## Abstract

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.

## Keywords

Linear algebra Systems of equations Augmented matrices Student reasoning## Notes

### Funding

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

## References

- Dorier, J. L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacles of formalism in linear algebra. In J. L. Dorier (Ed.),
*On the teaching of linear algebra*(pp. 85–124). Dordrecht: Kluwer.Google Scholar - Dorier, J.-L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level: An ICMI study*(pp. 255–273). Dordrecht: Kluwer Academic Publishers.Google Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education*. Dordrecht: Kluwer Academic.Google Scholar - Harel, G. (2017). The learning and teaching of linear algebra: Observations and generalizations.
*The Journal of Mathematical Behavior,**46,*69–95.CrossRefGoogle Scholar - Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.),
*On the teaching of linear algebra*(pp. 191–207). Dordrecht: Kluwer Academic Publishers.Google Scholar - Larson, C., & Zandieh, M. (2013). Three interpretations of the matrix equation Ax = b.
*For the Learning of Mathematics,**33*(2), 11–17.Google Scholar - Lay, D. (2012).
*Linear algebra and its applications*(4th ed.). Hoboken: Pearson.Google Scholar - Oktaç, A. (2018). Conceptions about system of linear equations and solution. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.),
*Challenges and strategies in teaching linear algebra*(pp. 71–101). Berlin: Springer.CrossRefGoogle Scholar - Possani, E., Trigueros, M., Preciado, J. G., & Lozano, M. D. (2010). Use of models in the teaching of linear algebra.
*Linear Algebra and its Applications,**432*(8), 2125–2140.CrossRefGoogle Scholar - Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress.
*Educational Studies in Mathematics,**88*(2), 259–281.CrossRefGoogle Scholar - Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A view of advanced mathematical thinking.
*Mathematical Thinking and Learning,**7,*51–73.CrossRefGoogle Scholar - Sandoval, I., & Possani, E. (2016). An analysis of different representations for vectors and planes in ℝ
^{3}.*Educational Studies in Mathematics,**92*(1), 109–127.CrossRefGoogle Scholar - Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.),
*On the teaching of linear algebra*(pp. 209–246). Dordrecht: Kluwer.Google Scholar - Steinberg, R. M., Sleeman, D. H., & Ktorza, D. (1991). Algebra students’ knowledge of equivalence of equations.
*Journal for Research in Mathematics Education,**22*(2), 112–121.CrossRefGoogle Scholar - Stewart, S., & Thomas, M. O. J. (2009). A framework for mathematical thinking: The case of linear algebra.
*International Journal of Mathematical Education in Science and Technology,**40*(7), 951–961.CrossRefGoogle Scholar - Trigueros, M. (2018). Learning linear algebra using models and conceptual activities. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.),
*Challenges and strategies in teaching linear algebra*(pp. 29–50). Berlin: Springer.CrossRefGoogle Scholar - Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra.
*Linear Algebra and its Applications,**438*(4), 1779–1792.CrossRefGoogle Scholar - Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative.
*CBMS Issues in Mathematics Education,**8,*103–127.CrossRefGoogle Scholar - Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example.
*The Journal of Mathematical Behavior,**25*(1), 1–17.CrossRefGoogle Scholar - Zandieh, M., & Knapp, J. (2018). Metonymy and metaphor: How language can impact understanding of mathematical concepts (Part I).
*MathAMATYC Educator,**9*(2), 23–27.Google Scholar - Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning.
*The Journal of Mathematical Behavior,**29*(2), 57–75.CrossRefGoogle Scholar - Zandieh, M., Wawro, M., & Rasmussen, C. (2017). An example of inquiry in linear algebra: The roles of symbolizing and brokering.
*PRIMUS,**27*(1), 96–124.CrossRefGoogle Scholar