Peggy, around three years
Human beings do not only interact with objects and natural phenomena...but also, and in a primary sense, with other human beings.Mme. Sinclair focusses 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.H. Sinclair (1989)
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.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.
Bob: Who's there?
Peggy: (Broad laughter).
Peggy: Knock-knock.At breakfast, Peggy's first words were "Knock-knock," and I responded appropriately. Then I said:
Peggy: Who's there?
Peggy: (Laughter).
Bob: Knock-knock.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.
Peggy: Who's there ?
Peggy: (Laughter).
Elements of the Example:
A learner with a relatively inferior comprehension is engaged socially with more comprehending people -- in this case focussed 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.
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.)
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:
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 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 ?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 ?
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.
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 ?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 focusses on this specific incident as a moment of insight.
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.
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.
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 :CHANGEMiriam 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)
FORWARD :SIDE
RIGHT :ANGLE
POLYSPI (:SIDE + :ANGLE) :ANGLE :CHANGE
END
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 3x5 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.This is the point at which Miriam brought together in a comprehensible relation the steps and result of a maze generating procedure.
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.
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 focussed 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 Parts | S-M Subsystem | Major Operations |
|---|---|---|
| Trunk | Somatic | Being here |
| Legs | Locomotive | Moving from here to there |
| Head-eyes | Capital/visual | Looking at that there |
| Arms-hands | Manipulative | Changing that there |
| Tongue/ears | Linguistic | Saying/hearing whatever |
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:
| Locomotive | Mediating | Visual |
|---|---|---|
| I move from | You (hand) move | That(thing) goes |
| here to there | from here to there | from here to there |
| Body | Hand | Eye |
| as agent | as remote agent | as active agent |
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]
What makes men happy is loving to do what they have to do.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.
This is a principle upon which society is not founded....
Helvetius
from De L'espirit
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....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.
from The Savage Mind, pp. 17, 21.
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:
References:
Publication notes:
Text notes:
