Math 28 - Winter 2006
Dartmouth College
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Math 28 is a course in combinatorial mathematics. Combinatorics is a branch of mathematics
that studies sets (usually finite) of objects that satisfy some properties. In particular,
it is concerned with "counting" the objects in a set (enumerative combinatorics), with
determining when an object with a required list of properties exist, with constructing and
analyzing objects meeting certain properties (as in combinatorial designs and matroid theory),
with finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial
optimization), and with finding algebraic structures these objects may have (algebraic combinatorics).
This year the course is being taught as a "guided discovery" course.
This means that the primary text for the course is a set of problems with just enough prose so that
the problems will make sense. It is available at the Copy center in
the form of course notes. By working these problems, students will discover
for themselves the main theorems of combinatorial mathematics (and their
proofs) and examples of how these theorems are used. Students are encouraged
to work in groups on the problems. Students are expected to work as many
of the problems as they can and submit them. Certain problems will be designated
each week in advance as "mandatory" problems. These problems will be graded
and returned to the student.
Class time will be devoted largely to a discussion of the problems and the broad themes in combinatorial mathematics that the problems illustrate. This discussion will be split between whole class discussion, including possible student presentations of their ideas, and small group discussions. Students will be graded on class participation. Unfortunately, coming late to class may disrupt small group discussions, and so this will have to be reflected in the class participation grades of students who habitually come late to class. Of course it is difficult to participate if one does not come to class at all, and so habitual absence will also be reflected in the class participation grade.
I hope the following quote will inspire you to participate in this course.
"Mathematics is not for spectators; in order to gain in understanding, confidence, and enthusiasm one has to participate." M.A. Armstrong