LC0aA1

Constructing Knowledge from Interactions

Human beings do not only interact with objects and natural phenomena…but also,
and in a primary sense, with other human beings.

H. Sinclair (1989)

Mme. Sinclair focuses our attention on the profound issue of how interaction
and self-construction relate to one another. In presenting an approach to this
issue which I have found productive, I will begin with a few general
observations and then go on to some concrete stories of development, drawn from
very detailed and meticulously analyzed corpora (in Lawler, 1985, 1986).[1]
My preferred descriptions, through which
I bring such general issues down to concrete cases suitable for examination, are
functionalist in orientation and ultimately computational in technique. Let me
illustrate the role of control knowledge in developing behavior with a simple
example before going on to consider two complex examples of mathematical
learning, involving the integration of disparate varieties of mathematical
knowledge.

Learning to Control Interactive Protocols

The Articulation of Complementary Roles

At the age of 2, my daughter Peggy imitated the other members of her
family. She began to imitate the knock-knock jokes of her sister Miriam (8 at
the time), this way:

Peggy: Knock-knock.
Bob: Who’s there?
Peggy: (Broad laughter).

That first night, Peggy plied her “joke” upon me time and again. Eventually,
for variety, I said “Knock-knock,” but she did not reply. I tried many times.
Even though she sensed something was expected of her, she did not
reply. I would say she could not.

Peggy, around three years

Peggy, around three years

Early the next morning I heard Peggy talking to herself in her crib:

Peggy: Knock-knock.
Peggy: Who’s there?
Peggy: (Laughter).

At breakfast, Peggy’s first words were “Knock-knock,” and I responded
appropriately. Then I said:

Bob: Knock-knock.
Peggy: Who’s there ?
Peggy: (Laughter).

That same afternoon, Miriam confirmed my observation, “Dad, Peggy can say
`Who’s there.'” I consider this a simple, lucid example of processes in the
articulation of complementary roles.

Elements of the Example:

A learner with a relatively inferior comprehension is engaged socially with
more comprehending people — in this case focused around what is literally a
script for a joke’s telling. During the engagement, social demands push at the
boundaries of comprehension of the person with the undeveloped perspective.
The learner attaches to herself uncomprehended “routines” of engagement (in
both the theatrical and programming senses). The process may be friendly or
not so — but it is more aptly and generally described by that wide ranging
class of intimate relationships that characterizes the interactions of a small
society, the home. This first type of process I call homely
(“home-like”) binding .

The second type of process, lonely discovery, occurs when the learner is
deprived of social engagement — left to her own devices — and uses those
devices to re-enact the uncomprehended experiences, compensating for the
solitude by simulating the role of the other actor. This simulation of the
other actors imposes a real demand for the distinction between roles and their
relations lacking in the initial engagement. My name for this pervasive and
repeated sequence of homely binding and lonely discovery is the articulation
of complementary roles
. In such cases, the relation between social
experience and personal construction is that more integrated discriminations
are required for controlling or directing multi-role enactment of interactive
protocols than are required for acting in them. These incidents provide a
succinct example of how the articulation of complementary roles creates new
control structures in the mind.[2]

This empirical material and its interpretation create a puzzle for instruction.
Learning occurred not because it was socially directed but as a compensating
adaptation to the deprivation of social interaction. Fantasy rescued the child
from loneliness; the more complex requirements of interaction with one agent
simulating another as well as acting out her own role engendered the
construction by the individual of skills of sufficient generality and lability
that they could function effectively in other domains of life. How one should
represent such knowledge and its changes is a complex question, one which these
observations by themselves provide insufficient guidance to permit us to
resolve.

Integrating Related Knowledge Structures

An Introduction to Paper Sums

This story is about a child’s learning to do additions whose unit sums
crossed a decade boundary. In this specific sense, it relates to “carrying”.
(In the development of the particular child, it also was crucial to her later
learning to do vertical form sums in the Hindu-Arabic notation.) At the
beginning of the study, Miriam then 6;0 (six years, zero months), was unable to
add 10 plus 20 on paper in the vertical form. When I asked her the question
“How much is ten plus twenty ?”, Miriam answered with confidence, “Thirty.”
Her response to the first sum presented in Figure 1 (a) was quite different.
“I don’t know… twelve hundred ?” (After this confusion, vertical lines were
used frequently to emphasize column alignment.)

