数学困难儿童的数学估计能力

对数学困难儿童(简称数困)的数学估计进行探讨有助于理解数困的成因和寻找适当的干预措施。本文在简要回顾数困和估计的概念基础上, 重点对近年来国内外关于数困儿童数学估计(主要是估算和估数)的表现及影响因素、神经基础和相关的干预研究进行了回顾和梳理, 并强调未来对数困儿童的数学估计研究应注重从选取被试、扩展研究范围和加强干预方案设计等方面开展探讨。     

数学成就的性别差异

数学成就的性别差异是多年来广受关注的问题。长期以来人们多关注男性、女性数学成就水平的高低, 而近年来研究结果一方面揭示出总体上男性和女性数学成就的平均水平差异很小, 呈现出相似性多于差异性的特点; 同时也显示男性内部变异比女性更大, 男性在高数学成就者中占多数。数学成就性别差异的大小和方向受到评分系统、测验组织形式、测验内容和难度的影响。数学成就性别差异的形成是心理、生物、社会文化等方面多因素综合作用的结果。近期研究探讨了年龄、遗传和进化、激素和脑、刻板印象威胁、社会性别公平和时代等因素在数学成就性别差异的形成中的作用。未来对数学成就性别差异的研究应注意开展追踪研究, 关注低数学能力者, 进一步探讨复杂数学加工机制的性别差异, 建立数学成就性别差异形成机制的综合模型, 并在更广阔的社会文化背景下开展研究。     

小学三年级数学学优生与学困生解决比较问题的差异

运用实验法和临床访谈法研究了数学学优生与学困生在解决比较问题时的差异及元认知对解题成绩的影响。被试为40名小学三年级学生。比较问题分为一致问题和不一致问题。元认知包括元认知知识和元认知监控技能。结果表明：(1)学优生与学困生解决比较问题的成绩差异显著,学优生在一致问题和不一致问题上的解题成绩均优于学困生。这种差异与其解题时所运用的表征策略有关。(2)学生在解决比较问题中出现的主要错误为转换错误,在不一致问题中出现的错误多于一致问题中出现的错误。(3)学优生与学困生在元认知知识和监控技能上均有显著差异,元认知监控技能对解决比较问题的成绩有显著预测作用。     

Distributing mathematical practice of third and seventh graders: Applicability of the spacing effect in the classroom

We examined the effect of distributed practice on the mathematical performance of third and seventh graders (N = 213) in school. Students first received an introduction to a mathematical topic, derived from their curriculum. Thereafter, they practiced in one of two conditions. In the massed condition, they worked on three practice sets in 1 day. In the distributed condition, they worked on one practice set per day for 3 consecutive days. Bayesian analyses of the performance in two follow‐up tests 1 and 6 weeks after the last practice set revealed a positive effect of distributed practice as compared with massed practice in Grade 7. In Grade 3, a positive effect of distributed practice was supported by the data only in the test 1 week after the last practice set. The results suggest that distributed practice is a powerful learning tool for both elementary and secondary school students in the classroom.     

高中生数学元认知水平调查问卷的设计与编制*

对于元认知能力的评价和测试一直以来都是研究的难点,特别是就数学学科而言,已有研究多围绕数学问题解决过程中元认知监控水平的评价以及能力的培养展开,鲜有全面评价学生数学学习元认知水平的问卷。同时,没有针对高中生编制的元认知水平调查问卷。故本研究在已有研究的基础上,对已有的问卷进行修正和完善,通过征求专家意见与样本测试和数据分析,形成《高中生数学元认知水平调查问卷》。该问卷具有较好的内容效度和结构效度。信度方面,问卷整体的内部信度为0.952,分半信度为0.931,重测信度为0.946,说明问卷具有很好的内部一致性和测量稳定性。     

高效率数学学习高中生数学成绩的影响路径

为探究高效率数学学习高中生的数学学习成绩的影响因素的不同作用效果,以及这些影响因素间存在的作用路径,通过目标抽样,选取102名高中生为被试开展调查研究。通过数据分析,并依托AMOS软件的模型界定搜寻功能进行路径分析,获得研究结论：（1）数学元认知、数学非智力因素、智力因素、数学学习策略和数学素养这5个变量对高效率数学学习高中生的数学学习成绩的作用效果值依次递减。（2）5个变量通过两条主要路径影响高效率数学学习高中生的数学学习成绩,一是智力因素与数学素养构成的影响路径,二是数学元认知、数学非智力因素以及数学学习策略构成的影响路径。     

Understanding linear and exponential growth: Searching for the roots in 6- to 9-year-olds

Previous studies have suggested that children as young as 9 years old have developed an understanding of non-linear growth processes prior to formal education. The present experiment aimed at investigating this competency in even younger samples (i.e., in kindergartners, first, and third graders, ages 6, 7 and 9, respectively). Children (N = 90) solved non-verbal inductive reasoning tasks by forecasting linear and exponential growth. While children of all ages forecasted linear growth adequately, exponential growth was also estimated remarkably well. Surprisingly, kindergartners and third graders showed similar high achievement concerning the magnitude and curve shape of forecasts, whereas first graders performed significantly worse. We concluded that primary knowledge of both linearity and non-linearity exists even in kindergartners. However, children's understanding is quite fragile, as their performance was strongly affected by task sequence: Children underestimated exponential growth when the previous task required a forecast of linear growth, and overestimated linear growth when the previous task required forecasting of exponential growth.     

Hilbert's epsilon as an operator of indefinite committed choice

Hilbert and Bernays avoided overspecification of Hilbert's ε-operator. They axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the ε-operator underspecified. After briefly reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the ε simplifies proof search and is natural in the sense that it mirrors some cases of referential interpretation of indefinite articles in natural language.     

SAD as a mathematical assistant—how should we go from here to there?

The System for Automated Deduction (SAD) is developed in the framework of the Evidence Algorithm research project and is intended for automated processing of mathematical texts. The SAD system works on three levels of reasoning: (a) the level of text presentation where proofs are written in a formal natural-like language for subsequent verification; (b) the level of foreground reasoning where a particular theorem proving problem is simplified and decomposed; (c) the level of background deduction where exhaustive combinatorial inference search in classical first-order logic is applied to prove end subgoals.

We present an overview of SAD describing the ideas behind the project, the system's design, and the process of problem formalization in the fashion of SAD. We show that the choice of classical first-order logic as the background logic of SAD is not too restrictive. For example, we can handle binders like Σ or lim without resort to second order or to a full-powered set theory. We illustrate our approach with a series of examples, in particular, with the classical problem .