Understandings of solutions to differential equations through contexts, Web-based simulations, and student discussion
Author: School Science and Mathematics

As in the case of elementary mathematics, the instruction of high-level mathematical concepts can often be sacrificed at the expense of a focus on algorithmic procedures. Computer-based simulations can expand an undergraduate mathematics instructor's opportunity to explore high-- level mathematical concepts in an applied environment. This study describes one instructor's approach to incorporating simulations and classroom discussions in a differential equations course and the subsequent effects on student learning attitudes and outcomes. Students made modest gains in the area of conceptualizing and applying ideas regarding solutions to differential equations in this learning environment. Implications of the study include the identification of specific gains relative to computer-mediated learning environments and recommendations for using simulations to support conceptual development.

As technology evolves, the use of specific instructional technology to support student learning is evolving, as well. However, a broad gap still exists between the rate of increase in technological power and educators' understandings of its educational uses (Moor & Zazkis, 2000). Flick and Bell (2000), in the context of science education, described how technologies might be used to facilitate and change the nature of student-instructor interactions. They have described specific guidelines for teaching with technology that focus on the need to intertwine the roles of the technology, the nature of the content being developed, and the instructional approach. These recommendations attempt to address the concerns raised by Moor and Zazkis by minimizing the distance between the nature of the technology and its use in a specific educational situation.

Educational applications of technology generally involve its use along three broad dimensions: (a) as a means of presenting and displaying information, (b) as a communication tool, and (c) as content-specific instructional technology (Slavit, 2000). In the case of the latter, mathematics educators are making use of commercial software, such as Mathematics and Derive, and graphing calculators in the instruction of elementary calculus and differential equations. However, others are utilizing more general software, such as Microsoft Office and the World Wide Web (WWW), to create mathematical environments that are technically simple to use and that can be individualized to meet directly the goals, composition, and style of a course or instructor. Software supportive ofthe WWW, such as Shockwave and Java, can further facilitate an instructor's efforts to enhance teaching and learning through the construction of simulations. Simulations allow for the enactment of classic learning theory, which suggests that students develop understanding through active participation in a meaningful context (Bruner, 1960; Dewey, 1990). Many of these simulations allow students to input conditions and manipulate an environment that can be analyzed with an accompanying mathematical representation of the situation. Kaput (1998), focusing on advanced computational mathematics, eloquently described how current technologies have facilitated real change in learning and instruction: