On phylogenetics of matrices in finite mathematics textbooks

Why study textbooks?

In the U.S., Finite Mathematics is a non-STEM-major course, typically covering matrices, finance, linear systems, and probability, and is a common alternative course to Business Calculus. Finite Math’s eclectic mix of topics makes textbook selection nontrivial. Textbook analysis studies benefit institutions by supporting the implementation of these courses (Mesa & Griffiths, 2012). Prior studies (e.g., Lockwood et al., 2017) qualitatively analyzed a combinatorics concept in FM texts. However, our study investigates how the matrix narrative is developed across textbooks through the lens of quantitative biology. 

Textbooks have family trees, too.

Phylogenetics examines organisms’ evolutionary history, often visualized as trees (Owen, 2008). While phylogenetic tree analysis has been applied in math and quantitative biology research (Owen, 2008), to our knowledge, it has not been applied to math textbook analysis. This study expands math textbook analysis to Finite Math and introduces phylogenetic methods as a quantitative approach. One of the authors teaches Finite Math and is interested in the alignment of matrices to the business curriculum, specifically, which topics are most emphasized. We begin to explore this by answering the research question: How similar are the structures of Finite Math textbooks’ presentation of matrices?  

How similar are the structures of Finite Math textbooks’ presentation of matrices?  

Methodology

In this study, we analyzed three popular Finite Math textbooks: Lial et al. (2021), Tan (2018), and Sekhon and Bloom (2016). We constructed (phylogenetic) trees to model the structure of each textbook’s chapter on matrices (Ch 2 in all three texts). We included section titles, subsection headers, and emphasized terms as nodes since these items indicate the author’s intended hierarchy of information. We analyzed subtrees corresponding to four topics common to the texts: Gauss-Jordan Method (Lial et al, Tan, Sekhon & Bloom: 2.2), Matrix Arithmetic (Lial et al: 2.3/2.4, Tan: 2.4/2.5, Sekhon & Bloom: 2.1), Inverses (Lial: 2.5, Tan: 2.6, Sekhon & Bloom: 2.4), and Input-Output Models (L, SB: 2.6, T: 2.7). In phylogenetics, the Robinson-Foulds distance (RFD) between two trees is the size of the symmetric difference between their edge sets, or the union of the edge sets minus their intersection (Owen, 2008). In this context, RFD quantifies the similarity between two texts’ structures. To account for size variation, we computed pairwise relative RFDs, dividing the RFD by the trees’ total number of edges.  

Results, Discussion, & Impacts

The pairwise RRFDs for all four topics are visualized below as triangles drawn to scale compared to an equilateral triangle of side length 1 (the max RRFD). Our results suggest that Finite Math texts differ substantially on matrix topics since the side lengths of the blue triangles are all ≥0.5 (50% dissimilar), but more work is needed to determine if this is statistically significant. Also, the triangles’ interior angles capture the consistency of the differences between texts; e.g., for inverses, Tan’s angle is smaller than the others, signifying that its approach diverges from the other two texts. To help reach mathematicians with this research, quantifying these differences may help inform institutions’ text selection for Finite Math courses. For instance, they could choose the cheaper option of two similar texts to benefit students. Since texts can influence how a course is presented (Mesa & Griffiths, 2012), text selection is impactful on math courses.   

References

Lial, M., Greenwell, R., and Ritchey, N. (2021). Finite Mathematics, Twelfth Edition. Pearson. 

Lockwood, E., Reed, Z., & Caughman, J. S. (2017). An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks. International Journal of Research in Undergraduate Mathematics Education, 3(3), 381–416. https://doi.org/10.1007/s40753-016-0045-y 

Mesa, V., & Griffiths, B. (2012). Textbook mediation of teaching: An example from tertiary mathematics instructors. Educational Studies in Mathematics, 79(1), 85–107. https://doi.org/10.1007/s10649-011-9339-9 

Owen, M. A. (2008). Distance Computation in the Space of Phylogenetic Trees [Doctoral dissertation, Cornell University]. Cornell E-Commons. https://ecommons.cornell.edu/server/api/core/bitstreams/4ec842c4-3aec-46ad-b8f6-dd6bdebe4da3/content  

Sekhon, R. & Bloom, R. (2016). Applied Finite Mathematics, Third Edition.  OER Commons. Retrieved March 27, 2026, from https://oercommons.org/courses/applied-finite-mathematics-3rd-edition 

Tan, S. T. (2018). Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach, Twelfth Edition. Cengage.  

Meet the Authors