zbrodoff & Logan, 1990). Second, we avoided the problem type. numbers that could represent operation errors." Filler items were introduced in order to make equal proportions METHOD of true even, false even, true odd, and false odd trials. These filler items were also simple multiplication problems, from the set of 100 pairs ofthe single digits 0 to 9 that were not used as experimental Subjects items. Since the 12 correct problems were presented four times, and Fifty undergraduate students at the Universite Catholique de since three quarters ofthem were even (32/48), we added 16 filler Louvain (16 males and 34 females) participated in this experiment odd correct problems. There were thus 64 correct equations. How­ to receive credit in an introductory course. Mean age was 19 years, 8 months (range: 18-23 years). ever, in order to respect the true/false proportion, we added 16 false items to the 48 experimental ones: Halfof them were odd and half Stimuli were even. Subjects thus had to answer to a set of 128 problems (32 The stimuli were multiplication problems presented in a standard ofeach true odd -even and false odd-even conditions), ofwhich 96 were experimental trials. These problems were presented twice, in form (a X b = c), where a and b were single-digit operands and c a two-digit number; the true products ranged in size from 27 to 72. two successive sessions (for a total of256 trials, ofwhich 192 were experimental ones). The basic set ofequations consisted of 12 multiplication problems (Table I). Four equations were selected for each type of problem (even X even, odd X odd, mixed). Procedure The equations were presented horizontally in the center ofa com­ All the selected problems were difficult, according to Campbell's puter screen (twinhead 400), via MEL/2.0. Each trial began with a (Campbell, 1987a, Appendix B) difficulty index, since they seem to 500-msec ready signal (a line of ***) in the center of the screen. be more sensitive to the odd-even effect (Lemaire & Fayol, 1995). The equations were then displayed horizontally in the center ofthe screen. They were presented in the standard form (a X b = c). Dig­ its and arithmetic symbols were separated by blank spaces equal to Table 1 Stimuli ofthe Verification Task the width of one character. Each character was 0.70-cm high X 0.50-cm wide. Each subject sat alone in a dark room, with the head FalseAnswer located 60-80 cm from the display screen. The equation remained True Answer Split on the screen until the subject responded, and trials were separated Type 0 2 3 4 by an interval of 2,500 msec. EXE Since the parity incongruence effect is maximum at short stimu­ 8X4 32 33 34 29 38 lus onset asynchronies (SOAs; Lemaire & Fayol, 1995), we chose 4X8 32 33 34 29 38 to present the proposed answer and the two operands simultaneously 8 X 6 48 47 46 51 44 (i.e., at O-msec SOA). Subjects were asked to respond "true" or 6X8 48 47 46 51 44 "false" as quickly and accurately as possible by pressing the ap­ OxO propriate key of the numeric keyboard, with the index finger of 9x7 63 62 61 66 59 each hand. For half ofthe subjects, the "true" response was on the 7 X 9 63 62 61 66 59 right side and the "false" response on the left side. The reverse in­ 3 X 9 27 26 29 22 23 struction was given to the other halfofthe subjects. 9X3 27 26 29 22 23 All subjects were individually presented with the same 192 ex­ Mixed perimental problems (4 pairs of operands X 3 types [E X E, 0 X 6 X 9 54 53 52 57 58 0, mixed] X 2 answers [true, false] x 4 levels ofsplits in the false 9 X 6 54 53 52 57 58 equations X 2 sessions), mixed with the 64 filler items. The prob­ 8x9 72 73 74 69 76 lems were presented in a pseudo-random order. Wetook care to avoid 9 x 8 72 73 74 69 76 repetition ofthe same problem or presentation ofthe correct answer Note-Type of problems:E x E = even x even;0 X 0 = odd x odd. for another equation (a minimum interval offour trials). The total EYENNESS FAMILIARITY EFFECT 361

