**A. ** Constructivism in its *moderate* (also called
*simple* or *trivial*) form is a philosophical position
characterized by the idea that knowledge is not passively received,
but rather actively constructed by an individual.
This central idea is applies to learning generally and is not
limited to any particular setting such
as problem solving or listening to lectures. "Knowledge" refers
to the contents of an individual mind, not to abstract or public
knowledge. "Constructed" refers to the result of mental activity,
such as reflecting on one's work or perceptions. It does not
necessarily include the physical or abstract aspects of construction,
such as assembling a jigsaw puzzle, constructing the rational numbers
from the integers, or making a proof, although doing or discussing
these might engender mental constructions.

New knowledge is seen as constructed out of older knowledge,
where "knowledge" refers to more than just remembered
"facts," e.g., the area of a triangle is one half its
base times its height or the Pythagorean theorem.
In addition, it refers to the largely tacit mental
structures that allow one to interpret the meaning of words,
bring ideas to mind, or effectively explore a new mathematical field.
Below we give two simple illustrations of knowledge construction. (There
are several views of knowledge and how it is constructed; however,
the Piagetian and the Vygotskian theories are too complex to
include here.)

Illustration 1. A beginning topology student's knowledge about
continuous functions can be viewed as a vast web of ideas, images, and
personal experiences, some based on mathematical relationships
and some not. The word "continuous" may bring to mind the
image of a sine wave or of a teacher saying "you can draw it
without lifting your pencil from the paper," which, in turn, might
bring to mind other associations, e.g., "but to prove it, you need
to use an epsilon-delta argument." Consider now how the student
might comprehend the proposition that the continuous image
of a connected set is connected and develop it into lasting, usable
knowledge. Suppose the proposition is presented by a teacher
together with a couple of examples. If the
student merely passively observes the presentation, he/she
might be able to repeat some of it for awhile, but later the
proposition might not come to mind when needed.
However, as the student works with the proposition - perhaps
examining its proof, applying it, even discussing its
significance - he/she might come to associate it with many
ideas and images, including aspects of his/her prior knowledge
of "continuous" and "connected." Thus, the student "constructs"
a mental structure that enhances the probability of the
proposition coming to mind when needed. This structure is
idiosyncratic and is neither directly observable nor fully
communicable.

Illustration 2. Consider an abstract algebra student who is fairly
familiar with groups, but has just met the definitions of field and
vector space. Suppose the student is searching for finite fields,
guided by his knowledge of how to find new algebraic structures --
a mainly tacit understanding based on his experiences with groups.
The student's way of searching for algebraic structures is not yet
likely to be very effective and may be altered, not necessarily
consciously, based on new experiences, perhaps including that
fields are also vector spaces which constrains somewhat their
possible orders. This alteration can be seen as knowledge
construction. Note that it might be more appropriate to
evaluate such knowledge, which amounts to a way of searching, as
effective or not, rather than true or false.

**B. ** Constructivism in its *radical* form can be
characterized by two additional ideas. (1) The function of cognition
is adaptive in the biological sense of tending toward "fit" or viability.
(Trees in a forest have more leaves where there is more light and runners
have strong legs. Both have adapted to fit a situation.) (2) Cognition
organizes the experiential world, but does not allow discovery
of objective reality. [Cf. E. von Glasersfeld, "An Exposition of
Constructivism: Why Some Like it Radical," in R. B. Davis,
C. A. Maher & N. Noddings (Eds.), *Constructivist Views
on the Teaching and Learning of Mathematics* (pp. 19-29).
Reston, VA: NCTM.]

This position does not deny the existence of the external world,
although it does deny the world can be mirrored objectively by
one's own knowledge. It is consistent with one's knowledge
appearing to approximately mirror much of the external world,
perhaps because of the tendency toward "fit."

Notes:

Currently much research in mathematics education is guided by, or
consistent with, a constructivist perspective. Piaget was perhaps
the best known constructivist. He did a great deal of empirical
work with children and developed theoretical explanations
including a description of the way knowledge construction occurs.
Vygotsky developed a view that depends more on social interaction.
Ernst von Glasersfeld, is known for his articulation and development
of the ideas of radical constructivism.

In the mathematics education community, constructivism appears
to have been very influential in legitimizing studies of students'
mental processes, such as how they learn to add or develop the
concept of function. It has facilitated moving away from
behaviorism, with its logical positivist viewpoint, which tended
to ignore the internal workings of the mind and thus effectively
discouraged research on many interesting mental questions.