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LC1b Text

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Source materials

This material consists of 21 protocols, primarily transcripts from audio recordings, some of which have been absorbed directly into the chapter “Sketches of Natural Learning,” and others used in “The Development of Objectives.” Some are in character more like the Vignettes of The Intimate Study, which derived from them. All are presented here, scanned from original typescripts, with addenda from Rob’s work and computer printouts, to the extent preserved. For pointers to source material about Rob in The Intimate Study, see LC1c-Text Rob in TIS Vignettes

Protocols 1-21, with supporting materials; Robby Lawler at 7-8 years







Publications Reference xref.


Detailed audio transcript: Introducing Rob to TG prograns; relating that to using Zoom; confusions






start using ZOOM; attempt to go from there to Logo TG
teaching bugs and problems
Miriam: adding (minimal)





Other approaches to programming intro; another teaching failure.






Triangle; Starting House




LC1bA5: Retrospective: a house that failed


Drawing a Fox; using TG to assemble shapes.




LC1bA5: Retrospective: a house that failed


Number represetations; Adding




LC1bA3: Complex Minds and Homely Circumstances

Adding right to left




LC1bA2: Third Encounter with Number
LC1bA3: Flexible Knowledge

(transcript year error)

Approaching programming by changing existing programs





Thanksgiving 1976

Instruction as Invention




LC1bA3: Instruction as Invention, Memorization and Engaging Games

Inventing Negative Numbers:
Proving that many exist.



LC1bA3: Complex Minds and Homely Circumstances

Christmas 1976

Place Value and Scale Conversions



LC1bA3: Natural Confusions

a few days later

Using Multiple Representations



LC1bA3: Mentioned but not quoted extensively.

Multiplication: the TIMES program




LC1bA3: Guiding Constructive Analogy
and The Obvious is Unknown

No sense of commutativity
Estimating in Multiplication



LC1bA3: The Obvious is Unknown

Feb. 1977

Showing a Solution



LC1bA3: The Urge to Instruct

Transfering Dimensions




LC1bA3: Concrete Action in a Non-square World

a month after protocol 14

Commutativity; the relation of money to concrete calculations




LC1bA3: Multiplying (by Analogy with) and Money

A Companion Inspires some musical interest




Not published.

Numbers and Letter codes
Beyond his grasp




Not published.

April 1977

Emphasizing a Problem



LC1bA3: The Urge to Instruct

The Multi-add Algorithm




LC1bA3: The Natural Confusions


LC1bT20 Protocol 20

Included Text Pages

RAL protocol 20

Included Materials



Vn 001.01

Everyday Calculation


Miriam suffered a queasy stomach today, so she didn’t join us at our evening meal. She lay on the loveseat near our dining area. The speakers for our radio are directly behind the loveseat, thus more enforcing Miriam’s attention than ours.

During a radio commercial we others chose to ignore, Miriam burst out in disbelief, “That would be 60 dollars.” When I asked what her reference was, she explained that if the four of us attended a certain fixed-price dinner (one we had no interest in) the total cost would be 60 dollars. “How did you figure that out?” I asked. “Did you learn how to multiply already?”

Miriam coupled a disclaimer of any knowledge about multiplication with one of her ‘dumb-Daddy’ looks and explained that each meal cost 15 dollars and she knew that 15 plus 15 were 30 (which, as she added in parentheses, accounted for 2) and another 15 and 15 make another 30. The two 30’s making 60 dollars gave her a conclusion and accounted for the four meals she knew we would want.





As I sat transcribing the dialogue from recent logo sessions, I heard Robby inquire of Gretchen, at work in the kitchen, how many were 5 twelves. Gretchen simplified the computation by elaborating the problem: 5 twelves is half of 10 twelves. How much is 10 twelves? As Robby worked away on that problem, Miriam, playing at a puzzle within earshot of that conversation, piped up: “the answer is 60.”

Poor Robby! How frustrating when working on a different problem to be prevented by some one else’s interjecting the ‘correct’ answer. And yet, Miriam did have it right. I was quite worried that she had computed the answer by summing twelves (which Robby could have done, albeit with some difficulty and uncertainty) while he wrestled with the transformed problem,

Gretchen had been watching Miriam. She saw Miriam compute 5 twelves by finger-counting thus: 5, 10, 15, 20, 25, / 30, 35, 40, 45, 50, / . . . 60. Thus Miriam’s procedure is more primitive than Robby’s but it is also more sophisticated. She makes use of the commutativity of basic arithmetic operations at every turn. Several weks ago, Miriam gave direct evidence of her use of commutativity in adding. Mimi Sinclair asked her: “How many is 17 plus 6?” ’23’ Miriam responded counting up from 17 on 6 fingers. When the query turned to 6 plus 17, Miriam responded with no hesitation, ’23, because it’s the same problem.

I speculate that she uses commutativity because it permits her to proceed to an answer which costs her little if wrong; Robby, more concerned with the correctness of his results than the unimpeded progress of the computation, is more inclined to ask for advice than to trust to a property, commutativity, which can give him an answer but one about whose correctness he is uncertain. This speculation may demean the actual extent of Miriam’s understanding.



Productive Cheating


Today was a difficult day. Snow in mid-May for a beginning. Before that problem appeared, Miriam came early with me in to Logo in our joint expectation of going to the Coop to buy a hula hoop. With that option closed by inclement weather, Miriam pushed me early in the afternoon to proceed with the day’s experiment. We proceeded as described in Logo Session 4.

Gretchen and Robby reached the lab later and Robby chose not to engage himself in my work with Miriam, preferring to play with SHOOT by himself in the central portion of the Children’s Learning Lab. Sam Lewis, another child frequently at Logo and a year older than Robby, played with him in the lab at that time. When Miriam declared a break from our work in writing a story, I discussed (with Gretchen) the children’s use of SHOOT and how I was awaiting their discovery of how to cheat. Instead of using the SHOOT : DISTANCE program to project the turtle into the target (which evaluates his location after movement and immediately judges the movement a ‘hit’ or a ‘miss’), one may locate the turtle within the target with a series of forward and turning commands; then, guaranteed of a bull’s-eye, execute SHOOT 0 to register one’s score. Such was my explanation. I noted that the most efficient cheat would be to execute a ‘HOME’ command (which puts the turtle in the target with a single command), then SHOOT 0.

Because of the snow and Miriam’s disinclination to proceed with writing a second story, I suggested Gretchen take the children home while I proceeded with some work they could not be involved in. Robby was most eager to stay and play with SHOOT. After a slow start in the first 3 Logo sessions, Robby was developing skill quickly. He had already, as he noted, scored 5 points that afternoon, and wanted to go on while doing well. I reluctantly agreed. I agreed because I believe the children should be allowed to follow active interests. My agreements was reluctant because I did not want Robby to make further significant advances without my observation. This is precisely what happened. As we discussed the day at supper, Robby noted that he had a good afternoon. His second use of SHOOT garnered him 9 points, giving him a total of 16 (? ). . . this may include in his calculation points from the 3 earlier sessions). Robby then added he had figured out how to score every time. “How?” Robby explained that after drawing the target, the turtle goes ‘Home’ before going somewhere [a setting of his heading and location to random values] and that if one were to key ‘H’ or ‘Home’, then SHOOT 0, he would score every time. To be certain Robby was saying what I thought I posed these questions.

Bob Suppose you key ‘H’, carriage return?
Robby The turtle goes to the center of the target.
Bob Like this?
Robby Yes. Then you say SHOOT 0. illustration:
Bob And what does the turtle say? target and turtle
Robby Ouch. Your score is 1.

I asked Robby if Sam had showed him that and received a negative answer and the claim that he had figured it out himself. I recall informing Robby, before his second terminal session of the day, that because of his squabble with Miriam in Logo Session 3, I changed the SHOOT program so that if the turtle were within the target after execution of GO-SOMEWHERE, he would be made to GO-SOMEWHERE-ELSE, i.e. land at a different location.

Miriam then confided to Robby in her most conspiratorial stage whisper: “Robby, you shouldn’t have told me; I’m going to do that every time.”

I pursued this question, asking Robby whether he had used this new idea to score all his points during the afternoon. Robby denied it, saying the trick didn’t work. I was surprised (it should work perfectly) and asked why not. Robby said the computer would respond ‘You didn’t tell me how to H or Home.’ I asked if he knew it wouldn’t work and how. The answer was that he hadn’t tried it, thus he couldn’t say why he knew it didn’t work.

Interpreting this incident depends on how open Robby is with me, generally, and on the extent to which his final comments were an attempt to delude Miriam by convincing her that his discovery isn’t worth attempting. Robby is usually quite open with me. Nonetheless, given the intellectual rivalry between the children, I would not be surprised at Robby’s attempting to throw Miriam off the track of a discovery he made which his revealing to her had made useless to him. An alternative explanation for Robby’s not trying the “Home SHOOT 0” cheat (and perhaps the impetus for it’s coming to his mind) is my explaining that I had modified SHOOT to forbid those lucky landings of the turtle within the target. He may have believed any time the turtle were found in the the circle at the beginning of executing the SHOOT procedure he would GO-SOMEWHERE-ELSE before being shot at the target. [Indeed, such is possible and is the way one would prohibit the ‘forward and turn commands/SHOOT 0’ cheat if one were so inclined.]

This incident promises further interest in that part of my intention is to guide Miriam’s concerns from getting a correct answer to attending to the process and operations by which one can achieve an answer. Her obvious engagement with the desire to succeed immediately will lead her to pursue Robby’s discovery. I expect and intend to have her succeed thus. My following countermove (which will be to relocate the target off center screen) may show how too simple “an answer” is inadequate and must give way to deeper comprehension of process by which “an answer” is developed. When, later, both children realize they can still succeed by deferring execution of SHOOT until the turtle has been relocated within the target circle through forward and turn commands, they will have extracted all the value they can get from the use of this introductory game.



Openness and Reticence


Miriam rarely discusses her activities in kindergarten unless directly asked. I was surprised today when Miriam volunteered that her class had put on a little play. I asked what part she played.

Miriam I didn’t play any, Daddy.
Bob Oh, didn’t you want to?
Miriam I read the story and the others did it.
Bob Was it hard to read?
Miriam Yes, very hard, but I did it good.

It may be that Miriam’s taking her reader Friends Old and New to school had something to do with this selection.





When collecting the data reported in ‘Pre-Readers’ Concept of the English Word,’ I was shown by one of the children, Tina, a 3×5 notebook her mother gave her for writing down any words she wanted to learn to read. I bought Miriam such a notebook (a green one) some weeks ago for such a purpose. She asked me how to spell ‘floccinaucinihilipilification’ and proceeded to print that in her notebook (today she refused to give me that datum because she still wants to learn to spell that word). Her uses of the notebook are more various than my plan. I find: a list of models she would like to have; an upper and lower case alphabet page (the first set done by Robby as a set of patterns, the second Miriam’s copy); the address and telephone number of her friend Maria. The most frequent use of that notebook has been to make greeting cards for her friends: typically Miriam draws a picture, then prints “For ___” and “Love Miriam”. (It also has many blank pages.) Clearly, Miriam’s use of this notebook has been richer than my conception

I recently purchased another such small notebook (a blue one) for jotting down notes about what Miriam does during the day. Miriam saw that notebook and asked if she could have it for making cards. A conjunction of things occurred. Miriam wanted the notebook; the time she spends in school is one wherein I can’t observe what she is doing; an observation several years ago by Sarene Boocock at an AAAS seminar that children themselves had access to data most researchers could not get at and that one should consider enlisting them as agents of data collection. I told Miriam she could have the blue notebook if she would write in it the things she did in kindergarten. Miriam refused, saying she wanted the notebook for drawing pictures. When I said she could not have the notebook for drawing pictures, Miriam responded, “If you don’t give me the notebook, I won’t even tell you what I do.

At the current round of negotiations, Miriam agreed to use the notebook as I wanted when I agreed to buy a replacement for her green notebook after she uses up all the pages. Her intentions are suspect though because of the final inquiry she made: “If I do a rotten job can I keep the notebook and not write down what I do?

Contrast now the relative openness of the two children, Robby and Miriam. For reference, consider the protocols and 21 (on adding units of English length and on multiplication, respectively) from the series of Robby’s arithmetic development. These protocols are exemplary of a child’s uninhibited exposition of his thought processes; at 7, Robby is a ‘loud thinker’. Contrariwise, Miriam’s characteristic behavior shows her inclination to display a perfect result, a reluctance to exhibit pre-competence confusion

One may laud such a stance as reflecting essential good taste (for who wants to overwhelm the possibility of communication with the mess inside every mind). However, for an ignorant child (is anyone at six years anything else) such a performance criterion is unrealistic and counterproductive. One of my objectives is to render Miriam less sensitive to revealing her imperfect comprehensions in the expectation tha her doing so will provide guidance to those who are willing to help her learn.



