三～六年级小学生数学能力及认知结构的发展

本文是对三～六年级昆明地区和西双版纳地区的汉族与傣族儿童所作的数学能力和认知结构发展的实验研究.结果指出:昆明三、四年级儿童处于具体运算阶段,能掌握维度间的双重协调,准确地完成直观加减乘除运算.从五年级开始,儿童数学能力和认知结构出现了显着变化,儿童运用抽象推理进行维度间精致协调,开始向形式运算转化.其发展的过程是从量的增长逐步地过渡到质的变化.西双版纳地区的两种不同民族儿童数学能力发展未出现明显的差异,但与昆明同年级儿童相比,都低于昆明儿童.低年级差异尤其明显,高年级差异趋于缩小.说明早期教育对儿童数学能力和认知结构的发展有特殊意义.     

5-13岁儿童对物种起源认知的发展

本研究采用临床访谈法,采用开放式和封闭式问题情境,测查了5-13岁儿童对物种起源认知的发展。结果显示儿童对物种起源的解释越来越符合科学的解释,这种认知发展过程并不是以一种起源认知替代另一种起源认知的过程,而是多种起源认知以某种方式共存于儿童的认知系统中。本研究结果既显示了儿童对物种起源认知发展的跨文化一致性,也体现了不同文化和宗教环境影响下的差异性。     

7─8岁儿童数学运算能力发展的实验研究

７─８岁儿童数学运算能力发展的实验研究一、引言儿童数学运算能力的发展是教育工作者和心理学家共同关心的问题。近二十年来的研究已经深入到对儿童解决数学问题时的策略、表征和机制的探讨。心理学的研究还发现生活在不同文化环境中的人,解决数学运算时既有共同规则,...     

态射的建构与发展:发生认识论的一种动态形式化

发生认识论主要研究认识的心理发生及其形式化。对于后者的研究,上个世纪30年代皮亚杰建构了“群集”结构来解释儿童前运算阶段的思维方式,70年代又引入了数学范畴论来解释形式运算阶段的认知发展,再一次对儿童思维的发展与科学发展两者间的连续性给出了明证。数学范畴论是对数学对象的结构间变换进行形式化的一个数学分支,建构的思想以及对所建构结构进行转换是范畴论的本质,而其中作为范畴对象的态射与态射组合体现了其建构本质,且能最为妥贴地描述认识的动态发生与发展。新的理论建模将对应与转换揉合在一起更好地阐释认知发展过程中每一转换过程的机制与结构发展。本文试图通过介绍认识发生过程中对应、转换的发展,以及所引入数学范畴论中态射的概念,来阐释认识发生中认知生成工具———态射———的建构与发展,以及认知结构的系统发展,从而对皮亚杰晚年这一形式化工作予以简单说明与评价。     

数学学习困难儿童的加工速度与工作记忆

以数学学习困难儿童为研究对象,系统考察了数学学习困难儿童与控制组的加工速度和工作记忆,比较了工作记忆和加工速度对数学学习困难影响的相对贡献.结果发现,与控制组相比,数学学习困难儿童在工作记忆和加工速度方面均存在明显的不足,但加工速度不能解释不同能力组之间的差异,数学学习困难儿童的缺陷在于工作记忆能力的下降.工作记忆缺陷在于数字工作记忆、视空间工作记忆和中央执行功能的整体不足,中央执行功能对数学学习困难儿童影响的解释量最大.     

超常儿童数学能力的因素分析

培养与发展超常儿童的智力与能力是教育心理研究的一个重要课题。本文将在智力与能力发展理论的基础上,重点对组成超常儿童数学能力的因素进行结构分析。通过因素分析的方法抽出了五个主因素．即综合运算能力、逻辑思维能力、抽象概括能力、空间想象能力、灵活的形象思维能力,并对各能力特点作了进一步分析。本文的结果将为超常儿童智能发展的理论研究提供可资参照。     

发展瀑布效应:儿童认知功能与社会适应领域的研究

发展瀑布效应指的是在复杂的发展过程中各种交互作用或相互作用的累积结果,这种发展的累积效应导致跨越不同水平、领域、系统或者不同世代的传播效应。本文阐述了研究者利用发展瀑布效应解释儿童认知功能与社会适应领域的发展时采用的不同模型,介绍了儿童认知功能与社会适应领域的发展瀑布效应,最后对发展瀑布效应在这两个领域的研究方法与应用价值进行了总结与展望。     

