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.     

心算加工中tie effect的加工机制

tie effect主要表现为tie比nontie更快更准确解决,以及tie/nontie与问题大小的交互作用,较大题目的tie比nontie具有更明显的反应时优势,tie的问题大小效应比nontie要小的多。有关tie effect加工机制的解释主要有基于编码的理论和基于结果通达的理论。目前的研究主要通过操作问题呈现方式以及报告加工策略的方法分别考察编码方式及加工策略的影响,今后研究应开拓新的研究范式以加深对其加工机制的认识。     

The arithmetic tie effect is mainly encoding-based.

Arithmetic tie problems like 6 + 6 or 7 x 7 can be solved much faster than non-ties. The present article contrasts two possible explanations for the tie effect, faster encoding of tie problems vs. faster access to arithmetic facts. For that purpose homogeneous (3 + 3, four x four) and heterogeneous (3 + three, four x 4) addition and multiplication problems had to be solved. For all participants the tie effect vanished with heterogeneous addition problems and for seven out of eight subjects the effect disappeared with heterogeneous multiplication problems. It is concluded that the tie effect is mainly encoding-based.     

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.     

The use of procedural knowledge in simple addition and subtraction problems

In a first experiment, adults were asked to solve one-digit additions, subtractions and multiplications. When the sign appeared 150 ms before the operands, addition and subtraction were solved faster than when the sign and the operands appeared simultaneously on screen. This priming effect was not observed for multiplication problems. A second experiment replicates these results on addition and multiplication and, moreover, shows that the priming effect in addition is observed for all problems, including very small ones such as 4+3. In fact, the only problems that were not primed by the addition sign were tie problems, which confirms that they have a special status in memory. Taken together, these results suggest that abstract procedures are pre-activated by the addition and subtraction signs and that these procedures are consequently used by adults to solve the problems. No such procedures would be pre-activated for multiplication, which are then most probably solved by retrieval of the result from memory. Moreover, while obviously two different strategies were used by individuals in order to solve addition and multiplication, solution times were similar when the problems were presented in their whole. These results, which question most of the conclusions of the current literature, support Anderson's model (1982) and Baroody's assumptions (1983) on the existence of compacted procedures that could be as fast as retrievals.     

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.     

The representation of multiplication facts: developmental changes in the problem size, five, and tie effects

In this study, we investigated the development of basic effects that have been found in single-digit multiplication arithmetic: the problem size, five, and tie effects. Participants (9-, 10-, and 11-year-olds and adults) performed a production task on simple multiplication. The procedure replicated study [Canadian Journal of Psychology, Vol. 39, pp. 338-366], but the results show that the gradual decrease of the problem size effect ends in sixth grade. We report analyses on raw latencies and state trace analyses that take into account reaction time scaling as a function of age. The results show that 11-year-olds do not differ significantly from adults on any of the three effects. Before 11 years of age, interesting developmental changes occur.     

On mental multiplication and age.

In 2 experiments, younger and older adults were presented with simple multiplication problems (e.g., 4 x 7 = 28 and 5 x 3 = 10) for their timed, true or false judgments. All of the effects typically obtained in basic research on mental arithmetic were obtained, that is, reaction time (a) increased with the size of the problem, (b) was slowed for answers deviating only a small amount from the correct value, and (c) was slowed when related (e.g., 7 x 4 = 21) versus unrelated (e.g., 7 x 4 = 18) answers were presented. Older adults were slower in their judgments. Most important, age did not interact significantly with problem size or split size. The authors suggest that elderly adults' central processes, such as memory retrieval and decision making, did not demonstrate the typical age deficit because of the skilled nature of these processes in simple arithmetic.     

