Collaborative Research: Integrating conceptual reasoning with mathematical formalism: Teaching and assessing mathematical sense-making in quantum mechanics

这个项目解决了物理学习中一个重要的领域：将数学理解和对物理的概念掌握联系起来。以前关于量子力学教与学的研究要么集中在概念理解上，要么集中在对数学形式主义和解决问题技术的能力上，但很少将这两个理解的关键要素联系起来。在量子力学中，将这两个元素结合在一起是必不可少的。我们将改进现有的工具，并提供相关的教学指南，以支持中级(大约二年级)和高年级(初级和高级)量子力学课程的强化学生学习。二年级教材的教学目标将部分取决于我们的高年级学生最需要的思维技能和习惯，这是通过对高年级学生在数学意识构建方面的优势和劣势的研究揭示出来的。这个项目的目标是采取一种基于研究和研究验证的方法来修改材料套件，明确地将学生对量子力学的概念掌握和数学理解联系起来。评估工具和教师资源的开发将与课程模块的完善同步进行，所有这些都是通过研究来促进对破坏或支持量子力学中数学意义形成的机制的理解。这项研究将结合对学生的大N调查和细粒度视频分析，以调查设计/修改的材料在课堂和临床环境中的有效性。精细化的课程材料将在5所规模和多样性不同的机构实施，直接影响到约400名学生。该小组将鼓励和便利其他教员和机构在知情的情况下对材料进行改编，以满足各自教学环境的需要。通过将数学意义创造和量子力学“隐藏课程”的其他元素公之于众，该项目将有助于让更多没有接受物理教育的学生更容易接触到高级物理。提高学生对量子力学的理解，这可能是物理学进步的障碍(例如，在博士资格考试中)，将增加对代表不足的人群的留存。最后，接受课程改革后成为教师的物理专业学生可能更倾向于寻找基于物理教育研究的材料和方法，这有利于物理课程的下一代学生。由于乘数效应，该项目的这种二阶更广泛的影响从长远来看可能是巨大的。

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.