The Clever Hack (2)


At dinner this evening we talked over some of the incidents of the day. It had been one of novelty for Robby. His friend John (the son of a former naval officer with whom Robby shares an interest in naval battles and model building) came to Logo with Robby as a stopping point on a visit to the Hart Nautical Museum (a museum of models) in MIT building 5. Miriam stayed at Logo with me (the events are recorded in Logo session 7) while Gretchen took the boys to the museum. By the time the boys had returned, Miriam and I had finished our work for the day. While we played otherwheres with a hula hoop and some tennis balls, Gretchen and the boys went into the music room where Robby introduced John to SHOOT. When Miriam heard of this at dinner, she said, “I should have shown John the clever hack.”

The immediate surfacing of this suggestion to Miriam’s mind made me curious about what role, if any, it had played in Robby’s introducing John to SHOOT. These selections, from the transcription made by the chance of the tape recorder’s having been left running, are extracted from Logo session 7.

GretchenWhy don’t you play SHOOT?
Robby That’s a good idea. (To John) Let’s play SHOOT.
John What is this SHOOT?
Robby [Robby logs into Logo, reads the file “SHOOT from secondary storage] You’ll want to see SETUP. Are you looking at the screen?
John Yeah.
Robby [executes the procedure SETUP. The procedure clears the terminal display, then puts a message on the logging portion of the display and creates a ‘screen’ (a movement domain for the turtle); the procedure increases the screen size in steps until it reaches the standard size, thereby pretending to simulate the distant first appearance of something coming into view. While the screen size increases, messaged are printed on the logging portion of the display. Robby reads them.]
Robby Look in the sky!
It’s a bird — it’s a plane.
No, it’s super turtle.
(The SETUP procedure then draws a bull’s eye on the screen and sets the turtle at random location and heading.)
Gretchen I never saw that before. Why don’t you explain to John what you are doing? (When Robby fails to respond, Gretchen continues), The object. . . this is the turtle, and the line in the middle shows which way he is pointing. The object is to get the turtle pointing towards the target. And then say SHOOT-
Robby Something.
Gretchen A certain distance. . . and see if he stops, if that gets him to the target.
Robby [keying]
GretchenRobby has just made a turn of 90 degrees. You try to get it lined up. Now he has to figure out how far to tell it to go.
Robby Yes! (meaning his shot hit the target)
Gretchen Just made it. If it’s a hit, the turtle says ‘ouch’. [exits]
John(laughs) . . . Can I try?
RobbyNo. I want to show you something. [keying] H . . . (then realizing he has omitted the carriage return) Oh . . . Now SHOOT O. It worked! Isn’t that a great trick?
RobbyIt’s sort of easy.
JohnHow do you do it? Show me how to do it. I want a turn.
Robby(after shooting successfully at the target, but in no way describing what he did). Here, John. You do everything now.
John Well, you have to help me.
Robby O.K. Right or left?
John Right. I want to do right . . . What do I do?
Robby I want to go home.
John What button do I push? This one? To aim it at this?
Robby Oh, . . . just do Home, SHOOT O.
John No, I don’t want to.
Robby [keying in Home, carriage return, despite John’s preferences] Just say SHOOT O. He’s already at Home . . . I want to go to my house, do you?
John What would we do there, Robby?
Robby Play soccer . . . play in the tree fort.
John SHOOT O. Oh, come on. (shortly after this point, the tape ran out)

Out of three cycles of SHOOT (from setting the target through hitting it), the clever hack was used for two, the basic turtle commands being used for the first cycle. In the second cycle, Robby clearly paraded for John a solution to a problem John could not appreciate. The second use of the clever hack was to short-circut John’s interest in finding out “how to do it”; by appealing to John’s interest in achieving the objective, Robby attempted to circumvent the more time consuming process of showing how to play the game. The mark of the clever hack in both uses is its salience; whenever little time is given to the problem solving process, either through motives of setting a dramatic effect or simply to reach a quick solution, the clever hack comes into its own.



Tic Tac Toe


Years ago I bought a tic-tac-toe game for playing with Robby. The board is 12 x 12; the X’s and O’s are large yellow pieces. At that time I taught Robby a single strategy for playing the game: look for two ways to win. Robby quickly became quite good at the game. Miriam, at age four, learned that one took turns and that winning was getting your three pieces in a row. She did not take the strategy instruction so readily. The children played with the game with different competences: Robby winning with his strategy; Miriam winning frequently enough when he made errors to be satisfied. As Robby came to make fewer errors, Miriam played less, the pieces falling to other uses: the dog chewed one of the X’s, the play group used an O as a hockey puck.

Tic-tac-toe came forward again as a game at a recent visit to the Children’s Museum. Miriam lost to the computer when it, moving first, chose the center square. When she moved first in the center square she never did better than a draw. Robby beat the computer with a first move corner choice (let Robby’s sequence be 1, 2, 3, 4, and the computer’s A, B, C, D):

 1  |     |  3
 |  A  |
 B  |     |  2

By move 3, Robby had forked the computer, had “two ways to win.” Since neither Miriam nor his friend John had done better than a draw, he gladly showed them the gambit on his next turn.

Tonight, after dinner, Miriam asked me to play tic-tac-toe with her. She was quite familiar with the terminology of two ways to win and implicit victory whenever she achieved a forking pattern. She understood and accepted the terminology of a forced move.

Whenever Miriam had the first move and chose the center square, and I chose a non-corner, she consistently won. This was true regardless of orientation of the board. Let these two games represent the rest as well (Miriam’s moves are the digits, mine the letters):

    |     |  2       2 |  A  |
 --------------    --------------
    |  1  |            |  1  |
 --------------    --------------
  B |  A  |  3       3 |     | B

In both, her “two ways to win” victories were not noted as being ‘tricks’ of any sort.

I gradually altered my responses to her first move center choice until I chose a corner square in responsse to every center square first move. Making no mistakes, we come inevitably to a draw, thus:

 2 |  C  |  5        3 |  B  |  5
 --------------     --------------
 D |  1  |  4        4 |  1  |  D
 --------------     --------------
 A |  3  |  B        A |  2  |  C 

Miriam began then a new gambit, the first move corner choice. When I responded with a center square choice (as did the computer at the Children’s Muuseum), Miriam had her two ways to win by the third move. Miriam described this new gambit as her ‘dirty trick.’ I was quick to tell her I thought it was not a ‘dirty trick’ but a ‘good trick,’ and we both later referred to it that way.

After being beaten several times by her new gambit, I blocked its effectiveness by refusing to make the center square response to her opening move. When I responded thus

  1 |      |
    |      |
    |      |  A   

Miriam complained vociferously. It was clear she did not know what to do. She eventually proceeded by placing her piece anywhere. In another variation, she was quite surprised at my victory:

  1 |  B  |  3
    |  C  |
    |  A  |  2

She was so intent on her ‘good trick’ she failed to see the simple victory she might have achieved on her third move. This was the only ‘mistake’ I recall Miriam making.

I conclude from these observations that Miriam’s strategies are very specific in nature. When consolidated, they may be orientation insensitive, but not even orientation insensitivity is immediate. Further, Miriam made her ‘mistake’ because she had not yet integrated her ‘good trick’ with the primary rule of tic-tac-toe: make all forced moves. One might better conceive of her “two ways to win” less as a strategy than as a more complex and various, more immediate objective, more immediate than the victory criterion of three pieces in a row.



Magic Words


While she was in the kitchen and returning to the dinner table, I asked Miriam to bring me something. She asked, “What’s the magic word?” My grandmother avers that the ‘magic word’ is ‘please,’ which can make so many things easy that would be impossible without it. We have never played such a game in our house; we try to use courtesy but do not require it.

I told Miriam I did not know the magic word and asked for a hint. “It begins with a ‘p’ and ends with an ‘e’.” This is the sort of hint Miriam has given in playing the game “I am thinking of a word” (described in ‘Pre-Readers’ Concepts of the English Word’). So I guessed: prime, purple, people, pupae, pleistocene, prune — all to no avail. “It’s ‘please,” Miriam chortled as she went to get what I wanted.



Magic Words


Miriam tried again her ‘magic word’ game, “It begins with ‘p’ and ends with ‘e’.” The choice of time was unfortunate; I was in a hurry to some other purpose, so I snapped back “please.” “That’s not it,” Miriam claimed, for who can tolerate any conundrum’s immediate solution? Sensing my impatience, she laughed and said, “It’s a pickle.”

Later in the evening, when we were both unhurried and glad of the other’s company, Miriam asked for a bedtime story. Since she’s learned to read, Miriam’s bedtime stories have changed. We read many stories to her before, so now she reads to us. This evening she climbed into my lap and read two library books by Sid Hoff: Who Will be my Friends and Thunderhoof (a book she much enjoyed my reading to her last year). Miriam read both books perfectly; then, as I told her it was bed time and she should go for her bath, she inquired as to what was the magic word. ‘Please’ was rejected outright. “No. It begins with a ‘t’.” When I complained that such was too weak a hint, that there were ever so many words beginning with ‘t’, Miriam added that it ended with an ‘e’. I tried, with little hope: tie, and twine, tweedledee and terpsichore. “No.” Another hint, please? Miriam replied, “It has to do with hands.” Three, perhaps? “And it makes people laugh.” So I tickled her until she fell off my lap and left, supposedly to her bath.

To interpret this series of events, one need know that Miriam’s
current favorite of rhymes is this doggerel couplet:

Tickle, tickle, rhymes with pickle.
If you laugh, give me a nickel.

Miriam enjoys being tickled and I oblige her.

So, please goes to pickle and pickle to tickle: obvious connections after the fact with sufficient data available. But who would dare predict the next magic word ?



A Willing Subject


Today was the most difficult experiment of the initial series, separating the variables implicated in the flexibility of bending rods (Cf. The Growth of Logical Thinking from Childhood to Adolescence, Inhelder and Piaget). Of the many experiments through which one may distinguish concrete from formal thinking, this one shows best the distinction between conceiving of things as objects with properties which may occasionally be of interest and conceiving of objects as instantiations of intersecting ranges of properties of which some may be relevant to its use in a specific context.

How could Miriam possibly be interested in distinguishing the contribution of different properties to rigidity? How many adults could, or would care to, distinguish between the relative importance of a rod’s thickness parallel to a force and its cross-sectional shape? How to quantify, or even balance, such factors is not obvious to the untrained intuition at any age. And Miriam is 6, not 16. Need one not subject a child to exorbitant pressure to have her work at such an experiment? If any experiment in this initial series is vulnerable to such a criticism (either through its very nature or through my execution) it is this experiment.

I was hurried into the bending rods experiment by Miriam’s declaration that she would not come to Logo today unless she could do it. Despite feeling imperfectly prepared, I acquiesced in her demand since the obvious problem was that she felt ‘left out’; I had done the experiment earlier with Robby for practice. During the preliminaries wherein I hoped to establish with certainty a common terminology for the experiment (for a description of this work see Miriam at 6: Bending Rods) Miriam was rambunctious. When not interested much in my questions, Miriam began to create chaos by throwing around dice and balls, which were the experimental materials. I stopped the experiment and we had an argument of this general form:

Bob I won’t tolerate your throwing things and causing this confusion; it will ruin the experiment.
Miriam Then I won’t do it. You’re asking me dumb questions. I’ll play SHOOT instead.
Bob That’s not fair. You demanded we do this experiment today and now we have to finish it.
Miriam I never promised I would do all your experiments. I only said I might do them.
Bob That’s not true.
Miriam I wish I was a baby again so I wouldn’t have to do these experiments.
Bob But you’re six now and doing something important.

Miriam reluctantly agreed to continue; she agreed because of the pressure I put on her, because, in effect, I made her do it.

Despite this grim start to the main experiment, Miriam quickly became engaged in working with the materials and finding “which one is the champion” for bendings. She exhibited no interest at all in separating the variables; consequently, the latter part of the experiment bored her silly.

Riding home from Logo, I told Miriam I felt bad because I thought I was pushing her too hard in the experiment. She said she didn’t want to do any more like that, and I replied that we would do no more of that sort for several months. I expanded that we should think of doing other kinds of things. That this thesis project could be not just going to Logo, but going to other places too. Part could be my coming to kindergarten with her; another part could be visiting friends and family and telling them about our project; another part could be trips to places around Boston. I noted that she had been to the Harvard Peabody Museum and I never had; she could take me there. Miriam said we should do that soon.



