a journal of the international council of
philosophical inquiry with children
Paradox and
Learning: Implications from Paradoxical Psychotherapy
and Zen Buddhism for Mathematical Inquiry with Paradoxes
Nadia Stoyanova Kennedy
If you have the staff, I will
give it to you. If you have no staff, I will take it away from you! Zen
Koan[1]
ABSTRACT
This paper argues
that paradox offers an ideal didactic context for open-ended group discussion,
for the intensive practice of reasoning, acquiring dispositions critical for
mathematical thinking, and higher order learning. In order to characterize the
full pedagogical range of paradox, I offer a short overview of the effects of
paradox, followed by a discussion of some parallels between the use of paradox
in paradoxical psychotherapy and the use of the koan in Zen Buddhist spiritual
training. Reasoning with paradoxes in a community of mathematical inquiry is
interpreted as a comparative if not isomorphic pedagogical and cognitive
phenomenon. Finally, some broad implications are drawn for mathematical inquiry
with paradoxes.
KEY-WORDS
learning,
paradox, community of mathematical inquiry, constructivist pedagogy.
Introduction
Historians trace the origins of paradox to the riddles of
Greek folklore and the Delphic Oracle. This ancient epideictic[2]
tradition was developed extensively in Greek tragedy and philosophy. An early
example of its literary forms—a kind of “rhetorical” paradox—can be found in
Sophocles’ play Oedipus the King, in
which wit, puzzlement, and ambiguity proliferate. According to the ancient
myth, Oedipus was the first to encounter the Sphinx and answer her riddle, and
for his accomplishment was rewarded with the throne of
Another rhetorical paradox comes to
us via
The ancient world had different
reactions to paradox. Pythagoreans, for example, who thought of order and
perfection as attainable only through the elimination of conflict, resorted to
denial of paradox. And when it invaded their idyllic mathematical harmonia through Hippasus’s discovery of
incommensurables (e.g.
), which was in direct conflict with the Greek theory of
rational numbers, the presented paradox was consigned to secrecy, and Hippasus
expelled from the Pythagorean school. On the other hand, Heraclitus’ idea of a
cosmic order was grounded in ontological paradox, and his idea of the Logos
encompassed notions both of unity and of a conflict of opposites. Similarly,
the Eleatic school and its most notable philosopher Zeno, by emphasizing
paradox developed a method of disarming his opponents by
leading their arguments ad absurdum,
and consequently became famous for his paradoxes, most of them concerned with
dichotomy and the mathematical problem of infinity. Through his formulation of
over forty paradoxes, Zeno touched upon fundamental issues connected with the
nature of mathematical objects and fatally
compromised the hitherto assumed- unshakable foundations of mathematics and
logic which, centuries later, spurred discussions among
mathematicians that further inspired the development of set theory and various
trends in philosophy of mathematics.
After Zeno, the epideictic
tradition—which up to then had been diffused throughout rhetoric, drama and
nature philosophy—took a different direction.
Plato’s dialogues, the most famous documented example of the ancient
Greek tendency to undertake rational discourse as a kind of play, initiates a
form of dialectic with different philosophical purposes. Socrates–that iconic philosopher whose method
of inquiry initiated the Western tradition of philosophical dialogue—promoted a
form of dialectic known as elenchus. In contrast to the
eristic form, Socratic elenchus
launched inquiry with a question, and then required that one “…submit nobly to the
argument as you would to a doctor….” and follow it where it leads. It aimed at
persuasion, not through lavish speech, but though the examination of one’s
opponent’s assertions in order to draw out the inherent contradictions within
the other's position. Socratic elenchus
is believed to be a refined form of Zenonean reasoning, and to be highly
influenced by Zeno’s reductio ad absurdum
approach (Edwards, 1967; Vlastos, 1983). This
characteristic of Socrates’ discursive style brings his interlocutor to a state
of aporia,[4]
whereby his system of propositions and beliefs collapse under the
insurmountable pressure of the derived contradiction. Socratic dialectic, then, offers an
idiosyncratic educational model which operates by exposing contradictions which
act as catalysts for dialogue
between opposites, and in which the facilitator functions maeutically[5]
rather than didactically. Thus in the Socratic method
paradox (contradiction) is assigned a central role, which is essential for the
thinking and learning process.
