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

Education

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

S. Li and L. Fan

This study, conducted by Shuhui Li and Lianghuo Fan, utilizes social network analysis to unveil the contrasting mathematical connections in U.S. and Chinese high school textbooks on quadratic relations. Discover how connection density and strength differ, offering fresh insights into students’ conceptual understanding.

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Playback language: English
Introduction
This research investigates the representation of mathematical connections within U.S. and Chinese high school mathematics textbooks, specifically focusing on problems related to quadratic relations (circles, ellipses, hyperbolas, and parabolas). The study's rationale stems from the observed disparity in mathematical achievement between Chinese and U.S. students, as highlighted by PISA, and the potential influence of textbooks on this difference. While national curriculum reforms in both countries emphasize fostering mathematical connections—defined as within-concept (connections between different representations of the same concept) and between-concept (connections between distinct concepts)—a detailed comparative analysis of how these connections are presented in textbooks is lacking. The study aims to address this gap by employing social network analysis (SNA), an innovative approach in mathematics education research, to map and analyze the networks of connections presented in two widely used textbook series: the University of Chicago School Mathematics Project (UCSMP) series in the U.S. and the PEP-A series in China. The central research question is: What are the similarities and differences in the network of connections represented in popular U.S. and Chinese high school mathematics textbook problems focusing on quadratic relations?
Literature Review
Existing research comparing U.S. and Chinese mathematics textbooks reveals significant differences in the integration of connections, particularly between-concept connections. Studies at the elementary and middle school levels show that Chinese textbooks tend to present bidirectional connections (typical and reverse connections presented with equal emphasis), while U.S. textbooks often lack this balance. However, less attention has been given to high school textbooks, particularly concerning the overall structure and network of connections rather than simply comparing the number or percentage of bidirectional connections. Traditional methods of textbook analysis, such as focusing on specific connections or using concept maps, are limited in their ability to comprehensively analyze the complex network of connections. Concept maps, for instance, struggle to manage a large number of concepts and connections and effectively represent their directionality. The study highlights connectivism as a learning theory that supports the use of SNA for connection analysis, emphasizing the importance of mapping the directed edges between concepts and representations within a network.
Methodology
The study employed a quantitative approach using social network analysis (SNA) to examine U.S. (UCSMP) and Chinese (PEP-A) high school textbooks focusing on quadratic relations. Data collection involved identifying all problems related to quadratic relations (including worked examples, exercises, and solutions in both student and teacher editions) from the chosen textbooks. A total of 537 problems were analyzed, with UCSMP having a slightly higher number of problems. Data coding involved creating a comprehensive connection table listing all possible connections between concepts and representations (written description, symbolic expressions, tables, graphs, and numerals). Each problem solution was coded step-by-step based on the relevant connections, their types (within-concept or between-concept), and directionality (typical or reverse). Two experienced mathematics teachers (one from each country) and two graduate students verified the coding reliability, achieving an agreement rate exceeding 80% across all dimensions (connection type, directionality, and source/target vertices). NodeXL software was utilized to generate digraphs representing the connection networks, allowing analysis of graph properties (order, total edges, distinct edges, density) and node properties (in-degree, out-degree, centrality, in-connection, out-connection, connectivity). Custom criteria based on the average number of vertices and edges were established to categorize digraphs as dense, moderate, or sparse. Additional metrics (ratio of in-/out-degree, ratio of in-/out-connection, bidirectional edges, bidirectional gravity, ratio of typical/reverse connections) were calculated to further characterize connections and identify influential, prominent, or dual concepts and representations.
Key Findings
The SNA analysis revealed significant differences in the structure and nature of mathematical connections presented in the U.S. and Chinese textbooks. **Digraph Analysis:** * **Between-Concept Connections (BCC):** The Chinese PEP-A series consistently presented denser BCC networks than the U.S. UCSMP series across all quadratic relation subtopics (circles, ellipses, hyperbolas, parabolas). While both series showed dense networks for circles and ellipses, the PEP-A series had significantly more connections. The PEP-A series also showed moderately dense networks for hyperbolas and parabolas, while the UCSMP series showed sparse networks for these topics. This suggests that Chinese textbooks foster more diverse connections between concepts. * **Within-Concept Connections (WCC):** The UCSMP series exhibited a moderately dense WCC network, significantly denser than the sparse network found in the PEP-A series. This difference suggests that U.S. textbooks may provide more opportunities for students to connect different representations of the same concept. **Vertex Analysis:** * The circle concept played a central dual role (influential and prominent) in both textbook series. However, the PEP-A series emphasized connections between the circle and linear function-related concepts (e.g., lines, slopes), while the UCSMP series focused on connections within the quadratic relations (circles and ellipses). * In the UCSMP series, ellipses and parabolas were predominantly influential (more outbound connections), reflecting an emphasis on their impact on other concepts. * The PEP-A series highlighted the prominence of lines, while the UCSMP series emphasized the prominence of “two intersections,” suggesting different approaches to presenting linear-quadratic systems. **Bidirectional Edge Analysis:** * A significant majority of connections (55% of total, 74% of distinct connections) in both textbook series lacked bidirectional representation. * The PEP-A series demonstrated a more balanced representation of typical and reverse connections for several key between-concept pairs, while the UCSMP series displayed an imbalance, particularly emphasizing typical connections for ellipse attributes. * The UCSMP series had a higher percentage of bidirectional within-concept connections. Overall, the findings highlight that Chinese textbooks (PEP-A) present a denser network of more balanced between-concept connections, while U.S. textbooks (UCSMP) prioritize within-concept connections and emphasize typical connections over reverse ones.
Discussion
The findings directly address the research question by highlighting the contrasting approaches to integrating mathematical connections within U.S. and Chinese high school textbooks. The denser networks and balanced bidirectional connections in Chinese textbooks suggest a greater emphasis on connecting mathematical concepts and fostering a deeper understanding of the interconnectedness of mathematical ideas. The U.S. textbooks, while showing strength in integrating within-concept connections, appear to prioritize the forward flow of information, potentially neglecting opportunities for students to develop flexible, bidirectional reasoning skills. The differences observed align with previous research indicating variations in teaching styles and emphasis on conceptual understanding between the two educational systems. The study's methodology contributes to the field by introducing SNA as a powerful tool for analyzing the complex network of connections presented in mathematics textbooks. The detailed quantitative analysis, supplemented by qualitative examples, offers a nuanced understanding of connection integration beyond simple counts or percentages. This innovative approach can be applied to analyze various mathematical topics, textbooks, and educational contexts, paving the way for more comprehensive curriculum analysis.
Conclusion
This study presents a novel method for analyzing mathematical connections in textbooks using social network analysis. The comparison of U.S. and Chinese textbooks reveals significant differences in the density and balance of connections, suggesting potential implications for student learning. The framework developed can be applied to assess students' own connection-making and to inform the design of more effective curriculum materials. Future research could extend this method to other mathematical topics, countries, and textbook types, including e-textbooks.
Limitations
The study's scope is limited to two specific textbook series from the U.S. and China. Generalizing the findings to all textbooks within these countries requires caution. The coding process, while reliable, involved a considerable amount of manual effort, potentially introducing some subjective bias. The focus on quadratic relations might not fully generalize to other mathematical topics. Future research could address these limitations by employing broader samples and automating the coding process.
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