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

本项目探讨了物理学习中一个重要的关键领域：将数学理解与物理概念掌握联系起来。先前对量子力学的教学和学习的研究要么集中在概念理解上，要么集中在数学形式主义和问题解决技术的能力上，但很少将这两个理解的关键要素联系起来。在量子力学中，把这两个元素结合起来是必不可少的。我们将完善现有的工具，并提供相关的教学指南，以支持增强学生在中年级（大约大二）和高年级（大三和大四）量子力学课程中的学习。二年级教材的教学目标部分取决于高年级学生最需要的技能和思维习惯，正如对高年级学生在数学意义构建方面的优缺点的研究所揭示的那样。该项目的目标是采用一种基于研究和研究验证的方法来修改一系列材料，这些材料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，所有这些研究都旨在促进对量子力学中破坏或支持数学意义构建的机制的理解。本研究将结合大n调查和学生的细粒度视频分析，以调查设计/修改后的材料在课堂和临床环境中的有效性。修订后的教材将在5所不同规模和种类的学校实施，直接影响约400名学生。该小组将鼓励和促进其他教员和机构对这些材料进行知情改编，以满足其自身教学环境的需要。通过将数学意义构建和量子力学“隐藏课程”的其他元素公开化，该项目将有助于使更大范围的学生更容易接触到高年级物理，而这些学生一开始并没有被物理所吸引。提高学生对量子力学的理解，这可能是物理学进步的一个障碍（例如，在博士资格考试中），将增加未被充分代表的人群的保留率。最后，接触过改革课程的物理专业学生日后成为教师后，可能会更倾向于寻找物理教育的研究性材料和方法，从而使下一代物理课程的学生受益。由于乘数效应，该项目的二阶更广泛影响可能在长期内很大。

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.