Skip to main content

Advertisement

SpringerLink
Log in

Learning the concept of eigenvalues and eigenvectors: a comparative analysis of achieved concept construction in linear algebra using APOS theory among students from different educational backgrounds

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

Abstract

International comparative studies of education have shown that the increase in mathematical attainment differs significantly among students in relation to the different school types in which their mathematical foundations were acquired. In tertiary education, the same differences are observed with respect to simple tasks that relate to the subject matter of secondary education. Therefore, we investigate the question of whether this observation holds equally for concept construction in the field of eigen theory, by applying Dubinsky’s action, process, object, schema (APOS) theory, and Tall’s worlds of mathematics. We focus on eigen theory because we regard it as particularly representative of linear algebra in terms of its expressibility in the embodied and symbolic worlds. In our empirical study, we investigated concept constructions as correlates of educational trajectories of 36 students who covered the entire course program of a first semester mathematics course for first-year engineering students. Although significant differences in procedural knowledge were observed, results indicate that the type of secondary school does not affect concept construction among students who complete the entire course program. In our final discussion, we present implications for teaching and learning of linear algebra in heterogeneous classes and the integration of deeper learning methods into the course design.

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

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. System for Teaching and Assessment using a Computer algebra Kernel

  2. https://educational-media.de/demos/darstellungsvernetzung-in-echtzeit/

  3. HTML5–Package, see https://h5p.org

  4. Example: https://educational-media.de/demos/learning-analytics/

  5. Example: https://educational-media.de/pbl/pbl-in-der-ingenieurmathematik/

References

  • Altieri, M., Paluch, D., Staupe, M., & Schirmer, E. (2019a). STACK trifft H5P—interaktives, audiovisuelles Feedback in STACK-Aufgaben basierend auf H5P. https://doi.org/10.5281/zenodo.2564881.

  • Altieri, M., Schellenbach, M., Schirmer, E., Opfermann, C., Kunze, J. E., Regnet, J., & Paluch, D. (2019b). Unreal Engine 4 trifft H5P und PBL—Integration einer virtuellen Realität mit interaktiven Erklärvideos in ein digitales Fachkonzept zur Unterstützung problembasierten Lernens. In M. Klinger, A. Schüler-Meyer & L. Wessel (Eds.), Hanse-Kolloquium zur Hochschuldidaktik Mathematik 2018: Beiträge zum gleichnamigen Symposium am 9. und 10. November 2018 an der Universität Duisburg-Essen. Münster: WTM-Verlag.

  • Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., et al. (2014). APOS theory: A framework for research and curriculum development in mathematics education. New York: Springer.

    Book  Google Scholar 

  • Barrows, H. (2002). Is it truly possible to have such a thing as dPBL? Distance Education,23(1), 119–122. https://doi.org/10.1080/01587910220124026.

    Article  Google Scholar 

  • Baumert, J., Trautwein, U., & Artelt, C. (2003). Schulumwelten—Institutionelle Bedingungen des Lehrens und Lernens. In Baumert et al. (Eds.), PISA 2000—Ein differenzierter Blick auf die Länder der Bundesrepublik Deutschland (pp. 261–331). Opladen: Leske + Budrich.

  • Becker, M., Lüdtke, O., Trautwein, U., & Baumert, J. (2006). Leistungszuwachs in Mathematik: Evidenz für einen Schereneffekt im mehrgliedrigen Schulsystem? Zeitschrift für pädagogische Psychologie,20(4), 233–242. https://doi.org/10.1024/1010-0652.20.4.233.

    Article  Google Scholar 

  • Beltrán-Meneu, M. J., Murillo-Arcila, M., & Albarracín, L. (2017). Emphasizing visualization and physical applications in the study of eigenvectors and eigenvalues. Teaching Mathematics and its Applications: An International Journal of the IMA,36(3), 123–135. https://doi.org/10.1093/teamat/hrw018.

    Article  Google Scholar 

  • Berman, A., & Shvartsman, L. (2016). Definitions are important: The case of linear algebra. European Journal of Science and Mathematics Education,4(1), 26–32.

