^{*}

^{a}

Three experiments were conducted in which adults practiced complex multiplication problems (e.g., 4 x 17). In Experiments 1 and 2, after practice participants completed a number-matching task in which two digits (cues) were followed by a single digit (probe) and had to determine whether the probe matched either of the cues. In simple arithmetic (e.g., 4 x 3), when the probe is the product of the cues (12), participants are slower/more error prone when determining whether there is a match. Results of Experiment 1 extended this effect to complex multiplication. In Experiment 2, participants practiced problems with the larger operand first (e.g., 17 x 4) or with the smaller operand first (e.g., 4 x 17). The number-matching interference effect from Experiment 1 was replicated, and was equal across the two groups whether cues were presented in their practiced or non-practiced order. Experiment 3 was conducted to determine if two additional simple multiplication effects, consistency and relatedness, could be documented for complex multiplication. After practice, in a verification task (4 x 13 = 56?) it was found that when presented answers shared a digit with the decade digit of the correct answer (consistency) or were a correct answer to another practiced problem (relatedness), participants rejected answers more slowly and/or less accurately. Together, findings from the three experiments support arithmetic models that posit that commuted pairs are not represented in long-term memory independently and that posit representations of two-digit multiplication answers are decomposed into decades and units during arithmetic processing.

After extensive practice with single-digit arithmetic problems (e.g., 8 x 6), adults mostly change from using deliberate and resource-demanding problem-solving strategies to retrieving answers for those problems directly from long-term memory (e.g.,

Criticisms of the verbal report technique have been raised, however, calling into question their validity (

Another technique to determine whether participants are solving problems via retrieval from long-term memory uses tasks for which fact retrieval is irrelevant. For example,

Since this seminal investigation that supported the obligatory activation of addition facts, subsequent studies have examined the obligatory activation of answers to other operations (e.g.,

For example,

While there has been a proliferation of research examining simple arithmetic processing, the research examining complex arithmetic processing is relatively sparse. Much of this research has focused on examining working memory involvement (e.g.,

In this first experiment, participants were asked to practice solving a small subset of complex multiplication problems that had one multi-digit operand between 14 and 19 and one single-digit operand between 3 and 6. After practice, participants completed a number-matching task modeled after that used by

Twenty-one undergraduates (16 female, 5 male) from a small Northeastern college in the USA participated and received extra credit toward their psychology classes; their mean age was 19.4 years. All had normal or corrected-to-normal vision. Three participants did not have English as their first language, and one reported a diagnosed math disability. Patterns of results did not differ when these four participants’ data were removed from analyses, so their data were retained in the analyses that are reported.

The computer tasks described below were administered using the SuperLab Pro experiment software (version 4.5) with responses made either verbally into a microphone (complex multiplication practice) or via a response-box button press (number-matching task). Stimuli in all tasks were displayed on a computer monitor in black Tahoma Regular 36-point font against a white background. Double-digit numbers were a width of approximately 12 mm and single-digit numbers a width of approximately 6 mm; both were 10 mm in height. Participants viewed these stimuli at a distance of approximately 60 centimeters from the computer monitor.

Five complex multiplication problems (16 x 3, 18 x 3, 17 x 4, 19 x 4, and 14 x 6) and their commuted pairs were presented on a computer screen for participants to solve mentally. In total, the multiplication problems covered a width of approximately 4.5 cm on the screen. These problems were presented in blocks of ten (each problem and its commuted pair), and within each of the blocks, problems were presented in a random order.

As noted above, the stimuli for this task were (mostly) modeled after those that have been used in number matching tasks examining the automatic activation of simple multiplication answers (e.g.,

There were two main categories of stimuli constructed for this task, non-matching stimuli and matching stimuli. Within the non-matching stimuli were three categories of stimuli including product, unrelated, and filler stimuli. The non-matching product stimuli included probes that were the correct answers to the product of the cues (e.g., cues: “16” and “3”; probe: “48”). Non-matching unrelated stimuli included probes that were not the correct answers to the product of the cues (e.g., cues: “16” and “3”; probe: “52”). The non-matching filler stimuli included double-digit and single-digit cues followed by a non-matching double-digit probe (e.g., cues: “67” and “8”; probe: “58”). These items were included so that non-matching item cues were not only the operands of the practiced problems and so there were items with double-digit cues beyond the teen numbers.