of 256 trials (192 experimental and 64 filler items) was separated F2(2,9) = 8.03,MSe = 2,162.62,p<.01]. When subjects into eight blocks of32 trials each, which allowed subjects to rest if were presented with a true answer, the E X E problems they needed to; these trials were preceded by a practice set ofsix tri­ were solved faster than either the a X a or the mixed als not included in the analysis. problems, but RT for the 0 X 0 and the mixed problems The whole experiment lasted approximately 30 min. Subjects did not receive any feedback on their accuracy or speed. did not significantly differ from each other. With a false answer, however, the E X E and the a X a types did not RESULTS AND DISCUSSION significantly differ and were verified faster than the mixed problems (Student Newman-Keuls comparisons at al­ General Analysis: True Versus False pha = .05). There was only a trend for this interaction on In the first analysis, trials were classified as true or error rates [F1(2,98) = 11.80, MSe = 0.0042,p < .0001; false, and size ofsplit was disregarded. A 2 (sessions) X F2(2,9) = 0.0026,p = 0.3]: No difference in function of 3 (problem type: E X E, 0 X 0, mixed) X 2 (answer: true answer was found for mixed problems, while for the E X vs. false) analysis ofvariance (ANOYA) was performed E and a X a types ofproblems, there were more errors on on the subjects' mean correct RT and percentage oferrors true answers than on false proposed answers. Since false (FI)' with repeated measures on the last three factors and answers can be either congruent or incongruent, we need Student Newman-Keuls comparisons. The corresponding to know what the effects are on congruency in order to 2 X 3 X 2 ANaYAs were performed on the item means interpret these interactions. The mixed problems tended to be verified more slowly (F2 ), treating types as between-items variables and the other factors as within-items variables. than the a X 0 problems, which tended to be verified The results indicated that subjects answered faster dur­ more slowly than the E X E problems. However,this effect ing the second session than during the first one (Table 2), did not reach significance in the items analysis [F1(2,98) which could be explained by a simple effect oftraining­ = 20.62, MSe = 126,411.82,p < .0001; F2(2,9) = 1.88, the subjects may have become more familiar with the task MSe = 112,103.03, P = .2]. This insignificant trend and the restricted range of problems [FI(1,49) = 48.68, might be due to some of the mixed problems being more difficult than the others> (e.g., 8 X 9). MSe = 321,892.37,p < .0001; F2(1,9) = 62.81, MSe = 22,675.19, p < .0001]. This was also reflected in error Finally, there was a trend toward a triple interaction rates, which tended to be smaller in the second than in between problem type, session, and answer, but it did not the first session, although this trend was significant only reach significance in the subjects analysis [F](2,94) = 2.63,p = 0.1; F = 4.56, MS = I940.94, P < .04]. in the items analysis [F,(1,49) = 2.49, MSe = 0.0056, 2(2,9) e p = 0.2; F2(1,9) = 82.9, MSe = 0.0002,p < .0001]. True equations were answered 123 msec faster than Analysis Restricted to False Trials false equations [F, (1,49) = 21.50, MS = 93,238.08, p < In order to study the so-called incongruence effect, e and to contrast the parity rule and the evenness familiar­ .0001; F1(1,9) = 84.35, MSe = 2,162.62, p < .0001], a classic finding presumably reflecting the fact that the true ity hypotheses, analyses were run on trials with a false answers primed the retrieval ofthe correct product in the proposed solution only. In the first analysis, false trials arithmetic facts network (Campbell, 1987b, 1991). The were classified according to their congruence with the subjects more often rejected a correct answer than they ac­ parity ofthe correct answer. In the second one, they were cepted a false one [FI(1,49) = 30.22, MS = 0.0056, p < classified according to the parity ofthe proposed answer e per se (i.e., odd vs. even proposed answers) . .000 I; F2(l ,9) = 5.20, MSe = 0.0027, P < .05]. Such an effect on error rates was previously reported by Krueger Congruenceofthe proposed answer. Subjects' mean (1986, Experiment 2), who proposed to account for it