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The conditions under which multiplication verification (3 × 6 = 12, true or false?) involves product retrieval and comparison or familiarity-based recognition judgements has not been clearly established. In two experiments examining verification of single-digit multiplication problems, we used Retrieval-Induced Forgetting (RIF), a signature of retrieval use, as an index of product retrieval in multiplication verification. In Experiment 1, 72 adults practiced multiplication either in a production format or in a verification format and then were tested on corresponding addition and control problems. The results showed RIF (i.e., slower answer production for addition problems whose multiplication counterparts had been practiced) in both the production-practice and the verification-practice groups, but RIF was stronger following true than false verification. Experiment 2 tested verification with related-false and unrelated-false products. Related-false equations produced longer RTs than unrelated false equations. Practice of true, related-false and unrelated-false multiplication equations all produced RIF of the addition counterparts but, overall, related-false multiplication equations produced relatively weak RIF. The results indicated that product retrieval mediates multiplication verification even when false answers are weak associative lures and suggest that a retrieve-and-compare process is the default strategy when false answers are at least plausible. We conclude that the presented answer in verification equations act as retrieval-priming stimuli with true equations priming correct answer retrieval and related-false answers interfering with correct answer retrieval.

The arithmetic verification task (8 × 4 = 24, true or false?) has been widely used and studied since the 1980s (e.g.,

Despite wide usage of the arithmetic verification task in diverse research, an important unresolved issue concerns what the normative or default strategy is for simple arithmetic verification. Do participants typically produce the correct answer during verification or rely on a familiarity-based recognition strategy or plausibility strategy? When false equations may be consistently identified as false on the basis of a salient characteristic or manipulation (e.g., parity or magnitude disagreement with the correct answer; e.g.,

There is a long history to this question. With respect to multiplication verification,

To explain these findings,

Furthermore, other phenomena raise doubts that product retrieval (e.g., as opposed to recognition of the equation without product retrieval) occurs during multiplication verification.

An alternative approach to investigating retrieval processes in product verification involves retrieval-induced forgetting effects observed in simple addition (e.g., 2 + 3 = ?) following retrieval practice of the multiplication counterparts (e.g., 2 × 3 = ?). Several experiments (e.g.,

The retrieval-dependence of multiplication-fact retrieval induced addition RIF provides a diagnostic test for the occurrence of answer retrieval during multiplication verification trials. If product verification entails answer retrieval, rather than only familiarity-based recognition, for example, then subsequent RIF of addition counterparts should be as robust following verification practice as following multiplication production practice. Furthermore, we may expect multiplication RIF to be more robust following true-verification practice than false-verification practice owing to retrieval priming. According to

In Experiment 1, two groups of participants received either six practice blocks of product verification trials (e.g., 4 × 6 = 28, true or false?) or product production trials (e.g., 4 × 6, state the product). False equations involved categorically-related answers: specifically, the correct answer if one operand is changed by +/-1 (e.g., 28 is an exemplar in the factor category 4). The multiplication practice phase was followed by an addition test phase with two addition-production blocks including addition counterparts of practiced multiplication problems and counterpart-unpracticed controls. The addition problems all had sums ≤ 10 or were so-called “tie” problems (2 + 2, 3 + 3, etc.), because larger, non-tie additions (e.g., 9 + 6) usually do not produce the RIF effect (

Seventy-two participants were recruited at the University of Saskatchewan and received course credit or $7.50. Participants were assigned alternately to the multiplication-production practice or multiplication-verification practice condition yielding two groups of 36. Recruitment materials stipulated English as the first language for elementary arithmetic because addition RIF is potentially sensitivity to linguistic or cultural factors. The effect is robust in English speakers (

Stimuli were presented using E-prime 2.0 (

Participants received six practice blocks of product-verification trials (e.g., 4 × 6 = 28, true or false?) or six practice blocks of product-production trials (e.g., 4 × 6 = ?, state the product) followed by a two-block addition production test phase (e.g., 4 + 6 = ?). The multiplication practice phase included primary problems and filler problems (explained further on), and the test phase included only primary problems. For both the verification and production practice groups in both the practice and test phases, problem order was independently randomized for each block.

