ON MAKING DISTINCTIONS PERMEABLE:
A View of Creativity as Learning

Jeanne Bamberger
Massachusetts Institute of Technology
 
 

INTRODUCTION: MAKING DISTINCTIONS

We make distinctions as a way of organizing and making clear, holding steady, the multiplicity of possibles in our experience.  However, once distinctions are made and their objects defined and accepted as common practice, deep learning and new knowledge occur particularly at moments when the limits of these distinctions become porous.   For it is at these moments that the permeable boundaries of distinctions allow multiple meanings to collide and stumble over one another.
Thinking about creativity in this way, I am plunged into enigmas.  How can I disentangle the distinctions—the meaning, the mythic quality of creativity from other acts of enlightenment, of coming to see in a new way: LEARNING, DISCOVERY, DEVELOPMENT, INVENTION, PROGRESS—a tangled thicket of natural acts. Can I extricate myself by making cleaner distinctions?  And, once made, is it then more useful, more generative, to allow the tangle to take on new form—to obscure the newfound boundaries, making them again porous and, with that, relieve the mythic values each distinction has accrued?
Sometimes boundaries blur in the moment of seeing one thing as another, or, as Wittgenstein calls it, "seeing as...."


Four dots?  Two pairs of dots?  One pair of dots inside another?
           (Wittgenstein, 1960, p. 168)

Or the evolution of a musical composition, where boundaries of structural entities, becoming permeable, transform as new distinctions emerge from the old.  Perhaps in its becoming, a composer and his composition also learn.
Or take another more everyday happening, but I think no less significant:  watch a child noticing—noticing, for instance, in the fleeting, ever present, that a ping-pong ball, left alone, bouncing across the floor, gets faster and faster

.       .     .     .    .   .  . ..

but then stops.   Such surprise only happens when there are expectations—allowing surprise is to tempt uncertainty by tampering with distinctions already made: faster is stumbling over stop.   But these moments of surprise, of noticing, quickly disappear, absorbed into the new present; we are tempted to let them go, preferring to take seriously only that which is stable, less fleeting—moments when we can measure, classify, and name distinctions.
Dewey puts the tension this way:
The realm of immediate qualities contains everything of worth and significance.  But it is uncertain, unstable and precarious.  The first consideration [immediate qualities] induces us to prize it supremely; the second [uncertain, unstable] leads us to deny reality to it as compared with alleged underlying things with their fixity and permanence. (Dewey, 1929, p. 110)
Now listen to Alice Munro, the creator of wonderful stories, as she responds to an interviewer's questions: 1
INT:    First of all, Ms Munro, how do you know so much?
MUNRO:   How do I know so much?  I know barely anything at all.
INT:   You know everything.
MUNRO:   You know, a friend of mine once said to me—a friend who is a writer—that she thinks that it's not that writers know more things than other people, or understand more things than other people, but that they are surprised by more things than other people.  And so things that other people take for granted, they are always saying, "Oh, God, really?"  and trying to figure them out.
INT:    Do you do that?  Do you say, "Oh, God, really?"
AM:   Well, a lot of things interest me that are fairly ordinary things in life…. And yet to me they seem really extraordinary and things that I want to explain…. In fact … every story is an investigation  for me.  And sometimes I'm a little surprised  by what I'm thinking  about it, and I see how it's going to turn out and I think, "Oh, is that what happens?" [italics added]
"Surprise" ; "investigation" ; "thinking"—creativity would seem to involve learning; but does learning involve creativity?
To test and provoke these enigmas and to better understand the possible role of such permeable transactions, I give two examples.  Both involve children working with the most ordinary of tunes—Hot Cross Buns, and Twinkle Twinkle Little Star, yet both involve moments of extraordinary learning.  In each case, we watch children grappling with moments when the boundaries of their self-made distinctions become surprisingly porous.  Each story has been an investigation and I, too, have been more than a little surprised by what happens.

THE CHILDREN

While both children work on a similar task, and both are much the same age, nine and ten years old, they otherwise appear to be quite different.  The first, a boy I'll call Brad, is a virtuoso at taking apart, designing, building complex systems (his bicycle, plumbing, motors, machines built of lego blocks and gears), and an expert at devising experiments to analyze and test problems confronted on the way.   But he is having trouble learning in school.  Described as having problems with common symbolic expressions—numbers, graphs, simple calculations, written language—instead of being seen as a virtuoso, he is seen as "failing to perform."
The second child, I call her Susan, is also a virtuoso—a virtuoso violinist.  A member of the Young Performers Program at a Cambridge, MA music school, she studies violin, plays in a string quartet, performed as soloist in a Mozart concerto with the student orchestra, and, of course, reads music fluently.   She is rarely if ever described as "failing to perform," in or out of school.

