How infants process addition and subtraction events

Three experiments are described that assess 5‐month‐old infants’ processing of addition and subtraction events similar to those reported by Wynn (1992a ). In Experiment 1, prior to each test trial, one group of infants was shown an addition event (1 + 1) while another group was shown a subtraction event (2 ? 1). On test trials, all infants were shown outcomes of 0, 1, 2 and 3. The results seemed to require one of two dual‐process models. One such model assumed that the infants could add and subtract but also had a tendency to look longer when more items were on the stage. The other model assumed that infants had a preference for familiarity along with the tendency to look longer when more items were on the stage. Experiments 2 and 3 examined the assumptions made by these two models. In Experiment 2, infants were given only the test trials they had received in Experiment 1. Thus, no addition and subtraction or familiarity was involved. In Experiment 3 infants were familiarized to either one or two items prior to each test trial, but experienced no actual addition or subtraction. The results of these two experiments support the familiarity plus more items to look at model more than the addition and subtraction plus more items to look at model. Taken together, these three experiments shed doubt on Wynn’s (1992a ) assertion that 5‐month‐old infants can add and subtract. Instead they indicate the importance of familiarity preferences and the fact that one should be cautious before assuming that young infants have sophisticated numerical abilities.     

MICROANALYSIS OF CONCRETE REASONING

Subjects with median age 5–11 were given 24 triads of elementary + and - operations with one unit (AS items), besides pre- and posttests of conservation of quantity. A strong initial resistance to use the 'equal' category was observed. A considerable proportion of the subjects acquired conservation. The difficulty of the AS items seemed to be mainly determined by the number of required intellectual transformations and non-redundant elements to be processed. Degree of perceptual support and codability of element combinations also appeared to be likely determinants.     

THE ACQUISITION OF CONSERVATION OF SUBSTANCE AND WEIGHT IN CHILDREN: II.External reinforcement of conservation of weight and of the operations of addition and subtraction

S medslund , J. The acquisition of conservation of substance and weight in children. II. External reinforcement of conservation of weight and of the operations of addition and subtraction. Scand. J. Psychol ., 1961, 2 , 71–84. —The subjects were 5–7 years old. Group I was allowed to observe empirical conservation of weight on a balance. Group II practised on additions and subtractions of material with controls on the scales. Group III took only the pre- and posttests. The data show no appreciable differences in frequency of acquisition of conservation between the three groups. It is concluded that none of the experimental conditions were necessary or sufficient for the acquisition of principles of conservation. The hypothesis is advanced that cognitive conflict may be the crucial factor involved.     

MICROANALYSIS OF CONCRETE REASONING II.

Groups of five-year-old subjects were given some items involving elementary combinations of addition and subtraction of one unit. Predictions from the following two hypotheses were tested and confirmed. (1) Difficulty is directly related to the number of transformations involved. (2) Difficulty is directly related to the number of non-redundant elements involved. Some variations in the exact phrasing of the questions were shown to have no appreciable effects.     

Do young children grasp the inverse relationship between addition and subtraction?: Evidence against early arithmetic

This research investigates young children’s reasoning about the inverse relationship between addition and subtraction. We argue that this investigation is necessary before asserting that preschoolers have a full understanding of addition and subtraction and use arithmetic principles. From the current models of quantification in infancy, we also propose that the children’s earliest ability to add and subtract is based on representations combining and separating sets of objects without arithmetical operations. In an initial study, 2- to 5-year-old children was tested on addition (2+1), subtraction (3−1) and inversion problems (2+1−1) by using Wynn’s procedure (1992b) of possible and impossible events. Only the oldest age group (4–5 years) succeeded on the inverse problem. In a follow-up study, 3- to 4-year-old children were given a brief training intervention in which they performed adding and subtracting transformations by manipulating small sets of objects without counting. The beneficial effects of the training support the claim that preschoolers respond to the inverse problem on the basis of object representations and not on the basis of numerical representations.     

Children's understanding of the relation between addition and subtraction: inversion, identity, and decomposition.

