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.     

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.     

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.     

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.     

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.     

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.     

Why 2×2=5 looks so wrong: On the odd-even rule in product verification

Subjects judged whether the proposed product for two multipliers was true or false. Each equation could be judged as plausible or implausible because a product must be even if any of its multipliers is even; otherwise, it must be odd. A proposed product that violates the required odd-even status of the product—that is, deviates from the correct product, whether odd or even, by an odd value (e.g., splits of ±l, ±3, ±5)—can be rejected as false before normal processing is completed (i.e., before the correct product is retrieved and compared with the proposed product). Subjects were indeed faster and more accurate in rejecting a split of +1 or +3 than a split of +2 or +4, and this effect increased as the number of even multipliers in the pair increased. Subjects did not use the odd-even rule when either multiplier was 0 or 1 (Experiment 2), perhaps because other rules are available to bypass normal processing in those cases. A similar odd-even rule is used in sum verification (Krueger &Hallford), and use of the odd-even rules may help to explain why oddness and evenness are such salient features of numbers as abstract concepts (Shepard, Kilpatric, & Cunningham).     

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。     

Verifying simple arithmetic sums and products: Are the phonological loop and the central executive involved?

In two experiments, we investigated the role of the phonological loop and the central executive in the verification of the complete set of one-digit addition (Experiment 1) and multiplication (Experiment 2) problems. The focus of the present study was on the contradictory results concerning the contribution of the phonological loop in the verification of true problems (e.g., 8 + 4 = 12 or 4 x 6 = 24) reported until now. The results revealed that this slave system is not involved in verifying simple arithmetic problems, in contrast to the central executive. Furthermore, our results indicated that the split effect is due to the use of two different arithmetic strategies.     

Synesthesia Affects Verification of Simple Arithmetic Equations

To investigate the effects of color–digit synesthesia on numerical representation, we presented a synesthete, called SE, in the present study, and controls with mathematical equations for verification. In Experiment 1, SE verified addition equations made up of digits that either matched or mismatched her color–digit photisms or were in black. In Experiment 2A, the addends were presented in the different color conditions and the solution was presented in black, whereas in Experiment 2B the addends were presented in black and the solutions were presented in the different color conditions. In Experiment 3, multiplication and division equations were presented in the same color conditions as in Experiment 1. SE responded significantly faster to equations that matched her photisms than to those that did not; controls did not show this effect. These results suggest that photisms influence the processing of digits in arithmetic verification, replicating and extending previous findings.     

Effects of cell-phone and text-message distractions on true and false recognition

This study examined the effects of electronic communication distractions, including cell-phone and texting demands, on true and false recognition, specifically semantically related words presented and not presented on a computer screen. Participants were presented with 24 Deese-Roediger-McDermott (DRM) lists while manipulating the concurrent presence or absence of cell-phone and text-message distractions during study. In the DRM paradigm, participants study lists of semantically related words (e.g., mother, crib, and diaper) linked to a non-presented critical lure (e.g., baby). After studying the lists of words, participants are then requested to recall or recognize previously presented words. Participants often not only demonstrate high remembrance for presented words (true memory: crib), but also recollection for non-presented words (false memory: baby). In the present study, true memory was highest when participants were not presented with any distraction tasks during study of DRM words, but poorer when they were required to complete a cell-phone conversation or text-message task during study. False recognition measures did not statistically vary across distraction conditions. Signal detection analyses showed that participants better discriminated true targets (list items presented during study) from true target controls (items presented during study only) when cell-phone or text-message distractions were absent than when they were present. Response bias did not vary significantly across distraction conditions, as there were no differences in the likelihood that a participant would claim an item as "old" (previously presented) rather than "new" (not previously presented). Results of this study are examined with respect to both activation monitoring and fuzzy trace theories.     

Perceptual processing of nontargets in an attention task

The capability of nontargets to qualitatively influence the semantic processing of coincident targets was investigated in three experiments. Subjects were aurally presented a series of word pairs and attempted to detect homonymic instances of a predesignated category (e.g., animals). The nontarget with which a target (e.g., ANT) was paired was appropriate (e.g., CRAWLING), inappropriate (e.g., UNCLE), or neutral (e.g., STRAW). Experiments 1 and 2 established that detection of targets can be facilitated by appropriate nontargets and inhibited by inappropriate ones. Thus, nontargets can influence the way in which targets are semantically represented. Experiment 3 showed that this effect is eliminated when subjects are precued as to the ear of entry of targets. Thus, precuing appears to curtail the perceptual processing of nontargets. The data run counter to theories that claim that focused attention does not entail the perceptual suppression of nontargets.     

The effects of realism on Wason's THOG problem

Summary In this series of experiments the effects of phrasing Wason's THOG problem in realistic terms were investigated. Experiments 1 and 2 used realistic materials of very different kinds, but neither version of the problem produced any facilitation compared with the original abstract version. Experiment 3 used a version of the problem in which the correct answer was cued in by the realistic material, and a significant improvement was found. Experiment 4 used a version of the problem similar to that used in Experiment 3 and again improved performance was found; since the subjects in this experiment were eight- and nine-year-old children, the facilitation almost certainly was not the result of improved logical ability. The results of Experiment 5, however, suggested that it was difficult to cue in adults to give the logically incorrect answer. It is concluded that realism improves performance on this problem only when the realistic material cues in the correct answer from memory. A review of the research that makes use of other reasoning paradigms suggests that this conclusion may hold true for these as well.     

