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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?

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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    $\begingroup$ First posted at academia.stackexchange.com/q/62266/12339, where I suggested that it might be worth asking here. That said, also noted that Stack Exchange generally discourages cross-posting. $\endgroup$ – J W Jan 26 '16 at 16:19
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    $\begingroup$ No research paper I know of, but I think it's obviously beneficial if the application answers a natural enquiry or has a natural objective and depends on results from multiple areas simultaneously. This makes it clear how useful the individual topics are, rather than being ad-hoc or isolated curiosities. However, you must make sure that the application is not itself some ad-hoc problem that is contrived to require using all the topics learnt, otherwise the only benefit would be getting students to combine the techniques and knowledge. It is still good but arguably just an isolated curiosity. $\endgroup$ – user21820 Feb 3 '16 at 9:56
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    $\begingroup$ Applications are so important that -- in the absence of more information -- the opposite approach could be just as good. You can use class time for the application, and let students learn or at least memorize the theory on their own by having a final which asks them to reproduce key proofs. If that seems unsatisfying, let us know what the class is, who the students are, and what options you see; then we can discuss this more seriously. $\endgroup$ – user173 Feb 3 '16 at 10:11
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    $\begingroup$ "However, I have also designed a long final assignment that introduces a problem covering roughly all of the topics in the course." This is quite different from what I thought of based on the word "applications" in the title. It seems like you're actually asking about the educational effectiveness of a summative assessment that requires the student to synthesize knowledge from throughout the course. I've done some searching just now and cannot find anything concrete, but to help you (and others) narrow down searches, avoid "applications" in the search terms and focus on these other terms. $\endgroup$ – Brendan W. Sullivan Dec 29 '17 at 4:23
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    $\begingroup$ Although no research citations, Motivation vs. Rigor asks a similar question. $\endgroup$ – Joseph O'Rourke Aug 12 at 1:10
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Be it undergraduate or professor, you need hands-on examples to "see" why some concept or technique is worthwhile. Sure, you can take it that more advanced people are better able to come up with their own examples and applications, or have a richer experience to which new material relates. So examples, applications, cross-connections are certainly needed for undergraduates. If you don't provide them, your audience will get lost. If you don't intersperse motivation for some more exotic proof technique, or don't lay out a overall plan for a proof, they will get bored.

In a Discrete Math class I've taught, I found out the hard way that it is important to give a general structure and format for e.g. proofs by induction, and stick to it. State each time "We use proof by contradiction/counterpositive/if and only if by proving implication in both directions/...". Students get lost otherwise. Same for proofs that some problem is NP-complete: need to prove in NP and NP-hard. Do it every time, even if it's obvious/one line. Write it down.

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As someone who probably took undergraduate courses more recently than most, I can attest to the fact that applications are necessary. If not given, students very often get lost - not understanding the big picture. They may be able to apply specific operations to specific problems, but it is the applications that allows them to see the forest instead of just the trees. I don't have a research paper to quote, but I talk from my own experience as a student - which I believe is your ideal case study.

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