What is Inquiry-Oriented Linear Algebra?The Inquiry-Oriented Linear Algebra (IOLA) project focuses on developing student materials composed of challenging and coherent task sequences that facilitate an inquiry-oriented approach to the teaching and learning of linear algebra. The project has also developed instructional support materials to help instructors implement the IOLA tasks in their classrooms.
What is Inquiry?We think about inquiry both in terms of what students do and what instructors do in relation to student activity. On the one hand, students learn mathematics through inquiry as they work on challenging problems that engage them in authentic mathematical practices. On the other hand, instructors engage in inquiry by listening to student ideas, responding to student thinking, and using student thinking to advance the mathematical agenda of the classroom community [1,2]. This approach to inquiry is closely aligned with the principles of inquiry-oriented instruction  and is compatible with how Inquiry-Based Learning (IBL) is characterized [4,5].
At present, three units and one bridging sequence comprise the IOLA student materials:
- Unit 1: Linear Independence and Span 
- Bridging Sequence on Systems of Equations and Row Reduction
- Unit 2: Matrices as Linear Transformations 
- Unit 3: Change of Basis, Diagonalization, and Eigentheory 
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 Rn would be explored.
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:
- 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 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.