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LC1bT04 Protocol 4

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RAL protocol 4.1

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LC1bT05 Protocol 5

Drawing a Fox (cf. discussion in Development of Objectives)
n.b. hand-written date at top of first page in error by 2 years.

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LC1bT08 Protocol 8

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LC1bT09 Protocol 9

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LC1bT10 Protocol 10

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LC1bT11 Protocol 11

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LC1bT13 Protocol 13

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LC1bT20 Protocol 20

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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.


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.


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.


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


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.


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


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.


Vn69.1 Chatterbox 8/19/77

In Vignette 3, I noted one of my objectives was to render Miriam
more willing to reveal her thoughts than was formerly the case. Such
a change has gradually but very definitely taken place. Gretchen now
complains that Miriam is never quiet, that she talks about every least
action she undertakes; for example, “I’m taking my dishes over to the
sink.” A more typical example is what Miriam said just now. (She is
making a “card” for a friend; Gretchen and I are sitting in the same
room, 10′ and 20′ away.


I am coloring the flower red. . . and blue. . . and now yellow. . . .
I am coloring the cloud white, Daddy, isn’t that a good idea?


Do you know why I am making the cloud cry?


Because the sun is very hot and it can’t rain.

This is a description of ongoing action, mixed with request for approval
and her explanation for the meaning of her drawing.

First ask is it a good thing for Miriam to be so open at this
point in her life? I believe it is good now and that she will eventually
learn when to bite her tongue. What is one to make of the very
pervasiveness of Miriam’s chatter? Is this a regression of sorts to
ego-centric speech? I choose to think of it differently, in a way
recently suggested to me by Laurie Miller. In this view, Miriam is
giving evidence that she has discovered self-description as an inter-
esting thing to do. . . and is overdoing it. (Recall G. B. Shaw’s asking,
in a paraphrase from the book of Proverbs, “How can you know what
enough is, unless you’ve had too much?”) Such self-description may
result from the reflection and explanation I have asked of her in the
Piagetian tasks of April’s experiments as well as from the rudimentary
debugging we have undertaken in our Logo sessions.

In the little snippet of dialogue above, Miriam was not using the
description of her actions for any purpose which is reflected back into
the action. However, to the extent that she articulates her actions,
it is clear that she can reflect upon them when that engages her interest.

This vignette notices the change Miriam shows in the public
description of her actions. This indicates she has available descriptions of
her action upon which she can reflect if she finds such an activity


Vn70.1 8/22/77

Over the past few weeks, Robby has shown an interest in playing
frisbee. Miriam has tried to play with us but has been so inept that
the game always became a squabble. Robby usually argued that since the
frisbee was his, he should choose the players for the game.

It was an obvious conclusion, then, that Miriam should have the
frisbee I received at the IJCAI registration. We three played in the
court yard in a 20′ triangle. Miriam was supposed to throw to Robby,
but even when she did her best she came nowhere near him:

Vn 70-1 Frisbee Bugs drawing

Robby tried to evict Miriam from the game for ineptitude, but could not
because the frisbee was hers. I asked if maybe we could fix the bug?
Miriam agreed. I described the bug as a ‘holding-on’ bug. We slowly
executed her throwing motion, and I noted the point in her swing (a
cross-body arm sweep with a wrist flick) at which she should let go of
the frisbee. On her second throw, and thereafter, Miriam was able to
aim the frisbee in Robby’s direction.

The second bug frequently manifest after fixing the ‘hold-on’ bug
was one Robby described as a ‘too-low’ bug. Miriam developed her own

This incident shows Miriam’s application of debugging to her own
actions. This way of talking is endemic in the Logo culture. It is
clearly accessible to this child and productive in actions she values.


Vn71.1 Tic Tac Toe (7) 8/25/77

This material provides Miriam with an opportunity to exhibit what she retained of instruction in the previous tic-tac-toe session (cf. vignette 61, 8/10/77). Where Miriam fails to elect a winning strategy (game 3), I subsequently demonstrate how she should have played, then provide the opportunity for her to turn the tables on me. (These data were recorded in Home Session 17.)

Game 1: Miriam moves first (letters)

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

After my first move, I ask Miriam:


Can you beat me if I move here?

I think so [moves B].

Oh ho. I’ve got a forced move. I bet you’ve got me already [moves 2]. Do you?

[shaking head ‘yes’, smiles and moves C]

You do. You’ve got two ways to win already.

