a journal of the international council of
philosophical inquiry with children
Philosophical
Methodology and the Mathematization of Pedagogy
Freeing
Children’s Imagination through Philosophy
John Roemischer
Socrates:
Do we not know that all this is no more
than an introduction to the
main theme which has yet to be
learnt? Surely you would not regard experts
in mathematics as masters of
dialectic?
Glaucon: Certainly not, except for a few of those I
have met.
Socrates: Well, can the knowledge we are demanding ever
be attained by
people who cannot give a
rational account of their statements or make others
give an account of theirs? (Plato.
Republic, vii.532.)
At Trinity and later, in a wide
experience of educational problems at
precisely the kind of education that
Plato approves. Mathematics must be
studied; philosophy should be
discussed. (Alfred North Whitehead.
Dialogues.)
Philosophy for Children and the controversy
over philosophic method:
Pedagogy
in Western society is rooted in a notion which evolved largely in the 17th
century as a reaction to Plato’s
attempt to strictly distinguish philosophic and
mathematical methodologies and,
at the same time, to the flamboyant speculations of
Medieval Scholasticism. This reaction, most clearly formulated in
Descartes’
Discourse on Method,*
made mathematical methodology the supreme
method of inquiry
________________________________________________________________
*The inspiration for the Discourse, as Descartes notes, is a
complete dissatisfaction with his university education; it is largely as
remediation for the misleading tendencies of the educational currents of his
time that moves him to write. Concerning the difference between mathematics and
philosophy, Descartes clearly prefers mathematical methodology as the vehicle
for determining truth: “…in Mathematics there are the subtlest discoveries and
inventions which may accomplish much, both in satisfying the curious, and in
furthering all the arts, and in diminishing man’s labour….Philosophy teaches us
to speak with an appearance of truth on all things, and causes us to be admired
by the less learned.” More specifically:
“I was delighted with Mathematics because of the certainty of its
demonstrations and the evidence of its reasoning…” In Philosophy, though
cultivated by the best minds, “no single thing is to be found in it which is
not subject of dispute, and in consequence which is not dubious….And also,
considering how many conflicting opinions there may be regarding the self-same
matter, all supported by learned people, while there can never be more than one
which is true, I esteemed as well-nigh false all that only went as far as being
probable.” (Descartes, 5ff.)
and, in the centuries that followed, the
litmus test for determining what constitutes acceptable academic subject
matter; the sufficient reason provided for this claim was that mathematical
methodology is simply the best vehicle for engaging and developing thinking;
however, in the years that followed, competence in such thinking required children to be subjected to a long stretch
of imitative learning—what Thomas O’Brien has referred to as “parrot math.”
Mathematics, long thought of as providing the model intellectual methodology, has
left generations of children with the sense that the intellectual life consists
of an endless expanse of arid rules and exercises; the free exercise of
imagination was expendable within the school context. The oft’ experienced
imbalance, within the context of schooling, between tendencies toward fanatical
resistance and erratically expressed freedom, when examined, could be traced
back to the mathematization of school subjects—except, of course, physical and
social play which has always served as the relief valve for many children.
There are obviously advanced areas of knowledge for which mathematics is
an indispensable tool, but that is generally not the reason contemporary
institutions of higher education require entrance examinations which consist in
large part of exercises in mathematics; a majority of students taking those
examinations do not move in the direction of the sciences. The main reason,
notwithstanding Plato’s reservations, is that such tests are supposed to
determine the capacity of candidates to engage in critical and abstract
thinking. One may note however that, with the exception of Descartes’ work in
analytic geometry and his invention of the Cartesian system of coordinates, and
Spinoza’s geometrized Ethics (though Spinoza is remembered more for his
philosophic pantheism than for his use of the geometric method), celebrated
philosophers have generally not been contributors to mathematics—an observation
which would not have surprised Descartes. It is in the last section of Kant’s
Critique of Pure Reason that we get a concentrated effort to provide an
explanation for this remarkable fact.
Though
Plato gave geometry a place of importance in the curriculum designed for his
Republic, its dependence both on propositional knowledge* and empirical content
left it far behind as a vehicle for the difficult “ascent” (aporia) to substantive Formal Knowledge.
Plato’s characterization of Philosophy as the infusion of “perfect freedom”
into thinking process, a process that is broad enough to make possible the
non-dialectical logic of mathematics, gave dialogical/dialectical thinking a
distinctive property: A
self-corrective capacity which allows for objectives that exceed the more
restrictive boundaries of mathematical thought. Philosophy, in essence, cannot
be formulaic.
Dialogical/dialectical thinking, thinking with “perfect freedom,” for
Plato, could not simply be an application of the methods of the geometers to
all questions of knowledge, notwithstanding the fact that, as methods of
thinking, there would have to be some connection between these fields. In the
18th century, Kant argued that “freedom” is a transcendental condition
for thought in general, and he did this precisely to free thought
from its mathematical constraints.
“Noumenal freedom,” a freedom tied up with the
___________________________________________
*Plato’s interest in “non-propositional knowledge” is explored
in F.J. Gonzalez, Dialectic and
Dialogue: Plato’s Practice of Philosophical Inquiry. Plato’s elaboration of the inadequacies of mathematical
methodology for philosophic inquiry is discussed at length in Collingwood’s Philosophical Method. Plato finds that
mathematical method falls short of philosophy’s interest in “idealization;” its
defects are not open to “remediation;” it fails at “epistemic understanding of
its objects;” and it depends on “images.” Referring to Plato’s famous allegory,
Gonzalez writes: “Those who have been outside the Cave and have returned are
said to have gained, not proofs nor an axiomatic system, but infinitely greater
understanding (vision) of the things within the Cave…” (Gonzalez, 377n 96)
“transcendental (ego) subject,” made
freedom a constitutive part of all thinking, and unfettered moral deliberation a possibility.
