A computational analysis of conservation

An approach to modeling cognitive development with a generative connectionist algorithm is described and illustrated with a new model of conservation acquisition. Among the conservation phenomena captured with this model are acquisition, the problem size effect, the length bias effect, and the screening effect. The simulations suggest novel explanations for sudden jumps in conservation performance (based on new representations of conservation transformations) and for the problem size effect (based on an analog representation of number). The simulations support the correlation-learning explanation of length bias (that length correlates with number during number altering transformations). Some conservation phenomena that so far elude computational modeling attempts are also discussed along with their prospects for capture. Suggestions are made for theorizing about cognitive development as well as about conservation acquisition. A variety of classic puzzles about cognitive development are addressed in the light of this model and similar models of other aspects of cognitive development.     

Application of piecewise hierarchical linear growth modeling to the study of continuity in behavioral development of baboons (Papio hamadryas)

In behavioral science, developmental discontinuities are thought to arise when the association between an outcome measure and the underlying process changes over time. Sudden changes in behavior across time are often taken to indicate that a reorganization in the outcome-process relationship may have occurred. The authors proposed in this article the use of piecewise hierarchical linear growth modeling as a statistical methodology to search for discontinuities in behavioral development and illustrated its possibilities by applying 2-piece hierarchical linear models to the study of developmental trajectories of baboon (Papio hamadryas) mothers' behavior during their infants' 1st year of life. The authors provided empirical evidence that piecewise growth modeling can be used to determine whether abrupt changes in development trajectories are tied to changes in the underlying process.     

Children's Criteria for Representational Adequacy in the Perception of Simple Sonic Stimuli

This article describes how children build increasingly abstract knowledge of linear measurement, emphasizing ways they relate space and number. Assessments indicate children struggle to understand measurement, especially concepts related to complex paths as in perimeter tasks. This article draws on developmental accounts of children's knowledge of measurement to describe the coordination of cognitive processes as a progression through increasingly abstract layers of strategy (Clements, 2003; Lehrer, 2003) within a constructivist perspective (Steffe &; Cobb, 1988; Steffe &; Thompson, 2000). This article reports 4 case studies from a 6-month teaching experiment with Grade 4 students to examine ways of promoting their strategy development. Attending to children's understanding of length along straight objects alone proved inadequate in predicting and validating models of children's developing strategies and concepts for perimeter. Setting perimeter tasks that emphasized integral relations among unit of length, sides of polygons and perimeter, and prompting children to integrate partitive, iterative, and counting schemes appeared to promote increasingly abstract length measurement strategies. The researchers came to distinguish between a naïve use of units to find length (Level 2a) and a more abstract strategy for unitizing and iterating (Level 2b), extending an earlier framework (Clements, Battista, Sarama, Swaminathan, &; McMillen, 1997). Finally, implications for instruction are drawn from the development of tasks and questioning sequences.     

Facilitation of attention to number and conservation of number

This study examined the relation between number conservation and attention to number, density, and length or area in kindergarteners (age 5 years, 8 months) and third graders (age 8 years, 11 months). Attention was assessed by the dimensional preference technique, using triads. Both the attention and conservation tasks included stimulus factors known to facilitate or hinder conservation: small or large numbers of objects, nonlinear or linear arrays, and trials ordered from easiest to hardest, hardest to easiest, or randomly. The number of objects and type of array affected both tasks similarly and there was an order effect in the attention task. There was an increase in attention to number from kindergarten nonconservers to kindergarten conservers to third-graders. Possible developmental interactions between attention and conservation were suggested.     