Paradox and Its Effects
Plato held that paradox in
philosophy acts to provoke reflection—to “compel the soul to be at a loss and
to inquire”—and thus impels human reason toward those reconsiderations which in
turn lead to a higher level of understanding. And in his Republic, Plato (1961) used contradiction (paradox)
in a similar way in his preface to the Allegory of the Cave (vi. 509) to
describe knowledge acquisition as a four-part process: beliefs are acquired,
then examined—a process in fact of coming to know what those beliefs are—then
tested to see whether they contradict each other (i.e. yield paradoxes), and,
finally, justified, which legitimates them as “knowledge.” So Greek aporia became so
pervasive in human inquiry that it came to be seen by many philosophers as
inherent in any cognitive enterprise.
Among later philosophers, we find the same
idea that contradiction/conflict not only initiates thinking, but guides its
course as well. Charles Pierce suggested a method of justifying belief which,
like Plato’s, involves the examination of beliefs that are contrary or
contradictory—but he placed a new emphasis on the role of a community as
opposed to the lone individual in the “fixation of belief” (Pierce, 1958,
1966). Quine and Ullian (1970) elaborate on this process of “the fixing of
belief” by suggesting that when, in a dialogical process, a set of beliefs is
found to be inconsistent, the necessity to assess the grounds for this set may
lead to the rejection of the least firmly supported of the contradictory
beliefs, and thereby restore the consistency of the “web of beliefs” (Quine
& Ullian, 1970).
Similarly for Dewey (1910),
thinking is launched by conflicts, which “often fuse into one conflict between
conditions at hand and a desired and intended result. . . . The object of
thinking is to introduce a congruity between the two” (Dewey, 1910). In his Logic:
The Theory of Inquiry he says: “Nothing is more important in inquiry than
the institution of contradictory propositions” (Dewey, 1939, p. 197). Dewey
sees the introduction of negation as inevitably bringing contradictions, and
thus forcing discrimination and producing differences. Although his idea seems
similar to Plato’s, Dewey’s institution of contradictories is conceived as one
step in an ongoing process of inquiry, which he understands as providing only
provisional conclusions, which are always liable to modifications. A supposed
contradiction may bring a revision of a generalization, concept or conclusion
which in itself is a result of the preceding inquiry--which is to say that the
contradiction and the respective modification may affect the current as well as
the preceding inquiry. In short, philosophers seem to consider contradiction
(paradox) to be the most pointed example of cognitive conflict and an essential
force in thinking and the search for truth.
Dialecticians,
however, understand contradiction to be at the very center of human development
itself. The dialectical moment begins with “being conscious of it [the thing]
as a unity of opposed determinations” (Lawler, 1975, p. 4) and of the
unfeasibility of adequately comprehending the thing in its given form—a notion
which in fact resembles Dewey’s idea of the starting point of inquiry. Thus the
dialectical approach seeks to grasp processes in the full complexity of their
interrelationships, and to understand them as comprising a dynamic,
self-organizing unity of internal contradictions and configurations of
oppositions which are in fact the internal forces for movement and change.
Furthermore, dialectical analysis does not regard the process of development as
a simple process of growth, but as a development which passes from
insignificant and imperceptible quantitative changes to fundamental qualitative
changes.
The two
giants of 20th century psychological theory—Piaget and Vygotsky—pay
special attention to the role of cognitive conflict in development, and both
offer an explanation of the transformations in cognitive structures, which
assumes the existence of irreducible contradictions and proceeds through
overcoming them. In other words, both
theorists understand cognitive development as a continuous process of
reconstruction of cognitive structures, whether understood as a movement from
disequilibrium to re-equilibration which in fact mirrors the mechanisms of
dialectics (Piaget, 1977, although it has never been acknowledged by Piaget),
or explicitly described as a dialectical process as in the case of Vygotsky
(1981). In both thinkers, cognitive development is understood as a process of
active adaptation to the external world which involves progressive changes, and
therefore learning.
