Numerical Estimation in Children for Both Positive and Negative Numbers

Numerical estimation has been used to study how children mentally represent numbers for many years (e.g., Siegler & Opfer, 2003). However, these studies have always presented children with positive numbers and positive number lines. Children’s mental representation of negative numbers has never been addressed. The present study tested children in the 2nd, 4th, and 6th grades to assess their mental representations of both positive and negative numbers using a standard numerical estimation task. We replicated the shift from a logarithmic to linear representation for positive numbers (0–1,000 scale) in that 2nd graders represented positive numbers logarithmically, but 4th and 6th graders represented the numbers linearly. Furthermore, children’s representation of negative numbers paralleled their representations of positive numbers and showed the same shift from a logarithmic representation at Grade 2 to linear representations at Grades 4 and 6. This is the first study to provide data on children’s representation of negative numbers, and the implications of these findings are discussed.     

The development of numerical estimation: evidence for multiple representations of numerical quantity

Abstract - We examined children's and adults' numerical estimation and the representations that gave rise to their estimates. The results were inconsistent with two prominent models of numerical representation: the logarithmic-ruler model, which proposes that people of all ages possess a single, logarithmically spaced representation of numbers, and the accumulator model, which proposes that people of all ages represent numbers as linearly increasing magnitudes with scalar variability. Instead, the data indicated that individual children possess multiple numerical representations; that with increasing age and numerical experience, they rely on appropriate representations increasingly often; and that the numerical context influences their choice of representation. The results, obtained with second graders, fourth graders, sixth graders, and adults who performed two estimation tasks in two numerical contexts, strongly suggest that one cause of children's difficulties with estimation is reliance on logarithmic representations of numerical magnitudes in situations in which accurate estimation requires reliance on linear representations.     

Do children "DRM" like adults? False memory production in children

The Deese/Roediger-McDermott (DRM) paradigm was used to investigate developmental trends in accurate and false memory production. In Experiment 1, DRM lists adjusted to be more consistent with children's vocabulary were used with 2nd graders, 8th graders, and college students. Accurate and false recall and recognition increased with age, but semantic information appeared to be available to all age groups. Experiment 2 created a set of child-generated lists based on the free associations by a group of 3rd graders to critical items. The child-generated associates were different from those generated by adults; long and short versions of the child-generated lists were therefore presented to 2nd, 5th, and 8th graders and college students in Experiment 3. Second graders exhibited few false memories, whereas 5th graders were similar to adults in low-demand conditions and more similar to younger children in high-demand conditions. Findings are discussed in terms of developmental changes in automatic and effortful processing and the use of semantic networks.     

Representational change and children's numerical estimation

We applied overlapping waves theory and microgenetic methods to examine how children improve their estimation proficiency, and in particular how they shift from reliance on immature to mature representations of numerical magnitude. We also tested the theoretical prediction that feedback on problems on which the discrepancy between two representations is greatest will cause the greatest representational change. Second graders who initially were assessed as relying on an immature representation were presented feedback that varied in degree of discrepancy between the predictions of the mature and immature representations. The most discrepant feedback produced the greatest representational change. The change was strikingly abrupt, often occurring after a single feedback trial, and impressively broad, affecting estimates over the entire range of numbers from 0 to 1000. The findings indicated that cognitive change can occur at the level of an entire representation, rather than always involving a sequence of local repairs.     

From algorithmic computing to direct retrieval: Evidence from number and alphabetic arithmetic in children and adults

A number of theories of mental arithmetic suggest that the ability to solve simple addition and subtraction problems develops from an algorithmic strategy toward a strategy based on the direct retrieval of the result from memory. In the experiment presented here, 2nd and 12th graders were asked to solve two tasks of number and alphabet arithmetic. The subjects transformed series of 1 to 4 numbers or letters (item span) by adding or subtracting an operand varying from 1 to 4 (operation span). Although both the item and operation span were associated with major and identical effects in the case of both numbers and letters at 2nd grade, such effects were clearly observable only in the case of letters for the adult subjects. This suggests the use of an algorithmic strategy for both types of material in the case of the children and for the letters only in the case of the adults, who retrieved numerical results directly from memory.     