儿童早期空间非标准测量能力发展研究进展

测量是数学的主题之一。空间非标准测量能力的发展对儿童数学认知能力的发展有重要作用,对儿童思维抽象性、逻辑性、创造性的培养,问题解决能力的提高有重要影响。文章通过对近四十年来儿童空间非标准测量能力发展研究的成果进行概括分析,发现研究者主要对儿童空间非标准测量的认知加工线索,长度测量策略与工具的使用,空间非标准测量知识的获得顺序以及相关的测量教学顺序,空间非标准测量能力与守恒和推理能力的关系等问题进行了研究。并在此基础上提出了儿童空间非标准测量能力发展研究需进一步探讨的问题。     

Conservation training in four-year-old children

In Experiment 1 two training procedures were used to teach four-year-olds to conserve. Verbal rule instruction consisted of providing verbal rules and demonstrations of the operations referred to by the rules. Feedback consisted of providing verbal feedback contingent upon the children's responses. One week after training on conservation of number and length the children were given a posttest which included tests of conservation of number, length, and mass. Children who were given verbal rule instruction conserved significantly more on the number and length posttest problems than children who were not. However, this learning did not transfer to the mass problems, possibly because mass is not naturally acquired until some time after conservation of number and length. The feedback training procedure had no effect on conservation performance. In Experiment 2, the verbal rule instruction procedure was used to train four-year-olds on conservation of length and mass. One week after training the children were tested on both conservation of number, which is typically acquired before length and mass, and conservation of weight, which is typically acquired after length and mass, as well as on conservation of length and mass. Children who were given training conserved more on all four types of problems than children in the control group.     

Using the longitudinal Guttman simplex as a basis for measuring growth

Many difficulties inherent in the measurement of growth stem from the use of traditional measurement methodologies. The longitudinal Guttman simplex (LGS), an alternative approach based on a model of growth, is discussed in this article. The LGS has several advantages over traditional methodology. First, interindividual differences in developmental rates are a part of the model. Second, the LGS procedure can easily handle any number of occasions of measurement. Third, the LGS is suited to nonlinear as well as linear monotonic growth. Fourth, a consistency index associated with the LGS methodology, CL, indicates the extent to which cumulative, unitary development characterizes a particular latent variable. Finally, and perhaps most important, because a model of the growth undergone by the latent variable being measured is incorporated in the LGS model the resulting instruments enjoy a high level of construct validity. The LGS is limited to cumulative, unitary development; additional measurement theories are needed for other kinds of development.     

The sentence-verification paradigm: A case study of two conflicting aproaches to individual differences

The contrasting approaches of differential psychology and cognitive psychology to the same individual differences data are outlined. Using illustrative data from the Clark and Chase (1972) sentence-picture verification task, four loci of conflict between these two disciplines are identified. These areas of conflict center around issues of (1) theory versus measurement, (2) meaningfulness versus reliability, (3) linearity of relationships, and (4) discontinuities in performance. We conclude on the basis of observed incompatibilities that a simple derivation of differential psychology from cognitive psychology is not likely, but separate development of complementary theories may be possible.     

The problem of dream length in analysis of content.

The present study investigated the linear as well as nonlinear relationships between scores on selected dream content scales and dream length. Analysis indicated that biases due to control solely for the linear components are almost negligible.     

Attention and cognition

Forty 4-year-old children were subjects in an experimental designed to determine whether learning to attend to relevant cues is a sufficient condition for acquisition of length and number conservation. Three groups of non-conservers were trained by means of an oddity-problem procedure to attend to the cues specifying either length, number, or both length and number. They were subsequently tested along with an untrained control group on tasks of length, number, mass and continuous quantity conservation. Some improvement in conservation was found, but it was neither impressive in magnitude nor specific to the cues of training. For example, the group trained to attend to length cues conserved length about 25% of the time, the same rate at which this group also conserved number, mass, and liquid quantity. This non-specificity is contrary to an attention hypothesis and suggests instead that training, to the extent it was effective, induced a general, abstract quantitative knowledge.     

Conservation of Physical Attributes of Animate and Inanimate Objects

Children's (N = 83 boys and girls in kindergarten and first grade) responses to a series of conservation tasks (number, length, mass, weight) which had animate and inanimate referent objects, in two experiments, showed significant differences in the proportions of conservation between animate and inanimate objects. Physical attributes were conserved more easily with inanimate objects than with animate objects, and significant conservation differences between various animate objects were found also.     