A Willing Subject


As the children left for school this morning, I asked Miriam if she wanted to come to Logo today. She said she would not come. When I asked why not, Miriam replied, “I just need a day off.” After the previous day’s tensions, it seemed a most reasonable request.

I planned at Miriam’s return from school a small shopping trip, with perhaps a detour to a favorite playground. Miriam came bounding in from the school bus. “Hey, Dad, let’s go to Logo.” I objected that I had nothing planned to do for the day because I thought she didn’t want to go. “That doesn’t matter,” she said. “Let’s go anyway; I’ll play with SHOOT or something.” When I asked why she changed her mind, Miriam said, “I don’t know. I just did.”



Hula Hoop Analogies

One of the most active foci of developmental incidents so far has been Miriam’s use of the hula hoop. Four separate incidents come together as activities centered on this toy.

How we came to buy a hula hoop
Our family was having dinner at the house of a Cambridge friend. My children had been playing with Jenner (my friend’s 5 year old daughter) during the late afternoon. When I arrived from the lab, I found the three children, two bikes, and a red-white-and-blue striped hula hoop on the sidewalk. Since it fell my lot to put the toys away, I noted it well.

Because Miriam suffered some confusion about right and left turns in using SHOOT (see, for example, Logo Sessions 1 and 3), I decided to undertake “playing turtle” with the game (i.e. playing SHOOT with the floor taking the place of the display screen and Miriam and me taking turns being the turtle and being the turtle commander) Which we did in Logo Session 3. With this intention, I manufactured a hoop from some polyethylene tube lying in a pile of oddments in the music room. The hoop was adequate for playing turtle, but not as a hula hoop (Miriam attempted to so use it). Miriam suggested that we buy one at the Coop. (We had walked there the day before to buy the puzzles for Logo Session 2). Miriam had not seen any hula hoops at the Coop. When I asked her if she knew whether or not they were sold there, she said, “Maybe.” Other lab members had seen them on sale there, so we agreed to get one from the Coop before our next session.

The following Monday snow kept us from walking to the Coop. I asked Miriam if she would mind my buying the hula hoop the next day before she came to MIT. Miriam agreed to that on condition that I buy one with red-white-and-blue stripes. I argued the Coop might not have such a kind. In that case, Miriam responded, she would have to pick one out. Luckily, Jenner’s hoop had been purchased at the Coop also.



The Lemon Twist


I had purchased the hula hoop in the morning and was setting up the music room for our later use when one of the boys in an on-going class from CAPS (the Cambridge Alternative School Program) asked if he could use the hula hoop. After doing a hula, he let the hoop fall to the floor, slipped a foot under the hoop, and rotated it about one leg, raising the other foot so that the rotating hoop would not strike him in the ankle. I was impressed; I had never seen anyone do that with a hula hoop. But I had seen Miriam do a similar thing with one of her toys, the Lemon Twist.

The Lemon Twist has been one of Miriam’s favorite active toys for some time. Having seen it advertised on a TV commercial, she bought one with her own money. (This was the first such purchase she ever made). The toy has a hard plastic lemon at one end, connected to a small loop at the other by a piece of tubing about 18″ long. A child slips one foot through the loop, then kicks in such a way as to cause the attached lemon to swing around that leg. I remember the day last spring when Miriam bought the toy, her first trials, her showing it to older friends, her watching them, and her slowly developing skill.

This afternoon Miriam was delighted to find her new hula hoop.
It was perfect, even having the marble inside as did Jenner’s. I mentioned to her the boy from CAPS, how he made it go around on his leg. Miriam put her foot under the hoop and kicked it a few times. “Like that?” Obviously not. “I don’t know how he did it, Miriam, but he made it work just like your lemon twist.” With two or three tries, Miriam was able to make the hoop circle her leg several times at each execution



Doing the Hula


Despite Miriam’s success with the foot centered hula twist, she was unable to keep the hoop from falling down. She holds the hoop with both hands in front and a part of it against her back. She throws it in one direction or the other, moving her trunk in no clear way (it’s very hard to see any pattern because the hoop falls so quickly).

Over the following days, several people showed their skill in the Logo foyer. (Playing with the hoop was a favorite pastime on Miriam’s breaks from our sessions). Sherry Turkle claimed having once been champion of Brooklyn and gave a demonstration. So did Donna and many of those who wandered through. Miriam improved rapidly. Her later description of how to keep the hoop from falling: “It’s easy. Just keep pushing your belly in and out,” I believe puts at the surface what she saw as significant in her observations of others’ practice.



The Bicycle Analogy


During a break from Logo Session 7, Miriam discovered that the hula hoop will stay upright if rolled. For the past several days, maybe the past two weeks, Miriam has attempted to ride her bicycle without training wheels. She received one hint, one good piece of advice from Jim, our neighbor: if you try to go fast on the bike, it will stay up. Miriam has succeeded through doing that.

When I asked her now why the hula hoop stays up instead of falling over, she said, “Well, because I make it go fast.” When I asked if there were anything else she knew like that, Miriam replied, “Yeah, sure. The bike.”



Ping Pong Balls


Miriam continued playing with the hula hoop at Logo throughout this week. Since she is willing to watch other people and listen, adults incline to show her the things they enjoy and can do. This has caused me a problem. I will elaborate.

At the beginning of the project, Miriam underwent a number of experiments to permit the probing of her skills and understanding. One of these experiments involved showing her how to make a ping pong ball slide away and then return as an initially imparted backspin overcomes the impetus of its forward projection. (This experiment is described in The Grasp of Consciousness, Piaget (1974 French, 1976 English)). Since the time of that experiment, Miriam has been, whenever she has a ping pong ball at hand, making it slide away and spin back to her. She has shown this game to friends in the play group. The back spinning phenomenon is clearly one that engaged her interest.

A secondary intention of mine in buying the hula hoop was to conduct with Miriam a follow-up experiment to explore how easily she could generalize her ping pong ball knowledge to the similar back spin phenomenon with a hula hoop.

As I passed through the foyer a few days ago I heard Donna say, “Miriam, did you ever see this?” as she set the hula hoop on the floor with its circumference vertical. I asked Donna not to show Miriam the back spinning. Today, before our session began, Miriam was doing the hula in the foyer. She and Glen were apparently too noisy for the good order of the office, so while Miriam joined me in the music room, Glen went into the Learning Lab to play with the hula hoop. When Miriam and I came out for a break, Sam (an 8 year old) said, “Hey, Miriam, did you ever see this?” Glen had just been demonstrating back spinning to Sam. I stopped Sam’s explanation, explaining to him and Sam that Miriam and I were going to do an experiment about that and I did not want them to explain it to her now. Miriam and I left for sodas.

A while later as we re-entered the Learning Lab, Miriam, whom I was carrying at the time, glanced through the opening door, then excitedly turned to me and said, “Daddy, did you see what Glen just did?” I put Miriam down in the music room and asked what Glen had done. Miriam explained clearly enough to show that she had seen his back spinning the hula hoop. I turned on the tape recorder beginning again the transcription of Logo Session 10.

BobWait a minute. No, I don’t understand. You said he rolled something and made it come back?
Miriam A hula hoop.
Bob He did. How did that happen?
Miriam I don’t know. I think it went (a gesture in the air–unclear) like this.
Bob It did what?
Miriam I think it went like that (gesture again), then it rolled and came back.
Bob . . .well, wait a minute. Let’s see if I can get the hula hoop and you can explain what happened. (Bob brings in the hula hoop) Now, what happened?
Miriam It went like that (here Miriam gestures with the ping pong ball back spinning gesture on the edge of the hula hoop). Like that (repeating the gesture). I don’t know how he did it. (This gesture represents the only procedure Miriam knows creating a comparable effect; Miriam assumes Glen used some such procedure but is uncertain).
Bob Why?. . . I saw you pushing on it, the back of the hula hoop.
Miriam Yeah. (Miriam repeats the gesture several times).
Bob I get it. Have you ever done anything else like that?
Miriam Yeah.
Bob What?
Miriam The ping pong ball.
Bob That’s absolutely right, Miriam. I find that very striking. Did you ever see anybody else do that with a hula hoop?
Miriam Unh-uh
Bob Glen, would you come here for a while please? Miriam saw you doing this (spinning the hoop) for the first time she has ever seen anybody doing it. She figured out how it worked and why. So it doesn’t matter if Miriam sees it happening all over, now. (spins the hula hoop). Did you see it go out the door and come back?
Miriam Yeah. (Miriam tries once and is interrupted by talk). Hold it. I know. I’m going to do it. (Miriam tries backspin and succeeds, laughing). It rolled backwards that time.
Bob That’s a direct, analogous extension of our work with the ping pong ball.


The problem I mentioned at the beginning of the last incident receives its fourth illustration; after the end of Logo session 7, while I gathered my paraphernalia for our trip home, Miriam played with the hula hoop outside the music room. Marvin saw Miriam playing and said, “Miriam, have you seen this good trick yet?”

Thus, over the course of a few days, while the materials were at hand and Miriam was sensitized to the phenomenon, in four separate cases she encountered situations of potential informal instruction (if you count Sam’s attempt and Glen’s demonstration as separate). Can one control such exposure? I believe such attempts would fail, as this attempt of mine failed, because a lively intelligence, sensitized to an engaging phenomenon, will notice its manifestation with only the slightest exposure. Since controlling exposure is not possible, especially in a rich environment and an active culture, the problem becomes methodological. How to be in the right place (for me, with Miriam) at the right turn (when an insight occurs); how to recognize a significant development and document its occurrence in detail sufficient to support subsequent analysis and interpretation. I believe the design of this project, as an intensive, protracted, naturalistic study of a bright child in a supportive environment during a recognized stage of rapid development, focusses on a rich domain of developmental data. The breadth of this study with respect to child’s life in the home, at play with friends, and under tutelage (at Logo), being both intrusive (thereby perturbing the structure of her mind) and extensive (opening to observation situations not usually attended to), offers a better hope of following the fine structure of developing ideas than does any method limited to sampling ideas in separate minds. The recognition of significant developments is circumscribed by my sensitivity: whether that is adequate remains to be seen. The coupling of selective observation with mechanical recording and immediate transcription is my best answer to the documentation aspect of the problem.

Beyond the issue of methodology highlighted by these incidents, raised to theoretical prominence are the issues of analogy (how what is learned as a concrete action is extended to situations where the same action control structure effects a comparable result), the importance of sensitivity to phenomena (that periphery of effects, as Piaget has it, from which cognition proceeds to the center of explanation through the hypothesis of a known action), and the contrast of learning through analogy with learning through the progressive elaboration of not-yet- adequately-structured descriptions. These issues are raised but not to be addressed here.



Tic Tac Toe (2)


Miriam asked Robby to play with her this afternoon, offering “Sorry,” “Raggedy Ann” and “Chinese Checkers.” All were refused. Robby finally agreed to playing TIC TAC TOE. I asked the children to come sit in the reading alcove. They did so while I got out my tape recorder.

Two games were played before I could get a cassette in the recorder. In game 1, Robby went first [let the letters be his moves, the numbers for Miriam], and quickly won with his computer beating gambit:

B  | 3 | C
   | 1 | D
2  | A |

Miriam should go first after being defeated, but she asked Robby to go first. He told her she must go first. I asked why she did not want to go first. Miriam: “I’m afraid he will take the place I want to go. I won’t get two ways to win.” This game was played when Miriam went first:

A | 3 | B
  | 1 | D
4 | C | 2

Robby again having the initiative. This game was played and the following dialogue was offered in explanation when I asked an unhappy Miriam how she lost:

 B |   | 2
   | C |
 1 |   | A

Miriam I put my X over there (move 2)
Robby She thought she could stop me from getting two ways to win, but I did that (move C in center square) because I already had one way to win.
Miriam ‘Cause I even saw that.
Bob Oh. You were trying to stop him from getting two ways to win.
Robby Yeah. But I did something else. O.K. Your turn to go first.
Miriam Are you going to block me? (i.e. put a counter in the diagonally opposite corner)
Robby No.
Miriam (puts an X in one corner)
Robby (puts his the the diagonal corner)
Miriam (shifting her piece to the common row corner)
Robby You took your hand off it! (outrage)
Miriam Liar, liar, your pants are on fire, your nose is as big as a telephone wire.
Robby Quiet! (Robby moves to the other diagonal corner)
Bob Miriam, please cut that out. What is all this switching and changing?
Robby You can’t do that.
Miriam He promised he wouldn’t go there.
Robby I didn’t promise.
Miriam You did!
Bob I think if you can’t play nicely together, you shouldn’t play together, you shouldn’t play together.
Miriam (moves her piece again)
Robby Miriam! (a shriek)
Bob Robby, leave the room. Miriam, put the toys away.