by correct RTs and mean error rates to reject false trials were assuming that "internal noise is more likely to make a cor­ analyzed by a 2 (sessions) X 3 (problem type: E X E, a X rect proposed product seem false than to make an incor­ 0, mixed) X 2 (congruence: congruent vs. incongruent) rect proposed product seem true" (Krueger, 1986, p. 143). ANaYA, with repeated measures on the last three factors There was an interaction between answer and prob­ (FI); the corresponding 2 X 3 X 2 ANaYAs were run on lem type [F)(2,98) = 3.42, MS = 58,639.48, p < .03; items mean correct RT and error rates, treating type as a e between-items factor and sessions and congruence as (F ). Table 2 within-items factors 1 Mean Correct Latencies (in Milliseconds), In regard to the effects of session and problem type, Standard Deviations, and Percent of Errors (PE) for results revealed similar tendencies to those observed in True and False Answers by Problem Type in the Verification Task the previous analysis. RTs were shorter (-364 msec) True Answer False Answer M during the second than the first session [FI(1,49) = 40.20,

Type RT SD PE RT SD PE RT SD PE MSe = 451,094.72, p < .0001; F2(l,9) = 65.15, MSe = EXE U52 233 10.5 1,504 209 7.1 1,428 228 8.8 24,435.34, P < .0001], and a trend for an effect of type OxO 1,496 236 10.8 1,543 311 4.3 1,519 268 7.6 was observed in the subjects analysis on RT [F1(2,98) = Mixed 1,571 241 6.7 1,741 276 6.5 1,656 265 6.5 13.97, MSe = 204,089.82, p < .0001; F2(2,9) = 1.92, M 1,473 244 9.3 1,596 278 5.9 1,534 291 7.6 MSe = 132,405.26,P = .2] and on error rates [F](2,98) = 362 LOCHY, SERON, DELAZER, AND BUTTERWORTH

Table 3 "degraded" rule, at least on the E X E problems, may Mean Latencies (in Milliseconds ), Standard Deviations, and well have been reinforced during the course of the task. Percent of Errors (PE) for the Incongruent and Congruent False A trend for an interaction between problem type and Proposed Answers by Problem Type in the Verification Task session was found in the items analysis only on error rates Incongruent Answer Congruent Answer M [F)(2,48) = 1.96, MSe = 58,955.5, p = .2; F2(2,9) = Type RT SD PE RT SD PE RT SD PE 5.26, MSe = 0.00014, p < .04]: While the decrease in Ex E 1,429 249 4.00 1,585 173 10.25 1,507 222 7.10 error rate between the first and the second sessions was OX 0 1,594 306 4.38 1,494 323 4.25 1,543 308 4.30 Mixed 1,665 227 6.25 1,817 337 6.60 1,741 288 6.40 about 5% for E X E and 0 X 0 types, it was only 2.3% M 1,563 270 4.80 1,632 308 7.04 1,597 289 5.90 on the mixed problems. Parity of the proposed answer. In order to test the hypothesis that even answers should be rejected more slowly than odd ones, whatever the type of problem, 7.26, MSe = 0.0059, p < .001; F2(2,9) = 4.61, MSe = ANOVAs with sessions (2), problem type (3: E X E, 0 0.0007,p < .05; see Table 3). X 0, mixed), and parity of the false answer (2: odd vs. Incongruent answers were rejected faster (-69 msec) even) were performed on RTs and error rates for false than congruent ones [F](1,49) = 11.14,MSe = 64,703.49, problems correctly rejected, using both subjects (F1) and p<.001;F2(1,9) = 19.94,MSe = 2,882.l2,p<.001],and items means (F2 ). In the subject analysis, all factors were more errors were committed on false congruent (7.04%) within-subjects factors, while in the items analysis, than false incongruent answers [4.8%; F](1,49) = 10.78, problem type was a between-items factor. MS e = 0.0065,p<.001;F2(1,9) = 9.54,MSe = 0.00059, These analyses revealed that even answers were re­ p < .01). However,this incongruence effect significantly jected significantly more slowly (121 msec) than odd interacted with problem type on RT [Ft(2,98) = 16.02, ones [F](1,49) = 45.16, MSe = 55,028.28,p < .0001; MSe = 58,955.50, p < .0001; F2(2,9) = 29.95, MSe = F2(1,9) = 77.17; MSe = 2,882.12,p < .0001], and that 2,882.12, p < .0001]. Contrasts revealed that incongru­ this evenness effect did not significantly interact with ent answers were rejected significantly more quickly than problem type [type X parity, F] < 1; F2(2,9) = 1.33, congruent ones for both the E X E problems [- 156 msec; MSe = 2,882.12,p = .3]. The other significant effects re­ F t(1,98) = 1O.31,MSe = 58,955.50,p<.001;F2(1,9) = vealed by these analyses were similar to those observed 50.66, MSe = 2,882.12, p < .0001] and the mixed prob­ when analyses were done with the congruence factor lems [-152 msec; F 1(1,98) = 9.797, MSe = 58,955.50, rather than the parity one. The evenness familiarity pre­ p < .01; F2(1,9) = 48.09, MSe = 2,882.12, p < .0001). diction is thus confirmed by this analysis, despite a po­ However, the opposite pattern was observed for the 0 X tential size effect going against the predicted trend: Even o problems: In these cases, incongruent answers were re­ answers were ofa smaller magnitude (mean size = 42.9) jected significantly more slowly (+100 msec) than con­ than odd answers (mean size = 52.1).6 gruent ones [F](1,98) = 4.24, MSe = 58,955.50,p < .05; A similar pattern emerged from the error analyses: F2(1,9) = 20.81, MSe = 2,882.12,p < .001]. On error Even answers gave rise to more errors (7.08%) than odd rates, a significant difference between congruent and in­ answers [4.8%; F,(1,49) = 9.35, MSe = .0081,p > .004; congruent answers was found only for the E X E type of F2(1,9) = 10.29, MSe = .00059, p < .01], and this was problems: Congruent answers were more often erro­ especially apparent for the E X E problems [10.25% of neously accepted (10.25%) than incongruent ones (4%). errors for even false answers against 4% for odd false Such a large difference did not appear for the 0 X 0 or answers; type X parity, F I ( 2,98) = 11.43, MSe = .0052, the mixed problems [F](2,98) = 10.38, MSe = 0.0061, p<.0001;F2(1,9) = 8.14,MSe = .00059,p<.009). The p < .0001; F2(2,9) = 8.52, MSe = 0.0005,p < .008]. contrasts revealed that the effect was significant for E X There was a significant session X problem type X E problems [Ft(1,98) = 18.78, MSe = .0052,p < .0001; congruence three-way interactionon RT [F](2,94) = 7.4, F2(1,9) = 39.73, MSe = .00059,p < .0001], while it did MSe = 53,106.88, p < .001; F2(2,9) = 4.8, MSe = not reach significance for the mixed problems [F] (1,98) < 6,715.61,p < .04]. In Session 1, the incongruence effect 1; F2(1,9) < 1] or the 0 X 0 problems [F](1,98 < 1; was larger for the mixed (- 264 msec) than for the E X F2(1,9) < 1]. E (- 89 msec) problems, whereas the opposite pattern Discussion. Ifa parity rule was used to verify the equa­ appeared in Session 2 (i.e., a difference of -41 msec for tions, all the incongruent answers should have been re­ mixed, and of -222 msec for EX E problems). No dif­ jected faster than congruent answers. This is not what we ferences were noted between Sessions 1 and 2 for the obtained: The incongruent false answers were rejected o X 0 problems: Incongruent false answers were re­ faster for the E X E and the mixed types of problems, jected more slowly than congruent ones in both sessions while on the contrary, they were rejected more slowly for (98 and 102 msec). While this last finding is in dis­ the 0 X 0 problems. When RTs were averaged on all agreement with the use ofthe parity rule and favors the problem types, a general incongruence effect appeared evenness familiarity hypothesis, the variations ofthe ef­ just because there are more E X E and mixed problems fect from the first to the second session for the E X E (two thirds) than 0 X 0 ones (one third), as in earlier and the mixed problems might suggest that some kind of studies (Krueger, 1986; Lemaire & Fayol, 1995). Indeed, EVENNESS FAMILIARITY EFFECT 363

