Sources of Mathematical Thinking

为了理解幼儿的数学认知发展和学校的数学认知教学，我们必须了解数学认知的构建系统，以及这些系统协调产生新概念和新技能的过程。 基于以前的研究，我们假设，小学数学建立在三个代表性的系统：一个系统，代表确切的小numerosities，一个系统，代表近似大numerosities，和自然语言系统的数字词和其他量词。 拟议的研究通过对人类婴儿，非人类灵长类动物，学龄前儿童学习计数，小学儿童学习算术和分数以及成年人的实验来调查这些构建块系统及其相互作用。 为了直接研究积木认知系统，实验研究了人类婴儿和未经训练的成年猴子的自发数字表征，在每个群体中使用相同的三种收敛行为测量：查找不同数值的数组和加法或减法事件的时间（基于婴儿和猴子都能更长时间地观察新的阵列或意外事件的发现），手动搜索（建立在婴儿或猴子在容器中搜索的次数取决于它在容器中代表的物体的数量的发现之上），以及对具有不同数量吸引物的容器的运动方法（建立在婴儿和猴子将接近具有更多数量物体的容器的发现上）。 进一步的实验研究如何学前儿童组装这些组件在学习数词和计数程序，通过使用口头和指向任务，以评估儿童的理解的发展变化的数词和计数程序。 为了揭示数学认知的神经基础，行为和神经成像实验都研究了成年人是否以及如何使用这三种表征系统进行数值比较和初等算术。 最后，本研究探讨国小学童数概念与算术学习之关系。 训练研究中，儿童被教导新的事实或概念，然后测试一系列相关的问题将有助于调查参与这种学习的子系统，探索这些子系统组装，以满足新的教育挑战的过程，并探讨如何提高小学数学学习。 这项研究有望通过协调的、以实验室为基础的研究来阐明数学的教学和学习，在这些研究中，猴子、婴儿、儿童和成年人被给予相同的刺激，而且往往是相同的任务。 这种协调的努力应该提供一个广泛的画像的来源数学思维，从其系统发育和个体发生的起源，其高潮在受过教育的成年人。

To understand mathematical cognition both as it develops in the young child and as it is taught in school, one must understand the cognitive systems from which it is constructed and the processes by which those systems are coordinated to produce new concepts and skills. Based on previous research, we hypothesize that elementary school mathematics builds on three representational systems: a system for representing exact small numerosities, a system for representing approximate large numerosities, and natural language with its system of number words and other quantifiers. The proposed research investigates each of these building block systems and their interactions through experiments on human infants, non-human primates, preschool children learning counting, elementary school children learning arithmetic and fractions, and adults. To study the building block cognitive systems directly, experiments investigate spontaneous number representations in human infants and in untrained adult monkeys, using in each population the same three converging behavioral measures: looking time to arrays of different numerosities and to addition or subtraction events (building on the finding that both infants and monkeys look longer at novel arrays or unexpected events), manual search (building on the finding that the number of times that an infant or monkey will search in a container depends on the number of objects it represents within the container), and locomotor approach to containers with different numbers of attractive objects (building on the finding that infants and monkeys will approach the container with the greater number of objects). Further experiments investigate how preschool children assemble these components in learning number words and the counting routine, by using verbal and pointing tasks to assess developmental changes in children's understanding of number words and counting procedures. To uncover the neural substrates underlying mathematical cognition, both behavioral and neuroimaging experiments investigate whether and how human adults use each of the three representational systems in performing numerical comparisons and elementary arithmetic. Finally, experiments investigate number concepts and arithmetic learning in elementary school children. Training studies in which children are taught new facts or concepts and then are tested on a range of related problems will serve to investigate the subsystems involved in this learning, to probe the processes by which those subsystems are assembled to meet new educational challenges, and to explore ways of enhancing mathematics learning in elementary school. This research promises to shed light on the teaching and learning of mathematics through coordinated, laboratory-based studies in which monkeys, infants, children and adults are given the same stimuli and often the same tasks. This coordinated effort should provide a broad portrait of the sources of mathematical thinking, from its phylogenetic and ontogenetic origins to its culmination in educated adults.