website is devoted to a course portfolio for a new class, Introduction
to Mathematical Problem Solving, that I created and then taught for
the first time at the University of California, Santa Cruz (UCSC) during
the Winter Quarter, 2000. I was moved to initiate this project out of a
personal sense that some essential aspects of what it means to "think mathematically"
were missing from my department's overall curriculum. As a consequence,
in my view, all too often our students leave the university with the impression
that mathematics is a fixed body of knowledge to be memorized rather than
a sphere in which they can learn to think for themselves.
Represented here is my attempt to create a course devoted to achieving these
higher-order skills. The course described was intended to be accessible
to all students with a mastery of the mathematical prerequisites for admission
to the University of California and at least a modicum of mathematical sophistication.
The course emphasized problem-solving in the context of learning basic concepts
from combinatorial theory, number theory, logic, graph theory and abstract
algebra. The principle activity in the class was students working on and
discussing novel problems which required them to formulate experiments,
work out cases, look for patterns in data, pose questions, make conjectures,
search for counterexamples and attempt to prove their assertions.
questions that I examine are: (1) Can students learn mathematical content
knowledge through problem solving without formal lectures/ can this mode
of teaching be used to make students' pre-existing content knowledge more
robust? (2) Can this mode of teaching strengthen students understanding
of mathematical proof and inculcate more sophisticated views of what it
means to do and think mathmatically?
I have collected lots of data. These include a record of all problems submitted
by students as well as their finals and my written responses to this work.
Some of that work is accessible here. All students filled out questionnaires
at the beginning of class and a majority turned in supplemental evaluations
at the end of the course. Summaries of those surveys are included. Additionally,
I interviewed twelve of the thirteen students during the middle of the subsequent
quarter (one student was abroad) and summaries of these, too, are available.
all the data I have collected, I make no pretense that this is scientific
research of any kind. Rather this is a report of a single offering of a
problem solving course, given for the first time by this instructor at this
particular institution and my thoughts about what was achieved.
this caveat I do think there are some genuine nuggets within what I report
and all is not fools' gold. For example, I offer some insights about getting
students to redraft and resubmit their work drawn from my efforts to encourage
this practice. Perhaps it is conceit, but I do think this report can benefit
others who are considering offering a course in problem solving or else
ways of strengthening student understanding in traditional upper division
courses and it is for that purpose that I make it public. I also do so in
a commitment to improve my own teaching practice and to benefit from the
wisdom of others as to how to better understand and capture evidence of
my students learning. In that spirit I invite reader's critical comments.