How I Came to Teach Problem Solving at UCSC
"To think that you can do the same thing over and over
again and get different results is crazy." Chinese proverb.
"To think that you can do the same thing over and over
again and get the same results is crazy." Bruce Cooperstein
idea for this course emerged after over two decades of teaching mathematics
classes at the University of California, Santa Cruz (UCSC). These courses
have included calculus, linear algebra, a two quarter sequence in abstract
algebra, a one quarter abstract algebra course for prospective secondary
math teachers, the departments initial course in number theory, a
transition course to the upper division, Introduction to Mathematical Proof,
and others. Increasingly I had come to the conclusion that the curriculum
was not adequately preparing students to think mathematically and that either
changes needed to be made in some of these courses or else new courses should
be added to our offerings.
those things that convinced me of this need was the very differing views
my students had of what constitutes a mathematical explanation or argument
a proof and what I expected. This was true even after students
had taken our proof course and even several upper division courses. Additionally,
I found in teaching courses such as abstract algebra and number theory that
many of my students encountered serious difficulty when faced with situations
in which they did not have immediate access to a method for solving a problem
and often gave up after a minimum of effort.
difficulties arise, I came to think, because of the near total emphasis
on students acquiring mathematical knowledge and procedures. Mastery of
the facts and methods is obtained by solving numerous "exercises" usually
by imitating the actions of the professor or examples found in a textbook.
To be sure being able to imitate the actions of a professor or book are
important, for example, to automate the access and application of certain
mastery of facts and procedures, though necessary is not sufficient for
higher order mathematical thinking. In my judgement, all the courses in
the lower division at UCSC are graded on the ability of students to do such
"exercises" on homework assignments, midterms and final examinations and
much of the work required in many upper division courses as well emphasize
such activity as the evidence of student learning. A corollary of this is
that all the courses in the mathematics department at the lower and upper
division, with one exception, are lecture courses.
noted exception to the lecture course format is a course, Senior Seminar,
which is one way that students can satisfy the comprehensive requirement
in mathematics (the other is through the submission of a senior thesis).
In this course the students choose, with the supervision of the instructor,
an advanced topic. With further guidance from the instructor the students
research their topics, often with some minimum requirement for number of
references to books and articles and prepare both an oral presentation and
a written paper. Beginning about the middle of the quarter, individual students
give lectures followed by questions and discussions.
capstone experience is exceedingly worthwhile and student evaluations indicate
that, after the initial anxiety of giving a public talk on an advanced topic
in mathematics is overcome, it is one of the highlights of their experience
at UCSC. However, for most students it is an extension of what they have
done in other classes: Rather than being engaged in their own investigations,
making conjectures, applying principles of heuristics to write a proof or
in other ways solve a novel problem they are reading and learning about
the already completed mathematics of others, albeit in a more sustained
fashion then previously. There is no doubt that in such a course these students
are far more actively involved in their own learning then their lecture
courses but this does not fill the gap in their learning that I think is
essential to understanding what it means to think mathematically.
decision to create and teach a course in problem solving has also been influenced
by my interest in the improvement in the teaching of pre-collegiate mathematics.
I am convinced that the most important element in improving student achievement
in mathematics is the preparation of better teachers of mathematics and
my energies have been directed towards this goal. One aspect of my work
towards that objective has led to extensive contact with students who are
considering careers as secondary teachers of mathematics, for example, my
creating and teaching an abstract algebra course for prospective teachers
and through my direction of an internship program for students interested
in teaching, the Santa Cruz Community Teaching Fellowship (now the Community
Teaching Fellowship in Mathematics and Science) Program. This led to my
leading a two quarter seminar in mathematics education.
conversations that I have had with these students have reinforced my thinking
that many of our students do not have a full sense of what it means to do
mathematics and have missed out on the most enjoyable and exciting aspects
of the enterprise. More than a few, despite the fact that they have majored
in mathematics, have told me that their mathematics courses are not stimulating;
to the contrary, some even have the courage to say that they are boring.
This is not meant as an indictment of my colleagues for this is no doubt
as true of the courses that I teach as any one else in our department.
first effort to do something about this, to give my students an enriched
mathematical experience, was during the teaching of Math 100, Introduction
to Mathematical Proof, in the Spring quarter of 1998. This course is a relatively
new addition to the UCSC mathematics curriculum, now about six years old.
It was conceived and introduced by Professor Geoffrey Mason as an elective
course out of frustration with students enrolled in courses such as abstract
algebra and real analysis who lacked knowledge essential for success in
most upper division courses, for example, properties of functions, equivalence
relations, but which were not taught in previous courses.
100 did not function well as an elective course for often the students who
enrolled were the very students who needed it the least. Subsequently, Math
100 became a required course in the mathematics curriculum though students
could opt out through examination. It was in the second year as a required
course that I taught it. In approaching this course I wanted to get students
out of a passive mode and involve them actively in class. As envisioned
this would be done in two different ways. First, each of the classes would
begin with a warm-up period intended to get the student "thinking mathematically."
These sessions were twenty minutes long and would involve the distribution
of problem sheets. After a minute of orientation the students would work
on these problems, either individually or in self-selected groups. After
fifteen minutes there was a call for volunteers to present solutions. Most
students enjoyed this aspect of the class and these periods were characterized
by animated conversation. It was also an arena to use the knowledge and
discourse learned in the other part of class.
remainder of the class, about 50 minutes was devoted to the content of the
course. This encompassed: logic, naïve set theory, relations and ordering,
functions, and mathematical induction, and cardinal arithmetic. Had time
permitted there would have been an application of these concepts to a topic
like elementary number theory or the construction of the real numbers, but
there was not. My intention was to teach this material Socratically, with
the students responsible for doing assigned reading prior to class. I would
then ask questions such as what new terms or concepts were introduced? What
theorems were proved? and so on. The text itself was conducive to this because
it only contained sketches of proofs and expected students to complete them
as exercises. However, after the initial few meetings of the class more
and more students fell behind and I was faced with lots of silence. At this
point I reverted to lecturing.
personally reviewing my overall dissatisfaction with the outcome of Math
100 I came to the conclusion that including problem solving as a small add-on
to an existing course did not work well and resolved to create and teach
a problem solving course at the earliest available time which, of course,
would be constrained by the resources of my department or my willingness
to teach an extra course.
following year, 1998-99, I took a full sabbatical leave and did no teaching.
During that year I applied for and was awarded a Pew
National Fellowship for Carnegie Scholars which is a program of the
Carnegie Foundation for the Advancement of Teaching. The problem solving
course and the effort to represent the teaching and learning that occured
in this course portfolio is he outcome of a project I undertook during my
active year as a Carnegie Scholar.