This study takes as its theoretical
framework the broad philosophical tradition briefly reviewed above, which
understands cognitive conflict and contradiction in their extreme form as
triggers for inquiry—a tradition which provided an epistemological basis for
conflict-theory in psychology, and which assumes the inherent ability of humans
to overcome or transcend constraints by creating new intellectual tools and
developing new modes of thinking. Paradox could be said to represent the most
pointed example of cognitive conflict, and thus merits inquiry as to its role
as a mediator in cognitive development and learning.
The
questions which naturally follow are:
what form of pedagogy is both appropriate and effective in creating this
form of cognitive conflict specifically in a mathematics classroom? Do we
regard all learning as equally important? And what kinds of problematic
situations can be engineered in order to support different types of learning?
In preparing to answer this question, it is necessary to examine some effects
of paradox and aspects of learning and for this purpose I will turn to a brief
description of two traditions that use paradox as a tool for triggering
cognitive change and promoting learning in their specific contextual
frames—paradoxical psychotherapy and Zen Buddhism.
Paradoxical Psychotherapy and Zen Buddhism
I will precede the description of paradoxical psychotherapy with
an explanation of what is understood by paradoxical communication—which in fact
is what usually brings people to paradoxical therapy. Paradoxical communication is defined as a
medium in which various communicational modes are consciously or unconsciously
mixed, and their differences are not recognized as such by some of the
communicative partners. Bateson’s Double Bind Theory was formulated as an
attempt to explain paradoxical communication and its therapeutic treatment. Schizophrenic patients in particular are
often unable to differentiate between message and context or message and
metamessage. To illustrate this, Jay Haley (1955)—a co-author of The Double Bind Theory--describes a
patient who always diligently knocked on the door of an office as he passed by
it—literally following the request on the door which read “Please knock.” And more generally, mixing different levels
of abstraction in communication tends to lead to paradoxes, as in humorous
discourses, which often take one message-mode for another—for example a literal
message for a metaphorical one, or visa versa, or choosing messages whose
ambiguity allows them to be read in different modes. In such cases, the moment
of discovery or “punch line” comes with the realization of the existence of these
alternative modes, which triggers a sudden reinterpretation of the message.
A double bind situation is created when: 1) at least two
messages of different levels of abstraction are communicated; 2) one of them is
a message about the other—that is, a metamessage; and 3) both messages are
contradictory. If they contradict each
other, they create a paradox similar to the Liar Paradox, or what Bateson and
his colleagues referred to as a double bind situation. In the sphere of human
relationships, Bateson (1972) finds the majority of double bind situations in
long-term relations in which one of the communicators is understood—consciously
or not— as more powerful than the other. He provides a prime example in
describing a relationship between a mother and a child or a teacher and a child
in which the mother/teacher is communicating a primary injunction of the form
“Do not do this or I will punish you,” and a secondary injunction which
communicates the message “Do not see this as punishment,” or even “Do not
submit to my prohibitions.” The child finds herself in a situation in which the
two messages produce a paradox. Such a communication system is characterized as
homeostatic—that is, as a system that perpetuates its communicative patterns,
responses, and roles—i.e. “a game without end” (Watzlawick, Beavin &
Jackson, 1967). A similar case is produced by the injunction “Be spontaneous!”
which one can only obey by disobeying, since complying with the order implies
the rejection of externally imposed rules and following one’s own internal
motivation. The concept of “child-centered” or “democratic” education might
also create paradoxes, since the requirement that students engage in studying
what really interests them sounds more like “We require that you wish to study
what we don’t require you to study.”
In the homeostatic communication system like the one described
above, there is no way out—whatever the response is, it will contradict one of
the initial messages, and will perpetuate the paradoxical communication pattern.
The only way to escape the situation is to recognize the paradox (the
psychotherapist usually helps the patient with such a recognition), which means
to become aware of the impossibility of establishing the consistency of the
series of communicational messages just by choosing an alternative response.