The primary multiplication and addition stimuli were composed from two sets of numerically small non-tie (sum ≤ 10) and tie (i.e., repeated) operand pairs. For example, the number pair 2 and 5 yielded 2 × 5 and 2 + 5 (or the complements) and the pair 44 yielded 4 × 4 and 4 + 4. Direct memory retrieval is the predominant strategy reported by educated adults for the small non-tie and tie multiplication and addition problems (

For the true-false verification-practice group, assignment of Set 1 and 2 to the multiplication-practiced and multiplication-unpracticed condition, and assignment of the problem subsets to the true-verification and false-verification practice conditions, were fully counterbalanced across participants. Each operand pair appeared consistently as a true or false equation throughout verification practice. For each false verification trial a related-false answer was assigned pseudo-randomly by either increasing or decreasing one operand by one and multiplying it by the other operand (e.g., 4 × 8 = 24). False answers were restricted to not equal one of the problem’s operands or a multiple of five when 5 was one of the operands.

For the production-practice group, exactly the same counterbalancing procedure with the problem sets and subsets was applied. For this group, however, who viewed a problem to be answered verbally (e.g., 2 × 5 = ?), rather than an equation to be verified (e.g., 2 × 5 = 10 or 2 × 5 = 8), the counterbalanced assignment of problem subsets to nominal true and false conditions allowed us to treat true vs. false as a properly counterbalanced factor (with respect to problem subsets) in analyses that combined the production and verification groups.

Additionally, during the practice phase both groups also received 10 large (sum > 10) non-tie multiplication problems including 2 × 9, 3 × 8, 3 × 9, 4 × 7, 4 × 8, 4 × 9, 5 × 6, 5 × 7, 5 × 8 and 6 × 7. These served as filler problems to interfere with verification participants noticing that the same small/tie problems were consistently true or false. In each block, five of the foil problems were randomly selected to be true problems and the other five were false problems.

Following the practice phase, all participants received two test blocks of 20 addition production problems made from all operand pairs in both Sets 1 and Set 2. Addition counterparts of the multiplication-practiced pairs (e.g., 2 + 5 is the addition counterpart of 2 × 5) were the RIF targets and the addition counterparts of multiplication-unpracticed pairs served as the control additions.

Participants were tested individually in a half-hour session that included a warm-up task preceding the main experimental task. Instructions encouraged both speed and accuracy. In the warm-up, the participant named the eight letters “a” through “h” appearing individually in a random order at the center of the screen. On each trial, a fixation dot appeared at the center of the screen and then flashed twice over a 1-sec interval. On what would have been the third flash, the letter to be named appeared on the screen at fixation. For experimental trials, the fixation dot display was the same as in the warm up task and the problem appeared with the operator (× or +) at fixation. Verification participants were instructed to verbally respond "true" or "false" for verification trials and production participants were asked to state the correct product. In the addition production test phase, all participants were instructed to state the correct sum. Response timing began when the problem appeared and stopped when the participant's verbal response triggered the voice-activated relay. The spoken response caused the problem to immediately disappear from the screen, which allowed the experimenter to detect and record spoiled RTs where the microphone had failed to detect response onset. After the experimenter entered the given answer or pressed the enter key, the fixation dot for the next trial appeared. There was no feedback about speed or accuracy.

