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In some universities, high schools, and summer programs, students are required to do their own research project in maths and write their own essays/research papers. At the same time, however, many other schools and colleges only require them to take examinations.

In high school and the first a few years of university, it is almost impossible to come up with something new in maths (unless you are a genius:), and those research papers are usually just presentations of existing mathematical knowledge. Although writing papers develop research skill, it is likely to be very time-consuming for both students and teachers marking them. If they do exams instead, a lot of time could be saved, which means students have more freedom and can study more content. But of course, content is not everything. Doing a summary of existing knowledge doesn't seem to be more related to problem-solving skills than exams, either.

So, is it better to just do the exams and learn as fast as possible, or slow down and do some research projects?

The answer to this question is very likely to be "it depends", but I am still asking this because there might be other reasons for/against doing mathematics research projects when one still doesn't have very much knowledge.

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    $\begingroup$ Doing research projects (even if it is something already known) still gives the student valuable experience. Something not gained by "taking exams". Indeed, in the experimental sciences, a (group of) students may take some classic experiment and try to replicate it. $\endgroup$
    – Gerald Edgar
    Commented Jan 3, 2020 at 14:34
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    $\begingroup$ Not a full answer, but Math 389 at the University of Michigan was a good introduction to research for me. Apparently it is modeled after a course at MIT. math.lsa.umich.edu/courses/389/index.html $\endgroup$
    – Chris Cunningham
    Commented Jan 3, 2020 at 16:22
  • $\begingroup$ A good undergraduate student, a good problem (and good luck) can produce real research, publishable in a high-quality journal. Example: Section 5 (in my opinion the hardest part) of "Divisibility of Dedekind Finite Sets" (J. Math. Logic 5 (2005)58-74 and math.lsa.umich.edu/~ablass/ddiv.pdf) was the result of a summer research project by David Blair in the "Research Experiences for Undergraduates" program. $\endgroup$
    – Andreas Blass
    Commented Jan 5, 2020 at 2:10

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I think it is very possible to have real, meaningful research projects at all levels.

As an example, I had 2 of my students in Calc 2 work on a project which started with the idea "Can we define a tangent circle instead of a tangent line?"

They started by finding a formula for the center and radius of a circle given three non-colinear points:

https://www.desmos.com/calculator/3aty1fcgdw

They then applied this formula to the points $(a,f(a))$, $(a+h,f(a+h))$, $(a-h,f(a-h))$ of a function $f$.

https://www.desmos.com/calculator/xowdomvxq3

They then needed to take the limit of the expressions for the center and radius which they had as $h \to 0$. As luck would have it, we covered Taylor series a few days before they needed it to take these limits!

https://www.desmos.com/calculator/vfgzav8ngx

I then let them know that they had rediscovered "curvature" and the "osculating circle" of a plane curve.

This really worked out their algebra and calculus skills, and felt like real research. They would get stuck for a week at a time, experiment, try it in Desmos, see it didn't work, go back to the drawing board. I think it was an extremely valuable experience for them.

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Perhaps a research project could be replaced by an independent study project. The idea is to learn more math, in a more independent way. The student would still have to produce something to show what they've learned (although if the teacher found it easier, they could instead require an oral explanation of the content learned).

I learned a lot on the independent project where I worked from the (tiny) book Surreal Numbers, by Knuth.

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Opinion: I think they get enough exposure to writing reports in other subjects. Think the time is better used for drill and exams, than for writing a report.

The one benefit might be learning to search the literature, but even that is probably much better done in history or science where, although rather difficult, the papers are still comprehensible. (Not so in math.)

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