Core Systems for Learning Mathematics

虽然人类婴儿很容易发展出一些表示数字和几何的能力，但儿童在学习学校数学时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的数学和科学学习和实践都至关重要，但他们的习得与年幼儿童先前存在的数字和几何概念的关系还没有得到很好的理解。这些实验的指导假设是，儿童通过使用符号（包括语言和视觉空间符号，如图片和地图）来学习学校数学，从而有效地将婴儿期出现的早期发展的数字和空间表征联合收割机结合起来。为了验证这一假设，拟议的实验探讨儿童的核心数字和几何能力的测试和他们的表现上的测试的符号算术，地图阅读，抽象几何推理的表现之间的关系。一些实验使用个体差异方法来测试儿童对核心数字或几何信息的敏感性的变化是否与掌握语言或符号或更抽象的数学概念的变化有关。进一步的实验使用训练方法，以调查是否行使儿童的核心能力或使用符号的任务，提高儿童的数字或几何推理在学校数学测试。训练实验探讨了核心数学能力和建构数学能力之间的关系，以及可能解释这些关系的潜在认知和动机过程。通过这些调查，实验的目的是有助于对数学认知的基本理解，并为提高儿童的数学学习和推理提供信息。首先，它旨在阐明基本的认知能力，使人类能够发展小学数学知识，特别是算术和欧几里得几何的抽象概念和规则。第二，研究的目的是开发更好的方法来帮助儿童的数学学习，建立在基本的研究成果的性质和早期发展的推理的核心认知领域，小学数学可能成立。为了实现这些目标，任务，以前被开发为工具，以探索基本的数字和几何能力的婴儿，儿童和成人在不同的文化部署既评估模式的变化，儿童的数学能力，并作为培训干预措施，有可能提高学校的数学能力。实验不仅探讨了早期和后期发展的数字和几何概念之间的直接关系，而且还探讨了儿童对符号，认知控制和学习态度的发展掌握可能会调节这些关系并影响儿童的学习和学校数学表现的方式。通过这些实验室实验，研究旨在更好地理解成熟的数学推理，并发现帮助儿童掌握这一关键知识领域的新方法。通过调查儿童数学推理的变异性来源和增强儿童推理的经验，建议的研究有望有助于促进所有儿童学习数学，并帮助在这一领域学习有困难的儿童。由于儿童的数学学习受到动机模式以及认知能力的影响，实验干预的目的不仅是提高数学推理的认知过程，而且儿童对数学的态度，相信数学技能的可塑性与实践，并意识到自己的能力作为数学学习者。由于儿童通常在社会环境中学习得最好，干预措施旨在创建一套儿童可以用来与他人玩耍的培训材料，实验将比较这些干预措施与个性化交互式计算机培训计划中的实例。这项研究的结果应有助于加强儿童在学校内外学习数学的努力，适用于4至10岁各种能力水平的儿童。

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.