Using social network analysis to investigate mathematical connections in U.S. and Chinese textbook problems

The study addresses how mathematical connections are represented in high school mathematics textbook problems and compares patterns between U.S. (UCSMP) and Chinese (PEP-A) series on quadratic relations (circles, ellipses, hyperbolas, parabolas). Connections are defined as directed typical (prior-to-new) and reverse (new-to-prior) within-concept (between representations of the same concept) and between-concept (across distinct concepts). The research question asks about similarities and differences in the network of connections in popular U.S. and Chinese textbooks on quadratic relations. Motivated by the importance of connections for conceptual understanding and evidence that balanced bidirectional connections (BC) support learning but are often limited in textbooks, the authors adopt social network analysis (SNA) to visualize, quantify, and compare connection networks.

Prior international textbook comparisons show differences in how connections are integrated, with Chinese textbooks often embedding more bidirectional between-concept connections (e.g., inverse operations, distributive property) while both U.S. and Chinese textbooks can lack bidirectional within-concept tasks. Most research has focused on elementary/middle school content; fewer studies analyze high school topics where patterns may differ. Traditional textbook problem analyses often count instances or categorize cognitive demand, which does not reveal network structure, directionality, or the interplay of concepts and representations. Concept maps typically handle small sets and are less suitable for large, directed networks. Connectivism frames learning as building directed connections, motivating use of SNA. Previous SNA in math education has mainly analyzed classroom interaction/arguments, identifying central ideas and shifts over time; applications to textbook problems are rare. The authors extend earlier work (Li and Fan, 2023) by examining both typical and reverse within- and between-concept connections using digraphs and new metrics (centrality, connectivity, directionality ratios, bidirectional gravity) in a three-level analytical framework (digraph, vertex, edge).

Design: Comparative textbook analysis using SNA to model connections as directed edges among vertices representing concepts and representations. The framework operates at three levels: digraph (order, total/distinct edges, density; aggregation patterns), vertex (in/out-degree, centrality; in/out-connection, connectivity; in:out ratios; ego-network roles as influential, prominent, dual), and edge (counts and percentages of bidirectional edges/pairs, bidirectional gravity, typical:reverse ratios, directionality rating). Materials: Chinese PEP-A (People’s Education Press, 2007): Compulsory 2, Chapter 4 (Grade 10) and Elective 2-1, Chapter 2 (Grade 11). U.S. UCSMP Advanced Algebra (McGraw-Hill, 2010), Chapter 12 (Grades 9–12). Sequences differ: PEP-A covers circle, ellipse, hyperbola, parabola (6 lessons across two chapters); UCSMP covers parabola, circle, ellipse, hyperbola (9 lessons in one chapter). Teachers’ editions were used to access additional worked examples and full solutions. Data: 537 problems (61% UCSMP). Problems were split into items by numbering (first- and second-level indices). A concepts table was compiled from chapter review/glossary; representations were categorized as W (written description), S1 (standard conic form), S2 (general quadratic Ax²+Bxy+Cx²+Dx+Ey+F=0), S (other symbolic), T (table), G (graph), N (numerals). A complete connection table listed all potential between-concept pairs and within-concept representation pairs. Each solution step was coded as a connection with type (BCC or WCC) and directionality (typical T or reverse R). Reliability: One U.S. and one Chinese math teacher (5+ years experience) reviewed and finalized coding. Two bilingual math education graduate students checked one randomly selected lesson from each series (74 items, <20% of lessons); agreement exceeded 80% across type, directionality, and source/target coding. Analysis: NodeXL generated digraphs and computed order, total edges (with multiplicity), and distinct edges (schema-based). Density categories (dense, moderate, sparse) were set by thresholds derived from average vertices and edges per between-concept digraph (dense cut-offs and 75%, 50% for moderate/sparse). Vertices were arranged by categories: quadratic relations, linear-function-related (L), attributes (A), relations between relations (SS), and other (O). Metrics were computed for vertex centrality/connectivity and bidirectional edge properties, including bidirectional gravity and typical:reverse ratios.

