儿童非符号数量表征的特点及作用探析

数量表征是人类重要的基础心理能力,包括了符号和非符号两种表征系统。非符号数量表征是个体不需要依赖符号知识对视觉、听觉或跨通道呈现的实物或实物记号的数量刺激进行表达和运算的过程,具有一定的种系进化基础,先于语言发生。在未接受正式数学教育之前,儿童在一定程度上就能在平面空间和三维空间内表征非符号数量,并表现出了在视觉-听觉跨通道条件下的抽象非符号数量表征能力。个体早期即可能拥有两种特点不同的离散量表征系统:对非符号小数量的精确表征和对大数量的近似表征,但对其产生机制还存在不同的理论解释。在符号系统出现后,非符号数量表征和符号数量表征在行为及神经机制既有区别又有联系,并可能通过映射过程在头脑内部相互转换。与符号数量表征相互作用发展成为个体高级数能力的重要基础。     

The role of submicroscopic and symbolic representations in chemical explanations

Chemistry is commonly portrayed at three different levels of representation – macroscopic, submicroscopic and symbolic – that combine to enrich the explanations of chemical concepts. In this article, we examine the use of submicroscopic and symbolic representations in chemical explanations and ascertain how they provide meaning. Of specific interest is the development of students' levels of understanding, conceived as instrumental (knowing how) and relational (knowing why) understanding, as a result of regular Grade 11 chemistry lessons using analogical, anthropomorphic, relational, problem‐based, and model‐based explanations. Examples of both teachers' and students' dialogue are used to illustrate how submicroscopic and symbolic representations are manifested in their explanations of observed chemical phenomena. The data in this research indicated that effective learning at a relational level of understanding requires simultaneous use of submicroscopic and symbolic representations in chemical explanations. Representations are used to help the learner learn; however, the research findings showed that students do not always understand the role of the representation that is assumed by the teacher.     

An analysis of Grade 11 learners’ levels of understanding of functions in terms of APOS theory

This article reports on a study of six Grade 11 learners’ levels of understanding of concepts related to the function definition and representation. Task-based clinical interviews were used to elicit the learners’ interpretations and reasoning when working with these function-related concepts. Indicators for Action–Process–Object–Schema (APOS) theory conception levels were formulated and used to categorise learners’ written and interview responses into conception levels of understanding of the function concept. According to the South African Curriculum Assessment Policy Statement (CAPS) curriculum, Grade 11 learners are expected to be operating at the object and schema levels after instruction. however, the results indicated that learners were operating at the action and process levels as their understanding was characterised by vague definitions of function-related concepts and memorisation of procedures for translating between symbolic and graphical representations.     

Students' mental load,stress, and performance when working with symbolic or symbolic–textual molecular representations

In science education, representations are necessary inter alia for the understanding of relationships between structures and systems. However, several studies have identified difficulties of students when working with representations. In the present study, we investigated students' responses (regarding their preference, test performance, mental load (ML), and stress) toward two kinds of representations: symbolic representations, which only use abstract symbols, versus combined symbolic–textual representations, which additionally comprise textual elements. Therefore, students were randomly assigned to one of two treatment groups: one group worked on test tasks accompanied by symbolic representations, and the others worked on the same tasks, but with symbolic–textual representations. Thereafter, the students' test performance and ML were assessed. The level of perceived stress and the salivary cortisol concentration were measured before and after the test and again a few minutes later. Additionally, heart rate variability parameters were assessed continuously. We found a strong preference of the test version with symbolic representations. Additionally, the students showed better test performance and lower ML when they worked with symbolic representations. However, the level of perceived stress was comparable between both groups and there was no strong physiological stress response: The cortisol concentration decreased in both groups and the heart rate was relatively similar. However, during the second half of the test, we observed a significantly higher ratio between low and high heartbeat frequencies in the group with symbolic–textual representations and we found an indirect influence of the kind of representation on test performance through its effect on ML. The poorer test performance and higher ML in connection with symbolic–textual representations confirm previous studies, which found that symbolic–textual representations pose major problems for students. Thus, teachers should enable students to understand symbolic–textual representations and consider carefully whether they can use symbolic representations instead, especially when they teach complex content.     

On fractions and non-standard representations: Pre-service teachers' concepts

The present study examined the reasoning strategies and arguments given by pre-service school teachers as they solved two problems regarding fractions in different symbolic representations. In the first problem, the pre-service school teachers were asked to compare between two different fractions having the same numerical representation, and in the second problem, they were asked to compare between different notational representations of the same fraction. Numeration systems in bases other than ten were used to generate various representations of fractions. All students were asked to provide justifications to their responses. Strategies and arguments relative to pre-service teachers' concepts of fractions and place value were identified and analyzed based on results of 38 individual clinical interviews, and written responses of 124 students. It was found that the majority of students believe that fractions change their numerical value under different symbolic representations.     

