iola.math.vt.eduInquiry Oriented Linear Algebra - IOLA

Team Resources Typical Day VideosNSF Publications Related Projects Sign in Request Access Welcome to IOLA! 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 [3] and is compatible with how Inquiry-Based Learning ( IBL ) is characterized [4,5]. The IOLA Student Materials At present, three units and one bridging sequence comprise the IOLA student materials: 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 [2] 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. The IOLA Instructor Support Materials 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. Classroom Motivation Up View details » Donec id elit non mi porta gravida at eget metus. Fusce dapibus, tellus ac cursus commodo, tortor mauris condimentum nibh, ut fermentum massa justo sit amet risus. Etiam porta sem malesuada magna mollis euismod. Donec sed odio dui.A Typical Day We realize that this approach to instruction is new for many people, so we encourage users to check out our Typical Day page. This resource highlights various classroom interactions (small group work, whole class discussion, partner talk, and telling) that together help foster a productive inquiry-oriented class environment. Our approach to whole class discussion relies on both the students and the instructor playing active roles. The Typical Day page provides five goals for productive whole class discussions and detailed suggestions of how to foster it. Realistic Mathematics Education The IOLA materials are guided by the instructional design theory of Realistic Mathematics Education (RME) [8]. A central tenet of RME is that mathematics is first and foremost a human activity. IOLA is guided by the two RME heuristics of guided reinvention and emergent models. The notion of guided reinvention emphasizes the active role an instructor plays in utilizing student ideas and justifications to move forward the mathematical development of the class. The notion of emergent models emphasizes that classroom endeavors should support students in developing models of their mathematical activity that can in turn be used as models for subsequent mathematical activity. In keeping with these heuristics, we have developed task sequences that are based on realistic starting points and are designed to support students in making progress toward a set of mathematical learning goals. In IOLA, students’ activity evolves toward the reinvention of formal notions and ways of reasoning about the mathematics initially investigated. Research-Based Curriculum These units are a product of our research over the last decade in the teaching and learning of linear algebra, which has been grounded in the design-based research paradigm of classroom-based teaching experiments [9,10]. This involves a cyclical process of (a) investigating student reasoning about mathematical concepts and (b) designing and refining tasks that leverage students’ mathematical ideas towards accomplishing the desired learning goals [11]. See the Publications page for some of our findings regarding how students think about concepts in linear algebra. Citations [1] Rasmussen, C., & Kwon, O. (2007). An inquiry oriented approach to undergraduate mathematics. Journal of Mathematical Behavior, 26 , 189-194. [2] 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. [3] Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2017). Inquiry-oriented instruction: A conceptualization of the instructional principles. PRIMUS. DOI 10.1080/10511970.2017.1338807 [4] Ernst, D. C. , Hodge, A., & Yoshinobu, S. (2017). What is Inquiry-Based Learning? AMS Notices, 64 (6), 570-574. [5] Rasmussen, C., Marrongelle, K., Kwon, O. N., & Hodge, A. (2017). Four goals for instructors using Inquiry-Based Learning. Manuscript submitted for publication. [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. PRIMUS , 22 (8), 577-599. [7] Andrews-Larson, C., Wawro, M., & Zandieh, M. (2017). A hypothetical learning trajectory for conceptualizing matrices as linear transformations. International Journal of Mathematical Education in Science and Technology, 48 (6), 809-829. [8] Freudenthal, H. (1991). Revisiting mathematics education . Dordrecht, The Netherlands: Kluwer Academic Publishers. [9] Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307-330). Mahwah, NJ: Lawrence Erlbaum Associates. [10] 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, the Netherlands: SLO. [11] Gravemeijer, K. (1994). Educational development and developmental research. Journal for Research in Mathematics Education, 25 (5), 443-471. How to Cite the IOLA Materials To cite the IOLA materials, please use the following citation: Wawro, M., Zandieh, M., Rasmussen, C., & Andrews-Larson, C. (2013). Inquiry oriented linear algebra: Course materials. Available at http://iola.math.vt.edu. This...