[laughing] I did the forced move and two ways to win.

That’s absolutely perfect, Miriam. You got it.

Game 2: Bob moves first (numbers)

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

This dull game is of interest only in Miriam’s avoiding the middle of the row response to a center opening.

Games 3, 4, and 5 —

Game 3: Miriam first (letters) Game 4: Bob first (numbers)

	 A | 4  | D 
	 1 |    | 4    
	 C | 2  | 1 

	 B | 3  | A    
	 3 | 5  | B	
         2 |    | C   

Upon my response (1) to Miriam’s corner opening, she had the opportunity to beat me directly and failed to do so. When she made her second move (B), I informed her of her oversight. She was angry and had to be cajoled to play game 4 with roles reversed. When she moved A in game 4, I review her move of game 3 (B) comparable to the one I then made (2).


You went down here, where the B is, next to the 1.
If you had gone over here, where my 2 is now —


You could have beat me. You know why?


‘Cause you’ve got a forced move between the 1 and 2.

Oh [she move B].

Now, what chances to win do I have? From the 2 across
the bottom; from the 1 across the top; from the 1, down through the middle;
from the 2 up through the middle. And I have to go in the middle because
you have one way to win. Now look at this —

I get it.

I take my forced move —

I get it.

Two ways to win. . . .

Miriam became very angry upon suffering this defeat. She cried a little, wanted to quit, and generally made me feel like a bad guy. When she was convinced to turn the tables on me, she played game 5 and beat me directly. With her compensatory victory achieved, she no longer wanted to quit.

Game 6: Bob moves first (numbers)

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

My opening gambit (1) I characterize for Miriam as “probably a pretty dumb move. I’ve never seen anyone go first here before.”

Game 7: Miriam moves first (letters)

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

I check at first to make sure we have not played this corner opening response in this session; then upon moving (1), ask Miriam:


Do you remember how to beat me?

Unh-huh [then she laughs and moves B].

Oh, you’ve got me now.

[gestures toward moving next in the center]

[stopping her] Show me your chances to win.

[gestures along the top and from B up through the center square]

If you want two ways to win, you have to move where the chances to win come together.

[gestures to move in the center square]

That’s wrong.

It is?

Where do the two chances come together?

Here [along the top], here [up through the middle]. Here [the intersection corner].
If I go here, you can block here [the center square], but I’ll go here.

O. K.

Miriam makes move C, getting her two ways to win.

This vignette continues the documentation of Miriam’s tic-tac-toe experiences. Her preferences suggest that she has begun to think of appropriate strategies selected by response to the opening move, and show she can think in terms of intersecting chances to win even though her first inclination is to move in an empty center square. (I myself played so before analyzing the game in the course of this work with Miriam.)


Vn72.1 Tic Tac Toe with Robby 8/25/77

Having seen Miriam play tic-tac-toe with me and feeling a little left out, Robby asked to play with me after Miriam went to bed.

Game 1: Robby moves first (numbers)

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

Robby originally made move 3 in the middle of the top row, belatedly recognizing his error, and asked to move instead in the middle of the bottom row. Such oversights appear to be characteristic. When I mentioned, before placing C, that I had a forced move, Robby noted, “This is probably going to turn out to be a draw.”

Game 2: Bob moves first (letters)

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

After Robby’s first move (1), I asked:


Do you believe I can beat you?


You don’t believe that? I’ll prove you wrong.

All right.

Watch. I put a B in that corner. Do you have a forced move?


How many chances to win do I have?

[gesturing across the top and up through the center from B]
This way and this way.

Two chances to win, right?


Do they come together?

Yeah. In that corner.

So I put my letter C up there and what do I have?

Two ways.

I had not in the past described play in such a manner with Robby. His finding it immediately natural is a sign he thinks of the game in such terms himself.

Game 3: Robby moves first (letters)

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

After Robby’s corner opening, I brag that I’m not so easy to beat as the computers at the Children’s Museum. He responds:


I also have a different technique if you do that [unclear referent;
perhaps: respond with center move to his corner opening as the computer did].

You think I’ll do that? Well, suppose I go over here. You think you can beat me
if I go there? . . . Son of a gun, you got me. Do you believe you have me?

[a less than absolutely confident smile]

You’re right. You know why?

Yeah. You’re forced to go there (2) and I can go there (C), then I have two ways to win.

