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.     

Groups perform better than the best individuals on letters-to-numbers problems: informative equations and effective strategies

One-hundred 3-person groups and 300 individuals solved 2 letters-to-numbers problems, requiring identification of the coding of 10 letters to 10 numbers by proposing an equation in letters, receiving the answer in letters, proposing a hypothesis, and receiving feedback on the hypothesis on each trial. There were 5 instruction conditions: (a). standard, (b). use at least 3 letters on all equations, (c). use at least 4 letters on all equations, (d). number 1 known before beginning problem, and (e). number 9 known before beginning problem. The groups had fewer trials to solution, proposed more complex equations, and identified more letters per equation than the best individuals. Performance was best under instructions to use at least 4 letters and with the number 9 known.     

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.     

Network interference and number-fact retrieval: evidence from children's alphaplication.

The origins of several phenomena of number-fact retrieval were investigated by having children in Grades 3 and 4 memorize alphaplication facts (arithmetic-like memory items composed of letters instead of numbers). Alphaplication performance paralleled memory for arithmetic facts in several important respects: Results showed (a) a large performance advantage for tie (e.g., E, E = j) over nontie problems (E, I = p), (b) that most errors involved answers from the correct alpha-table, (c) that response times and error rates were strongly correlated across problems, (d) that the correct answers to poorly learned problems tended to be the most common error responses, and (e) that performance was lower for problems introduced later in the learning sequence. Taken together, these findings support a network-interference approach (Campbell & Graham, 1985) to memory for arithmetic facts.     

Working memory demands of exact and approximate addition

We compared the working memory requirements of two forms of mental addition: exact calculation (e.g., 63 + 49 = 112) and approximation (e.g., 63 + 49 is about 110). In two experiments, participants solved two-digit addition problems (e.g., 63 + 49) alone and in combination with a working memory task (i.e., remembering four consonants). In Experiment 1, participants chose an answer from two alternatives (e.g., exact: 112 vs. 122; approximate: 110 vs. 140). In Experiment 2, participants responded verbally with exact or approximate answers. In both experiments, the working memory load impaired exact and approximate addition performance, but exact addition was affected more. Load also impaired performance on problems with a carry operation in the units (e.g., 28 + 59 or 76 + 57) more than on problems without a unit carry (e.g., 24 + 53 or 76 + 52). These results identify the carry operation as the source of the working memory demands in multidigit addition.     

Cultural Influences on Number Preferences: Christmas and Grading Systems

People consistently prefer numbers associated with themselves (e.g., birth dates) over other numbers. We argue that such number preferences are also shaped by cultural influences, such as customs regarding the day on which Christmas is celebrated and customs regarding ranking of numerals used in national school’s grading system. Across 6 different countries (N = 657), the day on which Christmas is celebrated had an influence on the preference for the numbers 24 vs. 25, and numbers used for better grades were uniformly rated better than numbers used for poorer grades. If grading did not involve numbers (e.g., letters A to F), then no effect appeared.

     

Groups perform better than the best individuals on letters-to-numbers problems: effects of group size

Individuals and groups of 2, 3, 4, or 5 people solved 2 letters-to-numbers problems that required participants, on each trial, to identify the coding of 10 letters to 10 numbers by proposing an equation in letters, receiving the answer in letters, proposing a hypothesis, and receiving feedback on the correctness of the hypothesis. Groups of 3, 4, and 5 people proposed more complex equations and had fewer trials to solution than the best of an equivalent number of individuals. Groups of 3, 4, and 5 people had fewer trials to solution than 2-person groups but did not differ from each other. These results suggest that 3-person groups are necessary and sufficient to perform better than the best individuals on highly intellective problems.     

Visual similarity effects on masked priming

We investigated the role of the visual similarity of masked primes to targets in a lexical decision experiment. In the primes, some letters in the target (e.g., A in ABANDON) had either visually similar letters (e.g., H), dissimilar letters (D), visually similar digits (4), or dissimilar digits (6) substituted for them. The similarities of the digits and letters to the base letter were equated and verified in a two-alternative forced choice (2AFC) perceptual identification task. Using targets presented in lowercase (e.g., abandon) and primes presented in uppercase, visually similar digit primes (e.g., 484NDON) produced more priming than did visually dissimilar digit primes (676NDON), but little difference was found between the visually similar and dissimilar letter primes (HRHNDON vs. DWDNDON). These results were explained in terms of task-driven competition between the target letter and the visually similar letter.     

