Collaborative Research: Project DIRACC: Developing and Investigating a Rigorous Approach to Conceptual Calculus

微积分是一个必不可少的工具，并提供了一个概念基础，在每个STEM学科。在第一次学习微积分时获得深刻的概念理解给学生带来了许多困难，也给教师带来了许多挑战。因此，微积分改革中的许多协同努力似乎未能使学生的理解产生他们的支持者所预测的教育上的重大差异。DIRACC项目：开发和研究一种严格的方法来概念微积分将建立在亚利桑那州州立大学的已经重新设计和已经部署的微积分I的成功记录的基础上，重新设计和实施微积分II，并将研究学生在重新设计和传统的微积分序列中的学习。此外，DIRACC计划亦会为微积分一及微积分二发展简答评核工具（又称概念清单），供其他院校用以评核、比较及对比学生对微积分核心概念的理解进度。学生采取这种重新设计的微积分序列将更好地准备，并将更有可能应用，他们的变化率函数和积累函数的理解在其他STEM课程，并学习微积分相关的材料在随后的数学课程。该项目编写的教科书将作为一个开放资源供其他人使用或借鉴。最后，学生微积分学习的结果，以及为调查学生微积分学习而建立的微积分概念量表，将为未来在这些领域的研究提供参考。DIRACC项目的成果将在地区、国家和国际会议上广泛传播，并发表在同行评审的数学教育期刊和会议记录上。微积分改革缺乏有效性的一个可能原因是，在教授这门学科的绝大多数情况下，基础课程的基本结构保持不变。具体来说，改革项目没有关注学生对变化率函数和累积函数的丰富联系意义的发展，这是任何微积分入门所必需的。项目DIRACC课程是高度概念化的，因为它们基于学生对整个微积分适用的概念的连贯意义的发展，并且也促进了微积分之外的数学思想的学习。第一门课程通过使微积分的基本定理成为学生在课程中微积分经验的各个方面来解决这一挑战。与此同时，DIRACC项目的演算序列将明确计算，因为在现有的第一门课程，学生将使用计算机来表示过程，定义函数作为动态情况的模型，然后成为研究对象本身。这些课程还将承认和解决学生准备微积分的已知弱点。微积分概念清单将使用标准的仪器开发技术，在各自的课程开始和结束时给出，其心理测量特性将建立在每门课程约600名学生。

Calculus is an essential tool, and provides a conceptual foundation, in each of the STEM disciplines. Acquiring deep conceptual understandings when first learning calculus poses many difficulties for students and creates many challenges for teachers. As a consequence, many concerted efforts in calculus reform appear to have failed to make educationally significant differences in student understanding that their proponents predicted. Project DIRACC: Developing and Investigating a Rigorous Approach to Conceptual Calculus will build on the documented success at Arizona State University's already-redesigned and already-deployed Calculus I to redesign and implement Calculus II, and will research students' learning in both the redesigned and traditional calculus sequences. In addition, Project DIRACC will develop short-answer assessment tools (also known as concept inventories) for Calculus I and Calculus II that other institutions can use to assess, compare and contrast their students' progress in understanding central ideas of the calculus. Students taking this redesigned calculus sequence will be better prepared for, and will be more likely to apply, their comprehension of rate-of-change functions and accumulation functions in other STEM courses and to learn calculus-dependent material in subsequent mathematics courses. The textbook produced by this project will be made available as an open resource for others to use or build upon. Finally, the results on students' calculus learning, and the calculus concept inventories built to investigate students' learning, will inform future research in these areas. Results from Project DIRACC will be disseminated widely at regional, national and international conferences and published in peer reviewed mathematics education journals and conference proceedings. One possible reason for the lack of effectiveness of calculus reform is that the fundamental structure of the underlying curriculum has remained unchanged in the vast majority of contexts where the subject is taught. Specifically, reform projects have not focused on students' development of richly connected meanings for rate-of-change functions and accumulation functions, which are essential to any introductory calculus. The Project DIRACC courses are highly conceptual because they are based upon students' development of coherent meanings for ideas applicable throughout calculus and that also facilitate the learning of mathematical ideas beyond calculus. The first course addresses this challenge by making the fundamental theorem of calculus central to every aspect of students' experience of calculus in the course. At the same time, the Project DIRACC calculus sequence will be explicitly computational because, as in the existing first course, students will use computers to represent processes that define functions as models of dynamic situations, which then become objects of study themselves. The courses will also acknowledge and address known weaknesses in students' preparation for calculus. The calculus concept inventories will be developed using standard instrument-development techniques, given at the beginning and end of their respective courses, and their psychometric properties will be established with approximately 600 students per course.