I congratulate Robby on being “pretty good at this” and inquire how he learned to be so good at tic-tac-toe. Robby explained that the 3 times we were at the Children’s Museum he played tic-tac-toe with the computer “quite a bit.” He suggested as many as 26 games.

At this point in recording Home Session 17 the tape recorder malfunctioned and the remainder of the conversation was lost.

Game 4: Bob moves first (letters)

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

This game exhibits use of the block I developed to counter the strategy Robby first employed against the computer at the Children’s Museum (cf. Vignette 5).

The remaining three games we played this evening were all center openings by Robby. When I responded with corner moves twice, we tied. When I responded with a middle row move, he beat me.

At the end of the games, we discussed the game generally. Robby, in response to a question of how many ways one could start out, explained that there were possible only 3 opening moves (center, side, and corner). He also knew that when responding to a center opening, a move in the middle of a row invariably led to defeat, whereas a corner move would guarantee a tie unless you made a mistake.

These data are collected for comparison and contrast with the more extensive collection of Miriam’s games. My general impression is that there are two main differences between the children’s grasp of the game. Robby appears to conceive of an entire game as a single entity, the sort of game it is being determined by the first 2 moves. I infer this from his being able to describe and discuss the games in a relatively abstract way: there are only three opening moves; there are only two responses to a center opening. This is a different way of thinking of the game’s symmetry from the way it is manifest in Miriam’s thought: she will recognize one game as equivalent to a second when both appear for judgment in that respect. Her response to such questions needs further probing.


Vn73.1 Not Being Ready; Logo vs. School 8/26/77

For the past week Miriam has been mentioning that she doesn’t ‘feel
ready for school.’ I’ve tried to find out what Miriam means by her
feeling ‘not-ready.’ In one case, she explained to me that she didn’t
know what they do there. In another incident, at the dinner table,
when Miriam mentioned not being ready for school, I pointed out to her
that she was surely ‘ready’ for Logo and asked both children if they
thought of Logo and school as being the same or different. Robby
answered first, that Logo and school are different.


How are they different?

You don’t learn anything at Logo.

Oh? And you do at school?


What do you learn? I know you have art, but you knew how to draw before you went to school.

You learn. . . ah. . . mathetating.


Mathetating; what you do with numbers.

Don’t you ever do adding at Logo?

Yeah, but all you learn at Logo is how to use computers.

I learned how to write.

A third incident showed a different perspective.


(To Robby) I wonder what school will be like? Was it very fun in second grade?

Pretty much fun if you have a teacher like Mrs. Johnson and Mrs. – – – [a student teacher]

Miriam, are you more concerned with school’s being fun or your being ready?

Fun. . . but I’m not sure I’m ready.

In what way?

They may be different people. I hope not. I want the same people again.

This last comment recalls the difficulty Miriam had in making friends
at the beginning of the last year. That September was the first major
upsurge of her hayfever allergy (previously only dust and mold had
been diagnosed); her reaction was so severe that she was physically
depressed for the first 8 weeks of school. I surmise she remembers
that time as a very bad time and has vague fears associated with the
returning to school.

These three notes touch on Miriam’s sense of being ‘not ready’ for
first grade and some contrast of what they do at school and at Logo.


Vn76.1 Where Do Ideas Come From? 8/29/77

In this hot, humid weather, Gretchen and the children have been
spending all day at Logo with me. This morning I found Sylvia Weir had
taken a desk in the room where Robby had just laid claim to an empty
desk. She seemed intent on reading, and knowing how distracting the
children can be, I asked Robby to move to a free desk in the adjacent
room. Later, when I asked him had he done so, Robby told me he had been
locked out of the office.

When Sylvia returned from lunch, she was as surprised as everyone
else that the door had been locked — and that was for her a problem,
because she needed to pick up her materials before leaving shortly. Did
Donna have the key? No. Greg or Eva? Perhaps, but neither was about.
George or Gordon, could they help — neither could. An impasse.

Recalling one of the avocations of students at Caltech had been
lock picking, I thought maybe Danny or Brian might have become similarly
skillful here. Going back into the computer room, I looked toward the
locked room and noticed a roof panel was out of place. Aha! Should
the lock picking be difficult (I had never developed skill at that),
one could go over the partition through the roof. Both lock and door
were sturdy, the lock not accessible to a knife edge or spatula prying
gambit (the only one I know). I looked again at the ceiling and worried
that it would be too tight to snake over except for a child, and I
wouldn’t risk one of the children’s falling from ceiling height. I was
standing on the floor; it is raised for the computer cabling and also
could be dismantled. I walked to the Logo foyer and told Sylvia not to
worry. We couldn’t open the door, but we could open the floor.

