Skip to content
Archive with last of tag-string LC2


3V0492.01 New Car Seat Opens up Peggy’s World (5/29/79)

Ever since the children got some real bargains at a tag sale last summer,
they have been followers of local tag sales. They take whatever cash they
can scrape up and spend it all, giving away their loot in case they can
not imagine a use for it and to justify the spending. Miriam bought
Peggy a crib toy and Robby bought her a set of little wheeled racing
animals some days ago. The next day, Miriam recalled seeing on sale
for $5. a car seat, which we need now that Peggy has outgrown her
infant seat. Gretchen purchased and I repaired the new car seat for
Peggy. A small thing this seems to be, but it has changed Peggy’s access
to the world significantly.

No longer does Peggy ride in a car facing backwards and below the level
of the window sill. She sits up, facing forward and looks out on the
world. Peggy has enjoyed coming outside to ride in her swing, play in
the sand box, or just walk about, say up the driveway to where Scurry
is tied. She has complained when brought in. But now her complaints
are getting more vehement. She even gestures inside, that she wants to
go outside. She has been so eager to go for rides that later on (June
4th) she rode all the way to Boston and back the next day without any
significant fussing.

Importance: This simple furniture addition, the new car seat, has
opened wider Peggy’s access to the world. When she goes shopping
with Gretchen, now she can see variety in the world about her as she
moves through it.


3V0495.02 Pretending; incorrect choice as a joke (6/01/79)

Late in the afternoon I found myself waiting at home for two telephone
calls while Gretchen took the cub scouts on a trip. Peggy played in my
care and during the hour and more the following incidents occurred:
Pretending: Peggy of pulls dishes and other utensils from a cabinet with
low shelves. She pulled out and emptied a coffee jar. The lid to that
specific jar has a lip on it. It’s general appearance is like the surface of
the shield for Peggy’s drinking cup./ Peggy picked up the jar, lifted it to
her lips and “drank” from it. She turned to me and smiled. Was she
pretending to drink ? Did she expect milk to come out of the empty jar
(it was a transparent jar – but her cup is opaque). Is it possible she was
trying on the chance that it might work ? Or just to be sure that it
would not work ?

If she were disappointed, would she have smiled when she put the jar
down and looked at me ? Could we see here a very early example of
“incorrect-choice-interpreted-as-a-joke: as in the examples of Miriam’s
“going-flying” bug in CECD ?


3V0502.02 Pure verbal interpretation overwhelms context: 6/08/79

Pick up Foxy
The older children have a bad habit (likely picked up from me) of
dropping wherever they are whatever they have no further need of.
when I try to get them to pick up after themselves they complain “I
didn’t have that” or “Shouldn’t (the other child) pick up that (other
thing) also ?” With considerable justice, they complain that Peggy
makes an absolute mess of the house, dropping her things, theirs, or
whatever comes to have wherever she is when something else
dominates her mind. Thus, when I asked Robby today to pick up some
clothes he had dropped in the kitchen I turned to Peggy who had
dropped the toy red fox near her high chair and said “Peggy, will you
pick up Foxy ?” pointing at the toy on the floor. Standing near me and
the toy (to which I pointed and which was in her sight), she looked up
at me then crossed the kitchen to the dog’s bed, grabbed Scurry by the
ear, and tugged at it three times.

Importance: Peggy’s reaction to this instruction was entirely
unexpected. No one has ever referred to Scurry as Foxy. Even though
Foxy (the name we all use for her toy red fox) was in plain view and
further specified by pointing, Peggy apparently considered Scurry the
intended referent of the name I spoke. Clearly, Scurry is the
outstanding exemplar of what a fox is — for Peggy has identified the
Scotty as a fox numerous times on videotape.

It would be a mistake to erect a theory of label fixation on the basis of
a single example, but I incline to see this “error” of interpretation as
similar to the hypothetical process I have otherwheres called the
“nucleation of microworld clusters.” Here, in place of an archetype,
the primary example of Peggy’s class of ‘Fox’, i.e. Scurry, is interpreted
as the referent for a term which has never been applied to her. If no
more, this incident is evidence and a lucid example of how thought
intervenes even in so “simple” a process as the association of names
with referents.


LC1bT06 Protocol 6

Included Text Pages (2)

RAL protocol 6.1

RAL protocol 6.2

Included Materials (2)

RAL protocol 6-A1

RAL protocol 6-A2


LC1bT09 Protocol 9

Included Text Pages (2)

RAL protocol 9.1

RAL protocol 9.2

Included Materials

RAL protocol 9-A1

RAL protocol 9-A2

RAL protocol 9-A3

RAL protocol 9-A4

RAL protocol 9-A5


LC1bT11 Protocol 11

Included Text Pages (2)

RAL protocoll 11.1

RAL protocoll 11.2

Included Materials



LC1bT13 Protocol 13

Included Text Pages (7)

RAL protocol 13.1

RAL protocol 13.2

RAL protocol 13.3

RAL protocol 13.4

RAL protocol 13.5

RAL protocol 13.6

RAL protocol 13.7

Included Materials (6)

Figure 1
RAL protocol 13 Figure 1

Addendum 1
RAL protocol 13-A1

Addendum 2
RAL protocol 13-A2

Addendum 3
RAL protocol 13-A3

Addendum 4
RAL protocol 13-A4

Addendum 5
RAL protocol 13-A5


LC1bT20 Protocol 20

Included Text Pages

RAL protocol 20

Included Materials




The Genesis of Symbolic Thought

Learning, in General

Let us begin by going beyond a “stimulus-response” couple to a stimulus-response arc. That “arc,” represented as a link between input and output (more generally stimulus — which may be entirely interior — and response — which may be entirely interior) is the site for attachment of interventions. In a simple case (e.g. Meltzoff’s experiment), the output of the SR arc mimics the input.

Interventions are, at first, interruptions of process, because of some sort of disruption (types might be failure, confusion, discordance, etc.); in this case, the links become sites for attachment of problem descriptions. When discriminations occur, these interrupted-links become the loci of extensions to the SR arc; how do the discriminations occur ? They follow Sussman’s formulation: problem descriptions are converted into prescriptions for change by local structure modification agents. (Minsky’s B-brains are intended to be capable of this functionality.) After a discrimination, the interruption has been repaired and is now an intervention in the preceding structure. Every link can be interrupted and the development of interventions occurs everywhere. As intervention-extended networks grow out of SR-arcs, they become slower in processing, more confusing for B-brains to manage, and ultimately, too complex for B-brains to change (this means they cease being capable of learning).

As these networks (societies of agents) grow, they compete with each other. Note well, the simplest processes (SR-arcs) still compete with them, and this can lead to the later replacement of a well established complex society of agents by a later developing but better fitting simple society.

Language-Specific Learning Theory

Following Peirce, we schematically represent three kinds of ways in which signs are involved with things signified: Iconic signs recall things signified, indexical signs indicate or “point to” things signified, and symbolic signs name things signified. (“Names” here implies conventional assignment of reference, variable by society and language groups.) The main issue to be explored is relations of the three kinds of signs, among themselves, and the way their interactions can be seen to explain important linguistic and psychological phenomena.

Iconic Signs as Fundamental Beginnings
In the simplest case, iconic signs are pristine SR arcs; they remind individuals of the things signified in that their intepretant recognizes No Significant Difference (NSD) between the icon and the thing signified. The interpretant for such an iconic sign is no more than a K-line which responds to the stimulus. The internal representations of the external sign or e-sign (which has in this case been taken as an iconic sign) becomes associated with the K-line; this association creates a change to the K-line where the counterpart “e-sign related modification to the K-line” serves as a personal-sign, or p-sign. This associated p-sign is used expressively to produce a vocalization intended to indicate the thing signified, e.g.:

  • the e-sign /”Scurry”/ used by family to refer to the dog [Scurry] is interpreted by Peggy to be associated with the dog [Scurry].
  • Peggy’s K-lines involving [Scurry] become associated with her p-sign for [Scurry] which, in her vocal expression, is manifest as /cul/di/.
  • when Peggy uses her p-sign { /cul/di/} to signify [Scurry], she is using the term as an indexical sign. This expressed p-sign functions as an indexical sign if and only if others recognize what it indicates.

How P-signs Become Indexical Signs
Knowing that Peggy expresses her p-sign for [Scurry] as /cul/di/, some people use the sounds /cul/di/ to refer to [Scurry] in the attempt to communicate with Peggy. It works. In this special case, the infant’s p-sign functions as the personal part of an indexical shared sign (s-sign) because the local Society accommodates to the infant. Consequently, association of the p-sign with the [Scurry] k-line is strengthened. The use by another person of /cul/di/ is not a p-sign itself (for them), but a transient, symbolic e-sign referring to the same entity [Scurry].

