Retrieval-induced forgetting of arithmetic facts but not rules

Retrieval of a multiplication fact (2×6 =12) can disrupt retrieval of its addition counterpart (2+6=8). We investigated whether this retrieval-induced forgetting effect applies to rule-governed arithmetic facts (i.e., 0×N=0, 1×N=N). Participants (n=40) practised rule-governed multiplication problems (e.g., 1×4, 0×5) and multiplication facts (e.g., 2×3, 4×5) for four blocks and then were tested on the addition counterparts (e.g., 1+4, 0+5, 2+3, 4+5) and control additions. Increased addition response times and errors relative to controls occurred only for problems corresponding to multiplication facts, with no problem-specific effects on addition counterparts of rule-governed multiplications. In contrast, the rule-governed 0+N problems provided evidence of generalisation of practice across items, whereas the fact-based 1+N problems did not. These findings support the theory that elementary arithmetic rules and facts involve distinct memory processes, and confirmed that previous, seemly inconsistent findings of RIF in arithmetic owed to the inclusion or exclusion of rule-governed problems.     

Target strength and retrieval-induced forgetting in semantic recall

Retrieval-induced forgetting (RIF) occurs when practice of a memory item impairs retrieval of related, unpracticed items. Here, we demonstrated that RIF in semantic memory is retrieval dependent. University students either studied (7 × 8 = 56) or retrieved (7 × 8 = ?) the answers to a set of multiplication problems for 40 blocks and then were tested on their addition counterparts (7 + 8 = ?). For the retrieval practice group, but not the study practice group, response time for the multiplication-practiced addition facts was about 100 msec slower, relative to control addition problems, in the first of five postpractice addition blocks. Subsequent blocks of addition were interleaved with retrieval blocks of all the multiplication counterparts, which permitted measurement of RIF for the control addition problems after only a single retrieval of their multiplication counterparts. The control problems presented RIF in excess of 200 msec, much larger than the RIF observed after massive practice. This is consistent with the hypothesis that inhibition of competitors should be weaker when target strength is high than when target strength is only moderate (Anderson, 2003; Norman, Newman, &; Detre, 2007). The evidence that RIF in semantic retrieval is both retrieval dependent and weaker following massive target practice than following moderate target practice provides strong support for inhibition-based theories of RIF.     

An evaluation of sex and cultural differences in arithmetic retrieval-induced forgetting

Retrieval practice of arithmetic facts (e.g. 2?×?3) can interfere with retrieval of other, closely related arithmetic facts (e.g. 2?+?3), increasing response time (RT) and errors for these problems. Here we examined potential sex and culture-related differences in arithmetic retrieval-induced forgetting (RIF). This was motivated by re-analyses of several published arithmetic RIF data sets that appeared to show that the effect occurred for women but not men. Experiment 1 (n?=?72) tested for possible sex differences in a diverse but predominantly Canadian university sample. Experiment 2 (n?=?48) examined potential sex differences in native Chinese participants, which previous research indicated may not be susceptible to the RIF effect for a particular subset of small addition problems (sum?≤?10). In Experiment 1, we found no evidence that the addition RIF effect differed between male and female adults. In Experiment 2, the Chinese adults showed RIF for tie problems (e.g. 2?+?2, 3?+?3, etc.) regardless of sex, but neither sex presented RIF for small non-tie addition problems. The results indicated that the RIF effect is not gender specific, and there might not be strong memory retrieval competition between addition and multiplication facts for non-tie problems in Chinese adults.     

Activation of number facts in bilinguals

Two experiments examined the effect of the presentation format of numbers—digits versus word format in the first and in the second languages of bilinguals—on mental arithmetic. Speed of number-fact retrieval and the presence of interference produced by numbers that were either numerically close to or associatively related to the correct answers of stored arithmetic problems (e.g., 2+5 and 7×8) were compared across formats. The verification of true problems was increasingly slower and less accurate from the digit condition to the second-language condition. Interference was produced by both types of incorrect answers in the digit and first-language conditions, whereas in the second-language condition, it was constrained to answers that were numerically close to correct answers. Together, the results suggest that the retrieval of arithmetic facts and the automatic spreading of activation within the network of numerical facts are not only language-sensitive, but format-sensitive in general.     

Interoperation transfer in Chinese–English bilinguals’ arithmetic

We examined interoperation transfer of practice in adult Chinese-English bilinguals' memory for simple multiplication (6 × 8 = 48) and addition (6 + 8 = 14) facts. The purpose was to determine whether they possessed distinct number-fact representations in both Chinese (L1) and English (L2). Participants repeatedly practiced multiplication problems (e.g., 4 × 5 = ?), answering a subset in L1 and another subset in L2. Then separate groups answered corresponding addition problems (4 + 5 = ?) and control addition problems in either L1 (N = 24) or L2 (N = 24). The results demonstrated language-specific negative transfer of multiplication practice to corresponding addition problems. Specifically, large simple addition problems (sum > 10) presented a significant response time cost (i.e., retrieval-induced forgetting) after their multiplication counterparts were practiced in the same language, relative to practice in the other language. The results indicate that our Chinese-English bilinguals had multiplication and addition facts represented in distinct language-specific memory stores.     

