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.     

The development of cynicism

Two experiments explored the development of cynicism by examining how children evaluate other people who make claims consistent or inconsistent with their self-interests. In Experiment 1, kindergartners, second graders, and fourth graders heard stories with ambiguous conclusions in which characters made statements that were aligned either with or against self-interest. Older children took into account the self-interests of characters in determining how much to believe them: They discounted statements aligned with self-interest, whereas they accepted statements going against self-interest. Experiment 2 examined children's endorsement of three different explanations for potentially self-interested statements: lies, biases, and mistakes. Like adults, sixth graders endorsed lies and bias as plausible explanations for wrong statements aligned with self-interest; younger children did not endorse bias. Implications for the development of cynicism and children's understanding of bias are discussed.     

Teaching Bayesian reasoning in less than two hours

The authors present and test a new method of teaching Bayesian reasoning, something about which previous teaching studies reported little success. Based on G. Gigerenzer and U. Hoffrage's (1995) ecological framework, the authors wrote a computerized tutorial program to train people to construct frequency representations (representation training) rather than to insert probabilities into Bayes's rule (rule training). Bayesian computations are simpler to perform with natural frequencies than with probabilities, and there are evolutionary reasons for assuming that cognitive algorithms have been developed to deal with natural frequencies. In 2 studies, the authors compared representation training with rule training; the criteria were an immediate learning effect, transfer to new problems, and long-term temporal stability. Rule training was as good in transfer as representation training, but representation training had a higher immediate learning effect and greater temporal stability.     

4~8年级学生分数数量表征的准确性及形式

在分数学习中,分数数量表征是非常重要的方面。本研究利用数字线估计任务对四到八年级学生分数数量表征的情况进行探索。研究结果表明：随着年级升高,被试分数数量表征的准确性也随之提高;被试对单位分数和非单位分数表征的准确性存在显著差异,表现为对单位分数表征的准确性显著高于对非单位分数的表征,这种差异在低年级显著,随着年级的增长,差异逐渐消失;四到八年级学生在0~3数字线上,对分数数量的表征表现为线性形式而非对数形式,且这种线性表征形式是在六年级开始出现并随着年龄增长逐渐发展起来的。中西方儿童在分数数量表征的准确性和形式上有相似的发展路径,但是在表征准确性上中国儿童更高、线性形式出现年级上中国儿童可能更早。     

3～6年级儿童整数数量表征与分数数量表征的关系

研究主要探讨了整数数量表征和分数数量表征的关系以及年级对两者关系的影响。实验对155名三至六年级儿童进行0~1分数数字线估计任务和0~1000整数数字线估计任务的测量。结果发现:(1)对于整数数字线估计,所有年级儿童均主要采取了线性表征;(2)对于分数数字线估计,五六年级儿童主要采取了线性表征,三四年级儿童没有明显的线性表征或对数表征的倾向;(3)整数数量表征和分数数量表征呈显著正相关,不过年级对两者的关系产生了影响,表现在只有五六年级儿童的整数数字线估计对分数数字线估计有显著预测作用。     

小学儿童社交焦虑与孤独感研究

本研究采用经修订的量表对大城市三所普通小学的569名小学生测查了与社会交往有关的情绪,结果表明：男生的孤独感显著地高于女生。被拒绝儿童的孤独感最高,其次是被忽视儿童。二年级和六年级儿童的社交焦虑显著高于四年级儿童;被拒绝儿童的社交焦虑水平显著高于其他儿童。     

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.     

Ecological rationality or nested sets? Individual differences in cognitive processing predict Bayesian reasoning

The presentation of a Bayesian inference problem in terms of natural frequencies rather than probabilities has been shown to enhance performance. The effect of individual differences in cognitive processing on Bayesian reasoning has rarely been studied, despite enabling us to test process-oriented variants of the two main accounts of the facilitative effect of natural frequencies: The ecological rationality account (ERA), which postulates an evolutionarily shaped ease of natural frequency automatic processing, and the nested sets account (NSA), which posits analytical processing of nested sets. In two experiments, we found that cognitive reflection abilities predicted normative performance equally well in tasks featuring whole and arbitrarily parsed objects (Experiment 1) and that cognitive abilities and thinking dispositions (analytical vs. intuitive) predicted performance with single-event probabilities, as well as natural frequencies (Experiment 2). Since these individual differences indicate that analytical processing improves Bayesian reasoning, our findings provide stronger support for the NSA than for the ERA.     

