Rule transition on the balance scale task: a case study in belief change

For various domains in proportional reasoning cognitive development is characterized as a progression through a series of increasingly complex rules. A multiplicative relationship between two task features, such as weight and distance information of blocks placed at both sides of the fulcrum of a balance scale, appears difficult to discover. During development, children change their beliefs about the balance scale several times: from a focus on the weight dimension (Rule I) to occasionally considering the distance dimension (Rule II), guessing (Rule III), and applying multiplication (Rule IV; Siegler, 1981). Because of the detailed empirical findings the balance scale task has become a benchmark task for computational models of proportional reasoning. In this article, we present a large empirical study (N = 420) of which the findings provide a challenge for computational models. The effect of feedback and the effect of individually adapted training items on rule transition were tested for children using Rule I or Rule II. Presenting adapted training items initiates belief revision for Rule I but not for Rule II. The experience of making mistakes (by providing feedback) induces a change for both Rule I and Rule II. However, a delayed posttest shows that these changes are preserved after 2 weeks only for children using Rule I. We conclude that the transition from Rule I to Rule II differs from the transition from Rule II to a more complex rule. Concerning these empirical findings, we will review performance of computational models and the implications for a future belief revision model.

It is one Thing, to show a Man that he is in an Error, and another, to put him in possession of Truth. John Locke

     

基于潜在类别分析的6~15岁儿童平衡秤任务认知规则探索

本研究使用自编平衡秤测验测试468名6~15岁儿童,通过潜在类别分析对认知规则进行分类。结果发现,被试使用了规则Ⅰ、规则Ⅰ'、规则Ⅱ、补偿规则、规则Ⅳ、距离优势规则等六种规则;6~9岁儿童主要使用规则Ⅰ;10~13岁儿童主要使用补偿规则;14岁以上儿童主要使用规则Ⅳ;13岁到14岁之间,使用规则Ⅳ的儿童数量呈跳跃式增加。与规则评估技术相比,潜在类别分析用于认知规则研究具有明显优势,最后对运用此方法的前提假设与局限进行了讨论。     

What response times tell of children's behavior on the balance scale task

Analysis of accuracy of responses to balance scale problems gives a global idea of the cognitive processes that underlie problem-solving behavior on this task. We show that response times (RTs) provide additional detailed information about the kind and duration of these processes. We derive predictions about the RTs from Siegler's (1981) model for the balance scale task, including the counterintuitive prediction that young adults are slower than children in solving particular balance scale problems. The predictions were tested in a study in which 191 6- to 22-year-old participants were presented with a computerized balance scale task. RTs were analyzed with regression models. In addition to qualitative differences between items, we also modeled quantitative differences between items in the regression models. Analyses supported the predictions and provided additional knowledge on the rules. Rule II was reformulated as a rule that always involves the encoding, but not always the correct application of the distance cue. RTs provided evidence for the use of a buggy-rule and not an addition-rule. Finally, a relation between rule inconsistency and increased RT was found.     

Modeling developmental transitions on the balance scale task

Periods of relatively stable, rule‐like behavior alternated with short transition periods characterize cognitive development on reasoning tasks like the balance scale task. Each transition gives rise to an improvement in behavior, until a phase is reached in which performance is flawless or improvement is not worthwhile given the necessary extra effort. Several computational models have been developed to capture the developmental phenomena associated with the balance scale task. These models, which originate from different computational traditions, explain the main phenomena of development. Recently, empirical phenomena have been reported that these models cannot easily accommodate. We propose a computational model that is implemented in ACT‐R and that is based on the evaluation of success of applied knowledge, combined with a mechanism to construct newknowledge by searching for differences between the left‐ and right‐hand sides of presented balance scale problems. This model accounts for the main empirical phenomena as well as for the recently reported empirical phenomena such as learning without feedback.     

不同任务类型条件下4～6年级儿童比例推理策略的表现

随机选取小学4~6年级被试86名,从辅助策略、比例推理策略的策略选择和策略效用三方面,通过五种类型的天平任务考察儿童比例推理策略的表现。结果表明:(1)儿童最常使用手指动作辅助加工基本数量信息。辅助策略的使用率随年龄增长而减少,五年级开始使用出声思维,反映出元认知能力的发展。(2)在正式学习比例知识之前,各年级儿童都能使用两种以上策略,也能根据任务难度自发产生新策略,具备策略选择的多样性和适应性。其中,三个年级均能使用定性比例推理策略(双维策略,IIIA策略,补偿策略),表明儿童初步认识了距离和重量两个维度的共变关系。此外,六年级儿童能使用"运货车策略"将冲突问题灵活地化解为简单问题。(3)儿童的错误策略表现为:在冲突任务中盲目使用补偿策略、简单策略或加法策略。(4)分层回归分析表明,在控制年龄后,儿童的一般推理能力越高,其对重量策略的依赖性越低,且可能更容易发掘距离维度的意义,其使用运货车策略的频次更多。此外,一般推理能力对解决冲突类天平任务的正确次数有正向预测作用。     

