Knowledge is a network of interconnected concepts. Yet, precisely how the topological structure of knowledge constrains its acquisition remains unknown, hampering the development of learning enhancement strategies. Here, we study the topological structure of semantic networks reflecting mathematical concepts and their relations in college-level linear algebra texts. We hypothesize that these networks will exhibit structural order, reflecting the logical sequence of topics that ensures accessibility. We find that the networks exhibit strong core–periphery architecture, where a dense core of concepts presented early is complemented with a sparse periphery presented evenly throughout the exposition; the latter is composed of many small modules each reflecting more narrow domains. Using tools from applied topology, we find that the expositional evolution of the semantic networks produces and subsequently fills knowledge gaps, and that the density of these gaps tracks negatively with community ratings of each textbook. Broadly, our study lays the groundwork for future efforts developing optimal design principles for textbook exposition and teaching in a classroom setting.

中文翻译：

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数学文本中语义网络的架构与演化

知识是相互关联的概念的网络。然而，知识的拓扑结构究竟如何限制其获取仍然未知，这阻碍了学习增强策略的发展。在这里，我们研究了在大学水平的线性代数课本中反映数学概念的语义网络的拓扑结构及其关系。我们假设这些网络将表现出结构顺序，反映确保可访问性的主题的逻辑顺序。我们发现网络表现出强大的核心 - 外围架构，其中早期提出的密集概念核心与整个展览中均匀呈现的稀疏外围相辅相成；后者由许多小模块组成，每个模块都反映了更窄的领域。使用来自应用拓扑的工具，我们发现语义网络的阐述演化产生并随后填补了知识空白，并且这些空白的密度与每本教科书的社区评分呈负相关。从广义上讲，我们的研究为未来在课堂环境中为教科书展示和教学开发最佳设计原则奠定了基础。