Removing the floor panel in front of the door, I could see that one
would have to crawl under the floor for a distance of at least 2 feet,
then lift the panel beyond the next to rise up inside the room. There
might be a desk inside on that second panel; this possible impediment
would make it too difficult for Robby to tackle — if the simple plan
failed he might feel trapped and become frightened. Danny Hillis,
declaring he had done so before, volunteered to crawl under the floor
and open the door. Thus Sylvia’s afternoon was saved and we all had a
good time solving a practical problem.

At dinner this evening, Miriam asked: “Daddy, how did you ever
think of going under the floor? Was it because you remembered how good
a time we had before when the floor was up?” (Cf. Vignette 42) I told
Miriam her guess was pretty good, and I set out my “problem solving
process” in the previous paragraphs to show how good her guess was —
for these notes show how local were the changes, stepwise, to the problem
as I perceived it, by which I arrived at a solution others saw as an
imaginative transformation. What I find most striking, however, is that
Miriam asked me how I got a particularly good idea. This implies she
is capable of reflecting not only on her own thought processes, but also
on mine as well, and even more, has formed her own hypothesis to explain
my thought process in this instance.

Miriam inquires how I generated a good idea and offers her
speculation on how I might have done that. This is as clear an example as
one could want of her sensitivity to and reflection upon the process of


Vn77.1 A Geometric Puzzle 8/29 & 31/77

8/29 Since Miriam’s completion of our work with picture puzzles (cf.
Logo Session 40, 8/1/77), it has been my intention to examine her
performance with geometric puzzles. In the past, she has played with a
puzzle, the Pythagorean puzzle, which I had made from wood.

The Pythagorean puzzle is of 5 pieces and fits together in two ways.
The first fitting provides a square whose sides are the hypotenuses of
4 congruent triangles. The second fitting may be seen as the contiguous
placement of two smaller squares whose sides are the same length as the
two sides of the four congruent triangles.

Vn 77-1 Pythagorean puzzle

Today I found Miriam and Robby playing on the floor of Glenn Iba’s
office with a small geometric puzzle. Robby played with a plastic
version and Miriam with a duplicate cut from cardboard. The pieces
below form squares in two ways:

Vn 77-2 Glenn's puzzle (1)

Pieces 1 through 4 fit together to form a square. Pieces 1 through 5
form a slightly larger square.
During their play in Glenn’s office, Robby accepted a proffered
hint (Glenn first showed him the outline of the square and the location
of one piece). Miriam first refused to look at the hint Glenn offered,
then got mad at him when later he refused to show it to her. I brought
the cardboard puzzle home and put it on my desk for later use.

8/31 I found Miriam working at the puzzle this morning. She succeeded
relatively rapidly at the 4-piece assembly. As Robby tried to show her
Glenn’s hint, the arrival of the mailman drew the children away from
that task. Gretchen picked up the pieces, assembled the four, and left
it on a chair near the reading alcove.

Later Miriam joined me and tackled the 5-piece assembly. She failed.
She went over to my book shelves and took out the Pythagorean puzzle as
she said, “I’m going to give myself a good hint.” Miriam successfully
assembled the Pythagorean puzzle in both forms, but did not find that
success useful with Glenn’s puzzle. She decided first to make a design,
then asked for my help.

I had seen Glenn’s hint. I recalled the orientation of piece 1
with respect to the square’s edge and showed it to her. I noted that
the edge with 5 pieces was bigger than the edge with 4 pieces and set
as a sub-task finding a combination with edge length equal to that of
piece 1 with piece 5 inset. Once we found the place of piece 2, thus,

Vn 77-3 Glenn's puzzle (2)

success was in reach. Miriam attempted piece 3 and failed repeatedly.
I recalled to her mind the picture puzzle hint: rotate the pieces.
Miriam then fit pieces 3 and 4 in place. Miriam is very happy and says,
“Robby thinks he’s the only one who can do this.” Miriam shows Gretchen
she can assemble the puzzle, then calls Robby to witness her success.

