Karen Allen Keene

Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles

Abstract Research has highlighted that inquiry-based learning (IBL) instruction leads to many positive student outcomes in undergraduate mathematics. Although this research points to the value of IBL instruction, the practices of IBL instructors are not well-understood. Here, we offer a characterization of a particular form of IBL instruction: inquiry-oriented instruction. This characterization draws on K-16 research literature in order to explicate the instructional principles central to inquiry-oriented instruction. As a result, this conceptualization of inquiry-oriented instruction makes connections across research communities and provides a characterization that is not limited to undergraduate, secondary, or elementary mathematics education.

Mathematics INstruction using Decision Science and Engineering Tools

MINDSET (Mathematics INstruction using Decision Science and Engineering Tools) is a multiyear project involving collaboration between engineers, educators, and mathematicians at three universities in two states. Through a partnership with secondary teachers, the MINDSET project staff has created a ground-breaking new mathematics curriculum for fourth-year high school students using mathematical-based decision-making tools from the fields of operations research and industrial engineering. High school students are presented a series of real-world problem contexts with the purpose of making the underlying mathematics more relevant to them and allowing them to model the situations and actively do mathematics. This innovative new integration of mathematics and engineering at the high school level provides new opportunities for students, who have not been successful in the abstract world of mathematics, to learn to enjoy and to want to pursue a more applied form of mathematics. This paper reports the results of the nationwide implementation of this engineering-based mathematics course.

Identifying and assessing relational understanding in ordinary differential equations

Is it possible to assess conceptual understanding of ordinary differential equations and their solutions? There is significant tension between students learning mathematics for understanding and students learning to drill a set of algorithms to solve standard ODE exercises. This paper presents the Framework for Relational Understanding of Procedures, a categorization of assessable conceptual knowledge. Example conceptual knowledge we examine in this project include relating a graphical representation of an ODE to a symbolic one, checking a result, and knowing why a particular solution method might be applicable. We applied this framework to three techniques taught in Ordinary Differential Equations: separation of variables, solving a first order linear ordinary differential equation, and Eulers method and developed a set of assessment items. These assessment questions were then transformed into knowledge pieces and moved into an online platform and elaborated. Thus, the assessment can be administered either in traditional pencil-and-paper form or through a learn-as-you-assess online web site.

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.