in E x E and mixed problems, congruent-incongruent ones for the other two types ofproblems. Such a result is answers are confounded with even-odd answers. not compatible with the parity rule since in that theoret­ These results strongly indicate that faster rejection of ical perspective, rejection times should have been faster false answers in a product verification task are not so for false incongruent answers (i.e., even ones) in the case much due to the incongruence ofthe false answer (even of0 X 0 problems. if it shortened RTs by 69 msec when averaged over all The replacement ofthe "congruence" variable by the types of problems), but to the fact that the answer is odd "parity" of the proposed answer confirmed that the fa­ (latencies are shortened by 121 msec in that case). We miliarity with even answers in multiplication influences thus conclude that it is not the use ofthe parity rule that the plausibility process. Even answers were rejected facilitates rejection offalse incongruent answers, but the 121 msec more slowly than odd ones, and this effect was use ofanother kind of arithmetic knowledge, the higher similar for all types ofproblem. However, subjects were proportion ofeven products in the multiplication tables. more prone to accept false even answers when proposed This evenness effect was larger when both the oper­ as answers to E X E type problems: This shows a greater ands and the proposed answer were even, on RTs as well effect of the plausibility of even answers in that case. as on error rate. This result was also found by Lemaire This might be due to the fact that it is the only type of and Reder (1999), who suggested that "noting evenness problem for which the parity status ofthe answer is the as a feature that will invoke the parity rule is twice as same in addition (E + E = E), which may reinforce the likely when there are two even operands." This might association with an even correct answer. Another inter­ also arise from a greater association of two even terms pretation ofthis effect is proposed by Lemaire and Reder and an even answer, maybe in part due to the addition (1999), who suggested that subjects are biased in their pattern of parity. The E X E type is the only one for strategy selection by both stimuli and contextual charac­ which the parity ofthe answer is the same for both oper­ teristics. The situation presented to subjects in the pres­ ations (E X E = E; E + E = E), while for the other types, ent experiment was a "neutral" one and did not favor any the parity ofthe answer is different from that in addition strategy to be used (the probability to see an even or an (E X 0 = E and E + 0 = 0; 0 X 0 = 0 and 0 + 0 = odd correct or false proposed answer was the same). On E). This effect grew from the first to the second session; the other hand, stimuli characteristics such as the num­ it suggests that exposure to the task may have reinforced ber ofeven operands influenced the effect: These authors this strong association, or that some subjects relied on suggested that problems with two even operands are twice some kind of rule for the E X E = E type of problem. as likely to invoke the evenness feature ofthe rule. How­ ever, even ifthe familiarity effect was stronger for E X E GENERAL DISCUSSION problems, it was the reverse for 0 X 0 problems, which is not explainable by the influence ofthe parity rule. This study's main purpose was to determine further Our interpretation does not rule out the particular pro­ the role ofparity during verification of simple multipli­ cesses involved in such a plausibility checking strategy cation and to test an alternative interpretation to the par­ (parallel-race competing processes; Krueger, 1986; ity rule checking strategy. Wepostulated that the so-called Lemaire & Fayol, 1995), but only the content of that incongruence effect is not the result ofa parity checking plausibility strategy. Instead ofthe parity rule (E X E = process based on a rule, linking the parity ofthe multi­ E, E X 0 = E, 0 X E = E, 0 X 0 = 0), it could be for­ plicands to the parity ofthe product, but that it could be mulated as an implicit known probability of the type more adequately interpreted as an evenness familiarity [P(correct product being even) > P(correct product being effect. Weproposed that an even number is considered as odd)]. Thus, the familiarity effect could account for the a more plausible correct product than an odd one because effects found previously, like the difficulty effect. As Le­ ofthe natural distribution ofodd (25%) and even (75%) maire and Fayol (1995) suggested, when the correct an­ products in the multiplication tables. Our results support swer is retrieved in memory, as for easy