Development of Mapping Processes in Mathematical Problem Solving

本研究探讨了数学问题解决中的中心加工过程的发展。该项目研究了正式问题解决理论中的一个核心问题：儿童和成人在遇到新问题时如何选择适当的数学运算。尽管学生学习特定的数学运算（例如，划分）在相对较少的问题域的上下文中（例如，平均分配糖果），他们最终可以应用该操作来解决各种各样的新领域中的问题（即，尚未为其识别解决方案操作的域）。本研究探讨数学问题解决的两个核心过程：领域映射和操作映射。域映射过程类似于类比映射，并如下展开。学生学习一个正式的操作（例如，分割）在特定基础结构域的上下文中（例如，平均分配糖果）。因此，基域和形式运算就联系在一起了。当遇到一个新的问题领域时，孩子会试图将新的领域类比地映射到基础领域。如果两个域可以映射，则应用形式运算。数学运算也可以通过运算映射来实现。孩子们在将数学运算应用于问题和练习计算过程时，会归纳出适用于数学运算的函数关系。在操作映射下，这些功能关系（即，数学运算完成了什么）直接映射到新问题的表示。在第一项研究中，发展变化的两个映射过程的使用将使用相关设计与儿童从4日至8年级进行评估。第二项研究将考察引入有理数对学生数学运算表征和随后解决问题的影响。有理数的引入有效地改写了适用于不同操作的函数关系。这应该会影响学生对数学运算的表示，并随后对他们使用运算映射过程的能力产生不利影响。第三项研究直接操纵领域和操作映射过程，并测试每个预测的因果关系。建议的项目将大大有助于研究解决问题的发展，并在成人解决问题的研究。这项研究还将为学生提供巨大的教育机会，并涉及与当地教师和管理人员的合作。

This research investigates the development of central processes in mathematical problem solving. The project examines a central issue in theories of formal problem solving: how children and adults select an appropriate mathematical operation when they encounter a novel problem. Despite the fact that students learn a particular mathematical operation (e.g., division) in the context of relatively few problem domains (e.g., equally distributing candy), they can eventually apply the operation to solve problems in a tremendous variety of novel domains (i.e., domains in which the solution operation has not been identified for them). The proposed research investigates two central processes in mathematical problem solving: Domain Mapping and Operation Mapping. The Domain Mapping process is similar to analogical mapping and unfolds as follows. Students learn a formal operation (e.g., division) in the context of a particular base domain (e.g., equally distributing candy). As a consequence, the base domain and the formal operation become linked. When a novel problem domain is encountered the child attempts to analogically map the novel domain to the base domain. If the two domains can be mapped, the formal operation is applied. Mathematical operations can also be reached through Operation Mapping. Children induce the functional relationships that hold for a mathematical operation as they apply it to problems and practice computational procedures. Under Operation Mapping, these functional relationships (i.e., what the mathematical operation accomplishes) are directly mapped to the representation of the novel problem. In the first study, developmental changes in the use of the two mapping processes will be assessed using a correlational design with children from 4th through 8th grade. A second study will examine the effects of the introduction of rational numbers on students' representations of the mathematical operations and on their subsequent problem solving. The introduction of rational numbers effectively rewrites the functional relationships that hold for different operations. This should affect students' representations of mathematical operations and have subsequent detrimental effects on their ability to use the Operation Mapping process. The third study directly manipulates the Domain and Operation Mapping processes and test the causal relationships each predicts. The proposed project will substantially contribute to research on the development of problem solving, and to the study of problem solving in adults. This research will also provide tremendous educational opportunities for students, and involve collaboration with local teachers and administrators.