Book:
Papert, S. (1993). The Children's Machine: Rethinking School in the Age of
the Computer. New York: Basic Books.
Quotes and Comments:
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).
"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.)
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).
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.
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.
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.
"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
Learning and Mathematics
The Math Forum is a research and educational enterprise of the Drexel School of Education.