The Developmental Foundations of Number and Operations Sense

摘要：数字和运算的发展基础阿瑟·J·巴罗迪老师儿童基于计数的数字和算术知识建立在他们在这些领域的非语言知识上。然而，这一计数前阶段的性质以及儿童如何过渡到计数阶段尚不清楚。根据Janellan Huttenlocher和他的同事提出的心理模型，儿童最初甚至不准确地代表1到4个物品的小集合，并不完全像目前的理论所表明的那样。在转换1中，孩子们培养了准确但非语言地表示集合的能力。这一点和计数的发展允许向精确的、基于语言的数字表示过渡。心理模型关注的是数字如何被表征，而劳伦·雷斯尼克的发展模型关注的是被表征的东西。根据这一模型，数学思维从具体的(上下文制约的)发展到抽象的(一般的)。在第一阶段，儿童形成对数不清数量的非语言理解(参与原量推理)。在第二阶段，他们构建了对计数的收藏品的理解(能够进行数量级的推理)。在第三阶段，儿童构建知识，并能在没有实际收集的情况下对特定数字进行推理(数字层面的思维)。在第四阶段，他们发现数字或算术关系，并能对共性进行推理(抽象的L级推理)。拟议的项目将需要评估一个整合了上述两个模型的模型。根据这个整合模型，儿童可能会经历原量思维和量思维之间的三个子阶段。在转换1之后，他们可能首先理智地但不精确地(定性地)推理关于数的准确表示(子阶段1)，然后精确地(定量地)推理它们(子阶段2)。在儿童学习数字名称之后，但在他们能够列举集合之前(可以使用计数来确定集合中的项目的数量)，他们可能能够对集合的准确表示进行定量推理并给它们贴上数字标签(子阶段3)。虽然综合模型与许多已有的研究一致，包括那些表明快速识别数字而不计数是量化水平思维的基础的研究，但该模型的具体含义需要检验。研究1和研究2将评估从综合模型得出的关于数字、言语计数和简单加/减的准确非语言表征的发展如何相互关联的预测。例如，根据这个模型，在子阶段1中预先计算孩子的数量应该是非语言地为以前看到但现在隐藏的集合创建一个匹配的集合。然而，这些儿童应该只能估计加法或减法对集合的影响，子阶段2预计数器可以在心理上准确地确定小和和差，子阶段3预计数器可以通过口头给结果贴上数字标签来进一步识别这样的结果。研究一将需要结合横断面设计、对儿童父母或教师的问卷访谈(以提供背景和观察数据)和反复重新测试(以检查测试幼儿经常产生的学习效果)。研究2将包括基于自然主义观察和微遗传学方法的长期个案研究(以规定的间隔以特定方式重复执行选定的任务，特别是在发育过渡阶段)。与使用任何单一方法相比，组合使用方法应该可以提供更丰富的关于数字和算术发展的数据。例如，了解儿童在微遗传学研究中的表现，可以通过详细了解他们在自然环境中的发育准备和表现(如自然主义数据所记录的那样)，大大提高他们的表现。反过来，微遗传学研究的重点性质可以为自然主义观察提供明确的方向(例如，直接关注关键行为或行为模式)。其他四项研究将涉及以下关键数字和算术概念的早期发展(非语言理解)：部分-整体知识(例如，整体大于任何单个部分)、相加合成(例如，和大于任何一部分)、逆原理(通过减去相同的量来取消一定数量的相加)和相加交换性(两个集合合并的顺序不会影响结果)。拟议的项目应该具有重要的理论、方法和实践价值。直接测试集成模型的含义应该会更好地理解数字和运算感的起源以及它是如何演变的。所开发的综合方法和针对具体任务的测试应有助于调查计数前的数学知识，探索向基于计数的知识的过渡，并衡量前者对后者的影响。更强大的发展框架和评估措施应该对那些计划、开发或实施幼儿数学课程的人有用。

AbstractThe Developmental Foundations of Number and Operations SenseArthur J. BaroodyChildren's counting-based knowledge of number and arithmetic builds on their nonverbal knowledge in these domains. The nature of this pre-counting phase and how children make the transition to the counting phase, however, are not clear. According to the mental model proposed by Janellan Huttenlocher and colleagues, children initially represent even small collections of 1 to 4 items inexactly, not precisely as much current theorizing suggests. With Transition 1, children develop the ability to represent collections exactly but nonverbally. This and the development of counting permit Transition 2 to an exact, verbally based representation of number. Whereas the mental model focuses on how number is represented, Lauren Resnick's developmental model focuses on what is represented. According to this model, mathematical thinking evolves from concrete (context-bound) to abstract (general). In the first phase, children form a nonverbal understanding of uncounted quantities (engage in protoquantitive reasoning). In the second phase, they construct understandings of counted collections (become capable of quantities-level reasoning). In the third phase, children construct knowledge and can reason about specific numbers in the absence of actual collections (numbers-level thinking). In the fourth phase, they discover numerical or arithmetic relations and can reason with and about generalities (abstract l-level reasoning).The proposed project will entail evaluating a model that integrates the two models discussed above. According to this integrated model, children may pass through three subphases between protoquantitative- and quantities-level thinking. After Transition 1, they may first reason sensibly but imprecisely (qualitatively) about exact representations of number (subphase 1) and then reason precisely (quantitatively) about