Writing on Kant,
…we have a sufficient
ground when we attribute a man’s use of A rather than
B to his use of freedom, even though
we then have no determining or
epistemic ground. When we
know that a man freely chose to do A rather than
B, though he could
have chosen either, we have an entirely adequate
explanation of his action,
which imports lucidity, not confusion, into our view
of the world. This very
illuminating conception of the spontaneous as a
species, not a violation, of
causality, is of course one that Kant employed in
his conception of noumenal
freedom, even though the conceptual terror
inspired in him by Newtonian
physics made it impossible for him to hold to it
in his conceptions of
phenomenal nature.
Continuing his statement,
But a law is senseless
unless it can be implemented, which brings in the
freedom of the elective
will. Beings endowed with a will cannot but think of
themselves as able to
implement its policies, and also by default, not to do so.
The possibility of such freedom cannot,
however, be illustrated phenomenally,
and hence not known
by us to exist. (
Whether what we have in Kant’s
transcendental idealism is a set of a
priori constitutive conditions for granting the availability of philosophic
thought to children depends on the extent to which a child’s cognition is
subject to the same transcendental freedom and “unity of consciousness” which
Kant grants to all agents of knowledge. And if it can be demonstrated that imaginative
thinking requires those same conditions, children may, a fortiori, enter the world of reasoned discourse. In his Way to Wisdom, Karl Jaspers provides an
empirical confirmation of what Kant could easily explain on transcendental grounds: “A marvelous indication of man’s innate
disposition to philosophy is to be found in the questions asked by children. It
is not uncommon to hear from the mouths children words which penetrate to the
very depths of philosophy.”
Mathematical
method and the advent of pedagogical “reductionism”:
Descartes’ 17th century reaction to the flamboyant
dialectical adventures of the Medieval Schoolmen set the basis for the
mathematization of all major fields of knowledge, and, hence, of all “modern”
curricula, pedagogy, and pedagogic training. In his work on philosophical
method, Collingwood attempts to trace this progression historically. He cites
Plato’s interest in mathematics as a vital field of study and indicates that
“Socrates had found in mathematics a model for dialectical reasoning,” but he
nevertheless argues that Plato’s
…theory
of method must be admitted defective through failure to drive deep
enough the distinction
established by himself between philosophy and
mathematics. The result is that
he splits philosophy into two parts: one an arid
waste of ingenious logic-chopping,
the other an intuitive vision of ultimate reality.
Descartes, disgusted
with the dialectic of the [Medieval] schools, went back to
the same model, and described
the lessons he learnt there under four heads: the
canons of evidence, division, order,
and exhaustion. Nothing was to be assented
to, unless evidently known to be
true; every subject-matter was to be divided
into the smallest parts, each
to be dealt with separately; each part was to be
considered in its right order,
the simplest first; and no part was to be omitted in
reviewing the whole.
It was from the study of
mathematics that Descartes learnt these rules, and it was
to the advancement of mathematics
that he first applied them; but he hoped from
the first that they would prove useful in a
far wider sphere, and by degrees he
applied them to the whole field
of knowledge as he conceived it: that is, not only
to mathematics but to metaphysics
and the sciences of nature; for divinity he
ruled out as a matter of faith,
poetry he considered a gift rather than a fruit of
study, and history he regarded as
a pastime full of interest and not devoid of
profit, but very far from the
dignity or utility of a science. (Collingwood, 1933, 16ff.
Italics added.)
Plato’s critical treatment of the relationship of philosophy to mathematics
was lost in later efforts to restrict the unbridled freedom implicit in
dialectical thinking. Descartes, in need of a way to make mathematics
methodologically applicable to knowledge determinations in general, found in
the Scholasticism that he criticized, a procedure which, largely due to his
influence, was soon identified as “mathematical methodology.” This method (outlined
above by Collingwood), largely because it was a step-by-step procedure, had an
enormous impact on educational theory and pedagogy. Dialectical adventures into
“possible worlds” of the imagination, promoted by some of the great
philosophers, yielded to the promotion of sets of reductive skills. This
reductionism, which seemed to square with
The
Progressive response to Cartesian methodology consists of a general critique of
the impact of “rationalism” on education: Dewey’s critique of Descartes’
dualism is clearly applied to pedagogy in Democracy
and Education; applied appropriately, from the standpoint developed here, in
his treatment of “The Nature of Method”:
…under the influence of the
conception of the separation of mind and material, method tends
to be reduced to a cut and dried
routine, to following mechanically prescribed steps. No one
can tell in how many schoolrooms children reciting in
arithmetic or grammar are compelled
to go through, under the alleged
sanction of method, certain preordained verbal formulae.
Instead of being encouraged to
attack their topics directly, experimenting with methods that
seem promising and learning to
discriminate by the consequences that accrue, it is assumed
that there is one fixed method to be
followed….Mechanical rigid woodenness is an
inevitable corollary of any theory which
separates mind from activity motivated by a purpose.
(Dewey,
Following Dewey, in a contemporary critique
of reductionist approaches to teaching literacy, Kieran Egan argues that the
typical pedagogical “method” used in the teaching of reading sacrifices
“imagination” on the altar of mechanistic coding and decoding processes:
Crude literacy tests often miss
the subtle problem which literacy has left us. They count
as unqualified successes many
cases where students can manage the coding and decoding
skills that open the big front
door of literacy’s storehouse without being equipped to go
into the further rooms where its
great delights and power are accessible. (Egan, xv.)