CKI Figure 1

Despite instruction that she should not “read” the individual digits but should
add within the columns and assemble a result from the columnar sums, for (b) of
Figure 1 Miriam summed the addends to “five hundred nine” [2+1+4+2 = 9]. She
received instruction for solving problems such as (c) of Figure 1 by a
procedure I call “order-free adding” — based on the very simple idea that it
doesn’t matter in what order one sums column digits so long as any column
interaction is accounted for subsequently. After preliminary instruction, the
typical problem presented two multi-digit addends in the vertical form. Her
typical solution began with writing down from left to right the column sums of
well known results. Next, Miriam would return to the omitted subproblems and
calculate them with her fingers. When this first pass solution produced
multi-digit sums in a column — a formal illegality, as I informed her —
Miriam had to confront the interaction of columns. I instructed her to cross
off the ten’s digit of such a sum and add it as a 1 to the next left column,
that is, to “carry the one.” With less than two hours of such instruction,
Miriam succeeded at solving sums with two addends of up to ten digits; but she
realized no significant gain, for the procedures were subject first to
confusion and then to forgetting.

An Analysis: Rules that Don’t Make Sense

Why were Miriam’s initial skills with paper sums vulnerable? Consider the
three representative solutions of Figure 2:

CKI Figure 2

The first, (a) of Figure 2, shows no integration of columnar sums; the second,
(b) of Figure 2, shows a confusion over which digit to “put down” and which to
“carry” (with an implicit rule-like slogan behind the action). If you don’t
already understand the meaning of the rule “put down the N and carry the one,”
why should you prefer that to a comparable rule, “put down the 1 and carry the
N” [as in (b) above]. Miriam was confusable in the sense that she chose, with
no regularity and no apparent reason, to apply both these rules. Although
frequently instructed in the former rule, she did not remember it. The
rule-like formulation made no direct contact with her underlying microview
structures. Without support from “below,” the rule could not be remembered.
Microview is a term I use to specify a particular species of schema, one whose
principal component is a collection of pattern matching procedures and whose
functions are executed by a cascade of activation when a pattern is adequately
matched. Microviews are postulated to embody very local knowledge and to
compete with one another in a race to solve problems as they interpret them.
For example, a verbal query “How much is 25 plus 10?” could be solved by
counting from 25 on fingers or in terms of US coin equivalences. The specific
character of solutions emerging as behavior provides evidence about which
structure among those known to exist won the race in a given instance. At the
time of this incident, Miriam’s arithmetic competence is describable as
embodied in a COUNT-view (based on mastery of one-to-one correspondence), a
MONEY-view (based on coin relationships), and a DECADAL-view (based on
manipulation of numbers as multiples of ten; this unusual knowledge derived
from her particular experiences with computer-based materials at the MIT Logo
project). These three microviews form a cluster, related as components of her
mental calculation repertoire. See Lawler 1985 for more data and analyses.

Miriam eliminated her confusion by inventing a carrying procedure that made
sense to her, shown in (c) of Figure 2. “Reduction to nines” satisfied the
formal constraint that each column could have only a single digit in the result
by reducing to a 9 any multi-digit column sum and “carrying” the “excess” to
the next left column. (38 plus 34 became 99 through 12 reducing to a 9 with a
3 carried.) Miriam’s invention of this non-standard procedure (at 6;3;6) I
take as weighty evidence characterizing her understanding of numbers and
addition in the vertical form. (The latter we will discuss shortly.) About
numbers we may conclude she saw the digits as representing things which ought
to be conserved, as did the numbers of the Count microview. The achievement of
columnar sums by finger counting or by recall of well-known results further
substantiates the relation of paper sums to numbers of the Count view. Let us
declare, then, that these experiences led to the development the
Paper-sums microview, a cognitive structure that is a direct descendent
of the Count view.

Miriam did not understand “carrying” as being at all related to place value.
The numbers within the vertical columns did not relate to those of any other
column in a comprehensible way. Despite my initial criticism of “reduction to
nines” — by asking if she were surprised that all her answers had so many
nines in them — Miriam was strongly committed to this method of carrying. For
Miriam, at this time, vertical form addition had nothing to do with the Money
or Decadal sums she achieved through mental calculation. “Right” or “wrong”
was a judgment applicable to a calculation only in the terms of the microview
wherein it was going forward. I conclude then that the Paper-sums microview
shows a line of descent from Miriam’s counting knowledge, diverging with
respect to those other microviews which involved mental calculation.