Social cognition is not reducible to theory of mind: When children use deontic rules to predict the behaviour of others

The objective of this paper is to discuss whether children have a capacity for deontic reasoning that is irreducible to mentalizing. The results of two experiments point to the existence of such non‐mentalistic understanding and prediction of the behaviour of others. In Study 1, young children (3‐ and 4‐year‐olds) were told different versions of classic false‐belief tasks, some of which were modified by the introduction of a rule or a regularity. When the task (a standard change of location task) included a rule, the performance of 3‐year‐olds, who fail traditional false‐belief tasks, significantly improved. In Study 2, 3‐year‐olds proved to be able to infer a rule from a social situation and to use it in order to predict the behaviour of a character involved in a modified version of the false‐belief task. These studies suggest that rules play a central role in the social cognition of young children and that deontic reasoning might not necessarily involve mind reading.     

Learning to Understand the Balance Beam

Twenty-two university students who did not initially know the quantitative rule for predicting whether a configuration of weights placed on a balance beam would cause the beam to balance, tip left, or tip right were asked to induce the rule in a training procedure adapted from Siegler (1976). For each of a series of balance beam problems, subjects predicted the action of the beam and explained how they arrived at their prediction. Protocols revealed that although all subjects realized on early on that both weight and distance were relevant to their predictions, they used a variety of heuristics prior to inducing the correct quantitative rule. These heuristics included instance-based reasoning, qualitative estimation of distance, and the use of quantitative rules of limited generality. The common use of instance-based reasoning suggests that learning to understand the balance beam cannot be described completely in terms of a simple rule acquisition theory. Also, the variability in the use of heuristics across subjects suggests that no simple theory that depicts subjects as linearly progressing through a hierarchy of levels can be adequately describe the development of balance understanding.     

Rule following and rule use in the balance-scale task

Quinlan et al. [Quinlan, p., van der Mass, H., Jansen, B., Booij, O., & Rendell, M. (this issue). Re-thinking stages of cognitive development: An appraisal of connectionist models of the balance scale task. Cognition, doi:10.1016/j.cognition.2006.02.004] use Latent Class Analysis (LCA) to criticize a connectionist model of development on the balance-scale task, arguing that LCA shows that this model fails to capture a torque rule and exhibits rules that children do not. In this rejoinder we focus on the latter problem, noting the tendency of LCA to find small, unreliable, and difficult-to-interpret classes. This tendency is documented in network and synthetic simulations and in psychological research, and statistical reasons for finding such unreliable classes are discussed. We recommend that LCA should be used with care, and argue that its small and unreliable classes should be discounted. Further, we note that a preoccupation with diagnosing rules ignores important phenomena that rules do not account for. Finally, we conjecture that simple extensions of the network model should be able to achieve torque-rule performance.     

“如果P,那么Q,……?”─—儿童充分条件假言演绎推理能力发展初探

以日常生活事件为内容的三套测验题探查了9－15岁儿童充分条件假言推理能力的发展。研究发现被试有关能力的发展可以区分出三种不同水平；儿童对充分条件假言推理规则的掌握没有固定的难易顺序，这取决于课题任务的性质和主体思维发展水平。研究还探查了发展的个体内部差异和个体之间的差异以及影响差异的各种内外因素。     

平衡秤任务复杂性的事前与事后分析

任务复杂性的分析和评估是心理测量学和认知心理学都非常关注的重要主题。以264名小学四、五、六年级儿童为被试,平衡秤任务为研究材料,考察任务在未旋转时第一个因素上的载荷(事后分析)能否作为评价任务复杂性的一个指标,以及关系-表征复杂性模型对平衡秤任务复杂性分析(事前分析)的有效性两个问题。结果表明:平衡秤任务施测后所得所有项目的因素载荷与其难度之间没有显著正相关,即因素载荷的高低没有反映平衡秤任务复杂性的大小;而基于关系-表征复杂性模型对任务的事前分析所确定的任务等级复杂性和知识经验对任务难度的解释率为95.0%,可见,关系-表征复杂性模型提供的分析任务复杂性的思路和方法是较为合理的。     

The development of an understanding of balance and the effect of training via stop-action video

