Division by multiplication

In two experiments, item-specific transfer was examined in simple multiplication and division with prime and probe problems separated by four to six trials. As was predicted by Rickard and Bourne's (1996) identical-elements model, response time (RT) savings were larger with identical (e.g., prime 63 divided by 7, probe 63 divided by 7) than with inverted (63 divided by 9 and 63 divided by 7) division problems, whereas identical (7 x 9 and 7 x 9) and inverted (9 x 7 and 7 x 9) multiplication problems produced equivalent transfer. Nonetheless, there was statistically significant transfer between inverted division problems. Furthermore, RT savings in the multiplication-to-division transfer conditions (e.g., prime 7 x 9, probe 63 divided by 7) indicated that multiplication mediated large-number division problems. These latter effects are not predicted by the identical-elements model but may be reconciled with the model by distinguishing associative transfer (facilitation owing to strengthening of a common problem node in memory) from mediated transfer (facilitation owing to mediation by a strengthened, related problem). Skilled adults can exploit the conceptual correspondences between multiplication and division facts in a highly efficient way to facilitate performance.     

Retrieval savings with nonidentical elements: The case of simple addition and subtraction

The identical elements (IE) theory of fact representation (Rickard, 2005) proposes that memorized facts that are composed of identical elements (e.g., 6 × 8 = 48 and 8 × 6 = 48) share a common representation in memory, whereas facts with nonidentical elements (e.g., 6 × 8 = 48 and 48 ÷ 8 = 6) are represented separately in memory. The IE model has been successfully applied to the transfer of practice in simple multiplication and division, in transition from procedure-based to retrieval-based performance, and in cued episodic recall. In the present article, we examined the effects of practicing simple addition problems (e.g., 3 + 6 = 9) on the performance of corresponding subtraction problems (9 − 6 = 3), and vice versa. According to IE theory, there should be no transfer of retrieval savings between addition and subtraction facts if performance is based on discrete IE fact representations. Cross-operation response time savings were observed, however, for both small, well-memorized problems (e.g., practice 3 + 2, test 5 − 2) and larger problems (6 + 8, 14 − 6), and they were statistically robust when trials that were self-reported as direct retrieval were analyzed. The transfer of retrieval practice savings between facts with nonidentical elements challenges IE theory as a comprehensive model of transfer in memory retrieval.     

Specificity of learning through memory retrieval practice: The case of addition and subtraction

The identical elements (IE) model (Rickard, Healy, & Bourne, Learning, Memory, and Cognition 32:734–748, 1994) of fact representation predicts that, in both verbal and numerical domains, performance gains with retrieval practice on multielement items will be specific to the practiced stimulus–response combinations, failing to transfer even to altered stimulus–response mappings of practiced items. In the case of arithmetic, the model predicts no transfer across either complementary operations (e.g., 4 × 7 to 28 / 4) or complementary division or subtraction problems (e.g., 28 / 4 to 28 / 7). Although that model has successfully described transfer effects in the domains of multiplication–division and episodic cued recall, it is challenged by a recent demonstration of positive cross-operation transfer for addition and subtraction (Campbell & Agnew, Psychonomic Bulletin & Review 16:938–944, 2009). We report results of a new addition–subtraction transfer experiment, the design of which closely matched that of a prior multiplication–division experiment that supported the model. The transfer results were consistent with the IE model. A two-component model of memory retrieval practice effects is proposed to account for the discrepant experimental results for addition and subtraction and to guide future work.     

The tie effect in simple arithmetic: An access-based account

Simple arithmetic problems with repeated operands (i.e., ties such as 4 + 4, 6 x 6, 10 - 5, or 49 / 7) are solved more quickly and accurately than similar nontie problems (e.g., 4 + 5, 6 x 7, 10 - 6, or 48 / 6). Further, as compared with nonties, ties show small or nonexistent problem-size effects (whereby problems with smaller operands such as 2 + 3 are solved more quickly and accurately than problems with larger operands such as 8 + 9). Blankenberger (2001) proposed that the tie advantage occurred because repetition of the same physical stimulus resulted in faster encoding of tie than of nontie problems. Alternatively, ties may be easier to solve than nonties because of differences in accessibility in memory or differences in the solution processes. Adults solved addition and multiplication (Experiment 1) or subtraction and division (Experiment 2) problems in four two pure formats (e.g., 4 + 4, FOUR + FOUR) and two mixed formats (e.g., 4 + FOUR, and FOUR + 4). Tie advantages were reduced in mixed formats, as compared with pure formats, but the tie x problem-size interaction persisted across formats. These findings support the view that tie effects are strongly related to memory access and are influenced only moderately by encoding factors.     

