# Fostering mathematical connections in introductory linear algebra through adapted inquiry

## Abstract

Mathematical connections are widely considered an important aspect of learning linear algebra, particularly at the introductory level. One effective strategy for teaching mathematical connections in introductory linear algebra is through inquiry-based learning (IBL). The demands of IBL instruction can make it difficult to implement such strategies in courses in which the instructor faces various constraints. The findings presented here are the product of an action research study in which IBL instructional materials were designed for a large-enrolled introductory linear algebra course with limited class time. This resulted in IBL being presented in a limited capacity alongside traditional lecture in what will be described as *adapted inquiry*. Specifically, these IBL materials were designed as vehicles through which students could form mathematical connections. This study was conducted with the goal of determining what mathematical connections students appear to be able to exhibit within the context of an adapted inquiry approach to IBL instruction.

## Keywords

Linear algebra Inquiry-based learning Mathematical connections Adapted inquiry## References

- Carlson, D. (1993). Teaching linear algebra: Must the fog always roll in?
*College Mathematics Journal, 24*(1), 29–40.CrossRefGoogle Scholar - Dancy, M., Henderson, C., & Turpen, C. (2016). How faculty learn about and implement research-based instructional strategies: The case of Peer Instruction.
*Physical Review Physics Education Research, 12*, 1–17. https://doi.org/10.1103/PhysRevPhysEducRes.12.010110.CrossRefGoogle 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). The Netherlands: Kluwer Academic Publishers.Google Scholar - Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics.
*Proceedings of the National Academy of Sciences in the United States of America, 111*(23), 8410–8415. https://doi.org/10.1073/pnas.1319030111.CrossRefGoogle Scholar - Harel, G. (1997). The Linear Algebra Curriculum Study Group Recommendations: Moving Beyond Concept Definition. In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.),
*Resources for Teaching Linear Algebra*(pp. 107–126). The Mathematical Association of America, Washington, DC.Google Scholar - Henderson, C., & Dancy, M. H. (2007). Barriers to the use of research-based instructional strategies: The influence of both individual and situational characteristics.
*Physical Review Special Topics - Physics Education Research, 3*, 1–14. https://doi.org/10.1103/PhysRevSTPER.3.020102.Google Scholar - Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2017). Inquiry-oriented instruction: A conceptualization of the instructional principles.
*PRIMUS*. https://doi.org/10.1080/10511970.2017.1338807.Google Scholar - Larson, C., Wawro, M., Zandieh, M., Rasmussen, C., Plaxco, D., & Czeranko, K. (2014). Implementing inquiry-oriented instructional materials in undergraduate mathematics. In T. Fukawa-Connelly, G. Karakok, K. Keene, & M. Zandieh (Eds.),
*Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education*(pp. 797–801). Denver.Google Scholar - Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study.
*Journal for Research in Mathematics Education, 45*(4), 406–418.CrossRefGoogle Scholar - Lay, D. C. (2011).
*Linear algebra and its applications*, 4th edn. Reading: Addison-Wesley.Google Scholar - Mertler, C. A. (2006).
*Action research: Teachers as researchers in the classroom*. Thousand Oaks: Sage Publications, Inc.Google Scholar - Plaxco, D., & Wawro, M. (2015). Analyzing student understanding in linear algebra through mathematical activity.
*Journal of Mathematical Behavior, 38*, 87–100.CrossRefGoogle Scholar - Rasmussen, C., & Kwon, O. N. (2007). An inquiry-oriented approach to undergraduate mathematics.
*Journal of Mathematical Behavior, 26*, 189–194.CrossRefGoogle Scholar - Richards, J. (1991). Mathematical discussions. In E. von Glasersfeld (Ed.),
*Radical constructivism in mathematics education*(pp. 13–51). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar - Selinski, N. E., Rasmussen, C., Wawro, M., & Zandieh, M. (2014). A method for using adjacency matrices to analyze the connections students make within and between concepts: The case of linear algebra.
*Journal for Research in Mathematics Education, 45*(5), 550–583.CrossRefGoogle Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics, 12*(2), 151–169.CrossRefGoogle Scholar - Turpen, C., Dancy, M., & Henderson, C. (2016). Perceived affordances and constraints regarding instructors’ use of Peer Instruction: Implications for promoting instructional change.
*Physical Review Physics Education Research, 12*, 1–18. https://doi.org/10.1103/PhysRevPhysEducRes.12.010116.CrossRefGoogle Scholar - Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra.
*ZDM - The International Journal on Mathematics Education, 46*(3), 389–406. https://doi.org/10.1007/s11858-014-0579-x.CrossRefGoogle Scholar - Wawro, M. (2015). Reasoning about solutions in linear algebra: The case of Abraham and the Invertible Matrix Theorem.
*International Journal of Research in Undergraduate Mathematics Education, 1*(3), 315–338.CrossRefGoogle Scholar - Wawro, M., Rasmussen, C., Zandieh, M., Sweeney, G. F., & Larson, C. (2012). An inquiry-oriented approach to span and linear independence: The case of the Magic Carpet Ride sequence.
*PRIMUS, 22*(7), 1–23. https://doi.org/10.1080/10511970.2012.667516.Google Scholar - Wawro, M., Zandieh, M., Rasmussen, C., Larson, C., Plaxco, D., & Czeranko, K. (2014). Developing inquiry oriented instructional materials for linear algebra (DIOIMLA): Overview of the research project. In T. Fukawa-Connelly, G. Karakok, K. Keene, & M. Zandieh (Eds.),
*Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education*(pp. 1124–1126). Denver.Google Scholar