Although the multiscale structure of many important processes in engineering is becoming more widely acknowledged, making this connection in the classroom is a difficult task. This is due in part because the concept of multiscale structure itself is challenging and it requires the students to develop new conceptual pictures of physical systems, and in part because the mathematical machinery used to analyze multiscale systems often obscures the essential features of the development. In this paper, Taylor-Aris dispersion in a capillary tube is developed using the method of volume averaging, but with a particular effort to keep the important features of the analysis as transparent as possible so that students who are unfamiliar with the concepts of upscaling can see its essential features through a clear and explicit example.