目标与非目标身份特征的范畴间差异对多身份追踪的促进作用

以数字和字母为实验材料, 考察了多身份追踪任务中目标与非目标的身份特征范畴性差异对目标追踪与身份识别的影响, 并探讨了其可能的认知加工机制。实验1A让被试追踪5个目标, 并操作目标与非目标的身份特征从无差异(目标与非目标为相同的数字)、范畴内差异(目标与非目标为各不相同的数字)到范畴间差异(目标为各不相同的数字, 非目标为各不相同的字母)变化, 发现随着目标与非目标差异水平的提高, 追踪正确率也显著提高。实验1B通过变化追踪目标数量为4和5操作追踪负荷, 发现中等负荷条件下(目标数量为5)实验结果与实验1A相同, 即目标与非目标身份特征的范畴内差异和范畴间差异均可促进追踪表现。实验2考察了目标与非目标身份特征的范畴间差异水平对追踪表现与身份识别的影响。结果发现随着范畴间差异水平的提高, 被试的追踪表现与身份识别表现都得到了显著提高, 但身份识别表现低于追踪表现。对可能解释以上现象的工作记忆理论、分组理论以及多身份追踪模型进行了分析讨论。     

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.     

Hedonic and instrumental motives in anger regulation

What motivates individuals to regulate their emotions? One answer, which has been highlighted in emotion-regulation research, is that individuals are motivated by short-term hedonic goals (e.g., the motivation to feel pleasure). Another answer, however, is that individuals are motivated by instrumental goals (e.g., the motivation to perform certain behaviors). We suggest that both answers have merit. To demonstrate the role instrumental goals may play in emotion regulation, we pitted short-term hedonic motives and instrumental motives against each other, by testing whether individuals were motivated to experience a potentially useful, albeit unpleasant, emotion. We found that (a) individuals preferred activities that would increase their level of anger (but not their level of excitement) when they were anticipating confrontational, but not nonconfrontational, tasks and that (b) anger improved performance in a confrontational, but not a nonconfrontational, task. These findings support a functional view of emotion regulation, and demonstrate that in certain contexts, individuals may choose to experience emotions that are instrumental, despite short-term hedonic costs.     

Masked transposition effects for simple versus complex nonalphanumeric objects

When two letters/digits/symbols are switched in a string (e.g., jugde–judge; 1492–1942; *?When two letters/digits/symbols are switched in a string (e.g., jugde-judge; 1492-1942; *?$&-*$?&), the resulting strings are perceptually similar to each other and produce a sizable masked transposition priming effect with the masked priming same-different matching task. However, a parallel effect does not occur for strings of pseudoletters (e.g., [letters in the text]); García-Orza, Perea, & Mu?oz, Quarterly Journal of Experimental Psychology, 63, 1603-1618, 2010). In the present study, we examined whether masked transposition priming is specific to alphanumeric stimuli or whether it also occurs with strings composed of other "objects"-namely, line drawings of common objects (Experiment 1) and geometrical shapes (Experiment 2). Results showed a significant masked transposition priming effect for geometrical shapes (e.g., [geometrical shapes in the text]), but not for line drawings of common objects (e.g., [symbols in the text]). These findings suggest that the mechanism involved in the coding of position in masked priming works only with perceptually simple, familiar "objects" (i.e., letters, numbers, symbols, or geometrical shapes), once their identities have been well ascertained.     

Middle-School Students' Understanding of the Equal Sign: The Books They Read Can't Help

This study examined how 4 middle school textbook series (2 skills-based, 2 Standards-based) present equal signs. Equal signs were often presented in standard operations equals answer contexts (e.g., 3 + 4 = 7) and were rarely presented in nonstandard operations on both sides contexts (e.g., 3 + 4 = 5 + 2). They were, however, presented in other nonstandard contexts (e.g., 7 = 7). Two follow-up experiments showed that students' interpretations of the equal sign depend on the context. The other nonstandard contexts were better than the operations equals answer context at eliciting a relational understanding of the equal sign, but the operations on both sides context was best. Results suggest that textbooks rarely present equal signs in contexts most likely to elicit a relational interpretation—an interpretation critical to success in algebra.     

Words, hemispheres, and dissociable subsystems: the effects of exposure duration, case alternation, priming, and continuity of form on word recognition in the left and right visual fields

Three experiments explore aspects of the dissociable neural subsystems theory of hemispheric specialisation proposed by Marsolek and colleagues, and in particular a study by [Deason, R. G., & Marsolek, C. J. (2005). A critical boundary to the left-hemisphere advantage in word processing. Brain and Language, 92, 251–261]. Experiment 1A showed that shorter exposure durations for lower-case words (13 ms) are associated with reduced right visual field (RVF) advantages compared with longer exposure durations (144 ms). Experiment 1B compared report accuracy for lower case and mixed case words at the same exposure duration (144 ms). The RVF advantage was reduced for mixed case words due to case alternation having more of an adverse effect in the RVF than in the LVF. Experiment 2 tested a different prediction of dissociable neural subsystems theory. Four-letter words were presented in mixed case in the LVF or RVF for 100 ms. They were preceded at the same location by a prime which could be in the same word in the same alternation pattern (e.g., FlAg–FlAg), the same word in the opposite alternation pattern (e.g., fLaG–FlAg), or an unrelated letter string in the same or opposite case alternation pattern (WoPk–FlAg or wOpK–FlAg). Relative to performance in the letter string prime conditions, which did not differ significantly between the two visual fields, there was more of an effect of word primes in the RVF than in the LVF. Importantly, the benefit of a word prime was the same whether the prime was in the same alternation pattern or was in the opposition alternation pattern. We argue that these results run contrary to the predictions of dissociable neural subsystems theory and are more compatible with theories which propose that a left hemisphere word recognition system is responsible for identifying written words, whether they are presented in the LVF or the RVF, and that letters are processed to an abstract graphemic level of representation before being identified by that system.     

