A teacher's knowledge base consists of knowledge of mathematics content, knowledge of student epistemology, and pedagogical knowledge. It has severe implications on the understanding of student's knowledge of content, and the learning context in general. The necessity to formalize the different content knowledge in approximate senses is recognized in the education research literature. A related problem is that of coherent formalizability. Responsive or smart AI-based software systems do not concern themselves with meaning, and trained ones are replete with their own issues. In the present research, many issues in modeling teachers' understanding of content are identified, and a two-tier rough set-based model is proposed by the present author. The main advantage of the proposed approach is in its ability to coherently handle vagueness, granularity and multi-modality. An extended example to equational reasoning is used to demonstrate these.

翻译：教师的知识基础由数学内容知识、学生认知知识和教学法知识组成。该知识体系对理解学生内容知识及整体学习情境具有重要影响。教育研究文献已认识到在不同近似意义上形式化各类内容知识的必要性，而连贯形式化能力是其中的关联问题。基于人工智能的响应式或智能软件系统并不关注意义层面，经过训练的系统则存在诸多自身缺陷。本研究辨识了教师内容理解建模中的多项问题，作者提出了一种双层粗糙集模型。该方案的主要优势在于能够连贯处理模糊性、粒度和多模态性。通过求解方程推理的扩展示例对此进行了论证。