For ANOVA tests, Greenhouse-Geisser corrected statistics were reported when Mauchly’s Test indicated violation of the sphericity assumption. Along with null hypothesis significance tests we also reported a Bayes Factor (BF) for each test, calculated using MorePower 6.0 (_{0} for small effect sizes making it conservative with respect to Type I errors (_{0}) over alternative hypothesis (H_{1}). For example, BF equal to 10 for a given ANOVA test indicates that the data favor H_{0} over H_{1} by 10 to 1, whereas a value of 0.1 indicates a 10 to 1 ratio in favour of H_{1.}^{i}

A total of 451 practice RTs (5.2%) were marked for exclusion by the experimenter because the voice-key failed to detect response onset, or were discarded as outliers more than 2.5 SD from each Block (1 to 6) × Problem Type (true practiced, false practiced) mean for each participant. The overall error rate (excluding foil problems) during multiplication practice was 3.4%. Mean RT for correct responses received a Practice task (verification vs. production) × Problem type (true vs. false) × Block (1 to 6) ANOVA with practice task as a between-participants factor and problem type and block as repeated-measures factors. True vs. false problem type was a pseudo-factor for the production-practice group. The corresponding means and SEs appear in ^{ii}

Block | Verification Group |
Production Group |
||
---|---|---|---|---|

True | False | “True” | “False” | |

RT |
||||

1 | 1081 (50) | 1229 (60) | 1077 (54) | 1054 (51) |

2 | 931 (35) | 1148 (42) | 954 (53) | 936 (48) |

3 | 896 (35) | 1146 (49) | 963 (53) | 969 (45) |

4 | 884 (39) | 1090 (48) | 896 (45) | 944 (47) |

5 | 893 (40) | 1070 (44) | 915 (45) | 912 (45) |

6 | 882 (38) | 1067 (49) | 881 (39) | 906 (36) |

% Errors |
||||

1 | 2.2 (1.1) | 8.3 (2.2) | 2.2 (1.1) | 4.4 (2.1) |

2 | 1.1 (0.8) | 6.7 (1.6) | 2.8 (1.8) | 3.3 (1.3) |

3 | 1.7 (0.9) | 5.6 (2.1) | 1.1 (0.8) | 3.9 (1.8) |

4 | 1.7 (0.9) | 6.1 (1.9) | 1.7 (0.9) | 1.7 (0.9) |

5 | 2.2 (1.1) | 6.1 (2.4) | 0.6 (0.6) | 2.8 (1.2) |

6 | 1.7 (0.9) | 7.8 (1.8) | 2.8 (1.4) | 2.8 (1.4) |

Mean RT followed a decelerating speed-up function across practice blocks (means of 1110, 992, 994, 953, 947 and 943 ms) with greater RT gains from Block 1 to Block 2 than across later practice blocks [^{2}_{p} = .23, BF < .0001]. There were no significant interactions involving the block factor (all ^{2}_{p} < .01, BFs > 75000). Overall, false problems were answered slower than true problems [^{2}_{p} = .32, BF < .0001], but practice task interacted with problem type [^{2}_{p} = .29, BF < .0001]. For the verification-practice group, mean RT for false equations (1125 ms) was slower than for true equations (928 ms), whereas for the production-practice group, for which true-false was a pseudo-factor, the nominally true and false problems had practically identical mean RTs (948 ms and 954 ms, respectively).

^{2}_{p} = .16, BF = .01], and there was weak evidence for the same Practice task × Problem type interaction observed in RT [^{2}_{p} = .07, BF = .74], with the verification group making more errors on false equations (6.8%) than true equations (1.8%), whereas the production group had more similar error rates for the nominally false (3.1%) and true (1.9%) problems. There were no other significant omnibus effects (all ^{2}_{p} < .02, BFs > 247000).

A total of 77 test-phase RTs (2.7% of trials) were marked for exclusion by the experimenter or discarded as outliers more than 2.5 SD from each Block (1, 2) × Problem Type (true practiced, false practiced, or unpracticed) mean for each participant. The overall error rate during the addition test phase was 2.1% (61 errors). Mean RT for correct responses received a Practice task (verification vs. production) × Problem type (true practiced, false practiced or unpracticed) × Block (1 vs. 2) ANOVA with practice task as a between-participants factor and problem type and block as repeated-measures.^{iii}

Mean RT (see ^{2}_{p} = .35, BF < .0001].^{4} RT differed across problem types with means of 905 ms, 897 ms and 856 ms for the true practice, false practice and unpracticed conditions respectively [^{2}_{p} = .11 for the omnibus test, BF = .04], with weak evidence for a linear component of the three-way interaction [^{2}_{p} = .06, BF = .88].