Such a realization initiates a reductio
ad absurdum line of reasoning (Haley, 1955), which facilitates a
transgression of this communication level, and the arrival at an inference
that, given the established rules, a
non-contradictory response is impossible, and therefore the “rules” must be
changed. Paul Watzlawick, John Weakland, and Richard Fish (1974)—a research
group from The Mental Research Institute in
A key psychotherapeutic
technique which is thought to facilitate “bisociation” and second-order change
is termed reframing which, according
to Watzlawick, Weakland, & Fish (1974), means “to change the conceptual
and/or emotional setting or viewpoint in relation to which a situation is
experienced and to place it in another frame which fits the ‘facts’ of the same
concrete situation equally well or even better, and thereby changes its entire
meaning” (p.95). Understood in this way, second-order change does not change a
system directly, but rather alters the concept of the system, which, in turn,
entails a system change. Reframing could also be described as the modification
of a concept by assigning it to a different class of members. The change in class membership introduces a
whole new set of conceptual relationships, a process somewhat similar to the
“ontological shift” which, according to Michelene Chi (1997), occurs when a
concept “changes the ontological tree to which [it] belongs” (p. 220).
Therapeutic interventions in pathological communication
usually aim at second-order change, and because the interventions themselves
employ paradox and are based on dialectical strategies, are often referred to
as “paradoxical psychotherapy” (Mozdzierz, Macchitelli, & Lisiecki, 1976).
Alfred Adler (1956) is known as the first Western psychotherapist to have
utilized—influenced by Hegelian dialectical thinking—paradoxical strategies in
the interest of behavioral change. He
was preceded, however, by the centuries’ old tradition of Zen Buddhist
philosophy in
Joshu asked the teacher
Nansen, “What is the true Way?”
Nansen
answered, “Everyday way is the true Way.”
Joshu asked,
“Can I study it?”
Nansen
answered, “The more you study, the further from the Way.”
Joshu asked, “If
I don’t study it, how can I know it?”
Nansen
answered, “The Way does not belong to things seen: nor to things unseen. It
does not belong to things known: nor to things unknown. Do not seek it, study
it, or name it. To find yourself on it, open yourself wide as the sky.” (Reps,
1970, p. 105)
The koan, it would
seem, initiates a form of deliberation in the seeker reminiscent of the
Hegelian oscillation between thesis and antithesis, and a leap—described by
Daisetz Suzuki (1973) as a result of having “exhausted everything belonging to
his intellect or his conscious deliberation” (p. 222)—into synthesis. Koans force one into a double bind, and
trigger transcendence toward deeper meanings beyond words, and beyond
object-thinking.
Shuzan
held out his short staff and said: “If you call this a short staff, you oppose
its reality. If you do not call it a short staff, you ignore the fact. Now what
do you wish to call this?”
Mumon’ commentary[6]: If
you call this a short staff, you oppose its reality. If you do not call it a
short staff, you ignore the fact. It cannot be expressed with words and it
cannot be expressed without words. (Reps, 1970, p. 124)
For Zen, truth is of a higher level of abstraction than words,
and can only be achieved through transcending the dualism which thinking
necessarily implies. Zen does not value abstraction or conceptualization, for
any conceptualization implies a division into categories and
compartmentalization. In fact, thinking and even perception itself are bound to
dualism, for conceiving or even perceiving an object means to set boundaries
between what the object is and what the object is not—that is to resort to the
standard logic of the excluded middle[7]. Contrary to this kind of
Western thinking, Zen preaches the transcendence of dualism through the
attainment of an imageless, speechless, holistic state of being (Suzuki, 1959).
Zen philosophy implies
understanding the world as a whole, without breaking it into pieces, and
without making of it an object of thought—a concept that might cause a great
deal of anxiety for the Cartesian mind. Words are
not trustworthy tools in the search for meaning and truth, and yet they are all
we have, for as the koan
says, “It cannot be expressed with words and it cannot be expressed without
words.” And here we arrive at a paradoxical statement, which leaves the Zen
devotee in a double bind. The role of the koan is to set up paradoxical situations like the one above in
an attempt to obviate the limits of conceptual thought and verbalization. It is
only by entering a state of bewilderment that one’s mind can break free and
make a leap to enlightenment. The leap is made, not as a result of a choice of
“yes” or “no,” but through the suspension of choice and the contemplation of
both sides of the paradox, which makes the transgression of its boundaries
possible. Like Double Bind theory, it requires “stepping
out” of the system, thereby changing the concept of the system and consequently
the system itself.