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.     

Retrieval-induced forgetting of arithmetic facts across cultures

Retrieving a single-digit multiplication fact (3×4 =12) can slow response time (RT) for the corresponding addition fact (3+4=7). The present experiment investigated effects of problem type (i.e., tie addition problems such as 3+3 vs. non-ties such as 3+4) and cultural background on this retrieval-induced forgetting (RIF) phenomenon in young adults. Canadians answering in English (n=36), Chinese adults answering in English (n=36), and Chinese answering in Chinese (n=36) received four blocks of multiplication practice and then two blocks of the addition counterparts and control additions. Tie addition problems presented a robust RIF effect that did not differ between groups, but only the Canadian group showed RIF for non-ties and only for small non-ties with sum≤10 (3+4). The Chinese groups' RIF effect for addition ties, but not small non-ties, converges with recent evidence that ties are solved by direct memory retrieval whereas small non-ties may be solved by highly efficient procedural processes in skilled performers.     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

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.     

Rote memory and arithmetic fact processing

The goal of the study was to examine the part played by skill in memorizing arbitrary sequences in the efficiency with which normal young adults perform simple arithmetic fact problems. The first experiment showed a clear independent role for sequence memory in all arithmetic fact processing, but a lesser role for semantic retrieval. This result was particularly true for large-answer multiplication problems and subtraction and division problems with large first operands. In a second experiment, which included a visuomotor processing control task, sequence memory predicted processing of all arithmetic problems apart from small additions independently of semantic retrieval, with the most robust independent contribution being to large-answer multiplication problems. The results, which are compatible with Dehaene and colleagues' triple-code model, suggest that rote learning may be a successful way for some people to process arithmetic facts efficiently.     

Simple arithmetic processing: Individual differences in automaticity

This study investigated individual differences in the ability to automatically access simple addition and multiplication facts from memory. It employed a target-naming task and a priming procedure similar to that utilised in the single word semantic-priming paradigm. In each trial, participants were first presented with a single digit arithmetic problem (e.g., 6+8) and were then presented with a target that was either congruent (e.g., 14) or incongruent (e.g., 17) with this prime. Response times for congruent and incongruent conditions were then compared to a neutral condition (e.g., X+Y, with target 14). For the high skilled group, significant facilitation in naming congruent multiplication and addition targets was found at SOAs of 300 and 1000?ms. In contrast, for the low skilled group, facilitation in naming congruent targets was only observed at 1000?ms. Significant inhibition in naming incongruent multiplication and addition targets at 300?ms, and addition targets at 1000?ms, was found for the high skilled group alone. This advantage in access to simple facts for the high skilled group was then further supported in a problem size analysis that revealed individual differences in access to small and large problems that varied by operation. These findings support the notion that individual differences in arithmetic skill stem from automaticity in solution retrieval and additionally, that they also derive from strategic access to multiplication solutions.     

Retrieval-practice task affects relationship between working memory capacity and retrieval-induced forgetting

Retrieving a subset of items from memory can cause forgetting of other items in memory, a phenomenon referred to as retrieval-induced forgetting (RIF). Individuals who exhibit greater amounts of RIF have been shown to also exhibit superior working memory capacity (WMC) and faster stop-signal reaction times (SSRTs), results which have been interpreted as suggesting that RIF reflects an inhibitory process that is mediated by the processes of executive control. Across four experiments, we sought to further elucidate this issue by manipulating the way in which participants retrieved items during retrieval practice and examining how the resulting effects of forgetting correlated with WMC (Experiments 1–3) and SSRT (Experiment 4). Significant correlations were observed when participants retrieved items from an earlier study phase (within-list retrieval practice), but not when participants generated items from semantic memory (extra-list retrieval practice). These results provide important new insight into the role of executive-control processes in RIF.     

Simple arithmetic processing: the question of automaticity

In adult simple arithmetic performance, it is commonly held that retrieval of solutions occurs automatically from a network of stored facts in memory. However, such an account of performance necessarily predicts a uniform reaction time for solution retrieval and is therefore not consistent with the robust finding that reaction time increases with problem size and difficulty. Additionally, past research into arithmetic performance has relied on tasks that may have actually induced and measured attentional processing, thereby possibly confounding previous results and conclusions pertaining to automaticity. The present study therefore, attempted to more reliably assess the influence of automatic processing in arithmetic performance by utilizing a variant of the well-established semantic word-priming procedure with a target-naming task. The overall results revealed significant facilitation in naming times at SOAs of 240 and 1000 ms for congruent targets i.e., targets that represented the correct solutions to problems presented as primes (e.g., 6+8 and 14). Significant inhibition in comparison to a neutral condition (0+0 and 17) was also observed at 120 and 240 ms SOAs in naming incongruent targets (e.g., 6+8 and 17). Furthermore, response times were found to vary as a function of both arithmetic fluency and problem size. Differences in performance to addition and multiplication operations and implications for cognitive research and education are considered.     

