Collaborative Research: Integrating conceptual reasoning with mathematical formalism: Teaching and assessing mathematical sense-making in quantum mechanics
This project addresses a key area of significance in physics learning: linking mathematical understanding and conceptual mastery of physics. Prior research on the teaching and learning of quantum mechanics has focused either on conceptual understanding or on competency with the mathematical formalism and problem-solving techniques, but only rarely linked these two critical elements of understanding. In quantum mechanics, integrating these two elements is essential. We will refine existing tools and provide associated instructional guides that support enhanced student learning both in the middle division (roughly sophomore) and upper division (junior and senior) quantum mechanics courses. The instructional targets for the sophomore-level materials will be determined in part by the skills and habits of mind that our upper-division students most need, as revealed by research on upper-division students' strengths and weaknesses with mathematical sense-making.
The goal of this project is to take a research-based and research-validated approach to modifying suites of materials that explicitly link students conceptual mastery and mathematical understanding of quantum mechanics. Development of assessment tools and of resources for instructors will happen in tandem with the refinement of the curricular modules, all informed by research designed to advance understanding of the mechanisms that disrupt or support mathematical sense-making in quantum mechanics. This research will combine large-N surveys with fine-grained video analysis of students to investigate the effectiveness of the designed/modified materials in both classroom and clinical settings. The refined curricular materials will be implemented at 5 institutions varying in size and diversity, directly impacting around 400 students. The team will encourage and facilitate informed adaptation of the materials by other instructors and institutions to meet the needs of their own instructional contexts. By bringing mathematical sense-making and other elements of the quantum mechanics "hidden curriculum" out into the open, this project will contribute to making upper-division physics more accessible to a wider range of students who do not start out enculturated into physics. Improving students' understanding of quantum mechanics, which can be a barrier to advancement in physics (e.g, on Ph.D. qualifying exams), will increase retention of underrepresented populations. Finally, physics majors exposed to reformed curricula who later become instructors might be more inclined to seek out physics education research-based materials and methods, benefiting the next generation of students in physics courses. This second-order broader impact of the project might be large in the long term due to the multiplier effect.