大数据环境下精准化教学的数学模型及应用研究

In education, the emergence of “big data” through new extensive educational media, combined with advances in computation holds promise for improving learning processes in formal education, and beyond as well. With the popularization of e-learning in the era of Internet, the precision education based on the educational big data has become an inevitable trend. This project aims to find out the inherent laws of teaching activities and cognitive behaviors by analyzing and mining historical and experimental data in teaching activities, establish a general and standardized mathematical model of teaching evaluation, quantify the individual's learning status, design and optimize based on multi-stage dynamic teaching programs. This project includes two parts: theoretical modeling and practical application, which runs through the entire process of education, including academic analysis, program design, individual implementation, process monitoring and effectiveness testing. Key research points include: 1. Mathematical model of teaching evaluation in precision education; 2. Evaluation and comparison of precision education regime, including prediction and test of teaching effectiveness; 3. Selection of variables that affect the design of teaching regime; 4. Design of optimal precision teaching regime and optimization of multi-stage dynamic teaching regime; 5. Evaluation and classification of individual students' learning states, and subgroup analysis of learning community; 6. Practical application of precision teaching regime. The work of this project is expected to improve the theoretical system of precision education under the big data environment, as well as enrich the research methods in educational big data, and also has a broad prospects for educational applications.

教育领域中，伴随着信息技术和新兴媒体的应用，“大数据”有望极大地促进教学活动的提高。基于教育大数据的精准化教学成为一种必然趋势。本项目旨在通过对教学活动中历史和实验数据的分析挖掘，发现教学活动和认知行为的内在规律，建立一个通用规范的数学模型，量化地分析个体的学习状态，科学地设计和优化基于个体的多阶段动态教学方案。本项目研究包括理论模型和实践应用两部分，贯穿从学情分析、方案设计、个体实施、过程监控到效果检验整个教学过程。重点研究的问题包括：1.构建精准教学中的教学评价数学模型；2.精准教学方案的评估和比对；3.影响精准教学方案的变量选择；4.最优精准教学方案的设计和多阶段动态教学方案的优化；5.学习个体状态的评价和分类、学习群体的亚组分析；6.精准教学方案的实践应用。本项目的工作是对大数据环境下精准化教育理论体系的完善，也是对相关教育大数据研究方法的丰富，同时具备广阔的教育应用前景。

项目组引入和改进认知诊断模型和机器学习算法，围绕教育大数据的三个基本科学问题“教学主体可测量、教学过程可计算、教学服务可定制”进行探索研究。构建了基于大数据精准教学的通用框架和数学模型，将精准教学干预的核心问题转化为多阶段动态决策问题，利用认知诊断实现精准测量，引入强化学习提炼最优决策。项目组还在普通高中、大学教学以及特殊教育中进行了教学干预实证研究，积累了大量有价值的数据。除了理论模块，项目组在分位数迹回归、Boosting 算法、变量选择加权Lasso方法、数据增强等大数据和机器学习的关键技术上取得研究成果，对相应方法的应用具有指导意义，例如完成了Boosting算法永远不会过拟合的理论证明等。