Menatal addition: A test of three verification models

Three explanations of adults’ mental addition performance, a counting-based model, a direct-access model with a backup counting procedure, and a network retrieval model, were tested. Whereas important predictions of the two counting models were not upheld, reaction times (RTs) to simple addition problems were consistent with the network retrieval model. RT both increased with problem size and was progressively attenuated to false stimuli as the split (numerical difference between the false and correct sums increased. For large problems, the extreme level of split (13) yielded an RT advantage for false over true problems, suggestive of a global evaluation process operating in parallel with retrieval. RTs to the more complex addition problems in Experiment 2 exhibited a similar pattern of significance and, in regression analyses, demonstrated that complex addition (e.g., 14+12=26) involves retrieval of the simple addition components (4+2=6). The network retrieval/decision model is discussed in terms of its fit to the present data, and predictions concerning priming facilitation and inhibition are specified. The similarities between mental arithmetic results and the areas of semantic memory and mental comparisons indicate both the usefulness of the network approach to mental arithmetic and the usefulness of mental arithmetic to cognitive psychology.     

Strategy choice for arithmetic verification: effects of numerical surface form

Canadian university students (n=48) solved simple addition problems in a true/false verification task with equations in digit format (3+4=8) or written English format (three+four=eight). Participants reported their solution strategy (e.g. retrieval or calculation) after each trial. Reported use of calculation strategies was much greater with word (41%) than digit stimuli (26%), and this difference was exaggerated for numerically larger problems. Word-format costs on reaction time (RT) were correspondingly greater for large than for small problems, but this Format×Size RT effect was bigger for true than for false equations. The results demonstrate that surface format affects central, rather than only peripheral, stages of cognitive arithmetic.     

Language-specific memory for everyday arithmetic facts in Chinese-English bilinguals

The role of language in memory for arithmetic facts remains controversial. Here, we examined transfer of memory training for evidence that bilinguals may acquire language-specific memory stores for everyday arithmetic facts. Chinese-English bilingual adults (n = 32) were trained on different subsets of simple addition and multiplication problems. Each operation was trained in one language or the other. The subsequent test phase included all problems with addition and multiplication alternating across trials in two blocks, one in each language. Averaging over training language, the response time (RT) gains for trained problems relative to untrained problems were greater in the trained language than in the untrained language. Subsequent analysis showed that English training produced larger RT gains for trained problems relative to untrained problems in English at test relative to the untrained Chinese language. In contrast, there was no evidence with Chinese training that problem-specific RT gains differed between Chinese and the untrained English language. We propose that training in Chinese promoted a translation strategy for English arithmetic (particularly multiplication) that produced strong cross-language generalization of practice, whereas training in English strengthened relatively weak, English-language arithmetic memories and produced little generalization to Chinese (i.e., English training did not induce an English translation strategy for Chinese language trials). The results support the existence of language-specific strengthening of memory for everyday arithmetic facts.     

Simple addition and multiplication: No comparison

Some models of memory for arithmetic facts (e.g., 5+2=7, 6×7=42) assume that only the max-left order is stored in memory (e.g., 5+2=7 is stored but not 2+5=7). These models further assume an initial comparison of the two operands so that either operand order (5+2 or 2+5) can be mapped to the common internal representation. We sought evidence of number comparison in simple addition and multiplication by manipulating size congruity. In number comparison tasks, performance costs occur when the physical and numerical size of numerals are incongruent (8 3) relative to when they are congruent (8 3). Sixty-four volunteers completed a number comparison task, an addition task, and a multiplication task with both size congruent and size incongruent stimuli. The comparison task demonstrated that our stimuli were capable of producing robust size congruity and split effects. In the addition and multiplication task, however, we were unable to detect any of the RT signatures of comparison or reordering processes despite ample statistical power: Specifically, there was no evidence of size congruity, split, or order effects in either the addition or multiplication data. We conclude that our participants did not routinely engage a comparison operation and did not consistently reorder the operands to a preferred orientation.     

Individual differences in the obligatory activation of addition facts

In two experiments, we found evidence for individual differences in the obligatory activation of addition facts. Subjects were required to verify the presence of a target digit (e. g., 4) in a previously presented pair (e. g., 5 + 4). Subjects rejected targets that formed the sum of the initial pair (e. g., 5+4 and 9) more slowly than they rejected unrelated targets (e. g., 5+4 and 7). This interference of the sum was largest for subjects who were relatively skilled at multidigit arithmetic. Less skilled subjects did not show statistically significant effects of obligatory activation. In comparison with less skilled subjects, skilled subjects showed differential interference on plus-one (e. g., 34 1) and standard (e. g, 2+3) problems when the plus sign was presented, and on ties (e. g., 22) when the plus sign was omitted. These results suggest that network models of arithmetic fact retrieval are appropriate for skilled subjects, but that alternative models need to be considered for less skilled individuals.     

Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.