BRAD

I worked with Brad in the context of The Laboratory for Making Things that we had developed in a public school in Cambridge, MA.   Teachers in the school usually brought their classes to the Lab for special projects.   However, Brad was a member of a group of 6-, 8- and 9-year-old children who had been invited by the Special Ed teacher, Mary Briggs, to work with us in the Lab once a week after school.
 The Lab was a large room in the school  "furnished" with a great variety of materials for designing and building structures that work. These materials included gears and pulleys, lego blocks, pattern blocks and large building blocks; cuisenaire rods, batteries and buzzers for building simple circuitry; as well as foamcore, wood, and glue for model house construction.
Together with making things, the children were used to inventing some kind of graphic instructions/notations that could help someone else build what they had built.   Children were also accustomed to informal conversations in which they explained to one another or to an adult how they were making sense of the material.  Such conversations were particularly cherished when they spontaneously arose in response to a child's surprising discovery or when an insight led to solving a particularly difficult problem.   In the course of this collaborative reflection, children learned from one another—rethinking their understanding, their subsequent descriptions, and even their work on later projects that involved quite different materials.

THE MATERIALS AND THE TASK
The Montessori Bells

The task, as presented to Brad and the other children was:
"Build the melody Hot Cross Buns with your bells, and then make some instructions so someone else can play the tune on your bells as you have them set up."2

In preparation for the task, each of the six children in the Lab was given five Montessori bells and a small mallet with which to play them.

Figure 1


 Five bells and a small mallet

The Montessori bells, originally invented by Maria Montessori with the help of her music specialist, Anna Macaroni, are a rather extraordinary technological invention in themselves.  Each individual mushroomed-shaped metal bell is attached to a wooden stem, with bell and stem, in turn, standing on a small wooden base.  Some of the bells stand on brown bases and others on white bases but this single difference has no immediate significance to this task.  Unlike any other pitch-making materials that I am aware of, the Montessori bells all look alike, and they are also free to be moved about.  Thus, the only way to distinguish the pitch of one bell from another is to play them and listen to the result.
Since the structure of Hot Cross Buns plays a prominent role in Brad's work, the sketch in Figure 2 shows the conventional structural distinctions.

 Figure 2

 3 part structure of Hot Cross Buns

BRAD'S WORK3

I have grouped Brad's work into four phases.   Each phase is marked by a significant shift in Brad's tune building strategy, in his view of the tune and its constituents, and, as a result, the emergence of a new notation.  Each of these, in turn, involves a process in which the boundaries of previously held distinctions among entities become permeable.  The critical questions in following Brad's work are: what events, surprises, investigations, thinking, trigger these transactions, and what is the evidence?   The entire session was video-taped, making it possible to trace Brad's work in detail.
 
 

PHASE I

A First Construction and Notation

Brad begins the task of building "Hot Cross Buns" by giving himself another task of his own invention—namely, labeling the bells.  Cutting out five paper squares, Brad writes on each square a number from 1 to 5.  Then, lining up his bells on the table in a row without playing them, he places a numbered square in front of each bell, ordering the squares 1-5, from right to left,  as shown in Figure 3.

Figure 3

Brad was presumably looking ahead—i.e., unlike any of the other children, he was anticipating making instructions so that he could show someone else how to play the tune on the bells the way he was going to set them up.
At this point, the numbers Brad placed in front of the bells were simply neutral labels: since he had not yet played the bells, they could not refer to their pitch properties nor to their position or function within the tune.  However, as long as Brad kept a number-label attached to a bell, the label could serve as an identifying name.
Building the tune

Playing the bells now with the small mallet, Brad searches for bells that match each tune-event as he needs it —i.e., in the order of their occurrence in the tune.  Finding the first bell in the tune at the end of his arbitrary line-up, Brad places the found bell on the table, carrying the label (5) along with it.  This is the beginning of a new bell path that will play the whole tune.   Playing this first bell again, as if starting from the beginning of the tune, Brad searches through the remaining bells to find a bell that matches the next event in the tune.  Placing it to the left of the first bell in his new tune line-up, he again moves its arbitrary label (1) along with it (see Figure 4).

Figure 4

 Labels move along with bells

Continuing this process, Brad lines up all the bells according to their chronological position in the tune, always being careful to move each number-label along with its bell. In this way Brad transforms his initially arbitrary line-up of bells into a bell path with a corresponding number path  made to play just the tune Hot Cross Buns—a unique, "one purpose" instrument (see Figure 5).

Figure 5





  Looking at the "number path" [5 1 3 4 2], it is clear that the numerals are arbitrary labels.  The numbers are not related, for instance, to the ordinal numbers in a number line—the bell named 5 is the first bell in the series, while the bell named 1 is the second in the series.  This use of numerals for arbitrarily naming will change significantly in the course of Brad's work.

Making an action path to play the tune

To play the whole tune, Brad plays ON the set-up bells in contrast to making the tune by moving the bells, themselves.   Picking out the sequence of tune events as he moves along his bell path, he makes a unique action path along the bells (see Figure 6).4

Figure 6


  Playing the whole tune: An action path

       Brad's strategy in building the tune and then playing it on his built bell path is basically to go "straight-ahead."  But, unlike most children of this age for whom "straight-ahead" goes left-->right (L-->R), Brad goes right-->left (R-->L) in all dimensions simultaneously: in the temporal sequence of tune events, in the chronological position of the bells lined up on the table, and in his actions on them.   There are, however, three notable exceptions to this straight-ahead, prevailing "onward"  direction. Most important, each exception is mediated by and simultaneously marks structural boundaries of groups of "figures" within the tune, itself.   The first exception is the repetition of the first three bells (5-1-3] which groups these bells and tune events together to form a first figure.  This boundary is marked in the action path by the "turn around," a move "back," as required by the repetition of the group of tune events and bells (see Figure 7).

