CAREER: Developing Undergraduate Combinatorial Curriculum In Computational Settings

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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.