READING MATHEMATICS REPRESENTATIONS: AN EYE-TRACKING STUDY

We use eye tracking as a method to examine how different mathematical representations of the same mathematical object are attended to by students. The results of this study show that there is a meaningful difference in the eye movements between formulas and graphs. This difference can be understood in terms of the cultural and social shaping of human perception, as well as in terms of differences between the symbolic and graphical registers.     

Differential effects on the achievement of males and females of teaching the particulate nature of chemistry

Chemistry concepts in three 2‐week modules were presented to a treatment and control group of high school students from 10 different high schools. The treatment group was taught using three representations of matter (macroscopic, particulate, and symbolic), whereas only two representations (macroscopic and symbolic) were used with the control group. The treatment group scored significantly higher across the combined score of all three modules. When the data were analyzed for gender–group interactions, females who were taught the particulate representation scored as well as males whereas those who were not taught the particulate representation scored significantly lower than males. Teaching males the particulate representation did not significantly affect males' achievement. © 2002 Wiley Periodicals, Inc. J Res Sci Teach 39: 911–927, 2002     

READING MATHEMATICS REPRESENTATIONS: AN EYE-TRACKING STUDY

We use eye tracking as a method to examine how different mathematical representations of the same mathematical object are attended to by students. The results of this study show that there is a meaningful difference in the eye movements between formulas and graphs. This difference can be understood in terms of the cultural and social shaping of human perception, as well as in terms of differences between the symbolic and graphical registers.

     

认知心理学关于时间推理研究

时间推理是人们对事件的时序、时距和时点进行推论的心理过程。时间推理是基于一定的表征来进行的 ,不同的表征系统对时间推理的效应不同。如词表表征方向效应和距离效应 ,数字表征产生越界效应。还可以进一步看到 ,对时间信息进行推理 ,实质上就是对时间信息的表征进行信息加工。例如 ,对于词表表征和数值表征 ,就分别有计数的加工和运算的加工两种方式。特别值得指出的是 ,即使对相同的表征 ,也可有不同的加工方式。当前有关时间推理的研究 ,都是从信息表征和内部操作的角度来进行研究的 ,最能体现当前认知心理学研究的特色     

Relationships among computational performance,pictorial representation,symbolic representation and number sense of sixth‐grade students in Taiwan

Twenty classes in ten schools with 627 sixth‐grade students in five cities in Taiwan participated in this study. The research provides information on the performance differences among written computation, pictorial representation, symbolic representation and number sense. The results of One‐way ANOVA analysis indicate that significant difference was found among WCT, PRT, SRT and NST tests, with F=536.327, p=0.000. The a posteriori comparisons show for each pair (WCT vs PRT, WCT vs SRT, WCT vs NST, PRT vs SRT and SRT vs NST) significant difference at the 0.001 level (p=0.000), except for the pair comparison between PRT and NST. This implies that these Taiwanese students were highly skilled in written computation but their written skills were not equally transferred to use of non‐computational paths that depended on symbolic representation, pictorial representation and number sense to solve similar problems.     

An exploration of the relationship between symbolic representation in dramatic play and art and the cognitive and reading readiness levels of kindergarten children

The purpose of this study was to explore the relationship between symbolic representation in dramatic play and art and the cognitive and reading readiness levels of kindergarten children. Specifically, the study attempted to determine if there is a significant difference in the performance of kindergarten children on the conservation of number tasks (reflective of the level of operational thought and cognitive development), the Metropolitan Reading Readiness Test, and their involvement and/or level of symbolic re resentation in dramatic play and/or art. Results indicated that children at diffrent cognitive levels (a) performed significantly differently on the Metropolitan Reading Readiness Test, (b) spent significantly different amounts of time involved in dramatic play, and (c) expressed significantly different levels of symbolic representations in dramatic play. No differences were found to exist, however, in their symbolic representation in art.     

Relations Between Secondary Pupils’ Conceptions About Functions and Problem Solving in Different Representations

The present study explores pupils’ constructed definitions of the concept of function in relation to their abilities in dealing with tasks of functions involving different forms of representations and problem solving tasks. A major concern is also to examine the interrelations between these three ways of thinking about or dealing with the concept of function. The sample of the study consisted of secondary school pupils in Cyprus. A test was developed which involved seven items: one item requested pupils to provide a definition of what function is and the other six items were developed in order to investigate pupils’ ability to transfer information from one representation to another and to solve problems on function. Findings revealed pupils’ difficulties in giving a proper definition for the concept of function and resolving problems on functions involving conversions between diverse modes of representation. Several inconsistencies among pupils’ constructed definitions, their competence to use different representations of functions and their problem solving ability, were also uncovered, indicating lack of flexibility between different ways of approaching functions.     

A STUDY OF SEMIOTIC REGISTERS IN THE DEVELOPMENT OF THE DEFINITE INTEGRAL OF FUNCTIONS OF TWO AND THREE VARIABLES

Tracing the path from a numerical Riemann sum approximating the area under a curve to a definite integral representing the precise area in various texts and online presentations, we found 3 semiotic registers that are used: the geometric register, the numerical register, and the symbolic register. The symbolic register had 3 representations: an expanded sum, a sum in sigma notation, and the definite integral. Reviewing the same texts, we found that in the presentation of double and triple integrals, not a single textbook continues to present the numerical register and the expanded sum representation of the symbolic register. They are implied and the expectation appears to be that students no longer need them. The omission of these representations is quite ubiquitous and correspondingly affects millions of students. Materials that present the missing numerical register representation and the expanded sum representation of the symbolic register throughout topics associated with double and triple integrals have been created. This paper presents the results of a clinical study on the improvement of student comprehension of multivariable integral topics when these representations are included.     

多元表征:探寻数学智慧课堂的一把密钥

将多元表征渗透到数学课堂教学中,一方面可以调动学生多感官的认知因素,促进知识的理解,培养学生的数学思维以及促进学生数学智慧的生长;另一方面通过对问题进行多元化的表征,为学生解决数学问题提供了新的平台,从而有助于提高学生对问题多角度的解释能力和创新能力。数学中多元表征的教学策略为:精深挖掘资源,探索数学表征的多元化;运用教育机智,达到多元表征的最优化。但要注意,多元表征不是每堂课都适用,也不是每个学生都能在课上得到最大的收获。     

The interplay of representations and patterns of classroom discourse in science teaching sequences

The purpose of this study is to examines the relationship between the communicative approach of classroom talk and the modes of representations used by science teachers. Based on video data from two physics classrooms in Singapore, a recurring pattern in the relationship was observed as the teaching sequence of a lesson unfolded. It was found that as the mode of representation shifted from enactive (action based) to iconic (image based) to symbolic (language based), there was a concurrent and coordinated shift in the classroom communicative approach from interactive–dialogic to interactive–authoritative to non-interactive–authoritative. Specifically, the shift from enactive to iconic to symbolic representations occurred mainly within the interactive–dialogic approach while the shift towards the interactive–authoritative and non-interactive–authoritative approaches occurred when symbolic modes of representation were used. This concurrent and coordinated shift has implications on how we conceive the use of representations in conjunction with the co-occurring classroom discourse, both theoretically and pedagogically.     

Symbolic Functioning in Very Young Children: Understanding of Pictures and Models

Before one can understand or use any symbol, one must first realize that it is a symbol, that is, that it stands for or represents something other than itself. This article reports 4 studies investigating very young children's understanding of 2 different kinds of symbolic stimuli--scale models and pictures. The data replicate previous findings that 2.5-year-old children have great difficulty appreciating the relation between a scale model and the larger space it represents, but that they very readily appreciate the relation between a picture and its referent. This result is interpreted in terms of the dual orientation hypothesis. Models are difficult for young children because they require a dual representation--a child must think about a model both as an object itself and as a representation of something else. Because pictures are not salient as real objects, they do not require a dual representation. Several kinds of evidence supporting the dual orientation hypothesis are presented. An additional result was the occurrence of a transfer effect: Prior experience with a picture task led to better performance on a subsequent model task. This finding suggests that experience with a symbolic medium they understand can help young children figure out a different, unfamilar medium that they would otherwise not understand.     

中大班幼儿统计数据表征能力的发展

以上海市两所幼儿园的120名中班和大班幼儿为被试考察幼儿统计数据表征能力的发展时发现:(1)中大班幼儿的统计数据表征可分为四种类型,分别是图画型、数字型、图文结合型和初级图表型,每种类型各可分为五个发展水平。(2)随着年龄的增长,幼儿对统计数据的表征类型从数字型、图画型逐渐向图文结合型和初级图表型过渡。     

时间信息表征方式研究概述

个体如何表征时间信息,是近年来时间研究领域的关注焦点。已有研究表明,个体对客观日历时间采用线性和网络层次的方式进行表征,将来研究应当关注个体对相同语义、不同表述方式的客观时间的表征方式。     

A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity, we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes. Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning. Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically gathered: some empirical evidence is presented in this paper.     

蛋白质序列的一种新的二维图形表示

基于氨基酸的五字母模型,提出蛋白质序列的一种二维图形表示方法,然后证明这种图是非退化的.最后用该方法给出了8个物种的β球蛋白序列的图形表示.