Children's rule-governed, scientific reasoning in relation to the physical principles involved in balance has been studied by many, for example, Inhelder and Piaget (1958), Karmiloff-Smith and Inhelder (1974), as well as Siegler (1978). Although Siegler criticized the Genevan research on both methodological and conceptual grounds, his work can be criticized for being narrow in scope thus making his data a result of his task and design, rather than a true representation of the rules children employ naturally as they develop an understanding of balance. The work described herein is an attempt to integrate the methodologies of the Piagetian School and Siegler by making use of current video technology to naturalistically assess children's rule-governed behaviors but to also empirically validate them. The first part of this study (N=128, ages 4–9 years) used video to microanalytically assess the strategies children used as they attempted to balance a set of symmetrical, asymmetrical, and weighted blocks on a fulcrum. After validating on ordinal progression of strategies, the study used stop-action video feedback, in a predict-consequence paradigm, to focus reflection on (a) the action of the block, (b) the placement of the block, and (c) the episode in general. A two-way interaction between entering level and type of video feedback was significant. The results are interpreted in terms of Piaget's theory of equilibration suggesting that reflection and cognitive conflict can improve understanding if the child assimilates the feedback to the confirmation or refutation of an initial assimilatory scheme regarding balance.     

四卡问题的视角效应发生机制：领域一般性还是领域特殊性？

视角效应（Perspective Effects)是指四卡问题解决中,当被试从不同的角度对条件规则进行检验时,其选择偏向会在卡片p,q和-p,q之间发生稳定变化的现象。视角效应对四卡问题的解决有重要影响,但其存在条件及发生机制仍存在争议。本研究通过3个实验,探讨四卡问题中的视角效应。实验1和实验2采用混合实验设计,结果表明,转换社会契约、因果规则的四卡问题中,发现视角效应,抽象规则中,提供反例比提供充分必要条件更有利于产生视角效应。实验3在不同视角下研究四卡问题与条件推理的关系。结果表明,四卡问题与条件推理之间不具有逻辑一致性,产生分离现象。总之,视角效应在抽象规则和特定规则中均存在,而提供反例更有利于视角效应的产生。领域一般性或领域特殊性均不能有效解释视角效应的产生,两种机制的综合调节为解释视角效应提供新的角度。     

Young children’s analogical reasoning across cultures: Similarities and differences

A cross-cultural comparison between U.S. and Hong Kong preschoolers examined factors responsible for young children’s analogical reasoning errors. On a scene analogy task, both groups had adequate prerequisite knowledge of the key relations, were the same age, and showed similar baseline performance, yet Chinese children outperformed U.S. children on more relationally complex problems. Children from both groups were highly susceptible to choosing a perceptual or semantic distractor during reasoning when one was present. Taken together, these similarities and differences suggest that (a) cultural differences can facilitate better knowledge representations by allowing more efficient processing of relationally complex problems and (b) inhibitory control is an important factor in explaining the development of children’s analogical reasoning.     

Measuring the ability of transitive reasoning,using product and strategy information

Cognitive theories disagree about the processes and the number of abilities involved in transitive reasoning. This led to controversies about the influence of task characteristics on individuals' performance and the development of transitive reasoning. In this study, a computer test was constructed containing 16 transitive reasoning tasks having different characteristics with respect to presentation form, task format, and task content. Both product and strategy information were analyzed to measure the performance of 6- to 13-year-old children. Three methods (MSP, DETECT, and Improved DIMTEST) were used to determine the number of abilities involved and to test the assumptions imposed on the data by item response models. Nonparametric IRT models were used to construct a scale for transitive reasoning. Multiple regression was used to determine the influence of task characteristics on the difficulty level of the tasks. It was concluded that: (1) the qualitatively distinct abilities predicted by Piaget's theory could not be distinguished by means of different dimensions in the data structure; (2) transitive reasoning could be described by one ability, and some task characteristics influenced the difficulty of a task; and (3) strategy information provided a stronger scale than product information.     

小学生图形推理策略发展特点的研究

本研究用我们修订的瑞文推理为材料,对145名儿童图形推理策略进行研究。结果发现：小学生在解决图形推理问题时使用六种策略,它们分别是分析策略、不完全分析策略、知觉分析策略、知觉匹配策略、格式塔策略和自主想象策略;不同年龄儿童在解决不同类型题目时的策略使用表现出不同的特点,儿童在解决数量规则题中,知觉分析策略在整个小学儿童阶段占主导地位,而在解决加减规则题中,分析策略占主导地位,随年龄增长而呈上升趋势;小学二年级开始出现图形推理能力发展的飞跃,二年级儿童开始能够同时观察到两种规则,五六年级儿童更能够不受题目形式的影响．而是从本质上把握逻辑规则。     

The emergence of inferential rules: The use of pragmatic reasoning schemas by preschoolers

