The Developmental Foundations of Number and Operations Sense

数感和运算感的发展基础

• 批准号: 0111829
• 负责人: Arthur Baroody
• 金额: $ 28.75万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-15 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0111829&HistoricalAwards=false
• 关键词: Developmental; Foundations; Number; Operations; Sense

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.

数字和运算感的发展基础阿瑟·J·巴鲁迪儿童的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之上。然而，这一点算前阶段的性质以及儿童如何过渡到点算阶段尚不清楚。 根据Janellan Huttenlocher及其同事提出的心理模型，儿童最初甚至不准确地表示1到4个项目的小集合，而不是像当前理论所建议的那样精确。 通过过渡1，儿童发展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过渡2到一个精确的，基于口头的数字表示。 心理模型关注的是数字是如何被表征的，而劳伦·雷斯尼克的发展模型则关注的是数字被表征的内容。 根据这一模型，数学思维从具体（上下文约束）到抽象（一般）。 在第一阶段，儿童形成对不可计数的数量的非语言理解（进行原数量推理）。 在第二阶段，他们构建对计数集合的理解（能够进行数量级推理）。 在第三阶段，孩子们构建知识，并能在没有实际收集的情况下对特定数字进行推理（数字水平思维）。在第四阶段，他们发现数字或算术关系，并能与一般性和有关的原因（抽象的l级推理）。根据这一整合模型，儿童在原始数量水平思维和数量水平思维之间可能会经历三个阶段。 在过渡1之后，他们可能首先理性地但不精确地（定性地）推理数字的精确表示（子阶段1），然后精确地（定量地）推理它们（子阶段2）。 在孩子们学习了数字名称之后，但在他们能够枚举集合之前（可以使用计数来确定集合中的项目数量），他们可能能够定量地推理集合的精确表示并为它们贴上数字标签（子阶段3）。 尽管整合模型与许多现有的研究一致，包括那些认为不计数的快速识别数字是定量思维的基础的研究，但该模型的具体含义需要进行测试。研究1和2将涉及评估整合模型的预测，这些预测是关于数字的精确非语言表征，语言计数，和简单的加法/减法是相互关联的。 例如，根据这个模型，子阶段1中的预先计数的孩子应该是非语言地为之前看到但现在隐藏的孩子创建一个匹配的集合。然而，这些孩子应该只能估计加法或减法对集合的影响，子阶段2的前计数器可以在心理上准确地确定小的和和差，子阶段3的前计数器可以通过口头标记数字来进一步识别这些结果。 研究1将结合一个横断面设计，基于访谈的儿童的父母或教师（提供背景和观察数据），并重复重新测试（检查学习效果往往导致测试幼儿）。 研究2将包括基于自然观察和微观遗传方法的长期案例研究（以特定方式在规定的时间间隔内重复执行选定的任务，特别是在发育过渡阶段）。 使用多种方法的组合应该比使用任何单一方法提供更丰富的数字和算术发展数据。 例如，了解儿童在微观遗传研究中的表现，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）来显着改善。 反过来，微观遗传学研究的重点性质可以为自然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检查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），添加剂组合物（例如，一个和大于任何一个部分）、逆原理（一定量的相加被相同量的相减所抵消）和加性交换性（两个集合合并的顺序不影响结果）。所提出的项目应该具有重要的理论、方法和实用价值。 直接测试的综合模型的影响，应导致更好地了解数和操作感的起源，以及它是如何演变的。 开发的综合方法和特定任务的测试应该是有用的，在调查计数前的数学知识，探索过渡到计数为基础的知识，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是有用的，为那些规划，开发，或实施幼儿数学课程。