I believe this vignette confirms the data of number 5 (while Miriam is with another player) by showing the same concreteness and vulnerability to conflicting objectives. What is most striking is that while Miriam tries to negotiate a victory using an effective but vulnerable gambit, she utterly fails to adopt Robby’s counter-measure for her own defense against the same attack.

The conclusion of this squabble is that when Miriam wants to play TIC TAC TOE she will play with me instead of Robby.



Necklaces [nil]

This vignette was planned but never written. I recall vaguely that it involved using beads from one of our combinatorial experiments to make necklaces.



Taking Hints


One of Miriam’s proudest achievements since her 6th birthday had been learning to successfully ride her bike without training wheels. Because it had been her custom to make a considerable fuss on the occasion of a small scrape (from tripping over the dog, for example), I was disinclined to help Miriam. She borrowed Robby’s crescent wrench and removed the wheels herself. For several days thereafter her procedure was as follows: Sit on the seat and push off; try to get both feet on the pedals before the bike falls over; at the first indication of instability, turn the wheel in the direction of fall and stick both feet out to catch oneself.

The procedure is not bad; it’s nearly perfect in fact. The only flaw was that the bike would fall over after going about 3 feet. Luckily for Miriam, at this point she received some good advice from our neighbor Jim: “If you start off fast you won’t fall over.” When Miriam recounted that advice to me, I reinforced its authority, noting that Jim’s advice was absolutely correct and that for problems that look hard or mysterious, if you get one good hint you find they are not hard at all. Miriam conjoined Jim’s advice and a lot of practice. The advice provided the breakthrough she needed and with practice, she has refined her skills so that she now rides ably.

This evening when she encountered Jim in the courtyard, Miriam exhibited her skill with the hula hoop at both waist and foot. (confer Vignette 10) After being praised for her considerable skill, Miriam went on to tell Jim he should see her ride her bike, she was really good, and his “one good hint” had taught her how to do it.


I consider these observations important because they reveal a central incident in Miriam’s developing view of learning. Two roles are defined: that of a person who is having trouble doing something he wants to do; and that of an advisor who gives advice with these qualities — the advice is directly applicable to the problem; the advice is abstract and non-directive, therefore leaving the person latitude to develop a personally satisfying particular solution to the problem to be solved. In general terms, the two outstanding features of this view are: the desire and execution are her responsibility and privilege; ideas (hints, good tricks) are effective and thus worth knowing. If Miriam can maintain this view, which I infer from her comment to Jim, the terms in which we talk, and from her behavior, her education promises to be a profoundly satisfying experience.



Making Beaded Bracelets


In our work at Logo, Miriam has recently tended to not do what I tell her. That’s too strong a statement. When I give Miriam advice, for example, that a certain distance is probably a hundred forty turtle steps, Miriam specifically commands the turtle with some similar but different operand, e.g. FD 130.

Miriam was unhappy with the bracelet she made for Kim and/or Sue (her student teachers at Baldwin, who are leaving on Wednesday). I took the liberty of arranging her beads in a bracelet thus:
2 red (R) 2 blue (B) 2 black (Bl) 6 orange (O)
7 brown (Br)

Br – O – Bl – Br – O – B – Br – O – R –
Br – O – Bl – Br – O – B – Br – O – R –

triplets with a single varying member and the extra bead in the center.

Later, I showed this new bracelet to Miriam and told her I thought I had made a pretty pattern. She untied the knot and removed the string, saying she would make her own pretty pattern. As Miriam proceeded to sort the beads by color, I pointed out a major difficulty. There was one extra brown bead. Miriam put one brown bead on the string. I asked if I could give her a hint. She was willing. “A good trick when you have an even number of beads is to put one of the same kind on at each end of the string and push them to the middle.” Miriam took this advice and produced the following string with which she is quite pleased:

R – Br – O – Br – O – Br – B – Bl – O –
R – Br – O – Br – O – Br – B – Bl – O –



Miriam Collecting Data


While the children were awaiting the school bus this morning, I noticed the little blue notebook in which Miriam was to record her kindergarten activities. I took it out and offered it to her. She said she didn’t want it when I reminded her what it was for and then confided, “I can’t spell all the words of the things I do, but I’ll remember them.”

In a short conference with her teacher, I inquired about Miriam’s role in the skit she mentioned in Vignette 3. I was informed that the skit was an impromptu performance, no adults had been working with the children’s group at the time, and that the other children selected Miriam to be the narrator and tell them what to do.

Miriam and I drove home from school (we stayed a little longer to play in the playground). While Gretchen was making lunch for us, Miriam came to sit beside me, saw the blue notebook, and asked if I would help her write down what she did. Four main events:
– housekeeping corner
– going to the Star Market (a four block walk) with Sue to buy provisions for Wednesday’s farewell party
– making peanut butter
– recess


Although this single collection of data is insignificant, such notes for the remaining 6 weeks of school will indicate Miriam’s foci while there. Because I go to the school freely, such data will help me select what parts of her school setting are most relevant to her development and worth detailing.





In discussions some weeks ago with a distinguished Genevan psycholinguist, the question of instructing children in phonetics came up. My earlier work on children’s conception of wordhood (‘Pre-Readers’ Concept of the English Word’) and pig-latin (not available) had convinced me that children should learn how to read before delving into phonetics. (Such a point of view is contrary to much current practice in late kindergarten and early first year instruction). My colleague considered it an aberration in American education to emphasize phonetics as an introduction to reading.

Miriam now reads well, at what is generally considered the second grade level (see ‘Miriam at 6: Reading’ for more detailed infor-mation). I speculate confidently that her reading vocabulary is several thousand words. She now knows enough to appreciate the value of knowledge about the correspondence (and lack thereof) between English phonemic and lexical structure. The following observations record the upsurgence of Miriam’s interest in phonemic variation:

In the spring and summer of last year, one of Miriam’s favorite games was bouncing on my knees. We would sing the familiar chant:

Ride a horse to Boston,
Ride a horse to Lynn.
Careful when you get there,
Don’t fall in!

The child rides facing the bouncer, whose knees are the horse. On the last line, the adult separates his knees, and the child tries not to “fall in.” Boston was a place I went to. Lynn was the neighbor who lived across the street from our Connecticut home. (Miriam has yet to discover the existence of Lynn, Massachusetts). The other common variant chant is:

Trot, trot to Boston
To buy a loaf of bread.
Trot, trot home again,
The old horse is dead.

I introduce this history to enforce the idea of how well known are these rhymes to Miriam.

Today at lunch, having finished earlier than Gretchen and me who sat talking and unattentive, Miriam picked up a large rag doll, put it on her lap, and chanted (quietly, to herself):

side a sorse to soston,
side a sorse to synn.
sareful sen sou set sere,
son’t sall sin.


ide a orse to oston,
ide a orse to ynn.
areful en ou et ere,
on’t all in.

And then:

fide a forse fo foston,
fide a forse fo fynn.
fareful fen fou fet fere,
fon’t fall fin.

It is clear that phonemic separation is an issue that engages Miriam now.


Although she knows that letter represent sounds, it is not clear that Miriam understands the way in which phonetic knowledge will help her with reading. She may. Do note, however, that Miriam was introduced to phoneme separation two weeks ago (confer Logo session 5) through using my pig-latin system to drive the voice box. Five days ago, Miriam selected that activity as one she wanted to engage in (confer Logo session 10) even though I attempted to dissuade her from it by arguing that the programs I had available were limited and the output boring. Today I asked Miriam what I should plan for tomorrow: story writing, drawing, voice box. . . we didn’t always have to play SHOOT or READY, AIM, FIRE. “Voice box,” she exclaimd. “That’s what I want for tomorrow.”



Housekeeping Corner

5/24 & 27/77

Since Miriam started recounting where she spends her time in kindergarten (see Vignette 12), it has become clear that she spends most of her time in the ‘housekeeping corner.’ When I’ve been in the kindergarten class, I’ve usually found myself playing with blocks, or making designs and elaborate towers from Cuisenaire rods. Miriam has frequently played otherwheres. I have seen her there, curled up in the baby carriage, but I’ve had no good idea of what games she and her friends invented for that location.

That question received a major clarification today while Miriam was stringing beads for necklaces. I had given her a Bic pen cap to poke the string through the beads. After she learned the cap was from an exhausted pen, Miriam began this conversation:

Miriam Can I keep the cap?
Bob Sure.
Miriam Thank you. Tomorrow I’m going to take it to school.
Bob The cap?
Miriam Yeah.
Bob What for?
Miriam Because we usually play Doctor, there in the housekeeping corner.
Bob Un-huh.
Miriam And we give each other pretend shots.
Bob Oh my goodness. You’r not really going to poke anybody with that, are you?
Miriam No (you silly Daddy).
Bob But that will be your needle?
Miriam Yes. Sometimes we use pencils.
Bob I hope nobody ever gets hurt.
Miriam We don’t.
Bob Good. . . I think that’s kind of funny. That you have a house-keeping corner and you play Doctor. Is that because everybody likes giving shots?
Miriam Yeah.
Bob Does anybody like getting shots?
Miriam No. We always run away from the Doctors.
Bob Well, who’s the Doctors. . .or does it change?
Miriam It changes. We run away ’cause we don’t want our shots.
Bob Yeah.
Miriam We always have it in the summer. We run away because we don’t want it and the door’s always open in the summer.
Bob You mean the door to the kindergarten? Or the housekeeping
corner, a pretend door?
Miriam A pretend door to the housekeeping corner. It’s always open so we run right out.

One fact, of possible relevance in initially suggesting the game to the children, though not at all accounting for its continued interest, is that Dara’s mother is a nurse. Dara and Maria are the two girls Miriam most plays with in kindergarten.

After we focussed a while on the beads, I resumed the theme of the housekeeping corner by attacking the game of ‘Doctor.’

Bob I still think it’s kind of silly that you play Doctor in the housekeeping corner. Do Doctors come to houses or something?
Miriam Sometimes they do.
Bob Don’t you ever play anything else? Or is it always shot giving?
Miriam We like the Doctor but sometimes we play House of the Wicked Witch.
Bob Wicked Witch? How’s that go? I never heard of that. Is that
like something from the Wizard of Oz? Or a different wicked
Miriam From the Wizard of Oz.
Bob Does anybody know the song or what?
Miriam I and Dara know the song.
Bob You and Dara?
Miriam And Maria
Bob How’s it go? “La la the wicked, la la the wicked witch, la la the wicked witch is dead”? No? That’s not the song?
Miriam No. It’s about the Wizard.
Bob Oh. We’re off to see the Wizard?
Miriam Yeah.
Bob The wonderful Wizard of Oz?
Miriam Yeah.

It’s clear that my wicked witch was she of the west upon whom did fall Dorothy’s house. In retrospect, I’m sure the children think more of the Witch of the East, she commander of flying apes and profoundly allergic to water. No dancing Munchkins for them.

6/27 Miriam arranged for Dara to come play at our house today. Because Miriam expected to come to Logo, I asked her if she intended to bring Dara with her and wondered whether Dara would want to come. Miriam responded that she could get Dara to come to Logo by telling her it was a good place to play Wicked Witch. I had no idea why this was so.

Dara and Miriam at lunch told me a little more of Wicked Witch, not clearly perhaps but enough to reveal what sorts of sides and tensions exist. They mentioned that the boys build spaceships in the kindergarten and should they be left unattended, the girls play Wicked Witch, swoop down on the spaceships, and keep the boys away. This, I saw too late, was relevant to Logo’s being a good place to play Wicked Witch. Robby and Sam have been playing war games in the Learning Lab, building barricades or trenches from unoccupied chairs. When Miriam and Dara seized the momentarily unoccupied trenches, I realized from the commotion how Wicked Witch was being applied at Logo.

Gretchen informs me that while the children played at home, most of their time was spent playing Doctor in the tree fort.


Both Doctor and Wicked Witch are highly mobile fantasies which appear to be role centered with improvised skits focused on dramatic actions: giving and getting needles; seizing somebody else’s place. From outside the kindergarten, the setting dependence of the games I speculate to be primarily in the nature of a space allocation. The girls play in the housekeeping corner. They use it as their home base for whatever fantasies they can construct with a sufficiency of roles for themselves.



Tic tac toe


Miriam emerged from her bath not at all ready for bed but looking for someone to play with her. I agreed she could stay up and that we could play together while Robby was getting ready for bed. The game was my choice. My objective was to induce Miriam’s copying my successful gambits and her re-applying them against me (cf. vignette 9).

