Generative Aspects of Mathematical Thinking: Formulating Conjectures and Posing Problems

这项研究项目将仔细研究和发展对数学思维某些生成性方面的认知描述和分析--提出问题和以待解决问题的形式提出猜想--因为这些过程被从六年级学生到中学数学教师的不同数学经验和成熟度的人使用。将认真注意不同受试者群体之间和内部的个体差异，以及对使用计算机软件支助或结对解决问题等支助机制过程的频率和(或)有效性的影响。从头两年的观察研究中获得的对这些过程的理解将在项目的第三年在一项教学研究中付诸实施，在这项研究中，一套基于描述性分析的教学活动被用于中学生，以促进这些生成过程的发展。这项拟议的研究针对与数学思维和问题解决有关的现有科学发现的重要扩展。此外，本文提出的工作与当前改革大学预科数学教学的努力有关，通过增加对学生猜测和调查的教学时间，使其对学生的智力更具启发性。因此，拟议的研究计划的结果不仅对不断增长的关于数学认知和问题解决的科学知识库做出贡献，而且对大学预科数学教育具有实际意义。

This research project will carefully examine and develop a cognitive description and analysis of certain generative aspects of mathematical thinking - posing problems and formulating conjectures in the form of problems to be solved - as these processes are used by persons at varying levels of mathematical experience and maturity, ranging from sixth-grade students to secondary school mathematics teachers. Careful attention will be paid to individual differences among and within the various subject populations and also to the effects on the frequency and/or effectiveness of use of the processes of support mechanisms, such as the use of computer software support or paired problem solving. The understanding of these processes obtained from observational studies during the first two years will be operationalized in the third year of the project in an instructional study in which a set of instructional activities, based on the descriptive analysis and devised to promote the development of these generative processes, is used with middle school students. The proposed research is directed at important extensions of extant scientific findings related to mathematical thinking and problem solving. Moreover, the work proposed here is related to current efforts to reform precollege mathematics instruction to make it more intellectually stimulating for students by increasing the amount of instructional time given over the student conjecturing and investigation. Thus, the results of the proposed research program should be significant not only for the contribution they can make to the growing scientific knowledge base concerning mathematical cognition and problem solving but also for their practical relevance for precollege mathematics education.