When the infant modifies the expression of her p-sign to accommodate to the Society, the p-sign becomes an indexical sign through a different process, as follows. Other people use the e-sign, e.g. /”Scurry”/ to refer to [Scurry]. Interpretants need to be developed to relate Peggy’s perception of /”Scurry”/ (already associated with her k-line for [Scurry]) through her p-sign { cul/di/} to a modified vocal expression similar enough to the conventional or common e-sign /”Scurry”/ to be recognized by others. In this general case, the infant’s p-sign functions as part of an indexical common sign (c-sign) because she accommodates her vocal expression to the conventional e-sign for the entity signified. In sum, every individual has p-signs as parts of k-lines associated with entities. When indexical signs are used in communication, it is because individuals negotiate the vocal expression of their signs to permit communication. In the special case above , we refer to these signs as s-signs; in the common case, we refer to these signs as c-signs. Both s-signs and c-signs are kinds of e-signs. The difference is that indexical s-signs are part of the infant’s idiolang. Indexical c-signs are part of the society’s public language. The need to associate infants’ idiolang-effective p-signs with others’ c-signs is a primary interior motor of the symbolic transition, as explained in the following.

The Theoretical Context of Modeling

We will model language development in the theoretical context of Minsky’s Society Theory of Mind and its suggested forms of representation. The central ideas used in setting the context of this model are that the general processing structure of the mind is represented by Minsky’s “ring closing” structures (p.205), with K-lines as the basic structural elements of memory (p.82 ff.). It is presumed that SR arcs grow into elaborate K-lines through the processes described below and may also grow into societies of agents, depending on the circumstances of learning. A first assumption is that one can think of the interior perceptions of “iconic” signs as Not-Significantly-Different from associated memories. A second is that incremental learning with respect to any SR arc proceeds in the Sussman paradigm, with B-brain structures modifying interrupted arcs, which leads to the recognition of e-signs for distinguished external things.

Indexical Signs and Learning Processes
The meaning of an iconic sign is determined entirely in the mind of the infant. The beginning of the indexical sign learning process is in the infant’s attempt to communicate something. Grant that the infant has associated some producible sounds, such as /cul/di/, with the entity [Scurry], through a listener agency mimicking what the infant perceives of the e-sign /”Scurry”/. {/Cul/di/} is then a p-sign which serves the infant’s expressive intentions. The e-sign/p-sign couple does not function as an indexical sign until there is negotiated a shared meaning between the infant’s talker agency and that of some other person who understands what the infant intends to communicate.

This can happen either through the infant improving her production of sounds to match better the conventional e-sign or by the other person changing his language to better communicate with the infant. In Peggy’s case, Bob started referring to [Scurry] by /cul/di/ when in her company. /Cul/di/ functioned as an indexical sign between Bob and Peggy because both used the same sounds to refer to the same thing signified by a convention, explicit here in Bob’s decision to adopt Peggy’s term for [Scurry]. This is a shared sign, or s-sign. Such signs are the main elements of the individual infant’s idiolang. Subsequently, Peggy modified the vocalization of her p-sign { /cul/di/} to conform to the sound /”Scurry”/ used by her mother and siblings when calling the dog or referring to her. This exemplifies the second process of negotiation of meaning between the infant and her local society, through which her idiolang p-signs are brought into correlation with the c-signs of the public language in her Society. Negotiating meaning is the common ground of these two different processes, even though the first is so transitory and the second is so dominant.

Negotiating meaning has been described as the primary means for learning to communicate with sounds. I argue that the portion of the process in the interior world — one linking modifiable responses to external signs (e-signs) in recognition processes with modifiable personal signs in expression processes — is the prototype for the symbolic use of words. Can we more precisely articulate this process to clarify the transition from using indexical signs to using symbols as a natural consequence of processes of communicating?

If the infant’s listener agency does not discriminate initially between what she perceives on hearing /”Scurry”/ and what her talker agency expresses as /cul/di/, there may be either improved discrimination with respect to /”Scurry”/ or better correlated expression by modifying /cul/di/ to become more like /skuh/ri/, or both. The first case is a modified discrimination on the recognition side of the SR arc. The second is one of modified production on the expression side of the SR arc.

We have already postulated that the infant’s p-sign generates a vocalization, but we have not previously noted that corresponding to the expressive aspect of the p-sign represented by the vocalization, there is a perceptual aspect which also needs to be represented. Let’s say that on hearing /”Scurry”/, the [Scurry] k-line perceives /skuh/ri/ but does not discriminate it as different from /cul/di/ through which it expresses p-sign {/cul/di/} . When subsequently a discrimination is made that perceived /skuh/ri/ is different from the normally produced /cul/di/ the infant’s listener and talker agencies have to negotiate an integration of the two aspects of the internal sign. It is this process that is the prototype of word definition by symbolic use of known words. Consider the following example:
We know that Scurry is a dog, and that /”dog”/ or /”doggie”/ are appropriate words to use in referring to [Scurry]. How could such a word enter the repertoire of indexical signs in Peggy’s idiolang ? We might say “Scurry is a kind of dog.” Peggy surely heard the word /”dog”/ before she had any well formed concept of [dog] as a category of kinds of entities. For her, when she understood that the term /”dog”/ was meant to indicate [Scurry], the sense she would have made through interpreting /”dog”/ would be that “dog” is another name for [Scurry]. /”dog”/ is thus a synonym for aspects of the p-sign represented either as { /skuh/ri/} or { /cul/di/} , depending on the time that the association was made.

Synonyms are common in an infant’s world. Mom and Dad have personal names. The infant herself may be Peg, Peggy, sweetie, or have any of a myriad of affectionate appellations. It is an open question as to the point in time at which anyone learns that what one assumes is a synonym for a well known entity’s name refers to some other entity. Recall, for example, that Peggy’s first two word sentence was /cul/di/va/va/! The context made it clear that she was referring to the barking of a remote dog. Did Peggy intend to communicate that ? Did she even know there were other dogs in the world besides Scurry ? No — this is a case of her marking no significant difference between her notion of [Scurry] and any other similar creature.
This reflection can help us understand one simple way that Peggy’s listener and talker agencies could integrate the perceived vocalization /skuh/ri/ of e-sign /”Scurry”/ with the expressive vocalization /cul/di/ of p-sign { /cul/di/} . Let it be the usual case to have multiple names for entities. Then /cul/di/ could be Peggy’s personal name for [Scurry] whom others may choose to refer to as either [/cul/di/] or /”Scurry”/. Some people might even refer to [Scurry] as /”dog”/ or /”chien”/. Development from the infant’s idiolang of indexical signs to a broadly flexibly public language then proceeds through elaboration of names for well known entities, where the relation of synonymity is the first wave of definition of new words (symbolic signs) in terms of preceding linguistic structures (indexical signs). It is in this specific sense that the integration of perceived and expressive vocalizations related to a specific k-line is the precursor and prototype of the development of symbolic thought, the key aspect of which is that words have meanings specifiable in terms of the network of meaning of other words.




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



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.



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.



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.


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(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 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 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.

Protected: Vn03301

This content is password protected. To view it please enter your password below:


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.


Vn036.01 Losing a Friend 6/25/77

Now that school has ended for the summer, Miriam will see much
less of her schoolmates. Of her close, new-found kindergarten friends,
most will be away for most of the summer. The child going farthest
away, unfortunately, is Maria, Miriam’s closest friend. Maria is going
to Spain, has in fact left already, and she will not be coming back.

Any adult would call Maria a very attractive child. She is a
pretty child of pale olive skin, dark hair, and dark brown eyes. All
her best features were accentuated by the particular care with which
she was dressed (typically I think of her wearing a frilly white blouse,
red skirt, and brighter red sweater, and black Mary Janes as shiny as
the gold rings in her ears). My first recollection of her was early in
the autumn as a leader of the other children: she ran to a specific
place in the school yard, turned, and called, “All the puppies come
over to me,” whereat most of the other children gathered around her,
barking loudly. I would characterize Maria’s demeanor as modest.