Gregory Bateson’s Theory of
Learning
Bateson (1972) associates cognitive change with learning, and
he recognizes ahierarchical ordering of learning. What he calls Learning I is
associated with habituation. Rote learning can be attributed to this
category: in this case, to learn is to
link a certain response with a specific stimulus/question/problem. The
classical example of Learning I is the Pavlovian type of conditioned response
based on instrumental reward. An instrumental “trial-and-error” problem solving
process, where a revision of choice is made within a set of possible answers,
could also be attributed to the same category. Multiple choice tests often call
for a simple trial and error check in order to find out which of the suggested
answers works.
Learning II, Bateson
holds, requires contextual evaluation of the problematic situation whereby a
response is chosen from an altered set of alternatives. This, according to
Bateson’s theory, represents higher-order learning than Learning I, and implies
the ability to reinterpret the context. Such a reinterpretation also allows for
a different response chosen from among a new set of alternatives--but although
a reinterpretation of the context can cause a change within the system, it
won’t change the system itself. On the other hand, Learning III is connected
with what has already been referred to as a second-order change, i.e. with the
change in the fundamentals of the system itself. This even higher-order learning than Learning
II Bateson calls “learning to learn,” which he regards as the highest category
of human learning, offering as it does the possibility of transcending double
bind situations and reaching dialectical synthesis. Each type of learning is
characterized by a certain degree of flexibility—that is, freedom from habitual
reactions–and the higher the category of learning, the greater the flexibility.
As we have seen, Bateson suggests
that the double bind is encountered in the face of a contradiction which by
definition is unresolvable, and forces what he calls a process of “learning to
learn,” which is realized through a complete reorganization of the
understanding of the context of the problem, and consequently a profound change
in the system or theory used to interpret it.
It appears to resemble a Hegelian synthesis or “sublation,” although it
is not interpreted by Bateson in this way. Learning III or “learning to learn”
seems to be a rare phenomenon, which coincides with profound emotional
realization, deep cognitive restructuring, and extensive meta-analysis. The
conditions that make Learning III possible resemble the koans practiced in the spiritual tradition of Zen Buddhism and
the “double bind” situations created in a homeostatic systems that are approached by
Paradoxical psychotherapy.
There appear to be parallels between
overcoming the barrier posed by Zen koans and the transcendence of the double
bind, or what, according to Bateson (1972), makes Learning III attainable.
Although the metalevel of Learning III-- to our Western minds—appears to be
“thinking about thinking,” and satori,
according to the Zen masters, is a state of not thinking and merging with the
universe, both experiences are end products of a chain which begins with
paradoxes of some kind that trigger a double bind situation, lead one to
recognize the double bind, and, if successful, trigger a transcendental leap
out of it. Analogously, practitioners of the paradigm of paradoxical therapy
hold that only after the realization of a “double bind” situation that
confronts them with an untenable absurdity are patients able to break out of a
self-perpetuating, pathological and homeostatic system by changing the system
and its rules (Weeks & L’abate, 1982; Watzlawick, Beavin, & Jackson,
1967). Similarly, the psychotherapist Hans Sachs, a colleague of Freud’s,
stated that as a general rule an analysis ends when the patient realizes that
it could continue endlessly (Watzlawick et al., 1967). What seems compelling about paradoxical
psychotherapy is that what is available as an immediate choice is the habitual
response that maintains the homeostasis, and what breaks it is a completely new
possibility born from the awakened mind.
The way out of a double-bind is not within the set of
possibilities offered within the system, but beyond it. It comes with an
awakening, which is an act of undoing, reversing, discontinuation, or
unlearning. It would appear that this undoing mode is only a precursor of
enlightenment, which is a continuous experience of the unity of subject and
object, and this indeed is a new form of knowledge. The process at least evokes the dialectical
spiral characteristic of Hegelian notions of development. Awakening is the
undoing, or the negation of the products of previous learning, whereby—in the
unity of both the previous learning and its negation, the synthesis, a new form
of learning, is born. The new form is qualitatively different, in that it both
retains aspects of the previous form and negates it at the same time.
