Architecture and evolution of semantic networks in mathematics texts

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.

Optimizing the Human Learnability of Abstract Network Representations

“…These networks exhibit core-periphery structure, indicating that they are composed of nodes that can roughly be divided into a densely-connected core and a periphery that is loosely connected to nodes within the core [46,47,48]. In addition, prior work has established that the periphery of these semantic networks possesses community structure [39]. Importantly, unlike modular networks, we find that the semantic networks are not very optimizable near β ≈ 0, but are significantly optimizable even for moderately Large values of β.…”

“…Still, it remains to be demonstrated that these results extend more generally to real-world information networks. To probe the optimal emphasis modulation strategies of real-world networks, we study semantic networks extracted from college-level linear algebra textbooks [39]. Specifically, nodes represent recurring concepts (e.g., "vector space", "invertible"), and edges between concepts are weighted by the number of sentences in which the two concepts co-occur.…”

“…This observation contrasts greatly with our prior findings in studying modular networks: that the greatest improvements in learnability occur near β ≈ 0, with significantly diminishing improvements at higher β values. One natural explanation for this difference is that the semantic networks do not possess Large-scale community structure, but rather can be characterized as possessing core-periphery structure with community structure within the periphery nodes [39]. Therefore, the learnability optimization strategies for modular networks-which mainly involved over-emphasizing edges within clusters and de-emphasizing crosscluster edges-are likely not applicable to increasing learnability of these semantic networks.…”

“…To understand which edges are reinforced or de-emphasized to increase learnability, for each semantic network, we follow the procedure outlined in Ref. [39] and classify nodes into core and periphery categories (see Methods). Then, for each network, we determine the community structure of the periphery nodes, and categorize the edges of each network into four categories (Fig.…”

Edgework: Viewing curiosity as fundamentally relational

“…The first study reports that science progresses as much through identifying uncharted gaps as through advancing solutions within scientific communities; the latter, filling-the-gap (or tessellating-the-cavity) discoveries are more frequently awarded Nobel prizes than discoveries adding edges at the boundaries of collective knowledge [74]. The second study reports that college-level mathematics textbooks are more likely to be rated highly on GoodReads when they progressively fill (rather than leave open) gaps in the knowledge network by connecting concepts introduced early with concepts introduced late [103]. Together, these studies suggest that there exist some domains of human inquiry where the act of placing a connection that fills a network cavity is more greatly valued than the act of placing a connection to the unknown at the frontiers of knowledge.…”