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Constructivist Learning and Teaching
Clements, Douglas H;Battista, Michael T
The Arithmetic Teacher; Sep 1990; 38, 1; ProQuest Central
pg. 34
RESEARCH
INTO
PRACTICE
Constructivist
Learning
and
Teaching
Douglas H. Clements
and
Michael
T.
BaHista
I
n reality,
n~
one can teach mathematics. Effective teachers are those
who can stlmulate students to learn mathematics. Educational re-
search offers compelling evidence that students learn mathematics well
only when they
construct their own mathematical understanding (MSEB
and National Research Council
1989,
58).
Radical changes have been advocated
in
recent reports
on mathematics education, such as NCTM's
Curricu-
lum and Evaluation Standards
for
School Mathematics
(National Council
of
Teachers
of
Mathematics
1989)
and Everybody Counts (MSEB and National Research
Council
1989).
Unfortunately, many educators are
fo-
cusing on alterations
in
content rather than the reports'
recommendations for fundamental changes
in
instruc-
tional practices. Many
of
these instructional changes
can best be understood from a
constructivist perspec-
tive. Although references to constructivist approaches
are pervasive, practical descriptions
of
such approaches
have not been readily accessible. Therefore, to promote
dialogue about instructional change, each "Research
into Practice" column this year will illustrate how a
constructivist approach to teaching might be taken for a
specific topic in mathematics.
What
Is
Constructivism?
Most traditional mathematics instruction and curricula
are based on the
transmission,
or
absorption, view
of
teaching and learning. In this view, students passively
"absorb"
mathematical structures invented by others
and recorded in texts
or
known by authoritative adults.
Teaching consists
of
transmitting sets
of
established
facts, skills, and concepts to students.
Constructivism offers a sharp contrast to this view.
Its basic
tenets-which
are embraced to a greater or
lesser extent by different
proponents-are
the follow-
mg:
1.
Knowledge
is
actively created
or
invented by the
child, not passively received from the environment.
This idea can be illustrated by the Piagetian position
that mathematical ideas are
made by children, not
found like a pebble
or
accepted from others like a gift
(Sinclair, in Steffe and Cobb
1988).
For
example, the
idea
"four"
cannot be directly detected by a child's
senses.
It
is a relation that the child superimposes on a
set
of
objects. This relation is constructed by the child
by reflecting on actions performed on numerous sets
of
objects, such as contrasting the counting
of
sets having
four units with the counting
of
sets having three and five
units. Although a teacher may have demonstrated and
numerically labeled many sets
of
objects for the stu-
dent, the mental entity
"four"
can be created only by
the student's thought. In other words, students do not
"discover"
the way the world works like Columbus
found a new continent. Rather they
invent new ways
of
thinking about the world.
2.
Children create new mathematical knowledge by
reflecting on their physical and mental actions. Ideas
are constructed or made meaningful when children in-
tegrate them into their existing structures
of
knowledge.
3.
No
one true reality exists, only individual inter-
pretations
of
the world.
The~'!
interpretations are
shaped by experience and soc
,1
interactions. Thus,
learning mathematics should be thought
of
as a process
of
adapting to and organizing one's quantitative world,
not discovering preexisting ideas imposed by others.
(This tenet is perhaps the most controversial.)
4.
Learning
is
a social process in which children
grow into the intellectual life
of
those around them
(Bruner
1986).
Mathematical ideas and truths, both in
use and in meaning, are cooperatively established by
Prepared by Constance Kamii and Barbara
A.
Lewis, University
of
Alabama at Birmingham, Birmingham,
AL
35294
Edited
by
Douglas H. Clements, State University
of
New York at Buffalo, Buffalo,
NY
14260
Michael T. Battista, Kent State University, Kent,
OR
44242
34
ARITHMETIC
TEACHER
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the members
of
a culture. Thus, the constructivist class-
room
is
seen as a culture in which students are involved
not only in discovery and invention but
in
a social dis-
course involving explanation, negotiation, sharing, and
evaluation.
5.
When a teacher demands that students use set
mathematical methods, the sense-making activity
of
students is seriously curtailed. Students tend to mimic
the methods by rote so that they can appear to achieve
the teacher's goals. Their beliefs about the nature
of
mathematics change from viewing mathematics as
sense making to viewing it as learning set procedures
that make little sense.