Implications for Teaching and
Learning with Paradox in a Community of Mathematical Inquiry
What implications for mathematics education can we draw from
Bateson’s theory of learning and the two paradigmatic modes of learning
experiences encountered in Paradoxical therapy and Zen Buddhist spiritual
training? Bateson’s conception of types of learning has obvious implications
for teaching and learning understood as problem-solving. Mathematics education is still notorious for
perpetuating Learning I—rote learning accompanied by a simplistic, instrumental
approach to problem-solving. Learning II and III are closely connected, as we
have already seen, with the concept of change—whether systemic and/or personal
–and therefore with the Hegelian ideal of mental development—or bildung. Both types of learning are
associated with an encounter of inadequacies, and conceptual change and
transformation as the outcome of overcoming these inadequacies. The latter are,
if we assume a dialectical perspective, inherent in every dialectical system,
whether communicative, cognitive, social, or some combination of the three. Learning,
then, is a characteristic and a dimension of these systems in all their
combinations, acting both as an adaptive mechanism and as an agent of system
change. I want to suggest that the
distinctive problematic of the paradox, which represents inadequacies
(contradiction in its “purest” form), when approached in a dialogical and
therefore dialectical pedagogical form—that is “Community of inquiry”—may act
in a powerful way to support Learning II and III.
Community of Inquiry
Community of inquiry as understood in this study may be
broadly and simply described as the collective execution of a dialogical,
language-based activity whose goal is to reach communal agreement through
argumentation. The model of community of mathematical inquiry (henceforth CMI)
so conceived is adapted from the model of community of philosophical inquiry
developed by Mathew Lipman and Ann Sharp in the 1970’s at the Institute for the
Advancement of Philosophy for Children at Montclair State University in New
Jersey, which is the pedagogical basis for the Philosophy for Children
curriculum (Lipman, Sharp & Oscanyan, 1980; Lipman, 1991). One main
objective of both is an emphasis on the construction of meaning and the
formation of concepts, not through transmission, solitary reflection or debate,
but through what is referred to as “building on each other’s ideas”— that is
through distributed thinking in a dialogical context (Kennedy, 1999). Communal
inquiry is understood to advance through reasoning, or “the giving of reasons,”
and each reasoning move ideally represents a reconstruction of the inquiring
system in the direction of a more sophisticated conception. The ideal inquiry
proceeds through a form of argumentation which is inherently dialogical and
thus by implication a dialectical process, which is to say a process which
moves forward through encountering and attempting to resolve
tensions/inadequacies/contradictions. In the
inquiring system as such, any given argument is built on or as a counter
argument to a previous one.
In an open, communicative system such as CMI, it is deemed
important that students fully orient to the problem from the very beginning, so
that they understood themselves as primary agents and “owners” of the process
(Gal’perin, 1980). Further, it is considered essential that the goals toward
which the activity is directed be negotiated collectively, and thereby at least
putatively accepted by each individual member (Davydov & Markova, 1983).
Such a negotiation represents one dimension of an implicit contract between
facilitator and students, which is based on the mutual understanding that each
individual is viewed as self-regulating subject who is responsible for her own
learning process very much in the way patients in paradoxical therapy and Zen
students come to understand themselves. Another dimension of the same contract
is the implicit agreement that the facilitator’s role is to support the
students in advancing with the activity only if they need help—that is, the
facilitator is expected to operate within the students’ zone of proximal
development, but not within their zone of actual development (Vygotsky, 1962).
Such support proceeds without providing direct answers or authoritative
perspectives, but more through a form of the Socratic elenchus—that is, through provocative questioning, reformulation,
and the offering of counter-examples and counter-perspectives. In fact the
constructive process can be influenced
by any single element of the system—for example by any single participant—as
well as by any element in the cognitive medium, for example the initial problem
under consideration, specific examples and counterexamples, or by the presence
of conscious or unconscious assumptions. The chief pedagogical significance of
this process of CMI is that it operates in the collective zone of proximal
development, which acts to “scaffold” concepts, skills and dispositions for
each individual. The scaffolding process functions through subprocesses such as
clarification, reformulation, summarization, and explanation, as well as
through challenge and disagreement.