Miriam began with her currently favorite opening to produce this game, recorded in the following dialog (her moves are letters, mine are numbers):

1.   B |  3  | C
       |  1  |
       |  2  | A
Miriam Me first, please.
Bob O. K. You first.
Miriam Will you go in front of me?
Bob What do you mean?
Miriam Like here, if I go here [at opposite diagonal].
Bob Well, let’s try it and see. . . . Suppose I go over there? [at opposite diagonal]
Miriam No. Don’t.
Bob Suppose I go there?
Miriam O. K.
Bob That’ll be number 1. . . . Now I’ll put 2 right there.
Miriam [placing her third X] Two ways to win!
Bob Um. Do you have any ways to lose?
Miriam Yeah [in a small voice]
Bob You’re going to lose.
Miriam I’ll put —
Miriam [complaint — wah wah wah!] You stupid.
Bob I’m not stupid.
Miriam Yes you are.
Bob No. I’m pretty good at tic-tac-toe. How did I beat you?
Miriam You went to, to, to [noises match her gestures to the places I moved].

Miriam’s description of my winning play was not illuminating to me. I hoped replaying game one in reversed roles would help decenter her focus. In game two Miriam refuses to replay game one, preferring to block my third corner move (contrast games two and one). Her putative blocking attempt fails because of the symmetry of the gambit. Game three replays game one with the original roles maintained. When I call attention to the place of forced moves in my play, Miriam follows that lead in modifying her failing three corner strategy.

2.    2  |     |  B        3.    B  |  C  |  3    
     ---------------            ---------------
      C  | A   |                 4  |  2  |  E     
     ---------------            ---------------
      3  |  4  |  1              D  |  1  |  A 
Bob You watch. I’ll play the same game you played. I’ll put my 1 there. Where are you going to put your piece? [center square X move] Oh. You don’t want to play my game, huh. How ’bout I put my 2 up here? [Miriam then puts 2nd X in opposite corner] Are you watching now? what have I got?
Miriam Two ways to win.
Bob How did I do that to you?
Miriam You went to, to, so you can have a way to win.
Bob Could you do that to me?
Miriam Yeah.
Bob Let’s try it again.
Miriam Me this time first.
Bob You want to go first?
Miriam Are you going to go in front of me?
Bob I don’t think I’ll let you beat me. . . . You’re afraid I’ll go over in this diagonal corner here? Right there? Well, I won’t do that. I’ll go some other place. But remember: in this game [1] I did not go in the diagonal corner and still had you, didn’t I? Yeah. I’ll go right here.
Miriam Oh. You’re trying to play your dirty trick.
Bob I don’t play and dirty tricks. I play good tricks. . . . Now. You have one way to win there. I am forced to move here.
Miriam [tooting noises — continuing intermittently]
Bob Do I have one way to win? Yes. You are forced to move down there. You have one way to win there. I am forced to move there.
Miriam X.
Bob So that’s a tie.

Game four proceeds as my attempt to show Miriam what is expected of a player whose initial plan is frustrated, i. e. one should not gripe nor negotiate turn takings at victory but should adopt the best expedient one can.

4.    A  | 4  |  C 
      D  |  3 |  B  
      2  |  B |  1 
Bob Let’s play game #4. I’ll go first now. I’m going to go right there. Are you going to go across from me? Are you going to block my move? Go ahead. Can you block me so I don’t do that? Oh phooey. Now I’ve got to figure out some other way, because I know I can’t use that good trick that you know, so I have to figure out some other trick. I will go here [2]. Now I have one way to win.
Miriam [blocking row] None way.
Bob O. K. You blocked me. Ha. I will go here [3]. Now I have one way to win. . . . Hum. Right here, I see you have a way to win. I will go there [4].

In game five, I attempt to exhibit the purposes behind each of my moves, specifically showing that I think of her responses to my moves as well as my own objectives. Instead of attempting to negotiate a victory, I assume she will move to block my plan and adopt a different gambit on that basis.

5.   A  |  C  | 3 
     4  |  1  |
     D  |  2  | B  
Miriam Me first. Will you block me?
Bob Maybe. But even if I don’t block you, it still seems I do pretty good, don’t I?
Miriam Yep.
Bob Did I block you here? [in gane 1] No. But I beat you. . . . If I go here [at perpendicular diagonal corner on 2nd move] you can block me and get two ways to win. Right?
Miriam Right.
Bob If I go here and you block me, do you get two ways to win? No, you can’t. I am going to go here [move 2] and I have one way to win. You made a forced move [C]. You have one way to win, so I am forced to move [3] [Miriam blocks 3 – 1] and you have one way to win again. So I have another forced move and it’s a draw.

In games six and seven, after defeating Miriam, I again attempt getting her to re-apply an opponent’s successful strategy against him. (My opening in game six, Miriam’s in game seven; dialog describes game 7):

6.        | A   |  2    
          |  1  |  B   
        C |  1  |  3  

7. 3 | A | C -------------- B | | 4 -------------- 2 | | D

Bob You move first. Let’s see if you can beat me the same way I just beat you. O. K. You’re starting with an X. I’m going to go right where you went. Let’s see if you can beat me just the same way I beat you.
Miriam Wish.
Bob Is that the same way?
Miriam Did you go here?
Bob Yes. O. K. So you’re going in the corner now. Now this [2] is a forced move, because you have one way to win, so I have to go here.
Miriam Two ways to win.
Bob Yes, you do. And you went over here [3]. So I will too, and you beat me. . . . O. K.
Miriam [cheering herself] Yaaaa. I won for the first time. Hooray.

The interest in game eight is that it shows Miriam more intent on blocking the opponent’s next move than winning directly. Her failing to notice a winning move leads into my codifying the order in which she should apply her decision principles.

8     B  |  1  |
      C  |  A  | D2
      3  |  D1 | 2  
Bob Would you like to first again, Miriam?
Miriam O. K. Yeah.
Bob That one’s yours. Let’s see if you can beat me a different way. I will go there again. But see if you can beat me some different way. Oh. O. K. I have a forced move. I have to go here [2].
Miriam [gets two ways to win]
Bob I have a forced move here. So I must go here [3].
Miriam [starts to block 2 – 3 row]
Bob No, no.
Miriam I blocked you.
Bob But look. Is it better to block me or better to win?
Miriam Win, win.
Bob But one of the things you have to figure, Miriam, every time, you have to ask yourself: does the other guy have a way to win? Can I beat him, first? ‘Cause if you can beat him, first, you don’t have to stop him from winning, ’cause you won already.
Miriam Right.
Bob So let’s see. The number 1 thing you look for [writing list], you say: can I win?
Miriam Can we stop for a while?
Bob Yeah. The second thing is: forced moves. And the third thing is what? Two ways to win! O. K.?
Miriam O. K. What’s the seventeenth thing?
Bob No, they’re the only three things you have to look for, Miriam. . . . Can you tell me what the three things are you look for?
Miriam Yeah
Bob The first thing is what?
Miriam Can I win.
Bob What’s the second?
Miriam Forced moves.
Bob And the third?
Miriam Two ways to win.
Bob Which one do you look for first?
Miriam Can I win.
Bob Second?
Miriam Forced moves.
Bob Third?
Miriam Two ways to win.
Bob You got it. That’s all there is to tic-tac-toe. If you always use those three rules, in that order, you’re going to be a winner. O. K.?
Miriam Yeah.
Bob Or else maybe you’ll come to a draw. I think you’d better wash your face and go to bed.
Miriam Good night.
Bob Good night, sweety.


I expect tic-tac-toe to serve Miriam as a simple model of a bi-polar activity, i. e. one wherein at each step of your activity you must attend to your previous actions and a response to that action. (By a model, I mean a framework in terms of which one may conceive of other activities, such as putting questions to nature.) The features of tic-tac-toe which I see as useful are: its interactivity; the opening gambit may be yours or your antagonist’s; there are a set of good tricks one can learn; there are pitfalls to avoid; when one does not see a sequence of forced moves to game end, there is an ordered set of heuristics to follow.

If Miriam can reflect on her own procedures in playing tic-tac-toe and uses tic-tac-toe as a model for exploring phenomena, reflexive abstraction will be a natural consequence .



Binet Test


Miriam has known for over a week that our next trip into Boston would be for taking a test. I had introduced her to the idea with the explanation that nearly everyone takes such a test some time and that she was simply taking this test earlier than most other children. So, after kindergarten and a rousing 2 hours with her playgroup (see Home Session 3), Miriam put on a dress and we took the Green Line into the center of Boston.

Miriam had earlier expressed concern that she didn’t know how to get ready for this test. (The only other test formally so defined to her was having her ears checked. She apparently does not think of our experiments at Logo as being tests.) This concern surfaced again as we waited for the trolley car. “Daddy, what kind of questions did they ask you?” I could recall only one question from an earlier intelligence test (25 years ago). “They asked me who was the president before Franklin Roosevelt.” “Who was it?” Thinking she now had the inside track, Miriam asked who was the president before Carter, and before him, and before that one. We stopped at Eisenhower when the trolley came.

It was a beautiful day as we strolled through the Common, stopped at an ice cream store, and continued to the testing center. Miriam was clearly content and relaxed when she went with the tester. She was also relaxed and pleased with herself when she had finished.

Although we need wait another week or so for a formal evaluation, the tester offered these general comments: since Miriam had just turned 6, she began with the age 6 series; Miriam had to be confronted with questions from the eleven year old series before she failed to get at least some of them correct; they have never had to go through so many series with such a young child in their laboratory. I believe the comments need be put in this perspective: the laboratory (Tufts-New England Medical Center Neuropsychology Section) typically is called upon to diagnose problems. Thus, they do not see as a matter of course so young a child as Miriam who has no immediately caused requirement for such a test.

In the evening before going to bed, I asked Miriam if the questions were difficult or easy and if she tried hard to answer them. She responded that she tried as hard as she could, but that some of the questions were just too difficult for her to answer. She was pleased with her performance — having overheard the comment about never having to go through so many tests with a child her age; she was proud that she “did better than anybody else.” Her only gripe was that they didn’t ask her a single question about the presidents.



Arithmetic Ripples


After the session in which I introduced Miriam to adding large numbers (see Home Session 4, 5/28), passing Miriam’s room I noticed in her open loose-leaf book a page of computation. Miriam later gave it to me and I include it as Addendum 17 – 1.

Note that the written form of the equations mimics the horizontal form used in our introduction (see addendum 1 in Home Session 4). Additionally, Miriam attempted here a subtraction with large numbers (i.e. 80 – 7 = 73), her suggestion which I turned down during Home Session 4. Place value, as a topic of interest to Miriam, appears not only in her large numbers, but also in the directly contrasting sums: 11 + 1 = 12 and 1 + 1 = 2.

When she gave me the page, Miriam explained her attempt to subtract 7 from 1; how 1 minus 1 was zero and 1 minus 7 was zero. I expect she will conceive of the negative integers soon.


These incidents document the ways computation crops up in Miriam’s world.

Addendum 17-1


Comments Off on Vn01701



Arithmetic Ripples


As Robby and Miriam came in from play for a little refreshing juice, I heard from the kitchen the squabbling one expects of near-aged siblings:

Miriam I can add big numbers.
Rob Oh brother!
Miriam I can. I can do one thousand and thirty five plus two thousand.
Rob Easy.
Miriam No. Three thousand and thirty five.

When I asked Miriam later where she got those numbers for adding, she replied, “From the adding you and I did the other day.”


These incidents document the ways computation crops up in Miriam’s world.



Arithmetic Ripples


Miriam was playing in the kitchen with Scurry this morning. Gretchen and I were discussing some topic, and I mentioned a division problem. Miriam piped up, “I can divide, Daddy. . . . 8 divided by 8 is 1.”

I congratulated her on her prowess. For Miriam the formula she recited constitutes division. The division problem is the one I executed in playing Dr. World’s computer game (in Home Session 5, 5/30). Despite her ability to divide sets concretely (see Miriam at 6: Arithmetic), Miriam does not appear to associate dividing with “division,” a process for which she has, I believe, this one example.


These incidents document the ways computation crops up in Miriam’s world.



Arithmetic Ripples


When she came home from school, Miriam asked to stay home today. I suggested we do her second adding session. Miriam then countered that we should do math at Logo and use the computer. I agreed.

At the lab, while I was logging in, Miriam began playing with a pile of bricks and set them in a row to mark off a “stage.” Instead of doing math, we did “Goldilicks,” her version of the 3 bears story.


These incidents document the ways computation crops up in Miriam’s world.