The non-matching unrelated items were constructed using several criteria to ensure that the hypothesis, interference from product probes stems from automatic activation of correct answers by the cues, could be tested without confounds stemming from the nature of the probe items.

First, the odd-even (parity) status between probes in the product and unrelated trials were matched—in fact, all probes were even numbers; previous research has shown that different processing strategies may be used during the processing of arithmetic verification problems when parity of the proposed answer does not match the parity of the correct answer (e.g.,

The other category of items in the number-matching task was the matching stimuli. These stimuli also included three sub-categories: probe-balancing stimuli, cue-balancing stimuli, and filler stimuli. The matching probe-balancing stimuli included the same probes as the non-matching product stimuli, but all double-digit probes matched the double-digit number in the cue (e.g., cues: “48” and “3”; probe: “48”). Matching cue-balancing stimuli were when the same cues as the non-matching product trials were used, and cues were followed by probes that matched the single-digit number in the cue (e.g., cues: “16” and “3”; probe: “3”). Lastly, the matching-filler stimuli were included to balance the number of matching and non-matching trials; each contained a double-digit number and single digit number as a cue, and the probes matched the double-digit number in the cue (e.g., cues: “67” and “8”; probe: “67”).

In total, 60 trials of stimuli (cues and probe), ten in each of the six different categories described above, were included. Half of the items in each category had cues with the larger number on the left, and half had cues with the larger number on the right. Fourteen blocks with 30 trials in each block were administered for a total of 420 trials; within each block, trials were presented in a random order, half of the trials were matching and half were non-matching, and half of the trials from each of the categories were those with the larger cue number on the left.

Participants signed up for two separate appointments that were not to be more than one week apart. During the first appointment, participants practiced solving the complex multiplication problems for 60 minutes, and during the second appointment they practiced the problems for an additional 15 minutes before completing the number matching task that took approximately 25 to 30 minutes.

Before data collection began for the complex multiplication practice task, a research assistant read a set of instructions that described the problem-solving strategy participants were to use. This procedure involved multiplying the single digit operand by the tens digit of the double-digit operand, multiplying the single digit by the ones digit of the double-digit operand, and then adding together the partial products to arrive at an answer. For example, to solve the problem 18 x 3, participants would use the following steps: 10 x 3 = 30; 3 x 8 = 24; 30 + 24 = 54; the name “tens strategy” was used to identify this procedure. After the instructions, participants completed a set of five warm-up problems that were different from the practice problems so that they could become accustomed to the use of the tens strategy, could become familiar with voicing answers into the microphone, and could ask questions about the procedure. During the data collection phase of the task, the multiplication problems remained on the screen until the microphone was tripped by participants’ vocalization of an answer. Once the problem had disappeared, the word “STRATEGY” was displayed for 1500 milliseconds (ms); participants responded “Yes” if they had used the “tens strategy,” or “No” if they had simply remembered the answer (retrieved the answer from long-term memory). A research assistant recorded participants’ answers and strategy use for each problem. Once the 1500 ms time limit had expired, a white screen was presented for 1000 ms followed by the next practice problem. After completing two blocks of problems (20 in total), participants were offered the opportunity to take a short break to help reduce visual and attentional fatigue. Once 60 minutes had elapsed, participants were released from the session.

As noted, during the first 15 minutes of their second appointment, participants solved the practice problems before completing the number-matching task. A research assistant read instructions that explained the nature of the number-matching task and instructed participants to respond as quickly and accurately as possible. Responses were made using the index fingers of each hand with roughly half of the participants’ randomly assigned to use the left key to indicate the probe matched one of the cues, and the other half to use the right key to indicate a match. A set of 35 warm-up trials (different from those used in the data collection portion of the task) were completed so participants could get accustomed to the speed of presentation of stimuli and become familiar with the appropriate key to press for matching versus non-matching trials.