Quality Vs. Quantity: Assessing Behavior Change over Time

Beyond the typical design factors that impact a study’s power (e.g., participant sample size), planning longitudinal research involves additional considerations such as assessment frequency and participant retention. Because this type of research relies so strongly on individual commitment, investigators must be judicious in determining how much information is necessary to study the phenomena in question; collecting too little information will render the data less useful, but requiring excessive participant investment will likely lower participation rates. We conducted a simulation study to empirically examine statistical power and the trade-off between assessment quality (as a function of instrument length) and assessment frequency across a number of sample sizes with intermittently missing data or attrition. Results indicated that reductions in power resulting from shorter, less reliable measurements can be at least somewhat offset by increasing assessment frequency. Because study planning involves a number of factors competing for finite resources, equations were derived to find the balance points between pairs of design characteristics affecting statistical power. These equations allow researchers to calculate the amount that a particular design factor (e.g., assessment frequency) would need to increase to result in the same improvement in power as increasing an alternative factor (e.g., measurement reliability. Applications for the equations are discussed.     

Effects of Differential Instructions and Anxiety Level on Discrimination Learning

Seventy-seven nonconserver-conserver dyads and 53 nonconserver-nonconserver dyads were given a conservation of length task which encouraged conflicting judgments from the nonconserving dyad. Ss were boys and girls aged 4 through 8. Control tasks for social dominance/compliance were also administered. All sessions were videotaped. The nonconserver-nonconserver dyads did not tend to produce the correct answer by perspective coordination and showed little posttest gain. Nonconservers who had been paired with conservers showed posttest gain on length, mass, liquid, and number conservation problems. Being the winner of an interaction session was associated with opposing the partner's judgment and with producing justifications in conserver-nonconserver dyads, but only with social compliance in nonconserver-nonconserver dyads. Justifications in support of conservation judgments were invariably by logical principles but very rarely so in support of nonconservation judgments. The results are discussed in terms of the relative importance of cognitive conflict and transmission as crucial social experiences in cognitive development, and in terms of the possibility that nonconservers may lack the notion of objective correctness relative to conservers.     

The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: evidence for a segmented linear model

This experiment aimed to expand previous findings on the development of mental number representation. We tested the hypothesis that children's familiarity with numbers is directly reflected by the shape of their mental number line. This mental number line was expected to be linear as long as numbers lay within the range of numbers children were familiar with. Five- to 9-year-olds (N=78) estimated the positions of numbers on an external number line and additionally completed a counting assessment mirroring their familiarity with numbers. A segmented regression model consisting of two linear segments described number line estimations significantly better than a logarithmic or a simple linear model. Moreover, the change point between the two linear segments, indicating a change of discriminability between numbers, was significantly correlated with children's familiar number range. Findings are discussed in terms of the accumulator model, assuming a linear mental representation with scalar variability.     

Apparent motion by edge discontinuities

When the eyes move vertically across a jagged diamond, a local shift (LS) of edge discontinuities and a global shape distortion (GD) (ie expansion/contraction opposite to that expected by the aperture effect) are perceived. These phenomena cannot be accounted for by a local motion signals integration rule based either on the intersection of constraint lines or on the velocity vector summation. The threshold for GD perception and the salience of LS and GD (1 to 10 scale) were measured in two experiments by different methods and displays. In experiment 1 we induced GD through mimicking LS with a kinetic pattern constituted of a set of circular illusory apertures revealing drifting gratings. The point of subjective equality for compression/expansion was reached for gratings the linear extrapolations of which form an angle of 94.4 degrees. In experiment 2 observers followed a dot moving along the vertical elongation axis of a static jagged diamond (with 70 degrees or 90 degrees angles), varying in the shape (triangular, wave, square), frequency, and amplitude of edge discontinuities. GD scores were correlated with LS scores that were inversely related to frequency/amplitude ratios of triangular edge discontinuities. Data are partially accounted for by averaging neighbouring local motion-capture vectors. Results prove that there are strong interrelations between phenomena in which visual motion affects visual localisation and phenomena involving apparent deformation of global shape.     