A revised identical elements model of arithmetic fact representation

The identical elements model of arithmetic fact representation (T. C. Rickard, A. F. Healy, & L. E. Bourne, 1994) states that, for each triplet of numbers (e.g., 4, 7, 28) that are related by complementary multiplication and division problems, there are 3 independent fact representations in memory: (4, 7, x) --> 28; (28/7) --> 4; and (28/4) --> 7. In this article, the author reviews the evidence for this model, considers alternative accounts, and proposes a simple and empirically motivated revision to the model that (a) accommodates conflicting results, (b) provides a novel account of the ties effect, and (c) makes new and nonintuitive predictions for the factoring operation (e.g., factoring of 28 into 4 and 7). The author reports 3 experiments designed to test these predictions and discusses implications for arithmetic instruction.     

Transfer Effects in Children's Recall of Arithmetic Facts

Predictions of the Identical Elements (IE) model of arithmetic fact representation (Rickard, 2005 Rickard, T. C. 2005. A revised identical elements model of arithmetic fact representation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 31: 250–257. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Rickard & Bourne, 1996 Rickard, T. C. and Bourne, L. E. 1996. Some tests of an identical elements model of basic arithmetic skills. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22: 1281–1295. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) about transfer between arithmetic facts were tested in primary school children. The aim of the study was to test whether the IE model, constructed to explain adult performance, also applies to children. The IE model predicts no positive transfer when the numerical elements of a test problem do not match exactly with those of a practice problem. In two experiments, children practiced addition or multiplication problems. A large positive transfer effect was found for problems with an operand order change; improvement was just as high as for practiced problems. Smaller transfer effects were found for problems where one operand was increased with one unit and for problems with operands more different from practice problems. Analogous results were found for addition and multiplication, suggesting that storage and retrieval processes in both domains are similar in children. The findings suggest that there is a gradual decrease in transfer as the test problems become more different from practice problems, indicating that a more sophisticated model than the IE model is needed to explain transfer effects in children's recall of arithmetic facts.     

The representation of multiplication and division facts in memory

Recently, using a training paradigm, Campbell and Agnew (2009) observed cross-operation response time savings with nonidentical elements (e.g., practice 3 + 2, test 5 - 2) for addition and subtraction, showing that a single memory representation underlies addition and subtraction performance. Evidence for cross-operation savings between multiplication and division have been described frequently (e.g., Campbell, Fuchs-Lacelle, & Phenix, 2006) but they have always been attributed to a mediation strategy (reformulating a division problem as a multiplication problem, e.g., Campbell et al., 2006). Campbell and Agnew (2009) therefore concluded that there exists a fundamental difference between addition and subtraction on the one hand and multiplication and division on the other hand. However, our results suggest that retrieval savings between inverse multiplication and division problems can be observed. Even for small problems (solved by direct retrieval) practicing a division problem facilitated the corresponding multiplication problem and vice versa. These findings indicate that shared memory representations underlie multiplication and division retrieval. Hence, memory and learning processes do not seem to differ fundamentally between addition-subtraction and multiplication-division.     

Cued recall from image and sentence memory: a shift from episodic to identical elements representation

The applicability of the identical elements (IE) model of arithmetic fact retrieval (T. C. Rickard, A. F. Healy, & L. E. Bourne, 1994) to cued recall from episodic (image and sentence) memory was explored in 3 transfer experiments. In agreement with results from arithmetic, speedup following even minimal practice recalling a missing word from an episodically bound word triplet did not transfer positively to other cued recall items involving the same triplet. The shape of the learning curve further supported a shift from episode-based to IE-based recall, extending some models of skill learning to cued recall practice. In contrast with previous findings, these results indicate that a form of representation that is independent of the original episodic memory underlies cued-recall performance following minimal practice.     