problems, the this interpretation since in a verification task, rejection subject's decision is based on that retrieved answer rather times were longer for false even answers whatever the than on the plausibility process. The latter would help de­ problem type. It is worth noting that when analysis was cision making as long as the correct answer is not com­ done on congruence, we replicated the finding ofthe pre­ pletely retrieved, as when difficult problems are in­ vious studies: Incongruent answers were rejected with a volved. Inour experiment, even ifwe did not manipulate 69-msec advantage relative to congruent ones. However, the difficulty level of the problems, we realized after­ this was probably an artifact since it was only true when ward that mixed problems were larger in size and thus the analysis was carried out on all types of problems were more difficult than the other types: Indeed, the par­ merged. Indeed, when detailing the different types of ity effects were larger for those difficult problems. An­ problems (E X E, 0 X 0, mixed), we found an inter­ other effect that may be accounted for by our interpreta­ action between congruence and types ofproblem that re­ tion is the age-related effect. Lemaire and Siegler (1995) vealed an interference of even answers. They were the found that the error patterns of second-grade children most slowly rejected, and they corresponded to incon­ emerged during the course ofthe school year: They first gruent answers for 0 X 0 problems and to congruent corresponded to the addition pattern and then grew to 364 LOCHY, SERON, DELAZER, AND BUTTERWORTH

correspond to the multiplication pattern. This may be un­ and the split effects are added, there should be a slight derstood as reflecting the constitution ofthe arithmetical difference favoring the rejection of congruent answers. fact network in multiplication, and the coding of parity Indeed, the split effect is a linear decrease in RT, while inside that network. the parity effect would not manifest when all problems A third effect that we replicated (Lemaire & Reder, are averaged (since there are 50% ofeach): Thus the ad­ 1999) is a change over the course ofthe experiment: We dition of these effects would give rise to a pattern of re­ indeed found interactions with the factor session involved. sults in which there is a slight decrease from split I to More specifically, the effect was reinforced on E X E split 4, and the average of the congruent versus incon­ problems on error rate. This means that subjects associ­ gruent answers would result in slower rejection times for ated more strongly an even answer with an E X E prob­ incongruent (mean distance = 2) than congruent (mean lem in the second part of the experiment, and thus that distance = 3) ones. Indeed, this is what Krueger and they learned the rule for this type ofproblem. However, Hallford (1984) obtained: The mean rejection time in­ the reverse incongruence effect on 0 X 0 was the same congruent answers was 1,375 msec, while that for con­ over both sessions, and opposite to the parity rule per­ gruent answers was slightly slower,with 1,362 msec. Thus, spective. Thus, the familiarity hypothesis has to integrate it seems that the results obtained by Krueger and Hall­ some modulations due to learning effects: Even answers ford could be explained by an alternative hypothesis ofa are more plausible in general, but it is more the case when general evenness familiarity. Further experiments are the two operands are even and when there is some practice. needed to test the clear predictions raised by these dif­ One could argue that the familiarity hypothesis could ferent hypotheses in relation to parity and addition, as account for the parity effects found in multiplication well as to assess the more general familiarity with even only, but not for those in addition, since there is no bias numbers in different tasks requiring number processing. toward even numbers in the distribution ofresults in ad­ The use ofplausibility strategies based on stimuli char­ dition. Indeed, E + E = E, 0 + 0 = E, E + 0 = 0, and acteristics has been