In order to understand addition and subtraction fully, children have to know about the relation between these two operations. We looked at this knowledge in two studies. In one we asked whether 5- and 6-year-old children understand that addition and subtraction cancel each other out and whether this understanding is based on the identity of the addend and subtrahend or on their quantity. We showed that children at this age use the inversion principle even when the addend and subtrahend are the same in quantity but involve different material. In our second study we showed that 6- to 8-year-old children also use the inversion in combination with decomposition to solve a + b - (b + 1) problems. In both studies, factor analyses suggested that the children were using different strategies in the control problems, which require computation, than in the inversion problems, which do not. We conclude that young children understand the relations between addition and subtraction and that this understanding may not be based on their computational skills.     

Adults' understanding of inversion concepts: how does performance on addition and subtraction inversion problems compare to performance on multiplication and division inversion problems?

Problems of the form a + b - b have been used to assess conceptual understanding of the relationship between addition and subtraction. No study has investigated the same relationship between multiplication and division on problems of the form d x e / e. In both types of inversion problems, no calculation is required if the inverse relationship between the operations is understood. Adult participants solved addition/subtraction and multiplication/division inversion (e.g., 9 x 22 / 22) and standard (e.g., 2 + 27 - 28) problems. Participants started to use the inversion strategy earlier and more frequently on addition/subtraction problems. Participants took longer to solve both types of multiplication/division problems. Overall, conceptual understanding of the relationship between multiplication and division was not as strong as that between addition and subtraction. One explanation for this difference in performance is that the operation of division is more weakly represented and understood than the other operations and that this weakness affects performance on problems of the form d x e / e.     

Identical elements model of arithmetic memory: Extension to addition and subtraction

The identical elements (IE) model of arithmetic fact representation (Rickard, 2005; Rickard & Bourne, 1996) was developed and tested with multiplication and division. In Experiment 1, we demonstrated that the model also applies to addition and subtraction by examining transfer of response time (RT) savings between prime and probe problems tested in the same block of trials. As is predicted by the IE model, there were equivalent probe RT savings for addition with identical repetition (prime 6 + 9 --> probe 6 + 9) or an order change (9 + 6 --> 6 + 9), but much greater savings for subtraction with identical repetition (15 - 6 --> 15 - 6) than with an order change (15 - 9 --> 15 - 6), and no savings with an operation change (15 - 9 --> 6 + 9 or 6 + 9 --> 4 15 - 6). In Experiment 2, we examined transfer in simple multiplication and division and demonstrated symmetrical transfer between operations. Cross-operation RT savings were eliminated, however, when the RT analysis included only trials on which both the prime and the probe problems were reportedly solved by direct retrieval. An IE model extended to accommodate savings associated with procedural strategies provides a coherent account of facilitative transfer effects in simple arithmetic.     

Intention superiority effect: A context-switching account

Intention superiority effect [J. Exp. Psychol. Learn. Mem. Cogn. 19 (1993) 1211; J. Exp. Psychol. Learn. Mem. Cogn. 24 (1998) 350] is the finding that the times to retrieve memory items related to uncompleted or partially completed intentions are faster than for those with no associated intentions. However, this relationship reverses when the intended tasks are completed [J. Exp. Psychol. Learn. Mem. Cogn. 24 (1998) 350; Mem. Cogn. 27 (1999) 320]. That is, the times to retrieve memory items related to completed intentions are slower than for those with no associated intentions. In this paper, we present a computational account of the intention superiority effect using the ACT-R cognitive architecture [Atomic components of thought. Mahwah, NJ: Lawrence Erlbaum Associates, 1998]. Our modeling approach is based on the idea that uncompleted or partially completed intentions are available as context in the current goal, and they prime related memory items while inhibiting unrelated memory items. However, once the intended tasks are completed, they are removed from the current goal, which produces an inhibitory effect on memory items associated with them. We describe an ACT-R model that is able to reproduce all of the effects reported in Marsh et al. [J. Exp. Psychol. Learn. Mem. Cogn. 24 (1998) 350].     

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.     

