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

微积分是一个重要的工具，并提供了一个概念基础，在每个STEM学科。初学微积分时对概念的深刻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，在微积分改革方面的许多共同努力似乎都未能像其支持者所预测的那样，在学生的理解方面产生具有教育意义的显著差异。DIRACC项目：开发和调查一种严格的概念微积分方法，将以亚利桑那州立大学已经重新设计和部署的微积分I的成功为基础，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微积分序列中的学习情况。此外，DIRACC项目将为微积分I和微积分II开发简答评估工具（也称为概念清单），其他机构可以使用它来评估、比较和对比学生在理解微积分核心思想方面的进展。学习这门重新设计的微积分课程的学生将更好地准备，并更有可能应用他们在其他STEM课程中对变化率函数和积累函数的理解，并在随后的数学课程中学习微积分相关的材料。本计划制作的教材将作为开放资源提供给其他人使用或在此基础上发展。最后，学生微积分学习的结果，以及为调查学生学习而建立的微积分概念清单，将为这些领域的未来研究提供信息。DIRACC项目的结果将在区域、国家和国际会议上广泛传播，并在同行评议的数学教育期刊和会议论文集上发表。微积分改革缺乏成效的一个可能原因是，在绝大多数教学环境中，基础课程的基本结构没有改变。具体来说，改革项目没有关注学生对变化率函数和积累函数的丰富关联意义的发展，这对任何入门级微积分都是必不可少的。DIRACC项目的课程是高度概念性的，因为它们是基于学生对适用于整个微积分的思想的连贯意义的发展，也促进了微积分以外的数学思想的学习。第一门课程通过使微积分基本定理成为课程中学生微积分体验的中心来解决这一挑战。同时，项目DIRACC微积分序列将明确计算，因为在现有的第一门课程中，学生将使用计算机来表示将函数定义为动态情况模型的过程，然后这些过程本身将成为研究对象。课程也将承认和解决已知的弱点，在学生准备微积分。微积分概念清单将使用标准的工具开发技术进行开发，在各自课程的开始和结束时给出，并且每门课程将建立大约600名学生的心理测量特性。

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.