Citing recent international literacy tests
as an example, Egan points out that the
British attempt to drive up literacy scores has not affected the “voluntary
reading” rates of British children. “Thus many countries that now score worse
than
Matthew Lipman’s important contribution to this discussion involves the
elaboration of the notion that children’s “reasoning” is precisely what’s
missing in contemporary literacy programs. The Philosophy for Children
curriculum is offered as a remedy. “When we try to teach children to read,”
Lipman states, “we tend to overlook how mechanical are our techniques, such as
those that stress grammar and phonics, and how close these techniques are to
what actually blocks the children from reading.” (Lipman, 101.)
Descartes’
effort to impose a “Scholasticized,” reductive method of mathematization on
knowledge determinations, when applied to pedagogy, had several lasting
results: of all of these results, one stands out as most influential in modern
pedagogy, viz., the conceptual confusion of “concreteness” with “simplicity” which
infected the teaching-learning process with the cognitive-sensory clutter that
has served to limit the intellectual
freedom which children exhibit when exposed
to dialogue. The corollary notion that “abstraction” is synonymous with
“complexity,” and that all pedagogy should therefore move from the simple to
the complex, that is from the concrete to the abstract, is precisely the reductionist
method which is defied by the dialogical thought of children. A contemporary
Philosophy for Children curriculum, by returning to Plato’s notion that
dialectic has larger purposes than mathematical methodology can support, could
be a significant remedy for the mathematization of pedagogy which pervades
contemporary education and learning theory. The fight is simply against a
pedagogy conceived and carried out as a process of “machine” construction and
implementation, a process which leads to the fragmentation of both knowledge
and the learning experiences of children.
Thus,
notwithstanding John Dewey’s criticism of the commonplace tendency in
contemporary pedagogy to make the knowledge relation ubiquitous by equating
concepts with perceptions, contemporary pedagogic texts tend to claim
that “concepts” so generated have the sort of “apodicticity” generally reserved
for mathematical intuitions. The result comes in the form of a reductive
concept of “experience,” a “naïve
empiricism,” and the vague claim that experience
is veridical, that “experience teaches”—and teaches correctly!
The pedagogical outcome of this ironic effort to crystallize what is
considered “experiential” and empirical is the reduction of children’s thinking
to a meager, but endless, diet of “simplistic, fragmented materials” from which
children supposedly construct “clear and distinct ideas.” Turning sense data into
“axiom pillars” in the process of concept formation, an ironic
“rationalization” of empiricism, is most notoriously evident in such 19th
century pedagogic methods as Herbartianism, a not-so-subtle attempt to
mathematize teaching methodology along Cartesian lines.*
Since much instructional
methodology, including mathematics instruction, revels in this confusion of
“concreteness” with “simplicity,” the result is an “additive” approach to
learning which provides children with the mere end products of knowledge and
makes knowledge a product of learning rather than learning a by-product of
knowledge determinations. Teachers who engage children in dialogue soon notice
that, for children, the experience of “concreteness” is actually an experience
of the enormous “complexity” of the world around them. Their curiosity seems
insatiable. Their job is much more one of interpretation of the world which
surrounds them than the mechanical construction of concepts from simplified
experiential intuitions—from sensory “axioms.” Dewey’s attempt to connect
concept development to hypothesis formation (How We Think) was an
_____________________________________________________
* “Principle XXX” in
Descartes’ “The Principles of
Philosophy” announces that “…all that we perceive clearly is true…” Our
God-given “faculty of knowledge…can never disclose to us any object which is not true…. inasmuch
as it apprehends it clearly and distinctly.” (Descartes, 270, et passim.)
attempt to fight this mathematized version
of empiricist epistemology. What is being characterized here as the negative
impact of the mathematization of pedagogy is underscored in Collingwood’s
attempt to specify what is limited and restrictive in mathematical methodology:
Mathematics and dialectic are so far alike
that each begins with an hypothesis: ‘Let so-and-so
be assumed.’ But in mathematics the hypothesis
forms a barrier to all further thought in that
direction: the rules of mathematical
method do not allow us to ask ‘Is this assumption true? Let
us see what would follow if it were not.’ Hence mathematics, although intellectual, is
not
intellectual a outrance; it is a way of thinking, but
it is also a way of refusing to think. In
dialectic we not only draw
the consequences of our hypotheses, but we recollect that they are
only hypotheses; that is, we are free to ‘cancel
the hypothesis’, or assume the opposite and see
what follows from that. (Collingwood,
1933.
It is Kant, Collingwood points out, who tries to provide a corrective to
the Cartesian view of mathematical and philosophical methodologies. Kant’s work
provides insight into the characteristics of philosophical methodology, as
Collingwood notes; however, since
philosophy is defined as the “freedom to think,” Kant sees fit to merely
inventory the negative aspects of mathematization
that limit intellectual freedom. But Kant’s task is twofold: he
also needs to contain the capricious uses of Pure Reason; and if we add to this
his profound respect for phenomenal experience, he might earn the arguable
distinction of being one of the first philosophers to view philosophy itself as
a cure for extravagant intellectual “methodization.”
[Kant’s] aim is not so
much to controvert but rather to correct Descartes, by
a careful distinction
between philosophical and mathematical thinking. He
argues in detail that, of
the special marks of mathematical science, not one is
to be found in philosophy,
and the adoption of mathematical methods there
can do nothing but harm.
Philosophy knows no definitions; or rather, their
place in philosophy is not
at the beginning of an inquiry but at the end; for
we can philosophize
without them, and if this were not so we could not
philosophize at all. Philosophy knows no
axioms: no truths, there, are self-
evident, any two concepts
must be discursively connected by means of a
third. Philosophy knows no
demonstrations: its proofs are not
demonstrative but
acroamatic [i.e., esoteric—only for the initiated]; in other
words, the difference
between mathematical proof and philosophical is that
in the former you proceed
from point to point in a chain of grounds and
consequents, in the latter
you
must always be ready to go back and revise
your premises when errors,
undetected in them, reveal themselves in the
conclusion. (Collingwood, 1933.