The more general final point is that what made sense to Miriam completely
dominated what she was told. Why is it that the rule she was given didn’t make
sense ? How can we recapture a sense of what that must have seemed like? To
her, a number represented a collection of things with a name: “12” was a name
by which reference could be made to a collection of twelve things. Digit
strings may have seemed to her as words do to us, things which cannot be
decomposed without destroying their signification. If you divide the word
“goat” into “go” and “at,” you have two other words not sensibly related to the
vanished goat. Similarly, from our common perspective, if you don’t see the
`1′ as a `10′ when you decompose a `12′ into a `1′ and `2′, you lose `9′.
Unless you appreciate the structured representation, the decomposition of 12
can make no more sense than cutting up a word. What appears as forgetting in
Miriam’s case is an interference from established processes; what makes sense
in terms of ancestral cognitive structures dominates what is inculcated as an
extrinsic rule. (I don’t claim to offer a theory of forgetting. Competition
from sensible ideas of long dependability is a very good reason, however, for
forgetting what one is told but can’t comprehend.)

The Carrying Breakthrough

The “carrying problem” was not restricted to Paper-sums and in fact
began its resolution through integrating the microviews of mental calculation.
Although she could add double digit numbers that involved no decade boundary
crossing, like 55 plus 22, Miriam’s Decadal view functions failed with sums
only slightly different, such as 55 plus 26. Sums of this latter sort
initially produced results with illegal number names, i.e. 55 + 26 = 70:11
(“seventy-eleven”). In playing her favorite computer game, however, precision
was not required. Miriam’s typical “fix” for such a calculation problem was to
drop one of the unit digits from the problem and conclude that 55 + 26 = 76 was
an adequate solution. She could, of course, cross decade boundaries by
counting, but for a long time this Count view knowledge was not used in
conjunction with her Decadal view knowledge. Miriam’s resolution of one
species of carrying problem became evident to me in her spontaneous
presentation of a problem and its solution (at 6;3;23). She picked up some of
her brother’s second grade homework and brought it to me (M: for Miriam; B: for Bob):

M: Dad, twenty eight plus forty eight is seventy six, right ?
B: How did you figure that out ?
M: Well, twenty and forty are like two and four. That six is like sixty. We
take the eight, sixty-eight (then counting on her fingers) sixty-nine, seventy,
seventy-one, seventy-two, seventy three, seventy-four, seventy-five,
seventy-six.

Here was clear evidence that Miriam had solved one carrying problem by relating
her Decadal and Count microviews. When and how did that integration occur ?

Integrating Disparate Microviews

We were on vacation at the time. I felt Miriam had been working too
hard at the laboratory and was determined that she should have a rest from our
experiments. I was curious, however, about the representation development of
her finger counting and raised the question one day at lunch (at 6;3;16):

B: Miriam, do you remember when you used to count on your fingers all the time?
How would you do a sum like seven plus two ?
M: Nine.
B: I know you know the answer — but can you tell me how you used to figure it
out, before you knew?
M: (Counting up on fingers) Seven, eight, nine.
B: Think back even further, to long ago, to last year.
M: (Miriam counted to nine with both addends on her fingers — leaving the
middle finger of her right hand depressed.) But I don’t do that any more. Why
don’t you give me a harder problem?
B: Thirty seven plus twelve.
M: (With a shocked look on her face) That’s forty-nine.

Something about this problem and result surprised Miriam. I recorded this
situation and her reaction in the corpus; I did not appreciate it as especially
significant at that time. My current interpretation focuses on this specific
incident as a moment of insight.