In the school Maria was one of the four other girls with whom
Miriam played most in the Housekeeping Corner (cf. Vignettes 14, 18,
19). Why did Miriam like Maria? What was special about her in Miriam’s
eyes? When I asked Miriam why she liked Maria, why she was her special
friend, Miriam replied, “I don’t know, I just like her.” One element
of their cameraderie derived from accident. Most children on the kin-
dergarten bus disembarked at a central point north of Route 9 or at the
Heath School (to participate in after school programs). Miriam and
Maria were left alone on the bus for another 10 minutes riding home.
I believe during these many small segments of time began an activity
Miriam mentioned every time she spoke of asking Maria to visit: “making
funny faces.” Some examples of thses ‘funny faces’ can be seen on
videotapes of Logo Sessions. The typical procedure for making a funny
face is to stick the fourth finger of each hand in the corner of the
mouth; bracing the hands with the thumbs by the ears and little fingers
near the chin; to deform the face around the eye sockets with the
second and third fingers; finally, to stick out the tongue. Thus it
was, in bringing Maria to our house or taking her home, the two girls
would amuse themselves in the back seat of the car.

As Maria’s departure approached, Miriam has been very sad and
said recently she knows that when Maria leaves she will never see her
again. I have tried to console Miriam as best I could without deluding
her. For her sorrow, I could only offer my sympathy and my assurance
that her sorrow was a praiseworty, mature, and appropriate feeling
and not something to be devalued.

This vignette documents how the end of the school year has been a
difficult time for Miriam. It puts in perspective the material of
Vignette 33. I believe it also suggests that Miriam’s work on this
project, providing a sense of continuity and access to her friends at
Logo, is generally supportive.

Addendum 37-1

Letter to Maria

Vn 37 Letter to Maria


Vn37.1 Explaining SHOOT 6/26/77

While visiting some friends with a summer place at Lake Winnepesaukee,
the nature of our work at Logo came up in conversation. When
I asked the children if they would like to explain any part of it, they
agreed to explain how SHOOT works.

They designated a mid-floor hot air register as the target and
said, “One guy has to be the turtle, the other guy the keyboard.”
After minor contention they agreed Miriam should be the turtle and
Robby give directions. Miriam at the command SETUP turned a circle
and did a GO-SOME-WHERE (she moved to a random place and turned away
from the target). Robby commanded ‘left turn 90, left turn 20’ for
alignment, then ‘SHOOT 400.’ Miriam walked to the target and announced,
“Ouch. Your score is 1.” Miriam then suggested Robby be the turtle.

Robby agreed and executed a GO-SOME-WHERE. Apparently Robby
agreed to be the turtle in order to make this joke: he went from the
target through an open bedroom door and closed it. “I’ve GONE-SOME-
WHERE!” This pretty much ended the game.

These notes are not meant to exhibit the children as articulate
expositors of the project; they do show the manner in which these two
children most naturally represent to others what we do.


Vn38.01 Robby’s Place in the Project 6/28/77

Robby raised a very difficult question today — how much of the
work he does at Logo will be a part of my doctoral thesis. The answer
Robby required, and it is a superficial answer, that the thesis will be
about Miriam’s development, was bound to disappoint him. My answer to
his question attempted to provide him with a perspective from which he
could see the value of his contribution to the project, could imagine
that contribution being adequately recognized in the future, and view-
point from which he could judge my preferring to study Miriam’s devel-
opment as a back-handed compliment.

The facts from which we began he knew well: that he was doing
precisely ‘the same experiments’ as Miriam; that the sessions with him
were being recorded as faithfully as were those with Miriam; that some-
times he did work that was beyond Miriam’s grasp (e.g. his understanding
of GUNSIGHT, an absolute coordinate variant of the SHOOT program).
The other outstanding fact was his seeing how hard I work: I sleep
little and spend the rest of my time transcribing the data and planning
future sessions. He sees every day that I have no free time. I ex-
plained to Robby that, for now, I was forced to choose; in effect I had
chosen to work with Miriam’s data first. Since I have also recorded
his work and can transcribe it later, that work is not lost although
little of it will appear in the thesis.

Here I suggested beyond the thesis lies the idea of a book, one in
which his work would appear as central as Miriam’s and even more so.
For Robby has worked at Logo longer than Miriam, and his sessions of
past years were for us the pioneering precursors of the more sharply
focussed study that this thesis work represents. I sketched for him
the theme of this book as our family’s involvement with computers and
the impact of that involvement. He could appreciate that our experience
now is unique, that his is a central role in that story, from its begin-
ning till whenever it ends, and that Miriam’s contributions follow his.

As for choosing to focus this study on Miriam, I explained my
intention was not to see how much she could learn (for Robby now appears
capable of learning more and more rapidly), but to understand the way
she learned things in detail. Further, I could not hope to understand
well how Robby learned new material because he already knew too much.
Robby recognizes that he knows far more about World War II than I do.
Referring to this as an example, I asked how I could hope to under-
stand his learning when he knew some things better than I knew them.

This issue touches a critical nerve of the project, for it is a family engagement
as well as being a focussed study of Miriam’s development.


Vn41.1 7/5-7/77

Whenever we ride to Logo in the MG, Miriam has a standing request
that we follow Memorial Drive down past the underpass at Massachusetts
Avenue. The children like the magnification of their voices provided
when they shout in a closed place. Over the past several years, we have
agreed that they may do such shouting when I am driving them about in
the MG, but not otherwise.

While we lived in Connecticut, the children introduced the ejaculation
“Daddy is a dum-dum” as their underpass chant. I don’t recall
the details but merely the impression that its use involved some sort
of trick (perhaps a promise, not to be kept, that if I let them shout
they would not broadcast what a dummy I am). The children believe this
annoys me, and they relish it as a way of teasing me.

When, two days ago, from the BU Bridge I preferred the Vassar Street
route to the Mass. Ave. underpass, Miriam claimed she was mad at me and
was going to quit my thesis project. I complained to her: “Do you think
I like to hear you shout that I’m a dum-dum? You always yell that.
Don’t you think that hurts my feelings?”

Today, as Robby, Miriam, and I drove home from Logo, we took the
scenic route — down Memorial Drive. Once again the cry was raised.
We continued down Mem. Drive and Miriam looked troubled. “Daddy, we
really don’t think you’re a dum-dum. But we like to shout under bridges
and don’t know anything else to say.”

This anecdote exemplifies how peculiarly specific is Miriam’s use
of speech. The phrase “Daddy is a dum-dumb” is thought of as a chant-
for-passing-under-bridges, but one devoid of semantic content.

Post Script

Miriam’s sensitivity to my feelings led her over the past few days
to attempt the development of a new chant. She came up with:

Daddy is a smart-smart.
Daddy is a smart-dumb.
Daddy is a dumb-smart.

Having asked Robby for help she received this suggestion (his view is
different from hers):


Is Daddy a dummy?

Is Daddy a smarty?

What is he?

An idiot!

This latter expression is clearly a relatively flexible variation on
some small script for a shouting-insult.


Vn43.1 Binary Counting 7/7/77

At dinner this evening, the topic of counting on fingers arose.
After performing some finger sum, Miriam turned to Robby with 2 fingers
of her left hand raised and all the fingers of her right and asked:

Miriam Robby, how much is this?
Robby 7.
Miriam No. It’s 25.

Tricked by this representation shift, Robby gave her an equally challenging
problem. Holding up both hands with 5 fingers extended on each:

Robby How much is this?
Miriam (Uncertain and not consistent) 10?
Robby No. 25. It’s 5 times 5. Get it?

With these fluid finger counting representations in the air, Gretchen
asked me to explain hexadecimal finger counting (I use such a procedure
to keep track of telephone ring counts so I can think of other things
while waiting for people to answer the telephone). Since Miriam had
just invented a second finger counting representation and Robby a third,
it seemed appropriate to show the children binary (Richard Feynmann
introduced this procedure to me in an informal chat when I was an under-
graduate). I held up three fingers of my right hand — pinky, fourth,
and index. “How much is this?” Knowing 3 was not my answer, Miriam
guessed that number. I believe Robby guessed 21. I said, “11. I have
a funny way of counting. Let me show you how.” I proceeded to count
from 1 to 31 on the five fingers of my right hand. When Miriam opined
that it sure was a funny way of counting, I told her there was some-
thing she used a lot that counted that funny way; could she guess what
it was? Miriam could not guess that computers count in binary. It
made no sense to her that they could add such a funny way and not take
forever to get a result.

Miriam, in order to trick Robby, invents (with one example only)
a 2 place finger counting representation. Robby counters with multi-
plication of the finger count of both hands. I show both a one hand,
five place binary counting representation.


Vn44.1 A Boring Session 7/12/77

Riding home after this morning’s session (Logo Session 38) Miriam
said she thought the work was boring today. When I asked why, she said,
“Oh, I don’t know.” I have to look otherwheres for an explanation.