Figure 7

Repeating the first three bells

A second disruption of the prevailing forward motion comes at the end of the tune.  Since the tune ends as it began, another turn-around is required, a move "back" to play the first figure again (see Figure 8).

Figure 8

Playing the first figure again

The third exception involves the repeat of tune events on a single bell.   This is the move to new bells (4-2) that coincides with the new, middle figure.   But, while the repetitions on each bell require Brad to disrupt his straight-ahead direction of motion so as to remain in one place while the tune goes on, he maintains his prevailing R-->L motion as he moves "onward" from bell-3 to bell-4 and bell-2 (see Figure 9).

Figure 9

 The middle figure: repeating and moving "onward"

In Phase II of Brad's work, new distinctions emerge as these initial boundaries of his bell path and action path become permeable.

A first notation

To make instructions, Brad slowly plays the tune again, carefully keeping track of the number-labels. As he plays each bell, he copies its label onto his paper in the order in which he plays them.  In this way, Brad's bell path and action path, his sequence of actions through the bell terrain, become a static sequence of numbers on paper—a notation path (see Figure 10).

Figure 10

 A first notation

 Notice that in copying the labels onto his paper, Brad groups the numbers spatially across and down the page. The boundaries of the grouped labels [5-1-3] and [4 4 4 2 2 2] correspond to the exceptions in his straight-ahead action path and to the structural boundaries marked by these exceptions. These include the "turn-backs" for repetition and return of the first figure, and the repetitions on a single bell for the middle figure.   In thus showing structural groupings in the spatial groupings of his notation, Brad makes the boundaries of the distinctions he is making in action static and visible for himself and for the reader.

 PHASE II

New  Distinctions Emerge

In Phase II, Brad's previous boundary distinctions are initially made penetrable quite by chance, but, once made, they become the source of his making a significant discovery.  The process is triggered by an entirely accidental discovery by another child, Celia, who is working on the same tune-building task across the room from Brad.   Celia has set up her bells in a different configuration from Brad's.  Seen from above, in Figure 11,  Celia's bells explicitly embody the boundaries of the tune's figural grouping structure—two separate rows of bells, one for the first figure of the tune and its return, the other for the middle figure.

Figure 11

     Celia's bell path

Playing around on the bells, Celia stumbles upon a discovery.  Calling the other children over, she shows how she can play the first part of Hot Cross Buns "in two different ways so it sounds just the same" (see Figure 12).   To do so, Celia crosses over the boundaries of her spatial/structural set-up.  For Celia this seemed to be a  kind of magic and she did no more with it.

Figure 12

Celia's two  ways that "sound the same"





Adapting Celia's route

But Brad, intrigued by Celia's discovery, returns to his own bells where he quickly adapts Celia's alternate action path to his configuration of bells (see Figure 13).

Figure 13

After successfully transporting Celia's action path for the first part of the tune to the shape of his bells, Brad pauses, then starts again from the beginning  and  constructs a whole new action path for the entire tune.  His new path includes the alternate route for the first part of the tune and also a new route for the middle part of the tune.  He completes the tune by going back to use his original route to play the return of the first figure (see Figure 14).

Figure 14

Brad's new action path

With his performance completed, Brad pauses, looking around as if surprised and puzzled.  Sounding bewildered, he says:

"Oh this is weird.  I can play it with just three bells!"

Like Alice Munro, Brad seems to be saying, "...sometimes I'm a little surprised by what I'm thinking about it, and I see how it's going to turn out and I think, 'Oh, is that what happens?' "
 
 

MAKING NEW BOUNDARIES

How do the initial boundaries of distinctions become porous and how do these events lead to his "just three bells" discovery?  Consider the following scenario:

In adapting Celia's alternate route, Brad must necessarily violate boundaries of the groupings made along his previous action path.  As a result of these shifting boundaries, he also necessarily displaces the function of particular bells within the tune and its figural grouping structure.   Specifically, following Celia's alternative path, Brad must cross over (violate) the boundary of the first figure in the tune, leaping (rather than stepping) from the beginning bell-5 over the remaining members of his initial bell group [5-1-3] to the pair of bells at the other end of the bell path [2-4].  He thus displaces bells [2-4] from their previous function as members of the middle figure, these bells now becoming constituent members of the first figure, instead.   He must also play these two new bells [2-4] in the opposite direction (L-->R) and also opposite to the prevailing direction of his previous bell path and action path (see Figure 15).

Figure 15

Breaking up a distinct and bounded entity

Going on to the middle figure, Brad uses the bell pair [1-3] displacing it from membership in the first figure and again reversing the prevailing R-->L direction  (see Figure 16).