Situation Trait Relevance, Trait Expression, and Cross-Situational Consistency: Testing a Principle of Trait Activation

An interactionist principle of trait activation is proposed, emphasizing situation trait relevance (i.e., opportunity for trait expression) as a moderator of trait–behavior relations and cross-situational consistency (CSC). One hundred fifty-six students completed trait measures and expressed intentions in 10 scenarios targeted to each of five traits (e.g., risk taking). Trait–intention correlations within scenario sets were themselves correlated with mean situation trait relevance ratings provided by 26 proficient judges; CSCs in intentions (45 correlations per trait) were correlated with an index of shared trait relevance in situation pairs. In support of trait activation, (a) trait–intention relations for three traits were higher in more relevant situations (e.g., second-order r = .66 for risk taking) and (b) CSCs were higher in scenarios jointly high in targeted trait relevance (e.g., second-order r = .55 for risk taking). Discussion highlights applications of trait activation in diverse research domains.     

Efficiency of visual selection in duplex and conjunction conditions in partial report

Factors that determine the efficiency of visual selection were investigated in two precued partial-report experiments. There were four selection conditions: acolor condition (e.g., selecting purple letters among blue letters), aclass condition (e.g., selecting letters among digits), aduplex condition (e.g., selecting purple letters among blue digits; that is, the targets differed from the distractors both in color and in alphanumeric class), and aconjunction condition (e.g., selecting purple letters among blue letters and purple digits; that is, the targets differed from the distractors either in color or in alphanumeric class). The efficiency of visual selection was highest in the duplex condition and lowest in the conjunction condition. Thedifference in performance across the different selection conditions within subjects was accounted for by variation in the value of a single parameter of the fixed-capacity independent race model (FIRM) proposed by Shibuya and Bundesen (1988), which assumes independent parallel processing, limitations in both processing capacity and storage capacity, and time-invariant selectivity. By assuming that the factors that determine the efficiency of selection in the color and class conditions also underlie the efficiency of selection in the duplex and conjunction conditions, an extension of FIRM, FIRM⁺, accounted for therelationship with respect to efficiency of selection among the different conditions.     

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).     

Transitive inference in pigeons: Control for differential value transfer

Transitive inference (TI) effects have been demonstrated in several nonhuman species using a nonverbal version of the task in which A is better than B is represented in a simple simultaneous discrimination, A+B?. Following five-term training (e.g., A+B?, B+C?, C+D?, D+E?), the choice of B over D on test trials is taken as evidence of TI. Recently, differential value transfer from the S+ to the S? in a simultaneous discrimination has been proposed as a noncognitive basis for these effects. Two experiments were conducted to control for differential value transfer. The results suggest that differential value transfer is not the only basis for nonhuman TI performance. An alternative account based on spatial mapping is discussed.     

Are transposition effects specific to letters?

Recent research has consistently shown that pseudowords created by transposing two letters are perceptually similar to their corresponding base words (e.g., jugde–judge). In the framework of the overlap model (Gomez, Ratcliff, & Perea, 2008), this effect is due to a noisy process in the localization of the “objects” (e.g., letters, kana syllables). In the present study, we examine whether this effect is specific to letter strings or whether it also occurs with other “objects” (namely, digits, symbols, and pseudoletters). To that end, we conducted a series of five masked priming experiments using the same–different task. Results showed robust effects of transposition for all objects, except for pseudoletters. This is consistent with the view that locations of familiar objects (i.e., letters, numbers, and symbols) can be best understood as distributions along a dimension rather than as precise points.     

The Development of Procedural and Conceptual Knowledge in Computational Estimation

Children in Grades 4, 6, and 8 and adults estimated answers to multiplication problems. The problems varied in the number of digits: 1 x 2 (e.g., 8 x 18), 1 X 3 (e.g., 5 x 144), 2 x 2 (e.g., 22 x 91), and 2 x 3 (e.g., 45 x 164). Few children in Grade 4 could estimate. Most sixth and eighth graders provided reasonable estimates, however, even on difficult (e.g., 2 x 3) problems. Estimation performance improved with age, with adults producing more accurate estimates than children, but the most striking developmental changes were in the conceptual knowledge used to perform this estimation task. From Grade 6, students seemed to understand the role of the simplification principle in estimation. Children reduced complex problems through rounding and prior compensation to produce reasonable estimates. Only adults seemed to have a good grasp of the principle of proximity, however, understanding that it is important for the estimate to be reasonably close to the actual answer. Adults produced exact-answer solutions on simple problems and adjusted their preliminary estimates closer to the actual answer postcompensation). We propose a process model of estimation based on Siegler's model of strategy choice in simple arithmetic.