The sequence and timing of events during number-matching trials was as follows (see

It should be noted that conventional null-hypothesis significance testing (NHST) was conducted throughout the manuscript; however, since many researchers have noted the shortcomings of NHST, Bayesian analyses also were conducted (using the freeware

To document the problem-solving skill development over the course of the practice sessions in all three experiments median RT data, accuracy rates, and reported use of retrieval was averaged across the first 20, middle 20, and final 20 problems participants solved; please refer to ^{2}_{p}s > .56, _{10}s^{2}_{p}s > .20, _{10}s^{2}_{p}s > .42, _{10}

Initial 20 Problems | Middle 20 Problems | Final 20 Problems | |
---|---|---|---|

Response Time | |||

Experiment 1 | 8488 (2977) | 4214 (1837) | 2760 (1195) |

Experiment 2 | 8597 (3753) | 3057 (1940) | 2239 (1137) |

Experiment 3 | 8075 (4957) | 3092 (1310) | 2380 (978.6) |

Accuracy | |||

Experiment 1 | .75 (.25) | .88 (.13) | .94 (.12) |

Experiment 2 | .77 (.24) | .93 (.11) | .96 (.08) |

Experiment 3 | .83 (.29) | .97 (.06) | .99 (.02) |

Retrieval Use | |||

Experiment 1 | .16 (.33) | .51 (.41) | .59 (.45) |

Experiment 2 | .11 (.19) | .71 (.37) | .81 (.31) |

Experiment 3 | .22 (.35) | .78 (.38) | .94 (.20) |

Turning to the number-matching task, only product and unrelated trials were analyzed, as they were the critical trials needed to test the hypothesis that product probes would automatically activate the correct product of the cues; this analysis procedure follows previous investigations of this type (e.g.,

The hybrid trimming procedure resulted in the removal of 3.3% of the product probe and 3.0% of the unrelated probe RTs. Data then were averaged by participant by probe type and submitted to a one-tailed correlated _{10}

A second one-tailed correlated _{10}

Finally, recall that two cues in the number-matching task (“14” and “6”) and (“6” and “14”) occurred with a product probe (“84”) that had a partial match to one of the cue numbers—the number “4” in the ones place of the two-digit cue and in the two-digit probe, match. Both RT and accuracy analyses were re-run after excluding RT and error data related to trials with the aforementioned cues and after using the hybrid process to trim RTs. The significant effects from the initial analyses were unchanged; participants completed product-probe trials 21 ms more slowly than unrelated-probe trials, _{10}_{10}

In summary, the results of the first experiment show that after a little over an hour of practicing complex multiplication problems, participants demonstrated an interference effect during a number-matching task that suggests retrieving the answers to the complex multiplication problems was obligatory. It is especially compelling because multiplication of the cue numbers is irrelevant to what participants were instructed to do in the task. This finding mirrors both the nature and size of the interference effects that have been documented in recent investigations of simple multiplication. The effects in the present experiment were a 19 ms slowing and a 1% error rate increase, and are similar to those found in other studies, which have ranged from 18 to 67 ms (many have been between 18 and 25 ms) and from 2% to 4% errors (

Given the findings from Experiment 1 that parallel effects from simple multiplication, the purpose of the second experiment was twofold. First, it was to replicate the number-matching interference effect from Experiment 1 with a new sample. Second, it was to use complex multiplication practice and the number-matching task to investigate/evaluate existing models of arithmetic. One model of how multiplication facts are stored in long-term memory is the Identical Elements (IE) Model (

Some predictions of the IE model related to multiplication problem-answer representations are that RT changes from practicing a particular operand order should transfer to the reverse operand order, and transfer should result regardless of the format in which problems are practiced (e.g., 8 x 3, eight x three, or auditory presentation of 8 x 3). For example,

In contrast, there are researchers who propose that multiplication problems’ commuted pairs are represented independently. In these models, problems and answers are typically characterized as being part of a network with the physical format of the problem (8 x 3 vs. eight x three) influencing not just perceptual processes, but central cognitive processes as well. For example, in Campbell’s encoding complex model (

To evaluate further the models of arithmetic processing just described, Experiment 2 used a design that included complex multiplication practice and a number-matching task. Participants again were tasked with practicing a small subset of complex multiplication problems before completing a related number-matching task. The one difference from Experiment 1 to Experiment 2 was the use of two practice conditions. Participants in one condition practiced only complex multiplication problems that had the larger operand first (e.g., 17 x 4—L x s problems), and in the other condition participants practiced the same problems that had the smaller operand first (e.g., 4 x 17—s x L problems). In the subsequent number-matching task, participants were shown cues in both orders (“17” and “4”, and “4” and “17”). The different arithmetic representation models outlined above lead to different predictions about the interference that will result in the number-matching task. The IE model predicts that in both practice conditions, a single problem-answer representation will be strengthened for each practiced problem. Therefore, the interference effect observed in Experiment 1 should occur for participants in both practice conditions equally on the L x s and s x L probes in the number-matching task. Independent operand representation models (e.g.,