When learning situations in CMI are structured around problems
which are broadly described as paradoxes, they offer a strong element of cognitive
surprise and, given their succinctness, a well-defined starting point for
discussion. Here the term paradox is loosely employed to denote the general
class of problems which hide either a real or a fictitious contradiction, or
“paradoxical problems” which offer a surprise that runs contrary to the
assumptions and inferences that we spontaneously subscribe to on first reading.
The latter we would denote as Type I paradoxes, and the former would be called
Type II paradoxes. An example of such a problem is: A clock strikes 6 times in
5 seconds. How long would it take to
strike 12 times?
The idea that the clock
would strike 12 times in 10 seconds is highly plausible, but not correct. Another competitor for a solution is the
supposition that the clock would strike 12 times in 11 seconds. This situation
is in fact characteristic of the entire group of Type I paradoxes—they provoke
at least two competing contrary inferences[8] that
immediately present a cognitive conflict, which provides a natural context for
argumentation.
Even a problem as simple as this one--A bottle and its cork cost $ 1.10. The bottle costs $1 more
than the cork. How much does the bottle cost?—usually provokes several contrary
propositions which presents the group with a “forked-road” situation that calls
for a review of the relevant (to the conflicting
propositions) set of beliefs, assumptions or premises, for reflecting on the
unwarranted ones, and for “weeding out” the incorrect ones (Quine & Ullian,
1978, p. 18). Thus,
the problem situation has the potential of triggering a modification of the
whole web of beliefs, assumptions, or premises into a more adequate one.
The Type I paradox is expected to
structure argumentation so as to allow for the emergence and juxtaposition of
contrary propositions/interpretations, which in turn create an urgent felt need
for re-evaluation. The exigency of this re-evaluation has been already
explained as one result of the experience of cognitive dissonance, and the felt
frustration which results in a disruption of an expected consistency (Jecker,
1964; Festinger, 1964). As has already been pointed out, the phenomenon of
juxtaposition of contrary interpretive frames has been studied by Köestler in
his work on creativity (1969), where he introduced the concept of bisociation, which he defines as “ …the
perceiving of a situation or idea, L, in
two self-consistent but habitually incompatible frames of reference. . .”
(Köestler, 1969, p. 35). Thus any
changes associated with Type I paradoxes can be attributed to a change in the
subjectively used interpretative framework for grasping the context of the
problem, which is necessarily associated with a change in underlying
assumptions and beliefs.
In comparison to Type I, Type II
paradoxes provoke contradictory propositions and contradictory interpretive
frameworks for the problem. In order not to complicate the matter, I will not
differentiate between the Type II paradox and the antinomy. The latter, an
example of which is the Liar paradox, creates a Batesonian “double bind.” Another example would be Cantor’s paradox,
which can be offered in the following version: Let’s consider the following
infinite sets of numbers: [ 1, 2, 3, 4, 5, 6, . . . ] and [ 2, 4, 6, . . . ]. Do
they have equal or a different number of elements? If you think the latter is
the case, which set has more elements? In fact these were two paradoxes
triggered a crisis in the foundations of mathematics, and caused a series of
modifications and innovations in set theory and classical logic. So
historically, they did create untenable situations of “double bind” variety,
which brought about major structural changes in the system. Mathematical
paradoxes, or what we termed Type II paradoxes, are perfect demonstrations of
“borderline cases” that can’t be resolved within the formal system, and thus
call implicitly for reorganization of the whole system of mathematical
knowledge.
