The Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.


Math Forum » Discussions » Math Topics » geometry.pre-college

Topic: Learning and Mathematics: Papert, mathetics (repost)
Replies: 29   Last Post: Nov 19, 1995 4:46 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
K. Ann Renninger

Posts: 96
Registered: 12/4/04
Learning and Mathematics: Papert, mathetics (repost)
Posted: Nov 3, 1995 11:54 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

RESEARCH SUMMARY: REPOST

LEARNING AND MATHEMATICS

PAPERT (1993)

In Chapters Five, Six, and Seven of his book, _The Children's
Machine_, Seymour Papert examines the art of learning, a topic
that he contends has been widely ignored by educational
researchers and practitioners. In Chapter Five he introduces the
concept of 'mathetics,' which he defines as the art or act of
learning, and discusses the issues that surround mathetics -- in
the school setting, theoretically, and in light of his own
experiences. Chapter Six presents a series of case studies that
build upon and draw from his discussion of mathetics, and which
demonstrate the utility of computers in promoting flexible,
personal, and connected learning. Chapter Seven contains a more
theoretical discussion of instructionist versus constructionist
viewpoints, as well as a defense of concrete knowledge and thought
in the face of educational trends that favor abstract reasoning.
Overall, Papert stresses support for personal variation in
learning styles, and for the increased acceptance by schools of
the ability of children to learn without assistance.

Book:

Papert, S. (1993). The Children's Machine: Rethinking School in
the Age of the Computer. New York: Basic Books.


Quotes and Comments:

o Students frequently don't realize that the principles of real
learning (mathetics) also include heuristics, the art of
intellectual discovery, and 'taking the time' to learn. As Papert
writes, "Attempting to apply heuristic rules checks students in
the rush to get done with a problem and get on with the next. It
has them spend more time with the problems, and my mathetic point
is simply that spending relaxed time with a problem leads to
getting to know it, and through this, to improving one's ability
to deal with other problems like it. It is not using the rule that
solves the problem; it is thinking about the problem that fosters
learning" (p. 87).

o "A central tenet of mathetics is that good discussion promotes
learning, and one of its central research goals is to elucidate
the kinds of discussion that do most good and the kinds of
circumstances that favor such discussions. Yet in most circles
talking about what really goes on in our minds is blocked by
taboos as firm as those that inhibited Victorians from expressing
their sexual fantasies. These taboos are encouraged by School, but
go far beyond it, and point to ways in which our general culture
is profoundly 'antimathetic'" (p. 89). (This emphasis on the
importance of discussion in the classroom, and especially in the
field of mathematics, is echoed by other prominent theorists such
as Schoenfeld; Brown, Campione, Reeve, Ferrara and Palincsar;
Resnick; Lampert; Collins, Brown, and Newman; and Cocking and
Chipman, all of whom have been the focus of previous Forum
discussions.)

o Papert also offers reasons for including the concept of the
'taboo' in his discussion of mathetics. He writes, "Exaggerated or
not, the suggestion of a taboo is intended to state emphatically
that getting people to talk about learning is not simply a matter
of providing the subject matter and the language. The lack of
language is important. But there is also an active resistance of
some kind. Thus advancing toward the goal of mathetics requires
more than technical aids to discussion. It also requires
developing a system of psychological support. The simplest form of
support system I can imagine is to adopt the practice of opening
oneself by freely talking about learning experiences" (p. 92).

o Papert continues, "On a pragmatic level, 'Look for connections!'
is sound mathetic advice, and on a theoretical level the metaphor
leads to a range of interesting questions about the connectivity
of knowledge. It even suggests that the deliberate part of
learning consists of making connections between mental entities
that already exist; new mental entities seem to come into
existence in more subtle ways that escape conscious control.
However that may be, thinking about the interconnectivity of
knowledge suggests a theory of why some knowledge is so easily
acquired without deliberate teaching. In the sense in which it is
said that no two Americans are separated by more than five
handshakes, this cultural knowledge is so interconnected that
learning will spread by free migration to all its regions. This
suggests a strategy to facilitate learning by improving the
connectivity in the learning environment, by actions on cultures
rather than on individuals" (pp. 104-105).

We might there understand the use of discussion techniques in
mathematics as providing the means to more connections. This is
not to suggest, however, that all mathematics is best taught
through discussion. Discussion can be used to bring an issue to
mind and can then be followed by direct instruction; It can also
be used following instruction to enable students to consolidate
their "real" understanding of what has been presented. Discussion
can also be used in evaluating how students have synthesized their
understanding of a new skill in combination with prior skills.

o In Chapter Six, Papert presents six case studies, each of which
has a learning moral. The first story again emphasizes the need
for connectedness, and for tying new learning to interests and
knowledge structures that are already in place. The second looks
at people who use math informally to adjust recipes. At the end of
this story, Papert summarizes the morals: "The central
epistemological moral is that we all used concrete forms of
reasoning. The central mathetic moral is that in doing this we
demonstrated we had learned to do something mathematical without
instruction -- and even despite having been taught to proceed
differently" (p. 115).

