Abstract
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.
Highlights
Knowledge has been distilled into formal representations for millennia [1,2]
Meso-scale structural analysis indicates that the semantic networks exhibit strong core–periphery structure, where a tightly knit group of concepts form a core, surrounded by sparsely connected periphery concepts that are grouped into communities
We extracted the knowledge gaps inherent in the exposition and found that the number of distinct connected components tends to decrease throughout the text, while topological cavities tend to increase
Summary
Knowledge has been distilled into formal representations for millennia [1,2]. Such efforts have sought to explain human reasoning and support artificial intelligence [3,4,5]. Recent work using highly stylized laboratory experiments provides some preliminary evidence that network structure may play a role in how humans process information [14,15,16] and acquire knowledge [17,18,19]. Extending these findings to the real world has proven difficult, and it remains unknown precisely how the network structure of knowledge in the form of science textbooks [20], science and mathematics topics on Wikipedia [21], and even formal scientific papers [22,23] impacts the learnability of these content domains. The education literature establishes that the order in which topics are introduced can help or hinder learning at this level [24,25], but a rigorous understanding of order and dynamic structure in knowledge acquisition has not been formalized in ecologically valid experimental settings
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