Characterizing the Insight

Precisely what was it that Miriam saw ? In the Decadal view, the problem
“thirty-seven plus twelve” would be solved thus, “thirty plus ten is forty;
seven plus two is nine; forty nine” — a perfect result. Miriam had recently
become able to decompose numbers such as “twelve” into a “ten” and a “two”.
This marked a refinement of the Count view perspective. If we imagine the
calculation “thirty seven plus twelve” proceeding in the Count microview —
with the modified perspective able to “see the ten in the twelve” — Miriam
would say “thirty seven [the first number of the Count view’s perspective],
plus ten is forty seven [then counting up on her fingers the second addend
residuum], forty eight, forty nine” — also a perfect answer. We are not
surprised that the Count view answer is the same as that of the Decadal view,
but I believe the concurrence surprised Miriam. One can say that Miriam
experienced an insight (to which her “shocked look” testifies) based on the
surprising confluence of results from apparently disparate microviews.
`Insight’ is the appropriate common word for the situation, and I will continue
to use it where no confusion is likely. Since its range of meanings is too
broad for technical use, I introduce a new term, the elevation of control, as
the technical name for the learning process exemplified here. The elevation
of control
names the creation of a control element which subordinates
previously independent microviews, in the sense of permitting their controlled
invocation; some experiences of insight are the experienced correlates of
control elevation.

The character of control elevation is revealed in the example. The numbers
involved were of the right magnitude to engage Miriam’s Decadal microview.
Also, she had just been finger counting (a Count view function). If both
microviews were actively calculating results and simultaneously achieving
identical solutions, the surprising confluence of results — where none should
have been expected — could spark a significant cognitive event: the changing
of a non-relation into a relation, which is the quintessential alteration
required for the creation of new structure.

The sense of surprise attending the elevation of control is a direct
consequence of a common result being found where none was expected. The
competition of microviews, which usually leads to the dominance of one and the
suppression of others, also presents the possibility of cooperation replacing
competition. So we see, in the outcome, Decadal beginning a calculation and
Count completing it. This conclusion, howevermuch based on a rich
interpretation, is an empirical observation. Where we expected
development in response to incrementally more challenging problems, we
found something quite different: cognitive reorganization from the
redundant solution of simple problems.

The elevation of control, a minimal change which could account for the
integration of microviews witnessed by Miriam’s behavior, would be the addition
of a control element permitting the serial invocation of the Decadal view and
then the Count view. Let us declare at this moment of insight the formation of
a new microview, the SERIAL view.[3] Although
the Serial view is achieved as a minimal change of structure, its integration
of subordinated microviews permits a significantly enhanced calculation
performance, one so striking as to support the observation that a new
functional level of calculation emerged from the new organization. This is
especially evident where knowledge is articulated by proof. Consider this
example (at 6;5;24).

Miriam and Robby, himself no slouch at calculation, were making a clay by
mixing flour, salt, and water. They mixed the material, kneaded it, and folded
it over. Robby kept count of his foldings. With 95 plies, the material was
thick. He folded again, “96,” then cutting the pile in half, flopped the
second on top of the first and said, “Now I’ve got 96 plus 96.” Miriam
interjected, “That’s a hundred ninety two.” Robby was astounded, couldn’t
believe her result, and called to his mother to find if Miriam could possibly
be right. Miriam responded first, “Robby, we know ninety plus ninety is a
hundred and eighty. Six makes a hundred eighty six. [Then counting on her
fingers] One eighty-seven, one eighty-eight, one eighty-nine, one ninety, one
ninety-one, one ninety-two.”

We can see the Decadal well-known-result (90 plus 90) as a basis for this
calculation and its relation to her counting knowledge. Both these points
support the argument that Miriam’s new knowledge was specifically of
controlling pre-existing microviews. Robby was astounded — and we too should
try to preserve a sense of astonishment in order to remain sensitive to how
small a structural change permits the emergence of a new level of
performance.

Integrating Knowledge From Diverse Sensory Modes

Early papers of the MIT Logo project claimed that design producing
procedures written in Logo would be more comprehensible to children because one
could simulate the drawing agent (the light turtle on the video display) by
moving through space with her/his own body. For many children, this was not
obvious. The light turtle lived in a vertical world, they in a horizontal one.
Miriam played in a variety of “turtle navigation” games which led to her
familiarity with a set of angle values and useful relations (90 plus 90 equals
180). She also spent considerable time playing with design generating
procedures, such as the well known Logo polyspiral procedure:

TO POLYSPI :SIDE :ANGLE :CHANGE
FORWARD :SIDE
RIGHT :ANGLE
POLYSPI (:SIDE + :ANGLE) :ANGLE :CHANGE
END

Miriam enjoyed making designs, coloring them, and sharing them with her
friends. She became familiar with specific values of angles that would make
her favorite designs; but these “angle” values bore no apparent relationship to
her other experiences. During the core six months of The Intimate Study,
Miriam did not give evidence of understanding how angles and movements of
turtle navigation related to angles and designs produced by repetition in the
video context. She could use repetition, but there was no evidence she
understood it as she so obviously did in this later incident: Turtle on the Bed
(6;11;15)

As I worked at my bedroom desk, Miriam offered to sit in my lap, but I turned
her down. She moped a little, then crawled onto my bed and began to move and
spin in a most distracting fashion. “What are you doing ? You’re driving me
batty!” I complained. Requesting a pen and a 3×5 card, Miriam drew on it a
right rectangular polygonal spiral to show what she was doing in her “crawling
on the bed game.” Her verbal explanation was that she was “making one of those
maze things.”

