Design research on inquiry-based multivariable calculus: focusing on students’ argumentation and instructional design

Abstract

from traditional teacher-centered and textbook-dominated approaches to new instructional approaches that are student-centered and inquiry-based (Holton 2001). As part of this effort, inquiry-based undergraduate mathematics courses have been examined by some researchers (Kwon et al. 2005;Smith 2005;Rasmussen et al. 2006;Dhaler 2008).Multivariable calculus is a mandatory mathematics subject in most Korean universities, since it prepares students to study at an advanced level in diverse majors such as natural sciences, engineering, and economics. As Rasmussen et al. (2014) point out in the previous issue of ZDM, however, there is a shortage of studies that go beyond basic topics of calculus into areas such as multivariable calculus and differential equations. Also, there is a lack of instructional tasks developed for inquiry based learning (IBL) and a lack of research dealing with classroom interactions and the instructor's role in multivariable calculus teaching/learning. Accordingly, if an instructor wants to apply an inquirybased approach to a multivariable calculus course, he or she might face the difficulty of preparing suitable tasks to cultivate students' knowledge and find that students need to spend a long time at inquiry.In order to overcome this difficulty and to develop iteratively an inquiry-based multivariable calculus course, this study attempts to transform the curriculum and instruction by introducing an innovative instructional model to cover the entire curriculum with inquiry-based mathematical activities. In the fall semester of 2013, a multivariable calculus course for first year students majoring in mathematics education was organized as a flipped classroom at a university in Seoul, Korea. The flipped classroom enabled instructors' explanatory lectures of great amounts of content to be replaced by online video clips in order to assign more time to student inquiry during the face-to-face offline Abstract In this study, researchers design and implement an inquiry based multivariable calculus course in a university which aims at enhancing students' argumentation in rich mathematical discussions. This research aims to understand the characteristics of students' argumentation in activities involving proof constructions through mathematical discussions, and to derive instructional design principles in relation to three sites of intervention: instructional design, classroom interaction, and the instructor's role. Over the course of 14 weeks, 18 freshmen mathematics education majors participated in this study. Multiple sources of data were collected, students' reasoning in the classroom discussions were analyzed within the Toulmin's argumentation structure, and the instructional interventions were gradually revised according to the iterative cyclic process of the design research. The students' argumentation structures presented in the classroom gradually developed into more complicated forms as the study progressed, and the researchers conclude that the interventions were effective at improving student...

Design research with a focus on learning processes: an overview on achievements and challenges

“…Gresalfi 2015;Lobato et al 2015;Prediger and Krägeloh 2015;Stephan 2015) to university (e.g. Kwon et al 2015;Rasmussen 2001).…”

“…But what is also noticeable is that nearly all of the papers combine a primary design challenge with a secondary one. Besides restructuring a mathematical topic, a second challenge involves more general instructional issues like inquiry-based learning (Kwon et al 2015), strategic scaffolding (Prediger and Krägeloh 2015), engagement (Gresalfi 2015), model-based reasoning (de Beer et al 2015), and the role of diagnostic assessment (Confrey and Maloney 2015).…”

“…Even projects with a strong focus on curriculum innovation like Kwon et al (2015) sometimes adopt a process focus: In their design research project on inquiry-based learning for a multivariable calculus course, the quality criteria for which the inquiry-based learning is optimized required the process focus: When quality of innovation is related to the deepening the structures of students' argumentation processes, it is necessary to develop analytic tools for investigate deeply the students' processes (Kwon et al 2015).…”

“…One example of such a domain-specific instruction theory is Brousseau's (1997) Theory of Didactical Situations which is instrumental in the French Didactical Engineering as descibed by Margolinas and Drijvers (2015). Another example is the theory of realistic mathematics education, RME, which is used in various contributions to this special issue (de Beer et al 2015;Kwon et al 2015;Margolinas and Drijvers 2015;Stephan 2015). RME is typical in that it was defined as "a domain-specific instruction theory" (Treffers 1987) from the outset.…”

A computational lens on design research

“…This Special issue includes some notable contributions in the area of computer-supported mathematics education, with the technology used in a variety of ways: Gresalfi (2015) uses an online, interactive videogame; Kwon et al (2015) use software that allows students to observe and manipulate graphs to explore partial differentiation; de Beer et al use a computer simulation of filling glassware; Confrey et al use a tablet-based diagnostic assessment system to generate feedback to the students and the teacher, and to exploit a chat-function to support collaborative work.…”