One of the things which became clear was that the mathematical expectations some of the instructors had of the students were unrealistic. Although the students had had, in general, some exposure to calculus, it seemed that this was just enough to render them immune to the concepts of derivative, etc., rather than enough to do them useful service when push came to shove.
The same problem arose when we studied spatial statistics. While we were inclined to present the mathematical definitions of the various spatial statistics, it seemed afterwards that the majority of the students would have been happier simply to learn the cases in which each statistic is useful, how to interpret the statistic, and how to use the software to calculate it. At the end of the course one of the epidemiologists among the instructors asked where, in the course of their educations, would the epidemiology students have encountered the idea of a matrix? This astounded the mathematician and spatial statistician in the group: where indeed? We had just finished teaching a course in spatial statistics, had referred to matrices right and left, relied on them, and yet no student had ever so much as let out a whimper: could it possibly be the case that these students hid this lack of understanding all term long?
If, in fact, all the students are aware of what a matrix is, then all is well in the world; if not, then the conclusions are more sinister: how could the students let that concept go by without asking for the definition? How could the training programs turning out these students let them go by without teaching them this fundamental concept? Spreadsheets are built on matrices, and yet the concept of a matrix might never be introduced to a student? The idea is absurd, and yet....
Ironically, however, many of the students liked the unit we did on compartmental models, which, by rights, should entail some understanding of differential equations, but which was taught with only mild incorporation of the mathematical ideas and focused instead on the idea of ``flows'' in the model building stage. The software then encouraged the students to think graphically, rather than analytically. To the mathematician, the dangers of such an approach are obvious (since the models may have nothing whatsoever to do with the concepts the students think they're addressing, and since the problem of solving differential equations numerically are well known (e.g. stiff differential equations). Students could easily construct systems which are stiff, or even chaotic, and never have the wherewithal to understand the behavior of these systems.