A reappraisal of the relation between performances of quantitative identity and quantitative equivalence conservation tasks

Previous studies of children's performance on quantitative identity and equivalence conservation problems in the number domain are suggested to be flawed because of the low numbers of items in the arrays. In Experiment 1, children were tested on three different number versions of identity and equivalence conservation tasks. Their performance varied with the number of items and no child was found to pass a large number version of an equivalence conservation task and fail the corresponding version of an identity conservation task. In Experiment 2 the notion that the contextual effect of asking two similar questions might underlie the difference in performance on the two tasks was tested and rejected. The results are discussed in relation to recent analyses of conservation problems.     

PERFORMANCE ON MEASUREMENT AND PSEUDOMEASUREMENT TASKS BY FIVE- TO SEVEN-YEAR-OLD CHILDREN

Subjects trained by a non-verbal reinforcement method to select the longer one of a pair of sticks were given two genuine measurement (G) and two pseudomeasurement (P) tests. The first P-test involved the use of two measuring sticks, clearly different in length, and the second test addition or subtraction of a piece of the measuring stick between measurements. A single factor seemed to generate much of the performance in all four tests, suggesting that, since the P-tasks had to be solved by means of non-transitive inferences, the G-tasks were solved in the same way. It is concluded that non-verbal reinforcement methods may not yield a valid diagnosis of transitivity.     

Interrupting intentions: Zeigarnik-like effects in prospective memory

In two experiments, we examined the effects of task interruption on memory for intentions. Participants studied a series of anagrams, of which they solved one-half (Exp. 1) or two-thirds (Exp. 2), whereas the solution of the remaining items was interrupted by the experimenter. Furthermore, four anagrams (prospective cue items) differed from the remaining anagrams in that the third letter of each item was underlined. Participants were instructed to decide whether a subsequently presented (target) anagram contained the same or a different third letter as the underlined letter of the cue item. The results of both experiments showed Zeigarnik-like effects in prospective memory, so that cue items that were associated with interruption in the anagram task were better reminders than were items that were associated with completion. These findings suggest that interruption of an ongoing activity facilitates subsequent prospective memory performance, possibly by increasing the level of activation of the underlying intention representation that, in turn, increases the individual's sensitivity to identify the target event.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

Deformation of the useful visual field with state of vigilance,task priority,and central task complexity

In a dual-task situation, peripheral perception can deteriorate due to different variables related to the experimental situation, such as central task complexity and the relative priority given to the two tasks. Recent studies have also shown that individual characteristics (such as the state of vigilance) can alter the useful visual field. The ability to perceive signals presented at different places in the visual field decreases in situations that lead to episodes of low vigilance. In Exp. 1, 27 subjects (18-29 years) had to perform a monotonous and prolonged dual-monitoring task involving central and peripheral vision. The test was carried out with two different sets of instructions regarding priority: the more important task was either the central task or the peripheral one. In Exp. 2, 18 subjects (18-30 years) performed the same test, but this time with two levels of complexity of the central task, either a simple monitoring activity or the same monitoring task coupled with a counting activity. Elliptical interpolation of the data enabled assessment of the deformation of the useful visual field in both experiments. The analysis of the parameters of the elliptical model indicated that the useful visual field mainly shrinks on the vertical axis, as the individual becomes drowsy. The useful visual field also shrinks on the horizontal axis when the peripheral task has priority and the monitoring activity is prolonged, or when the central task is complex.     

Watching films with magical content facilitates creativity in children

Two experiments examined the possible link between magical thinking and creativity in preschool children. In Exp. 1, 4- and 6-yr.-old children were shown a film with either a magical or nonmagical theme. Results indicated that the mean scores of children shown the magical film was significantly higher than that of children watching the nonmagical film on the majority of subsequent creativity tests for both age groups. This trend was also found for 6-yr.-olds' drawings of impossible items. In Exp. 2, Exp. 1 was replicated successfully with 6- and 8-yr.-old children. Exposing children to a film with a magical theme did not affect their beliefs about magic. The results were interpreted to accentuate the role of magical thinking in children's cognitive development. Classroom implications of the results were also discussed.     