Evidence of more immanent justice responding in adults than children: A challenge to traditional developmental theories

Three studies investigated developmental changes in immanent justice responding by asking participants to respond to vignettes in which a person's bad behaviour was followed by a negative consequence. Study 1 consisted of 152 sixth graders and 128 college students and presented participants with a vignette that examined the notion of bad people deserving to get ill. Study 2 consisted of 185 sixth graders and 154 college students and examined whether children and adults reasoned that that bad behaviour would actually cause the illness. Study 3 consisted of 96 third graders, 115 fifth graders, and 114 college students, and presented participants with vignettes that examined negative behaviours and consequences. Contrary to expectations based on traditional and contemporary developmental theories, all three studies demonstrated more evidence of immanent justice responding among adults than among elementary school children. These results call into question the comprehensiveness of traditional and contemporary developmental theories, and suggest the need to examine cognitive reasoning in adulthood when constructing developmental theories.     

Solving probabilistic and statistical problems: a matter of information structure and question form

Is the human mind inherently unable to reason probabilistically, or is it able to do so only when problems tap into a module for reasoning about natural frequencies? We suggest an alternative possibility: naive individuals are able to reason probabilistically when they can rely on a representation of subsets of chances or frequencies. We predicted that naive individuals solve conditional probability problems if they can infer conditional probabilities from the subset relations in their representation of the problems, and if the question put to them makes it easy to consider the appropriate subsets. The results of seven studies corroborated these predictions: when the form of the question and the structure of the problem were framed so as to activate intuitive principles based on subset relations, naive individuals solved problems, whether they were stated in terms of probabilities or frequencies. Otherwise, they failed with both sorts of information. The results contravene the frequentist hypothesis and the evolutionary account of probabilistic reasoning.     

Aspects of Code-Specific Memory Development

We investigated the development of code specific representations of different kinds of information in long term memory. Forty second graders, 40 sixth graders and 40 adults learned the associations between 12 pictures and one position each in a 4 × 3 grid of squares, 12 pictures and 1 of 12 monosyllabic words each or 12 pictures and 1 of 12 faces. After a 3 min distractor task, a picture was presented in the retrieval phase, and the associated position, word or face had to be selected. Performance in the verbal condition improved as a function of age, while performance in the spatial condition turned out to be independent of age, and the performance in the facial condition showed a difference between both child groups and the adults. The results revealed a developmental difference of code specific representation of different kinds of information.     

Why are rocks pointy? Children's preference for teleological explanations of the natural world

Teleological explanations are based on the assumption that an object or behavior exists for a purpose. Two studies explored the tendency of adults and first-, second-, and fourth-grade elementary-school children to explain the properties of living and nonliving natural kinds in teleological terms. Consistent with the hypothesis that young children possess a promiscuous teleological tendency, Study 1 found that children were more likely than adults to broadly explain the properties of both living and nonliving natural kinds in teleological terms, although the kinds of functions that they endorsed varied with age. Study 2 was an attempt to reduce children's broad teleological bias by introducing a pretrial that described, in nonteleological terms, the physical process by which nonliving natural kinds form. In spite of this attempt, Study 2 replicated the effects of Study 1, with only fourth graders showing any shift in preference for teleological explanation.     

Social skill in third and sixth grade children: A moderator of lifetime stressful life events and behavior problems?

We investigated relationships among stressful lifetime life events, social skill, and behavior problems in children. Participants were 159 third graders, 138 sixth graders, and their mothers. Mothers reported on their children's life events, social skill, and behavior problems. Children self-reported on their social skill and behavior problems. Increased numbers of stressful life events significantly predicted increased behavior problems in third and sixth graders. Increased social skill significantly predicted decreased behavior problems in third and sixth graders. Social skill served a stress-protective role for third but not sixth graders; third graders with increased life events and better social skill had fewer behavior problems than their less socially skilled peers. Practical and theoretical implications are discussed, and suggestions for future resilience research are presented.     

Children's strategy preferences for resolving contradictory story information under two social conditions

This paper attempted to examine how children, when confronted with a literary implicature, resolve this implicature under two types of social conditions. More specifically, this paper identified five types of strategies which people employ to resolve text anomalies arising when old information in a story setting is incompatible with new information in a story ending. Two experiments demonstrated that third graders (8.4 and 8.6 years) and sixth graders (12.5 and 12.8 years) consistently selected certain strategies for resolving old and new, empirical and value, contradictory information. Although third and sixth graders demonstrated a similar strategy preference for resolving contradictory old and new information in formal conditions, the principal difference was that third graders modified old information to fit new information while sixth graders modified new information to fit old information. In contrasting the formal and informal conditions, third graders shifted their strategy preferences so as to minimize the amount of text restructuring in the formal condition; sixth graders, on the other hand, shifted their strategy preferences so as to maximize the amount of text restructuring in informal conditions. These findings suggest that story schema structures are more interpretive than story grammar psychologists presently assume.     