Figure 16

New route for middle figure

Finally, Brad uses his original action path (5-1-3) to play the return to the first figure again.  But, in this new context, the old route is an alternative to Celia's alternative route.
It is clear that the boundaries of the initial distinctions within Brad's bell paths, action paths, and figural groupings have been made permeable, leading to the emergence of new grouping boundaries.   But how do these transformations spawn Brad's surprising  "three-bell" theory?
As in most on-the-spot learning, Brad's discovery was emergent in real time, that is, almost entirely embedded in his actions. But in response to a question from Mary, Brad  does make an attempt at an accounting.  Mary asks: "How'd you discover it?  I want to know what happened.  All of a sudden you said, 'Wait a minute, I can do it with three.'"   Brad's rather halting but thoughtful account may on first reading seem a bit obscure and difficult to understand.  But with the help of his gestures and the expression in his voice and on his face, he does indeed tell us "what happened."
  He starts at the point when he has just adapted Celia's alternate route for the first figure to his own bells:
 
 
 

BRAD EXPLAINS
 COMMENTS
"...I was realizing that if I 
could  play it one way--like 5 1 3..."
 He makes a gesture along the bells to show how he could 
 "play it one way."
"...then I realized (pause) that if you could do 
all of them one way..."
 He starts again adding, "do all of them…." 
 This must refer   elliptically to the fact that, at that point, he 
 was using all five bells.

Then I realized that two of these 
(circling the pair [1 3]) could be 
used in a different way instead of these 
two (points to the pair [4 2])." 

                 Figure 17 


 Here, Brad describes, in much condensed form, his actual 
 process of inference in making the alternative path for the 
 middle figure (see Figure 17).

An unpacked version of Brad's compact account might go something like this:
Using all five bells, the two pairs of bells, 1-3 and 2-4, work equally well for playing the beginning figure—either  5-1-3 or 5-4-2.  These same two bells, 2-4 and 3-1, can be swapped to play the middle figure, too.   And since these two bell pairs can substitute for one another to make both the first and middle figures, they must be functionally equivalent pairs! (see Figure 17a).

 Figure 17a

Functionally equivalent pairs

And if this is so, then you don't need both pairs.  Just the 5-bell and one or the other pair will do it.  That makes "just three bells" in all!
 
 

PHASE III

Making the 3-Bell Theory Work

Playing just three bells

Having proposed his 3-bell theory, Brad puts it into action.   Pushing aside the two "extra" bells (2 and 4) and integrating his several alternate routes, he successfully plays the whole tune using just the three remaining bells (see figure 18).  Permeable boundaries have yielded to new distinctions, new entities, and new functions.

Figure 18

Just Three Bells

Notation 2

Watching Brad's 3-bell performance, Mary asks Brad, "Interesting …  but Brad, can you write down how you'd do it that way?"  Singing to himself and looking at the bells but not yet playing them, Brad said, "The first two are the same" and he writes:

Figure 19

"...the first two..."

Brad's comment, "The first two are the same," obviously refers to the first two groups of bells—i.e., the repetition of the first figure [5-1-3][5-1-3].   This is clear evidence that for Brad, an entity, a "thing," is a bundle of bells and an associated bundle of actions.   With boundary distinctions becoming porous, these initial grouping boundaries remain comfortably intact.
Playing the bells, now, testing the 3-bell route, Brad completes Notation 2 for the 3-bell route (see Figure 20).

Figure 20

Notation 2

Paradoxically, the [4-2] bell pair that has been set aside, has remained intact as a group as well, but it has also become irrelevant, superseded by the new bell pair [3-1] (see Figure 21).

       Figure 21

An irrelevant pair
 
 

PHASE IV

Brad's Reflections Produce a Third Notation

Turning back to his actions, his previous notations, and each of the transformations they have spawned—alternate action paths, coupled bells that are functionally equivalent, "just 3 bells"—the permeability of boundaries now gives way to a whole new entity.   Brad almost says to himself,  "I see a pattern," and, in response to a question from Mary about the "pattern" he sees, Brad says, "Well, you really could number them one-two-three."   And he continues, "So you kinda say ... you went up...," and he gestures across the bells, R-->L (see Figure 22):

Figure 22

"...you went up"

Brad continues: "I mean, let's say this was '1-2-3' " and he quickly repositions his numbered paper squares in front of the three bells, still moving  R-->L (see Figure 23).

Figure 23

Notation 3

Following the bell and number paths, Brad continues to the middle figure.  And as he says the numbers aloud, gesturing along the series, the numbers almost magically take on new meaning.  He says:
Then you could go 3 3 3, which is high; 2 2 2, which is a little lower; then 1 2 3.... So you go up, and then up again …. Then you go down, and then up again."