This study contrasts the pragmatic view with the natural logic view regarding the origin of inferential rules in conditional reasoning. The pragmatic view proposes that pragmatic rules emerge first, and the generalizations of these produce formal rules. In contrast, the natural logic view proposes that the formal rules emerge first and serve as a core that is then supplemented by pragmatic rules. In an experiment, scenarios involving conditional rules in different contexts, permission and arbitrary, were administered to independent groups of preschool children. To rule out the matching bias [Evans, J. St. B. T., & Lynch, J. S. (1973). Matching bias in the selection task. Br J Psychol 64, 391–397] as a possible explanation of reasoning performance, children were given conditional rules with a negated consequent. The results show that in the arbitrary context modus tollens (MT) was unavailable, and the use of modus ponens (MP) was unstable. In contrast, children in the permission context reliably used both MP and MT. In addition, they realized that a conditional rule does not imply a definite answer when the consequent holds. These findings suggest that, in their explicit forms, pragmatic rules emerge earlier than formal rules and in particular, even as basic a rule as MP is generalized from a context-specific form to a context-general one in preschool children.     

小学生图形推理策略个体差异

选取一至六年级儿童145名为对象,根据项目中图形关系的规则,把瑞文测验的项目分成6类,选择出14道题目为实验材料,探讨小学生图形推理策略的个体差异.结果发现,数学能力不同的小学生在解决简单图形推理问题时,基本上使用知觉算法策略,没有表现出差异.但在较难图形推理问题时差异显著,中高数学水平儿童主要使用分析策略和知觉分析策略;而低数学水平儿童基本上使用知觉匹配策略;除了格式塔类型的题目以外,推理水平高的儿童在解决这五类题目时主要使用分析策略和知觉分析策略,而推理水平低的儿童主要使用知觉匹配策略;推理水平高的儿童在解决较为简单的图形推理问题时的策略使用很集中,随着题目难度的加大,策略变得越来越分散,而推理水平低的儿童则无论题目难易如何变化,他们的策略主要使用知觉匹配策略,而且表现出随机性特点.     

Neurocognitive development of relational reasoning

Relational reasoning is an essential component of fluid intelligence, and is known to have a protracted developmental trajectory. To date, little is known about the neural changes that underlie improvements in reasoning ability over development. In this event‐related functional magnetic resonance imaging (fMRI) study, children aged 8–12 and adults aged 18–25 performed a relational reasoning task adapted from Raven's Progressive Matrices. The task included three levels of relational reasoning demands: REL‐0, REL‐1, and REL‐2. Children exhibited disproportionately lower accuracy than adults on trials that required integration of two relations (REL‐2). Like adults, children engaged lateral prefrontal cortex (PFC) and parietal cortex during task performance; however, they exhibited different time courses and activation profiles, providing insight into their approach to the problems. As in prior studies, adults exhibited increased rostrolateral PFC (RLPFC) activation when relational integration was required (REL‐2 > REL‐1, REL‐0). Children also engaged RLPFC most strongly for REL‐2 problems at early stages of processing, but this differential activation relative to REL‐1 trials was not sustained throughout the trial. These results suggest that the children recruited RLPFC while processing relations, but failed to use it to integrate across two relations. Relational integration is critical for solving a variety of problems, and for appreciating analogies; the current findings suggest that developmental improvements in this function rely on changes in the profile of engagement of RLPFC, as well as dorsolateral PFC and parietal cortex.     

The Case for Rules in Reasoning

A number of theoretical positions in psychology—including variants of case-based reasoning, instance-based analogy, and connectionist models—maintain that abstract rules are not involved in human reasoning, or at best play a minor role. Other views hold that the use of abstract rules is a core aspect of human reasoning. We propose eight criteria for determining whether or not people use abstract rules in reasoning, and examine evidence relevant to each criterion for several rule systems. We argue that there is substantial evidence that several different inferential rules, including modus ponens, contractual rules, causal rules, and the law of large numbers, are used in solving everyday problems. We discuss the implications for various theoretical positions and consider hybrid mechanisms that combine aspects of instance and rule models.     

小学儿童两维空间方位传递性推理能力的发展

研究了小学儿童两维空间方位传递性推理能力的发展水平及认知策略,同时,对心理模型理论进行了检验。被试为城市中等小学7岁、9岁、11岁儿童各24名,男女各半。单模型和双模型两种实验任务均为四前提。采用个别实验,儿童在前提呈现的情况下进行推理。主要研究结果：（1）7岁儿童开始萌发了两维空间方位传递性推理能力,9岁儿童和11岁儿童的推理能力处于发展和提高过程中;（2）随着年龄增长,使用模型建构策略解决问题的儿童人次越来越多,绝大部分11岁儿童都能使用这一策略进行推理。但即使儿童使用了模型建构策略,他们的推理成绩也没有反映出模型数量所造成的任务难度差异,即不符合心理模型理论关于模型数量的主要预期;（3）即使儿童能自发生成模型建构策略,儿童在两维空间方位传递性推理中使用这一策略的有效性程度并不高。