Before lunch, Miriam encountered the puzzle disassembled on the
dining room table. She talked to herself as she tried to assemble the
5-piece variation: “I’ve got a forgetting bug about this puzzle. . . . That
can’t be right. . . yep.” Miriam gives up and gets a snack.

In the afternoon, Miriam retries Glenn’s puzzle. She clearly
remembers the relation and placement of pieces 2 and 3. She also states
explicitly that piece 5 must be inset at the corner in piece 1, yet she
can not see how to fit the pieces together as she tries to place the 4th
piece adjacent to pieces 2 and 3. She is about to quit when I advise
her to rotate piece 4 once, then again, arriving at this arrangement:

Vn 77-4 Glenn's puzzle (3)

at which point she sees how to fit the 1 – 5 combination into the 2-3-4

Miriam’s puzzle assembly skill does not seem to generalize easily
from picture to geometric puzzles, nor from one geometric puzzle to
another. She knows when she is frustrated that she needs a ‘good hint’
and can apply it when given specific advice (note, however, that she
had to be directed to rotate piece 4 two times; she interpreted the
hint as: turn piece 4 so the next edge is adjacent to the 2-3 assembly,
instead of turn piece 4 until the configuration can accommodate sub-
assembly 1-5).


Vn79.1 Sums Over a Hundred 8/29/77-9/1/77

8/29 While we sat at lunch today, Miriam introduced the topic of adding
with this claim: “Daddy, if you live for another hundred years, I know
how old you’ll be.” When I expressed surprise Miriam demonstrated:
“A hundred 37.” Two complications derailed this discussion. Robby
introduced my birth on February 29th with its implication of quadrennial
birthdays. Before we entered more complicated computations on this
basis, I noted that I would be dead before a hundred more years and
that one stops counting a person’s birthdays when he dies. Both children
looked at me blankly, and we proceeded to a discussion of what death is like.
(If curious, confer the note appended at the end of this vignette.)

9/1 This evening, I read aloud to Gretchen an excerpt from a draft-
section of Seymour Papert’s Logo book, a sardonic description of the
class structure of the mathematics education world:

Mathematicians create mathematical knowledge, math education
researchers package the material for children, teachers deliver
the packaged stuff, evaluators measure how badly the whole
process worked.

When Gretchen laughed, Miriam, out of sight in the adjacent area of the
loft, commented, “I don’t get it. I don’t think that’s funny.” Although
in one sense this is not at all funny, in another way it is, and so I
told Miriam. She replied, “What do you mean?”


How much is a hundred 70 plus 27? [original has a hundred 7]

97. . . a hundred 97. Did I do it right?

Yes. Did you use your fingers?

You want to know how I did it?


I said 70 plus 20. That’s 90, so I have the 97.

Where’d the hundred come from?

It was a hundred 70. . . . Did I do it right?

You did it beautifully. . . and that’s more important than doing it right.

I know that.

You also did it correctly.

Miriam went back to playing at what had occupied her before the dis-
traction of my reading aloud, so I did not explain why this problem she
solved, documenting as it does her ongoing progress in constructing her
own algorithms for addition, shows how ‘funny’ in another sense are the
best efforts, even the well-intentioned efforts, of the mathematics
education establishment.

Since Miriam’s forgetting how to add multi-digit addends and her
subsequent reconstruction of adding procedures on a different basis,
I have let her curiosity guide our discussion of the algorithms she
employs for computation. This vignette records Miriam’s recrossing
of the hundred barrier with her own method of adding.

* For the curious: when I elaborated somewhat further, I said,
“You don’t count birthdays ’cause you can’t think at all when you’re
dead. You don’t eat or breathe either, but that doesn’t matter because
you can’t feel anything at all.” Robby came back: “Oh, I get it now.
Being dead is like you blew a fuse.” I agreed: “And each of the major
organs in your body — your heart, your lungs, your liver — each of
those is like a fuse and when one of ’em goes, you die.” Robby has
spent time since building two models, the Invisible Man and Invisible
Woman, attempted over a year ago and judged too complicated then.


Vn81.1 Imitating Machines 9/3/77

Ever since their first encounter with the Votrax Voice Box back in
May (Logo Session 5, 5/22/77), both children have thought it funny to
imitate the peculiarly mechanical tone of that speech generator. I have
suspected some correlation between my asking them questions they consider
stupid and their adopting this mode of reply, but that speculation
has never been clearly tested. Today, in between the sessions for Robby
and Miriam, Robby entered the room I was in and said something in Votrax
mode. I have felt generally uneasy about this imitation and I complained:
“You are not a Votrax Voice Box.” Robby responded (in Votrax
mode): “I am too a Votrax Voice Box. But I can do other things besides
talk. I can walk. And think. And poke.” (Here Robby poked me in the
stomach). I grab him: “And get tickled.” “And run away,” he concluded
as he broke away from me.