Today I tried to exhibit for Miriam the relation between closed
polygons and in-going spirals sufficiently regular to be judged ‘mazes’
rather than ‘pretty pictures.’ (Cf. Addenda 1 and 2). Yesterday Miriam
suggested for today that she would like to try to get more good numbers
for making mazes. I believe she had in mind a result like that of Logo
Session 27 (where we made a list of the members found with the ANGLE
procedure for making ‘pretty pictures.’) I made such a result our ob-
jective, but Miriam showed little interest in the work.

Note that Miriam was feeling sick this morning before we came to
MIT and also during the session. She ws disinclined to come in today
but agreed when I pointed out that we would be away from the lab for
the next 2 weeks. It may be that this was just a ‘bad day’ for her,
but I incline to believe I’ve been pushing her too hard in one direction .
(Turtle Geometry variable separation).

After we finished trying to find good mazes, Miriam began drawing
at my desk. She asked, “Hey, Daddy, how much is 14 and 14?” “Let’s
ask Logo,” I replied and keyed the expression. This captured her
interest. “I want to do some numbers.” Miriam keyed addends of about
20 digits each. Logo produced an answer in floating point format.
Miriam said, “That’s funny. It’s got a dot in it. That can’t be right.
I guess Logo doesn’t add very good.”

After Miriam complained about the session on the way home, I asked
the children what we could do to make the sessions better. Robby said
the day would have been OK if the printer worked, if we had been able
to make pictures out of designs. Miriam said she would just rather do
some adding instead.

This vignette discusses the circumstances surrounding a Logo
Session Miriam found boring. I suspect I’ve been pushing her too
hard. Though the conclusion is uncertain, I feel it’s appropriate
to go easy for a while.

Post Script

Miriam decided to take off the next 2 days, so we did not go into
the lab again until the 15th of July.

Addendum 44-1

My files no longer contain this figure, if they ever did.
I suppose it was intended to show the collection of the
regular polygons (triangle, square, pentagon, etc.) to be
followed by Addendum 44-2 below, as an example of a “maze.”

Addendum 44-2

Hexagonal Maze

Vn 44-2 Hexagonal Maze


Vn46.1 Rotten Hints 7/19/77

Two years ago, Miriam took swimming lessons. She was in the class
of ‘Blueberries.’ Their course of instruction amounted to splashing at
the edge of the lake. Their most advanced achievement was to say their
names with faces held in the water. Last year, in our move from
Connecticut to Massachusetts, Miriam and Robby missed out on swimming
lessons. With both children wanting to learn to swim, it seemed good
fortune that the summer swimming lessons at our lake were offered
during our 2 week vacation.

Robby, declaring the swimming lessons would interfere with his
visiting Raymond, decided not to enroll. Even though I was not willing
to spend much time at it, he figured I could teach him to swim. Miriam
was anxious to take the lessons. At registration, she was judged by
the teacher to be ready for ‘Kiddy 2,’ the class preceding beginners.
She seemed pleased enough.

Tuesday morning her class began with ‘Ring around the rosy.’ The
group of 8 joined hands, bounced around in waist-deep water, and on the
chant’s conclusion ‘we all fall down’ the children were supposed to sit
in the water, getting their heads completely wet while holding hands.
The next element of the lesson was the ‘dead man’s float’: one takes a
deep breath and floats face down in the water. Miriam refused. At the
end of the session they had another round of ‘Ring around the rosy.’
Miriam did not sit down as expected of her. One of the instructor’s
assistants approached me after the class and suggested that “we” might
try getting “our” face wet in the wash basin between swimming classes.

Miriam doesn’t like getting her face wet. Neither do I. My
version of the crawl (which I rarely employ) keeps my face out of the
water, as do the other strokes I prefer. Despite the ultimate limit
this may place on my speed or furthest reach, as a youth I achieved
swimming and lifesaving merit badges in the Scouts. I see no reason
why ‘face wetting’ should dominate early swimming instruction. This
strikes as even more forcefully true for a child whose allergies render
breathing difficult.

As we left the beach, I asked Miriam how she enjoyed her swimming
lesson. Her response was very direct. “That was terrible. She wants
you to get your face wet all the time. I’ll never learn to swim from
her. She can’t give me any good hints. All she knows is get your face
wet. What rotten hints.” I agreed she should not continue instruction
unless she wanted to. Miriam asked to go to the beach on the third day,
but once there refused to join the swimming class.

This vignette describes an instruction situation which Miriam
judged to be especially bad. Her formulation of the badness was that
the teacher could only give ‘rotten hints’ for learning.


Vn47.1 Losing a Tooth 7/20/77

Miriam lost her first baby tooth today. The fact is easily stated,
but to show how Miriam considers this a watershed defining event in her
life requires some elaboration. About a month ago, Miriam visited the
dentist. The occasion was the existence of a small abscess above a dead
tooth (both her top front teeth were killed by a fall she took 2 years
ago). Our dentist in Connecticut had warned us to look for signs of
abnormal eruption when the deciduous teeth should fall and advised us
to see a dentist at once should such a thing occur. An X-ray made
clear that the development was normal, and the dentist predicted, in
response to Miriam’s query, that she should lose some of her teeth
very soon.

In kindergarten a log had been kept of who had lost how many teeth,
and each tooth had become a local event, cause for discussing the exis-
tence of the tooth fairy and her munificence in exchanging money for
ejected teeth. Miriam had felt herself lagging behind her peers and
was overjoyed at the assurance her time had come. At that point, Miriam
began worrying her teeth and showing how loose they were.

It was otherwise with Robby. Some two years ago his first tooth
came out and was launched into the world with this gripe: “Hey, I’ve
got a gristle in this banana!” This family story led Miriam to the
frequent observation “If my tooth comes out now I’m going to have a
gristle in my potato,” or chewing gum or whatever. After a month of
such repetitions, she pushed the tooth over and pulled it out with her

Having told everyone she met today how good it was that her tooth
came out, Miriam came to see me when ready for bed, just wearing the shorts
from her pyjamas. With a big smile, she said, “Daddy, I’m really a big
girl now,” and pulled in her stomach. “See!” Surprised at first, I
caught on: “Oh, your boobs are getting big now, too?” Miriam laughed
and said, “Yeah!”


No, sweetheart. You’ll have to wait til you’re about 12.

(Surprised and a little disappointed) Oh.

This vignette shows a small event, losing a tooth, in the focus of
a protracted and persistent concern. Losing the first tooth is to Miriam
a sign that she is no longer a baby but on the verge of woman-hood.


Vn48.1 Tenable Explanations 7/23 & 25/77

7/23 We drove to town late in the afternoon. In the clear sky, Miriam
could see a bright half moon (I could not from my seat). “If you were
on the moon, Daddy, what would happen when it got skinnier and skinnier?
Would you get bumped off?” I couldn’t understand her question. Miriam
referred to the half moon in the sky, then restated her question: “Would
you get bumped off. . . or does part of the moon become invisible?” When
I returned the question to her, she decided that part of the moon becomes
invisible. I believe her use of the term is such that she conceives of
a part of the moon as becoming transparent in contrast with the (now)
standard view that it is not able to be seen because of our perspective.
When I asked her later, Miriam did not confirm this speculation. I
asked, “When part of the moon becomes invisible, can you see through
it?” She replied, “No.” When asked how it becomes invisible, Miriam
replied, “I don’t know.”

7/25 Miriam found a golf ball in the basement a day or two ago. I was
surprised to hear her complain of Scurry that she had put a lot of
“holes” in it by chomping on the ball. This suggests that she may not
yet divide all small white balls into two classes: ping pong (hollow,
of smooth surface) and golf (solid, with concave ‘bumps’ on the surface).

This ball entered play in a familiar way. I returned from other
engagements to find Miriam showing Gretchen how to make a ball go for-
ward and return. She said, “You do it like this,” and attempted to
backspin the ball. She was not able to control the ball effectively
(I speculate that she was unable to compensate for the differences of
weight and friction both). When I asked Miriam how backspinning had
worked with the ping pong ball, she offered to show me and cautioned:
“But, Daddy, don’t think about it.” I believe this showed no admission
on her part of my (false!) explanation of backspinning during the
experiment recorded in Miriam at 6, but had a more complicated purpose:
Miriam, confident of her ability to backspin and intending to disabuse
me of my incorrect notion, warned me not to “think about it” so I could
not offer that explanation of the phenomenon. Backspinning failed.
Miriam rolled the ball to Gretchen, who kicked it back to her. She
noted of the golf ball, “This doesn’t work too good,” then continued
the explanation with an excuse, “because the ball’s too heavy.” Miriam
tried again with a different ball, one she described as ‘lighter’: the
ball is solid rubber foam 2 1/2 ” in diameter; it is heavier than the golf
ball but smooth and compressible. When her attempt to backspin the
rubber ball failed, Miriam’s interest waned and she went off to some
other activity.

