Mathematical Cognition and Affect During Non-Routine Problem Solving

拟议工作的目标有两个方面。在个人层面上，目标是促进对问题解决过程中认知和情感的复杂性的理解。另一个目标是在社区层面：在大学数学课堂文化的背景下，认知和影响互动。在追求这些目标的过程中，非常规问题解决过程中情感和认知之间的关系应该被证明是特别重要的。该提案结合了迄今为止在大学数学教育研究中开发的最有效的方法之一：由研究生以及初级和高级研究人员组成的研究工作组。工作组的组成确保了研究方法的广泛性：定量方法(例如调查)、实验方法(例如教学实验)和深度方法：定性方法(例如访谈)、经验方法(例如案例研究)。调查什么问题之间的关系和因果关系？解决者在数学中思考他们的感觉、态度、信念和自我，以及这些方面如何影响解决问题的认知努力，当然，需要比下一年更多的时间。这项提案规定的是围绕上述目标组织研究工作小组。这种合作的一个结果将是设计、实施和完成至少一项试点研究。第二个结果将是创建一项研究设计，以继续研究个人和社区认知并影响数学学习的各种背景下的互动。在个人层面上，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，往往是失败的)。然而，什么是有希望的，是元？情感？自我控制。不是坚持让学生不再感到焦虑，而是一项个人层面的先导研究，将探索鼓励赋能型元影响的方法。这方面的大部分工作都是在学校层面上进行的。拟议的试点研究将在大学一级进行。一项针对社区层面目标的试点研究可以提供一种振兴课程改革的方法。许多数学课程改革失去了动力，因为他们的“一刀切”的方法未能影响到足够多的教师。社区一级的试点研究可以调查现有改革在哪些认知和影响方面似乎是有效的，然后通过教学实验和研究来创造几个“规模”，即根据最初改革努力的目标为课堂和/或远程学习模式量身定做课程。拟议的研究途径解决了当今高等教育面临的一个大问题：远程学习。互联网教育网站的形式隐含着关于认知和情感的假设，即认知和情感档案可以从网站的形式和过程中汇编而成。在许多远程学习环境中，教师和学生之间的互动并没有消失，而是通过网站(或其他媒介)的调解而变得更加复杂。尤其是在这个国家，远程教育正在从主要关注程序性知识转向教授概念性知识(例如，在非常规问题解决或定理证明中)。作为一个研究和开发领域，这种学生、教师和中介网站之间的三方关系(在数学教育中)很少被研究，而且非常有前途。拟议的工作将有助于为今后对这一专题的调查奠定基础。

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.