The third study shows how students must be allowed to cross the
gender/cultural divide in their own time and manner. As Papert
writes, "Knowing that one can exercise choice in shaping and
reshaping one's intellectual identity may be the most empowering
idea one can ever achieve" (p.123). The fourth story stresses the
need for humor in learning, and demonstrates how jokes can be used
as learning tools. The fifth story emphasizes the need for
'bricolage' or tinkering in learning -- namely, giving children
the opportunity to explore problems and projects freely, and to
develop their goals as they work. The last story differentiates
between 'clean' (unconnected) and 'dirty' (connected) learning,
and discusses the school's preference for 'clean' learning.

o Papert argues for a constructionist philosophy that will promote
teaching "in such a way as to produce the most learning for the
least teaching" (p. 139). He contrasts this view with that of
instructionism, in which "the route to better learning must be the
improvement of instruction" (p. 139). According to Papert,
constructionism is tied to mathetics in the sense that children
can often learn without the benefit of schooling, and, if given
the incentive to learn it independently, will even learn a subject
better than they would have learned it in school.

As Papert writes, "On some level we know that if we become really
involved with an area of knowledge, we learn it -- with or without
School, and in any case without the paraphernalia of curriculum
and tests and segregation by age groups that School takes as
axiomatic. We also know that if we do not become involved with the
area of knowledge, we'll have trouble learning it with or without
School's methods. In the context of a School-dominated society,
the most important principle of mathetics may be the incitement to
revolt against accepted wisdom that comes from knowing you can
learn without being taught and often learn best when taught least"
(p. 141).

It should be noted, however, that instituting discussions or
moving toward more student-directed classes requires that teachers
be ready to take up questions to which they do not know the
answers, come prepared with a variety of resources, and in
general, be well-versed in the content that they are teaching.

o "The important mathetic skill is that of constructing concrete
knowledge" (p. 143). Papert builds on this statement by stressing
the importance of bricolage (tinkering) in creating concrete
knowledge. "The basic tenets of bricolage as a methodology for
intellectual activity are: Use what you've got, improvise, make
do" (p. 144). He also brings computer use into the picture: "The
computer imply, but very significantly, enlarges the range of
opportunities to engage as a bricoleur or bricoleuse in activities
with scientific and mathematical content" (p. 145).

Finally, Papert emphasizes the importance of concrete thinking at
all ages, and argues against popular trends that favor abstract
reasoning over concrete knowledge. He writes, "My strategy is to
strengthen and perpetuate the typical concrete processes even at
my age. Rather than pushing children to think like adults, we
might do better to remember that they are great learners and to
try harder to be more like them. While formal thinking may be able
to do much that is beyond the scope of concrete methods, the
concrete processes have their own power" (p. 155).

-- summarized by Jane Ehrenfeld





Date Subject Author
11/3/95
Read Learning and Mathematics: Papert, mathetics (repost)
K. Ann Renninger
11/6/95
Read Re: Learning and Mathematics: Papert, mathetics
Problem of the Week
11/6/95
Read Re: Learning and Mathematics: Papert, mathetics
Tom Davis
11/7/95
Read Re: Learning and Mathematics: Papert, mathetics
Michael Keyton
11/7/95
Read Re: Learning and Mathematics: Papert, mathetics
Brian Hutchings
11/9/95
Read Re: Learning and Mathematics: Papert, mathetics
Ken Wood
11/9/95
Read Re: Learning and Mathematics: Papert, mathetics
Andrea Hall
11/9/95
Read Re: Learning and Mathematics: Papert, mathetics
Katie Laird
11/9/95
Read Re: Learning and Mathematics: Papert, mathetics
Katie Laird
11/10/95
Read Re: Learning and Mathematics: Papert, mathetics
Steve Means
11/10/95
Read Re: Learning and Mathematics: Papert, mathetics
Steve Means
11/10/95
Read Re: Learning and Mathematics: Papert, mathetics
Andrea Hall
11/10/95
Read Re: Learning and Mathematics: Papert, mathetics
Jim LaCasse
11/10/95
Read Re: Learning and Mathematics: Papert, mathetics
Steve Means
11/11/95
Read Re: Learning and Mathematics: Papert, mathetics
Andrea Hall
11/11/95
Read Re: Learning and Mathematics: Papert, mathetics
Steve Means
11/11/95
Read Re: Learning and Mathematics: Papert, mathetics
John Burnette
11/12/95
Read Re: Learning and Mathematics: Papert, mathetics
Lou Talman
11/13/95
Read Re: Learning and Mathematics: Papert, mathetics
Tarin Bross
11/13/95
Read Re: Learning and Mathematics: Papert, mathetics
Katie Laird
11/13/95
Read Re: Learning and Mathematics: Papert, mathemaatics
Steve Means
11/13/95
Read Re: Learning and Mathematics: Papert, mathetics
Kristina Lasher
11/14/95
Read Re: Learning and Mathematics: Papert, mathetics
David Weksler
11/14/95
Read Re: Learning and Mathematics: Papert, mathetics
Jane Ehrenfeld
11/15/95
Read Re: Learning and Mathematics: Papert, mathetics
Andrea Hall
11/16/95
Read Re: Learning and Mathematics: Papert,...
Robin J Healey
11/16/95
Read Re: Learning and Mathematics: Papert,...
Pat Ballew
11/17/95
Read Re: Learning and Mathematics: Papert, mathetics
Johnny Hamilton
11/17/95
Read Re: Learning and Mathematics: Papert, mathetics
Steve Means
11/19/95
Read Re: Learning and Mathematics: Papert, mathematics
Tarin Bross

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.