极值图论课程提升研究生综合素质的改革路径

Reform Pathways for Enhancing Graduate Students’ Comprehensive Competencies through an Extremal Graph Theory Course

摘要: 针对研究生极值图论课程中数学抽象度高、英文文献进入门槛高以及课堂知识与科研任务衔接不足等问题，本文以36学时《极值图论(双语)》为载体，构建“一核双链四阶”教学框架。课程以研究能力形成为核心，使“对象–定理–方法–问题”数学认知链与“术语–命题–论证–交流”学术语言链同步推进，并通过双语导学、证明重构、专题探究和成果表达实现能力递进。教学内容围绕Turán型问题、正则性与消去方法、Ramsey理论、依赖随机选择及极值集合论重组，以多证法比较、量词结构转述、论文片段研读和微型研究任务协同训练数学推理、学术英语、创新迁移与协作表达。进一步建立以学习档案为证据的评价–反馈–再评价闭环。首轮课程材料与科研转化个案显示，该路径能够打通经典理论学习与前沿问题探索；其普遍成效仍需通过后续多届学生评价持续检验。

Abstract: To address the high level of mathematical abstraction, difficulties in engaging with English-language literature, and limited connections between classroom learning and research tasks in graduate-level extremal graph theory, this paper develops a teaching framework comprising “one core, two chains, and four stages” for a 36 teaching hours bilingual course in Extremal Graph Theory. With the development of research competence at its core, the framework advances two learning chains in parallel: a mathematical cognition chain of “objects - theorems - methods - problems” and an academic language chain of “terminology - statements - argumentation - communication”. Progressive competence development is supported through four stages: guided bilingual learning, proof reconstruction, thematic inquiry, and presentation of outcomes. Course content is reorganized around Turán-type problems, regularity and removal methods, Ramsey theory, dependent random choice, and extremal set theory. Comparing multiple proofs, paraphrasing quantified statements, closely reading excerpts from research papers, and undertaking small-scale research tasks are integrated to develop mathematical reasoning, academic English proficiency, the creative application of knowledge in new contexts, and collaborative communication. An assessment-feedback-reassessment cycle is further established, with learning portfolios serving as the evidence base. Materials from the first course offering, together with case studies of the application of course learning to research, suggest that this approach can connect the study of classical theory with the exploration of frontier problems. Its broader effectiveness, however, requires continued evaluation across subsequent student cohorts.