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的一个重要领域：将数学理解和物理概念掌握联系起来。之前关于量子力学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，整合这两个元素是必不可少的。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间师（大致大二）和上师（初中和高中）量子力学课程的学习。小学一级教材的教学目标将部分取决于我们高年级学生最需要的技能和思维习惯，本研究的目的是通过对高年级学生在数学意义建构方面的优势和劣势的研究，经过验证的方法来修改套件的材料，明确联系学生的概念掌握和量子力学的数学理解。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，所有这些都是通过旨在促进对量子力学中破坏或支持数学意义的机制的理解的研究来了解的。本研究将结合联合收割机大N调查与细粒度的视频分析的学生，调查设计/修改后的材料在课堂和临床设置的有效性。经过改进的课程材料将在5所规模和多样性不同的机构实施，直接影响约400名学生。该小组将鼓励和促进其他教员和机构根据自己的教学需要对教材进行知情的改编。通过将量子力学“隐藏课程”中的数学意义和其他元素公开化，该项目将有助于使更广泛的学生更容易接触到高年级物理学，这些学生一开始并不熟悉物理学。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，博士学位）。资格考试），将增加任职人数不足人口的保留。最后，物理专业学生接触到改革后的课程谁后来成为教师可能更倾向于寻求物理教育研究为基础的材料和方法，受益于下一代的学生在物理课程。由于乘数效应，从长远来看，该项目的这种二阶更广泛的影响可能很大。

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.