, Volume 51, Issue 7, pp 1183–1197 | Cite as

Symbolizing while solving linear systems

  • Michelle ZandiehEmail author
  • Christine Andrews-Larson
Original Article


Solving systems of linear equations is of central importance in linear algebra and many related applications, yet there is limited literature examining the symbolizing processes students use as they work to solve systems of linear equations. In this paper, we examine this issue by analyzing final exam data from 68 students in an introductory undergraduate linear algebra course at a large public research university in the United States. Based on our analysis, we expanded our framework (Larson & Zandieh, 2013) for interpretations of matrix equations to include augmented matrices and symbolic forms commonly used in solving linear systems. We document considerable variation in students’ symbolization processes, which broadly occurred along two primary trajectories: systems trajectories and row reduction trajectories. Row reduction trajectories included at least five symbolic shifts, two of which students executed with a great deal of success and uniformity. Students’ symbolizing processes varied more in relation to the other three shifts, and these variations were often linked to trends of variable renaming, variable creation, or imagined parameter reasoning. Students were more flexible in their solution strategies when solving systems involving lines than for systems involving planes.


Linear algebra Systems of equations Augmented matrices Student reasoning 



Funding was provided by National Science Foundation (Grant no. DUE 1712524).


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Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.Faculty of Sciences and Mathematics, College of Integrative Sciences and ArtsArizona State University, Polytechnic CampusMesaUSA
  2. 2.Florida State UniversityTallahasseeUSA

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