them (subphase 2). After children learn number names but before they can enumerate collections (can use counting to determine the number of items in a collection), they may be able to reason quantitatively about exact representations of collections and attach number labels to them (subphase 3). Although the integrated model is consistent with much existing research, including that which suggests the rapid recognition of number without counting is a basis for quantitative-level thinking, specific implications of the model need to be tested.Studies 1 and 2 will involve evaluating predictions that follow from the integrated model about how the development of an exact nonverbal representation of number, verbal counting, and simple addition/subtraction are inter-related. For example, according to this model, pre-counting children in subphase 1 should be to nonverbally create a matching collection for one previously seen but now hidden. Whereas, these children should can only estimate the effects of addition or subtraction on a collection, subphase 2 pre-counters can mentally determine small sums and differences accurately, and subphase 3 pre-counters can further identify such results by verbally labeling it with a number. Study 1 will entail combining a cross-sectional design, questionnaire-based interviews of children's parents or teachers (to provide context and observational data), and repeated re-testing (to examine the learning effects often induced by testing young children). Study 2 will consist of long-term case studies based on naturalistic observations and microgenetic methods (repeatedly administering selected tasks in a specific manner at a prescribed interval, particularly during a developmental transition phase). Using a combination of methods should provide richer data on number and arithmetic development than using any single method. Understanding children's performance in a microgenetic study, for instance, can be significantly improved by a detailed knowledge of their developmental readiness and performance in their natural environment (as documented by naturalistic data). In return, the focused nature of a microgenetic study can provide naturalistic observations with a clear direction (e.g., direct attention to key behaviors or patterns of behavior). Four other studies will involve examining the early development (nonverbal understanding) of the following key number and arithmetic concepts: part-whole knowledge (e.g., a whole is larger than any single part), additive composition (e.g., a sum is larger than either part), the inverse principle (addition of a certain amount is undone by the subtraction of the same amount), and additive commutativity (the order in which two collections are combined does not affect the outcome).The proposed project should have important theoretical, methodological, and practical value. Directly testing the implications of the integrated model should lead to a better understanding about the origins of number and operation sense and how it evolves. The integrated methodologies and task-specific tests developed should be useful in investigating pre-counting mathematical knowledge, exploring the transition to counting-based knowledge, and gauging the effects of the former on the latter. The more powerful developmental framework and assessment measures should be useful for those planning, developing, or implementing early childhood mathematics programs.