**Constructivist
Learning and Teaching**

**Douglas
H. Clements and**

**Michael
T. Battista**

** **

Reprinted
with permission from Arithmetic Teacher, copyright September 1990 by the
National Council of Teachers of Mathematics. All rights reserved.

** **

In reality, no
one can *teach *mathematics.
Effective teachers are those who can stimulate students to *learn *mathematics. Educational research 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 focusing on alterations in content rather
than the reports' recommendations for fundamental changes in instructional
practices. Many of these instructional changes can best be understood from a *constructivist *perspective. 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 following:

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 student, 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 integrate them
into their existing structures of knowledge.

3. No one true
reality exists, only individual interpretations of the world. Their
interpretations are shaped by experience and social 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 the members of a culture. Thus, the
constructivist classroom is seen as a culture in which students are involved
not only in discovery and invention but in a social discourse 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
Major 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
capable of solving a wide variety of meaningful problems.

Second, students
should become autonomous and self-motivated in their mathematical activity.
Such students 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 mathematics 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 mathematics.

**Teaching
and Learning**

** **

Constructivist
instruction, on the one hand, gives preeminent value to the development of
students' personal mathematical ideas. Traditional instruction, on the other
hand, values only established mathematical techniques and concepts. For
example, even though many teachers consistently use concrete materials to
introduce 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 constructivist 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
sophisticated techniques, all methods are valued and supported. Through
interaction with mathematical tasks and other students, the student's own
intuitive mathematical thinking gradually becomes more abstract and powerful.

Because the role
of the constructivist teacher is to guide and support students' invention of
viable mathematical ideas rather than transmit "correct" adult ways
of doing mathematics, some see the constructivist approach 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 dialogue, guides the focus of students' attention, thus
unobtrusively 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 individual 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" column will present specific
examples of the constructivist approach in action. Each will describe how
students think about particular mathematical ideas and how instructional
environments can be structured to cause students 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 differences? Most important, what instructional
methods are consistent with a constructivist view of learning?

**References**

Bruner, Jerome. *Actual
Minds, Possible Worlds. *Cambridge,
Mass.: Harvard University Press, 1986.

Cobb, Paul.
"The Tension between Theories of Learning and Instruction in Mathematics
Education.- *Educational Psychologist 23 *(088):87103. 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.

*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, OH 44242'*

* *