It should be remembered that the
Type II paradox is a necessary but not a sufficient condition for Learning III,
since it requires cognitive maturity and knowledge; but it at least creates
conditions for meta-analysis, and for deliberation about restructuring the set
of interpretive frames under consideration. In fact my own research so far
(Kennedy, 2005) suggests that communal discussions in which paradoxes are
deconstructed in an attempt to restore the consistency broken by the appearance
of a logical contradiction echo the historical struggle of mathematicians in
their efforts to salvage the foundations of mathematics after they had been
seriously shaken by various paradoxes. And such discussions also implicitly
challenge the old belief in “certainty,” and the idea that a math statement has
a “stamp of incontestability” (Wittgenstein, 1969, par. 655). As a pedagogical
device, paradox is useful, not as a means of devaluing standard logic and
logical reasoning, but to demonstrate the limits of standard logic, and to open
students to alternatives. The benefits of perspectival flexibility are clear
enough—if one need not conform to one particular method or mode of reasoning
because it is the accepted standard, he or she is in a position to make a
better-informed decision after critical examination of the alternatives. The
possibility of cultivating such flexibility means, of course, the
transformation of our understanding of teaching from a matter of delivering
facts and/or “the true method”—whatever that happens to be—to understanding it
as the business of conducting critical inquiry within a spectrum of methods and
dimensions of reasoning.
Cognitive scientists Francisco Varela, Evan Thompson &
Eleanor Rosch (1999) integrate Buddhist phenomenology and Western cognitive
science in their development of the idea of “embodied mind.” A central aspect
of their description of this form of cognition is mindful, open-ended reflection, one which is not just on experience, but is a form of experience itself—and such A reflective form of
experience precludes habitual thought patterns and responses based on
preconception. The goals of being mindful of one’s own mental processes
expressed in the notion of “embodied mind” and of “enlightenment” in Zen
Buddhism are progressively reached, according to their proponents, through
observation and consequent unexpected discovery—or to put it in more secular
terms, through systematic doubt and inquiry. Contrary to Köestler (1969), who
sees the phenomenon of “bisociation” as unpredictable and rare, a phenomenon so
elusive that to “manage” it is inconceivable, Zen doctrine claims that the
cultivation of paradox, combined with meditation, can not only confound the habit of discursive thought and shock the
mind into awareness, but also alter the habitual mind-body relation. The mystery of the koan appears as a
precondition--much like the untenability of the personal paradox in paradoxical
psychotherapy, or the radical puzzlement encountered in classroom inquiry into paradoxes—which
compels students to reduce (but not necessarily to eliminate) uncertainties, a
phenomenon which, on these accounts, is associated with awakening and insight,
and with the consideration of possibilities
which may not otherwise have been conceived of.
Conclusion
This paper explores the effects of paradox and the mediation
of cognitive conflict it creates in a variety of topoi, including double-bind theory in psychotherapy and
learning—in particular Bateson’s three-tiered theory of learning—and the role
of the koan in Zen Buddhist practice. It argues that paradox—which plays a
major role in all of these forms of theory and practice, promises as well to
support and stimulate the conditions for higher order learning in the
mathematics classroom. The study has
concerned to justify the use of paradoxes quite specifically in the pedagogical context of a
community of mathematical inquiry—a discursive location in which participants
are invited to justify their positions and to examine their claims in the light
of the claims and positions of others, and to balance both justification and
negotiation processes in a sort of “third way.”
The
experience of being exposed to radical contradiction can lead to significant
restructuring of participants’ understanding through the alteration of
reasoning frameworks, and result in learning which is of a higher order than
rote or instrumental. Understood as the unavoidable self-presentation of
opposing ideas, of cognitive confusion, and in its ultimate form as a
contradictory statement, paradox offers an ideal didactic context for
open-ended group discussion, for the intensive practice of reasoning, acquiring
dispositions critical for mathematical thinking, and learning associated with
cognitive change.
Community of inquiry pedagogy
suggests the benefits of a form of practice in mathematics education
significantly different from the traditional teaching and learning paradigm. It
takes the notion of distributed learning and thinking with the utmost
seriousness, which amounts to the epistemological claim that knowledge
constructed in an inquiring system—a group whose chosen activity is carried out
through collaborative, dialogical deliberation—has qualitative differences from
knowledge attained individually, or even as a result of a dyadic
interaction. CMI and its forms of
learning demand a form of pedagogy informed by positive humanistic belief that
unites theory and practice, philosophy and application, argumentation and
calculation, conflict and its mediation in the concrete, problem-based context
of the classroom. The application of
this learning and teaching model poses a profound challenge to mathematics education,
given both the nature of the discipline and the pedagogical traditions which
still dominate it, but it is capable of developing into a form of classroom
practice which has the potential of transforming the field.
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