Whence came this connectedness in her knowledge of serial physical action to
pattern? My best answer is as follows.

Cuisenaire Rods and Polyspiral Mazes

When one day the children pestered me to play with some Cuisenaire rods
I had brought home from the lab, I agreed on condition that we begin with a
project of my choosing. My proposal was this: after they sorted the rods by
color (and thus by length as well), I would begin to make something; their
problem was to describe what I was making and what my procedure was. I began
to construct a square maze of Cuisenaire rods. After I placed four rods, I
asked the children what I was making. Robby answered immediately, “A swirl, a
maze.” Miriam chimed in with his answer. At that point, I asked Robby to hold
off on his answers until I discussed my questions thoroughly with Miriam.
Having placed eight rods, I asked the children if they could describe my
procedure. Miriam could not, at first, but when I focussed her attention on
the length of each piece, she remarked: “You’re growing it bigger and bigger.”
Upon questioning, she noted the increment was “one.” After Robby added rods
of length nine and ten, Miriam justified his action by arguing, “It goes in
order…littlest to biggest,” and finally described my rod selection rule as
“every time you put a rod in, it should be one bigger than the last one.”
Miriam understood well the incrementing of length, but she showed considerable
difficulty with the role of turning in the angles in my rods maze.

When I set down the eleven-length (the orange and white pair of rods), I did
not orient it perpendicularly to the previous length. Miriam declared the
arrangement incorrect but had trouble specifying precisely what was wrong.
When she rearranged the rods to place them correctly, she simply interchanged
the location of the orange (10cm) and white rods (1cm). From this action, I
infer Miriam considered the placement incorrect because two rods of the same
color were adjacent to each other — but not because the one rod was colinear
with the preceding one. Here I asked Robby to explain what I should have
done:

R: You should go a right 90. It could be orange, right 90, white orange.
B: And what should I do after the next orange?
R: You probably could do an orange and red.
B: (Placing the new rods colinear with those preceding)
R: Hold it ! You should do a right again.
B: Oh. Miriam, what should I do next?
M: A right 90, green and orange.
B: Next?
M: A right 90, purple and orange.

This is the point at which Miriam brought together in a comprehensible relation
the steps and result of a maze generating procedure.

Several aspects stand out. Miriam received extensive guidance. Second, Miriam
worked with a familiar objective and familiar objects, and applied familiar
operations. (This experience was clearly important for Miriam,
specifically in establishing this sort of knowledge as very personally owned:
in later years, whenever offered Cuisenaire rods to play with, constructing a
polyspiral maze surfaced regularly as her objective of choice.) These
experiences of the rods-maze and turtle on the bed appear to have integrated
and thus culminated the development of Miriam’s knowledge about iteration. The
preceding incident about addition focused on microviews which had much in
common. The turtle on the bed incident presents a concrete linking experience
as a possible basis for interconnection between essentially remote clusters of
microviews. Essentially remote refers here to Turtle Navigation’s
being related primarily to walking and Computer Design’s being related
primarily to seeing, thus being descended from different sensori-motor
subsystems, ie. locomotive and visual.

The central issue of human cognitive organization is how disparate and
long-developing structures become linked in communication to form a partially
coherent mind — such as we experience personally and witness in others. The
framework used here discriminates among the major components of the
sensori-motor system and their cognitive descendents, even while assuming the
preeminence of that system as the basis of mind. Imagine the entire
sensori-motor system of the body as made up of a few large, related, but
distinct sub-systems, each characterized by the special states and motions
of the major body parts, thus:

Body PartsS-M SubsystemMajor Operations
TrunkSomaticBeing here
LegsLocomotiveMoving from here to there
Head-eyesCapital/visualLooking at that there
Arms-handsManipulativeChanging that there
Tongue/earsLinguisticSaying/hearing whatever

Much of the activity of early infancy specifically involves developing
coordinations between these five major sensori-motor sub-systems. Such a
fundamental organization in the development of coordinated systems might be
assumed to ramify through all descendent cognitive structures developed from
interacting with the experiences of later life.

