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I am currently designing a proof-based Math course for my University. I already designed and ordered all of the theoretical content in the course and included some ad hoc exercises for practicing each of the particular topics in the course. However, I have also designed a long final assignment that introduces a problem covering roughly all of the topics in the course.

My question is on how helpful is it for the student to work on these kind of general assignments. Assuming he already understood each of the topics individually, will it be beneficial spending some weeks analyzing an application involving almost all of the topics seen? Is there any research on the educational benefits of these type of exercises?

EDIT: I would be grateful if someone could cite a research paper that talks about the benefits of showing these kind of general applications involving many topics in undergraduate courses.

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    It's almost self-evident that this is beneficial, when there is an application that ties the course concepts together and gives them context, and that fits into a reasonable time frame. The trouble is that such an application isn't always forthcoming. For example, if I'm teaching introductory analysis, it's hard to motivate the material on metric spaces without saying "trust me, you'll need this next semester." – user37208 Jan 26 '16 at 0:52
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    Although cross-posting is discouraged on Stack Exchange sites, this question might have been worth asking on Mathematics Educators Stack Exchange, especially given your specific situation. – J W Jan 26 '16 at 8:55
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I can't point to any specific research, but it is a common observation that (i) students have difficulty putting individual topics learned in class into a context that would allow them to see where they are used, (ii) students who don't understand why a topic is relevant tend to forget it rather quickly. This is as true in mathematics as in many other areas, and I think it is widely thought that this limits the ability of students to apply what they learned to new situations.

Consequently, at least to me, it is of great importance to provide students with the context in which each topic I cover in my classes lives. Why do we care about a particular statement? What implications does a particular theorem have on areas they are already familiar? I know that this approach is not universally shared in mathematics (or at least others may define the context and the applications to be rather narrowly as "related areas of mathematics"), but at least in my view, it is important to show students what the material they learn can be used for. Otherwise, mathematics is no more than an idle game of moving symbols from one side of an equation to another.

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    I'm pretty sure there's been a bunch of math ed research on (i). I think the name it goes by is something like "transfer of knowledge." – Kimball Jan 26 '16 at 3:48
  • I'm certain there is research on this. I'm just not familiar with this area, sorry :-( – Wolfgang Bangerth Jan 26 '16 at 18:51

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