Housekeeping Corner (3)


In her race to the school bus this morning, Miriam left behind a present for her friend Maria and scripts for the play rehearsal (see Vignette 19). When I entered the kindergarten about 9 (which means the children had finished their quiet reading time and the general class meeting), I saw Miriam and several friends playing in the “math” area. This area is defined by a table and a cupboard which is laden with toys such as cuisenaire rods, blocks, geometric puzzles, et cetera. The children were playing with Willy Fangel’s “Connector Set,” a sort of oversized tinker toy collection of rods and 4″ wooden disks. All the disks were in use — as lollipops. Three or four children pretended to eat these wooden lollipops and were squabbling over the distribution. As the newest entrant to their group I was given one by Elizabeth, a friend I knew from Miriam’s birthday party, visits to our house, and my previous visits to the kindergarten. I took another disk and then claimed my lollipop was great because it had candy on both ends. A few imitations later the children were using their dumbbells to show how strong they were.


When I gave to Miriam the present she had made for Maria, that interrupted the play. Miriam explained it was a going away present she had made for Maria (her closest friend and a frequent visitor at our house) who will soon be returning with her family to Spain. The present was, of course, to be opened immediately. After Maria saw her U-Bake-It Owl, she and Miriam discussed what presents Maria would give her in return and when.

A migration then began to the housekeeping corner. The children asked me to join their play to be the Daddy. I agreed on condition that I not be made to eat any more wooden lollipops (which the girls carried over). I found myself seated on a small chair at a small table. The housekeeping corner has a set of toy sinks across the wall, a refrigerator, a cupboard wherein is kept the food, and shelves. A large (big enough for two 6 year olds to crowd into) baby carriage and a pile of clothes were the other main items in the area. The children immediately began arguing over roles. “You be the mother. I’m the baby.” To reduce the conflict I noted we might have twins. That was acceptable and Michelle, who didn’t seem to mind too much, was constrained to be the mother. She began struggling with a short nightgown– long enough for her– while Miriam and Elizabeth climbed into the baby carriage. Since there were two blankets, this seemed a nicely balanced situation.

For a moment only. The children, still carrying their collection of wooden lollipops, began playing GOO-GOO-GA-GA. This game has three main features: the children say nothing but GOO-GOO-GA-GA; they fight with each other over the blankets; they throw to the floor whatever they can reach. The additional element they explained to me is that the parents get to pick everything up. I refused to play any such parental role. Michelle then acted her part. Going into the cupboard, she took out all the plastic fruit and boxes of cereal and dumped them over the babies — who proceeded to throw them to the floor.

Miriam, pulled out of the baby carriage by Michelle, crawled under the table to maintain possession of the blanket. Enter Meg, the largest child in her class. She too joined GOO-GOO-GA-GA. The noise level and contention increased. When Michelle rid herself of the nightgown there were four girls playing tug-of-war with two blankets amidst unspeakable noise. The one English word I was to hear come up: “share.” Elizabeth declared she would take a bath, hopped on the sink with her end of one blanket, and invited Michelle to join her in the bath. Then Meg and Miriam joined the bath. Four girls in the bath jabbering GOO-GOO-GA-GA and laughing like crazy.

I picked up some of the plastic fruit and walked away juggling. While some of the other children took an interest in juggling (and Maria boasted that her father can juggle 10 eggs at once), the game in the housekeeping corner came to an end and the girls decided to rehearse a play (as described in Vignette 19).


This incident exemplifies a game’s functioning as an excuse to engage in a small repertoire of satisfying actions. It appears to be the case that some peripheral factor (such as having a type-cast Daddy available) may evoke and not at all constrain these shared actions.





Last night Miriam asked me to come to kindergarten this morning to help out with a rehearsal of ‘Goldilocks.’ (see Logo Session 17, 5/1). When Miriam began showing an interest in plays (cf. Vignette 3), I mentioned to her that those years I spent at Yale were in the Drama School, that I had also done the kind of thing she was describing to me. I can not justify it by having impressed Miriam. She told me she believed I could be of help to her because I had a good book about plays for little people [A Dozen Little Plays, Parents’ Magazine Press 1965]. (She has read that book and finds its costuming of major interest; she asked if I would make for her Fox and Duck costumes. That book is a piece of flotsam remaining from a Master’s thesis project on ‘role rotation’ I once proposed and dropped.)

When I retreated from the housekeeping corner (as described in Vignette 18), Miriam’s intention to stage a rehearsal came to the fore. She asked her teacher if the children could go rehearse in the gym but that was not possible so her cast gathered near the clothes closet. Miriam brought out the scripts of ‘Goldilocks’ we had created the previous day at Logo. The potential cast was 6 in number: Miriam, Maria, Michelle, Elizabeth, Meg, and me. Given the dramatis personae of the 3 bears and Goldilocks, though I was immediately type cast as Daddy Bear I declined and elected myself to the audience. Then the squabbling began. Counting remaining noses, I was quick to suggest that we needed an “announcer.” Miriam declared she would do that and assumed directorship: “Meg, you be Daddy Bear.” (Meg is the largest child in the class). “No. I was Daddy Bear last time. I want to be Baby Bear.” “Michelle, you be Mommy Bear.” “No. I was the Mommy already.” (confer Vignette 18). To Elizabeth: “Here.” (as she gave her a script). When I asked Elizabeth what she thought of the script, she said it was nice but gave it to me because she can’t read. Maria made a cone-shaped hat of hers: “I’ll be the witch.” (a probable reference to the “Wicked Witch” game). Miriam tried to cast Maria as Goldilocks — Maria quit. Elizabeth and Michelle also refused the role. We were at this impasse in casting:

Miriam Self-declared narrator and director
Meg grudgingly willing to be Daddy Bear (and practicing growls)
Elizabeth both bears but neither willing to be the Mommy Bear
Michelle both bears but neither willing to be the Mommy Bear
Goldilocks persona non grata

The problem was solved by ignoring it. “We need costumes,” said Meg. I suggested shopping bags with head and arm-hole cut-outs as making good ‘bear suits.’ “No. Masks is what we need.” Seizing on this suggestion of Meg’s, the children got paper plates to make masks. My contribution was to mark the position of their fingers when each held a plate to her face and located her nose, eyes, and mouth. The children had finished cutting out the necessary holes and made the faces those of bears with brown crayon. Miriam was told there was no time for putting on the play before gym, but they could do it after. So the children closed off their activities and got ready for gym.

After gym, it was clean up time. Put away the wooden lollipops and those fruits Michelle had dumped out in Vignette 18. The children did not go further with their rehearsal that day and I left.


This incident is important as an example of the cross fertilization of ideas from different domains of Miriam’s life: kindergarten and Logo. It also hints at some of the constraints: what good does it do a 6 year old to make a script if none of her friends can read it? This last question is obviously rhetorical — one may distinguish between a project’s being immediately effective in achieving a goal and its value in a person’s development.



In the Gymnasium


Twenty four children filing into the gym, a space about 20 by 40 with a wall-wide, wooden climbing rack at one end. During the half hour, when any children did not want to play some game or other he might climb around on that climbing rack, come sit on the sawhorse where I was standing, or sit against the opposite wall. It was common for several children not to join each game.

During the period, I recall 5 games being played:
ICICLES — a few children are ‘it’ in the center of the gym; the others at command run lengthwise to the gym’s other end; anyone tagged should ‘freeze.’

COWBOYS AND INDIANS — a chase game with two teams; one team hides its eyes against the wall; as the second team sneaks up on the first, the teacher intones, “Oh, you sleepy cowboys, you’d better wake up. Here come the INDIANS!!!”; with this cry the cowboys chase the indians across the gym.

BATTLESTATIONS — a general exercise game in which the children leap to stereotyped postures (e.g. at attention, saluting for “here comes the captain”) or actions (e.g. swimming gestures for “here comes a shark”) when the teacher gives commands. These were usually given in pairs, the exercise value in going from one state to the other — port/starboard here comes the captain/all hands on deck naptime/chowtime

FIRE CHIEF — one child is the chief, getting to select which of three teams responds to the teacher’s alarm: “Fire, fire, fire: alarm at station. . .” When the chief yells his number, the selected team members race to the climbing rack, climb far enough to tag the top, then race to the opposite end of the gym. The winner of the race gets to be the new chief.

ANIMAL FARM — this final game of the period is the decrescendo of the hectic excitement of gym. The children sit in a circle. The teacher, in the center, covers the eyes of one child who must guess the identity of another, selected from the several who volunteer to make animal sounds.

After gym, the children spilled back into kindergarten, where
they began, relatively slowly, to pick up the variety of materials they
had so liberally scattered before gym.


The observations focus on an important part of the world of Miriam’s peers, and her standing apart from that. I infer that, because of her limitations and specific experiences, she has a different framework for thinking about gym activities from her peers.



In the Gymnasium (2)


This evening I asked Miriam whether she usually stood aside from the games in gym as she did yesterday (except for Animal Farm). She replied that she doesn’t like running around (her allergy to dust and its chronic incipient wheezing make her feelings quite understandable). But Miriam justified her alternative further: “I was on the balance beam.” She had been but I had not recognized it.

In the summer of 1975, Robby was enrolled in a gymnastics class. Miriam and I took him there and either waited inside, watching the older children exercise, or walked about outside. Miriam felt left out (she was too young for that class). When we arrived early one morning, I held her hand as she walked along the elevated balance beam. Back at our house she found a ‘balance beam’ (a 14′ 4×10 under the porch) and had me set up the timber so she could use it. The next summer brought the Olympics when those young girls from eastern Europe were so impressive in tumbling and on the balance beam. Miriam asked me to set up her balance beam again. Of the activities available in the after school program, Miriam chose gymnastics as the one she most wanted to pursue.


The observations focus an important part of the world of Miriam’s peers, and her standing apart from that. I infer that, because of her limitations and specific experiences, she has a different framework for thinking about gym activities from her peers.


Vignette 21

Miriam’s Room


Miriam is suffering a change about which she is unhappy but which I believe is for the best. Until last night, she and Robby shared a bedroom. Yesterday Robby moved into the third bedroom of the carriage house in which we live.

Miriam has complained that it’s lonely in her room now with Robby gone. It surely must be — for last night it was quiet after the children went to bed: none of the common fights over whether the night light should be on (Miriam’s position) or off (Robby’s); over who has taken whose favorite toy animal or reneged on a trade; no complaints that Miriam wanted to sleep while Robby wanted to watch Victory at Sea on TV or some even later special program. Instead, Miriam went to bed accompanied only by Foxy, two stuffed horses, 3 Peanuts books, Babar and the Wully Wully, and Richard Scarry’s Busy, Busy World. Miriam reappeared an hour later, spent a little while with Gretchen and me, then went off to bed and sleep.

Miriam does have trouble sleeping. Her profound allergic reaction to household dust causes her difficulty in breathing. During the day, her wheezing is suppressed effectively by a medication taken every six hours. If her room is dusty, she wakes up in the middle of the night (when the medicine’s effect has reduced) short of breath and fearful. Despite Miriam’s having a work table, shelves, and her toys in the kitchen and living room, Miriam and Robby together manage to create a terrible clutter in their bedroom. This persistent clutter made keeping the room dust-free near impossible. When Robby asked to move out (which has other unrelated benefits for him), I decided the benefits Miriam could not appreciate outweighed the drawbacks and the move would be good for us all.


Since Robby is Miriam’s closest playfellow, a competitor, and occasional instructor, their separate bedrooms will reduce the stimulation they each provide the other outside the bounds of observation’s scope.


Vignette 22.1

Emberley’s Faces


Miriam’s wheezing was so severe this evening she couldn’t sleep. About 10:30 she came from her room to ask if she could sit up with me because it was so lonely in bed and she didn’t want to read any more.

I brought from my briefcase Ed Emberley’s Drawing Book of Faces, a book we had used in Logo Session 9 (an unsuccessful attempt to engage Miriam in the use of an introductory drawing program). Miriam was delighted to draw with the aid of this book. The first face she drew was “Tired Tillie” from page 5. (How appropriate for a child 2 1/2 hours past her bed time). The second figure (at the top of page 4) has the face of “Happy Harriet.” (Notice the two eye circles were added late, when the hair bows were being colored in). The 6 and 7 fingered hands appear to be a free, somewhat controlled extension of the hair scribble motif. The body is merely indicated and the message is the common one appearing above Miriam’s name on all the notes she prepares and gives to friends. Miriam showed the page to me as she closed the book and returned my red pen: “Nice, Daddy?” I agreed. “Hey! I’ll do the Queen.” Then Miriam proceeded to copy the drawing from the back cover of the book (omitting the eyelashes and the collar at the neck).

I asked Miriam to write the date on this page in her notebook. She complained that she didn’t know how to spell ‘June.’ I suggested the number-slash-number representation and the date would be 6-slash-6. Miriam produced 6-back slash-6 for my examination. I said it was fine, but the more usual slant was the other way. Rather than abandon her work, with a simple elaboration much in the spirit of the earlier 3 faces and with the good humor of making a joke, Miriam created her own back slash face. Next, dating the work with the common form, 6/6, Miriam created the contrasting (and not so happy) front slash face, then elaborated the face with body and limbs.