The rods-maze microview closed the unbridgeable gap between turtle geometry
Navigation microviews and the Design cluster by playing a mediating role. The
local character, the task-specific binding of Miriam’s learning in the
rods-maze incident, implies that it was not developed analogically (i.e. from
her turtle geometry experiences) but de novo from more primitive
components of the sensori-motor system. If descended directly from the
coordinating scheme which results in hand-eye coordination, the rods-maze
microview was effective as mediator for two reasons, which can be brought
forward in this simple comparison expressing the activity of the primary agents
in these microviews:

LocomotiveMediatingVisual
I move fromYou (hand) moveThat(thing) goes
here to therefrom here to there from here to there
BodyHandEye
as agentas remote agent as active agent

The primary difference between the active programs of the human locomotive and
visual subsystems is the level of aggregation which is significant for their
functioning. The body lurches forward, step by step. The eye recognizes an
image as an entity by circulating repeatedly in the pattern of a closed loop, a
“feature ring,” which defines that object in memory. The feature ring
is a complex recognition procedure, which represents the saccades of eye focus
and the possibility of recognizing features. Its primitive elements can be
described as similar to the movements of the locomotive system, going forward
and turning right. [4]. Because of years of
developed hand-eye coordination, the eye can recognize the pattern that emerges
from what the hand does whereas it can not recognize so simply (if at all) the
pattern that emerges from the path of body movement through the plane. The
rods-maze experience was able to function mediatively between descendents of
the locomotive and visual subsystems because the hand, as the familiar agent
for manipulating remote objects (say little toy dolls some of whom may be
thought of as self or other), can make the bridge between an action of movement
which a body might make and one which can be coordinated with visual results.

The Channelled Description Conjecture

The body-parts mind proposal serves the function here of separating
groups of cognitive structures on a large scale. Some cognitive structures are
descended from ancestors in the locomotive sub-system and others from ancestors
in the visual sub-system. If there is body-based disparateness, what leads to
subsequent integration? The progressive organization of disparate structures
and subsystems proceeds from the needs of the individual as a complete being.
The achievement of an individual’s goals requires the cooperation of disparate
cognitive structures and subsystems of such structures, e.g. crawling to get
some desired object requires the use of arms, legs, and vision. Focussing as
it does on the descent of cognitive structures from ancestors in the motor
subsystems, the body-parts mind proposal definitely favors the activity of the
subject in the creation of cognitive structures over the impression of
sensations on the mind. In this specific sense, the proposal is fundamentally
compatible with Piagetian constructivism.

Even if the mind is a network of information structures comprised of the same
types of elements, one need not conclude that it is uniform. Microviews are
shaped both by their specific descent from body-defined sub-systems and by
their interconnection possibilities in terms of those sub-systems. The
connections between late-developed cognitive structures mirror — and are
guided by — the interconnection possibilities of the sensori-motor system
which are first explored and described in the motor programs developed during
the sensori-motor period of infancy. This idea, which I name the channelled
description conjecture
, is not an hypothesis which was posed for
experimental confirmation; rather, it is a ground of explanation found useful
in making sense of knowledge Miriam developed and failed to develop during her
many encounters with geometry during The Intimate Study.

The Power of Ideas and Cognitive Structure

The question of what constrains the possibility of some ideas being
powerful and others not so is the crux of the channelled description
conjecture. Concrete embodiments of ideas are personally owned because they
are not remote from the shaping structures of the soma itself. Experiences
such as those of the rods-maze are powerful precisely because they provide the
links between late developed structures and the coordinating schemata (the
primary integrations of the sensori-motor subsystems achieved during the
sensori-motor period). They are important because they link the concrete
structures of body knowledge to the more abstract descriptions of external
things that blossom in maturity as the cognitive network of the mind.