The two incidents cite a class of explanations or descriptions
which most adults would think silly but which Miriam still accepts as
serious, albeit mistaken, explanations.


Vn49.1 Finger Counting 7/24/77

At lunch, I inquired of Miriam how she used to add on her fingers
numbers like 2 plus 7. After saying ‘9’ and my refusing that answer,
she counted up, i.e. Miriam said ‘7’ then lifting her pinky and fourth
finger on the right hand, ‘8, 9.’

I again rejected the answer: “Try hard to remember when you couldn’t
do any sums greater than 10, how did you add 2 plus 7 then?” Miriam
counted from 1 to 7 using her right hand and the pinky and fourth finger
of her left hand; she then raised her thumb and index finger, saying
‘1, 2’ thus leaving her middle finger unused.

Miriam complained that she no longer enjoyed doing such easy sums,
so I asked her to add 37 and 12. She looked shocked — then said ’49.’
When asked, she explained: “I knew it had to be more than 40 ’cause it’s
like 30 plus 10, so I said ’47,’ then ‘8, 9’ because of the 2 left over.”
(Miriam counted upon her hand for the last 2).

This vignette confirms a speculation (cf. How Miriam Learned to
Add) that Miriam’s early use of commutativity is an artifact of her
finger counting procedure in that selecting the larger of two addends
to first represent is less confusing where the sum approaches 10.


Vn50.1 The Go-Cart 7/25/77

Kept inside on a rainy day and with me working in the living room,
Robby and Miriam were constrained to play quietly (more or less) in the
kitchen-dining area of our Connecticut house. Since we vacationed in
an unfurnished house, they had few of their usual toys and a limited
selection of books.

During the afternoon, I discovered them playing with empty boxes,
and shortly after Robby entered with the drawing of Addendum 50 – 1
inquiring whether or not it was a good plan for a go-cart. The vehicle
is to be propelled by pedaling. (The long hair on the front figure
indicates Robby thinks of Miriam performing that function.) The
‘steering wheel’ is for holding on to, for steering is to be provided
by a tiller at the back of the cart which turns the ‘tail’ wheel. I
admitted it as a good start but cautioned that more detail would be
needed before it would be a plan for construction.

The project was a joint one. The children planned to construct
and use it together. Miriam’s sense of construction was different from
Robby’s. She took one box, opened to a single flap on the top, and
declared this the front of the go-cart. Another box, ripped apart,
provided the rest of the carriage. She jumped in and was “off,” driving
the cart around while Robby explained to me how the pedals would be
mounted (drawing therein the ancillary figures below the side view).
Miriam seized Scurry to take her for a ride, put her in another empty
box, and declared it a ‘rumble seat.’ Robby redrew his plan as a three
seater with a ‘rumble seat assembly.’

Robby took his play very seriously and eventually found a set of
wheels in the garage I had salvaged from a junked garden tractor. He
began to talk of going to the lumber yard and to wrestle with the design
of a brake. (The final drawing of Addendum 50 – 2 is my advice; his
original idea had the shoe forward of the pivot). To Robby, building
a real, usable go-cart is an achievable objective. To Miriam, the idea
of a go-cart is a focus for a fantasy. Its symbolic realization is as
adequate to her use of the idea as she requires. A real go-cart, some-
where else at some later time, would be much less satisfying to her than
the play construct of the moment.

This vignette describes the joint efforts of Miriam and Robby in
a go-cart “project.” The children play together in the intersection of
fantasies that are worlds apart.

Addendum 50-1

Addendum 50-2


Vn51.1 Paper Ships 7/25/77

This has been a rainy, midsummer day with both children at home in
an acoustically live house. Having slept ill last night, under pressure
of the noise and our common confinement, I went to bed early. When the
children failed to fall silent instantly, I “yelled” at them, i.e. I
told them quite specifically that I had suffered too much of their noise
and commotion, that I needed sleep and they must be quiet.

Because of the rainy day bedlam, I had failed during most of the
day to make headway in my thinking about Miriam’s computations and my
understanding thereof. As I drifted into sleep, some imperfectly
remembered lyric from my early school days entered my mind:

. . . put down 6 and carry two —
Oh oh oh. Oh oh oh.
Gee, but this is hard to do
Oh oh oh. Oh oh oh. . . .

No greater fragment remains of that song, but I imagined that situation
and the woman conducting that song, and then another:

Some folks like to cry,
Some folks do, some folks do.
Some folks like to sigh,
But that’s not me nor you.
Long live the merry, merry heart
That laughs by night or day.
I’m the queen of mirth —
No matter what some folks say.

This ditty carried me along to a better feeling, one wherein I was
capable of feeling ashamed of my ill behavior to the children and happy
that our relationship was one where I could apologize to them and they
be capable of accepting that apology.

I called Robby. He entered my bedroom quietly and was obviously
relieved when I told him I was feeling better and was sorry I had been
so crabby. He asked if I would help him with a problem. When I agreed,
Miriam entered and pounced on me. (This was easy since my ‘bed’ was a
sleeping bag on the floor.) Robby returned with the book Curious George
Rides a Bike
. Both children had been attempting to make paper boats
following the instructions on pp. 17-18 (Cf. Addendum 51 – 1, 2). Robby
was stalled at step 5 and Miriam at step 3 of this 10-step procedure.

Both children were working with small (tablet size) pieces of paper.
I was sleepy and unfamiliar with the procedure, so instead of looking
at their problems, I first made a boat myself. A nearby newspaper pro-
vided paper of size large enough to escape folding-small-pieces-of-paper
bugs. When I reached step 3, Miriam noted that as the locus of her
impediment. When I asked, “Oh, you’ve got a bug there, sweety?” she
responded, “Yes. An I-don’t-know-what-to-do-next bug.” I slipped my
thumbs inside the paper and pulling at the side centers, brought the
ends together. Miriam said, “Oh, I get it now,” and continued with her
folding. (She had not been able to identify that transformation, failing
most likely to interpret the arrows and -ING STAR, that portion of the
newspaper masthead still visible after the folding as a clue.)

When Miriam some time later attempted step 7 (bringing the ends together
a second time), her construct disassembled. After I suggested she
hadn’t tucked in the corners carefully, Miriam described it as a ‘no-
tuck-in bug.’

In the transformation from step 9 to 10, because the central crease
must suffer a perpendicular crease in the opposite sense, one usually
has trouble pulling down the ends without the assembly’s failing. When
both children had made several boats, I asked Miriam what bugs she had
uncovered. She cited the original two and a third, the ‘last-pull-apart

The construction expanded. The newspaper pieces made battleships
(and stopping half-way, hats). Miriam made life boats and Robby, by
unfolding a newsprint page before beginning the folding procedure, made
a large, flimsy craft he dubbed an aircraft carrier. It was a small
step to carrier war in the Pacific (my bed as Pearl Harbor) and the
pillow fight which ended this war.

These observations show Miriam using the word ‘bug’ to describe
the difficulties she encounters in executing a complex procedure, both
with some direction and more nearly spontaneously.

Addendum 51-1

Vn 51-1 Curious George paper folding

Addendum 51-2

Vn 51-2 Curious George Paper Ship procedure


Vn53.1 A Birthday Party 7/28/77

Robby’s birthday comes in August. Connecticut friends whom he
would like to have at a party can not come to Boston. When he suggested
an early party during our vacation, we agreed. Preparations for the
party focused on choosing activities and procuring treats and prizes.

If you have ‘prizes’ at a party, you must have one for everyone
and the question devolves to one of who gets first pick. The ‘activities’
became a means of deciding the order of selecting prizes. Robby
suggested a foot race and pin-the-tail-on-the-donkey. Miriam, younger
than all his friends and predictably last in a race, objected. She was
declared ‘judge’ and assured she would receive a prize for that office.
Robby took some cardboard (left over from manufacture of the go-cart of
Vignette 50) and drew thereupon a donkey. He made a selection of tails
to be affixed with tape (I balked at the idea of children pinning tails
on the timber walls of our house). When Robby decided the prizes should
be “matchbox racers” (at $1.20 apiece), it was clear he had proposed
enough games. The party was to conclude with an ice cream cake and a
selection of favors. (The items selected were the same as those
distributed at a party for Raymond’s brother — Hershey bars, bubble gum,
a balloon). The chosen hours were 2 to 4 pm. (These hours had been
the standard for parties attended by Miriam and Robby that year).

