Spring 1998
Vol 6 Issue 2
IN THIS ISSUE...
New Alignments in Calculus Instruction
Change: Do We Really Have a Choice?
Change, Learning, and the Future.
Using a Student's Fund of Knowledge to Guide Discovery
You Say You Want and Evolution?
SEE ALSO...
The Labyrinth
Maricopa Center for Learning and Instruction
CHANGE: Do We Really Have A Choice?
Edgar M. Chandler, SCC
My preparation for a career teaching two-year college mathematics, like others
in the 60's, was traditional: study and receive a bachelor's and master's degree
in mathematics. The emphasis was on finding exact algebraic solutions along with
formal and rigorous presentation. I was introduced to proofs in the calculus course
and required to produce them throughout the rest of my time at college. Practical
applications were an afterthought, if considered at all, and I never saw a function
defined or modeled from numerical data. Graphs in textbooks were scarce, and I laboriously
drew the few geometric representations that time permitted.
I was well prepared to teach traditional mathematics during my early years in the
classroom. But in the 70's, I began to see the need for changes in both the content
and methodology of calculus and differential equations. First, the number of topics
we were required to teach increased each year, which forced instructors to cover
a new topic each class period. By the mid-eighties, calculus courses harbored the
totality of what all teachers of the subject and researchers in the field deemed
useful and attractive. A common calculus textbook in 1985 had over 1000 pages while
the calculus textbook I used as a student in the early 60's had fewer than 400 pages.
I came to realize that students must consider important ideas from various viewpoints
and need time to reflect and broaden their perspective of important concepts. I was
frustrated, as my students were, that we couldn't spend time considering topics in
depth and exploring interesting applications. In addition, many of the students were
already employed as engineers or engineering aides, and they would tell me about
the many topics we studied that were never used in the real world nor ever used in
the manner in which they were presented in the text.
But time and technology have given me choices. The choices, however, often spark
conflict. For example, when the simple electronic calculators of the 70's eliminated
the need for slide rules and numerical tables for roots and values of transcendental
functions, many predicted the change would be detrimental to student's minds. They
argued that students needed the exercise of looking up these numerical values in
tables and the use of which linear interpolation to glean more accuracy.
I decided I needed to pursue the choices, however, because of some distasteful answers
to a simple question: "What are the alternatives to change?" If I didn't
change, I would be teaching topics and techniques that technology has made obsolete.
I would be asking students to spend hours deriving closed-form algebraic solutions
that they could generate in seconds using readily available computer software, and
they wouldn't learn about numerical and graphical representations of relations and
functions that are so prevalent in practical applications. If I didn't change my
curriculum and methods, my students would be using primarily algorithmic approaches
to problem solving instead of developing more valuable critical thinking skills that
are applicable to a wider variety of situations. If I didn't change, my students
would not have the advantage of new and improved materials and methods and they would
be deprived of seeing mathematics as a lean and lively subject that has real importance
and interesting applications in a variety of fields.
Through a process of reading, studying, working with colleagues, interviewing perceptive
professors, and creating materials for students, I have found a richer mathematical
world that is full of interesting ways of thinking about mathematical ideas and their
applications. Indeed, my definition of mathematics has changed and my perception
of mathematical rigor has broadened.
I respect tradition, but I am willing to break with it to improve my teaching and
increase student learning. Although the world of mathematics in which I was trained
was restricted to algebraic methods, I have come to understand that geometric and
numerical models not only have a place in mathematics, but they are the primary source
of applied problems. Without denying the place and importance of training in derivations
and proofs for some students, I have come to understand that training in critical
thinking skills that allows one to recognize patterns and draw conclusions from numerical
data, or recognize certain geometric property such as symmetry, is valuable as well.
I believe students are better served by first being introduced to the beauty and
simplicity of the fundamental concepts of calculus and differential equations from
intuitive and practical points of view that involve analyzing numerical and graphical
information. I believe that rigorous definition is the end, rather than the beginning
of a subject. And, most importantly, I believe that mathematics and engineering students
can now be challenged and empowered in ways that we could not previously imagine.
For example, the introductory course in Ordinary Differential Equations (O.D.E.),
like many courses across the curriculum, is in a state of change in both content
and methods. Reasons for these changes include: 1) new and powerful technologies
are now available; 2) the emphasis in a traditional O.D.E. course is on finding exact
closed-form solutions that often have no practical value; 3) students need to investigate
differential equations that have no explicit solution; 4) the prerequisite calculus
course has been changed. Leaders in the field describe the need for a "big picture
course" -- a course that presents content from numerical, symbolic,
and graphical points of view and requires students to interpret outcomes and predict
long-term behavior through written descriptions and oral presentations. Implicit
in the "big picture" recommendation are that today's new and powerful technologies
will be fully utilized by students and teachers and that the teacher's role will
evolve into one of a mentor and guide rather than the primary source of content.
A specific example of this type of change is in the presentation of differential
equations to model the actions of a swaying building in Blanchard, Devaney, and Hall's
new text Differential Equations. In older books, the problem could only be considered
with an "idealized model" that overlooked many subtle, but important, considerations.
Now, the problem comes from an actual experience in which costly mistakes were made
during the construction of the John Hancock Tower in Boston. Powerful software, that
allows for a more complete model and a more realistic analysis and predictions of
the physical phenomenon, accompanies it.
Options like these are based on what we now know about student learning and motivation.
Although changes like these contradict a dominant belief among college teachers that
knowledge is "transmitted" from knower to learner, focusing on what and
how students learn provides meaningful instruction students can use because they
have "helped" to construct it. Duke University professor David Smith describes
these basic ideas of constructivism as "the belief that all learning is constructed
by the learner in response to challenges to refine or revise what is already 'known'
in order to cope with new situations." There is a simple, highly convincing
proof that contructivism better explains what's in students' heads than does "transmissionism":
Ask your students to explain their thinking as they work through a problem. They
will undoubtedly describe ideas and thought patterns they didn't learn from your
lecture or any text. Since those ideas didn't come from the external sources, their
owners must have constructed them.
The options that time and technology have presented have been full of confusion and
rife with controversy. But many courageous and insightful leaders across the nation
have provided direction for changing curriculum and pedagogy, and the rewards are
enormous. The rewards for me are in observing students actively learning -- engaging
in projects, interacting with each other, using the computer to analyze and explore.
By empowering students with new tools and methods, they can experiment and investigate
in ways that were previously unimaginable.
This period of rapid change that we are experiencing in higher education means, as
Uri Treisman stated, "We are expected to teach things we have never learned
in ways that we have never experienced." To overcome the tumultuous effects
of such fundamental changes, we must all approach change with collegiality and openness
to new ideas and innovative methods. We need to help each other. This happens occasionally
with criticism and suggestions but always with the intent of encouraging each other
through these sometimes difficult, but mostly exhilarating times.
And, when we have doubts about the need to change and move ahead in content, methods
and the use of technology, we just need to ask ourselves, "Would I go to a doctor
today who insisted on treating her or his patients the same way they were treated
twenty years ago?"