In strong form, the channelled description conjecture proposes that ONLY those
concrete embodiments of ideas which link together descendents of disparate
sensori-motor subsystems can be powerful; it claims that such models are the
correlate in concrete thought of the correspondence schemata of the
sensori-motor period and that on them depends the developing coherence of the
individual’s cognitive structure. Further, such microviews provide the bases of
construction of the more extended cognitive nets of developed minds,
functioning as the ancient cities, the geographic capitals of personal
importance. In contrast with a goal-oriented attempt to link feelings and
thoughts — as upon a basis of disparate need systems proposed in ethology, or
with a Freudian focus on the conflict between competing, even conflicting
homunculi in the mind — the channelled description conjecture proposes a third
model of basically disparate structure: the mind is not uniform because the
body, the effector agent of the sensori-motor system, is not uniform. This
view is better characterized by a pun of Wallace Stevens, “my anima likes its
animal,” than by either the needs or conflicts of the other mentioned
alternatives.

The role assigned to coordinating schemata bears on VonGlasersfeld’s
observation (1989) of their role in the naïve assumption of the reality of
external things. In his view, the correspondence of schemata in diverse
modalities leads to the unwarranted inference that we can know about external
things themselves. In my view, the later descendents of these coordinating
schemata are primary mediators in the construction of cognitive coherence. If
the assumption of the knowability of external things is an illusion (as we have
all believed since Kant), it is a very strong weakness, one perhaps partly
explicable by the coherence creating function which I ascribe to multi-modal
correspondences. [5]

Where Do Our Ends Begin ?

What makes men happy is loving to do what they have to do.
This is a principle upon which society is not founded….
Helvetius
from De L’espirit

How do we begin to think about the challenge of fitting society’s goals to
those of learners ? How can we instruct while respecting the
self-constructive character of mind
? Here is a view of the development of
goals I derived years ago (from Lévi-Strauss and François Jacob)
as an extension of the notion of bricolage.

Claude Levi-Strauss describes the concrete thought of not-yet-civilized people
as bricolage, the activity of the bricoleur –a sort of
jack-of-all-trades, or more precisely, a committed do-it-yourself man. The core
idea is looseness of commitment to specific goals, with the consequence that
materials and competences developed for one purpose are transferable to the
satisfaction of alternative objectives:

The bricoleur is adept at performing a large number of diverse tasks; but,
unlike the engineer, he does not subordinate each of them to the availability
of raw materials and tools conceived and procured for the purpose of the
project. His universe of instruments is closed and the rules of his game are
always to make do with ‘whatever is at hand’…. In the continual
reconstruction from the same materials, it is always earlier ends which are
called upon to play the part of means…. The bricoleur may not ever complete
his purpose but he always puts something of himself into it….
from The Savage Mind, pp. 17, 21.

One can appreciate the opposition of planning (the epitome of goal-directed
behavior) and the opportunism of bricolage. Of course, the two are not
discontinuous; all activities can be seen as a mixture of the polar tendencies
represented here. Second, the relationship is not directional: there is no
reason to suppose that planning is a more nearly perfect form of bricolage.
One could easily view planning as a highly specialized technique for solving
critical problems whose solutions demand scarce resources.

Bricolage and Cognition

Students of anatomy have named the adaptiveness of structures to
alternative purposes functional lability. Such functional lability is
the essential characteristic of the bricoleur’s use of his tools and materials,
so bricolage can serve as a metaphor for the relation of a person to the
contents and processes of his mind. This emphasizes the character of the
processes in terms of human action and can guide us in exploring how a
coherent mind could rise out of the disparateness of specific experience. What
are the practical advantages of discussing human activity as bricolage in
contrast to goal-driven planning?

first: bricolage presents a human model for the development of
objectives; it is a more natural, thus a more fit description of everyday
activity than planning.

second: it is more nearly compatible with a view of the mind as a
process controlled by contention of multiple objectives for resources than is
planning.

third: the most important advantage is a new vision of the process of
learning
. Bricolage can provide us with an image for the process of the
mind under self-construction in these specific respects:

    – if the resources of the individual’s mind are viewed as being like the tools
    and materials of the bricoleur, one can appreciate immediately how they
    constrain our undertaking and accomplishing any activity.
    – not only constraint comes from this set of limited resources; also comes
    creativity, the production of new things — perhaps not exactly suited to the
    situation but of genuine novelty.
    – the mind, if seen as self-constructed through bricolage, presents a clear
    image of the uniqueness of every person:
    * each will have developed his own history of conceptions and appreciations of
    situations through which to make sense of the world.
    * each will have his personal “bag of tricks,” knowledge and procedures useful
    in his past.
    * each will have his own set of different, alternative objectives to take up as
    chance puts the means at his disposal.