Six children were to be present. Miriam had to be there. Raymond
was his best friend. David and Vi were friends from a baby-sitting
playgroup he had been a member of. Who else should come? On the way
to Guilford, Robby said he might not have anyone to play with because
he couldn’t remember his schoolmates very well. On our first day in
Guilford, Robby encountered Michael on a walk and the 2 played that day.
After playing with John, a boy who lives across the street, he decided
playing with him was boring. Thus Michael was weakly preferred to John.
Robby called his friends and made the arrangements. David would arrive
late because of a conflict with his swimming class.

The day of the party I picked up Raymond by car and returned home
by 2:05. Robby and Miriam were awaiting guests at the end of the drive.
Raymond joined them while I put the car away and went inside. Before
the party, when he started wondering what presents he would get, I
asked Robby what was more important to him — that his friends come to
play or that he get presents. Robby said he really didn’t care about
the presents. Raymond came to the party without a present; he had
thought he was just coming over to play. I had told him not to worry
about it. He was Robby’s best friend and it was most important that
he come.

About 2:30 the 3 children entered the house. No other guests were
coming, a dreadful situation. Robby called Vi, who had forgotten about
the party and promised to come right over. No one answered the tele-
phone at Michael’s house. With Vi now definitely expected and David
known to be coming later, the 3 children occupied the interval by exam-
ining the prizes. They decided that half the 12 prizes (matchbox
racers) should be reclassified as favors and allocated them accordingly.
Robby asked me: “If Michael doesn’t come, can I have his two racers
because he won’t be bringing me a present?” This seemed reasonably
fair to the other 2 children and to me. Robby tried calling Michael
again with no response, and declared the two left-overs to be his.

Vi entered with the first present, a nicely wrapped package con-
taining a bottle of bubble bath in the shape of a brontosaurus. Robby
was pleased. Shortly after, David arrived. His present, the second
and last, was a nicely wrapped package containing a bottle of bubble
bath in the shape of a rocket. Robby: “Oh well, I guess I’ll have to
take lots of baths.” The 2 most recent arrivals inspected their favors
and prizes. All 5 children then fell to making their balloons scream
by letting the air escape through the neck stretched flat. At my
suggestion, the children took Miriam’s beach ball to play in the yard.
The game of choice was ‘keep away.’ I forbade them to keep the ball
away from Miriam (their original plan, since she objected to the game,
probably suspecting that end). Their alternate game pitted Robby and
Raymond against Vi and David. Miriam sulked and sat on her swing.

After a half hour’s play, the children came in for the cake. At
4:10, expecting the party to end with the last of the cake, I was
surprised to hear cries that I had promised to take the children over to
the playground for the prize selection race. I did so, but warned the
children that their stay would be very short because Raymond had a 4:30
deadline at home. The race was run, prizes were distributed, and all
were content except Miriam; David chose the racing car she wanted.
After we dropped David, Vi, and Raymond off at their houses, I told
Robby how unhappy Miriam was. He agreed to work out with her some
distribution of their six racers which she would consider satisfactory.

This party was one arranged by the children according to their
ideas and reflecting the way they coped with unexpected contingencies.
Robby has said since how much he enjoyed the party. Miriam suffered
the younger child’s burden of being left out and left behind.


Vn56.1 TicTacToe 7/19/77

These games of tic-tac-toe followed immediately the arithmetic of Home Session 13. The focus of the session is on the bipolar (i. e. competitive) quality of tic-tac-toe. This focus is maintained by contrasting the game with playing SHOOT around the issue of clever tactics. (My moves are numbers; Miriam’s are letters.)
Game 1: Miriam first

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

After Miriam’s move C:

B Do you know any clever tactics for tic-tac-toe? . . . Do you think it’s easier to win at SHOOT or tic-tac-toe?
M [points to tic-tac-toe frame]
B It’s easy to win at tic-tac-toe?
B Do you notice anything special about the way your markers are?
M Two ways to win.
B Did you just see that after I told you?
M No.
B You knew it all along?
M I had a forced move, and I wanted to move there.
B They came together, your wanting and the forced move?
M Miriam Yeah.

Game 2: Bob first

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

When Miriam responds to a center opening with a mid-row move (as I had done in game 1), I introduce the theme of turning the tables on your opponent.

B I know what I’ll do. I’ll play the game you played. I’ll use your own clever trick to beat you.
M Yeah? [I don’t believe you can]
B Just like that [move 2], ’cause you have a forced move now.
M [moves B]
B I’m going to use your clever trick to beat you.
M [moves C]
B I’ll win anyway. I turned the tables on you.
M I know.

Game 3: Miriam first

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

The game was to provide contrast with normal competitive play by my taking Miriam’s direction about where to move. It harks back to her earlier proclivity for negotiation in the game (cf. vignette 5) and induces a resurgence of that style. We act out the fairy tale motif of the child (Miriam) defeating the ogre (me) by making a promise, then escaping from it by a quibble (not, in fact, necessary in the move configuration).

B Where should I go?
M Not there [center square]. Don’t. Don’t.
B You tell me where to go. I’ll go where you tell me.
M Here [upper left corner].
B Over here in the corner?
M No. No. There.
B [moves 1]
M B [moves].
B Now I have a forced move [center square].
M I don’t want you to go there.
B I’m going to go in the center.
M No no. No no. I’m not going to move there. I promise. A million dollars.
B Where should I move?
M There.
B You want me to go up here? [moves 2]
M [moves C] Two ways to win [laughing].
B Yeah. But what about this? [center square] You could have won right away by going there.
M Yes. But I promised you I wouldn’t a million dollars.
B Oh boy.
M That’s why.
B It looks like you’ve got 3 ways to win, but if you go that way [center square], you lose a million dollars, so I’ll put my 3 down here.
M [moves D] I mean just for that once [laughing].
B Oh, you stinker! . . . Do you think it’s easier to win if I do what you tell me?
M Yeah.
B What is it about my moving where I want that makes it harder for you to win?
M [no response]

Game 4: Bob first

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

After Miriam moves A:

B You have frustrated my tactic.

M [laughs]

B I had a plan all set up, but you frustrated it.

M I always like to frustrate your plans.

B You do! Well. . . that’s what tic-tac-toe is all about. Stop the other guy from winning. . . . I’ll go here [moves 2 in center square].

M [moves B]

B You frustrated my — I was planning on going there. I was going to get two ways to win. Oh well, I’ll go over on this side [moves 3]. I’ve got you now. 2 ways to win.

M No. You made a mistake [laughing]. [moves C]

B Oh no. . . . It looks as though I didn’t have a good plan for getting 2 ways to win. I had one way to lose.

Game 5: Miriam first

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

The previous game exemplified losing by focusing on a winning tactic instead of attending to the opponent’s moves. Here, we try to exemplify how knowing a clever trick in an opponent’s repertoire permits frustrating it. After Miriam’s opening, she requests that I not move in the lower left corner.

B I’ll put a 1 right here in the center.

M [moves B]

B What’s going on here? . . . I remember now, you have a clever tactic in mind. ‘Cause if I go there [the other currently unoccupied corner], then you will have 2 ways to win, and I’ll have a way to lose.

M Yeah.

B I will frustrate your tactic.

M How?

B I will put my 2 here.

M Oh. [disappointed, she makes forced move C]

Game 6: Bob first

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

B I’m kind of tired of going in the center, so I’ll go someplace I hardly ever go.

M [moves A]

B There’s only one problem with your going in the center.

M What?

B It’s kind of hard for me to get 2 ways to win. I can go over here [move 2].

M [moves B]

B You’ve just blocked me by doing a forced move. Hmmm. Now I have a forced move too [move 3].

M [moves C; makes noises of discontent when I gesture to the square where D is later]

B You tell me where to move.

M Here.

B Shouldn’t I make a forced move?

M Unh-uh.

B How come? You want me to lose by making a stupid move?

M Yeah.

B O. K. [moves 4]

M [moves D]

B You won, ’cause I did what you told me.

This vignette focuses on the contrast between SHOOT and tic-tac-toe as a 2 person game. “Turning the tables” is articulated as a clever trick. Frustrating tactics is exemplified 2 ways.


