When it hurts to be misled: A Stroop-like effect in a simple addition production task

In four experiments, subjects saw simple addition equations (e.g., 3 + 4 = 9) and produced the sums while ignoring the presented answer. If the presented answer was false, subjects took longer to produce the sum, as compared with when the presented answer was true (Experiment 1), when there was no answer presented (blanks; Experiment 2), when a letter was presented (Experiment 3), and when a symbol was presented (Experiment 4). The results suggest that subjects were unable to ignore the presented answers, which raises problems for theories of arithmetic verification (i.e., deciding whether 3 + 4 = 9 is true or false) that claim that subjects verify equations by first producing the sum and then comparing the produced sum with the presented answer. Our results are more compatible with theories that claim that in verification and production, an arithmetic knowledge base is used in different ways.     

Conditions of error priming in number-fact retrieval

Analysis of errors in simple multiplication has shown that answers retrieved on previous trials are initially inhibited (negative error priming) but later are promoted as errors to subsequent problems (positive error priming). Two experiments investigated whether error priming is associated either with problem-specific retrieval processes or with representations of answers that can be manipulated independently of problems. In Experiment 1, answers were primed by visually presenting products for 200 msec prior to problems. Correct-answer primes facilitated retrieval, related-incorrect primes interfered with retrieval more than unrelated primes, and both effects were greater for more difficult problems. Primes affected only the trial on which they were presented, however, whereas both negative and positive error priming from previous problems were observed across trials. In Experiment 2, subjects named and retrieved multiplication products on alternating trials. Just-named products were inhibited as errors to the following multiplication problem (i.e., negative error priming), but, compared to positive priming from previous retrieved products, positive error priming from previously named numbers was weak. The results indicate that positive error priming is due mainly to an encoding or retrieval bias produced by previous problems, whereas negative error priming entails suppression, or de-selection, of answer representations.     

Verification of multiplication facts: an investigation using retrospective protocols

Retrospective verbal protocols collected throughout participants' performance of a multiplication verification task (e.g., "7 x 3 = 28, true or false?") documented a number of different strategies and changes in strategy use across different problem categories used for this common experimental task. Correct answer retrieval and comparison to the candidate answer was the modal but not the only strategy reported. Experiment 1 results supported the use of a calculation algorithm on some trials and the use of the difference between the candidate and correct answers (i.e., split) on others. Experiment 2 clearly demonstrated that participants sometimes bypassed retrieval by relying on the split information. Implications for mental arithmetic theories and the general efficacy of retrospective protocols are discussed.     

Effects of multiplication practice on product verification: Integrated structures model or retrieval-induced forgetting?

In theintegrated structures model of simple multiplication (Manly &; Spoehr, 1999), it is proposed that retrieval of a multiplication fact (e.g., 2 × 7=?) activates and strengthens operand multiples representations (e.g., 4, 6, 8, etc.; 14, 21, 28, etc.). In contrast, in the phenomenon ofretrieval-induced forgetting (RIF; M. C. Anderson, Bjork, &; Bjork, 1994), it is suggested that operand multiples ought to be suppressed with practice. Participants (N=72) performed 40 blocks of practice trials in which they generated answers to a subset of simple multiplication problems. We then measured response times and errors in a true—false product verification task. Both true and related false equations with practiced operand multiples as presented products were solved relatively poorly in the verification task. In agreement with RIF, this suggests that operand multiples were suppressed, rather than strengthened, during the practice phase.     

Retrieval processes in arithmetic production and verification

To investigate whether arithmetic production and verification involve the same retrieval processes, we alternated multiplication production trials (e.g., 9 × 6 = ?) with verification trials (4 × 9 = 36, true or false?) and analyzed positive error priming.Positive error priming is the phenomenon in which errors frequently match correct answers from preceding problems. Production errors were strongly primed by previous production trials (the error-answer matching rate was about 90% greater than expected by chance), but production errors were not strongly primed by previous verification trials (≈13% above chance). Conversely, false-verification errors were primed by previous verification trials (≈25% above chance), but not by production trials. The results indicated that arithmetic production and verification were mediated by different memory processes and suggest a familiarity-based over a retrieval-based model of arithmetic verification.     