The process of verifying affirmative and negative sentences against pictures

Experiments 1 and 2 investigated sentence context effects on picture encoding in a sentence-picture verification and a picture naming task, respectively. The results showed that a picture following a sentence was encoded faster when the two were congruent than when they were incongruent. Experiment 3 compared two conditions: Under one condition, true affirmatives, false affirmatives, true negatives, and false negatives were mixed in each block of presentation. Under the other condition, different sentence types were presented in different blocks. The results showed (l) that errors committed in verification were largely negation errors, but seldom falsification errors, and (2) that there was a decrease of falsification time, but a resistance to change in negation time from the mixed to the blocked presentation. These results were interpreted to mean that falsification time results from a longer time to encode the picture and to confirm or disconfirm the truth index value by the polarity of the sentence in false affirmatives and true negatives than in true affirmatives and false negatives, whereas negation time and negation errors result from a response-suppression operation.     

教条悖论：行为心理学家的新挑战

在各种专业或私人情境中, 人们经常需要整合不同的意见来判断某一观点的对错。当观点的形式类似于将多个论点组合而成的逻辑公式(使用“且”、“或”等逻辑连接词)时, 容易产生教条悖论(doctrinal paradox)。即, 虽然整体意见支持该观点是正确的(或错误的), 但是分析这些整体意见中所包含的各个论点, 却会得到相反的结论。教条悖论是判断整合研究中关注的重要问题, 已在各科学领域引发大量的规范性研究。行为心理学家需在这个重要问题上开展系统研究。本文简要介绍了教条悖论及其过往研究, 总结已有的行为数据, 并指出未来行为研究的方向和视角。     

Explorations in the social contagion of memory

Four experiments examined social influence on the development of false memories. We employed the social contagion paradigm: A subject and a confederate see scenes and then later take turns recalling items from the scenes, with the confederate erroneously reporting some items that were not present in the scenes; on a final test, the subject reports these suggested items when instructed to recall only items from the scenes. The first two experiments showed that the social contagion effect persisted when subjects were explicitly warned about the possibility that confederates' responses might induce false memories and when they were tested via source-monitoring tests that explicitly gave the choice of attributing suggested items to the other person. Levels of false recall and recognition increased with the number of times the misleading information was suggested (Experiment 3), and subjects were more likely to incorporate the erroneous responses of an actual confederate on a recognition/source test as compared with those of a simulated confederate (Experiment 4). Collectively, the data support the claim that false memories may be transmitted between people and reveal critical factors that modulate the social contagion of memories.     

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.     

On the Mental Representation of Conditional Sentences

Four experiments are reported which attempt to externalize subjects' mental representation of conditional sentences, using novel research methods. In Experiment 1, subjects were shown arrays of coloured shapes and asked to rate the degree to which they appeared to be true of conditional statements such as 'If the figure is green then it is a triangle'. The arrays contained different distributions of the four logically possible cases in which the antecedent or consequent is true or false: TT, TF, FT, and FF. For example, a blue triangle would be FT for the conditional quoted above. In Experiments 2 to 4, subjects were able to construct their own arrays to make conditionals either true or false with any distribution of the four cases they wished to choose. The presence and absence of negative components was varied, as was the form of the conditional, being either 'if then' as above or 'only if': 'The figure is green only if it is a triangle'. The first finding was that subjects represent conditionals in fuzzy way: conditionals that include some counter-example TF cases (Experiment 1) may be rated as true, and such cases are often included when subjects construct an array to make the rule true (Experiments 2 to 4). Other findings included a strong tendency to include psychologically irrelevant FT and FF cases in constructed arrays, presumably to show that conditional statements only apply some of the time. A tendency to construct cases in line with the 'matching bias' reported on analogous tasks in the literature was found, but only in Experiment 4, where the number of symbols available to construct each case was controlled. The findings are discussed in relation to the major contemporary theories of conditional reasoning based upon inference rules and mental models, neither of which can account for all the results.     

Inferences from disclosures about the truth and falsity of expert testimony

Participants acting as mock jurors made inferences about whether a person was a suspect in a murder based on an expert's testimony about the presence of objects at the crime scene and the disclosure that the testimony was true or false. Experiment 1 showed that participants made more correct inferences, and made inferences more quickly, when the truth or falsity of the expert's testimony was disclosed immediately after the testimony rather than when the disclosure was delayed. Experiment 2 showed no advantage for prior disclosure over immediate disclosure. Experiment 3 showed that the pattern of inferences when there was no disclosure mirrored the pattern when it was disclosed that the expert's testimony was true rather than false. Participants made more correct inferences from true conjunctions than disjunctions, and from false disjunctions than conjunctions. We discuss the implications for theories of the mental representations and cognitive processes that underlie human reasoning.