*What is Inquiry-Oriented Linear Algebra?*

*What is Inquiry?*

- Unit 1: Linear Independence and Span [6]
- Bridging Sequence on Systems of Equations and Row Reduction
- Unit 2: Matrices as Linear Transformations [7]
- Unit 3: Change of Basis, Diagonalization, and Eigentheory [8]

At present, three units and one bridging sequence comprise the IOLA student materials:

All materials focus on developing deep conceptual understanding of particular mathematical concepts and how the concepts relate to each other. Each unit is composed of a sequence of four tasks. The units are independent from each other in the sense that an instructor could use one without using another; however, if an instructor chose to use all three plus the bridging material, the majority of topics that one would expect to address in an introductory level linear algebra course in R^{n}would be explored.

*Learning Goals and Rationale*: Addresses how the task contributes to meeting instructional goals and what kinds of thinking are meant to be evoked, leveraged, or challenged;*Student Thinking*: Elaborates ways in which students might think about or approach the task, answers/strategies they will likely develop, and difficulties they are likely to have; and*Implementation*: Includes suggestions for implementing the task, what kinds of discussion topics might be most productive, and what types of student ideas that instructors should anticipate.

The IOLA website aims to make research-based task sequences more accessible to instructors interested in an inquiry-oriented approach to teaching linear algebra. For each task, three main components comprise the instructor support materials:

The instructor support materials also contain a lesson overview, editable task sheets for students’ use, implementation video clips, homework suggestions for after the lesson, and a discussion board for website users to leave comments or questions for the IOLA team.

[2] Zandieh, M., Wawro, M., & Rasmussen, C. (2017). Inquiry as participating in the mathematical practice of symbolizing: An example from linear algebra.

[3] Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2017). Inquiry-oriented instruction: A conceptualization of the instructional principles.

[4] Ernst, D. C. , Hodge, A., & Yoshinobu, S. (2017). What is Inquiry-Based Learning?

[5] Rasmussen, C., Marrongelle, K., Kwon, O. N., & Hodge, A. (2017). Four goals for instructors using Inquiry-Based Learning.

[6] 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.

[7] Andrews-Larson, C., Wawro, M., & Zandieh, M. (2017).

[8] Zandieh, M., Wawro, M., & Rasmussen, C. (2017). An example of inquiry in linear algebra: The roles of symbolizing and brokering.

[9] Freudenthal, H. (1991).

[10] Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. E. Kelly & R. A. Lesh (Eds.),

[11] 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.),

[12] Gravemeijer, K. (1994). Educational development and developmental research.

This material is based upon work supported by the National Science Foundation under grant numbers DUE-1245673/1245796/1246083. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.