工作记忆在简单指数乘法等式判断中的作用

选择大学二年级数学系的学生作为被试,要求他们完成控制条件、语音任务、手动任务和随机间隔决策任务条件下的简单指数乘法等式判断任务,实验结果表明：（1）三种次级任务在真假等式判断的反应时上的干扰效应非常显著;（2）三种次级任务显著增加了假等式判断的错误率,但没有显著增加真等式判断的错误率。这说明,语音环路、视空间模板和反应选择成分参与简单指数乘法等式判断,而且真假等式判断具有不同的注意资源需求。     

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.     

Mental addition in third,fourth, and sixth graders

Research on mental arithmetic has suggested that young children use a counting algorithm for simple mental addition, but that adults use memory retrieval from an organized representation of addition facts. To determine the age at which performance shifts from counting to retrieval, children in grades 3, 4, and 6 were tested in a true/false verification task. Reaction time patterns suggested that third grade is a transitional age with respect to memory structure for addition—half of these children seemed to be counting and half retrieving from memory. Results from fourth and sixth graders implicated retrieval quite strongly, as their results resembled adult RTs very closely. Fourth graders' processing, however, was easily disrupted when false problems were presented. The third graders' difficulties are not due to an inability to form mental representations of number; all three grades demonstrated a strong split effect, indicative of a simpler mental representation of numerical information than is necessary for addition. The results were discussed in the context of memory retrieval versus counting models of mental arithmetic, and the increase across age in automaticity of retrieval processes.     

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.     

What affects strategy selection in arithmetic? The example of parity and five effects on product verification

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.     

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.     

Automatization through Practice: The Opportunistic-Stopping Phenomenon Called into Question

As a theory of skill acquisition, the instance theory of automatization posits that, after a period of training, algorithm-based performance is replaced by retrieval-based performance. This theory has been tested using alphabet-arithmetic verification tasks (e.g., is A + 4  = E?), in which the equations are necessarily solved by counting at the beginning of practice but can be solved by memory retrieval after practice. A way to infer individuals’ strategies in this task was supposedly provided by the opportunistic-stopping phenomenon, according to which, if individuals use counting, they can take the opportunity to stop counting when a false equation associated with a letter preceding the true answer has to be verified (e.g., A + 4  = D). In this case, such within-count equations would be rejected faster than false equations associated with letters following the true answers (e.g., A + 4  = F, i.e., outside-of-count equations). Conversely, the absence of opportunistic stopping would be the sign of retrieval. However, through a training experiment involving 19 adults, we show that opportunistic stopping is not a phenomenon that can be observed in the context of an alphabet-arithmetic verification task. Moreover, we provide an explanation of how and why it was wrongly inferred in the past. These results and conclusions have important implications for learning theories because they demonstrate that a shift from counting to retrieval over training cannot be deduced from verification time differences between outside and within-count equations in an alphabet-arithmetic task.     

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.     

Priming effects of arithmetic signs in 10- to 15-year-old children

In this research, 10- to 12- and 13- to 15-year-old children were presented with very simple addition and multiplication problems involving operands from 1 to 4. Critically, the arithmetic sign was presented before the operands in half of the trials, whereas it was presented at the same time as the operands in the other half. Our results indicate that presenting the ‘x’ sign before the operands of a multiplication problem does not speed up the solving process, irrespective of the age of children. In contrast, presenting the ‘+’ sign before the operands of an addition problem facilitates the solving process, but only in 13 to 15-year-old children. Such priming effects of the arithmetic sign have been previously interpreted as the result of a pre-activation of an automated counting procedure, which can be applied as soon as the operands are presented. Therefore, our results echo previous conclusions of the literature that simple additions but not multiplications can be solved by fast counting procedures. More importantly, we show here that these procedures are possibly convoked automatically by children after the age of 13 years. At a more theoretical level, our results do not support the theory that simple additions are solved through retrieval of the answers from long-term memory by experts. Rather, the development of expertise for mental addition would consist in an acceleration of procedures until automatization.     