Sixty-four undergraduates from a small Northeastern college in the USA volunteered to participate in the experiment. Four participants did not attend their second appointment leaving a total of 60 participants (47 females and 13 males) with a mean age of 21.7 years (range was 18 to 48 years) who received extra credit toward their psychology classes for completing the experiment. All had normal or corrected-to-normal vision. Five participants did not have English as their first language, and none reported a diagnosed math disability or attention deficit disorder. Patterns of results did not differ when removing the five participants’ data, so their data were retained in the analyses below. The apparatus and computer tasks were the same as in Experiment 1, although there were minor differences in stimuli used in Experiment 2, and those are explained below.

The same complex multiplication problems from Experiment 1 were used with the exception of the problems 14 x 6 and 6 x 14, which were replaced by the problems 15 x 6 and 6 x 15. This was done so that none of the non-matching product trials in the number-matching task would have a probe that was a partial match to either of the cue digits. Problems were presented in blocks of five, and within each of the blocks, they were presented in a random order.

The stimuli for this task were the same as in Experiment 1 except for the product, probe-balancing, and cue-balancing trials related to the two practice problems that were replaced (see

Participants signed up for two appointments that were not more than one week apart. After signing the consent form, participants were assigned randomly either to the condition in which practice problems were presented larger operand first (L x s condition) or smaller operand first (s x L condition). During this first appointment, participants practiced solving the complex multiplication problems for 60 minutes, and during the second appointment practiced the problems for 15 minutes before completing the number-matching task that took approximately 25 minutes. The remaining procedures for the experiment and within the experimental tasks (e.g., sequence of stimuli in number-matching task) were identical to Experiment 1.

Once again, only product and unrelated trials were analyzed because they were the critical trials needed to test the hypotheses. Due to computer issues, 11 participants did not complete the full number-matching task, which would have resulted in 35 trials per condition, but each of these participants had at least 22 trials per condition and therefore remained in the data analysis. Overall, in 1.0% of the trials participants did not make a response before the 2500 ms cut-off—1.2% for the s x L product condition, .8% for the s x L unrelated condition, .8% for the L x s product condition, and 1.1% for the L x s unrelated condition. The hybrid trimming procedure was used again resulting in the removal of 3.5% of RTs in the s x L product, 2.7% in the s x L unrelated, 3.0% in the L x s product, and 2.9% in the L x s unrelated condition. The means of those trimmed RTs then were submitted to a 2 (Practice Type: s x L vs. L x s) x 2 (Cue Type: s x L vs. L x s) x 2 (Probe Type: product vs. unrelated) ANOVA with repeated measures on the last two factors. For a summary of the means and standard deviations related to the ANOVA analyses, please see

Practice | s x L Product | s x L Unrelated | L x s Product | L x s Unrelated |
---|---|---|---|---|

RTs | ||||

s x L | 554 (98.6) | 551 (106) | 571 (110) | 559 (113) |

L x s | 578 (126) | 562 (142) | 602 (150) | 566 (116) |

Accuracies | ||||

s x L | .95 (.11) | .97 (.08) | .95 (.09) | .97 (.09) |

L x s | .94 (.10) | .96 (.08) | .94 (.09) | .95 (.09) |

The 20 millisecond difference in number-matching RTs between the s x L and L x s practice conditions resulted in a non-significant main effect of practice type, ^{2}_{p} < .01, _{10}^{2}_{p} = .08, _{10}^{2}_{p} = .17, _{10}_{10}^{2}_{p}s < .07, _{10}s^{2}_{p} < .01, _{10}^{2}_{p} = .08, _{10}

Separate follow-up 2 x 2 ANOVAs for each practice group resulted in null effects for cue type for both groups, ^{2}_{p}s < .10, _{10}s^{2}_{p} = .26, _{10}^{2}_{p} = .06, _{10}^{2}_{p}s < .07, _{10}s

Due to experimenter error, one participant’s accuracy data were lost resulting in one fewer degree of freedom in the analysis below. Accuracy on the number-matching task was, in general, high across participants. The mean accuracy data were submitted to the same 2 x 2 x 2 ANOVA that the RT data were. Neither the main effects of practice nor cue type were significant, ^{2}_{p}s < .01, _{10}s^{2}_{p} = .18, _{10}^{2}_{p}s < .01, _{10}s