22ff. Italics are added.)
If
Kant’s interpretation of methodology is being correctly characterized by
Collingwood, and if it can be empirically demonstrated that children can think
philosophically (as Matthew Lipman has attempted to do), then Piaget’s developmentally
restrictive notion that formal thinking requires a capacity to reverse
operations would have to be qualified so far as children thinking
philosophically is concerned: Reversibility
in mathematical thinking would be strictly linear when compared with the
“free-ranging” work of philosophy, a work caught up with the determination
rather than the elaboration of pre-suppositions; philosophic thinking would
serve to loosen the imagination to do
its formidable intellectual projections uninhibited by restrictive “stages of
development.”
But even Kant, in the end, due to his
involvement with “transcendental objects,” failed to carry forth the critical
work of the imagination into an elaboration of metaphysics. Nevertheless, as
Collingwood points out, “Kant solved rightly the problem which Plato had solved
wrongly, the problem of the methodological difference between philosophy and
mathematics, and so laid a firm foundation for all future inquiries into the
nature of philosophical method.” (Collingwood, 1933. 25.)
Kant’s final section of his Critique
of Pure Reason, “Transcendental Doctrine of Method,” is reserved for the study of methodology: The first section of
this intense analysis is the source of Collingwood’s discussion of the
differences between philosophic and mathematical methods. Perhaps the most
significant idea which Kant unfolds, an idea which strikes at the very heart of
the mathematization of pedagogy, is that philosophy must sacrifice the infallibility
which mathematics provides (Descartes, of course, agreed). Hardnosed Behaviorist
pedagogues, who, on the basis of “science,” claim the infallibility of
behavioral learning processes, are paradoxically governed by a philosophy of learning
which provides a rather mechanized version of the reflex arc concept. (Both for
Descartes and Hegel, the intrinsic connection between Mathematics and Mechanics
lends itself to the mechanization of whatever is colored by mathematical
methodology—i.e. wherever “quantification” reigns.) Technically speaking, a true Behaviorist, if
we applied Kant’s position, could never “teach” some of our most important
concepts, since such concepts forever lack the clarity and distinctness which
mathematical “definition” provides. Strict Behaviorists respond by counting
such concepts as “meaningless.” Kant recognized that the complex “meanings”
embedded in human knowledge, which are the subjects of philosophic study,
cannot be grasped as products of Descartes’ mathematized method. In a footnote to his discussion, Kant states:
“In mathematics definition belongs ad esse,
in philosophy ad melius esse. It is
desirable to attain an adequate definition, but often very difficult. The
jurists are still without a definition of their concept of right.” (Kant, 588a)
If
the mathematization of pedagogy is responsible for the stresses and heated
discussions of contemporary educators, it is because, as Kant noted, of its
exclusive reliance on “quantification.” Some educators, adopting the mathematical
method as the basic instrument of pedagogy, are subject to Kant’s criticism:
“Current, empirical rules, which they borrow from ordinary consciousness, they
treat as being axiomatic.” (Kant, 584.) And, paradoxically, because such
pedagogic perspectives are not truly grounded in the synthesis of concepts and
experience, contemporary educators are ironically prepared to axiomatize one base
after another when their views are subjected to empirical, qualitative
critique. The effect is the see-saw movements of pedagogy in the history of education.
Kant’s
theme can be applied broadly to contemporary issues: “Philosophical knowledge is the knowledge
gained by reason from concepts; mathematical knowledge is the knowledge
gained by reason from the [a priori] construction of concepts.”(Kant, 577.)
Conceivably, this accounts for the success which pedagogical “Constructivism”
has had in mathematics instruction. But it also refers back to the Socratic
realization that philosophy is best served by “community of inquiry” pedagogy,
since “knowledge gained by reason from concepts” requires the kind of dialectic
which demands broad confirmation Thus,
if philosophy is best served by a community-based pedagogy, so, a fortiori, would all other disciplines
which involve the elaboration of concepts. Teaching reading as a process of
phonic analysis, when contrasted with an approach which makes the quest for
“meaning” the core of the reading experience, can be characterized and
paraphrased through Kant’s language: “Meaningful reading involves knowledge
gained by reason from concepts. Mathematized reading involves skill gained by
reason from the construction of terms.”
Concerning the methodological difference between philosophy and
mathematics, though Collingwood recognizes Hegel’s debt to Kant on this matter,
he fails to highlight the rather strong position that Hegel himself takes. In
the rather extended “Preface” to his Phenomenology
of Spirit, first published in 1807, Hegel’s flamboyant language graphically
draws the distinction between these two methods; arguing that mathematical
method, concerned not with a dialectical involvement with actualities but with
things “merely as units,” ends up with “an unmodifiable and lifeless fixity”
(Findlay, 1958, 56, ). Thus speaking of mathematics, Hegel states:
The evident character of this defective cognition of which mathematics
is
proud, and on which it plumes itself before philosophy, rests solely on the
poverty of its purpose and
the defectiveness of its stuff, and is therefore of a
kind that philosophy must
spurn….The actual is not something
spatial, as it is
regarded in mathematics; with non-actual
things like the objects of
mathematics, neither concrete
sense-intuition nor philosophy has the least
concern. In a non-actual
element like this there is only a truth of the same sort,
i.e. rigid, dead
propositions. (Hegel,
The
rest of Hegel’s discussion consists of a brilliant demonstration of the
limitations of mathematics from the standpoint of cognition. A critique of the
most serious “proofs of propositions,” which Hegel briefly demonstrates, “would
be as noteworthy as it would be instructive, partly in order to strip
mathematics of these fine feathers, partly in order to point out its
limitations, and thus show the necessity for a different kind of knowledge.”
