Talking about proportion: Fraction labels impact numerical interference in non‐symbolic proportional reasoning

Across two experiments, we investigated how verbal labels impact the way young children attend to proportional information, well before the introduction of formal fraction education. Five‐ to seven‐year‐old children were introduced to equivalent non‐symbolic proportions labeled in one of three ways: (a) a single, categorical label for multiple fractions (both 3/4 and 6/8 referred to as “blick”), (b) labels that focused on the numerator [e.g., 3/4 labeled as “three blicks” (Experiment 1) or “three‐fourths” (Experiment 2)], or (c) labels that had a complete part‐whole structure (“three‐out‐of‐four”). Children then completed measures of non‐symbolic proportional reasoning that pitted whole‐number information against proportional information for novel proportions. Across both experiments, children who heard the categorical labels were more likely to match non‐symbolic displays based on proportion than children in any of the other conditions, who demonstrated higher levels of numerical interference. These findings suggest that fraction labels have the potential to shape children's attention to proportional information even in the context of non‐symbolic part‐whole displays and for children who are not familiar with formal fraction symbols. We discuss these findings in terms of children's developing understanding of proportional reasoning and its implications for fraction education.     

整数构成对分数加工的影响

本研究通过两个实验考察了整数构成对分数加工的影响。实验一重复了Bonato等(2007)的研究,要求25名被试比较1/5与1/1、1/2、1/3、1/4、1/6、1/7、1/8、1/9等分数的大小,结果发现被试采用了成分加工策略,即通过比较分母判断分数的大小。实验二改变了分数的整数构成,让24名被试比较1/5与1/1、2/4、3/9、4/16、4/24、3/21、2/16、1/9等分数的大小。这些分数与实验一中的分数实数值相等但整数构成不同。结果表明被试既采取了成分策略,又采取了整体策略,即比较分数的实数值,不过成分策略比整体策略的作用更大。这些结果表明,分数加工具有成分和整体两种策略,具体使用哪种策略与分数的整数构成密切相关。     

等值分数概念的理解

等值分数是表示具有相等值的分数,它建立在两个量具有确定比例关系的基础上。研究表明,儿童在接受正式教学之前,就具有了等值分数的非正式知识,但仍然在概念理解上存在很大的困难,主要有两方面的原因:一是受自身运算思维发展水平的制约,未获得乘法思维和守恒观念;二是缺乏对等值分数不同语义的理解。在今后研究中,需进一步探讨从非正式知识到正式概念之间的发展路径,尝试开展等值分数的早期教学实验,并需要结合多种语义背景来考查儿童的概念发展水平。     

不同数字线下儿童与成人分数估计的表征模式

以44名小学六年级儿童与40名大学生为被试,通过0~1和1/100~1/10两种数字线的NP（数字位置）和PN（位置数字）估计任务系统考察儿童与成人的分数估计的表征方式。结果显示：（1）儿童和成人在0~1数字线的NP和PN任务上都呈线性表征,但在1/100~1/10数字线下,两组被试在NP任务上却呈对数表征,在PN任务上呈指数表征;（2）NP任务的错误百分比均高于PN任务,且儿童在两数字线下的准确性均明显低于成人。     

Counting by Identity: A Reply to Liebesman

David Liebesman (AJP, 2015) argues that we never count by identity. He generalizes from an argument that we don't do so with sentences indicating fractions, or with measurement sentences on their supposed count readings. In response, I argue that measurement sentences aren't covered by the thesis that we count by identity, in part because they don't have count readings. Then I use the data to which Liebesman appeals, in his argument that we don't count by identity using measurement sentences, in order to rebut his argument that we don't count by identity using sentences indicating fractions.     

Goal equivalent manifold analysis of task performance in non-specific LBP and healthy subjects during repetitive trunk movement: Effect of load,velocity, symmetry

Motor abundance allows reliability of motor performance despite its variability. The nature of this variability provides important information on the flexibility of control strategies. This feature of control may be affected by low back pain (LPB) and trunk flexion/extension conditions.Goal equivalent manifold (GEM) analysis was used to quantify the ability to exploit motor abundance during repeated trunk flexion/extension in healthy individuals and people with chronic non-specific LBP (CNSLBP).Kinematic data were collected from 22 healthy volunteers and 22 CNSLBP patients during metronomically timed, repeated trunk flexion/extension in three conditions of symmetry, velocity, and loading; each at two levels. A goal function for the task was defined as maintaining a constant movement time at each cycle. Given the GEM, flexibility index and performance index were calculated respectively as amounts of goal-equivalent variability and the ratio of goal-equivalent to non-goal-equivalent variability.CNSLBP group was as similar as healthy individuals in both flexibility index (p = 0.41) and performance index (p = 0.24). Performance index was higher in asymmetric (p < 0.001), high velocity (p < 0.001), and loaded (p = 0.006) conditions.Performance and flexibility in using motor abundance were influenced by repeated trunk flexion/extension conditions. However, these measures were not significantly affected by CNSLBP.     

On portfolio management with value at risk and uncertain returns via an artificial neural network scheme

This paper focuses on the computation issue of portfolio optimization with scenario-based Value-at-Risk. The main idea is to replace the portfolio selection models with linear programming problems. According to the convex optimization theory and some concepts of ordinary differential equations, a neural network model for solving linear programming problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns. Several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.     

The direct product model for the mtmm matrix parameterized as a second order factor analysis model

The composite direct product model for the multitrait-multimethod matrix is reparameterized as a second-order factor analysis model. This facilitates the use of widely available computer programs such as LISREL and LISCOMP for fitting the model.Bruce Bloxom. Paul Horst and Karl Jöreskog contributed helpful comments to an earlier version of this paper. Their suggestions are gratefully acknowledged.     

Equivalent models in covariance structure analysis

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified modelsM ₁ andM ₂ when fitting the more restricted modelM ₀. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters ofM ₁ andM ₂ (which characterizes modelM ₁₂), is a necessary and sufficient condition for the local equivalence ofM ₁ andM ₂. This condition is satisfied, in practice, when the analysis dealing with the fitting ofM ₀, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting ofM ₁ orM ₂, orM ₁₂ are all equal for any set of sample data, except on differences due to rounding errors.This research was supported by the Foundation of Social-Cultural Sciences which is subsidized by the Dutch Scientific Organization (N.W.O.) under project number 500-278-003. The author wishes to thank Anne Boomsma, Ivo Molenaar, Albert Satorra, and Tom Snijders for their stimulating and crucial comments during the research, and the Editor, Paul Bekker, Henk Broer, and anonymous reviewers for their helpful suggestions.     

On convergence of the generalized Maxwell scheme: conductivity of composites containing cubic arrays of spherical particles

The Maxwell concept of equivalent inhomogeneity generalized to account for the interactions between the particles in the cluster and combined with recently reported results on the polarizability of a cube is used to evaluate the effective conductivities of the materials reinforced by cubic arrays of spherical particles. New numerical results demonstrate that the estimates of the effective properties based on the generalized Maxwell scheme with the equivalent inhomogeneity of cubic shape converge to the accurate periodic benchmark solutions, unlike spherical shape-based estimates.