We study differential equations that model contaminant flow in a semi-infinite, fractured, porous medium consisting of a single fracture channel bounded by a porous matrix. Models in the literature usually do not incorporate diffusion in the porous matrix in the direction parallel to the fracture, and therefore they must omit a no-flux boundary condition at the edge, which, in some problems, may be unphysical. Herein we show that the problem usually treated in the literature is the outer problem for a correctly posed singular perturbation problem which includes diffusion in both directions as well as the no-flux boundary condition.
This chapter reviews teachers' perceptions of the collaborative learning experiences when enrolled in an online course to determine strategies for engaging teachers in active learning and meaningful collaboration in an online learning environment. A survey was designed to solicit feedback from mathematics teachers of Grades 6-12 who have completed online mathematics content courses at the University of Nebraska – Lincoln (UNL) for professional development or for graduate credit. The survey specifically addresses the teachers' perceptions of the collaborative learning experiences during their online course. Combined with feedback from numerous course evaluations and the experiences of several online mathematics instructors from UNL's Department of Mathematics, results of the survey are utilized to determine strategies for engaging teachers in active learning and meaningful collaboration in an online learning environment.
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