分类数据测量等价性检验方法及其比较：项目阈值(难度)参数的组间差异性检验

测量工具满足等价性是进行多组比较的前提, 测量等价性的检验方法主要有基于CFA的多组比较法和基于IRT的DIF检验两类方法。文章比较了单维测验情境下基于CCFA的DIFFTEST检验方法和基于IRT模型的IRT-LR检验方法, 以及多维测验情境下DIFFTEST和基于MIRT的卡方检验方法的差异。通过模拟研究的方法, 比较了几种方法的检验力和第一类错误, 并考虑了样本总量、样本量的组间均衡性、测验长度、阈值差异大小以及维度间相关程度的影响。研究结果表明：(1)在单维测验下, IRT-LR是比DIFFTEST更为严格的检验方法; 多维测验下, 在测验较长、测验维度之间相关较高时, MIRT-MG比DIFFTEST更容易检验出项目阈值的差异, 而在测验长度较短、维度之间相关较小时, DIFFTEST的检验力反而略高于MIRT-MG方法。(2)随着阈值差值增加, DIFFTEST、IRT-LR和MIRT-MG三种方法的检验力均在增加, 当阈值差异达到中等或较大时, 三种方法都可以有效检验出测验阈值的不等价性。(3)随着样本总量增加, DIFFTEST、IRT-LR和MIRT-MG方法的检验力均在增加; 在总样本量不变, 两组样本均衡情况下三种方法的检验力均高于不均衡的情况。(4)违背等价性题目个数不变时, 测验越长DIFFTEST的检验力会下降, 而IRT-LR和MIRT-MG检验力则上升。(5) DIFFTEST方法的一类错误率平均值接近名义值0.05; 而IRT-LR和MIRT-MG方法的一类错误率平均值远低于0.05。     

Detection of spatial discontinuities in first-order optical flow fields

We investigated the extent to which the human visual system can detect discontinuities in firstorder optical flow fields. We constructed two types of spatial discontinuities: a circular split field with a straight edge and a disk with annular surround. We used two different first-order optical flow components: an expansion and a rotation. We found an intriguing difference in the detection thresholds for straight and circular discontinuities. Whereas straight discontinuities yielded thresholds of 10%— 50% difference in expansion or rotation, circular discontinuities could, at first, only be detected at extreme differences (>>100%). After a learning period, thresholds for such stimuli decreased, but they remained significantly higher than thresholds for the straight edge. Thresholds rose for stimuli that formed a gradual transition between a circular and a straight edge, and they decreased with increasing eccentricity of the circular discontinuity. Results suggest that symmetry in the stimulus, defined by the coincidence of the center of expansion or rotation and the center of the circular discontinuity, was responsible for the difference in thresholds for circular and straight discontinuities.     

Orientation,length and exposure duration in the identification of line segments

Summary Single line segments titled with 0, 45, 90, and 135 degrees orientation were briefly exposed as stimuli and the subjects had to identify the orientation of a presented line segment. The length of the lines and the exposure duration were varied. The multicomponent theory (Rumelhart, 1970, 1971) was applied to the observed frequencies of a correct response and of some incorrect responses given to a stimulus with a particular orientation, length, and exposure duration. Within the framework of this theory several hypotheses concerning the effects of orientation, length and exposure duration in identification were tested. The main results are: (1) The rate of component detection was higher for the horizontal and vertical orientations than for the two oblique orientations. (2) The number of components in a line segment quite accurately followed a linear function of the line's physical length, and the negatively valued additive constant in this function could not be neglected. (3) The information in the visual information store decayed more rapidly after longer exposure durations and more slowly after shorter exposure durations. These findings are discussed in relation to other research on the subject.     

Categories of Large Numbers in Line Estimation

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time are expected near the boundaries between categories. In contrast to previous work (Landy, Silbert, & Goldin, 2013) that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers.