Characteristics of good undergraduate research problems

Characteristics of good problems for undergraduates in mathematics

Directing research with undergraduate students in mathematics requires a good programmatic environment such as 

1. quality time with students, 
2. strong faculty mentors, 
3. institutional support,

etc.  But the foremost requirement is good research problems that undergraduate students can attack. 

Below I give six qualities of a good undergraduate research problem.  (Future posts will attempt to describe how to find these problems.)

I have directed mathematical research at the undergraduate level, at the masters level and at the doctoral level (supervising four doctoral students.)  Directing research with undergraduates is enjoyable but has a unique set of challenges.

A good research problem for undergraduates should be accessible

This is obvious.  Any research problem should be accessible to the researcher.  But mathematicians often underestimate the variety of good undergraduate research problems in their discipline.  A good research problem involves several good entrypoints, places where someone can begin their exploration and have some chance of early elementary success.  

It is an art to finding these good problems; after twenty years of directing research with undergraduate math students, I still struggle to gauge a good entrypoint and I often misjudge the difficulties students have getting into the problem.

In looking for good entrypoints, I often look for "toy" problems, smaller problems (possibly extended exercises) which are solvable and have a good chance of providing the intuition necessary for looking at deeper problems.  Questions about 2x2 matrices can provide intuition for questions in linear algebra or operator theory; questions about modular arithmetic or symmetries of polygons may provide intuition about questions in modern algebra or group theory; questions about finite graphs can lead to more sophisticated problems in combinatorics.

Students who use a computer software program to attack a good math problem will feel more comfortable exploring and making conjectures.  If there is a place for computers and mathematical software in your project, use them!

The difficulties of finding "toy" problems varies by discipline, but if one can explain the problem to an undergraduate audience (you can, can't you?) then one can probably find a small problem that generalizes to deeper questions in higher mathematics.  If the right problem is found, students will achieve success with the early toy problem and will then be hooked on the drug of mathematics by the time the problems get hard.  (By that point, in order to continue the mathematical high, the student will have no choice but to move on to the higher mathematics!)

A good research problems for undergraduates has "legs"

A good research problem is not a mere exercise or merely a reading project, but it has "legs", it runs ahead of the researchers, unraveling more and more mathematics.  It is the nature of mathematics that many questions have that effect; once begun, their solutions reveal a larger realm of inquiry.  Questions about the quadratic formula led into cubic polynomials and complex numbers; questions about Euclid's fifth postulate led into nonEuclidean geometry and eventually relativity; questions about constructible numbers led to field extensions; Fermat's Last Theorem led on to algebraic number theory and elliptic curves.

It is probably clear -- but should be stated -- that it will be easier to find problems with "legs" if the supervising professor already works in the research area and has considerable experience with related problems.  (It is hard to direct research if one is not already active in research!)

Good research problems for undergraduates lead to higher math

This remark parallels the last one.  Along with a good problem that runs ahead of the researchers, leading to more open questions, a good research problem for undergraduates should begin to provide a glimpse of previously unknown realms of mathematics.  The bright undergraduate who says, "Yeah, I know all about matrices!" should get a glimpse of operators on infinite dimensional vector spaces or be given the opportunity to play with linear algebra over finite fields or p-adic rings.  

A major goal of undergraduate research should be to open the students' minds to the truly wild nature of higher mathematics.  If done well, the undergraduate will respond by wanting to go a little further and explore one small piece of that wilderness.

Good research problems for undergraduates dream of mathematical heights

I find it useful to have a "dream" result or open conjecture that I want the students to see.  It may be faraway, off on the horizon, but maybe we could get close to that result?  Probably not... but ... the students should realize that there are some BIG problems nearby that no one has solved.  

If you are hiking in the Himalayas, don't you want to at least see Mt. Everest on the horizon?  

Students working in posets of graphs might ask, "How does my work relate to the famous Graph Reconstruction Conjecture?"  Students working in difference sets or circulant matrices should understand the statement of the Circulant Hadamard Conjecture.

By getting a chance to dream about proving a BIG result, students glimpse a large profession devoted to exactly these ideas; they see a profession reaching out for hard problems and attacking them.

It is good for my students to see themselves following in the tradition of Euler, Gauss, Hilbert, von Neumann...

A good research problem for undergraduates uses "magic"

I use "magic" here to mean a result given to a student without proof or defense.  

Most mathematicians visualize mathematics as developed linearly, an edifice of axioms, lemmas, theorems, proofs.  This is true.  But in this philosophy, we sometimes forget that we can discuss a field of mathematics without uncovering every prerequisite concept.  A colleague may shake his head and say, "You cannot discuss finite fields until students have had at least one higher algebra class."  Or "operator theory cannot be taught to undergrads."    I disagree.

I understand (and I teach) the axiomatic development of mathematics but it is OK if occasionally one says, "We will use the following result whose proof is beyond the scope of this course."  Skip some of the prerequisite concepts; provide some results as "magic" and move on!

When directing projects in difference sets, I briefly describe group representations (to a student who has had higher algebra) but I claim Maschke's theorem as a piece of magic ("Trust me!")  I make similar statements about the sums of squares of degrees of irreducible representations.  I provide some computational tools, without defense, and my students use them successfully.

Feel free to provide "magical" useful results to your students, without defense.

One advantage of "magical" results (besides its practical aspects directing undergrads) is that a curious student may eventually ask you to "backfill", to go back and prove parts of the magic.  If done at the appropriate later moment, this can be a good experience since the student is now highly motivated to understand that piece of magic.

A good research problem offers alternatives

So your students won't climb Mt. Everest -- or prove the Graph Reconstruction Conjecture. But are there some other smaller summits they might be able to reach?  If so, hold that out in front of them, as motivation when the going gets rough.  "Yes, Mt. Everest is still 100 miles away. But that ridge over there, across the valley, looks very interesting and I bet if we got to that ridge, we would have an awesome view!"  Tired hikers in the mathematical mountains might be willing to go just a little further if they can see a nearby ridge to top, a nice small result to solve.  Find some good partial results that you can direct your students towards, as the project winds down.

Next time: Friday's post will compare tackling a good research problem with a walk in the woods (or a hike in the Himalayas.)

[Mathematical prerequisites: This article requires no mathematics beyond high school.]

Follow Ken W. Smith on Twitter @kenwsmith54