Calculation,culture, and the repeated operand effect

A basic phenomenon of cognitive arithmetic is that problems composed of a repeated operand, so-called "ties" (e.g. 6+6, 7 x 7), typically are solved more quickly and accurately than comparable non-tie problems (e.g. 6+5, 7 x 8). In Experiment 1, we present evidence that the tie effect is due to more efficient memory for ties than for non-ties, which participants reported solving more often using calculation strategies. The memory/strategy hypothesis accounts for differences in the tie effect as a function of culture (Asian Chinese vs. non-Asian Canadian university students), operation (addition, multiplication, subtraction, and division), and problem size (numerically small vs. large problems). Nonetheless, Blankenberger (Cognition 82 (2001) B15) eliminated the tie response time (RT) advantage by presenting problems in mixed formats (e.g. 4 x four), which suggests that the tie effect with homogenous formats (4 x 4 or four x four) is due to encoding. In Experiment 2, using simple multiplication problems, we replicated elimination of the tie effect with mixed formats, but also demonstrated an interference effect for mixed-format ties that slowed RTs and increased errors relative to non-tie problems. Additionally, practicing non-tie problems in both orders (e.g. 3 x 4 and 4 x 3) each time ties were tested once (cf. Cognition 82 (2001) B15) reduced the tie effect. The format-mismatch effect on ties, combined with a reduced tie advantage because of extra practice of non-ties, eliminated the tie effect. Rather than an encoding advantage, the results indicate that memory access for ties was better than for non-ties.     

Retrieval of incidental stimulus-response associations as a source of negative priming

Priming effects of ignored distractor words were investigated in a task-switching situation that allowed an orthogonal variation of priming and response compatibility between prime and probe. Across 3 experiments, the authors obtained a disordinal interaction of priming and response relation. Responding was delayed in the ignored repetition condition if different responses were required for identical stimuli in the prime and probe (negative priming). Repeating the prime distractor in the probe facilitated responding if the same response was required in the prime and in the probe (positive priming). The same pattern of results was replicated in a letter-matching task without task switching (Experiment 4). Findings lend support to a new model that explains negative priming in terms of an automatic retrieval of incidental stimulus-response associations.     

The role of phonological and visual working memory in complex arithmetic for Chinese- and Canadian-educated adults

Two experiments were conducted to test cultural differences in the role of phonological and visual working memory in complex arithmetic. Canadian- and Chinese-educated students solved complex subtraction problems (e.g., 85 − 27; Experiment 1) and complex multiplication problems (e.g., 6 − 13; Experiment 2) under phonological and visual working memory loads. Problem complexity (i.e., borrow or carry operations) and presentation format (i.e., horizontal vs. vertical) were also manipulated. The results showed that both Chinese- and Canadian-educated participants relied on both phonological and visual working memory resources when solving complex subtraction and multiplication problems. Selective involvement of phonological and visual working memory as a function of operation (Lee & Kang, 2002) or presentation format (Trbovich & LeFev re, 2003) was found only for Chinese-educated participants and not for Canadian-educated participants, calling into question the generalizability of these findings across arithmetic operations and cultural groups.     

Excitatory and inhibitory priming by attended and ignored non-recycled words with monolinguals and bilinguals

Experiments examining identity priming from attended and ignored novel words (words that are used only once except when repetition is required due to experimental manipulation) in a lexical decision task are reported. Experiment 1 tested English monolinguals whereas Experiment 2 tested Twi (a native language of Ghana, Africa)-English bilinguals. Participants were presented with sequential pairs of stimuli composed of a prime followed by a probe, with each containing two items. The participants were required to name the target word in the prime display, and to make a lexical decision to the target item in the probe display. On attended repetition (AR) trials the probe target item was identical to the target word on the preceding attentional display. On ignored repetition (IR) trials the probe target item was the same as the distractor word in the preceding attentional display. The experiments produced facilitated (positive) priming in the AR trials and delayed (negative) priming in the IR trials. Significantly, the positive and negative priming effects also replicated across both monolingual and bilingual groups of participants, despite the fact that the bilinguals were responding to the task in their non-dominant language.     

“2 x 3” primes naming “6”: Evidence from masked priming

It is a common assumption for multiplication-solving models that single-digit multiplications are automatically retrieved. However, the experimental evidence for this is based on paradigms under suspicion. In this research, we employed a new procedure with the aim of assessing the automatic retrieval of multiplication more directly. In two experiments, multiplication automatism was studied using briefly presented primes (stimulus onset asynchrony = 48 msec) in a number-naming task. In Experiment 1, in the congruent conditions, the target and the prime were the same numbers (e.g., prime, 6; target, 6) or the target was the solution to the multiplication prime (e.g., prime, 2×3=; target, 6). In the incongruent conditions, no relationship existed between the primes and the targets (e.g., prime, 32; target, 6; or prime, 4×8=; target, 6). Experiment 2 explored the relevance of the equal sign for the multiplication-priming effect. Data showed that naming was faster when the solution of the multiplication prime matched the target, as compared with the incongruent condition (multiplication-priming effect), and that these effects were found irrespective of the presence of the equal sign. The fact that this priming effect was found even though the participants were unaware of the presentation of the primes supports the automatic character of single-digit multiplication. We conclude by arguing that this procedure is highly valuable for exploring the mechanisms involved in simple arithmetic solving.     

Bidirectional associations in multiplication memory: conditions of negative and positive transfer

A variety of experimental evidence indicates that the memory representation for multiplication facts (e.g., 6 x 9 = 54) incorporates bidirectional links with a forward association from factors to product and a reverse association from product to factors. Surprisingly, the authors did not find evidence in Experiment 1 of facilitative transfer-of-practice from multiplication (6 x 9 = ?) to factoring (54 = ? x ?); in fact, multiplication practice produced item-specific interference with factoring. Similarly, the authors found no evidence in Experiment 2 that repetition of specific factoring problems (54 = ? x ?) facilitated performance of corresponding multiplication problems (6 x 9 = ?). In Experiment 3, participants practiced both multiplication and factoring and presented facilitative transfer in both directions. Thus, bidirectional facilitation occurred if both operations were practiced, but interference occurred when only one operation was practiced. We propose that this seemingly paradoxical behavior occurs because it is adaptive for the bidirectional retrieval structure to retain operational flexibility in the context of practicing both operations, whereas it is adaptive to specialize the memory representation for the practiced operation (i.e., factoring or multiplication) when only one operation is practiced.     

Behavioural evidence for sex differences in the overlap between subtraction and multiplication

The present study aims to identify factors that may influence the dissociability of number magnitude processing and arithmetic fact retrieval at the behavioural level. To that end, we assessed both subtraction and multiplication performance in a within-subject approach and evaluated the interdependence of unit-decade integration measures on the one hand as well as sex differences in the interdependence of performance measures on the other hand. We found that subtraction items requiring borrowing (e.g. 53–29 = 24, 3 < 9) are more error prone than subtraction items not requiring borrowing (e.g. 59–23 = 34, 9 > 3), thereby demonstrating a borrowing effect, which has been suggested as a measure of unit-decade integration in subtraction. Furthermore, we observed that multiplication items with decade-consistent distractors (e.g. 6 × 4 = 28 instead of 24) are more error prone that multiplication items with decade-inconsistent distractors (e.g. 6 × 4 = 30 instead of 24), thereby demonstrating a decade-consistency effect, which has been suggested as a measure of unit-decade integration in simple multiplication. However, the borrowing effect in subtraction was not correlated with the effect of decade consistency in simple multiplication in either men or women. This indicates that unit-decade integration arises from different systems in subtraction and multiplication. Nevertheless, men outperformed women not only in subtraction, but also in multiplication. Furthermore, subtraction and multiplication performance on correct solution probes were correlated in women, but unrelated in men. Thus, the view of differential systems for number magnitude processing and arithmetic fact retrieval may not be universal across sexes.     