Separate follow-up 2 x 2 ANOVAs for each practice group indicated that a probe-type error effect was found for the L x s practice group (1% more errors for product versus unrelated probes), ^{2}_{p} = .13, _{10}^{2}_{p} = .26, _{10}^{2}_{p}s < .01, _{10}s^{2}_{p}s < .04, _{10}s

In summary, the first finding of note is that the interference effect shown both in longer RTs (driven by the L x s group) and higher error rates (driven by both practice groups) for product vs. unrelated trials was replicated in Experiment 2. Partial eta squared values suggest that these effects were of moderate size, as they explained 17% of the within subject variance for RTs (in separate group analyses, 26% for L x s and 8% for s x L groups) and 18% of the within subject variance for accuracy (13% for L x s and 26% for s x L groups). In addition, results of the Bayesian analyses showed that the likelihood of the effects were very high; the Bayes factors indicated strong/very strong evidence for the effect of probe type on number-matching RT and substantial/strong evidence for the effect of probe type on number-matching accuracy (according to language used to report Bayes analyses that is outlined in

The second finding of note concerns the prediction related to the independent vs. IE models of arithmetic fact representation. Recall that if the independent model of arithmetic fact representation were true, there would be a three-way interaction effect in the number-matching task where the probe type effect for the L x s practice group would be present (or at least, larger) for the L x s cue trials and not present (or at least, smaller) for the s x L trials, and the reverse would be true for the s x L practice group. This interaction effect did not materialize; in fact, the resulting Bayes factors from the interaction analyses ranged between .14 and .30 (specifically, .14 for both three-way interactions), which corresponds to the null model being from 3.37 to 7.14 times more likely to occur under a model that does not include the interaction effects than one that does. This is positive/substantial evidence for the null hypothesis (

The focus of Experiment 3 was to extend to complex multiplication another set of effects that have been documented in recent investigations of simple multiplication. According to the Interacting Neighbors (IN) Model of single-digit multiplication, how many “neighbors” an arithmetic problem has will have an impact on the speed and accuracy with which people can retrieve its answer (e.g.,

Other activation spreads in the semantic field as well. For example, the problem 6 x 3 activates the 1-node in a decade field because there is a “1” in the decade of the correct answer. Similarly, the 8-node in the ones field is activated because there is an “8” in the ones place of the correct answer. In light of this activation of the different numerical components of answers, the IN model predicts that problems that have the most neighbors that activate the decade and unit numbers that are in the correct answer, the faster the correct answer will be retrieved. To demonstrate with the 6 x 3 example, because its answer shares the decade digit with the answers 12 and 15, and because 6 x 3 will activate nearby problems 6 x 2, 4 x 3, and 5 x 3, the 1-node for the decade part of the answer will be activated highly. A problem such as 9 x 7 won’t have the same degree of facilitation as 6 x 3 because only one problem has an answer in the same decade (64) and will only be activated weakly by 9 x 7 because the problem with that answer, 8 x 8, does not share an operand with 9 x 7. When neighboring problems share decades and/or unit numbers of their answer (e.g., 6 x 3 = 18 and 6 x 2 = 12), they are said to be “consistent;” when they do not (e.g., 6 x 3 = 18 and 6 x 4 = 24), they are termed “inconsistent.”

Recent studies using connectionist modeling (

Given the review of the IN model and findings concerning relatedness and consistency effects, the purpose of Experiment 3 was to determine if the findings would extend to complex multiplication problems. To test this, participants who were not involved in the previous two experiments were asked to practice a new set of complex multiplication problems. Subsequently, a verification task was administered in which incorrect answers varied in their consistency and relatedness. The hypothesis is RTs and error rates will increase in a post-practice verification task when the answers to be verified are consistent and/or related.

Forty undergraduates from a Northeastern college in the USA signed up for the experiment. Four of them did not return for their second appointment, leaving 36 participants (25 females and 11 males) with a mean age of 20.3 years who completed the experiment and received extra credit toward their psychology class. All participants had normal or corrected to normal vision. Three participants reported English was their second language, one participant reported a diagnosed attention disorder, and none of the participants indicated that they had a diagnosed math disability. As in the previous two experiments, data analysis including or deleting the four participants’ data resulted in the same pattern of results, so their data were retained in the analyses below. The apparatus was the same as was used in Experiments 1 and 2.

