Skip to main content
SpringerLink
Log in

Symbolizing while solving linear systems

  • Original Article
  • Published:
ZDM Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

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.

    Article  Google 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.

    Chapter  Google 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.

    Article  Google Scholar 

  • Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics,88(2), 259–281.

    Article  Google 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.

    Article  Google Scholar 

  • Sandoval, I., & Possani, E. (2016). An analysis of different representations for vectors and planes in ℝ3. Educational Studies in Mathematics,92(1), 109–127.

    Article  Google 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.

    Article  Google 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.

    Article  Google 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.

    Chapter  Google Scholar 

  • Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra. Linear Algebra and its Applications,438(4), 1779–1792.

    Article  Google Scholar 

  • Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. CBMS Issues in Mathematics Education,8, 103–127.

    Article  Google 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.

    Article  Google 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.

    Article  Google 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.

    Article  Google Scholar 

Download references

Funding

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

Author information

Authors and Affiliations

  1. Faculty of Sciences and Mathematics, College of Integrative Sciences and Arts, Arizona State University, Polytechnic Campus, 6073 S. Backus Mall MC 2780, Mesa, AZ, 85212-2780, USA

    Michelle Zandieh

  2. Florida State University, Tallahassee, USA

    Christine Andrews-Larson

Authors
  1. Michelle Zandieh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christine Andrews-Larson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michelle Zandieh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zandieh, M., Andrews-Larson, C. Symbolizing while solving linear systems. ZDM Mathematics Education 51, 1183–1197 (2019). https://doi.org/10.1007/s11858-019-01083-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-019-01083-3

Keywords

Search

Navigation