This vignette documents the incorporation of ideas immediately available: both those in a structure of availability (i.e. the book) and those extremely accidental.

a Sample of Miriam’s Work


Vignette 23.1

Arithmetic Ripples

6/5 & 11/77

Miriam does not yet recognize the existence of negative numbers. The typical problem this causes her was shown as we rode home from buying a Sunday paper (the children go with me to buy chewing gum). Miriam was discussing making change with Robby. She knew that paying for a 15¢ pack of gum with a quarter involved a ‘take-away’ problem. She asked Robby (getting the formulation backwards):

Miriam How much is 15 take away 25?
Robby 10.
Miriam That’s not right. I made a mistake. I said 15 take away 25.
Robby Minus 10, like 10 below (cf. Protocol from the series on Robby’s arithmetic development).
Bob Does that make any sense to you, Miriam?
Miriam No. You can’t do that. That’s like 1000 take away 7000. You can’t do it.
Robby 6000 below.
Bob Does that make any sense to you?
Miriam No.

6/11 Today was one of those terrible days. Gretchen and I had bad headaches. The weather was foul, rain for two days running when the forecast had been for a bright weekend. The children played inside all day; they played chase with the dog. And finally, Miriam is mad at me.

Late in the afternoon, she came to me: “Daddy, I’m mad at you for two reasons. You didn’t do any arithmetic with me today, and you told me it was going to be sunny.” I promised to do some adding (she said then both adding and subtracting) on the morrow and disclaimed all responsibility for the weather.

A little later, Miriam found Robby willing to talk about arithmetic. The two entered our reading alcove with this conversation:

Miriam 10 times 10 is 35.
Robby No, Miriam (counting on his fingers), ten 10’s are a hundred. Isn’t that right, Mommy? (Gretchen confirmed his result).
Miriam It can’t be. 5 times 5 is 25, so 10 times 10 is 35.

As Robby went on to other affairs, Miriam asked me, isn’t that a big number? I can add three thousand and thirty five (cf. Vignette 17, 5/30). Upon my responding that the number was something like that, she suggested we look in my notebook. We found there the number 3132 as an addend (cf. Home Session 4). I promised that she could learn to add some more big numbers.


These three incidents point to three separate themes that will be developed in future arithmetic sessions with Miriam. I intend to confront her, gradually, with situations which will require her inventing the negative integers. I intend to introduce her to ‘times’ as counting in non-unary increments. I intend to reveal to her that what she has learned of adding already (in Home Sessions 4 and 6) permits her to add all big numbers.



Writing Stories

Robby called me from Miriam’s bedroom: “Dad, come see the puppet show.” They have played with, even made, hand puppets for a while and enjoy giving shows — whose typical script has been, “Hello, my name is Owl. Goodbye.” I was in high spirits after a very successful arithmetic session with Miriam (see Home Session 7). Sucker that I am, I came along. Walking through the door, I found the children were playing “Ambush” — both lying under covers on the top bunk and crying “BAM BAM” as I walked through the door. Riotous laughter.
Suffering only flesh wounds, I managed to return their fire, then said I thought it was a dirty trick for them to call me to see a puppet show, then shoot me. Miriam responded, “This was our puppet show —

Once upon a time,
There were two guns.
Bang. Bang.
The end.

Her joke was a spontaneous expression of the WRITER program’s story format. Her use of it in this explanatory way shows her recognition that it is a shared model of what a story structure is.

The puppet-show play continued; the children agreed to put on a real puppet show. I had to leave the bedroom, and when ready, they summoned the audience with toots on a toy horn.

Act I The fierce Owl (Robby) swooped down and pounced upon the dancing ballerina (Miriam’s doll).,
Act II The fierce Owl, joined by Eeyore, climbed a tree, and both pounced upon whom? The ballerina had become a teacher and had to be pounced upon because the owl and donkey didn’t want to go to school.
Act III The owl and donkey climbed a tree whence better to pounce upon the teacher, now riding upon two toy horses because of injuries received in act II. (Miriam plays such an action herself because she is so big and her favorite toy horses so small). Miriam argued that it was O.K. for Robby to pounce upon the teacher, but he shouldn’t bite “little horsey.”
Act IVThe horns again called me into the room. “BAM BAM BAM” — ambushed again.

When I had resettled in the reading alcove of my bedroom, the children rushed upon me with cried of “Act V.” We engaged in a general rough house and pillow fight.



Writing Stories(2)


After dinner this evening, Miriam, who has been making late Mother’s Day presents for Gretchen, brought me “an early Father’s Day present.” The present, duplicated as Addendum 24 – 1, shows combined a typical drawing with another story in the WRITER model. Miriam could not spell the words, so she dictated it to Robby.


This material shows the expansion of the story form of WRITER into Miriam’s non-computer world. This conventional use of its story format shows its final liberation from the constraint of P, A, and Q stories. After writing the story about Scurry (see Logo Session 23), Miriam has apparently accepted the freedom of the form, and knowing she can get help with her spelling, will be now able to use simple text manipulators without the restraints of an excessive concreteness.

Addendum 24 – 1
Writing Stories: sample of Miriam’s work

Vn 24 story writer model


Vignette 25.1 TicTacToe (4) 6/3/77

Miriam wanted to play a few games of tic-tac-toe before going to bed this evening. After vignette 15’s instruction in the proper ordering of three rules,

    1. Look for a way to win (complete a row of three)
    2. Look for a way to lose (make all forced moves)
    3. Look for two ways to win (fork your opponent)

I wanted to document how well or ill Miriam could absorb those notions. Thus the game was recorded (on the tape of Home Session 6) but not transcribed in detail.

Miriam moves first (her moves are letters, mine are numbers) in this game where I make the sure-to-lose response of a non-corner move after she takes the center square:

	  C  |  1  |  2 
	  D  |  A  |    
	  B  |     |  3 

Miriam recognized my second move and her ‘C’ as forced moves and knew she had two ways to win before my third move.

After my defeat, I move first in this tied game:

	  B  |  4  |  5  
	  3  |  1  |  C  
	  A  |  D  |  2 

Miriam asked if I were going to try my ‘good old trick’ (a corner opening); instead I took the center. She replied “Not usually” to my question of whether she ever moved ‘in these side places’ (as I had just been defeated doing). When I asked “How come?”, she simply said, “I don’t know.” Then she revealed her plan of going on both sides of my center move, and also a third corner, thereby getting two ways to win. Such is a terribly unrealistic plan as it neither recognizes the importance of being one move ahead of the opponent nor makes allowance for the opponent’s possible moves to block such a plan. I made move 2, telling Miriam I would not let her complete such a plan. She said, “Then I won’t do it. I’m thinking of something else.” She made her forced moves.

We had been taking turns drawing the grids in which our games were played out. I introduced the word frame to Miriam as a label for the grid with our third game.

B I get to write the frame. I will call this the frame, O. K.?
M Alright.
B And that’s just the little tic-tac-toe thing but we will call it the frame.
M Me first.
B ‘Cause the frame tells us where to go and what to do.

Miriam begins with a corner move.

	  3  |  C  | B  
	  E  |  1  | 4   
	  A  |  2  | D  

After Miriam’s first two moves, I asked her advice. “Go anywhere.” She said then, “Oh, shucks, I’ve got a forced move,” and she followed all her forced moves appropriately. I requested her to draw the next from and she did so.

In game four, as I went first, I attempted the three corner strategy to see if Miriam was able to block it yet.

	 3   |  C  |  2  
	 4   |  B  |      
	 1   |     |  A 

After my move at 2:

M Oh, no.
B What did I do?
M You’ve got some plan.
B Does this look like a familiar plan?
M Unh-uh.
B Two ways to win [making move 3].
M [move C]
B That was a nice gambit, Miriam. You really did good work there. That was a good idea: trying a different way to beat me.

This is interesting in showing an attempt to block the corner opening on the first move.

	  3  |  C  |  B 
	  E  |  2  |  4 
	  A  |  1  |  D

Noting her last attempt at a non-center response to a corner opening and its failure, I moved 1. Her attempt to continue the three corner strategy failed with her forced move 'C' and its sequels. Miriam drew the next frame.

In game 6, I refer back to Miriam's earlier statement that she had a plan to get on both sides of a middle move. The center opening again:

	  5  |  C  |  2 
	  4  |  1  |  D 
	  A  |  3  |  B 

Since I have no forced move at 4, I discuss with Miriam whether I should put it in the corner (which can never win) or at the side (where I get 1 way to win). Miriam is happy to block it.

Miriam begins game seven with a center move.

	 2   |  4  |  E  
	 C   |  A  |  3  
	 1   |  D  |  B	 

I note how every time I go on the side, I end up losing, so I move 1. When Miriam remarks she has no forced move D, I note it was like the game we just played. I ask Miriam where she can go to win and not win; can she tell the difference. She moves D.

I note how every time I go on the side, I end up losing, so I move 1. When Miriam remarks she has no forced move D, I note it was like the game we just played. I ask Miriam where she can go to win and not win; can she tell the difference. She moves D.

	 C   |  B  |  4   
	 D   |  1  |     
	 A   |  2  |  3  

Move 3 forces Miriam's C. I remark on having a forced move at 4. Miriam responds to a 'forced' move between 3 and 4. We both discover together that she has overlooked the A - C win. Miriam changes her move to D.

B Oh well. You had two ways to win and didn't even know it. How did that happen?
M I went here [I had a forced move 'B'] so I have a move here ['C'] also. Then I couldn't see that because I was trying to keep my eye on if you were going to win.
B Hum. O. K.

In game 9, I request that we play the last game again because it was so tricky. Instead Miriam starts with a corner move.

	 D  |     |  1 
	    |  B  |  2   
	 A  |  3  |  C   

I play out this game in such a way as to duplicate the lesson, though not the form, of the last game. By move 2, Miriam is forced to move 'C' which also gives her two ways to win.

We stopped playing as it was near bedtime, and Miriam wanted to show Gretchen the videotape of the plays she and Meg had made at Logo today (Logo Session 19).

One clear conclusion is that Miriam now subordinates her strategy of finding two ways to win to that of making all forced moves. Game 8 shows an instance of her failing to make a winning move while responding to a forced move. I value game 4 for showing specifically how far advanced from its initial rigidity (cf. vignette 5) is her response to the three corner strategy employed by an opponent. Games 8 and 4 exhibit for Miriam how one can be forced into a series of moves that forks the opponent.

The word frame is introduced to Miriam to name the grid upon which the game is played. The idea is that it is a structure with implications for action.


Vignette 26.1 The Clever Hack (3) 6/13/77

After not using the SHOOT programs for nearly a month, today (in
Logo Session 24) Miriam returned to playing with that game. She started
using the Clever Hack to run up her score (keying ‘H’ followed by
‘SHOOT 0’; the former locates the turtle inside the origin-centered
target, the latter guarantees a hit). I showed her then that in the
interim I had added a new feature to SHOOT, the option (under control
of a switch) of having the target relocate as well as the turtle after
every hit.

This fact came up in our conversations after dinner. Robby was
quite pleased with the letter he had written (using the LETTER program,
Logo Session 24) to a friend in Connecticut. Miriam interjected, “You
know what Daddy did today. He made SHOOT so tricky the clever hack
doesn’t work any more.” What struck me was Miriam’s tone — she was
imparting to Robby some shocking news.

My intention is to lead Miriam to the discovery that she can get
the turtle inside the target area using forward and turn commands
(deferring execution of the SHOOT procedure until certain of a hit).
I will describe such an action as a clever tactic. My objective is to
introduce to her a set of distinctions which focus on the particularity
of a problem’s solution: the ‘hack’ (like the gambit) being the most
context dependent; the ‘tactic’ being a set of specific actions which
may be catenated to solve any member of a class of well-understood
problems; and the ‘strategy,’ a set of actions one employs where the
goal is clear but the appropriate operations and intermediate states
are not obviously limited.


Vignette 27.1 Emberley’s Faces (2) 6/17/77

A large portion of Miriam’s drawings during the past year have
taken the form of presents she makes to others. She has spoken of them
as presents many times. One formal element of these notes reflects
that character. Each typically bears a tag of the form: “TO _______/
LOVE, MIRIAM” (see Addendum 1 for an example). Her initial tags were
of the form: “TO ________/FROM MIRIAM,” reflecting, I believe, the
format of tags she read on presents received at Christmas and so forth.
Their great-grandmother sends both children postcards whenever she goes
on trips with a valediction “Love, G.G.”