Overall network: 1129 connections coded (483 distinct), with 91% BCC (89% distinct). Ten digraphs visualized networks by subtopic (BCC) and for WCC. Between-concept (BCC) density: Circles and ellipses formed dense digraphs in both series; PEP-A was denser for circles (vertices 48 vs 49; distinct edges 96 vs 78; total edges 240 vs 175) and ellipses (46 vs 38; 71 vs 66; 135 vs 128). Hyperbola and parabola were moderate in PEP-A but sparse in UCSMP; PEP-A had many more vertices, distinct edges, and total edges for these subtopics (e.g., hyperbola 42 vs 24 vertices; 51 vs 30 distinct; 110 vs 82 total; parabola 32 vs 20; 56 vs 33; 128 vs 72). Aggregation patterns: Both series emphasized attributes of quadratic relations. PEP-A gave additional attention to links with linear function-related concepts (e.g., lines, slopes), whereas UCSMP emphasized quadratic–quadratic systems and special-circle relations (e.g., semicircle, interior/exterior, circle–ellipse via scale change). Within-concept (WCC) density: UCSMP showed a denser WCC network (moderate) than PEP-A (sparse). UCSMP: 51 vertices, 44 distinct edges, 79 total edges vs PEP-A: 12 vertices, 8 distinct edges, 21 total edges. Both series emphasized symbolic→graphical connections, with relatively few reverse graphical→symbolic tasks. Vertex roles: In the original BCC networks, circle was the most central vertex in both series (approx. 9.7–9.9% of all BCCs lead to/from circle). In PEP-A, all four conics (circle, ellipse, hyperbola, parabola) tended to play dual roles; line was prominent (in:out ratio ~2.6). In UCSMP, ellipse and parabola were more influential (lower in:out ratios, e.g., ellipse ~0.3), and “two intersections” was prominent (in:out ~9.5). In schema-based view, ellipse had highest connectivity in both series (about 7.4–8.0% of distinct BCCs to/from ellipse). UCSMP’s circle showed greater diversity than PEP-A’s circle, suggesting PEP-A’s circle was less integrated with other conics (possibly due to chapter separation). Bidirectional edges: Majority of connections were not bidirectional (about 55% of total and 74% of distinct edges were non-bidirectional). Total bidirectional edges: 503 (127 distinct), mostly BCC (95% total; 88% distinct). UCSMP had slightly higher bidirectional percentages for BCC (48% vs 46% total; 28% vs 25% distinct) and much higher for WCC (30% vs 0% total; 34% vs 0% distinct) than PEP-A. High-gravity BCC pairs in both series included circle–center and circle–radius (e.g., PEP-A: circle–center 14.4% of bi pairs; circle–radius 11.2%; UCSMP: 9.0% and 9.5%, respectively). Typical:reverse balance differed: PEP-A balanced circle–center (~0.8) and circle–radius (~1.1), whereas UCSMP emphasized typical over reverse (ratios ~0.4–0.5 for reverse direction). UCSMP strongly emphasized ellipse→intercepts over reverse (typical:reverse ~9.0 for x-intercept; ~8.0 for y-intercept). Both series highlighted ellipse/parabola attribute→concept (typical) more than reverse, but hyperbola–foci was more balanced. Representation findings: Graphical representations were prominent for circles and parabolas in PEP-A and circles, ellipses, hyperbolas in UCSMP; symbolic forms tended to be influential. UCSMP frequently used circle standard form (x−a)²+(y−b)²=r², while PEP-A often used the general quadratic form Ax²+Bxy+Cx²+Dx+Ey+F=0, making reverse determinations of center/radius more demanding and thus more balanced in PEP-A.

Findings indicate PEP-A provides denser and more balanced between-concept networks, particularly across all conics, while UCSMP provides richer within-concept networks but with notable imbalances favoring typical directions. Network density appears related to topic sequencing and chapter organization: PEP-A separates circle from other conics, potentially limiting diversity of circle’s connections; UCSMP collocates all conics in one chapter and includes dedicated lessons (e.g., circle–ellipse), enhancing those links. Both series place linear functions far from conics, which may reduce linear–quadratic connectivity. Within-concept imbalances (scarcity of graphical→symbolic) align with documented student difficulties, suggesting textbooks may not provide enough reverse-direction opportunities. UCSMP’s SPUR approach likely fosters WCC density, though often with repetitive single-step tasks (e.g., ellipse attributes), which may skew typical:reverse ratios. Cross-series borrowing could improve balance: UCSMP could adopt PEP-A’s deliberately designed linear–quadratic problems; PEP-A could incorporate UCSMP-like quadratic–quadratic system variations and circle–ellipse links via transformations. Implications include designing problems to purposefully balance typical and reverse directions, integrating subtopic-bridging tasks (e.g., scale changes connecting circles and ellipses; eccentricity linking ellipses and hyperbolas), and expanding representation diversity (especially reverse graphical→symbolic), leveraging digital tools to support bidirectional WCC.

The study introduces a novel SNA-based framework to analyze and visualize networks of mathematical connections in textbooks, defining typical and reverse directions and applying a three-level analysis (digraph, vertex, edge). Applying the method to U.S. UCSMP and Chinese PEP-A conics chapters revealed that PEP-A offers denser, more balanced between-concept networks, whereas UCSMP offers denser within-concept networks but often emphasizes typical over reverse directions. The approach clarifies where imbalances exist, highlights influential/prominent vertices, and surfaces high-gravity bidirectional pairs, offering actionable insights for improving textbook problem design and sequencing. Future work can extend this methodology to other units, topics (e.g., statistics), subjects (e.g., physics), countries, and digital textbooks, with potential for automated data collection and analysis platforms. The framework can also be used to visualize and assess students’ constructed connection networks to support instruction and track understanding.

Generalizability is limited because only two specific, though popular, series were analyzed (PEP-A and UCSMP). Coding reliability checks covered fewer than 20% of lessons (one randomly selected lesson per series), although agreement exceeded 80%. The study analyzes intended textbook content rather than enacted classroom practice, and gaps may exist between textbooks and actual instruction. Some problem types (projects, readings, explorations, reviews, self-tests) were excluded due to irregular sequencing and frequency.