Metaphors, Meanings, and Notations

What has happened here?   With the bells lined up in their order of entry into the tune, Brad sees a "pattern."   To reflect the order of entry pattern, he labels the bells with a line-up of ordinal numbers: 1-2-3 or first, second, third.   But, once the numbers are laid down, Brad sees the number list as a self-animated number line: "you go up" and "you go down."    However, in the process, the boundary distinctions between number line and pitch become porous and the similar conventions of number lines and music notation stumble over one another.
Like the number line, pitch is also conventionally described as going from low to high or "going up," and like the number line the scale can also be represented by an ordinal numbers series— 1-2-3-4-5 for C D E F G.   Brad's new notation reveals, in its making, the permeable boundaries of distinctions hiding in our conventional notations.
Animating the numbers in a number line, putting them into motion and giving them direction, the boundaries of distinctions between graspable objects that literally move in space, and numbers which do not move at all, have been irretrievably collapsed.  We have created a "dead metaphor."    Perhaps a "dead metaphor" is the purest example of distinctions whose boundaries have become so porous as to have entirely crumbled.
But dead metaphors and crumbled distinctions come alive under conditions of uncertainty and confusion.   While Brad sees the bells entering the tune, next-next-next, 1-2-3, as "going up," by conventional music terminology the beginning of Hot Cross Buns is not going up but down, E-D-C or 3 2 1.     Moreover, music notation (like text in Western culture),  goes "onward" in time from left to right.   Thus, if Brad is to make his notation match the dead metaphors embodied by conventional music notation, his sequence of numbers under the bells would have to be doubly reversed—his "up" would become "down," and his R-->L for "onward" would become L-->R  (see Figure 24).6

Figure 24

Crossed metaphors

 The moment when we suddenly see one thing as another, is when boundaries of distinctions have become porous.  These are moments of instability, uncertainty, and confusion, but, in turn, they often yield unexpected moments of intense insight.   Perhaps this potential for conflict and confusion within the crumbled boundary distinctions of dead metaphors helps to account for Brad's difficulties in school.   But, at the same time, his ability to see one thing as another and to grasp the potential in moving across the boundaries of distinctions, might help to account for his depth of innovation and creative learning, as well.

"Out of that tense multiplicity of vision {comes} the possibility of insight" (Bateson, 1994).

SUSAN

The task I gave to Susan and the other young violinists was similar to Brad's, but with certain important differences.  First, I asked the children to work with the longer and somewhat more complex tune, Twinkle Twinkle Little Star  rather than Hot Cross Buns.  Second, I gave the children eight bells rather than only five, since Star includes more pitches (see Figure 25).

Figure 25

8 bells for Star

Finally, instead of telling the children right at the outset that I was going to ask them to make instructions, I did so only after they had finished building the tune.  While these differences are certainly significant, the issues of distinctions, and how their boundaries become permeable, turn out to be surprisingly similar.
The evolution of Susan's work, like Brad's, is much influenced by the structure of the tune (see Figure 26).7

Figure 26

The structure of Star

Like Hot Cross Buns, Star groups into three sections but each section is more extended, including within it two subsections or phrases.  Thus, the  beginning A section includes two phrases, a.1 and a.2, the middle, B section includes a repetition of phrase b.1, and the A section returns as the ending.
Because Susan makes use of her more extensive musical experience, I also give a skeletal view of the tune's pitch contour, including primary pitches (here, C and G) and the movement among these pitches.   Landmark events that generate more complete boundaries at the ends of sections are shown as  while the interim boundaries of phrases within sections are shown as (see Figure 27).

Figure 27

Skeleton pitch contour of Star

 In building the tune with bells, with which the children were unfamiliar, the direction of motion presented a special potential for conflict among these musically more experienced children. Musicians associate moving to the left on an instrument with going down in pitch, while going to the right is associated with going up in pitch (as on the keyboard).  But time moving on, moving "forward" is also associated with spatial direction—moving R-->L. Thus, an event that happens later than or after an earlier event, happens to the right of it (as in the Western conventions for writing and music notation).  This generates an interesting conflict between, on the one hand, body motion on an instrument and on the other, symbolic motion in a score. This conflict was made palpable as the young violinists, all of whom played the piano at least a little, worked at the task. In particular, in building the tune with the bells, if tune events went down (to the left) and at the same time onward (to the right), the young performers were presented with potential conflict. In fact, all of the gifted children, including Susan, confronted this conflict, and each instance led to uncertainty and to boundaries of entities becoming fragile.

Building Section A

At the beginning of Susan's construction of Star, her strategy looks much like Brad's.  She searches for each of the first three bells in turn, adding them to her cumulating bell path in order of occurrence.   As shown in Figure 27, Susan's bell path and action path on it, go together in synchrony, L-->R for "forward" in time and also up in pitch.

