Encyclopedia of Mathematics Education

Living Edition
| Editors: Steve Lerman

Linear Algebra Teaching and Learning

  • Maria TriguerosEmail author
  • Megan Wawro
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-77487-9_100021-1


Linear algebra has been recognized as a difficult subject for students, due mainly to the abstract nature of the concepts of this discipline (e.g., Dorier and Sierpinska 2001). The two main research themes regarding the teaching and learning of linear algebra are characterizations of student understanding of particular linear algebra concepts and different didactic approaches to improve its teaching and learning. After a brief summary of preliminary work, we describe research results related to three strands of interest.

Preliminary Work

Research on students’ learning of linear algebra started in the late 1980s. At that time, in France, several researchers called the attention of the international community to students’ difficulties in understanding concepts, even after they had taken one or two university courses on this subject. Dorier’s (2000) edited book was particularly important. His studies, based on a thorough analysis of the historical and epistemological...


Linear algebra Role of abstraction Learning linear algebra Didactic approaches Using models in the teaching of linear algebra 
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  1. Andrews-Larson C, Wawro M, Zandieh M (2017) A hypothetical learning trajectory for conceptualizing matrices as linear transformations. Int J Math Educ Sci Technol 48(6):809–829CrossRefGoogle Scholar
  2. Britton S, Henderson J (2009) Linear algebra revisited: an attempt to understand students’ conceptual difficulties. Int J Math Educ Sci Technol 40(7):963–974CrossRefGoogle Scholar
  3. Carlson D, Johnson C, Lay D, Porter AD (1993) The linear algebra curriculum study group recommendations for the first course in linear algebra. Coll Math J 24:41–46CrossRefGoogle Scholar
  4. Dogan H (2018) Differing instructional modalities and cognitive structures: linear algebra. Linear Algebra Appl 542:464–483CrossRefGoogle Scholar
  5. Dorier JL (2000) Epistemological analysis of the genesis of the theory of vector spaces. In: Dorier (ed) On the teaching of linear algebra. Kluwer, Dordrecht, pp 3–81Google Scholar
  6. Dorier J-L, Sierpinska A (2001) Research into teaching and learning of linear algebra. In: Holton D (ed) The teaching and learning of mathematics at university level, an ICMI study. Kluwer, Dordrecht, pp 255–274Google Scholar
  7. Dubinsky E (1997) Some thoughts on a first course in linear algebra at the college level. In: Carlson D, Johnson CR, Lay DC, Portery RD, Watkins A (eds) Resources for teaching linear algebra, MAA notes, vol 42, pp 85–105Google Scholar
  8. Figueroa AP, Possani E, Trigueros M (2018) Matrix multiplication and transformations: an APOS approach. The Journal of Mathematical Behavior 52:77–91CrossRefGoogle Scholar
  9. Gol Tabaghi S, Sinclair N (2013) Using dynamic geometry software to explore eigenvectors: the emergence of dynamic-synthetic-geometric thinking. Technol Knowl Learn 18(3):149–164CrossRefGoogle Scholar
  10. Gueudet-Chartier G (2006) Using geometry to teach and learn linear algebra. Res Coll Math Educ 13:171–195Google Scholar
  11. Harel G (2017) The learning and teaching of linear algebra: observations and generalizations. J Math Behav 6: 69–95CrossRefGoogle Scholar
  12. Klasa J (2010) A few pedagogical designs in linear algebra with cabri and maple. Linear Algebra Appl 432: 2100–2111CrossRefGoogle Scholar
  13. Kú D, Trigueros M, Oktaҫ A (2008) Comprensión del concepto de base de un espacio vectorial desde el punto de vista de la Teoría APOE. Educ Matemática 20:65–89Google Scholar
  14. Maracci M (2008) Combining different theoretical perspectives for analyzing students difficulties in vector spaces theory. ZDM Int J Math Educ 40:265–276, Springer Berlin/HeidelbergCrossRefGoogle Scholar
  15. Nyman MA, Lapp DA, St John D, Berry JS (2010) Those do what? Connecting eigenvectors and eigenvalues to the rest of linear algebra: Using visual enhancements to help students connect eigenvectors to the rest of linear algebra. International Journal for Technology in Mathematics. 17(1):33–41Google Scholar
  16. Oktaҫ A (2018) Understanding and visualizing linear transformations In: Kaiser G et al (eds) Invited Lectures from the 13th international congress on mathematical education, ICME-13 monographs.  https://doi.org/10.1007/978-3-319-72170-5_26CrossRefGoogle Scholar
  17. Oktaҫ A, Trigueros M (2010) ¿Cómo se aprenden los conceptos de álgebra lineal? Rev Latinoamericana Matemática Educ 13:473–485Google Scholar
  18. Parraguez M, Oktaç A (2010) Construction of the vector space concept from the viewpoint of APOS theory. Linear Algebra Appl 432(8):2112–2124CrossRefGoogle Scholar
  19. Possani E, Trigueros M, Preciado JG, Lozano MD (2010) Using models in the teaching of linear algebra. Linear Algebra Appl 432:2125–2140CrossRefGoogle Scholar
  20. Plaxco D, Wawro M (2015) Analyzing student understanding in linear algebra through mathematical activity. Journal of Mathematical Behavior, 38, 87–100CrossRefGoogle Scholar
  21. Rasmussen C, Wawro M, Zandieh M (2015) Examining individual and collective level mathematical progress. Educ Stud Math 88(2):259–281CrossRefGoogle Scholar
  22. Salgado H, Trigueros M (2015) Teaching eigenvalues and eigenvectors using models and APOS theory. J Math Behav 30:100–120CrossRefGoogle Scholar
  23. Sandoval I, Possani E (2016) An analysis of different representations for vectors and planes in R3: learning challenges. Educ Stud Math 92:109–127CrossRefGoogle Scholar
  24. Selinski N, Rasmussen C, Wawro M, Zandieh M (2014) A methodology for using adjacency matrices to analyze the connections students make between concepts: The case of linear algebra. Journal for Research in Mathematics Education 45(5):550–583CrossRefGoogle Scholar
  25. Stewart S, Andrews-Larson C, Berman A, Zandieh M (eds) (2018) Challenges and strategies in teaching linear algebra. ICME-13 monographs. Springer International Publishing, Cham. SwitzerlandGoogle Scholar
  26. Thomas M, Stewart S (2011) Eigenvalues and eigenvectors: embodied, symbolic and formal thinking. Math Educ Res J 23:275–296CrossRefGoogle Scholar
  27. Trigueros M, Possani E (2013) Using an economics model for teaching linear algebra. Linear Algebra Appl 438:1779–1792CrossRefGoogle Scholar
  28. Wawro M (2014) Student reasoning about the invertible matrix theorem in linear algebra. ZDM Int J Math Educ 46(3):1–18CrossRefGoogle Scholar
  29. Wawro M, Rasmussen C, Zandieh M, Sweeney G, Larson C (2012) An inquiry-oriented approach to span and linear independence: the case of the magic carpet ride sequence. Primus 22(8):577–599CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departmento de Matemáticas, División Académica de Actuaría, Estadística y MatemáticasInstituto Tecnológico Autónomo de México (ITAM)México CityMéxico
  2. 2.Mathematics DepartmentVirginia TechBlacksburgUSA

Section editors and affiliations

  • Michèle Artigue
    • 1
  1. 1.Laboratoire de Didactique André Revuz (EA4434)Université Paris-DiderotParisFrance