“直观–应用–探索”三位一体的线性代数课堂教学模式研究

Research on the “Intuition - Application - Exploration” Trinity Classroom Teaching Model in Linear Algebra

摘要: 当今社会对创新型、应用型人才的需求日益迫切，高等教育课堂教学模式亟需从知识传授向能力培养作出系统性调整。文章以线性代数课程为研究对象，以“直观–应用–探索”三位一体为主线，设计围绕学生全面成长的课堂教学模式，既满足社会对高素质人才的需求，又提升学生的综合素养。针对目前教学中有关抽象概念理解困难、理论脱离实践、学生缺少主动性等问题，研究提出：将几何直观融入概念讲解，借助可视化手段降低认知门槛；将行业典型案例引入教学实践，增强知识的应用性与迁移能力；利用探究式方法促进知识的主动形成，激发创新与合作精神。为了检验所提模式的实际效果，研究开展准实验研究(分实验组与对照组)，通过前后测、问卷和访谈进行观察，并以“特征值与特征向量”为例，给出一节完整“直观–应用–探索”的课程设计，涵盖可视化工具、PageRank实例、探究性问题以及学生成果分析。教学实践已经证明该模式显著提升了教学内容的可理解性与实践性，促进了学生自主学习能力的发展，拓展了知识传授的多样化路径，优化了教学评价过程。这些措施不仅有利于学生牢固地把握线性代数的核心知识与思想方法，还能有效增强其解决复杂实际问题的能力、创新思维及团队协作能力，对学生全面发展具有重要价值。

Abstract: Society today has an increasingly urgent need for innovative and application-oriented talent, and higher education classroom teaching models urgently need a systematic shift from knowledge transmission to competency development. This paper takes the linear algebra course as its subject of study and adopts the “intuition - application - exploration” approach as a three-pronged framework to design a classroom teaching model centered on students’ comprehensive development, thereby meeting society’s demand for high-caliber talent while simultaneously enhancing students’ overall competence. Addressing current challenges in teaching, such as difficulties in understanding abstract concepts, a disconnect between theory and practice, and a lack of student initiative, this paper proposes integrating geometric intuition into conceptual explanations and utilizing visualization techniques to lower the cognitive barrier. By incorporating typical industry case studies into teaching practice, the model enhances the applicability of knowledge and students’ ability to transfer it. Furthermore, inquiry-based methods are employed to promote active knowledge construction and foster a spirit of innovation and collaboration. To further verify the effectiveness of the model, a quasi-experimental study is designed and implemented, with an experimental group and a control group, and data are collected through pre- and post-tests, questionnaires, and interviews. Meanwhile, taking “eigenvalues and eigenvectors” as an example, a complete teaching design under the “intuition - application - exploration” framework is presented, including visualization tools, the PageRank example, inquiry questions, and analysis of student outputs. Teaching practice has demonstrated that this model significantly enhances the comprehensibility and practicality of the course content, fosters the development of students’ independent learning abilities, expands the diversity of knowledge transmission pathways, and optimizes the teaching evaluation process. These measures not only help students firmly grasp the core knowledge and methodological approaches of linear algebra, but also effectively enhance their ability to solve complex real-world problems, their innovative thinking, and their teamwork skills, thereby holding significant value for students’ all-round development.