Mathematical Cognition and Affect During Non-Routine Problem Solving

拟议工作的目标是两个？折在个人层面上，目标是促进对认知的复杂性的理解，并在问题？解决另一个目标是在社区层面：认知？然后呢？大学数学课堂文化背景下的情感互动。在追求这些目标，情感和认知之间的关系，在非？日常问题解决应证明是特别重要的。2该建议纳入了一个最有效的方法发展到目前为止的研究在大学数学教育：研究工作组，包括研究生以及初级和高级研究人员。研究方法的保证，使？工作组的方法既有广泛的定量方法（如调查）、实验方法（如教学实验），也有深入的定性方法（如访谈）、经验方法（如案例研究）。当然，解题者在解题过程中思考他们的情感、态度、信念和自我，以及情感的这些方面如何与认知努力相互作用，这需要比明年更多的时间。这一建议所规定的是研究工作的组织？围绕上述目标。这一合作的一个成果将是设计、实施和完成至少一项试点研究。第二个结果将是创造一个研究设计，继续对个人和社区认知的工作？然后呢？影响数学学习的各种背景下的互动。在个人？众所周知，显著降低数学焦虑学生的焦虑是具有挑战性的（充其量，而且经常失败）。然而，什么才是Meta呢？情感自我？控制而不是坚持一个学生不再感到焦虑，个人层面的试点研究将探讨如何鼓励一个授权Meta？影响。这方面的工作大多是在学校一级进行的。计划中的飞行员？研究将在学院一级进行。一项针对社区的试点研究？一级目标可以为振兴课程改革提供一种途径。许多数学课程改革失去了动力，因为他们的“一？“一刀切”的做法未能影响足够的教师。一个社区水平的初步研究可以调查哪些认知？然后呢？影响概况现有的改革似乎是有效的，然后通过教学实验和研究来创造几个“规模”，即为课堂和/或远程定制课程？学习形式的基础上的目标，原来的改革努力。拟议的途径研究解决了一个大问题，高等教育面临的今天：远程学习。互联网教育网站的形式隐含着对认知的假设？然后呢？影响;即认知？然后呢？可从网站上的表格和过程中编制影响概况。在许多远程学习环境中，教师和学生的互动并没有消失，而是通过网站（或其他媒介）的中介变得更加复杂。这是特别如此，因为在这个国家的远程教育正在从关注主要与程序知识的教学概念知识（例如，在非？常规问题解决或定理证明）。作为研究和开发的一个领域，这三个？在数学教育中，学生、教师和中介网站之间的关系很少被研究，但很有前途。拟议的工作将有助于为今后研究这一专题奠定基础。

The objective of the proposed work is two?fold. At the level of the individual the objective is to advance understanding of the complexities of cognition and affect during problem?solving. The other objective is at the level of community: cognition?and?affect interaction in the context of collegiate mathematics classroom culture. In pursuing these objectives, the relationships among affect and cognition during non?routine problem solving should prove particularly important.The Proposal incorporates one of the most effective methods developed to date for research in collegiate mathematics education: research working groups involving graduate students as well as junior and senior researchers. The research methodology ensured by the make?up of the working groups is both broad: quantitative methods (e.g. survey), experimental methods (e.g. teaching experiment) and deep: qualitative methods (e.g. interview), experiential methods (e.g. case study).Investigating the relationships and causalities among what problem?solvers think of their feelings, attitudes, beliefs, and selves in mathematics and how those aspects of affect interact with cognitive efforts during problem solving would, of course, take more than the next year. What this Proposal provides for is the organizing of research working?groups around the objectives given above. One result of this cooperation will be the design, implementation, and completion of at least one pilot study. A second result will be the creation of a research design for continued work on both individual and community cognition?and?affect interaction in the various contexts of mathematics learning.At the individual?level, it is well known that significantly reducing the anxiety of a mathematically anxious student is challenging (at best, and often fails). What does show promise, however, is meta?affective self?control. Rather than insisting a student no longer feel anxious, an individuallevel pilot study would investigate ways to encourage an empowering meta?affect. Most work in this area has been at the school level. The proposed pilot?study would be at the collegiate level. A pilot study aimed at the community?level objective could offer a way to revitalize curricular reforms. Many curricular reforms in mathematics lose momentum because their "one?size fits all" approach fails to influence enough instructors. A community?level pilot study could investigate for which cognition?and?affect profiles an existing reform appears to be effective and then work through teaching experimentation and research to create several "sizes," i.e. tailor curricula for classroom and/or distance?learning formats based on the goals of the original reform efforts.The proposed avenue of research addresses one of the large issues confronting higher education today: distance learning. The format of an internet education site has, implicitly, assumptions about cognition?and?affect; i.e. a cognition?and?affect profile can be compiled from the forms and processes at a website. In many distance learning settings, the interaction of teachers and students does not disappear, but rather is made more complex by the mediation of the website (or other vehicle). This is especially so as distance learning in this country is shifting from a concern mainly with procedural knowledge to the teaching of conceptual knowledge (for example, in non?routine problem solving or theorem proving). As an area for research and development, this three?way relationship between students, teachers, and a mediating website has been little studied (in mathematics education) and is very promising. The proposed work would contribute to the foundation for future investigations of this topic.