Vn57.1 Desserts 8/3/77

When I was a small child, there was rarely dessert in my house.
On special occasions my mother might make some rice pudding or tapioca
(when cooked, the large size tapioca became transparent balls we children
pretended were the eyes of frogs). When my children pester after
every meal for dessert, I bolster my refusals by the argument that I
have ‘spoiled’ them and retreat with what little grace I can to limiting
their desserts to 1 a day.
They love ice cream and most especially those popsicles known
locally as dreamsicles. These are vanilla ice cream with an orange-ice
coating. Popsicles are prized because the children don’t have to sit
down to eat them; and they frequently make their own from orange and
grape juice. After lunch today, Robby and Miriam offered us this
proposition: they should have dreamsicles then and not after supper this
evening. Who could refuse such an innocent and fair proposal? I did,
expecting they would forget by supper their agreement at noon, or even
more likely, attempt to roll backward their allocation from the morrow
and embroil me in accountings I wish to avoid.

Bob You may not have any dreamsicles now.
Children (In chorus) Rats.
Bob Oh. You mean you want rat-sicles.
Children (General responses of feigned disgust: making faces, cries of
“Yuk!” and “Bleah!”)
Bob What would be wrong with a rat-sicle? They would be much
easier to make than popsicles. You catch a rat and pop him
in the freezer. You use the tail as a handle instead of a
popsicle stick.
Gretchen Scurry [our Scotty] would love to have a rat-sicle, though
maybe a mouse-sicle would be better for her size.
Children (Continue objecting, laughing, and feigning revulsion)
Robby That’s terrible. It would just be raw meat.
Miriam And drip blood. Yuk.
Bob I get the problem now. If you don’t like the blood and guts,
maybe you should try a motor-sicle; that would be covered with

Recognizing this impasse, Robby laughed roundly at the joke and roared
off on his motorcycle, and Miriam followed him to play out in the court

This vignette concluded with an exposition of a situation in which
the children find themselves. They are confronted by an argument of
disguised force, i.e. they can’t do what they want because I won’t let
them. The disguise (in this case) is one of joking and absurd argument.
I believe both children recognize that if, and when, they outwit me in
this sort of absurdity, I may well relent and let them have what they


Vn58.1 Owning an Angle 8/4/77

As far back as the end of June (in Logo Session 32) making hexagonal
mazes has been a part of both children’s Logo work. Before our Connecticut
vacation both children worked together generating pictures of mazes
(7/8/77: Logo Session 36). During that session, Miriam “discovered” the
60 degree angle input creates a hexagonal spiral. During today’s session
Robby generated a “family of mazes,” including the hexagonal form with
the other regular spirals of integer angles (120, 90, 72, 60, 45, 30).
Both Robby and I were quite pleased with his work of the day and hung
on the wall the pictures made by the spiral procedure with those inputs.

While we were preparing to leave, Miriam entered my office (now
dubbed the ‘little learning lab’). Robby, naturally enough, showed her
his pictures — at which she complained vigorously that he had used
“her” angle of 60 degrees. One could dismiss the complaint as a
manifestation of sibling rivalry or a more general jealousy that I praised
his work. Nonetheless, it is clear that Miriam saw “her” hexagonal
maze as a unique object in a collection of other objects.

Miriam’s complaint has been repeated frequently in the weeks
following its surfacing.


Vn61.1 Tic Tac Toe (5) 8/10/77

This material shows Miriam accepting instruction at corner opening play through a process of “turning the tables” on me after my exemplary victory. (The data were transcribed as Home Session 15.) A corner opening in tic-tac-toe is the strategy of choice, since its use nearly guarantees victory for the player moving first. Nonetheless, because it is possible to lose through failing to recognize opportunities or through one tie-forcing response by the second player, the power of the corner opening is not excessively obvious.

At the beginning of our play I introduced to Miriam as an extension of “ways to win” the notion of “chances to win.” You have a “chance to win” when you have only a single marker in a particular line and there is no blocking marker. The first game, wherein Miriam moved first, was a tie of the center-opening/corner-response sort. It was during the execution of this game that the “chances to win” terminology was introduced. At the beginning of game 2, I proposed teaching Miriam a good trick. Since the gambit begins with a corner opening, Miriam believed and asserted that she already knew it. She is aware of at least three corner-opening games:

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

B.        |  A  | 4            |  3  | D         D |  A  | 5   

C.      B |     |  2         C |  4  | A         4 |  B  | 2 

The A game represents Miriam’s good trick, and B and C represent ways of blocking A which she can’t circumvent. In the games that follow where my move is first, Miriam attempted 3 different responses to my corner opening. In the other games, she “turns the tables” on me by using my play as a model to defeat me in turn.

Game 2: Bob moves first (numbers)

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

Miriam makes move A at my direction and after my move 3, recognizes not only that I have 2 ways to win but also that A has no chances and B 2 chances to win.

Game 3: Miriam moves first (letters)

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

Miriam here follows my advice to “turn the tables” on me by employing the same good trick (move 2 after response A to opening 1). During her role switch in applying this strategy, Miriam also switched from using X symbols as markers (which she had done in game 2) to literally copying the numbers I had used in that game (cf. games 2 and 3 in Addendum 61 – 1).

Game 4: Miriam moves first (letters)

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

Miriam moves first (out of turn) at my request to confront the challenge of turning the tables despite my choosing the corner response opposite to that of game 3. I asked her opinion:

Bob Is moving here [upper right corner] the same or different from moving there [lower left corner]?
Miriam Different.
Bob Can you play the same game even though I’ve moved in the opposite corner.
Miriam I think I can.

As we continue, Miriam comments, “I’m playing the same trick on you.” Miriam again uses numbers for her markers but disguises the copying by using numbers (9, 6, 5, 10) different from those I had used in game 2. After commenting that move 2 was a forced move as is move C, I emphasize that what is most important to see is that the single move C converts 2 chances to win into 2 ways to win.

Game 5: Bob moves first (numbers)

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

I warn Miriam after move 1 that I will probably beat her. She believes she can frustrate my plan by making move A (notice in the typical and familiar game B the outcome was a tie).

Bob In game 5 I am probably going to beat you —
Miriam Yeah.
Bob If you move where I tell you the first time, and after that —
Miriam I might not move where you tell me [laughing, she moves A; I had wanted her to move to the middle of the right column].
Bob Do you think I can beat you after that move?
Miriam Yeah [Miriam has not seen this game before, to my knowledge].
Bob I can. I will show you how.

After Miriam made her forced move B, I described my deciding where to move in terms of where I had chances to win and looking for a move where 2 chances to win come together. This game is one where selecting a usually valuable move (the center square) is not the optimal strategy.

Bob I can’t win this way [the 1 – 2 line is blocked by B]. I have a chance to win this way [in the row from number 1]. Do I have another chance anywhere? . . . Yes, I have a chance from 2 up through the center. And I have a chance along the top. So if I put my number 3 where the two chances come together, what do I get?
Miriam Two ways to win?
Bob That’s right, sweety.

Game 6: Miriam moves first (letters)

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

Miriam turns the tables on me successfully. The symbols she used in the actual game show her slipping over into direct copying of my previous game.

Game 7: Bob moves first (numbers)

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

Although I wanted her to go first (for another variation on game 6), Miriam insisted that I go first because it was my turn. After Miriam’s response A to the corner opening I proceeded, describing my reasoning at each step.

Bob I put my 2 here. Now watch. You have a forced move, don’t you [between 1 and 2].
Miriam Uh-huh [moves B].
Bob What chances to win do I have? I have one from the 1 along the top. I have one from the 2 along the bottom.
Miriam Two.
Bob I have one from the 2 through the center. . . but. . . I also have a forced move in the center. Right? . . . So I have to go in the center. But when I go in the center, how many ways to win will I have?
Miriam One?
Bob Watch. I have a way to win from the 2 and a way to win from the 1.

At this point Miriam confided to me that she would try to get Robby to move where she had placed her A, then she would make another move and try this trick on him.

I attempted to review with Miriam all the possible responses to corner openings, but she was tired and inattentive, and the session ended.

This vignette describes my introducing to Miriam the idea of “chances to win,” seeing the forking move as placing a marker where chances to win intersect. The method was that of her “turning the tables” on me, i. e. using a tactic I showed as effective against me.

Addendum 61-1

from Home Session 15

Vn 61-1 Addendum from Home Session 15


Vn62.1 Multiplication 8/7 & 11/77

8/7 Robby has many times now seen Miriam on my lap receiving some
instruction in addition. Complaining of feeling left out, he has asked
for help in math. Robby said he needs help with addition of numbers
such as 9 plus 6 and 8 plus 7. I found him a set of flash cards for
practicing with. Robby looked through them, declared he knew them all,
and set them aside. Miriam picked up the box of cards and has reviewed
them once or twice. Robby also specifically asked for help with mul-
This afternoon he inquired how much is 24 times 42. Gretchen told
him the answer. I suggested Robby estimate the answer as 20 times 40
and showed him how to factor the product thus:

		20		2 x 10
	      x	40		4 x 10
				8 x 100	800

with Robby doing the intermediate products and the final multiplication.
I posed for him the problem of multiplying 20 times 400. Under the
previous work Robby wrote

		20		2 x 10
	      x	400		4 x 10
				8 x 100

After I inquired whether or not he had left out a zero, Robby made the
lower product 4 x 100, looked in puzzlement at his product of 10 times
100 being 100, changed it to a thousand and the result to 8000.