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.     

“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.     

Syntactic presupposition in sentence comprehension

Two experiments investigated the role of syntactic presupposition in sentence comprehension. In Experiment I subjects verified cleft, pseudocleft and factive complement sentences with respect to preceding context paragraphs, which contradicted either the assertion or the presupposition of the target sentence. Subjects took significantly longer to verify sentences with false presuppositions than sentences with false assertions. In Experiment II subjects verified cleft and pseudocleft sentences with respect to subsequently presented pictures. Once again, verification times for sentences with false presuppositions were significantly longer than verification times for sentences with false assertions. It was argued that these findings are more adequately explained by a “structural” hypothesis, than in terms of strategies designed to locate given and new information.     

An investigation of false memory in perceptual implicit tasks

Reports of critical lure priming in perceptual implicit tasks [e.g., McKone, E., & Murphy, B. (2000). Implicit false memory: Effects of modality and multiple study presentations on long-lived semantic priming. Journal of Memory and Language, 43, 89-109] using the Deese-Roediger-McDermott [Roediger, H. L., III, & McDermott, K. B. (1995). Creating false memories: Remembering words not presented in lists. Journal of Experimental Psychology: Learning, Memory, and Cognition, 21, 803-814] procedure have suggested availability of the lexical form of lure items at study. Three experiments were conducted to further explore "false" implicit priming in perceptual tests. In Experiments 1 and 3, implicit and explicit stem completion tests were given in the DRM procedure with semantic lists; in Experiment 2, a graphemic response test was used in a similar design. For all experiments, explicit instructions resulted in reliable false memory, while implicit instructions resulted in priming for list items and no priming for lure items. Priming for lure items was evident for "test-aware" subjects only in Experiment 1 and in a combined analysis for all three experiments. These results establish boundary conditions for priming for critical lures and indicate that access to the lexical form of critical lures may not occur under incidental learning conditions when strong controls against explicit retrieval are implemented.     

工作记忆在简单整式和判断中的作用

选择20名大学二年级数学系的学生作为被试,要求他们完成控制条件、语音任务、手动击键任务和随机间隔决策任务条件下的简单整式和等式判断任务,实验结果表明:(1)三种次级任务均非常显著影响真假等式的反应时,(2)三种次级任务对真假不等式判断的错误率产生了不同程度的影响。这说明,语音环路、视空间模板和反应选择成分参与简单整式和判断,而且真假等式判断需要不同的注意资源。     

Dissociations between word priming effects in normal subjects and patients with memory disorders: Multiple memory systems or retrieval?

Repetition priming was measured in two different tasks within a single experiment--one in which subjects named briefly (tachistoscopically) presented words, and one requiring naming of visually fragmented/degraded words. Thirteen amnesic patients, 12 patients with Alzheimer's disease (AD), and 15 normal control subjects were tested under 4 experimental conditions involving the factorial combination of two variables: delay of test (10 minutes and 24 hours), and number of prior occurrences of the primed items (1 and 4). The two tasks produced very different patterns of priming effects, despite the fact that common study phases were employed. In one task the priming effect showed no decay and virtually no effect of the number of prior occurrences of the primed items, whereas both these variables affected priming in the other task. The AD patients evidenced impaired priming in both tasks. However, in the degraded-word-naming task the deficit was only apparent under some experimental conditions. The amnesics produced priming effects that in absolute terms were similar to those produced by control subjects. However, when group differences in overall performance level were taken into account in the tachistoscopic task, these patients also showed clear evidence of impaired priming. It is argued that the complex pattern of priming effects obtained is best explained by the characteristics of the retrieval cues provided in the tasks, and, generally, that such characteristics may determine whether or not experimental variables will affect measured priming.