Figure 28

L-->R goes onward and up in pitch

But with the next event Susan deviates from the straight-ahead strategy. Instead of adding a new bell to the right to continue onward, she switches-back her action path to go left and back down in pitch, and also to play again the G-bell that is already present in her bell path (see figure 29).8

Figure 29

 Switch back (left) to go down and play the G-bell again

With this move Susan uses the same G-bell for two different places in the tune and for two different functions in the phrase—as a middle event and as the ending event within the first phrase.   Giving this kind of double function to a single bell was something musically untrained children rarely did.  Indeed, recall that Brad gave a double function to bells only after adapting Celia's alternate path when his initial boundaries had become porous.
 Going on with her construction of Star, Susan starts a second row of bells on the table above her a.1 row.  This is phrase a.2.    Adding bells in order of occurrence, Susan again goes L-->R even though the pitches of this phrase are descending. Thus, with regard to direction of motion, Susan gives priority to going on in time rather than going down in pitch  (see Figure 29).

Figure 29

Two phrases

    Susan's two rows of bells exactly embody the two, self-contained phrases of the A section:  one for phrase a.1 and another for phrase a.2.    (In this respect, Susan's configuration of bells is reminiscent of Celia's configuration for Hot [Figure 11].)  But there is an exception: as before, Susan re-uses the same bell for different functions.  Rather than adding a C-bell to the end of her a.2 path, she re-uses the C-bell with which the tune begins.  Thus, the bell that has a beginning function in phrase a.1 also has an ending function in phrase a.2.   Of particular importance is that in giving dual function to a single bell, Susan's action path must cross over and thus violate the boundaries of the otherwise self-contained structural entities embodied by her bell path (see Figure 30).

Figure 30

Bell path and action path for Section A

As a result of her strategies, the configuration of bells and her action path on them have the interesting feature of showing the circular and complementary structure of Section A:  two phrases, the first turning back on itself, the second turning  back on the first, ending where the tune began.

Completing the Tune Construction

Pausing only a moment after completing Section A, Susan goes directly to the G-bell in the middle of her a.1 path to begin Section B.  Playing it twice now as the beginning of phrase b.1, she crosses the phrase boundary of a.1 now to go back into phrase a.2.   And, once there, she simply follows the descending pitch series down (F-E-D) to complete phrase b.1—the completion of b.1 is a.2 (see Figure 31).

Figure 31

Phrase b.1

Repeating her b.1 action path, and again crossing phrase boundaries, Susan completes Section B, and returns to Section A, playing it exactly as she did before.
 Susan's work clearly demonstrates strategies that differ significantly from Brad's.  Most particularly, in making her bell configurations, and her action paths on them, she moves easily among dimensions and across distinctions within the task and the tune.  Throughout her construction of Star she is giving priority sometimes to pitch direction, sometimes to temporal sequence, building bell groupings that match structural entities.   But, in her action path, these structural boundaries are porous—she easily crosses over them.  Moreover, by re-using the same bell for different positions and functions within a single phrase as well as for tune events that belong to separate phrases, Susan maintains the identity of the bell as an object while simultaneously allowing that identity to change its function and character.  Could these expert characteristics of Susan's, gained from musical experience and probably at least in part from formal instruction, perhaps be likened to Brad's self-taught expertise in making and fixing complex mechanical systems that work?

Boundaries of Previous Distinctions Become Permeable;
New Distinctions Emerge

After Susan had completed building the tune, I asked her to: "… make some instructions so a friend who came in the door right now could play the tune on the bells just as you have them set up."  Looking up at me, she responded to my request by saying,  "That's ridiculous!"
Susan's construction strategy and the trace it left behind might indeed help her to see this new task as "ridiculous."   Given the task of building Star with the bells, Susan organized the bells as if they were events unfolding in time and coalescing into structural entities—much as when one sings or plays a tune.   The new task apparently called up different associations.
Asked to make instructions, she flips to her familiarity with the conventions of written music notation.  In a first quick move, Susan picks up, as a group, the bells that had formed the bell path for the a.2 phrase.   With one motion, she attaches them to the end of her a.1 bell path, thus making a single, at least visually neat, line-up.  However, at this moment, the a.2 group still maintains its integrity as a graspable entity, but its boundaries have become fragile (see Figure 32).

Figure 32

a.2 bell-group attached to a.1 bell-group

Then, with only a moment's pause, Susan again picks up the a.2 bells as a group, and inserts them, still as a group, between the bells in her a.1 path (see Figure 33).   The functional boundaries of a.2 have now been virtually absorbed into the new, singular line-up.

Figure 33

Inserts a.2 bells between C and G in the a.1 group

Finally, reversing the position of the D and F bells (rather than rotating the whole group), she also reverses the directionality of the a.2 phrase:  as long as it was a separate bell path functioning as a phrase, it was going down in pitch (F E D) and  L-->R for onward in time;  it is now an ascending sub-group (D E F) going L-->R for upward in pitch (see Figure 34).

Figure 34


Upward in pitch

     With this move, the boundaries of the once distinct structural entities, a.1 and a.2, disappear entirely and a whole new object emerges—a single row of bells no longer ordered by their entrance in the tune, but ordered by pitch property alone—low-to-high! (see Figure 35).

