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The following is an expectation for our "course-based research initiative". I'll include the complete wording so you can best understand my question.

Designing a Research Proposal/Project

  1. Embark on a research proposal/project by being able to:

    -select an area of interest

    -develop a topic

    -identify a problem

    -and/or develop a research question

  2. Clarify by conducting a literature or artifact review or developing a main argument

  3. Find needed information/data by:

    -collecting relevant sources/evidence (including, but not limited to opposing or supporting arguments, frameworks, methodologies, and/or theories)

  4. Evaluate the information by identifying:

    -the credibility of sources

    -feedback from peers and professionals in the field to improve the proposal/project

  5. Organize the presented content and/or project timeline in a manageable, usable, and discipline appropriate format that can result in implementation of the research proposal or project.

  6. Critically analyze the strengths and weakness of the information collected, including, but not limited to:

    -supporting and opposing arguments

    -methodologies

    -the proposed process

  7. Synthesize by constructing or summarizing all information/data into a research proposal/project that includes rigorous, researchable questions based on new understandings.

My school apparently wants all of this integrated into an existing course in the Math major. My question is this:

Do you know any school where all undergraduate Math majors in a regular course are expected to do research in the sense outlined above ?

What is the best way to explain the distinction between Math and other disciplines where the above "research" might be plausible. Am I correct to say this is not a reasonable request if we accept the term research has its usual meaning in Mathematics ?

I appreciate an advice and/or experience you might have to share.

Extra Note: this is subtle, but this question is less specific than my own. My question concerns modifying an existing course, not creating a new one with extra resources and/or a special subcategory of students.

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    $\begingroup$ IMHO research projects in undergrad pure math should be a separate, truly optional course. The student should already have fairly well developed idea for a project BEFORE being accepted into such a course. $\endgroup$
    – Dan Christensen
    Commented May 31, 2018 at 15:21
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    $\begingroup$ Question: Are other departments at your school receiving the same required wording? It's possible that this works better for those other departments. $\endgroup$
    – Chris Cunningham
    Commented Jun 1, 2018 at 4:47
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    $\begingroup$ @ChrisCunningham this is university wide. It is possible it works better outside math. However, even there I'm not sure it's really fair to insist this be shoe-horned into an existing course with no compensation or real load reduction for what is obviously a rather involved project if we adhere to the letter of the law here. $\endgroup$
    – James S. Cook
    Commented Jun 1, 2018 at 14:48
  • $\begingroup$ Sounds like an administrator screwing stuff up because she has no idea what happens in the rubber meets the road trenches of teaching. Looks good on paper. Like NCLB or Common Core or any of the random "pass a decree to require things be better" plans of action that show no evidence of tradeoffs or cost/benefit analysis. $\endgroup$
    – guest
    Commented Jun 1, 2018 at 14:56
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    $\begingroup$ @BenjaminDickman I'm interested in both the existence of other schools doing and the opinion of people if it should or could be done in the fashion which is implied from the university verbiage. Honestly, I'm almost certain it's only happening at my university because from what I've read course-based research is not the preferred method. Indeed, personally, I believe in the apprentice model which involved a more substantive investment in just one or two people at a time. Quality over quantity. $\endgroup$
    – James S. Cook
    Commented Jun 2, 2018 at 0:01

3 Answers 3

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I think the term research causes a lot of confusion here. When mathematicians think about research, we immediately think about proving new theorems.

But that doesn't seem to be what's asked here. As I read the outline, the end product is supposed to be a research proposal, not research itself. That is, students are asked to propose a suitable question for research, including investigating prior work to determine whether the question is an appropriate and interesting one.

I don't see anything unreasonable about including that that in a suitable upper level undergraduate math course. Taken literally, the outline seems to call for students to skim some recent papers on a related subject in enough detail to identify what is and isn't known and make a conjecture supported by results in the literature. That doesn't require students to prove anything new, or even understand the proofs in those papers in any detail.

I'm not aware of a course with an assignment framed that way, but the main steps are similar to assignments I've seen, and given. I've had my discrete math class do a writing project where they had to mostly-independently read a few outside sources and synthesize enough information to present the material to the class. That's a similar level of "research" to what this outline asks for - the point is to start learning to parse information and context from the academic literature (as opposed to things pre-digested in textbooks).

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    $\begingroup$ But, I think it is really hard to know what is and isn't known in Math. Am I wrong? I mean, I can see some sort of writing assignment on say the Riemann Hypothesis. But, an assignment for them to decide what are open questions in commutative algebra? Or what's the state of the art in whatever mathematical. I really am lost how to assess or guide students in such an endeavor. $\endgroup$
    – James S. Cook
    Commented May 31, 2018 at 20:28
  • $\begingroup$ @TheChef: I'd think the bigger problem is that it's too easy to just pick an open question from some recent paper and check its citations to be sure it hasn't been answered. The hard part of setting up the assignment is going to be suggesting topics where current research is undergrad accessible - most of commutative algebra, for instance, isn't. But plenty of topics in discrete math and bits number theory are accessible; if I had to set up the assignment, I'd look at things like REU's to find topics. $\endgroup$
    – Henry Towsner
    Commented May 31, 2018 at 21:11
  • $\begingroup$ @TheChef: Here's what I'd do if I had to teach that. First, I'd interpret that outline to find goals I believe in; to me those are interacting with un-curated literature, and making conjectures, both of which I think are valuable skills that math curricula underdevelop. (You might prioritize different skills.) If it were me, I'd make them find and explain a theorem from the literature and propose a way the theorem could be extended; as long as they complete it, I'd assess it like a writing assignment based on giving a clear explanation of the theorem and their conjecture. $\endgroup$
    – Henry Towsner
    Commented May 31, 2018 at 21:24
  • $\begingroup$ It's likely I'll accept this answer. I dislike the fact that we are not really following what is literally asked. That said, you're approach is probably the best way to make the best of a ill-thought situtation. $\endgroup$
    – James S. Cook
    Commented Jun 1, 2018 at 23:06
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Do some applied projects: statistics or linear programming optimization or diffy Qs. Not pure math research. Doesn't need to be publication type finding a new law either. Could be analyzing a physical problem or a company or an industry or social issue. Would do some good to have that outreach anyways.

All that said, I think there is enough going on in a regular math course (majors or even pre-major) so that incorporating research is not a good idea.

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    $\begingroup$ I teach 5 courses. I have 10 required office hours a week. Beyond that I am doing real research with a student individually. I definitely do not have time for that. I do agree it would be good for the applied students and not for everybody. $\endgroup$
    – James S. Cook
    Commented May 31, 2018 at 20:30
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    $\begingroup$ I think you are doing plenty of work. I would kind of ignore these efforts to get you to do extra. Play passive aggressive. $\endgroup$
    – guest
    Commented Jun 1, 2018 at 3:55
  • $\begingroup$ I hadn't read the question careful (mea culpa). I was more thinking along the lines of assigning a project or paper (as is done in history class for instance) than this sort of research proposal topic. It's not that it is a bad idea (we did one in my grad school as a formal exercise). But it is probably a stretch for undergrads. Especially those not headed to grad school. $\endgroup$
    – guest
    Commented Jun 1, 2018 at 15:03
  • $\begingroup$ Oh, I very much believe in undergraduate research for graduate school bound students. But, not within the framework this question outlines. I have yet to find a way to use this research initiative to help with my own research projects with students. I try to get students to work with me and/or go to REUs if I think they're going to graduate school. That said, they need mathematical maturity for it to work well. $\endgroup$
    – James S. Cook
    Commented Jun 1, 2018 at 15:10
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    $\begingroup$ Recommendation: Don't write "diffy Qs". It's nonstandard, looks unprofessional, and doesn't really make sense. Standard abbreviation would be "DE" or just write "differential equations" to be clear. allacronyms.com/differential_equations/abbreviated $\endgroup$
    – Daniel R. Collins
    Commented Jun 3, 2018 at 16:29
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I have seen other schools and extracurricular programs aspire to include math "research" at younger levels without really understanding what that entails. Many programs don't understand that this usually doesn't happen until the end of a doctorate. Also (as you brought up in the comments below), focusing on the part where one just sets up the research question is not a good way to simplify the task because that part is typically done by the senior researcher or a doctoral student's advisor.

I think that a modified version of this might be a valuable activity for very advanced and motivated students. As written, I agree that it would be excessive to require every student to do the program you described. I also think it's unrealistic to expect every teacher of being qualified to carry it out. An alternative I would suggest is to first have the teacher find a handful of math papers that seem accessible (I would look only at expository journals, not research journals), then have a student pick one of those papers, and the project would simply be to read it (very carefully, over a long time), and write a review of it explaining the idea and what they think about it.

You also asked about explaining the distinction between math and other disciplines when it comes to research. I agree that is probably the source of the problem here. I'm including some comments on that referencing the items from your list.

  1. "Clarify by conducting a literature or artifact review or developing a main argument."

Math research literature is incredibly specialized. Not only does one usually need an advanced degree to be able to read it, but one needs to have specialized in a similar area as the author. And if we are talking about literature pertaining to open problems, it would be very difficult for a student (probably even the teacher) to do this review.

  1. Evaluate the information by identifying:

    -the credibility of sources

    -feedback from peers and professionals in the field to improve the proposal/project

In math, the credibility of an article is provided by whether or not it has been published in a peer reviewed journal. The math research community has a rigorous revision and refereeing process for articles before publication. So I would interpret this part as: restricting the student's sources to peer reviewed journals, as opposed to Wikipedia, blogs, or other information. But I think it would be more appropriate for students to be looking at those "less credible" sources because they are friendlier and potentially accessible.

  1. Organize the presented content and/or project timeline in a manageable, usable, and discipline appropriate format that can result in implementation of the research proposal or project.

Nobody, not even seasoned research mathematicians, can foresee the twists and turns that happen along the process of making a new mathematical discovery. For example, sometimes some trivial thing you overlooked at first turns out to be significant enough that it gets its own paper. It's not uncommon for a doctoral student to think he's a year away from graduating, then run into a theoretical problem that pushes that back a couple of years (or even indefinitely). The path is very unpredictable by nature.

  1. Critically analyze the strengths and weakness of the information collected, including, but not limited to:

    -supporting and opposing arguments

    -methodologies

    -the proposed process

There are no supporting or opposing arguments among proven math theorems. It's not like the social sciences where we are relying on evidence. We are relying on proofs of abstract logical statements. (There are some cases where mistakes end up published, but that's not what we're talking about here.) A known, checked math result has the consensus of the mathematical community, making this step basically irrelevant.

  1. Synthesize by constructing or summarizing all information/data into a research proposal/project that includes rigorous, researchable questions based on new understandings.

That is essentially what one does as a doctoral student after completing all their qualifying exams, and even then it is done under the guidance of a professor.

In the end, I would be happy that the school is expressing interest in something like this, but I would try to (in a friendly way) gear them toward something more realistic.

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  • $\begingroup$ Indeed many points require some real creativity to interpret for Math. I must disagree with you on one point though. I really think the fact that is a research "Proposal" as opposed to conducting research is not a merit. If we take a look at REUs then I think by in large you'll find students who are conducting research that has been proposed by a professor. In fact, the proposal phase is in many ways more difficult than the conduction. For example, I'm currently conducting research with a collaborator who proposed research that probably would not have occurred to me anytime soon. $\endgroup$
    – James S. Cook
    Commented Jun 3, 2018 at 1:42
  • $\begingroup$ @TheChef That's a good point. It is usually a PhD student's advisor who chooses the problem for a dissertation. I'm going to edit my post and delete that comment. $\endgroup$
    – j0equ1nn
    Commented Jun 3, 2018 at 13:45
  • $\begingroup$ I don't mind you leaving it. It is a misconception which is held by the people who wrote the learning outcome. $\endgroup$
    – James S. Cook
    Commented Jun 3, 2018 at 18:14

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