Block | Verification-Practice Group |
Production-Practice Group |
||||
---|---|---|---|---|---|---|

Unpracticed | ΔTrue | ΔFalse | Unpracticed | ΔTrue | ΔFalse | |

RT |
||||||

1 | 841 (30) | 85 (21)*** | 50 (15)** | 930 (41) | 31 (25) | 44 (33) |

2 | 801 (30) | 21 (20) | 3 (18) | 853 (28) | 59 (22)* | 67 (21)** |

821 (29) | 53 (14)*** | 27 (12)* | 892 (33) | 45 (18)* | 56 (22)* | |

% Errors |
||||||

1 | 2.2 (0.8) | -1.7 (0.9) | 0.6 (1.5) | 1.4 (0.6) | 3.1 (1.6) | 1.9 (1.7) |

2 | 1.4 (0.6) | -0.8 (0.6) | 1.9 (1.7) | 1.4 (0.7) | 1.9 (1.4) | 1.4 (1.2) |

1.8 (0.5) | -1.3 (0.5) | 1.3 (1.3) | 1.4 (0.4) | 2.5 (1.2) | 1.7 (1.2) |

*

To pursue this, we computed for each participant the mean difference between the true practiced and unpracticed (i.e., baseline) condition, and between the false practiced and unpracticed (i.e., baseline) condition. These two difference scores represent potential RIF effects generated by true and false practice trials, respectively. Positive differences correspond to longer RT in the practiced condition (i.e., RIF). A Block × Problem Type ANOVA was conducted for each group (i.e., verification or production practice task). The true vs. false factor is a pseudo-variable for the production practice group. The corresponding means and SEs appear in

The analysis of the verification-group data provided weak evidence for a larger RIF effect in Block 1 (68 ms, ^{2}_{p} = .14, BF = .44], which has been a common finding in arithmetic RIF (^{2}_{p} = .12, BF = .63]. The corresponding analysis of the production-task group indicated no significant effects of block or problem type (all ^{2}_{p} = .25, BF = .03] and 40.0 ms for the verification group [^{2}_{p} = .27, BF = .02]. Thus, the current experiment provided strong evidence that both the multiplication production and multiplication verification tasks induced RIF of the addition counterparts expressed in verbal production RTs.

The finding of robust addition RIF from practicing multiplication counterparts in a verification task implies that the multiplication verification problems were solved using a retrieve-and-compare strategy (e.g.,

Experiment 1 provided evidence that verification of simple multiplication equations was solved by a retrieve-and-compare strategy, but is this finding owed to using closely related false answers? A plausible consequence of using related-false products is that discrimination of true and false equations based only on familiarity was not a viable strategy because both trial types would produce a strong familiarity response (e.g., 2 × 8 = 14 might seem initially plausible). This could discourage use of familiarity information to perform the verification task and promote predominant use of a retrieve-and-compare strategy. Experiment 2 pursued RIF of addition fact retrieval manipulating the relatedness of false verification answers during the multiplication practice phase. One group constituted a replication of the verification condition in Experiment 1 in which false multiplication answers were categorically related (i.e., a multiple of one of the factors and the correct answer if one operand was changed by ±1; e.g., 2 × 8 = 14). For the second group in Experiment 2, the false answers were unrelated in that they were not a multiple of either operand; e.g., 2 × 8 = 21). Participants find it much easier to reject such unrelated-false answers compared to related-false answers (