Children and adults' distance estimations in a large-scale environment: effects of time and clutter

In two experiments adults (mean age = 19-5), sixth graders (mean age = 11-8), fourth graders (mean age = 9-8), and second graders (mean age = 7-8) walked a straight line distance through a large-scale environment. Subjects were then asked to estimate the time taken to traverse each half of the walk and to estimate the distance between objects seen along the walk. In Experiment 1 each half of the walk was traversed in the same amount of time but contained a different number of objects (clutter). Time and distance estimates were related, but were not affected by the number of intervening objects encountered between locations. In Experiment 2 subjects again encountered a different number of objects along each half of the walk but each half was traversed in varying amounts of time. Again, time and distance estimates were related, and there was no clutter effect. There were no consistent developmental differences across the two experiments. It was concluded that (1) Thorndyke's clutter effect does not occur across all types of spatial cognition tasks, and (2) children and adults tend to relate time and distance across a variety of distance estimation tasks.     

Response latency as a predictor of the accuracy of children's reports

Researchers have explored various diagnostic cues to the accuracy of information provided by child eyewitnesses. Previous studies indicated that children's confidence in their reports predicts the relative accuracy of these reports, and that the confidence-accuracy relationship generally improves as children grow older. In this study, we examined the added contribution of response latency to the prediction of children's accuracy over and above that of confidence ratings. In Experiments 1 and 2, 2nd and 5th graders studied picture-event pairs and were tested using forced-choice, 2-alternative, or 5-alternative questions. In Experiment 3, children watched a slideshow depicting a story and were tested by 5-alternative questions about story details. The children indicated their confidence in each response, and response latency was measured. The results of all experiments suggested that children in both age groups relied on response latency as a cue for confidence, and this reliance contributed to the success with which they monitored the accuracy of their reports. When the test format was easy (Experiment 1), 2nd graders were as accurate as 5th graders in monitoring the accuracy of their answers, and the latency of their responses was no less predictive of accuracy. When the task was more difficult, age differences emerged. Nevertheless, in all experiments and for both age groups, response latency was found to have added value for predicting accuracy over and above that of confidence. Theoretical and practical implications of these findings for predicting the accuracy of children's reports are discussed.     

Costs and benefits of representational change: effects of context on age and sex differences in symbolic magnitude estimation

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of numerical magnitude. Specifically, children learn that their default logarithmic representations are inappropriate for many numerical tasks, leading them to adopt more appropriate linear representations despite linear representations being inappropriate for estimating fractional magnitude. In Experiment 1, this hypothesis accurately predicted a developmental shift from logarithmic to linear estimates of numerical magnitude and a negative correlation between accuracy of numerical and fractional magnitude estimates (r = −.80). In Experiment 2, training that improved numerical estimates also led to poorer fractional magnitude estimates. Finally, both before and after training that eliminated age differences in estimation accuracy, complementary sex differences were observed across the two estimation contexts.     

The irrelevant sound effect in short-term memory: Is there developmental change?

With two experiments, effects of irrelevant speech and classroom noise on serial recall of common nouns presented pictorially were investigated in children and adults. Experiment 1 used fixed list lengths for children (first graders) and adults. Experiment 2 used list lengths adjusted to participants' (second–third graders, adults) individual spans. In both experiments, children and adults were equally impaired by irrelevant speech. This contrasts with a related study (differences in methodology) by Elliott (2002), who reported severe increase in the detrimental impact of irrelevant speech with decreasing age. In both experiments, classroom noise had no effect in overall analyses. For Experiment 1, however, separate group analyses revealed impairment in children. Results suggest that effects of irrelevant sounds on serial recall stem from two separate mechanisms: Specific interference due to the sounds' automatic access to short-term memory, and/or attention capture. Only for the latter there is developmental change.     

The effect of spacing repetitions on the recognition memory of young children and adults

In Experiment 1, preschoolers, first graders, and third graders were presented a list of pictures that included twice-presented items separated by varying numbers of intervening items. Performance on a subsequent recognition test improved as the spacing between repetitions increased, but the effect of spacing did not interact reliably with grade level. In Experiment 2a, we replicated the spaced-repetition effect in young children and found a similar effect in college students. In Experiment 2b, we varied the conditions under which lists were presented to college students and again found a spacing function that was comparable to that of very young children. The results are consistent with the hypothesis that spaced-repetition effects in recognition are produced by fundamental memory mechanisms that are operational at a very early age and which undergo little change with development.     

