CAREER: Developing Undergraduate Combinatorial Curriculum In Computational Settings

数学中的组合学问题是有效分析和开发有效算法的基础，它们在计算机科学和概率等领域有着宝贵的应用。该项目将通过调查学生对组合数学关键主题和思想的学习和理解，促进国家对与提高劳动力计算思维和解决问题能力有关的基础研究的需求。然而，有大量的证据表明，学生努力理解和解决组合问题。这样的斗争给数学教师带来了长期的困难，他们发现教授组合数学是一项挑战。目前，创新的组合课程材料培养计算方法不存在本科课堂。为了改善组合数学的教学，该项目将调查计算活动（如编写程序或开发算法）如何影响本科生解决和推理组合问题的方式。该项目将开发利用计算活动的课程材料，以引起丰富的学生参与组合问题，并改善组合问题解决的教学。该项目将促进对各种人群的数学内容的教学和理解，包括数学学生，计算机科学学生和教师，它将建立数学和计算机科学教育之间的跨学科联系。该项目将包括迭代设计实验，以开发需要计算活动来解决组合问题的教学任务。这个项目的创新研究思想是利用该领域当前的计算兴趣作为提高学生组合问题解决能力的一个背景。这项工作建立在先前的结果，证明学生的好处，当他们系统地列出结果，因为他们解决组合问题。通过参与编程和编写算法的计算活动，学生将有机会列出各种问题和背景下的结果，特别是那些结果太大而无法手工枚举的问题。课程材料将以模块的形式在多所大学的数学教室中实施。该项目的成功将根据学生在计算环境中的组合推理所获得的定性见解以及模块在提高学生组合问题解决能力方面的功效进行评估。该项目至少在三个方面推动了数学教育领域的发展：（1）它将深入了解学生在计算环境中的组合思维，这需要学生对组合数学领域进行认知推理的方式;（2）它将提供研究-基于课程材料的本科组合学教学通过计算活动，已通过课堂实施测试和完善;（3）它将使数学界更好地理解计算思维所涉及的广泛原则，以及计算活动与数学思维和学习之间的关系。

Combinatorics problems in mathematics are fundamental to the effective analysis and development of efficient algorithms, and they have invaluable applications in fields such as computer science and probability. This project will contribute towards the national need for basic research related to improving computational thinking and problem solving abilities in the workforce by investigating students' learning and understanding of key topics and ideas from combinatorial mathematics. However, there is abundant evidence that students struggle to understand and solve combinatorial problems. Such struggles create a perennial difficulty for mathematics instructors, who find it challenging to teach combinatorics. Currently, innovative combinatorial curricular materials fostering computational approaches do not exist for undergraduate classrooms. To improve the teaching and learning of combinatorics, the project will investigate how computational activity (such as writing programs or developing algorithms) affects the ways in which undergraduate students solve and reason about combinatorial problems. The project will develop curricular materials that leverage computational activity to elicit rich student engagement with combinatorial problems and to improve instruction of combinatorial problem solving. The project will promote teaching and understanding of mathematical content to a variety of populations, including mathematics students, computer science students, and instructors, and it will establish interdisciplinary connections between mathematics and computer science education.The project will consist of iterative design experiments to develop instructional tasks that require computational activities to solve combinatorial problems. The innovative research idea underlying this project is to leverage the current computational interests of the field as a setting to improve students' combinatorial problem solving. The work builds upon prior results that demonstrate benefits for students when they systematically list outcomes as they solve combinatorial problems. By engaging in the computational activities of programming and writing algorithms, students will gain opportunities to list outcomes on a wide range of problems and contexts, especially problems for which the set of outcomes is too large to enumerate by hand. The curricular materials will be in the form of modules that will be implemented in mathematics classrooms at multiple universities. The success of the project will be evaluated based both on the qualitative insights that are gained about students' combinatorial reasoning in computational settings, and the modules' efficacy in improving students' combinatorial problem solving. The project stands to advance the field of mathematics education in at least three ways: (1) it will offer insight into students' combinatorial thinking in computational settings, which entails the ways in which students cognitively reason about the mathematical domain of combinatorics; (2) it will deliver research-based curricular materials for the teaching of undergraduate combinatorics via computational activity that have been tested and refined through classroom implementation; and (3) it will lead to the mathematics community gaining a better sense of broad principles involved in computational thinking, and the relationship between computational activity and mathematical thinking and learning.