解释水平视角下的乐观偏差效应

研究基于解释水平理论,考查心理距离对乐观偏差效应的影响。实验从心理距离的事件概率维度、时间距离维度、空间距离维度入手,采用自编的材料对677名被试进行测量,探讨心理距离的远近对乐观偏差产生的影响。结果表明,被试对未来生活事件的判断存在总体上的乐观偏差。在较远的心理距离条件下,被试表现出更大的乐观偏差;而在较近的心理距离中,乐观偏差效应明显减小。但在时间距离维度以及消极事件的概率维度上,表现出与总体乐观偏差不一致的现象。     

5～8岁儿童对模糊信息具有多重解释的理解

参照Carpendale和Chandler的实验范式,研究儿童对于“人们可能对同样信息给出不同解释”这一现象的理解,考察5~8岁儿童的解释性心理理论的发展。结果表明,5岁儿童不能理解心理过程的解释性,认为同样的信息只有一种合理的解释。从6岁开始,儿童才认识到,模糊信息可以有多种解释,但6、7岁时的这种理解并不完善,成绩随着任务要求而变化。8岁儿童才有比较稳定的解释性心理理论。     

小学生运算法则掌握的实验研究

许多研究已表明,运算能力是儿童数学能力的重要组成部分.但长期以来,人们对儿童运算能力的培养更多地集中在运算技能和数的结合知识上.在我国小学数学教学中,存在着不重视运算法则教学的现象,表现为教材中缺乏表达完整的运算法则,教学中以运算法则的记忆代替学生掌握运算法则的智力活动.     

Mental movements without magnitude? A study of spatial biases in symbolic arithmetic

McCrink (McCrink, Dehaene, & Dehaene-Lambertz (2007). Moving along the number line: Operational momentum in nonsymbolic arithmetic. Perception and Psychophysics, 69(8), 1324-1333) documented an “Operational Momentum” (OM) effect - overestimation of addition and underestimation of subtraction outcomes in non-symbolic (dot pattern) arithmetic. We investigated whether OM also occurs with Arabic number symbols. Participants pointed to number locations (1-9) on a visually given number line after computing them from addition or subtraction problems. Pointing was biased leftward after subtracting and rightward after adding, especially when the second operand was zero. The findings generalize OM to the spatial domain and to symbolic number processing. Alternative interpretations of our results are discussed.     

Over-estimation in numerosity estimation tasks: More than an attentional bias?

Over- and under-estimation have been observed in numerosity estimation and approximate arithmetic tasks. Two different models have been proposed to account for these reverse patterns of performance: 1) the bi-directional mapping account (Crollen, Castronovo, & Seron, 2011); 2) the operational momentum hypothesis (McCrink, Dehaene, & Dehaene-Lambertz, 2007). Our study was designed to examine whether the operational momentum could account for the over-estimation found in numerosity estimation tasks. To this aim, a series of 3 experiments involving a symbolic to non-symbolic numerical mapping and a rightward or leftward displacement along the mental number line were designed. Over-estimation was observed in these three tasks irrespective of the direction and size of the displacement to be done on the mental number line. These results thus clearly demonstrated that overestimation was not merely due to an attentional bias, but rather relied on the cognitive operation of mapping two differently scaled numerical representations.     

L’outil « estimateur », la ligne numérique mentale et les habiletés arithmétiques

This report presents the effects of learning study based on the Estimator program to learn the addition and subtraction operations on children selected for mathematical difficulties. The Estimator is designed to link the magnitudes of the mental number line with the verbal representations of exact arithmetic. Experiment shows that using the Estimator for five 30-minute sessions increases not only the children's arithmetic capacities but also other numerical knowledge assessed with Zareki-R. By taking account of the limits of the sample, the results are discussed in terms of (re) educational implications.     

Heuristics and biases in mental arithmetic: revisiting and reversing operational momentum

ABSTRACT

Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum (OM) effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems (e.g., 3+0, 3?0) but reverse OM with non-zero problems (e.g., 2+1, 4?1). In a third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving.     