Two
Maior
Goals
Although it has many different interpretations, taking a
constructivist perspective appears to imply two major
goals for mathematics instruction (Cobb
1988).
First,
students should develop mathematical structures that
are more complex, abstract, and powerful than the ones
they currently possess so that they are increasingly ca-
pable
of
solving a wide variety
of
meaningful problems.
Second, students should become autonomous and
self-motivated in their mathematical activity. Such stu-
dents believe that mathematics
is
a way
of
thinking
about problems. They believe that they do not
"get"
mathematical knowledge from their teacher so much as
from their own explorations, thinking, and participation
in
discussions. They see their responsibility in the math-
ematics classroom not so much as completing assigned
tasks but as making sense of, and communicating about,
mathematics. Such independent students have the
sense
of
themselves as controlling and creating mathe-
matics.
Teaching
and
Learning
Constructivist instruction, on the one hand, gives pre-
eminent value to the development
of
students' personal
mathematical ideas. Traditional instruction, on the
other hand, values only established mathematical tech-
niques and concepts.
For
example, even though many
teachers consistently use concrete materials to intro-
duce ideas, they use them only for an introduction; the
goal
is
to get to the abstract, symbolic, established
mathematics. Inadvertently, students' intuitive thinking
about what is meaningful to them
is
devalued. They
come to feel that their intuitive ideas and methods are
not related to
real mathematics.
In
contrast,
in
con-
structivist instruction, students are encouraged to use
their own methods for solving problems. They are not
asked to adopt someone else's thinking but encouraged
to refine their own. Although the teacher presents tasks
that promote the invention
or
adoption
of
more sophis-
ticated techniques, all methods are valued and sup-
SEPTEMBER
1990
ported. Through interaction with mathematical tasks
and other students, the student's own intuitive mathe-
matical thinking gradually becomes more abstract and
powerful.
Because the role
of
the constructivist teacher
is
to
guide and support students' invention
of
viable mathe-
matical ideas rather than transmit
"correct"
adult ways
of
doing mathematics, some see the constructivist ap-
proach as inefficient, free-for-all discovery.
In
fact,
even
in
its least directive form, the guidance
of
the
teacher
is
the feature that distinguishes constructivism
from unguided discovery. The constructivist teacher,
by offering appropriate tasks and opportunities for dia-
logue, guides the focus
of
students' attention, thus un-
obtrusively directing their learning (Bruner
1986).
Constructivist teachers must be able to pose tasks
that bring about appropriate conceptual reorganizations
in
students. This approach requires knowledge
of
both
the normal developmental sequence in which students
learn specific mathematical ideas and the current indi-
vidual structures
of
students in the class. Such teachers
must also be skilled in structuring the intellectual and
social climate
of
the classroom so that students discuss,
reflect on, and make sense
of
these tasks.
An
Invitation
Each article in this year's "Research into Practice" col-
umn will present specific examples
of
the constructivist
approach in action. Each will describe how students
think about particular mathematical ideas and how in-
structional environments can be structured to cause stu-
dents to develop more powerful thinking about those
ideas. We invite you to consider the approach and how
it relates to your
teaching-to
try it
in
your classroom.
Which tenets
of
constructivism might you accept? How
might your teaching and classroom environment change
if you accept that students must construct their own
knowledge? Are the implications different for students
of
different ages? How do you deal with individual dif-
ferences? Most important, what instructional methods
are consistent with a constructivist view
of
learning?
References
Bruner, Jerome.
Actual
Minds, Possible Worlds. Cambridge, Mass.: Har-
vard University Press,
1986.
Cobb, Paul.
"The
Tension between Theories
of
Learning and Instruction
in
Mathematics Education." Educational Psychologist
23
(1988):87-
103.
Mathematical Sciences Education Board (MSEB) and National Research
Council.
Everybody Counts: A Report to the Nation on the Future
of
Mathematics Education. Washington, D.C.: National Academy Press,
1989.
National Council
of
Teachers
of
Mathematics, Commission on Standards
for School Mathematics.
Curriculum
and
Evaluation Standards for
School Mathematics.
Reston, Va.: The Council, 1989.
Steffe, Leslie, and Paul Cobb. Construction
of
Arithmetical Meanings
and
Strategies. New York: Springer-Verlag, 1988.
35