(Hegel, 27.)
Before
getting to mathematics as a topic in his work on Logic, Hegel
distinguishes three methods for “ascertaining truth” and indicates that each of
these methods is “no more than a form;” that is, in each instance mind must
bring something to that form in order to know truth: (1) experience, (2) reflection,
and (3) the pure form of thought are
the three methods available to man. Philosophic method, or “philosophic
cognition” as Hegel calls it, is that pure form of thought which brings perfect
freedom into inquiry. The first of these, “experience,” is the stage of
“human innocence;” the other two open an involvement with the binary stresses
which subject man to the level of the problematic in knowledge and morality.
(Hegel, 1873,
If we inquire with Hegel concerning our proximate interest, the impact
of giving priority to “the mathematical categories” on pedagogy overall, what
he offers in an esoteric passage in his Logic is a prescient insight, viz.
that mathematization introduces materialism:
and, indeed, it is the
overwhelming materialization in contemporary pedagogic methodology and its
resulting fragmentation of knowledge that is the bete noire in this discussion. Given Hegel’s notion that
mathematization gives priority to “quantification,” it is understandable that
he would view it as a limited “category of the Understanding” (using Kant’s
language).
If quantity is
not reached through the action of thought, but taken uncritically
from our generalized image
of it, we are liable to exaggerate the range of its
validity, or even to raise
it to the height of an absolute category….Our
knowledge would be in a
very awkward predicament if such objects as
freedom, law, morality, or
even God himself, because they cannot be
measured or calculated, or
expressed in a mathematical formula, were to be
reckoned beyond the reach
of exact knowledge, and we had to put up with a
vague generalized image of
them, leaving their details or particulars to the
pleasure of each
individual, to make out of them what he will…..And this
mere mathematical view,
which identifies with the Idea one of its special
stages, viz. quantity, is no other
than the principle of Materialism.
(Hegel, 1873,
Perhaps the most troublesome effect of the mathematization of pedagogy
has been the tendency on the part of educators to operate as if all pedagogical
problems have a solution—indeed, a single solution to the exclusion of all
other possibilities. This utopian approach to teaching seems to separate
pedagogy from all other practices—medical, political, legal. It is time for
pedagogy to adopt the first principle of all other practices, and that is,
above all, to “Do no harm.” The
commonplace labeling, sorting, and consigning of children, supported by a
reductionist, psychologized epistemology, suggests a profound neglect of this principle,
a neglect made evident by the subsumption of the human child to pedagogic
mechanization and bureaucratization. Seeing this as a methodological issue, an issue which reflects
the larger philosophic question raised here, calls
for a form of pedagogic practice which requires “reflective, philosophic practitioners, ” not mere scribblers of “My Philosophy of Education.”
Has reductionist
technology served to mathematize children’s play?
Is the contemporary child’s “wired play” having
a negative impact on children’s cognitive and social development? Has the loss
of the constructional play materials of the past (blocks, Erector Sets, model
airplanes, dolls and doll houses) become an impediment to the development of
the kind of imagination in children which, if Kant was correct, would support
the intellectual freedom required for “critical” thinking? Are children now
entering the adult world increasingly involved with passive enjoyment rather
than productive work? Having lost the “dialectical instrumentation” of
imaginative play, has play given way to linear reductionism and structurally
patterned thinking?
Psychology, long given to dialectical conflict, has come down on both
sides of this issue. There is the claim that the long stretches of constructive
imaging of traditional play, which put children into direct contact with their
physical and social worlds, has been lost in the world of TV and computers. But
does this notion, identified as the 19th century preference for the “protected child,” isolate
the child from the kind of play which the contemporary world, into which the
child must grow, prefers? Intellectually speaking, is a Philosophy for Children
curriculum, built on dialogical interchange, an anachronism? And is that the
reason why many teachers find such a curriculum irrelevant? Were the Greek rhetoricians, who were
attacked by Socrates on methodological grounds, correct after all—shouldn’t
teachers prepare children to win at the social games played in the world that
surrounds them? But this is not a
problem for mathematized methodology to resolve; it is, in the end, the “perennial
problem of philosophy.” Dewey’s “pedagogic creed” put it succinctly: the
conservatism of one period is the liberalism of an earlier period. The Erector
Sets of the mid-20th century were still home-bound and did not
activate the growing child more than the farm work or factory job of an earlier
period. The problem of philosophy is to guide each era to re-institute the
conditions for the development of dialogue between the child and his or her
world; so must it be for the electronic age. As philosophy has been
historically, so a Philosophy for Children curriculum can become the
overarching educational enterprise which takes up the challenges of each era
and connects them to the free and imaginative thinking of children.
In
the end, if Kant and Hegel are correct, it would not be surprising to hear some
disquieting thoughts of teachers concerning the matter under discussion: In the
general context of education, students of mathematics are not necessarily
better thinkers or writers than other students; furthermore, students who claim
that they were never really successful in their mathematics studies often turn
out to be good and productive thinkers in a variety of fields. Nevertheless,
from mechanical approaches to lesson plans and texts; from lectures to course
outlines; from the concentration on patterned and repetitive learning rather
than imaginative/dialogical thought; the reductionist formalization of
instructional process and learning practice has been the effect of a utopian mathematization
of pedagogy—a pervasive effort to provide teachers (and, ultimately, students)
with a uniform and universally applicable methodology for overriding all of the
“qualitative difficulties” inherent in so complex a practice as teaching,
learning, and intellectual problem-solving.