A new set of six complex multiplication problems and their commuted pairs was constructed so that both relatedness and consistency could be varied systematically in the multiplication verification task described below (see

These stimuli were constructed to control many confounding variables, largely following how

Because incorrect answers can be rejected without actually calculating a correct answer when odd-even (parity) status of problem and answer do not match and/or the split (distance from the correct answer) of the answer to be verified is large, these variables were controlled (e.g.,

As much as possible, lures did not contain digits that matched in congruent positions with the problem (e.g., 8 x 4 = 34). The only exception to this was the problems 4 x 18 and 18 x 4; for these, there was a match for both the related inconsistent (68) and unrelated consistent (78) lures. Because interference due to this match is uncommon when the first operand matches a number in the answer, minimal additional interference from the problem 18 x 4 to reject either of the lures should occur (e.g.,

The sign-up and length/scheduling of practice were the same as in Experiments 1 and 2. After practice, participants completed the complex multiplication verification task that took approximately 25 minutes. Procedures in the practice task were the same as in Experiment 1, including the instructions, completion of warm-up problems that were different from those in the data-collection portion of the task, the use of the STRATEGY screen, and the timing and sequencing of the stimuli.

For the verification task, participants were instructed to use their two index fingers to make the button-press responses to indicate an incorrect or correct answer. Participants were randomly assigned to use the left or right button to indicate a “correct” answer and the other button to indicate an “incorrect” answer. The task began with a set of instructions that a research assistant read aloud while a participant read them silently. Participants ran through a set of 36 warm-up problems to get accustomed to the verification task and the correct use of the two response buttons. The sequence and timing of the stimuli in each of the trials almost exactly mirrored those used in the long-SOA condition in

A total of 384 trials were presented. Of these, 192 were correct trials, and 192 were incorrect trials, the latter of which were divided evenly into 48 trials of each of the incorrect answer types. This meant that each problem and its commuted pair were presented 16 times with their correct answer and four times with each of the different types of lures. These 384 trials were separated into eight blocks; during each block, 24 correct trials (two repetitions of each commuted pair) and 24 incorrect trials (one of the commuted pairs from each lure category) were presented. After each block, participants were offered a break to ensure that visual/attentional fatigue would not compromise performance. Once the verification task was completed, the research assistant debriefed each participant before releasing him/her from the experiment.

Response times to correct trials and error rate data were collected from participants in the verification task. In 3.0% of the trials, participants did not make a response before the 2500 ms cut-off. The no-response rate was 2.7% for correct answers as well as consistent related and inconsistent related lures; 4.0% for consistent unrelated lures; and 3.2% for inconsistent unrelated lures. As a result the hybrid RT trimming procedure, 1.9% of correct answer, and 2.0% of consistent related, 1.7% of inconsistent related, 2.0% of consistent unrelated, and 2.1% of inconsistent unrelated lure RTs were removed. Four participants were excluded from the data analysis below because their combined error and no-response rates were very high (43%, 47%, 51%, and 61%), suggesting that they were simply guessing when verifying answers.

A 2 x 2 repeated-measures ANOVA was conducted with relatedness (related vs. unrelated) and consistency (consistent vs. inconsistent) as factors. For the descriptive statistics related to this analysis, refer to ^{2}_{p} = .35, _{10}^{2}_{p} = .08, _{10}^{2}_{p} < .01, _{10}

Correct | Related Consistent | Related Inconsistent | Unrelated Consistent | Unrelated Inconsistent | |
---|---|---|---|---|---|

Response Time | 820 (167) | 958 (194) | 926 (161) | 891 (174) | 871 (182) |

Accuracy | .89 (.09) | .81 (.13) | .88 (.14) | .89 (.12) | .92 (.10) |

A second 2 x 2 ANOVA with the same factors was conducted on the accuracy data. Both of the main effects were significant. Error rates were 6% higher for related than for unrelated lures, ^{2}_{p} = .49, _{10}^{2}_{p} = .61, _{10}^{2}_{p} = .12, _{10}