Children's Illusory Correlations: The Role of Attentional Bias in Group Impression Formation

Two experiments examined 2nd and 5th graders' tendency to perceive illusory correlations between a minority group and infrequent behaviors. Experiment 1 examined perceptions of fictitious majority and minority groups in which 67% of each group's behaviors were positive and 33% were negative. Second and 5th graders formed illusory correlations between the minority group and negative behaviors on group attribution and frequency estimation tasks, and frequency estimations predicted differential evaluations of the majority and minority groups. Experiment 2 replicated findings from Experiment 1 under conditions in which 67% of the target groups' behaviors were negative and 33% were positive. In this case, children perceived an illusory correlation between the minority group and positive behaviors, and frequency estimations again predicted differential group evaluations. The results are discussed in terms of their implications for understanding judgment biases and stereotype formation and for stereotyping and prejudice interventions.     

Influence of recall procedures on the modality effect with numbers and enumerated stimuli

Experiment 1 extended J. S. Nairne and W. L. McNabb's (1985) counting procedure for presenting numerical stimuli to examine the modality effect. The present authors presented participants with dots and beeps and instructed participants to count the items to derive to-be-remembered numbers. In addition, the authors presented numbers as visual and auditory symbols, and participants recalled items by using free-serial written recall. Experiment 1 demonstrated primacy effects, recency effects, and modality effects for visual and auditory symbols and for counts of dots and beeps. Experiment 2 replicated the procedure in Experiment 1 using strict-serial written recall instead of free-serial written recall. The authors demonstrated primacy and recency effects across all 4 presentation conditions and found a modality effect for numbers that the authors presented as symbols. However, the authors found no modality effect when they presented numbers as counts of beeps and dots. The authors discuss the implications of the results in terms of methods for testing modality effects.     

123s and ABCs: developmental shifts in logarithmic‐to‐linear responding reflect fluency with sequence values

When placing numbers along a number line with endpoints 0 and 1000, children generally space numbers logarithmically until around the age of 7, when they shift to a predominantly linear pattern of responding. This developmental shift of responding on the number placement task has been argued to be indicative of a shift in the format of the underlying representation of number (Siegler & Opfer, 2003 ). In the current study, we provide evidence from both child and adult participants to suggest that performance on the number placement task may not reflect the structure of the mental number line, but instead is a function of the fluency (i.e. ease) with which the individual can work with the values in the sequence. In Experiment 1, adult participants respond logarithmically when placing numbers on a line with less familiar anchors (1639 to 2897), despite linear responding on control tasks with standard anchors involving a similar range (0 to 1287) and a similar numerical magnitude (2000 to 3000). In Experiment 2, we show a similar developmental shift in childhood from logarithmic to linear responding for a non‐numerical sequence with no inherent magnitude (the alphabet). In conclusion, we argue that the developmental trend towards linear behavior on the number line task is a product of successful strategy use and mental fluency with the values of the sequence, resulting from familiarity with endpoints and increased knowledge about general ordering principles of the sequence.A video abstract of this article can be viewed at: http://www.youtube.com/watch?v=zg5Q2LIFk3M     

Is knowledge base development a necessary and sufficient condition for memory development?

The hypothesis that knowledge base development is an important condition for memory development was tested in two experiments. In Experiment 1 subjects from grades K, 6, and college participated in an incidental learning task where they analyzed words according to one of four featural systems: physical, acoustic, semantic, and imagery. Large developmental differences in recall were found in the imagery and semantic conditions, small differences were observed in the acoustic conditions, and no developmental differences were found in the physical condition. Experiment 2 involved presenting college students and third graders words either from categories highly salient to third graders, e.g., cartoons, games, etc., or from corresponding Battig and Montague categories generated by college students. Free recall and clustering performances showed significant interactions, with third graders superior on the third-grade salient items, and college students superior on Battig and Montague items.     