The historical reason for this contemporary utopian interest in
pedagogical reductionism; the reason that mathematization has given pedagogy an
air of certitude—a certitude based on what Hegel called “rigid, dead
propositions,” is simply because pedagogy has never given up the classical
interest in “completeness,” in what the
19th century referred to as
“complete education.” Because Plato could not achieve a vision of such
completeness; because the formal world of Ideas eluded him and could not
ultimately be completely captured and integrated with the experiences in the
Cave; for that reason alone Plato settled for an ongoing dialogical pedagogy
which celebrated the moral significance of the process itself—what Plato called
“protrepein” (Euthydemus). The
“protreptic” teacher needed to achieve the moral stance which was
characteristic of dialogical pedagogy—a communal “turning toward” truth which
was already infused with the Good and the True, even if incomplete. If
Descartes thought he could achieve “complete education” through the
mathematization of knowledge, complete in spite of the failure of the derided
Medieval schoolmen to do so, he ended with as many unanswered questions as did
the Socratic teacher who thought it was only possible to do so through
dialogical/dialectical inquiry. Though Dewey sought to resolve this issue,
which he inherited from 19th century philosophy, by the pedagogic
promotion of the more modest “complete
act of thought;” though Hegel fought to find a solution through a thoroughgoing
idealization of reality, and Marx through the materialization of Hegel’s
“notions,” contemporary teachers are still pursuing the utopian ideal of
complete education, but now through a mechanized reductionism, a naïve
confusion of “teaching” with “telling,”
knowledge with simplified sense experiences, meaning with reference, and
the “possible worlds” of the imagination with facticity.
By far, the most criticized tool of reductionist teaching is the modern
textbook: “But strange as it seems, these books just scratch the surface, and
that’s because they contain too much
material….In the drive to include everything, key ideas fade into the
background, or are never successfully communicated, or simply don’t stick with
students.” (Daniels and Zemelman, 39.)
Mathematical “Methodization”: The de facto
business of contemporary schooling:
It
needs to be noted, in passing, that Hegel’s Logic did not involve a
wholesale disparagement of mathematics or a rejection of the significance of
mathematical method. He simply wanted to point out that “quantity,” which is
the category of mathematical inquiry, is a “stage of the Idea” (a concept which
needs to be read in the context of Hegel’s “Idealism”). Its importance for the
natural as well as spiritual worlds is underscored in Hegel’s philosophy. But
“to seek all distinction and determinateness of objects merely in quantitative
considerations” is “in the interest of exact and thorough knowledge, one of the
most hurtful prejudices.” Contextualizing our knowledge of the world around us
provides us with the “qualitative” insights which serve our knowledge and our
capacity to individuate actualities (Hegel, 1873,188.). Notwithstanding this
Hegelian position, the contemporary pedagogic concentration on “skills”—even
skills of “critical thinking”—in isolation from practice has been the result of
a philosophic inattention to the qualitative difference between “practice” and
“exercise,” a difference swallowed up by an indifference to the mechanically
fixed or completion-driven condition of school mathematics which then serves as
the idealized model for all other fields of study. Furthermore, the
methodological significance of “counterinduction” and “unsupported hypotheses”
for the qualitative thought contained in fields of practice gets completely
overlooked. That indifference, when generalized to all forms of practice, is
responsible for that negative flavor of “abstractionism” which children often identify
with schooling. The seeming apodicticity of school learning is bought at the
expense of a serious awareness of the limitations of theoretical perspectives. It
is, therefore, not surprising that some scholar would, like Descartes himself,
once again pick up the weapon of intellectual anarchism, and once again find
that methodology is the culprit. Nothing could say this better than Paul
Feyerabend’s book title: Against Method,
which unfolds a critical examination of the place of method in the history of
philosophy and science, and which needs to be read as a corrective for pedagogy
as well as science and philosophy:
Not
only are facts and theories in constant disharmony, they are never as neatly
separated as everyone makes them
out to be. Methodological rules speak of
‘theories’, ‘observations’ and
‘experimental results’ as if these were well-defined
objects whose properties are
easy to evaluate and which are understood in the
same way by all scientists. (Feyerabend, 51)
Feyerabend’s critique of the general “quest
for certainty” is not only reminiscent of Dewey’s work, but it brings us back
to the beginning: Descartes’ Discourse on
Method; but even here, in the one
place in modern philosophy where “certainty” seems assured because reality is
ultimately open to mathematization, Descartes utters some last minute
reservations:
“I must confess that the
power of nature is so ample and so vast, and these
principles [the theoretical principles he
had developed for his mechanical
universe] so simple and so general, that I
almost never notice any particular
effect
such that I do not see right away that it can [be made to conform to these
principles]
in many different ways; and my greatest difficulty is usually to
discover
in which of these ways the effect is derived.” (Cited in Feyerabend, 49n.)
Fearing
the possible appearance of intellectual anarchism, Feyerabend raises the
obvious question: Does the attempt to restrict mathematization in the form of “rationalist”
methodology, whose inspiration has always involved what Hegel felt was a
process that loses the “deeper affinities or relations” that things have with
one another (Findlay, 1962, 56), move us in the direction of intellectual and
cultural relativism? In the context of
educational thought, the promotions of multiple, exclusivistic philosophies and
philosophies of education have created that sense of foundational relativism
which has turned teachers away. The result has been the adoption of the utopian
alternative which mathematization promises in the form of a single solution to
all aspects of pedagogy. A single-solution pedagogy, then, produces what
Feyerabend refers to as a “guided exchange” in contradistinction to an “open
exchange.” (At a much earlier time, Dewey distinguished between an “apprenticeship”
approach to teaching and a “laboratory” approach. It is well known which of
these he preferred.) Citing
one of his earlier books, Feyerabend states:
A guided exchange adopts ‘a
well-specified tradition and accept[s] only
those responses that correspond to its
standards. If one party has not yet
become a participant….he
will be badgered, persuaded, ‘educated’ until he
does—and then the exchange
begins.’ ‘A rational debate’, I
continue, ‘is a
special case of a guided
exchange.’ In the case of an open exchange ‘the
participants get immersed
into each other’s ways of thinking, feeling,
perceiving to such an
extent that their ideas, perceptions, world-views may
be entirely changed—they become different
people participating in a new
and different tradition.
