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This study investigates students' retention of mathematical knowledge and skills in two differential equations classes. Posttests and delayed posttests after 1 year were administered to students in inquiry‐oriented and traditional classes. The results show that students in the inquiry‐oriented class retained conceptual knowledge, as seen by their performance on modeling problems, and retained equal proficiency in procedural problems, when compared with students in the traditionally taught classes. The results of this study add additional support to the claim that teaching for conceptual understanding can lead to longer retention of mathematical knowledge.

This paper provides an overview of the Inquiry-Oriented Differential Equations (IO-DE) project and reports on the main results of a study that compared students' beliefs, skills, and understandings in IO-DE classes to more conventional approaches. The IO-DE project capitalizes on advances within mathematics and mathematics education, including the instructional design theory of Realistic Mathematics Education and the social negotiation of meaning. The main results of the comparison study found no significant difference between project students and comparison students on an assessment of routine skills and a significant difference in favor of project students on an assessment of conceptual understanding. Given these encouraging results, the theoretical underpinnings of the innovative approach may be useful more broadly for undergraduate mathematics education reform.

Summary:The Interactive Multisensor Analysis Training (IMAT) project is aimed at improving the preparation of operational users of undersea-warfare sensor systems. The effort has focused on training at all levels from initial individual training ashore through team, platform, and collective training at-sea, at all skill levels from apprentice sensor operators to senior tactical commanders. Operators and tacticians at all levels need a deep and scientifically accurate, but not necessarily formal, understanding of the physical principles that underlie tactical employment of their sensors. IMAT systems use model-based scientific visualizations, including three-dimensional graphics and animations, to illustrate complex physical interactions in mission-relevant contexts, and to provide interactive virtual laboratories in which the principles can be explored. Concepts in instruction include radiated acoustic characteristics, propagation in range-dependent environments, and sensor properties. Training systems provide exploratory environments in which operators and tacticians can examine the effects of change in any of the variables involved in the end-to-end sequence of emission, transmission, reflection and detection. Sensor settings, environmental conditions, and target characteristics can all be modified through a "what-if" simulation approach. These technologies have been applied effectively in basic and advanced sensor operations/employment courses; in individual and team training simulators, and in on-board training. At the battle-group and theater level, new-technology systems are used for decision support during at-sea exercises and operations, and for post-event reconstruction and performance analysis. This paper describes the IMAT training philosophy and approach, the design of training systems, and training effectiveness.