    Google Scholar 

  • Bouhjar, K., Andrews-Larson, C., Haider, M., & Zandieh, M. (2018). Examining students’ procedural and conceptual understanding of eigenvectors and eigenvalues in the context of inquiry-oriented instruction. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 193–216). Cham: Springer.

    Chapter  Google Scholar 

  • Caglayan, G. (2015). Making sense of eigenvalue-eigenvector relationships: Math majors’ linear algebra—Geometry connections in a dynamic environment. The Journal of Mathematical Behavior,40, 131–153. https://doi.org/10.1016/j.jmathb.2015.08.003.

    Article  Google Scholar 

  • Carlson, D., Johnson, C. R., Lay, D. C., & Porter, A. D. (1993). The Linear Algebra Curriculum Study Group recommendations for the first course in linear algebra. The College Mathematics Journal,24(1), 41–46.

    Article  Google Scholar 

  • Destatis. (2017). Schnellmeldungsergebnisse der Hochschulstatistik zu Studierenden und Studienanfänger/-innen. https://www.destatis.de/DE/Publikationen/Thematisch/BildungForschungKultur/Hochschulen/SchnellmeldungWSvorlaeufig5213103188004.pdf?__blob=publicationFile. Accessed 3 Feb 2018.

  • Dochy, F., Segers, M., Van den Bossche, P., & Gijbels, D. (2003). Effects of problem-based learning: A meta-analysis. Learning and instruction,13(5), 533–568.

    Article  Google Scholar 

  • Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, M. Artigue, U. Kirchgräber, J. Hillel, M. Niss, & A. Schoenfeld (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273–280). Dordrecht: Kluwer.

    Google Scholar 

  • Gijbels, D., Dochy, F., Van den Bossche, P., & Segers, M. (2005). Effects of problem-based learning: A meta-analysis from the angle of assessment. Review of Educational Research,75(1), 27–61.

    Article  Google Scholar 

  • Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning,1(2), 155–177.

    Article  Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. (2016). Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking. Teaching Mathematics and its Applications: An International Journal of the IMA,35(4), 216–235.

    Google Scholar 

  • Larson, C., Rasmussen, C., Zandieh, M., Smith, M., & Nelipovich, J. (2007). Modeling perspectives in linear algebra: A look at eigen-thinking. http://www.rume.org/crume2007/papers/larson-rasmussen-zandieh-smith-nelipovich.pdf. Accessed 22 Mar 2018.

  • Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspective on mathematics teaching, learning, and problem solving (pp. 3–33). Mahwah, NJ: Lawrence Erlbaum Associates.

    Chapter  Google Scholar 

  • Lingel, K. (2013). Metakognitives Wissen Mathematik—Entwicklung und Zusammenhang mit der Mathematikleistung in der Sekundarstufe I. Dissertation, Universität Würzburg.

  • Lingel, K., Neuenhaus, N., Artelt, C., & Schneider, W. (2014). Der Einfluss des metakognitiven Wissens auf die Entwicklung der Mathematikleistung am Beginn der Sekundarstufe I. Journal für Mathematik-Didaktik,35(1), 49–77. https://doi.org/10.1007/s13138-013-0061-2.

    Article  Google Scholar 

  • Meneu, M. J. B., Arcila, M. M., & Mora, E. J. (2017). A teaching proposal for the study of eigenvectors and eigenvalues. Journal of Technology and Science Education,7(1), 100–113. https://doi.org/10.3926/jotse.260.

    Article  Google Scholar 

  • NMC. (2017). NMC Horizon Report > 2017 Higher Education Edition. http://cdn.nmc.org/media/2017-nmc-horizon-report-he-EN.pdf. Accessed 11 Feb 2018.

  • Plaxco, D., Zandieh, M., & Wawro, M. (2018). Stretch directions and stretch factors: A sequence intended to support guided reinvention of eigenvector and eigenvalue. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 175–192). Cham: Springer.

    Chapter  Google Scholar 

  • Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS theory. The Journal of Mathematical Behavior,39, 100–120. https://doi.org/10.1016/j.jmathb.2015.06.005.

    Article  Google Scholar 

  • Sangwin, C. (2013). Computer aided assessment of mathematics. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Schneider, W., Lingel, K., Artelt, C., & Neuenhaus, N. (2017). Metacognitive knowledge in secondary school students: Assessment, structure, and developmental change. In D. Leutner, J. Fleischer, J. Grünkorn, & E. Klieme (Eds.), Competence assessment in education (pp. 285–302). Cham: Springer.

    Chapter  Google Scholar 

  • Stewart, S. (2018). Moving between the embodied, symbolic and formal worlds of mathematical thinking with specific linear algebra tasks. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 51–67). Cham: Springer.

    Chapter  Google Scholar 

  • Stewart, S., Andrews-Larson, C., Berman, A., & Zandieh, M. (Eds.). (2018). Challenges and strategies in teaching linear algebra. Cham: Springer.

    Google Scholar 

  • Stewart, S., & Thomas, M. O. J. (2006a). Process-object difficulties in linear algebra: Eigenvalues and eigenvectors. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 185–192). Prague: PME.

    Google Scholar 

  • Stewart, S., & Thomas, M. O. J. (2006b). Student thinking about eigenvalues and eigenvectors: Formal, symbolic and embodied notions. In P. Grootenboer, R. Zevenbergen & M. Chinnappan (Eds.), Identities, cultures and learning spaces (Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 487–495)). Canberra: MERGA.

  • Stewart, S., & Thomas, M. O. J. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology,41(2), 173–188.

    Article  Google Scholar 

  • Tall, D. (2004). Building theories: The three worlds of mathematics. For the Learning of Mathematics,24(1), 29–32.

    Google Scholar 

  • Tall, D. (2010). Perceptions, operations and proof in undergraduate mathematics. CULMS Newsletter,2, 21–28.

    Google Scholar 

  • Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking. Mathematics Education Research Journal,23, 275–296. https://doi.org/10.1007/s13394-011-0016-1.

    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). Cham: Springer.

    Chapter  Google Scholar 

  • Van Ophuysen, S., & Wendt, H. (2010). Zur Veränderung der Mathematikleistung von Klasse 4 bis 6. In J. Baumert, K. Maaz, & U. Trautwein (Eds.), Bildungsentscheidungen (pp. 302–327). Wiesbaden: VS Verlag für Sozialwissenschaften.

    Chapter  Google Scholar 

  • Walker, A., & Leary, H. (2009). A problem based learning meta analysis: Differences across problem types, implementation types, disciplines, and assessment levels. Interdisciplinary Journal of Problem-based Learning,3(1), 12–43.

    Article  Google Scholar 

  • Warburton, K. (2003). Deep learning and education for sustainability. International Journal of Sustainability in Higher Education,4(1), 44–56.

    Article  Google Scholar 

  • Wawro, M., Rasmussen, C., Zandieh, M., & Larson, C. (2013). Design research within undergraduate mathematics education: An example from introductory linear algebra. In T. Plomp & N. Nieveen (Eds.), Educational Design Research—Part B: Illustrative Cases (pp. 905–925). Enschede: SLO.

    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. https://doi.org/10.1080/10511970.2016.1199618.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Natural Sciences, Ruhr West University of Applied Sciences, Duisburger Straße 100, 45479, Mülheim an der Ruhr, Germany

    Mike Altieri & Evelyn Schirmer

Authors
  1. Mike Altieri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evelyn Schirmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mike Altieri.

Additional information

Publisher's Note

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

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (PDF 511 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Altieri, M., Schirmer, E. Learning the concept of eigenvalues and eigenvectors: a comparative analysis of achieved concept construction in linear algebra using APOS theory among students from different educational backgrounds. ZDM Mathematics Education 51, 1125–1140 (2019). https://doi.org/10.1007/s11858-019-01074-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-019-01074-4

Keywords

Search

Navigation