An open exchange respects the partner whether he is
an individual or an entire
culture, while a rational exchange promises
respect only within the framework of a
rational debate An open exchange
has no organon though it
may invent one; there is no logic though new
forms of logic may emerge
in its course.’ In sum, an open exchange is part
of an as yet unspecified
and unspecifiable practice. (Feyerabend, 269.)
Pedagogy has for too long a time been distracted by the quest for “the philosophy
of
education” which will serve all traditions and individuals, and this
search, which reflects Feyerabend’s “rational debate,” has often been
accelerated in the context of a pluralistic society. But what Socrates first
pursued, and what we can infer from Feyerabend’s discussion, is the need for a “philosophy
in
education” pedagogy—the very pursuit which a Philosophy for Children curriculum
would underwrite. Here “good teaching” can be redefined: Using Robert Nozick’s
terminology, one consistent with Feyerabend’s discussion, good teaching strives
for a “value-theoretic situation” rather than a “game-theoretic
situation”: It strives for the dialogical transaction which constitutes the “open
exchange” recommended by Feyerabend as an alternative to both absolutism and relativism—the
two contentious extremes which have always emerged as the result of the
mathematization of method; it strives for an acknowledgement of the value of
the minds of all children and teachers as members of a dialogically operational
“community of inquiry.”
The
philosophic significance of this move “against method” is that it undercuts all
of those positions which generate absolutism/relativism controversies—however
subtle: Instead of raising to a philosophic level the question of the relevance
for pedagogy and pedagogical methodology of
such prescriptive concepts as
Howard Gardner’s concept of “multiple intelligences,” a notion that
psychologizes learning and gives priority to Hegel’s least significant method
for ascertaining truth—the method of experience—the method which
“depends upon the mind we bring to bear upon actuality,” schools of education
turn these into the definitive “subject matter” of pedagogic study. The
fragmentation in pedagogy which results from this, Hegel would have argued,
must ultimately end in scepticism or finally be overcome by the philosophic
approach to knowledge which, paradoxically, involves the “attitude…of entire
freedom” in the quest for “absolute truth.” Hegel’s formalism, evident in his
notion that these methods for ascertaining truth are “only forms,” was an
effort to avoid the sort of reductionist fragmentation and pedagogic scepticism
which plagues contemporary education.
Once
dialogical/dialectical “reflective and philosophic cognition” are removed from
the work of the pedagogue, that is, once “reflective cognition” is sacrificed
in the face of the quest for “the
method,” the exigencies of practice soon test the “innocence” which the
pedagogic student brings to the field, an innocence which remains uncontested
in his professional training; in fact, it is the ultimate source of the teacher’s
pessimism which sooner or later makes its way to consciousness: The failure of
the promise of methodology ultimately becomes unforgiving. Looking through
Hegelian eyes, one can see that contemporary teachers haven’t abandoned the
pursuit of the domain of innocence—of “immediate cognition” given “absolute”
status, a notion which becomes inhibitive to the freedom implicit in “reflective
and philosophic cognition;” that is, they have not yet faced the critical
question: What if Feyerabend is right? What would happen if teachers were
“against method?” What would be left to do? Would teachers do what they were
taught to do—turn the “against method” disposition into a new method? In the Phi Delta
Kappan (1998, 2000), Martin Bickman asked for the recuperation of the
“tradition of the active mind for teacher education.” His plea was then, as it
is now, germane to this discussion: “Educational structures are more in flux
now than they have been in recent memory, and before they freeze into new
rigidities and simplicities, there may be a chance to restore thinking—the
continual act of mind—as the central activity of schooling.” He cites John
Dewey’s concern that his own point of view had been converted by teachers
colleges “into a fixed subject matter of ready-made rules, to be taught and
memorized according to certain standardized procedures and, when occasion
arises, to be applied to educational problems externally, the way mustard
plasters, for example, are applied.”
Hegel said it almost two centuries earlier: “The two other forms [other
than the “method of experience”], reflective and philosophical cognition, must
leave that unsought natural harmony behind” (Hegel, 1873, 53). The inadequacy of innocence, which is unaddressed
in professional pedagogical studies, can be seen in those statements by students
of pedagogy which are notoriously paradoxical—for example such commonplace
statements which romantically claim that it is “absolutely true that no two
children know things or learn things in the same way,” and that “no two
teachers teach alike, since they all have their own methods.” But these are the
two propositions most clearly violated in contemporary schooling. And all of
these statements are nurtured by the most outrageous claim of all, viz. that
each student of pedagogy has his very own philosophy of education.
We need to underscore these latter
observations: That sense which teachers have that their pedagogic training and
the actualities of schooling are out of sync is largely due to the fact that
their “philosophies of education,” so dear to schools of education, are really
nothing more than examples of Hegel’s “innocent” expressions endowed with the
status of certitude by the “method of experience;” these vague generalizations
that make up their thinking are not filtered through Hegel’s “free use of
reflective and philosophic cognition.” “Reflective teachers” are the exception,
not the rule. The top-down “professionalization” of pedagogy is perhaps the
underlying culprit. Schools of education protect themselves by both formalizing
and indefinitely extending the innocence of Eden, by letting Hegel’s “natural
harmony” become crystallized through the reductive “method of [naïve]
experience,” and by protecting their students from confrontations with the
Serpent.