In summary, as predicted, after complex multiplication practice, participants were slower and more error prone when rejecting lures that were the correct answer to another practiced problem (related), and were more error prone when the lures shared the decade number with the correct answer to the problem (consistent). The sizes of the effects were fairly large, as the variance accounted for according to partial eta squares were 35% (relatedness effect for RTs), 49% (relatedness effect for accuracies), and 61% (consistency effect for accuracies). These effects, given that the Bayes factors were greater than 150, can be characterized as very strong/decisive (

Recall that the purpose of the experiments was twofold. The first was to document that the obligatory activation of arithmetic answers extends from simple to complex multiplication problems. Both Experiments 1 and 2 demonstrated that this activation does extend to participants who, through recent practice, have become skilled at complex multiplication problem solving. As noted previously, in these two experiments, this obligatory activation was indexed using a number-matching task which does not require multiplication processing. Results showed after roughly 75 minutes of practice on a subset of problems that product probe-trial RTs/error rates were significantly longer/larger than unrelated probe-trial RTs/error rates.

The second goal of the set of experiments was to show that complex multiplication could be used to evaluate existing models of arithmetic fact representations. In Experiment 2, this was achieved by having participants practice the same subset of complex arithmetic problems with two practice groups employed—one that only saw problems with the smaller operand first (s x L) and the other that only saw problems with the larger operand first (L x s). The number-matching task was used again to determine whether the practice groups showed different levels of interference when the cues presented matched, versus did not match, their practiced order. The probe interference effects across the different cue types (s x L, L x s) did not differ as a function of practice group—in other words, no three-way interactions were found, and Bayes factors provided positive evidence for the null hypothesis. These findings support single-representation models of arithmetic that predict that practice of either problem order should strengthen the same single representation (

Turning to Experiment 3, in a multiplication verification task that followed complex multiplication practice, participants were slower and/or more error prone when they had to reject incorrect answers that shared the decade digit of the correct answer (consistency effect) or were a correct answer to a different practiced problem (relatedness effect). The replication of these effects for complex multiplication problems supports the IN Model of arithmetic that multi-digit numbers, in particular answers to multiplication problems, are processed in a decomposed (decade and units digits separately), rather than holistic fashion (

As mentioned previously, probe type, consistency, and relatedness effects across the three experiments were very similar to those found in simple multiplication investigations. This is particularly noteworthy given that in the present investigations the number of presentations of the complex problems was limited and occurred over a short time frame, while adults have encountered simple multiplication problems much more extensively and over a very long period of time. Recall that a medium-length SOA was used in the number-matching tasks, and the one used in the arithmetic verification task in Experiment 3 matched the longer SOA in

Give the findings of the three experiments, follow-up experiments using complex multiplication practice and number-matching could be used to further our understanding of how arithmetic problems and their answers are represented in long-term memory. For example, recently there has been a proliferation of preferred-representation models of arithmetic (a different type of single-representation model). These suggest that addition and multiplication problems may have a more privileged addend/operand order representation in long-term memory that corresponds to speed and error rate differences in solving problems presented in the preferred versus less-preferred operand order (e.g.,

For example, a recent set of experiments showed that Italian adults solved L x s problems faster than s x L problems but only when at least one of the operands was smaller than five; when both were larger than five, s x L problems were solved faster (

There are two issues that cloud the results and related causal explanations of

A second issue is the repeated addition strategy and learning order explanations offered by

In conclusion, the set of experiments conducted here have shown that the use of a complex multiplication practice design along with number-matching tasks have extended several well documented effects from the simple multiplication literature. Designing future experiments that employ complex arithmetic practice as a component should assist researchers in evaluating models of arithmetic processing. Practice experiments using pseudo number-arithmetic tasks have been conducted previously to evaluate arithmetic models (alphaplication such as “I, E = p,” e.g.,

Product | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Cue | 16 3 | 18 3 | 17 4 | 19 4 | 14 6 | 3 16 | 3 18 | 4 17 | 4 19 | 6 14 |