widely reported in various domains o + E = 0: There are thus 50% even answers and 50% (Ashcraft & Battaglia, 1978; Lemaire & Reder, 1999; odd answers. Thus if the familiarity toward even num­ Reder, 1982, 1988; Siegler, 1987, 1988; Siegler & Le­ bers is strictly linked to the network configuration, no maire, 1997; Siegler & Shrager, 1994). In arithmetic, se­ difference is expected between odd and even answers. mantic information about numbers such as parity or But even numbers could also be more familiar in gen­ magnitude has been shown to become quickly available eral, not only because oftheir higher proportion in mul­ during the processing ofa stimulus and to influence sub­ tiplication, but because they would be processed more jects' decisions (Krueger, 1986; Lemaire & Fayol, 1995; often in arithmetical tasks (e.g., 9 out ofthe 15 possible Lemaire & Reder, 1999). What has been clearly estab­ combinations of odd and even numbers across the four lished is a sensitivity of subjects toward regularities. In operations have an even result [E + E = E; E / 0 = E ...]; regard to parity, the subject ofthe present paper, our view we are more used to counting by two with even than with differs from earlier ones in that we propose a familiarity odd numbers, etc.). Thus the evenness familiarity hy­ effect that corresponds to a very global regularity pattern pothesis is either strictly limited to each operation or a in multiplication: The proportion of even numbers is very general property ofarithmetic and numbers. higher, and they are thus are more plausible. Earlier au­ The effect ofa parity rule on addition has never been thors considered a more subtle sensitivity, linking the par­ replicated since the original experiments ofKrueger and ity ofthe operands and results (Krueger, 1986; Lemaire Hallford (1984), where it was found in one of two exper­ & Fayol, 1995; Lemaire & Reder, 1999). Our data may iments. Moreover, biases in the item selection could in­ be explained by the evenness familiarity perspective, ex­ fluence the interpretation of the results (e.g., sums with cept for the largest effect we observed on E X E problems. one of the addends equal to 0 or I-no need to retrieve This could perhaps be due to a reinforcement of the as­ a result; sums equal to 10, or ties-very easy; etc.). Fur­ sociation between even terms and even answers because thermore, a close examination ofthe pattern ofresults re­ ofthe addition pattern ofthe rule. Or, the sensitivity link­ veals that it is not so much "2 + 2 = 5" that looks strange, ing operands and answer is greater in this case, because but "2 + 2 = 7." Indeed, the only downward dip is from evenness is a salient feature, as suggested by Krueger split 2 to split 3. The authors interpreted their result by and Lemaire and Reder. What our data show is actually the conjoint action ofthe parity rule and ofthe split effect, a kind of"degraded rule": The plausibility ofeven num­ which go in the same direction only from split 2 to split 3, bers seems to be greater than odd ones in general, and and thus reinforce each other, whereas they are opposed at contrary to the mathematical rule in the case of0 X O. the other splits. But it is also greater for E X E than for the other types. In terms ofthe congruent-incongruent answers, ifthe The level of expertise of subjects is certainly a variable parity rule and the split have additive effects, as sug­ that should be taken into account and studied further. In­ gested by Krueger and Hallford (1984), there should be deed, expert subjects, who process multiplication prob­ no difference at all between incongruent and congruent lems very often, would probably be more sensitive to the answers (split 1 = split 2; split 3 = split 4; thus the mean exact association between operands and answers. This of split 1 + split 3 would equal the mean of split 2 + view corresponds to Siegler's ASCM (Lemaire & Siegler, split 4). If the (general) familiarity with even numbers 1995; Siegler & Shrager, 1994), where the parity infor- EVENNESS FAMILIARITY EFFECT 365

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