Figure 35

Ordered from low to high

It is important to note that Susan made the whole series of transformations, up until the last move, from a bell path grouped by phrases to bells ordered as a scale, while still keeping the initial a.2 bells intact as a group.   Here we have a specific, perfectly concrete example of the boundaries of distinctions gradually becoming so porous as to finally disappear entirely into a new and different structure—a single, general purpose instrument.
It is also important to note that the transformations occurred and the new structure emerged in response to the request to "make instructions."   With this newly constructed and, for her, more familiar instrument, where each bell/pitch is more easily found and perhaps named, Susan is now prepared to make instructions for playing the tune.

Making Instructions

But surprisingly, these seemingly obvious assumptions about the advantages of the new structure are not Susan's.  She does not begin her instructions by naming the bells 1 2 3, as Brad tried to do.  Rather, she begins by drawing six, undifferentiated squares on her paper (see Figure 36).

Figure 36

Six squares





 Susan plays the first phrase of Star on her newly ordered bells (see Figure 37).

Figure 37


Susan's action path for phrase a.1

Stopping to write down numbers under some of the squares, she makes instructions for playing the first phrase (see Figure 38).

Figure 38

Instructions for phrase a.1

Without having watched Susan create this notation, it would indeed seem puzzling.   At first glance, Susan's drawn squares are reminiscent of Brad's cut-outs used for labeling the bells.   But it quickly becomes clear that Susan's squares have a quite different meaning and purpose. The squares are the bells, one square for each bell.  She has created a pretend instrument to play on.    The numbers below the squares tell the player where and in what order to go on this instrument to play the first phrase—a kind of trail map.   Continuing on to phrase a.2, Susan completes Section A as shown in Figure 39.

Figure 39

Instructions for Section A

With bell path and action path no longer synchronized in order of occurrence in the tune, Susan must show the player how to navigate on her low-to-high ordered instrument.   This includes going first to (1), the first bell,  jumping over to (2), then to the right (3),  and also making clear when to re-use the same bell again (4) for different "places" in the sequence of tune events. Thus, the fifth square in her pretend instrument, for example, is both the 2nd and the 4th (2,4) place along the route and the first bell is both the 1st and the 8th (1,8) place along the route.
     Susan completes the whole tune by adding numbers for the sequence of places to go to play Section B (see Figure 40).

Figure 40

Action path for Section A and phrase b.1

It is significant that Susan purposefully re-organizes the bells from low to high in the service of making instructions, thereby obliterating boundaries of the embodied phrases in her first bell path.  But, having done so, Susan does not use the ordered set to label the bells in the conventional manner—i.e., 1 2 3 4 5 6.   And this despite the fact that she has certainly learned to use conventional labels to refer to the pitches in the scale-ordered series.  Instead, seeing the bells as an instrument, she uses numbers to mark a temporal sequence of places to go on the instrument—an action path in contrast to a symbolic notation path.    Paradoxically, Brad, who is described as having trouble using symbolic expressions in school, invents and uses a symbolic representation to make instructions in contrast to Susan's iconic trail map.9
It seems quite clear that the task of making instructions was "ridiculous" for Susan as long as the configuration of bells on the table remained a trace of the process of building the tune—a series of cumulating events grouped as structural entities as she went along. And yet, her instructions are also a series of temporally cumulating events, but without any hint of the structural groupings of her initially built bell path. As the boundaries of distinctions become permeable, directions in time and space shift and stumble over one another—like the ping-pong ball that gets faster and stops.
Still, by making a configuration mappable onto her familiar violin, the new task becomes not so ridiculous. Indeed, the strategy Susan uses in her instructions is reminiscent of the Suzuki approach with which Susan began her violin studies. In this approach, children are taught to play tunes by being told where to put their fingers on the violin strings—essentially an action path on the instrument. For instance, instructions for how to play the first phrase of Star on the violin are:
"open A string,10
open E string,
first finger on E string,
open E string,"
And for the second phrase, a.2:
"third finger-second finger-first finger on the A string,
open A string...."
     Indeed, tipping Susan's pretend instrument so the squares run vertically, and mapping Susan's numbers and sequence of places on her pretend instrument onto corresponding places on the violin strings, the mapping fits quite perfectly (see figure 41).

Figure 41

Susan's instructions mapped to the violin strings

Susan's multiple representations

Susan's earlier phrase-embodied bell path and her new, scale-ordered bells along with her instructions, gives us a multiplicity of representations.   Each representation asks us to focus on different kinds of distinctions, different kinds of entities, and different relations among them (see Figure 42).
 
 

Figure 42

Multiple representations

     But to make the multiplicity useful, we must be able to choose among them depending on when and for what purpose we are asking them to serve. To do that, we must be able to move from one representation to another, and that involves making permeable the boundaries of the respective distinctions. If we succeed, we also gain a greater potential for coming to hear more complex music in new ways.   And perhaps most important, as we watch the children's own multiple inventions unfold and recognize the multiplicity of possibles, we are also developing a basis for appreciating creativity as a process of deep learning.