Philosophical methodology
embedded in a new pedagogy of mathematics: A challenge to the “single solution”
bias in contemporary teaching.
A modest attempt to reverse the deleterious
effects of the “machine” model of pedagogy, a mathematized and reductivist
pedagogy constructed along lines developed by such philosophers as Descartes,
is gradually finding its way into mathematics instruction itself. It is
suggested here that this somewhat new development impacts on Philosophy for
Children in a special way: It brings the mainspring of philosophic thinking
into the one field which has been resistant to engaging philosophically with
children and which is the basis of the pedagogy under review in this paper—viz.
mathematics. The notion that “conceptual understanding” requires a long process
of implanting isolated skills and details onto the “blank slate” which is the
mind of the child is now being challenged precisely in the way that Socrates
did in his response to Protagoras: Philosophical methodology does not work,
cannot work, from isolated part to whole; in fact, Socrates would have argued,
any and all methodologies involving the use of reason require a sense of
direction which only a reference to the whole—to the “solution,” even if only
hypothetical—provides. This romance with classical teleology has never died; it
lives and thrives in the pragmatism of Dewey and Rorty. In the Socratic method,
this sense of the whole needs to be hypothetical; it opens those diverse
possibilities which nurture dialogue, and, as such, is philosophically indispensable.
The child’s eagerness to promulgate and to make declamatory statements convinced
Socrates that the child is innately connected to knowledge—that the child has
“a mind of its own” which cannot be disregarded pedagogically.
Socrates would have seen contemporary pedagogic reductionism as a return
to the position of the Sophists: viewed from the point of view of the “machine”
model, the child is initially “blind,” cognitively speaking, and is therefore
utterly dependent on those who guide him.
Just as in our ordinary lives we act
virtuously by being inspired with some awareness of what virtue is, so is our
pursuit of the good guided by similar inspiration. The existence this kind of
inspiration grounds the theory of education…: education does not, as some people
(presumably the sophists) arrogantly profess, introduce knowledge into the mind
as sight into blind eyes but rather turns in the right direction a mind already
in possession of latent knowledge. (Gonzalez,
211)
The purpose of this move for Socratic teaching is precisely to avoid the
reductionism which otherwise makes a mess of the drive toward general concepts.
The virtue of the kind of explanation Socrates advocates here is its
avoidance of reductionism.
The explanation of a thing’s beauty
in terms of its possession of a certain shape or color reduces beauty to
something that at best contributes to it. Such an explanation can therefore be
easily refuted by citing something that either has a completely different shape
or color and is equally beautiful, or has the same shape or color and is not beautiful.
(Gonzalez,
We do not need to examine critically the five or so historical
interpretations of the Socratic position on the “ascent to” and “descent from”
general concepts; these are carefully evaluated by Gonzalez. Nor do we need to
examine those views of the Socratic position, criticized by Gonzalez, which attempt
to collapse “the distinction between methods of mathematics and dialectic, and
thus between dianoia and noesis.” (Gonzalez, 221.) But it seems possible to build through Socratic
methodology a pedagogic approach to mathematics that might avoid the reductive
mechanization commonly experienced by students; it would need to adopt the
whole-to-part approach which, if Socrates was correct, governs our thinking in
general and our philosophic thinking in particular. This approach avoids both
the cognitive dependency which mechanization requires and the “blindness” of
the student who finds himself in a forest of particularities.
In recent work on mathematics
instruction, Sullivan and Lilburn have moved in this direction. Simply put,
they argue that mathematics instruction should involve reflective problem
solving rather than mechanical exercises, and to do this the teacher must
reverse the standard procedure; this reversal opens the possibility of
hypothetical thinking in a field which is typically taught in a cut-and-dried fashion.
These mathematics educators give many types of examples, but one should
suffice: Mathematics teachers who teach “averaging” will generally provide students
with—let us say—the ages of five children, and then ask for their “average age.”
Students must then simply apply this commonplace procedure: add the ages and
divide by five. Let’s say the resulting average age is nine. A simple
mechanical process produces one possible solution, and no conceptual indication
as to what “averaging” means. Sullivan and Lilburn complain that this approach involves
the use of “closed questions” and, we might add, these give mathematics that single-solution character which Hegel
decried and Feyerabend called a “guided exchange” in contradistinction to an
“open exchange.” The approach which Sullivan and Lilburn recommend would begin
with what they refer to as a “good question.” Their approach would introduce
the same five children, but it would provide the solution by indicating that
their average age is nine. The “good question,” as in a Socratic dialogue which
starts with a proposed definition, would then ask for the discovery of their possible ages—i.e. the possible conditions
which make the solution possible (Sullivan and Lilburn, 2002). It does not require a great stretch of mind to
see the dialogical possibilities, internal or communal, in this procedure. And
though it deals with “quantification,” this approach might have satisfied even
Hegel. Indeed, had Socrates tried it, Meno’s servant boy might have had a more conceptually
significant mathematical revelation than in Socrates’ geometric excursion; furthermore,
Socrates’ theory of learning as “recollection” would have been more
dramatically served by this demonstration. Did Plato’s attempt to separate
philosophical and mathematical methodologies result in his leaving mathematics in
the more mechanistic single-solution domain? Was he so concerned with what
mathematics has to offer philosophical thought that he failed to inquire
whether philosophical methodology has anything to offer mathematics? That is, pedagogically
speaking, did the Socratic-Platonic epistemological alliance not go far enough?
In the Sullivan-Lilburn approach, it is not difficult to see the spirit of philosophical
methodology plowing new pathways in the garden of mathematics. Requiescat Descartes.
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