Seventy-two participant who did not participant in Experiment 1 were recruited in the same way as in Experiment 1. The sample included 51 women and 21 men with a mean age of 22.7 years (

A total of 199 practice RTs (2.3%) were excluded as in Experiment 1. The overall error rate during multiplication practice was 4.5%. Mean RT for correct responses received a False-answer type (related practice group vs. unrelated practice group) × Problem type (true vs. false) × Block (1 to 6) ANOVA with false-answer type as a between-participants factor and problem type and block as repeated-measures factors. The corresponding mean RTs and

Block | Unrelated False Group | Related False Group | ||
---|---|---|---|---|

True | False | True | False | |

RT |
||||

1 | 958 (44) | 1079 (42) | 940 (33) | 1191 (57) |

2 | 838 (39) | 987 (44) | 873 (36) | 1044 (44) |

3 | 797 (38) | 986 (57) | 822 (32) | 1042 (40) |

4 | 814 (34) | 945 (43) | 782 (31) | 1006 (49) |

5 | 820 (38) | 932 (41) | 794 (28) | 950 (37) |

6 | 784 (30) | 896 (40) | 783 (30) | 965 (40) |

% Errors |
||||

1 | 5.6 (1.7) | 7.8 (2.0) | 3.3 (1.3) | 7.8 (2.7) |

2 | 0.6 (0.6) | 5.6 (2.2) | 1.7 (0.9) | 7.2 (2.0) |

3 | 2.8 (1.4) | 5.0 (1.9) | 2.2 (1.1) | 8.9 (2.8) |

4 | 2.8 (1.2) | 3.9 (1.6) | 3.3 (1.5) | 7.8 (2.9) |

5 | 2.8 (1.4) | 3.3 (1.5) | 1.1 (0.8) | 5.6 (2.2) |

6 | 1.7 (0.9) | 5.6 (1.9) | 3.9 (1.8) | 7.2 (2.3) |

As in Experiment 1, mean RT followed a decelerating speed-up function across practice blocks with means of 1042, 935, 912, 887, 873, and 857 ms [^{2}_{p} = .35, BF < .0001]. There were no significant interactions involving the block factor (all ^{2}_{p} = .74, BF < .0001], but the between group false-answer type factor interacted with problem type [^{2}_{p} = .09, BF = .24]: The two groups had similar mean RTs for true equations (835 ms and 832 ms for the unrelated-false and related-false groups, respectively), but related-false equations were answered slower on average (1033 ms) than unrelated-false equations (971 ms). There were no other significant omnibus effects (all

^{2}_{p} = .09, BF = .06; all other

A total of 77 test RTs (2.7%) were discarded as outliers as in Experiment 1. The error rate was 3.3% of trials. Mean RT for correct trials was analyzed as in Experiment 1 and the means and SEs appear in ^{2}_{p} = .44, BF < .0001]. There was weak evidence that mean RT differed across problem types with means of 822 ms, 816 ms and 790 ms for the true-practiced, false-practiced and unpracticed conditions respectively [^{2}_{p} = .06, but with BF = 2.20, the Bayesian analysis slightly favored H_{0}]. This was qualified by the linear component of the three-way interaction [^{2}_{p} = .12, BF = .10].

Block | Unrelated False Group |
Related False Group |
||||
---|---|---|---|---|---|---|