类推理的发展序列与年龄特点

类推理主要涉及类别层级包含关系的理解和判断。使用修订的类包含、替代包含和二元律任务,以162名二至五年级儿童为被试,利用Raseh模型分析方法,旨在揭示儿童逻辑包含能力的发展序列。此外,探讨了儿童类推理发展的年龄特点。结果表明：1）类推理发展中儿童最先掌握类包含推理,其次是替代包含,最后是二元律推理;2）三、四、五年级儿童已能解决多数类包含任务,其成绩显著优于二年级儿童。四、五年级儿童能解决多数替代包含任务,其成绩均显著优于二年级儿童;3）二元律任务对二至五年级儿童来讲均比较困难。     

Representation facilitates reasoning: what natural frequencies are and what they are not

A good representation can be crucial for finding the solution to a problem. Gigerenzer and Hoffrage (Psychol. Rev. 102 (1995) 684; Psychol. Rev. 106 (1999) 425) have shown that representations in terms of natural frequencies, rather than conditional probabilities, facilitate the computation of a cause's probability (or frequency) given an effect--a problem that is usually referred to as Bayesian reasoning. They also have shown that normalized frequencies--which are not natural frequencies--do not lead to computational facilitation, and consequently, do not enhance people's performance. Here, we correct two misconceptions propagated in recent work (Cognition 77 (2000) 197; Cognition 78 (2001) 247; Psychol. Rev. 106 (1999) 62; Organ. Behav. Hum. Decision Process. 82 (2000) 217): normalized frequencies have been mistaken for natural frequencies and, as a consequence, "nested sets" and the "subset principle" have been proposed as new explanations. These new terms, however, are nothing more than vague labels for the basic properties of natural frequencies.     

Wayfinding behavior and spatial knowledge of adults and children in a virtual environment: The role of landmarks

This study investigated the effect of different organizations of landmark-location pairings as fine-space information on wayfinding behavior and spatial knowledge on a total of 90 participants: 30 second graders, 30 sixth graders, and 30 adults. All participants had to find their way to a goal in a virtual environment with either randomized or categorical landmarks, or without any landmarks. Thereafter, they had to find the shortest way from the start position to the goal in two consecutive trials (wayfinding performance), and they had to solve a number of spatial knowledge tasks. The results showed that independent of their categorical function, the existence of landmarks influenced the way-finding performance of adults and children in the same way. Whereas the presence of landmarks had no effect on spatial survey knowledge, landmark knowledge itself was influenced by the categorical function of the landmarks presented. Moreover, second graders showed limited achievement compared to adults independent of the existence of landmarks. The main results implicate firstly that children at school age indeed are able to use landmark-location pairings as fine-space information like adults during learning an unknown environmental space, and secondly that a dissociation between wayfinding behavior and spatial knowledge might exist.     

Children's judgments of event upsettingness and personal experiencing of stressful events

Five-hundred and three urban and suburban 4th- to 6th-grade schoolchildren judged event upsettingness and reported the occurrence of 22 life events. Judgments of event upsettingness ranged considerably, some corresponding to, others differing from, adult judgments. Children reported experiencing an average of seven events during their lifetimes. Girls judged events to be more upsetting than boys, and fourth and fifth graders judged events to be more upsetting than sixth graders. Urban children reported having experienced more stressful events than suburban children, and sixth graders experienced more events than fourth graders.     

Frequency interpretation of ambiguous statistical information facilitates Bayesian reasoning

The idea that naturally sampled frequencies facilitate performance in statistical reasoning tasks because they are a cognitively privileged representational format has been challenged by findings that similarly structured numbers presented as chances similarly facilitate performance, on the basis of the claim that these are technically single-event probabilities. A crucial opinion, however, is that of the research participants, who possibly interpret chances as de facto frequencies. A series of experiments here indicate that not only is performance improved by clearly presented natural frequencies, rather than chances phrasing, but also that participants who interpreted chances as frequencies, rather than as probabilities, were consistently better at statistical reasoning. This result was found across different variations of information presentation and across different populations.