  REVISITING

These two brief episodes give us the privilege of glimpsing learning as it is happening. In following the children's work, we have also been witness to creativity—the creativity of learning. We tend to speak of such moments of creativity as intuitive—emergent, elusive, even mysterious. Perhaps this is because such moments also conspire towards uncertainty, which we fear, along with the unexplainable and the unpredictable (Meyer, 2000). So we seek resolution by contriving stable distinctions—measuring, ordering, classifying, finding patterns, on the belief that this is how to "make sure."
But in watching the children's creative learning, we see learning not as an arrival at certainty, but as a continuous process of becoming: boundaries of distinctions become permeable, uniquely prefacing transformation and the emergence of new distinctions. Momentarily holding time still in static space, new invented modes and media of representation happen. And these, in turn, generate new distinctions, spawning new kinds of entities, and new sorts of relations among them.
 Christopher Hasty, in talking about musical experience, might also be talking about creative learning:

To take measurements or to analyze and compare patterns we must arrest the flow of music and seek quantitative representations of musical events.  But music as experienced is never so arrested. To the extent we find it comprehensible, music is organized; but this is an organization that is communicated in process and cannot be captured or held fast.  What we can hold onto are spatial representations (scores, diagrams, time lines) and concepts or ideas of order—fixed pattern, invariance, transformation, hierarchy, regularity, symmetry, and proportion.  Certainly such ideas can usefully be drawn from musical organization presented as something completed and fully formed.  However, a piece of music or any of its parts, while it is going on, is incomplete and not fully determinate—while it is going on, it is open, indeterminate, and in the process of becoming a piece of music or a part of that piece. (Hasty, 1999, p. 4)
 

REFERENCES

Bamberger, J. (in press).  Restructuring Conceptual Intuitions Through Invented Notations:  From Path-Making to Map-Making, in S. Strauss (ed.) The Development of Notational Representations, New York: Oxford University Press.
Bamberger, J. (2000). Developing Musical Intuitions. New York: Oxford University Press.
Bamberger, J. (1991/1995). The Mind behind the Musical Ear. Cambridge: Harvard University Press.
Bamberger, J. & Schön, D.A.  (1978). The figural<-->formal transaction; Working paper #1.  Cambridge: MIT Division for Study and Research in Education.
Bateson, M.K.  (1994). Peripheral Visions: Learning Along the Way. New York: Harper Collins.
Peter Gzowski, Interview with Alice Munro.  Montreal: The Globe and Mail, Sept. 29, 2001.
Hasty, C. (1999) Rhythm  as Meter. New York: Oxford University Press.
Lakoff, G. (1980).  Metaphors We Live By (with Mark Johnson) Chicago: University of Chicago Press.
Lashley, K.S. (1969). The problem of serial order in behavior, in  K. H. Pribram (ed) Perception and Action. Baltimore, MD: Penguin Books.
Meyer, L.B. (2000). Music and emotion:  Distinctions and uncertainties, in P.N. Juslin & J.A. Sloboda (eds.)  Music and Emotion. Oxford: Oxford University Press.
Schön, D.A. (1963). Displacement of Concepts. New York: Humanities Press.
Schön, D.A.  (1979).  Generative metaphor, in A. Ortony (ed.) Metaphor and Thought. New York: Cambridge University Press.
Wittgenstein, L. (1960). The Blue and Brown Books. New York: Harper and Row.



1. Peter Gzowski, Interview with Alice Munro, The Globe and Mail, Sat. Sept. 29,
  2001.
2. For those who are unfamiliar with the tune, I show "Hot Cross Buns" in standard music notation, below.  However, like all notations, this one makes assumptions, assumptions quite different in kind from Brad's.  Being privy to this assumed knowledge, readers are also at risk of missing the cognitive work and the transformations that Brad's understanding undergoes.

3. For a different and more complete version of Brad's work, see Bamberger (in press).
4. An X below a bell stands for a strike on that bell. Brad and many of the other children played the middle figure with just 3 repeats instead of 4 as given in the tune.
5. Evidence that Brad was quite capable of distinguishing what is conventionally termed "up and down" in pitch was clear when, on hearing (not seeing) the beginning of the same tune played by the computer synthesizer, he said quite spontaneously, "Oh, it goes down."
6. K. S. Lashley in his classic paper of 1951 says, "The syntax of the act can be described as an habitual order or mode of relating the expressive elements (symbols); a generalized pattern or schema of integration which may be imposed upon a wide range and a wide variety of specific acts.  This is the essential problem of serial order; the existence of generalized schemata of action which determine the sequence of specific acts, acts which in themselves or in their associations seem to have no temporal valence (Pribram, 1969, p. 525).
7. Of course, in describing this structure, I am defining the boundaries and making the distinctions that I have internalized, and have come to take for granted.
8. I am assuming that the reader realizes from the skeleton diagram of Star that bells within a phrase are struck twice while the bell at the end of a phrase is struck once.  I have not indicated bells that are struck twice along Susan's action path partly because Susan includes neither repetitions nor the rhythm of the tune in her subsequent "instructions."
9. For more on the differences between symbolic and iconic representations of music, see Bamberger, 1995.
10."Open string" means to just bow the specified string without fingering it.  The pitch contour in this disposition is the same as in Figure 24 and the actual pitches are:  A E F# E; D C# B A.


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