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Constructivism in Mathematics Education -- What Does it Mean?

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Abstract

The most influential and widely accepted philosophical perspective in mathematics education today is constructivism. This view, which holds that individuals construct their own knowledge, can be traced back to Piaget and beyond. It sees the learner as an active participant, not as a blank slate upon which we write. Cognition is considered adaptive, in the sense that it tends to organize experiences so they "fit" with a person's previously constructed knowledge. As a consequence, both researchers and teachers ask, "What is going on in students' minds when . . . ?", rather than speaking of behavioral outcomes and asking, "Which stimulus will elicit a desired response?" The term "constructivism" includes this view of how people learn, and constructivist teaching often simply means taking students' views and background into account so as to engender active, meaningful learning. However, constructivism comes in a variety of "flavors." There is a "moderate" version, compatible with the way most mathematicians see mathematics, and a social constructivist version, inspired by the work of Vygotsky, which takes into account sociocultural perspectives. There is the radical constructivism of von Glasersfeld, and beyond that, the sociology of scientific knowledge, which replaces the idea of truth with that of utility. We will describe these views and place them along a (increasingly radical) continuum.
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Constructivism in Mathematics
Education --
What Does it Mean?
John Selden
Mathematics Education Resources Company
Annie Selden
Tennessee Technological University
Research Conference
in
Collegiate Mathematics Education
CENTRAL MICHIGAN UNIVERSITY
SEPTEMBER 5-8, 1996
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WHAT'S GOING ON HERE?
"There is no such thing as constructivist
teaching . . ."
(math. ed. researcher)
" . . . with one commentator identifying 21
varieties of constructivism."
(from a review of The Content of Science:
A Constructivist Approach to its
Teaching and Learning)
3
"When I was visiting Australia, people kept
asking me to justify constructivism."
(senior ed. researcher)
"Constructivism is evil."
(speaker at a conference on the
history and philosophy of science,
and education)
4
We will try to sort out some of the different
meanings of constructivism relative to math ed.
We will discuss:
1) moderate constructivism
2) kinds of constructivism
3) implications of constructivism for
teaching
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I. MODERATE CONSTRUCTIVISM
Most researchers in math. ed. today share these
views (influenced by Piaget):
People construct their own knowledge.
This is done via mental processes, including
reflection (perhaps on actions).
This allows them to adapt to their environment.
People's old knowledge is used in constructing
their new knowledge.
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WHERE IS KNOWLEDGE?
(for constructivists in math ed)
It is in one's mind.
Constructivists are usually concerned with
what individual students know, not with
societal knowledge or knowledge residing in
language or the external world.
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WHAT KIND OF KNOWLEDGE?
Constructivists are interested in conceptual
knowledge, e.g., in students knowing that
(and why)
differentiable functions are continuous,
or understanding
the concept of function.
As opposed to, procedural knowledge, e.g.,
knowing how to (automatically)
find the derivatives of polynomials.
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WHAT KINDS OF
(MENTAL) CONSTRUCTIONS?
A Personal View*
For both concepts and propositions, there seem
to be two kinds:
"Insightful" constructions: coming to have a
concept, knowing a proposition, etc.
"Connective" constructions: coming to connect
a concept, proposition, etc., with others in one's
"knowledge network."
-----
*Constructivists, following Piaget, speak of constructing
"action schemes" and "cognitive structures."
(v. Glasersfeld, 1995)
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An Illustrative Vignette
Scenario: Ph.D. oral exam
'Easy' question to student (expecting something
more complex):
"Well, Mr. X, could you just give me a field of
order 6?"
[Very long silence, as he tries to find one]
Answer: "Apparently not."
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Somewhat after the exam
Mr. X realizes:
Fields are also vector spaces.
They can be seen as n-tuples over a field of
prime order.
So, they must have order pn and 6 is not of that
form.
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Mr. X has constructed knowledge in both
senses:
1. ("insightful") He now knows there are no
fields of order 6, etc.
2. ("connective") He has added to his network
of connections (between field, vector space, n-
tuple, etc.)
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Effects of Constructivism
It has encouraged and legitimized two shifts in
emphasis:
In math ed research, from directly observable
features to implicit ones involving what's going
on in students' minds.
In teaching,
from answers, automated procedures, and
transmitting information to problem-solving
processes, conceptual understanding, and
facilitating learning.
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II. KINDS OF CONSTRUCTIVISM
(affecting mathematics education)
moderate constructivism
radical constructivism
sociocultural views
sociology of scientific knowledge
(SSK)
social constructivism
(a philosophy of mathematics)
14
We discuss these kinds of constructivism
from two perspectives:
The influence of others on individuals'
knowledge construction
The relationship between knowledge and the
external world (including mathematics).
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Moderate Constructivism
Has nothing to say about the influence of others
on an individual's knowledge construction
and
the relationship between knowledge and the
external world.
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Radical Constructivism
Knowledge is in not passively received, but
rather actively constructed by the individual.
The function of cognition is adaptive in the
biological sense, tending toward fit or viability.
Cognition organizes the experiential world, but
does not allow discovery of objective reality.
(von Glasersfeld, 1990)
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Radical constructivism concentrates on the
individual cognizing subject. Others are one of
many influences on knowledge construction.
It does not deny the existence of an external
world, but denies that one's knowledge can
objectively reflect it.
It replaces truth with viability, but accepts
certainty in abstract mathematics.
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Sociocultural Views
Learning and understanding are seen as
inherently social and cultural.
Influenced by Vygotsky who maintained
knowledge appears twice --
once between people and then internally.
Sociocultural views can be seen as
complementary to radical constructivism. Each
"tells half a good story." (Cobb, 1994)
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Sociology of Scientific Knowledge
(a.k.a. SSK or social constructivism)
Somewhat similar to radical constructivism, but
concerning societal or public knowledge.
Regards science and mathematics as inherently
fallible.
Denies the existence of objective truth and
replaces it with social agreement.
Probably inconsistent with the views of many
mathematicians and scientists:
"I am among those who have found the claims
of the strong program [SSK] absurd: an
example of deconstruction gone mad."
(Kuhn, 1992 Harvard lecture)
20
Social Constructivism
(a philosophy of mathematics)
Not a view of knowledge construction.
Like SSK, it regards mathematics as
inherently fallible, denies the existence of
truth, and redefines "objective" as socially
agreed upon.
(Ernest, 1991)
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WHAT INFLUENCE DO THESE VIEWS
HAVE ON MATH ED RESEARCH?
Most researchers would investigate the same
questions, whatever they assume about the
relationship between knowledge and the
external world.
However, their view of others' influence on
knowledge construction has effects:
Radical constructivist researchers tend to focus
on an individual's learning, using case studies,
clinical interviews, teaching experiments
(researcher one-on-one with student), etc.
Socioculturalists tend to focus on how
knowledge is produced when people interact,
using ethnographic methods, observations
within classrooms (or other social settings).
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III. IMPLICATIONS OF
CONSTRUCTIVISM FOR TEACHING
People construct their own knowledge.
suggests
Teaching includes arranging situations to
facilitate construction.
and
Teachers can't "transmit" knowledge directly to
students.
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This is done via mental processes, including
reflection (perhaps on actions.)
suggests
Anything encouraging more or better reflection
may be helpful.
such as
cooperative learning
journal writing
(novel) problem solving
extended projects
and
engendering cognitive conflict
developing student autonomy
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People's old knowledge is used in constructing
their new knowledge.
suggests
It is important to keep track of what students
really know and understand.
which, in turn, suggests
the following three points:
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1. Teaching cannot be separated from, or
ignore, learning.
so
"The teacher's job is to teach and the students'
job is to learn" is misleading.
and
Teachers cannot really "cover" material
independent from students.
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2. Teacher-student communication should be
two-way, not one-way.
so
Lectures should be interspersed with
opportunities for teachers to find out what
students know and for students to encounter
constraints on their
constructions.
(Bettencourt, 1993)
3. It is useful to assess students' conceptual
understanding and problem-solving processes,
in addition to the correctness of their answers.
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Additional implications for teaching
(von Glasersfeld, 1993)
Students are sense-makers. Their answers
should be taken seriously.
Asking students how they came to their
answers
is a good way of getting at their thinking.
Successful thinking should be rewarded, even
when based on unacceptable premises.
Let students struggle with problems of their
own
choice.
The teacher must show a student what's
inadequate (about a piece of work) and
why.
Never present a solution as the only solution.
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It's no use presenting students a verbal
definition
unless they have an opportunity to have
some
kind of relevant experience.
who maintained knowledge appears twice --
once between people and then internally.
Data
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