颜色特征加工任务间的注意瞬脱研究

本研究以两个实验对颜色特征加工任务间的注意瞬脱效应进行探讨。实验一要求被试对首先呈现的两个色块(记忆项)与随后呈现的两个色块(检测项)是否相同进行比较。结果发现,在记忆项与检测项的SOA为220ms的条件下,记忆项加工对认知资源的持续占用使检测项的加工绩效下降。实验二含两个需相继执行的任务(T1和T2),二者均为比较同时呈现在屏幕左右两侧的两色块颜色是否相同。结果发现,对颜色特征的觉察即可产生注意瞬脱效应,其大小及时程与经典的采用字母识别任务的研究获得的注意瞬脱效应相当。上述结果表明,特征搜索的完成需有限认知资源的参与。然而,注意瞬脱过程中,特征加工与需对知觉对象细节信息进行提取的辨别任务(如字母辨别任务)在消耗认知资源的量上并无差别。     

数字概念的视知觉无意识语义启动效应

研究了数字概念大小判断任务中的无意识语义启动效应。被试为天津师范大学本科生和研究生 5 6名。仪器为 pentiumIII高分辨率计算机 ,程序用E prime心理实验软件系统编制而成。在实验一中 ,用信号检测论技术发现 ,当启动数字呈现为 30ms时 ,被试是不能觉知到启动数字的 ;在实验二中 ,启动数字和靶数字使用相同的刺激序列 ,用MANOVA分析了启动数字和靶数字属于不同字体时的启动效应 ,发现启动数字对靶数字加工有促进或抑制作用 ;在实验三中 ,启动数字和靶数字使用两个不同的刺激序列 ,同样发现类似的启动效应。实验证据提示 ,无意识知觉能够达到语义水平上的加工     

The whole is indeed more than the sum of its parts: perceptual averaging in the absence of individual item representation

We tested Ariely's (2001) proposal that the visual system represents the overall statistical properties of sets of objects against alternative accounts of rapid averaging involving sub-sampling strategies. In four experiments, observers could rapidly extract the mean size of a set of circles presented in an RSVP sequence, but could not reliably identify individual members. Experiment 1 contrasted performance on a member identification task with performance on a mean judgment task, and showed that the tasks could be dissociated based on whether the test probe was presented before or after the sequence, suggesting that member identification and mean judgment are subserved by different mechanisms. In Experiment 2, we confirmed that when given a choice between a probe corresponding to the mean size of the set and a foil corresponding to the mean of the smallest and largest items only, the former is preferred to the latter, even when observers are explicitly instructed to average only the smallest and largest items. Experiment 3 showed that a test item corresponding to the mean size of the set could be reliably discriminated from a foil but the largest item in the set, differing by an equivalent amount, could not. In Experiment 4, observers rejected test items dissimilar to the mean size of the set in a member identification task, favoring test items that corresponded to the mean of the set over items that were actually shown. These findings suggest that mean representation is accomplished without explicitly encoding individual items.     

Age differences in collaborative memory: the role of retrieval manipulations

In two experiments, we examined age differences in collaborative inhibition (reduced recall in pairs of people, relative to pooled individuals) across repeated retrieval attempts. Younger and older adults studied categorized word lists and were then given two consecutive recall tests and a recognition test. On the first recall test, the subjects were given free-report cued recall or forced-report cued recall instructions (Experiment 1) or free recall instructions (Experiment 2) and recalled the lists either alone or in collaboration with another subject of the same age group. Free-report cued recall and free recall instructions warned the subjects not to guess, whereas forcedreport cued recall instructions required them to guess. Collaborative inhibition was obtained for both younger and older adults on initial tests of free-report cued recall, forced-report cued recall, and free recall, showing that the effect generalizes across several tests for both younger and older adults. Collaborative inhibition did not persist on subsequent individual recall or recognition tests for list items. Older adults consistently falsely recalled and recognized items more than did younger adults, as had been found in previous studies. In addition, prior collaboration may exaggerate older adults’ tendency toward higher false alarms on a subsequent recognition test, but only after a free recall test. The results provide generality to the phenomenon of collaborative inhibition and can be explained by invoking concepts of strategy disruption and source monitoring.