From 1999-2002, the Hard Math Café provided a place for the Making Mathematics
community to converse about topics related to
mathematics research and teaching. We've included a sample of Café articles below.
Fundamentally, research is an act of discovery. It is also an act of creation.
You may wonder at my choice of words, but the analogy is surprisingly apt.
When encountering a research problem for the first time, there are very few
guideposts that you can use to find your way to "the end" of the problem.
Indeed, oftentimes you don't even know what questions to ask, or even what
"the end" ... is. You will have to discover these questions on
your own ... during the research process. When doing research in physics,
you are at least guided by a few immutable physical principles;
in mathematics you often don't even have that. It is at times scary and
frustrating. It is also, as my aunt who is a mathematician told me a
long time ago, ultimately a path you make alone.
So what do you do when you first encounter a research problem? You play with
it and perhaps solve simpler problems similar to the one you are working on it.
As you play with the problem you slowly develop an intuition or "feel"
about it, and you start to see patterns in what you are doing.
After you play with the problem long enough, which may take a long time
depending on your abilities, you reach a point and say, "aha, this seems
to be true"; then you go and try to prove your assertion.
You may also start to ask, "well if this is true, then perhaps such
and such is true", and go ahead and try to prove that.
This cycle of exploration-assertion (conjecture)-proof lies at the
heart of the research process and goes on continuously. Each cycle, after
it has been completed, deepens your understanding of the problem and leads
you to new avenues of exploration. Which questions, out of all the possible
questions [you] can ask, should you ask when you start a research project you may
wonder? ... this is where art and mathematical intuition comes in.
The great majority of the questions you ask, however, ultimately have their
roots in answering the question "why"? This underscores the fact that
research is ultimately driven by an innate sense of curiosity that you
have about the problem.
Research is ultimately a creative process, and the part I enjoy most about the
creative process comes in constructing the conceptual framework specific to the
problem at hand. This framework will be used to not only communicate your
results to others where it takes the form of mathematical notation, but it is
also as a tool that you use to develop your own understanding of the problem.
However, without this exploration cycle, creation of this conceptual framework
would not be possible. You may certainly try to impose a conceptual framework
on the problem, but this would short circuit the discovery process and do more
harm than good.