Production,verification, and priming of multiplication facts

In the arithmetic-verification procedure, subjects are presented with a simple equation (e.g., 4 × 8 = 24) and must decide quickly whether it is true or false. The prevailing model of arithmetic verification holds that the presented answer (e.g., 24) has no direct effect on the speed and accuracy of retrieving an answer to the problem. It follows that models of the retrieval stage based on verification are also valid models of retrieval in the production task, in which subjects simply retrieve and state the answer to a given problem. Results of two experiments using singledigit multiplication problems challenge these assumptions. It is argued that the presented answer in verification functions as a priming stimulus and that on “true” verification trials the effects of priming are sufficient to distort estimates of problem difficulty and to mask important evidence about the nature of the retrieval process. It is also argued that the priming of false answers that have associative links to a presented problem induces interference that disrupts both speed and accuracy of retrieval. The results raise questions about the interpretation of verification data and offer support for a network-interference theory of the mental processes underlying simple multiplication.     

加减法问题大小效应的加工机制

有关问题大小效应的加工机制存在两类不同的解释。初期的提取理论认为,问题大小效应是由算术知识的储存表征的各种特征所决定的。后期研究开始关注如计数、分解这样的非提取策略在问题大小效应形成中的作用,认为大问题任务中更低的提取效率和非提取加工效率以及更多采用非提取策略都会造成算术任务的问题大小效应。由于加法和减法中使用的各种加工策略的比重不同,所以表现为不同形式的问题大小效应。今后研究应该考虑采用不同的研究手段,并统一划分大问题和小问题的标准     

Effects of response time deadlines on adults' strategy choices for simple addition

In this investigation of adults' solution strategies for simple arithmetic, participants solved addition problems (e.g., 2 + 3, 8 + 7) under fast and slow response deadlines: The participants were instructed either to respond before a 750-msec warning beep, or to wait for a 2,500-msec beep before responding. After each trial, they indicated whether they had solved the problem by direct memory retrieval or by using a procedural strategy (e.g., counting, transformation). It was predicted that the fast deadline condition should curtail the use of procedural strategies, which generally are slower than direct retrieval. Furthermore, this deadline effect should be exaggerated for numerically larger problems because procedural strategies are especially slow for the larger problems. As predicted, we observed a deadline x size interaction whereby the fast deadline increased reported use of retrieval, especially for large problems. The results confirm that reported use of direct retrieval decreases systematically with elapsed time, and they provide additional evidence that young, educated adults rely substantially on procedural strategies even for simple addition.     

Cognitive arithmetic: comparison of operations

Adults' performance of simple arithmetic calculations (addition, multiplication, and numerical comparison) was examined to test predictions of digital (counting), analog, and network models. Although all of these models have been supported by studies of mental addition, each leads to a different prediction concerning relations between the times required for addition, multiplication, and numerical comparison. Pairs of single-digit integers were presented and reaction times (RTs) for adding, multiplying, and comparing the stimuli were collected. A high correlation between RT for addition and multiplication of the same digits was obtained. This result is consistent with a network model, but presents difficulties for both analog and counting models. A "ties" effect of no increase in RT with increases in problem size for doubles such as 2 + 2 has been found in previous studies of addition using verification procedures, but was not found with the production task employed in the present study. Instead, a different kind of ties effect was found. Reaction time for both addition and multiplication of ties increased more slowly with problem size than did RT for non-tie problems. This ties effect, and the finding that probability of making errors contributes independently of problem size to RT support a distinction between location and accessibility of information in a network.     

Effects of problem size,operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

Adult's simple-arithmetic strategy use depends on problem-related characteristics, such as problem size and operation, and on individual-difference variables, such as working-memory span. The current study investigates (a) whether the effects of problem size, operation, and working-memory span on children's simple-arithmetic strategy use are equal to those observed in adults, and (b) how these effects emerge and change across age. To this end, simple-arithmetic performance measures and a working-memory span measure were obtained from 8-year-old, 10-year-old, and 12-year-old children. Results showed that the problem-size effect in children results from the same strategic performance differences as in adults (i.e., size-related differences in strategy selection, retrieval efficiency, and procedural efficiency). Operation-related effects in children were equal to those observed in adults as well, with more frequent retrieval use on multiplication, more efficient strategy execution in addition, and more pronounced changes in multiplication. Finally, the advantage of having a large working-memory span was also present in children. The differences and similarities across children's and adult's strategic performance and the relevance of arithmetic models are discussed.     