8/11 Miriam, aware that Robby is interested in learning multiplication,
is turning her attention to that. Today Miriam told me, “I know how to
do it, that other thing, not adding or take away. . . . 10 times 1 is like
10 ones.” I asked her how much is 2 times 4. Miriam answered ‘8.’

Bob How much is 3 times 6?
Miriam (after a long pause) 36.
Bob How did you compute that?
Miriam 12 plus 12 is 24 and 10 more is 34 plus 2 is 36.

Miriam then asked, “Is 20 times 20 equal to 60?”

Bob That’s a big number but not very close.
Miriam 40?
Bob That’s a lot closer, Miriam.
Miriam Is it 20?
Bob No. That’s not the way to get a good answer, Miriam. We’ll
talk about multiplication later.

Because Robby and Miriam spend more time with each other than with
anyone else and because they compete with each other for their mother’s
and my attention and approval, they both view each other’s activities
for comparative advantage.


Vn63.1 Another Birthday Party 8/12/77

This was a party for Robby’s Boston friends, boys he has met while
at school here. With respect to planning, this party was pretty much a
rerun of the party in Guilford (cf. Vignette 53). The party favors were
the same: Hershey bars, bubble gum, and balloons. Match box racers were
still Robby’s ‘prizes’ of choice and the game to decide priority of
choosing the racers was again to be ‘Pin the tail on the donkey.’ A new
wrinkle was added by Robby’s attending the party last week of his friend
John. Then, the children played ‘Pin the ear on the Snoopy.’ The idea
was adopted here. The children waited impatiently while Robby opened
the presents. He was delighted to get several ship models and a game.
The boys were astounded that Miriam had made Robby 9 birthday cards.

Most of Robby’s friends were out of town on vacation. The three
boys who did attend were brought by their parents and picked up by them.
The suburban distances and the parents’ schedules provided a more rigid
time frame than that of the party in Guilford. One child had to leave
early; thus the cake eating ceremony was moved forward in time. This
circumstance helped fill the gap created by having no other games planned
for inside play on this sporadically rainy day. When Reese left early,
Robby showed the other 2 boys his collection of models, and they decided
to play outside even though the sky was overcast and the court yard
flooded. So the game of the day was kickball, with a huge puddle for
first base.

Miriam sulked inside. I believe she was jealous of the attention
Robby received (2 birthday parties is excessive!) and she was mad at me.
Her attempt to pin an ear on Snoopy was a dismal failure; the ear not
just missed Snoopy, but was pinned on the perpendicular wall. Since I
had been the spinner of children, the fault was mine. After Miriam’s
persistent complaints, this evening, Robby advised her that there was
a good trick she had not yet learned: when you play ‘pin the tail on the
donkey,’ you don’t start walking right after the spinning; you wait until
you’re no longer dizzy, then walk straight forward.

These two vignettes on birthday parties indicate the balance of
plan/script driven behavior and a general coping with whatever comes up.
Miriam found herself very much on the periphery of this party as of the
other. Robby’s advice indicates that he and Miriam both find it possible
to communicate in the language of ‘good tricks’ for coping with trouble-
some situations.


Vn64.1 Jumping Rope 8/13/77

Miriam began jumping rope after we moved to Massachusetts. Earlier
she had played a game ‘Angels/Devils’, a group rope jumping game in
which a child in the center of a ring turns, saying alternately ‘angels
devils angels devils. . .’ until one of the children in the peripheral ring
fails to jump up as the rope comes to his place. If that child is hit
by the rope while ‘devils’ is being said, he takes over in the center
of the ring; otherwise the child in the center starts the rope spinning

At kindergarten, the children apparently jumped with a long rope
(with a person to turn at each end). Miriam asked to have such a rope.
I bought some rope and we played with it in the court yard and at Logo.
Jumping with this rope was one of Miriam’s favorite activities on the
‘breaks’ she took in the course of Logo sessions. Inasmuch as I was
maladept at turning a rope with the proper rhythm and clearance,
Margaret Minsky and Ellen Hildreth were frequently attached for this
service. Margaret got caught up enough in Miriam’s enthusiasm to buy her
a book on jumping rope (Jump Rope, Peter Skolnik, Workman Publishing
Company). During this period of jumping rope at Logo, Miriam gradually
increased her skill to the point where her counting becomes confused
before her jumping fails.

Yesterday at Robby’s party Miriam attempted for the first time to
jump with the rope traveling backwards. Today she has been achieving
3 or more jumps per attempt. When I asked her why she was doing it
backwards and had she ever seen anyone else do that, Miriam replied,
“Just because I want to,” and “Lisa Larson.” Lisa, a former playmate
in Connecticut, was that daughter of Miriam’s baby sitter and her
senior by two years. After the rope jumping of today, this evening
Miriam was reading her jump rope book. I saw her with her arms crossed
on her leap and a puzzled look on her face as she apparently tried
figuring out from pictures how to jump “crossie.”

Rope jumping was an activity which much engaged Miriam at the
beginning of our project, which was put aside for about two months,
and is now coming back as Miriam considers attempting procedures more
complex than those she mastered before.


Vn65.1 Arithmetic Ripples 8/13/77

1. Miriam brought me this morning a dime she had found in the laundry.
My speculation is that it had been left in some pocket, fell out in the
wash, and was pinned in and nearly cut in half by the washing machine.
How it got on the floor where she found it I don’t know. When she
showed it to me, Miriam said, “See, Dad, somebody tried to cut this dime
in half. I bet they thought they would get two things. . . two nickels.
What a silly idea.”

2. Miriam’s most common purchase is chewing gum. She knows the going
rate for Care Free and Trident packs is 15 cents. When Gretchen went
shopping recently, Miriam gave her a quarter and a dime, placing an
order for two packs of gum.
Expecting a nickel change, Miriam asked Gretchen for it on her
return. There was no change. Gretchen explained that where she had
purchased the gum the price had been 20 cents per pack, 40 cents for
both. Miriam looked worried: “Do I owe you a nickel?” Gretchen told
her not to worry about it. Miriam muttered to herself, “20 cents, that’s
5 cents tax on each pack.”

These data are further examples of Miriam’s assigning any non-
explainable variation in price to the category of tax and her puzzle-
ment over the idea of dividing coins into fractional parts. (She does
not know about the obsolete piece-of-eight, a Spanish peso or dollar
designed to be cut into eight reals or ‘bits,’ whence our expression
‘two bits’ designating a quarter.)


Vn66.1 Pre-History 8/16/77

For some unknown reason (“I just wondered,” she says), Miriam asked
me who was the first person to sail around the world. Remembering the
Straits of Magellan and that it must have been the major obstacle to the
western passage, I speculated that Magellan must have been the man.
Gretchen, drawing on her deep fund of facts as she brought the rest of
supper to table, said decisively that Magellan himself died on the first
circumnavigation but that one of the ships originally under his command
completed the voyage.

I explained to myself and any who might be listening that this fact
was one of many of which I was ignorant, but that such information could
be found in our encyclopedias, that it had been written down. I con-
tinued that there were other great achievements, great discoveries made
before people had learned to write and make books. The example I offered
was the discovery of fire, that this was one of the greatest milestones
in human culture, but that since no one knew how to write when fire was
discovered no one knows who was the first to control fire. Miriam, I
believe, asked how fire was discovered. I admitted no one knew, then
proposed a commonplace scenario: lightning caused a forest fire; roast
flesh was found to be good enough for early men to brave the danger and
experiment with coals as fires burned out.

Miriam said she thought she knew: holding her two index fingers
perpendicularly, she explained (and demonstrated) that they rubbed two
sticks together and made a fire that way.

Only Gretchen had a sensible idea — that one of those luckless
buffoons, our not so remote ancestors, while sitting on a soft pile of straw,
chipping flint, gave himself, most accidentally, a royal hot seat and made
man king of the material world.

This casual dinner conversation exemplifies the way we adults,
because we are who we are, even with minimal didactic purpose, draw
along our children into an intellectual space foreign to their initial
concerns but accessible by a few simple steps from whatever catches
their interest.