Unpracticed | ΔTrue | ΔFalse | Unpracticed | ΔTrue | ΔFalse | |

RT |
||||||

1 | 838 (36) | 62 (23)* | 70 (25)** | 794 (27) | 25 (22) | 16 (17) |

2 | 795 (28) | -9 (16) | 9 (22) | 732 (25) | 51 (16)** | 13 (14) |

816 (31) | 26 (16) | 40 (19)* | 763 (25) | 38 (16)* | 14 (11) | |

% Errors |
||||||

1 | 4.7 (1.4) | 3.1 (2.5) | 0.8 (1.5) | 1.1 (0.5) | < 0.01 (0.8) | 4.4 (2.0) |

2 | 4.4 (1.2) | -0.6 (1.4) | -2.2 (1.3) | 1.4 (0.7) | 0.8 (1.1) | 0.3 (1.1) |

4.6 (1.2) | 1.3 (1.4) | -0.7 (0.9) | 1.3 (0.5) | 0.4 (0.7) | 2.4 (1.3) |

*

As in Experiment 1, to pursue the three-way interaction a Block × Problem Type ANOVA was conducted for each group (i.e., unrelated-false verification and related-false verification groups) on RT estimates of RIF for each cell (i.e., mean addition RT for true-multiplication practiced minus unpracticed, and mean RT for false-multiplication practiced minus unpracticed). The analysis of the unrelated-false group data indicated a larger RIF effect in Block 1 (66.1 ms) [^{2}_{p} = .26, BF =.02] than Block 2 (-.01 ms) [^{2}_{p} < .001, BF = 6.00], with the test for the main effect of block indicating ^{2}_{p} = .25, BF = .03. There was no main effect of problem type or Block × Problem Type interaction (both

The corresponding analysis of the related-false group indicated no significant effects, although the test for the main effect of problem type (true practice vs. false practiced) was in the same direction as observed in Experiment 1 with RIF of 38 ms for true-practiced problems [^{2}_{p} = .13, BF = .45] compared to 14 ms for related-false practiced problems [^{2}_{p} = .04, BF = 2.67] with ^{2}_{p} = .08, BF = 1.28, for the main effect of problem type. There was weak evidence of greater RIF following true than following related-false multiplication verification for this group in Block 2 (51 ms for true vs. 13 ms for false; ^{2}_{p} = .12, BF = .59).

During the multiplication-practice phase, unrelated-false equations were answered faster on average than related-false equations (971 ms vs. 1033 ms), but this difference between the groups was not observed for true equations (835 ms vs. 832 ms). Thus, as expected, unrelated-false equations were relatively faster to be identified as false compared to related-false equations. Nonetheless, in the test phase, having answered unrelated-false multiplication equations produced quite robust RIF of the addition counterparts in Block 1 (70 ms), suggesting that retrieval of correct products occurred for the unrelated-false multiplication problems in the practice phase. The related-false replication group produced significant RIF only for true equations but the evidence was weak that the addition RIF effect from practice of true multiplication equations was statistically greater than from practice of related-false equations and only observed in test Block 2. When the two related-false verification groups from Experiments 1 and 2 were combined, the observed addition RIF averaged across blocks was approximately twice as large following practice of true product-verification equations (46 ms, ^{2}_{p} = .10, BF = .20]. Thus, the combined experiments provided positive evidence that verification of true multiplication equations induced stronger RIF in addition counterparts than did practice of related-false equations.

Arithmetic verification is a widely-used experimental task, but what type of cognitive skill does it measure? Two experiments were designed to use RIF of addition fact retrieval as a diagnostic tool to assess whether multiplication product retrieval or a familiarity-based recognition strategy was used to solve true-false multiplication verification equations. Selection between these two strategies has been assumed to depend on familiarity or plausibility of the false answers used. We proposed that addition RIF from multiplication retrieval practice is indicative of answer retrieval rather than only a familiarity/plausibility check for multiplication verification because addition RIF has repeatedly been demonstrated to be retrieval dependent (e.g.,

Experiment 1 used related-false verification products (e.g., 6 × 4 = 28), which are products categorically related to one of the problem factors (i.e., operands). These would be expected to be relatively difficult to identify as false based on familiarity alone and therefore likely to promote a retrieve-and-compare strategy. Strong addition RIF was observed from verification practice of multiplication counterparts and it was not statistically different in effect size compared to the RIF produced by answer-production multiplication practice. Nonetheless, there was evidence in Experiment 1 that true-verification produced a larger addition RIF effect on addition counterparts than did false-verification multiplication practice. The related-false replication group in Experiment 2 did not produce as strong evidence as Experiment 1 that true multiplication verification yielded stronger addition RIF than related-false verification, but the effect was present for this group in Block 2 (51 ms for true vs. 13 ms for false), and evidence for the effect was positive when the two related-false verification groups from Experiments 1 and 2 were combined, with RIF averaged across blocks approximately twice as large for true as for related-false equations (46 ms vs. 21 ms).

