-- the Forum December 1993 --
New Paradigms for Mathematics Instruction
Betty Field, PVCC
The paradigm for mathematics instruction is shifting as a reflection
of current movements in educational reform. The new paradigms call
for the creation of student-centered environments and for the
promotion of active team learning, participatory knowledge, and
interdisciplinary approaches. The new models for mathematics
instruction promote a shift in both the content taught and how the
content is taught. The focus is on the teaching of concepts rather
than skills and on connecting students to the content through
interdisciplinary applications. The move is to place the emphasis on
concept-based learning and on the application of concepts and much
less on symbolic manipulation.
The roles of both the instructor and the students are also redefined
in the new learning environments. The instructor is no longer the
"expert" pouring knowledge into the "novices," or a "coach" who shows
students how to do things. The instructors role is to participate in
curricular decisions and then guide students doing important
mathematics. That is, conceptual based mathematics that uses
modeling, is descriptive of real life, involves the analysis of data,
requires consideration of assumptions, and involves writing logical
arguments and conclusions. This implies that every instructor knows
the mathematical world with both breadth and depth. The instructor's
role is to facilitate learning by providing a stimulating environment
and access to a variety of resources.
Students are no longer viewed as the "recipients" of information, who
can absorb knowledge as if by osmosis. Students are expected to
engage in the learning process and to establish partnerships with
each other and with the instructor.
Why is the Paradigm Shifting in Mathematics?
- -- Understanding the Nature of Learning Mathematics.
- Traditionally, we have made the assumption that the learning of
mathematics is a vertical ladder: one must learn one rung before
moving to the next. What we know now is that this is not true. Uri
Treisman, mathematics professor at the University of Texas at Austin,
has said that mathematics is a "bushy bush" with many branches, not a
vertical ladder. We have been withholding from our students the
power and enjoyment of mathematics because we make them climb the
ladder. In reality, once you have a small set of skills, you can go
in many different directions, use your skills, apply your knowledge,
and see the utility, power, and even beauty of mathematics.
- -- Understanding Student Learning.
- Perhaps we now have a
better understanding of students and how they learn. The mathematics
community acknowledges that student learning needs to be active,
collaborative, and meaningful. Research evidence supports that
cooperative learning can be appropriate and effective in the learning
- -- Technological Tools.
- We have access to technological tools
that can enable the shift. The traditional approach to the teaching
of mathematics has been primarily analytical, the manipulation of
symbols. Now a hand-held calculator or computer can even manipulate
symbols. Technological tools (i.e., calculators and spread sheets)
allow us to approach mathematical concepts graphically and
numerically. The power of graphing tools is changing how and what we
are teaching. These tools enable us to connect students back to real
- -- Diversity of Learners.
- Our student population is diverse;
students have different goals and purposes for taking mathematics
courses. Mathematics education is for "all" students and not for an
elite group. Our national and international need is for a literate
citizenry who has the mathematical skills to function productively in
an information-based society. The College Sciences Education Board
reminds us, a quality mathematics education for all students is
essential for a healthy economy."
- -- Societal Pressures are helping mathematics education
- As a society we are in the information age, which is definitely
changing the way we think and, hence, the way we learn. Every
message we get from society, from businesses, from the government is
that we, as educators, must prepare students to function in the
The feminist movement has changed our understanding of how some people
learn. Research indicates that some people are more comfortable
learning in a collaborative environment, want to understand the
concepts rather than just apply formulas, and need to create their
own meaning in mathematics. The information presented has to make
sense so they can put it into their own words and fit it into their
The citizenry is beginning to get very involved in community life,
both local and global. The "me" era seems to be over. People now
want to make commitments to solve the world problems. This has
opened the doors to the use of real-life applications in mathematics,
issues such as environment, toxic waste, population growth, can be
used when presenting concepts and solving problems.
Challenges Facing the Paradigm Shift
- -- Facilitating Student Change.
- Even if as educators we
acknowledge and accept the new learning models, we need to help
students understand the learning shift so they can become efficient
and effective participants in an active learning environment. We
must be aware that some students will not/do not want this shift.
Many want a passive environment, where the instructor, as "the
expert," lectures and the student passively listens and hopes s/he
can regurgitate back enough content to pass the course. Sometimes we
believe that what we have done in the past is the right way to doing
things. Students are not different.
Students who may have difficulty with the new environments are those
who have been successful in the traditional settings; those who are
successful symbol manipulators; those who thrive in a competitive
environment; those who "want" a passive environment; and those who
cling to the ways they have been taught in the past.
- -- Providing Instructor Support.
- Comprehensive faculty/staff
development is one of the keys to a successful paradigm shift to a
new mathematics learning and teaching environment. Teachers need
support as they begin to function in the new environment. As
Treisman has said, "We are asking teachers to teach content they have
not learned in ways they have never experienced." Teachers need to
be taught new content through interdisciplinary applications, i.e.,
population models from biology, supply and demand models from
economics, etc., and the new teaching strategies, such as writing for
learning, use of technology, cooperative learning, and alternative
assessment. An environment that promotes collegiality should be
provided so that instructors are in contact with fellow educators who
are experts in other disciplines.
- -- Enabling System Changes.
- We must rethink how our system is
structured and how it functions. The traditional role of teachers
and curriculum developers should be revised. The concepts of
loading, learning units, meeting times, length of meeting times,
hours per week, etc., must be redefined.
The system has to change to allow for interdisciplinary teaching in
active and collaborative learning environments with appropriate
technological support. Faculty must be given time and resource
support to learn, to attempt pilot projects, to evaluate, and to
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