The reasoning skills and thinking dispositions of problem gamblers: a dual‐process taxonomy

We present a taxonomy that categorizes the types of cognitive failure that might result in persistent gambling. The taxonomy subsumes most previous theories of gambling behavior, and it defines three categories of cognitive difficulties that might lead to gambling problems: The autonomous set of systems (TASS) override failure, missing TASS output, and mindware problems. TASS refers to the autonomous set of systems in the brain (which are executed rapidly and without volition, are not under conscious control, and are not dependent on analytic system output). Mindware consists of rules, procedures, and strategies available for explicit retrieval. Seven of the eight tasks administered to pathological gamblers, gamblers with subclinical symptoms, and control participants were associated with problem gambling, and five of the eight were significant predictors in analyses that statistically controlled for age and cognitive competence. In a commonality analysis, an indicator from each of the three categories of cognitive difficulties explained significant unique variance in problem gambling, indicating that each of the categories had some degree of predictive specificity. Copyright © 2006 John Wiley & Sons, Ltd.     

Spacing and the transition from calculation to retrieval

Many arithmetic problems can be solved in two ways—by a calculation involving several steps and by direct retrieval of the answer. With practice on particular problems, memory retrieval tends to supplant calculation—an important aspect of skill learning. We asked how the distribution of practice on particular problems affects this kind of learning. In two experiments, subjects repeatedly worked through sets of multiple-digit multiplication problems. The size of the trained problem set was varied. Using a smaller set size (with shorter average time between problem repetitions) showed faster responses and an earlier transition to retrieval during training. However, in a test session presented days later, the pattern reversed, with faster responses and more retrieval for the large set size. Evidently, maximizing the occurrence of direct retrieval within training is not the best way to promote learning to retrieve the answer. Practical implications are discussed.     

More on the relation between division and multiplication in simple arithmetic: Evidence for mediation of division solutions via multiplication

Adults (N = 32) solved simple multiplication (e.g., 8 x 7) and corresponding division problems (e.g., 56/8). Self-reports of solution processes were given by half of the participants. Latency patterns and error rates were closely related across operations and were similar in self-report and no-report conditions. Solution of division problems, however, facilitated solution of multiplication problems more than the reverse. On large division problems, participants reported that they "recast" problems as multiplication (e.g., 56/8 as 8 x = 56). These results support the hypothesis that multiplication and division are stored in separate mental representations but that solution of difficult division problems sometimes involves access to multiplication.     

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.     

Stability and change in children's division strategies

Age-related changes in children's performance on simple division problems (e.g., 6/2, 72/9) were investigated by asking children in Grades 4 through 7 to solve 32 simple division problems. Differences in performance were found across grade, with younger children performing more slowly and less accurately than older children. Problem size effects were also found in that children were faster and more accurate on small problems than on large problems. Two strategies changed across age, with children in Grade 4 relying heavily on the strategy of "addition" (adding the divisor until the dividend was reached) to solve the problems and children in Grades 5 through 7 relying primarily on the strategy of "multiplication" (recasting the division problem as a multiplication problem) to solve the problems. Surprisingly, the frequency of direct retrieval (retrieving the answer directly from memory) did not increase across grade and never became the dominant strategy of choice. Reasons for why retrieval use remains infrequent and age invariant are discussed. Overall, the results suggest that division is a unique operation and that the continued study of division may have implications for further understanding of how procedural and conceptual knowledge of arithmetic develops.     

Operation-specific encoding in single-digit arithmetic

Solving simple arithmetic problems involves three stages: encoding the problem, retrieving or calculating the answer, and reporting the answer. This study compared the event-related potentials elicited by single-digit addition and multiplication problems to examine the relationship between encoding and retrieval/calculation stages. Results showed that the operation effect appeared as early as the encoding of the first operand and continued to the retrieval/calculation stage: compared to addition, multiplication elicited larger negative potentials in the left anterior electrodes and larger positive potentials in the right posterior electrodes. The consistency of this operation effect across the first two stages of arithmetic processing suggests that encoding of arithmetic problems can be modulated by the nature of representation of the to-be-retrieved arithmetic facts, and thus these two stages are additive rather than interactive.