The note presented in Addenda 1 and 2 is to a fellow of Miriam’s
play group. Brian is a boy whose entire family is committed to foot-
ball, so it’s most appropriate that Emberley’s ‘Football Fred’ face
(page 11) should be a present for him. The face inside is ‘Sleeping
Simone’ (page 7). Because the note lay on her desk for several days,
I had the opportunity to ask Miriam how she came to draw such a nice
picture. She replied that she copied Football Fred’s face but made
the body up herself. The shoulders of that body come from Football
Fred’s face. The ‘bar’ arms, ‘stick’ legs, and circular body are
typical of her earlier drawing. The striped shirt is the costume of
a rugby player ( a very popular shirt style now, which is also found
in Emberley’s book on page 28).

The note/present Miriam made for her friend Brian shows in
high contrast the appearance of sophistication which she is
developing from copying faces in conjunction with the quite
primitive body-drawing she invented herself.

Addendum 27-1

Football Fred

Vn 27-1 Emberley's Football Fred

Addendum 27-2

Sleeping Simone

Vn 27-2 Sleeping Simone


Vn029.01 Making Puzzles 6/18/77


Vn29-2 Addendum1

Vn29-3 Addendum2


Vignette 30.1

temporary upload

Original Fair Copy, Scanned page 1

Vn 30-1 Original Fair copy; temporary upload

Original Fair Copy, Scanned page 2

Vn 30-2 Original Fair copy, temporary upload


Vn031.01 Collecting Tolls 6/19/77

Robby and Miriam each receive a nominal ‘allowance’ weekly,
regardless of whether they’ve been ‘good’ or ‘bad’ or done what we
parents have wanted them to. They know it is computed by multiplica-
tion: for each child the ‘allowance’ is 5¢/year times the child’s age.
Thus Miriam recently began receiving 30¢/week. Robby receives 35¢ and
will soon receive 40¢. Such is a small amount of money, enough to buy
one stick of chewing gum a day with only a little left over.

Upon our moving to Boston, Robby took over the chore of mopping
the floors, the frequent necessity of which derives from Miriam’s dust
allergy. Because I consider it my responsibility, and not one I can
manage easily and directly, I pay Robby to do this work for me. He
saves his money andy buys models of boats when his hoard is large enough.
Because Miriam cannot perform any similar work, this difference has
become another element of sibling competition and has intensified the
children’s general interest in money as an instrument of power.

After an early spring trip to Connecticut (which included
travel on the toll-collecting Connecticut Turnpike), I found Robby had
set up a toll booth at the entrance to the secretaries’ office at Logo.
(At one point he claimed Greg owed him $18.) I objected to that game
and told Robby to play it no more at the lab.

This Friday Gretchen brought home a stack of cut yellow paper,
the pieces being about 3×8″. When he first saw them, Robby referred
to the papers as ‘tickets.’ In fact, they are about the same size and
shape as the parking tickets I have collected with distressing regu-
larity at MIT. With this minimal suggestion of tickets (and paying
fines), Robby conceived and both children executed a plan to establish
toll booths in our carriage house, Miriam at the entrance to our general
living area and Robby at the entrance to our bathroom.

This seemingly harmless game was a good answer to the recurrent
question of what to do on a rainy day. The game was one of the chil-
dren’s invention, a simultaneous practicing at being grown up and an
expression of their concerns and ideas. It had the ultimate value, the
sine qua non, of absorbing their time and energies in a direction-free
project. The children made signs for the toll booth, tally sheets of
accrued obligations, and collections (typed) of commutation tickets
(examples may be seen in Addendum 31-1).

This toll-collection project finally spanned several days.
Aside from increasing the general clutter, the only flaw – and this a
fatal one – was that the children confounded the toll collection ‘debt’
with what we considered real obligations, i.e. the providing of their
weekly allowances and the daily snack allotment (50¢) the children
receive when they come to work with me at Logo. Robby began computa-
tions such as this: if I save 2 fifty centses, mop Miriam’s bedroom
and the hall, and you pay your tolls, with what’s in my bank I’ll have
enough to get a model of the King George V.

Direct confrontation was the only way to disabuse Robby of the
notion that we would really pay his tolls. I began by charging him a
quarter for a glass of soda (the commercial rate at MIT). His strenuous
objections were exacerbated when he found the evening meal would cost
him 3 dollars. When he countered that the price was too high, so he
and Miriam would ‘make their own,’ I announced the 1 dollar refriger-
ator opening fee and my 50 cent kitchen entry toll. Robby accepted,
albeit with little grace, the collapse of his scheme. Miriam persisted
for days thereafter making tickets charging me 99 cents for opening
the refrigerator (if you can charge me a dollar, I can charge you
99 cents).

These notes document the spontaneous generation and working
out of a small project at home with no grown up intervention.

Addendum 31-1

Toll Collection Records

Vn31 Toll Collection Records


Vn032.01 The Word Box 6/20/77

As her term end in kindergarten approaches, Miriam brings home
more of materials from school. Today came the Word Box — a small
plastic case for holding 3×5 cards. As a reading-readiness activity,
the children are, occasionally, asked to select some word whose spelling
they would like to learn. The teacher prints it on a card; the child
then copies the word onto other cards and places the words in his box.
If one must teach spelling to children (and one must, eventually), this
is an eminently sensible approach.

Telling us about her word box over the noon meal, Miriam
explained a problem and her little joke in solution. Miriam was asked
to pick some words she wanted to spell; she didn’t particularly want
to learn to spell any words she didn’t already know. Noting her box
had nothing at all in it, deciding ’empty’ was a good beginning amused
her. ‘Word box’ followed. The other words — in no significant order
I can tell — are:

wonder wonder April
wonder post office Indian

Post office, as a member of her word list, may derive from the near
term end walk she took with her student teacher Sue to buy supplies for a
farewell party. Miriam also noted she had asked Mrs. Badger, her
kindergarten teacher, how to spell ‘floccinaucinihilipilification’ . . . .
And thereby hangs a tale.

Last summer, while we were moving from Connecticut to our
quarters in Massachusetts, the whole family made the trip several times
(we did some renovating of those quarters and performed the thorough
cleaning Miriam’s dust allergy required). To amuse the children we
often played the game ‘I am thinking of a word.’ In this game, the
selector informs the others of the initial and final letters of some
word as hints for their guessing. It fell my lot to select a word once
under difficult driving conditions. I said: “I am thinking of a word.
It begins with ‘F’ and ends with ‘N’ and it’s one you’ll never guess.”
This permitted me to deny without thinking FAN, FIN, FLOWN, FAUN, FUN,
FON (the ruler of an African kingdom; Gretchen was with us. Cf. Gerald
Durrell’s book The Bafut Beagles). Traffic improved; I was ready for
the outrage then when I announced ‘floccinaucinihilipilification’ (which
is the habit of making small of things. [The word does not appear in
Webster’s Third International but may be found in the Oxford English
Dictionary.]) A few days later Miriam and I drove back to Massachusetts
in my MG (a 1953 TD). Above the roar of the engine, Miriam began, “I
am thinking of a word. It begins with ‘F’ and ends with ‘N’.” In my
turn I guessed the obvious monosyllables. When they were rejected I
tried those with long vowels marked in spelling by the terminal ‘e’.
(Miriam was just beginning to read at that time). When I finally gave
up, I listened quite attentively as Miriam burst out laughing and said,
“Floccinockihilification.” (She mispronounced one vowel, a diphthong,
and omitted 3 of the 12 syllables). Since that time of her great
surprise, Miriam has wanted to learn to spell the word.

It is my intention that Miriam should continue using the word
box during the course of this project for words she encounters at Logo
(both Logo-words and others — for example, those she needs help with
in writing letters). The incidents here illuminate some of the non-
pragmatic use of words that elevate their interest for Miriam.

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Vn034.01 Candle Fire Crackers 6/23/77

We usually dine by candlelight. We enjoy making candles and
using them, and the ill distribution of light in our dining area makes
this practice a useful enjoyment. Having agreed that he will not play
with fire, Robby has the responsible job of candle man: he brings the
candles to the table, lights them, and when the penny candles in old
bottles burn down, he replaces them. Having made a 1 stick candelabrum
in school (a ring of cardboard with pasted-on, brightly painted maca-
roni shells), Miriam after giving it to the family as a present reserves
its use to herself and the responsibilities pertaining thereto (lighting
it and blowing it out).

For some reason during the dinner Robby blew out a candle (per-
haps to replace one burned dowm). Miriam took this as her cue to blow
out hers. To minimize the air pollution Gretchen wet her fingers and
doused the smoke producing embers in the wick. Shortly thereafter, when
she attempted to re-light her candle, Miriam heard the sputtering
crackle made by the flame on the wet wick. “That sounds like a fire
cracker!” Questions immediately arose: what makes the candle sputter?
why doesn’t it light? It does now? Oh. Why didn’t it light before?
Because Mommy spit on it, the water. Miriam, Seymour, and I had just
been discussing the Piagetian experiments done earlier in the project.
I allowed that I thought Miriam most enjoyed the conservation of con-
tinuous quantity experiment because of the water play in pouring the
liquids from one container to another. (Miriam corrected my misappre-
hension: she most enjoyed the experiment of constructing tracks [cf.
Miriam at 6]). Thus it was a natural continuation that we indulge in
a little water play, even at supper. Seymour asked Miriam if she
thought she could make it happen again. I got her a small glass with
water in it. Miriam took her candle and inverted it inside the glass
slightly above the water. It went out. When she brought it to the
flame, the candle lit immediately without sputtering.

Miriam Hey! Why didn’t it work?
Seymour Did it go in the water?
Miriam It went out.
Seymour Try it again, just to be sure the end goes in the water.

Miriam dunked her candle in the water and upon the attempt to relight
it sputtered and crackled before catching fire. Miriam tried the
dunking again and it still worked. She remained curious as to why
the candle went out at first. Robby suggested that with the candle
inverted, the flame wanted to go up, but had no place to go, so it
went out. I suggested we make sure it wasn’t the water by holding the
candle about 2″ above the surface. Miriam did so, watching carefully.
“It’s the wax that does it!” Seymour asked, “Does it need to be in the
glass at all?” Miriam proved that it did not by inverting her re-lit
candle over a napkin.

This vignette highlights the role of engaging phenomena, e.g.
the surprising crackling sound from a candle, and the supportive
milieu in leading a child into those discoveries that constitute his
knowledge. The rich environment is less one rich in objects than it
is one rich in surprise, in the stepwise exploration of which the
child confronts alternative plausible explanations of those phenomena.
Obviously, since this surprise derives from the child’s ignorance,
what engages one child need not engage another.


Vn035.01 Hatch Who? 6/24/77

The jokes and puns a child is capable of creating and under-
standing are a fine developmental index of his ability to conceive of
some object or idea in multiple contexts. (For examples of Miriam’s
earlier, limited capability see the discussion in Pre-Readers’ Concepts
of the English Word). Precursors of the pun are typically insulting or
vulgar statements which are humorous to the extent that they are clever
in the context of the conversation. A vulgar example:

Some few weeks ago, having seen on TV a program on the flight
of the ‘Enola Gay’ over Hiroshima, the children asked me why the A-bomb
had a special name, then whether or not there might be a B-bomb, and so
forth. I explained as best I could the A-bomb and the H-bomb, that
although there was no B-bomb, there was a cobalt bomb that could be
called a C-bomb. When asked about the C-bomb I noted it was especially
damaging to people. Miriam interrupted here with her invention of a
bomb that wouldn’t hurt the buildings but would chase away all the
people: the F-bomb. When she had our full attention, Miriam burst out
laughing and said, “It’s a fart bomb.”

This morning, the last day of school, I went to kindergarten
with Miriam with a request that her class sing ‘Little Rabbit Foo-Foo’
so I could make an audio tape recording. We arrived early so Miriam
asked me to read to her and her friends. I read one book. Miriam then
produced a small book about a panda bear from China, Ah Choo. I asked
her to read it, which she did. The simple plot has the panda sneezing
on every page.

At lunch, Miriam began this dialogue:

M Knock, knock.
Br Who’s there?
M Ah.
B Ah who?
M No. Choo.
B Choo who?
M No, no. Ach.
Br Ach who?
M God bless you.

Miriam now clearly understands (and I believe came to understand through
the mediation of the panda bear joke) the knock-knock/booby-hatch joke
I invented months ago in response to her earlier insult:

M Knock, knock.
speaker Who’s there?
M Booby.
speaker Booby who?
M Booby you.

This vignette documents Miriam’s developing understanding of a
pun and contrasts it with her earlier insulting and vulgar jokes.