Experiment 2 introduced an unrelated-false multiplication practice condition. Although unrelated-false verification was easier (i.e., faster) than related-false verification, which might have induced a familiarity-based recognition strategy that would not produce addition RIF, we nonetheless observed addition RIF from practice of unrelated-false multiplication equal in magnitude to true-verification equations. This is consistent with correct-product retrieval mediating both true and unrelated-false verification equations. Perhaps correct answer priming on true trials promotes retrieve-and-compare more generally, at least when false answers are somewhat plausible. The similar RIF effect size for true and unrelated-false multiplication equations suggests that retrieve-and-compare is the default strategy under these experimental conditions. Finding addition RIF from verification practice of true, related-false and unrelated-false trials does not imply that a retrieve-and-compare strategy was used exclusively for multiplication verification in the present studies. Indeed, Romero et al. (2006, Experiment 1) found that North American university students reported using retrieve-and-compare on 72% of multiplication verification equations involving ties (e.g., 6 x 6 = 36) and small non-ties with a sum ≤ 10). Consistent with this, our results imply an answer-retrieval-based strategy was used for a sufficiently large proportion of true, related-false and unrelated-false multiplication verification trials to induce a robust RIF effect on the addition counterpart problems.

Arithmetic RIF and error priming in arithmetic both reflect associative competition in retrieval but the two phenomena may arise through different mechanisms. As explained previously,

Thus, although error priming and RIF in number-fact retrieval are both indicators of retrieval competition, they apparently arise from distinct mechanisms. Error priming is sensitive to the conditions of problem encoding (verification equation vs. production problem; Arabic digit vs. number word format) whereas arithmetic RIF is less sensitive to these factors. Error priming has been shown to be stronger when the priming problem and error-primed problem have the same operand in the same left or right position. For example, solving 6 × 4 is more likely to prime its answer (24) as an error to 8 × 4 than to 4 × 8 (

The present experiments provided strong evidence that multiplication verification produced RIF in addition counterparts expressed in slower addition RTs. Multiplication-induced RIF of addition counterparts has been repeatedly shown not to occur when multiplication equations are studied and no answer is generated, as would normally be the case if multiplication verification was solved by a familiarity-based recognition strategy to discriminate true from false equations. Consequently, the present results are strong evidence that product retrieval occurred in multiplication verification, even for false equations with weakly associated presented answers (i.e., the unrelated false products used in Experiment 2). We propose that except under conditions in which participants are induced to use logical criteria (e.g., odd even agreement of presented and correct answer) or that answers are very implausible (remote in numerical magnitude from correct) that retrieval of the product is normally a routine stage of the verification process (

This research was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) to Jamie Campbell.

Bayes factor is a continuous scale but a conventional interpretation is that a BF greater than 10 or less than 0.1 provides relatively strong evidence for H_{0} or H_{1}, respectively, whereas BF between 3 and .33 provides little evidence one way or the other for H_{0} or H_{1} (e.g., _{0} or H_{1}, respectively. The MorePower 6.0 program used to calculate BF in the present article is freely available at

We used mean rather than median RT because the true and false conditions and unpracticed condition were not based on the same number of observations (5 and 10 trials respectively). As sample size decreases, the sample median RT increasingly overestimates the population median RT (

In

The authors have declared that no competing interests exist.

As researchers we did not have research ethics approval at the time the paper was accepted for publication to publish individual participant’s data. Please contact the authors for further information.

The authors have no support to report.