Development of Mapping Processes in Mathematical Problem Solving

本研究探讨数学解题中心过程的发展。该项目研究了形式问题解决理论中的一个中心问题：儿童和成人在遇到新问题时如何选择适当的数学运算。尽管学生在相对较少的问题领域（例如，平均分配糖果）的背景下学习特定的数学运算（例如，除法），但他们最终可以将该运算应用于解决各种各样的新领域（例如，尚未确定解决运算的领域）中的问题。本研究探讨了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，并展开如下。学生在特定基域的背景下学习一种形式运算（如除法）（如平均分配糖果）。因此，基域和形式运算是联系在一起的。当遇到一个新的问题域时，子程序试图将新域类比地映射到基本域。如果两个域可以映射，则应用形式化操作。数学运算也可以达到通过操作映射。当孩子们将数学运算应用于问题和实践计算过程时，他们会归纳出适用于数学运算的函数关系。在操作映射下，这些函数关系（即，数学运算完成的内容）直接映射到新问题的表示。在第一项研究中，使用两种绘图过程的发展变化将通过对四年级到八年级儿童的相关设计进行评估。第二项研究将检验引入有理数对学生的数学运算表示和随后的问题解决的影响。有理数的引入有效地改写了适用于不同运算的函数关系。这将影响学生对数学运算的表述，并对他们使用运算映射过程的能力产生不利影响。第三个研究直接操纵领域和操作映射过程，并检验各自预测的因果关系。拟议的项目将大大有助于研究解决问题的发展，以及研究成人解决问题的能力。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.