Sources of Mathematical Thinking

要理解数学认知，无论是在幼儿发展的过程中，还是在学校教授的过程中，都必须了解构成数学认知的认知系统，以及协调这些系统以产生新概念和技能的过程。基于前人的研究，我们假设小学数学建立在三个表征系统上：一个表示精确小数值的系统，一个表示近似大数值的系统，以及由数字词和其他量词组成的自然语言系统。这项拟议的研究通过对人类婴儿、非人类灵长类动物、学龄前儿童学习算术、小学生学习算术和分数以及成年人的实验，研究了这些积木系统及其相互作用。为了直接研究积木认知系统，实验调查了人类婴儿和未受过训练的成年猴子的自发数字表征，在每个种群中使用了相同的三种收敛行为测量：查看不同数值阵列和加减事件的时间(基于婴儿和猴子都会更长时间地关注新阵列或意外事件的发现)、手动搜索(基于婴儿或猴子在容器中搜索的次数取决于它在容器中表示的对象的数量)，对装有不同数量的吸引物体的容器采取移动方法(基于婴儿和猴子会接近带有更多物体的容器的发现)。进一步的实验研究了学龄前儿童如何在学习数字单词和计算程序中组装这些组成部分，通过使用言语和指向任务来评估儿童对数字单词和计算程序的理解的发展变化。为了揭示数学认知背后的神经基础，行为和神经成像实验都调查了人类成年人是否以及如何使用这三个表征系统中的每一个来执行数字比较和基本算术。最后，实验考察了小学生的数字概念和算术学习。培训研究向儿童传授新的事实或概念，然后对一系列相关问题进行测试，这将有助于调查这种学习所涉及的子系统，探索这些子系统的组装过程，以应对新的教育挑战，并探索加强小学数学学习的方法。这项研究承诺通过以实验室为基础的协调研究来揭示数学的教与学，在这些研究中，猴子、婴儿、儿童和成年人被给予相同的刺激，通常是相同的任务。这种协调的努力应该为数学思维的来源提供一个广泛的图景，从它的系统发生和个体发生起源到它在受过教育的成年人中的顶峰。

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.