Stroop interference is a composite phenomenon: Evidence from distinct developmental trajectories of its components

Only one previous developmental study of Stroop task performance (Schiller, 1966) has controlled for differences in processing speed that exist both within and between age groups. Therefore, the question of whether the early developmental change in the magnitude of Stroop interference actually persists after controlling for processing speed needs further investigation; work that is further motivated by the possibility that any remaining differences would be caused by process(es) other than processing speed. Analysis of data from two experiments revealed that, even after controlling for processing speed using z‐transformed reaction times, early developmental change persists such that the magnitude of overall Stroop interference is larger in 3rd‐ and 5th graders as compared to 1st graders. This pattern indicates that the magnitude of overall Stroop interference peaks after 2 or 3 years of reading practice (Schadler & Thissen, 1981). Furthermore, this peak is shown to be due to distinct components of Stroop interference (resulting from specific conflicts) progressively falling into place. Experiment 2 revealed that the change in the magnitude of Stroop interference specifically results from joint contributions of task, semantic and response conflicts in 3rd‐ and 5th graders as compared to a sole contribution of task conflict in 1st graders. The specific developmental trajectories of different conflicts presented in the present work provide unique evidence for multiple loci of Stroop interference in the processing stream (respectively task, semantic and response conflict) as opposed to a single (i.e. response) locus predicted by historically – favored response competition accounts.     

Three aspects of cognitive development

An attempt was made to characterize and explain developmental differences in children's thinking, specifically in their understanding of balance scale problems. Such differences were sought in three domains: existing knowledge about the problems, ability to acquire new information about them, and process-level differences underlying developmental changes in the first two areas. In Experiment 1, four models of rules that might govern children's performance on balance scale problems were proposed. The rules proved to accurately describe individual performance and also to accurately predict developmental trends on different types of balance scale problems. Experiment 2 examined responsiveness to experience; it was found that older and younger children, equated for initial performance on balance scale problems, derived different benefits from identical experience. Experiment 3 examined a potential cause of this discrepancy, that younger children might be less able than older ones to benefit from experience because their encoding of stimuli was less adequate. Independent assessment procedures revealed that the predicted differences in older and younger children's encoding were present; it was also found that these differences were not artifactual and that reducing them also reduced the previously observed differences in responsiveness to experience. It was concluded, therefore, that the encoding hypothesis explained a large part of the developmental difference in ability to acquire new information.     

Developmental changes in the effects of types of self-corrected elaboration on incidental memory

The present study investigated the developmental changes in the effects of two types of self-corrected elaboration, namely generated correction and chosen correction, on incidental memory of words. Second and sixth graders performed an orienting task involving two types of correction followed by an unexpected recall test. They were presented with a target and its sentence, and were asked to correct the target to a congruous word in the generated-correction condition, or to choose one of the alternative congruous words in the chosen-correction condition. For second graders, chosen correction led to a better recall than generated correction, whereas for sixth graders the reverse relationship between the two corrections was observed. These results were interpreted as showing the developmental change in the effects of types of self-corrected elaboration on incidental memory.     

Individual differences in 2D:4D digit-ratios and otoacoustic emissions: Do they share a common developmental origin?

Differences across individuals in both 2D:4D, the ratio of the lengths of the 2nd-4th digit, and in otoacoustic emissions (OAEs), have been hypothesized to arise from differences in androgen exposure prenatally. Contrary to this hypothesis, however, McFadden and Shubel (2003) found no significant correlation between digit-ratios and OAE production. To further investigate the hypothesis of a common developmental origin, the current study assessed correlations between individual differences in 2D:4D and OAEs in 131 normally-hearing adults. All possible ratios of the 2nd, 3rd, 4th, and 5th digits were computed and OAEs, both spontaneously produced and in response to audible clicks, were measured. Observed correlations between digit length ratios and OAEs were largely nonsignificant. No significant correlations were found between 2D:4D and any of the five OAE measures. Fundamental differences in developmental timing and duration, postnatal influences, or factors other than prenatal androgens may explain the lack of associations.     

两位数的整体与局部加工

关于两位数的加工方式有整体加工说和局部加工说,实验证据主要来自数字数量控制/主动加工任务。本研究主要考察在数字数量自动加工任务中两位数的加工方式。实验一要求被试完成数量大小比较和物理大小比较两个任务,实验二只要求被试完成物理大小比较任务。结果是在数量比较任务和物理比较任务中都存在显著的个位十位一致性效应和数量物理一致性效应,这表明在两位数的数量主动和自动加工任务中均存在整体加工和局部加工两种方式。