Core Systems for Learning Mathematics

尽管人类婴儿很容易发展出一些表示数字和几何的能力，但当孩子们学习学校数学时，他们的思维远远超出了这些能力。儿童在小学学习的算术和几何的基本概念和技能对于以后所有数学和科学的学习和实践都是至关重要的，但他们的习得与幼儿先前存在的数字和几何概念之间的关系还没有得到很好的理解。拟议的实验是在这样一个假设的指导下进行的，即儿童通过使用符号来学习学校数学--包括语言符号和视觉空间符号，如图片和地图--有效地结合了婴儿时期出现的早期发展的数字和空间表征。为了验证这一假设，本实验探索了儿童在核心数字和几何能力测试中的表现与他们在符号算术、读图和抽象几何推理测试中的表现之间的关系。一些实验使用个体不同的方法来测试儿童对核心数字或几何信息的敏感度的变异性是否与语言或符号或更抽象的数学概念掌握的变异性有关。进一步的实验使用训练方法来调查锻炼孩子核心能力或使用符号的任务是否能提高孩子在学校数学考试中的数字或几何推理能力。训练实验既探索了核心数学能力和构建的数学能力之间存在的关系，也探索了可能解释这些关系的潜在认知和动机过程。通过这些调查，这些实验的目的既是为了促进对数学认知的基本理解，也是为了促进儿童的数学学习和推理。首先，它旨在阐明使人类发展小学数学知识的基本认知能力，特别是算术和欧几里德几何的抽象概念和规则。其次，这项研究旨在开发更好的方法来帮助儿童学习数学，通过建立在小学数学基础上的核心认知领域中关于推理的性质和早期发展的基本研究成果。为了实现这些目标，以前被开发为探索不同文化中婴儿、儿童和成年人基本数字和几何能力的工具的任务被用于评估儿童数学能力的变异性模式，并作为可能提高学校数学能力的训练干预措施。这些实验不仅调查了早期和后来发展的数字和几何概念之间的直接关系，而且还调查了儿童对符号、认知控制和学习态度的发展可能调节这些关系的方式，以及对儿童学习和学校数学表现的影响。通过这些实验室实验，本研究旨在更好地理解成熟的数学推理，并发现新的方法来帮助儿童掌握这一关键领域的知识。通过调查儿童数学推理中的变异性来源和增强儿童推理的经验，本研究有望有助于促进所有儿童的数学学习，并帮助在这一领域学习困难的儿童。由于儿童的数学学习受到动机模式和认知能力的影响，实验干预的目的不仅是加强儿童数学推理的认知过程，而且还旨在加强儿童对数学的态度，相信数学技能在实践中是可塑性的，并增强他们作为数学学习者的自身能力。由于儿童通常在社交环境中学习效果最好，这些干预措施的目的是创建一套儿童可以用来与他人玩耍的培训材料，实验将把这些干预措施与个性化互动计算机培训程序中的干预措施进行比较。这项研究的结果应该有助于加强儿童在校内外的数学学习，使4至10岁的儿童在所有能力水平上都能学习。

Although human infants readily develop some capacities for representing number and geometry, children's thinking extends far beyond these capacities when they learn school mathematics. The basic concepts and skills of arithmetic and geometry that children learn in elementary school are critical for all subsequent learning and practicing of mathematics and science, but their acquisition, in relation to younger children's preexisting numerical and geometrical concepts, is not well understood. The proposed experiments are guided by the hypothesis that children learn school mathematics by using symbols- both language and visual-spatial symbols such as pictures and maps- to combine productively the early-developing numerical and spatial representations that emerge in infancy. To test this hypothesis, the proposed experiments probe the relationship between children's performance on tests of core numerical and geometrical abilities and their performance on tests of symbolic arithmetic, map reading, and abstract geometrical reasoning. Some experiments use individual difference methods to test whether variability in children's sensitivity to core numerical or geometric information is associated with variability in the mastery of language or symbols or of more abstract mathematical concepts. Further experiments use training methods to investigate whether tasks that exercise children's core abilities or use of symbols enhance children's numerical or geometrical reasoning in tests of school mathematics. The training experiments probe both the existence of relationships between core and constructed mathematical abilities and the underlying cognitive and motivational processes that may account for these relationships. Through these investigations, the experiments aim both to contribute to basic understanding of mathematical cognition and to inform efforts to enhance children's mathematical learning and reasoning.The proposed research has two complementary goals. First, it aims to shed light on the fundamental cognitive capacities that allow humans to develop knowledge of elementary school mathematics, especially the abstract concepts and rules of arithmetic and Euclidean geometry. Second, the research aims to develop better ways to aid children's learning of mathematics, by building on basic research findings concerning the nature and early development of reasoning in the core cognitive domains on which elementary school mathematics may be founded. To accomplish these goals, tasks that previously were developed as tools to probe the fundamental numerical and geometrical abilities of infants, children, and adults in diverse cultures are deployed both to assess patterns of variability in children's mathematical abilities and to serve as training interventions with the potential to enhance school math abilities. The experiments investigate not only the direct relations between early and later developing numerical and geometrical concepts, but also the ways in which children's developing mastery of symbols, cognitive control, and attitudes toward learning may modulate those relations and impact on children's learning and performance of school mathematics. Through these laboratory experiments, the research aims to achieve a better understanding of mature mathematical reasoning and to discover new ways to aid children's mastery of this critical domain of knowledge.By investigating the sources of variability in children's mathematical reasoning and the experiences that enhance children's reasoning, the proposed research promises to contribute to efforts both to foster all children's learning of mathematics and to aid children who experience difficulty with learning in this domain. Because children's learning of mathematics is affected by motivational patterns as well as cognitive abilities, the experimental interventions aim to enhance not only the cognitive processes underlying mathematical reasoning but also children's attitudes towards mathematics, belief in the malleability of mathematical skills with practice, and sense of their own competence as mathematical learners. Because children often learn best in social contexts, the interventions aim to create a set of training materials that children can use to play with others, and experiments will compare these interventions to those instantiated in individualized interactive computer-training programs. The findings of this research should inform efforts to enhance children's learning of mathematics both in and outside of school, for 4- to 10-year-old children at all ability levels.