DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

Preface
Much of your experience in lower division mathematics courses probably had
the following flavor: You attended class and listened to lectures where theory
and examples were presented. Your text usually gave a parallel development.
Then your turn came. For every assigned problem or proof there was a
method already taught in class that was the key, and your job was to decide
which method applied and then to apply it. In upper-level math major
courses, your goal should be to discover some of the ideas and methods for
yourself – as many as you can. These notes are intended as an introduction to
discrete mathematics and also as an introduction to mathematical thinking
within a classroom mode of learning called Guided Discovery.
Guided Discovery approaches mathematics very much like a mathematician
does when on unfamiliar ground: Looking at special cases, trying to
discover patterns, wandering up blind alleys, possibly being frustrated, but
finally putting it all together into a solution of a problem or a proof of
a theorem. This thinking process is as important for you as the discrete
mathematics you will learn in the process. The notes consist principally of
sequences of problems designed for you to discover solutions to problems
yourself. Often you are guided to such solutions through simplified examples
that set the stage. As you work through later problems, you will recall
earlier techniques that can either be used directly or be slightly modified to
get a solution. The point of learning in this way is that you are not just
applying methods that someone else has developed for you but rather you
are learning how to discover ideas and methods for yourself. Understanding
small points and taking small steps is the usual way of doing mathematics,
and is the usual path to all mathematical results – including very significant
ones.
The notes are designed to be worked through linearly, with the problems
in the first chapter introducing you to the habit of thinking for yourself as
well as introducing you to discrete mathematics. During class you will work
in groups, with some class discussion that will help give an overall context.
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