Preschoolers prior formal mathematics education engage numerical magnitude representation rather than counting principles in symbolic ±1 arithmetic: Evidence from the operational momentum effect

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4–7 years old) who know Arabic numerals and counting principles (CP), but without prior school math education, solved addition and subtraction problems presented as videos with one as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered CP and the concept of an exact number.     

Moving attention along the mental number line in preschool age: Study of the operational momentum in 3‐ to 5‐year‐old children's non‐symbolic arithmetic

People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3–5 years old were tested with non‐symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give‐a‐number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial‐directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial‐directional and magnitude OM. This undermines the canonical version of the number line‐based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial‐numerical associations in numerical development are proposed.     

Number line estimation and complex mental calculation: Is there a shared cognitive process driving the two tasks?

It is widely accepted that different number-related tasks, including solving simple addition and subtraction, may induce attentional shifts on the so-called mental number line, which represents larger numbers on the right and smaller numbers on the left. Recently, it has been shown that different number-related tasks also employ spatial attention shifts along with general cognitive processes. Here we investigated for the first time whether number line estimation and complex mental arithmetic recruit a common mechanism in healthy adults. Participants’ performance in two-digit mental additions and subtractions using visual stimuli was compared with their performance in a mental bisection task using auditory numerical intervals. Results showed significant correlations between participants’ performance in number line bisection and that in two-digit mental arithmetic operations, especially in additions, providing a first proof of a shared cognitive mechanism (or multiple shared cognitive mechanisms) between auditory number bisection and complex mental calculation.     

When combined spatial polarities activated through spatio-temporal asynchrony lead to better mathematical reasoning for addition

Several recent studies have supported the existence of a link between spatial processing and some aspects of mathematical reasoning, including mental arithmetic. Some of these studies suggested that people are more accurate when performing arithmetic operations for which the operands appeared in the lower-left and upper-right spaces than in the upper-left and lower-right spaces. However, this cross-over Horizontality × Verticality interaction effect on arithmetic accuracy was only apparent for multiplication, not for addition. In these studies, the authors used a spatio-temporal synchronous operand presentation in which all the operands appeared simultaneously in the same part of space along the horizontal and vertical dimensions. In the present paper, we report studies designed to investigate whether these results can be generalized to mental arithmetic tasks using a spatio-temporal asynchronous operand presentation. We present three studies in which participants had to solve addition (Study 1a), subtraction (Study 1b), and multiplication (Study 2) in which the operands appeared successively at different locations along the horizontal and vertical dimensions. We found that the cross-over Horizontality × Verticality interaction effect on arithmetic accuracy emerged for addition but not for subtraction and multiplication. These results are consistent with our predictions derived from the spatial polarity correspondence account and suggest interesting directions for the study of the link between spatial processing and mental arithmetic performances.     

Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect

When we add or subtract, do the corresponding quantities “move” along a mental number line? Does this internal movement lead to spatial biases? A new method was designed to investigate the psychophysics of approximate arithmetic. Addition and subtraction problems were presented either with sets of dots or with Arabic numerals, and subjects selected, from among seven choices, the most plausible result. In two experiments, the subjects selected larger numbers for addition than for subtraction problems, as if moving too far along the number line. This operational momentum effect was present in both notations and increased with the size of the outcome. Furthermore, we observed a new effect of spatial-numerical congruence, related to but distinct from the spatial numerical association of response codes effect: During nonsymbolic addition, the subjects preferentially selected numbers at the upper right location, whereas during subtraction, they were biased toward the upper left location. These findings suggest that approximate mental arithmetic involves dynamic shifts on a spatially organized mental representation of numbers. Supplemental materials for this study may be downloaded from app.psychonomic-journals.org/content/supplemental.     

心算加工中tie effect的加工机制

tie effect主要表现为tie比nontie更快更准确解决,以及tie/nontie与问题大小的交互作用,较大题目的tie比nontie具有更明显的反应时优势,tie的问题大小效应比nontie要小的多。有关tie effect加工机制的解释主要有基于编码的理论和基于结果通达的理论。目前的研究主要通过操作问题呈现方式以及报告加工策略的方法分别考察编码方式及加工策略的影响,今后研究应开拓新的研究范式以加深对其加工机制的认识。