If viewed as claims, such statements are not easy to prove. However, they
provide a framework for investigating learning which could be valuable by
not demeaning human nature through assuming it is more simple than we
know to be the case.

References:

  1. Jacob, François (1977) Evolution and Tinkering. Science,
    196, pp.1161-1166, 10 June. Republished in The Possible and the Actual. New
    York: Pantheon Books, 1982.

  2. Lawler, Robert W.(1979) One Child’s Learning. Unpublished PhD Thesis.
    MIT.

  3. Lawler, Robert W. (1985) Computer Experience and Cognitive Development. John
    Wiley, NY.

  4. Lawler, Robert W. et al. (1986) Cognition and Computers. John Wiley,NY.
  5. Lawler, Robert W. (1989) Sharable Models: The Cognitive Equivvalent of a
    Lingua Franca
    . The Journal of Artificial Intelligence and
    Society, Vol. 3, 1. Springer, NY.

  6. Lawler, Robert W. (1990) Thinkable Models. The Journal of Mathematical
    Behavior, Ablex, Norwood, NJ. Forthcoming.

  7. Langer, Susanne. (1967) See Idols of the Laboratory. In Mind: An
    Essay on Feeling
    . Johns Hopkins , Baltimore.

  8. Lévi-Strauss, Claude. See The Science of the Concrete. In The
    Savage Mind
    . University of Chicago Press, 1966.

  9. Lewin, Kurt (1935) The Conflict between Galilean and Aristotelian Modes of
    Thought in Contemporary Psychology
    . In Dynamic Psychology: Selcted Papers
    of Kurt Lewin
    . McGraw-Hill.

  10. Noton and Stark. (1971) Eyes Movements and Visual Perception. In
    Scientific American, 224, 6.

  11. Sinclair, Hermine. (1989) Learning: The Interactive Re-Creation of
    Knowledge.
    In Steffe and Wood.

  12. Steffe, Leslie & Wood, Terry (1989) Transforming Early Childhood
    Mathematics Education
    , Lawrence Erlbaum, Hillsdale, NJ.

  13. VonGlasersfeld, E. (1989) Environment and Communication. In Steffe and
    Wood.

Publication notes:

  • Written in 1988 as a summary of earlier research and analyses.
  • This paper draws heavily on examples published in Lawler, 1985. It was published in
    “Transforming Early Mathematics Education,” (Steffe and Wood, eds.) along with other papers
    presented at the Sixth International Conference of Mathematics Educators, Budapest, 1988.
  • Published in the Journal of Mathematical Behavior, V.9, No.2 (Ed, Robert B. Davis)

Text notes:

  1. Central arguments bearing on the importance of the case method may be found in Lewin
    (1935) and in Langer(1967).

  2. One can not argue coercively that this single incident must have been the
    sole generator of such a change. If, however, particular experiences are the foundation for
    cognitive development, then some one among them must have been the first. This experience
    clearly exhibits a set of characteristics which seem essential to the process.

  3. One wants to avoid the creation of something from nothing. See in this connection the
    discussion of “relational conversion” in Lawler 1985 (chptr. 7). In Lawler 1979, I advanced
    the same argument, first that the boundaries between microviews are defined by networks of
    “must-not-confound” links which function to suppress confusion between competing, related
    microviews; second, that the conversion of these repressive links, established by experience,
    to more explicit relational links, generates “new” control structure at moments of insight.
    The creation of inhibiting relations between microviews to suppress confusion does the real work
    of structural creation. The relational conversion, in which an inhibiting relation is turned into
    one of richer semantic content, permits the smooth transformation in functional capability to
    another behavior over what otherwise would appear to be an unbridgeable gap.

  4. See chapter 5 in Lawler, 1985, or Noton and Stark (1972).
  5. For an attempt to apply such ideas directly to educational issues, see Lawler (1989) or (1990).

After Thoughts


Subjects of three case studies

These are the subjects of my three case studies. LC1, published in Cognition and Computers, focuses on Rob’s learning at the MIT Logo Laboratory between the ages of 5 and 8. LC2, published in Computer Experience and Cognitive Development, focuses on Miriam’s learning around the age six years. LC3, still under analyses, focuses on six years of Peggy’s learning, here seen while still an infant.

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