Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram

Abstract

This article provides an overview of a modeling sequence that culminates in student reinvention of a bifurcation diagram. The sequence is the result of years of classroom-based research and curriculum development grounded in the instructional design theory of Realistic Mathematics Education. The sequence of modeling tasks and examples of student work tendered offers a fresh perspective on modeling and how an inquiry-oriented sequence of tasks can lead to the reinvention of significant mathematics.

Keywords:

• Inquiry
• bifurcation
• reinvention
• realistic mathematics education
• emergent models

Notes

1 Details on these research-based curriculum development efforts, including empirical analyses and comparison studies of student learning we refer readers to the 2007 special issue of the Journal of Mathematical Behavior (see [6]). The curriculum itself is available at https://iode.wordpress.ncsu.edu/. This paper specifically references units 6, 7, and 8. Other RME-inspired curricula in linear algebra and abstract algebra are available at http://times.math.vt.edu.

2 Contact Justin Dunmyre ([email protected]) with questions about the applets or for more information on using the GeoGebra applets.

Additional information

Funding

Support for this work was funded by the National Science Foundation under grant No. 1431641. The opinions expressed do not necessarily reflect the views of the Foundation.

Notes on contributors

Chris Rasmussen

Chris Rasmussen is a professor of mathematics education in the Department of Mathematics and Statistics at San Diego State University. He received an undergraduate degree in mechanical engineering, a master’s degree in mathematics, and his Ph.D. in mathematics education at the University of Maryland. His research focuses on the learning and teaching of undergraduate mathematics. He is currently studying departmental change and factors that influence student success over the entire progression of the introductory mathematics courses.

Justin Dunmyre

Justin Dunmyre earned his Ph.D. in mathematics at the University of Pittsburgh where he focused his research in mathematical neuroscience. His first encounter with active learning was as a visiting assistant professor at the University of Michigan. He later became a Project NExT fellow (Brown '13), and now focuses most of his academic energy on teaching, especially in inquiry-oriented ways. He is currently an assistant professor in Frostburg State University's department of mathematics.

Nicholas Fortune

Nicholas Fortune earned his Ph.D. in mathematics education at North Carolina State University. He received his bachelors and masters in applied mathematics from Rensselaer Polytechnic Institute. His research centers on instructional change in undergraduate mathematics and how the mathematics and mathematics education communities can collaborate to support instructional change of undergraduate mathematics.

Karen Keene

Karen Keene is an associate professor of mathematics education at North Carolina State University. She conducts research in undergraduate mathematics education, primarily concerning differential equations teaching and learning. Additionally, she researches the social construction of mathematical meaning in undergraduate classrooms. Her second area of research is lies within secondary teacher education focusing on teachers’ content knowledge and how it connects to their teaching and curriculum development. She is currently serving as a program officer at the National Science Foundation.