Spatial and identity negative priming in audition: Evidence of feature binding in auditory spatial memory

Two experiments are reported with identical auditory stimulation in three-dimensional space but with different instructions. Participants localized a cued sound (Experiment 1) or identified a sound at a cued location (Experiment 2). A distractor sound at another location had to be ignored. The prime distractor and the probe target sound were manipulated with respect to sound identity (repeated vs. changed) and location (repeated vs. changed). The localization task revealed a symmetric pattern of partial repetition costs: Participants were impaired on trials with identity-location mismatches between the prime distractor and probe target-that is, when either the sound was repeated but not the location or vice versa. The identification task revealed an asymmetric pattern of partial repetition costs: Responding was slowed down when the prime distractor sound was repeated as the probe target, but at another location; identity changes at the same location were not impaired. Additionally, there was evidence of retrieval of incompatible prime responses in the identification task. It is concluded that feature binding of auditory prime distractor information takes place regardless of whether the task is to identify or locate a sound. Instructions determine the kind of identity-location mismatch that is detected. Identity information predominates over location information in auditory memory.     

Arabic digit naming speed: task context and redundancy gain

There is evidence for both semantic and asemantic routes for naming Arabic digits, but neuropsychological dissociations suggest that number-fact retrieval (2x3=6) can inhibit the semantic route for digit naming. Here, we tested the hypothesis that such inhibition should slow digit naming, based on the principle that reduced access to multiple routes would counteract redundancy gain (the response time advantage expected from parallel retrieval pathways). Participants named two single digit numbers and then performed simple addition or magnitude comparison (Experiment 1), multiplication or magnitude comparison (Experiment 2), and multiplication or subtraction (Experiment 3) on the same or on a different pair of digits. Addition and multiplication were expected to inhibit the semantic route, whereas comparison and subtraction should enable the semantic route. Digit naming time was approximately 15ms slower when participants subsequently performed addition or multiplication relative to comparison or subtraction, regardless of whether or not the same digit pair was involved. A letter naming control condition in Experiment 3 demonstrated that the effect was specific to digit naming. Number fact retrieval apparently can inhibit Arabic digit naming processes.     

Response conflict reverses priming: a replication

Subjects named target words that followed a masked prime word of 33-msec (Experiments 1A and 1B) or 200-msec (Experiment 2) duration. The target word was either presented alone or accompanied by an interleaved distractor word. Targets presented alone were named more quickly following an identical prime than following an unrelated prime (repetition priming). In Experiment 1A, targets with distractors were named more slowly following an identical prime than following an unrelated prime (negative priming), replicating Milliken, Joordens, Merikle, and Seiffert (Psychological Review, 1998). In Experiments 1B and 2, repetition priming was reduced, although not reversed, for targets with distractors. The results of all three experiments are opposite to the usual finding of enhanced priming for perceptually degraded targets and suggest that response conflict engages retrospective mechanisms that counteract the facilitatory effects of priming.     

Cross-domain Repetition Priming in Person Recognition

Three experiments examining repetition priming of personal names are reported. In each experiment, faces are used as prime stimuli and people's names as the test stimuli. Experiment 1 fails to demonstrate priming from faces to names when the same task-a familiar/ unfamiliar judgement-is made in prime and test phases. Experiment 2 shows that priming is observed when the same semantic judgement (British/ American) is made in prime and test phases. Experiment 3 shows that priming is observed when different semantic judgements (dead/ alive, British/ American) are made at prime and test phase. These results suggest that transfer appropriate processing cannot provide the sole account of repetition priming in person recognition. Instead, the results are interpreted in terms of a structural account of priming, embedded within an interactive activation and competition model of person recognition.     

Differential effects of prime-probe duration on positive and negative location priming: Evidence for opponent facilitatory and inhibitory influences in priming tasks

A series of four spatial localization experiments is reported that examined the effects of display duration and presentation mode on positive and negative priming using an attended-repetition and an ignored-repetition paradigm, respectively. Experiment 1 showed larger positive priming with response-dependent than with 150 ms display durations while negative priming remained unaffected. Experiments 2-4 were performed to further elucidate the effects of prime-probe durations. Data suggest largely independent effects of prime and probe duration on priming effects. Manipulation of prime duration affected facilitation due to repetition of the prime distractor location as well as inhibitory effects associated with ignored repetition. Furthermore, anticipated probe duration modulated the effectiveness of inhibition of return. Findings are discussed within a framework proposing two major components of priming effects—a stimulus-driven or automatic component, and a strategic component related to the participant's expectations towards the probe.