Probe | 48 | 54 | 68 | 76 | 84 | 48 | 54 | 68 | 76 | 84 |

Unrelated | ||||||||||

Cue | 16 3 | 18 3 | 17 4 | 19 4 | 14 6 | 3 16 | 3 18 | 4 17 | 4 19 | 6 14 |

Probe | 52 | 46 | 92 | 58 | 72 | 52 | 46 | 92 | 58 | 72 |

Fillers | ||||||||||

Cue | 67 8 | 34 9 | 23 7 | 54 8 | 72 3 | 8 67 | 9 34 | 7 23 | 8 54 | 3 72 |

Probe | 58 | 68 | 46 | 36 | 82 | 58 | 68 | 46 | 36 | 82 |

Probe-balancing | ||||||||||

Cue | 48 3 | 54 3 | 68 4 | 76 4 | 84 6 | 3 48 | 3 54 | 4 68 | 4 76 | 6 84 |

Probe | 48 | 54 | 68 | 76 | 84 | 48 | 54 | 68 | 76 | 84 |

Cue-balancing | ||||||||||

Cue | 16 3 | 18 3 | 17 4 | 19 4 | 14 6 | 3 16 | 3 18 | 4 17 | 4 19 | 6 14 |

Probe | 3 | 3 | 4 | 4 | 6 | 3 | 3 | 4 | 4 | 6 |

Fillers | ||||||||||

Cue | 86 7 | 98 6 | 62 3 | 46 9 | 38 6 | 7 86 | 6 98 | 3 62 | 9 46 | 6 38 |

Probe | 86 | 98 | 62 | 46 | 38 | 86 | 98 | 62 | 46 | 38 |

Sequence of events in the number-matching task used in both Experiments 1 and 2.

Product | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Cue | 16 3 | 18 3 | 17 4 | 19 4 | 15 6 | 3 16 | 3 18 | 4 17 | 4 19 | 6 15 |

Probe | 48 | 54 | 68 | 76 | 90 | 48 | 54 | 68 | 76 | 90 |

Unrelated | ||||||||||

Cue | 16 3 | 18 3 | 17 4 | 19 4 | 15 6 | 3 16 | 3 18 | 4 17 | 4 19 | 6 15 |

Probe | 52 | 46 | 92 | 58 | 72 | 52 | 46 | 92 | 58 | 72 |

Fillers | ||||||||||

Cue | 67 8 | 34 9 | 23 7 | 54 8 | 72 3 | 8 67 | 9 34 | 7 23 | 8 54 | 3 72 |

Probe | 58 | 68 | 46 | 36 | 82 | 58 | 68 | 46 | 36 | 82 |

Probe-balancing | ||||||||||

Cue | 48 3 | 54 3 | 68 4 | 76 4 | 90 6 | 3 48 | 3 54 | 4 68 | 4 76 | 6 90 |

Probe | 48 | 54 | 68 | 76 | 90 | 48 | 54 | 68 | 76 | 90 |

Cue-balancing | ||||||||||

Cue | 16 3 | 18 3 | 17 4 | 19 4 | 15 6 | 3 16 | 3 18 | 4 17 | 4 19 | 6 15 |

Probe | 3 | 3 | 4 | 4 | 6 | 3 | 3 | 4 | 4 | 6 |

Fillers | ||||||||||

Cue | 86 7 | 98 6 | 62 3 | 46 9 | 38 6 | 7 86 | 6 98 | 3 62 | 9 46 | 6 38 |

Probe | 86 | 98 | 62 | 46 | 38 | 86 | 98 | 62 | 46 | 38 |

Practiced Problems | Proposed Answer |
||||
---|---|---|---|---|---|

Correct Product | Related-consistent | Related-inconsistent | Unrelated-consistent | Unrelated-inconsistent | |

13 x 4 and 4 x 13 | 52 | 56 | 60 | 58 | 48 |

14 x 4 and 4 x 14 | 56 | 52 | 60 | 50 | 62 |

15 x 4 and 4 x 15 | 60 | 68 | 56 | 66 | 48 |

17 x 4 and 4 x 17 | 68 | 60 | 72 | 66 | 80 |

18 x 4 and 4 x 18 | 72 | 76 | 68 | 78 | 66 |

19 x 4 and 4 x 19 | 76 | 72 | 68 | 78 | 80 |

Sequence of events in the verification task used in Experiment 3.

I would like to thank the following undergraduate research assistants who were instrumental in collecting data for the three experiments: Alexia Levin, Andrea DeCusati, Anthony Corvino, Amy Benvenuto, Tiffany “Tippy” Shaffer, and Rob Gambardella. Further, I would like to thank Dr. John Towse and two anonymous reviewers for their incredibly helpful comments that significantly improved the quality of the manuscript.

The author has no funding to report.

The author has declared that no competing interests exist.