At the end of our day’s work, Robby was lying on a desk whereon was
a pencil sharpener. Miriam entered, sharpened her first of six pencils,
and held it up for examination. Robby blew the wood and carbon dust off
the pointed end. Miriam told him to stop and he did. With the next
pencil, at the appropriate time, Miriam commands Robby to “blow”; he does.


I’ll push your thumb. That will be your stop button. . . . Blow.

(Blows on pencil end and stops when Miriam presses his thumb.)
(Robby then gets up, stands beside Miriam, holding up two thumbs —
one for starting, apparently.)

Hey. Instead, this button can be for sort of running in place.
Your nose will be the start button. (Miriam raises a pencil before him
and presses his nose.)

(Blows on pencil and runs in place.)

(Presses his ‘stop’ thumb.)


(Presses ‘start,’ ‘stop,’ and ‘run in place’ buttons all at the same time.)

Arrgh. How did I ever get mixed up in this?

This game of imitating machines, like ‘Follow the Turtle’ of
Vignette 42, is a direct outgrowth of the children’s experiences art
Logo. Does Robby seriously think of himself as a machine? If he does,
he is also articulate about highly specific differentiae. . . and maybe
that’s not too wrong.


Vn82.1 Hanging Designs 9/3/77

After today’s session was complete, I asked Miriam why she had not
pinned on the wall — as she had said she intended — those designs made
in yesterday’s session (Logo 58, 9/2/77). She explained that she had
started to do so earlier but needed help.

I separated the designs from the interleaved blank pages in the
pile on her desk, then asked where to hang (“Up there.”) and how.
Miriam’s directions: “In alphabetical order, by the numbers.” When I
found this opaque, Miriam explained, “Like the way Robby did it.”

At Miriam’s direction, we set up a display of poly spirals varying
from the base of 60 degrees (we had originally called such a shape a
‘maze’) in order by the turtle’s angle of turning up to 67 degrees.
Miriam had created this complete set of designs with considerable direction
from me (cf. Session 58), and she used Robby’s arrangement of designs
as a model. Nonetheless, the creation of this family of shapes was her

We came to a last design. All the others had been made with an
increment (‘delta’ we call it) of 2 turtle steps. At the angle of 67
degrees, we made a design with delta = 1. (This was done because I had
been too directive earlier in the session, requiring Miriam to hold
delta constant.) I asked: “Where do we want to put this one? We have
a 67 degree design already, but this one’s got a different delta; should
we just put it under like the others?” Miriam instructed me (by placing
the design in this place) to tack the design on the wall at the side of
the other 67 degree design and “we may want to make another family later
like the other one.”

In the directions Miriam provides for how her poly spiral designs
should be hung on the wall, one can see her beginning to organize them
into groups defined by the changing of one variable while the others
are held constant.


apparently, this file needs to be recreated, from earlier sources.
The tags attached to the source suggest it is important.


Vn84.1 Go Cart Demon; Knock-Knock Jokes 9/5/77

The third-floor tenant in our landlord’s mansion was moving out
today. Robby and Miriam went to help. One comment of Miriam’s came
floating up from the court yard. When she chanced upon a collection
of records brought down in a wooden case, Miriam said, “Hey, Robby,
let’s ask Bill if we can have that box. If we get our wheels, it’s
just what we need for our go cart.” (Cf. Vignette 50). From this
comment, with the availability of ‘found’ material now rendering less
than fantastic for Miriam the construction of a real go cart, I see
Miriam thinking more in the style of a bricoleur than does Robby on
this project. (Recall his engineer-like inclination to draw up a materials
list for purchases to be made at the lumber yard.)

On this day, the children also encountered a book about which we
have heard since — a book of knock-knock jokes. Robby introduced this:


Knock knock.

Who’s there?


Robin who?

Robbin’ you. Gimme your wallet.

Miriam recalled a second:


Knock knock.

Who’s there?


Ivanitch who?

I’ve an itch I can’t scratch.

While this theme was before us, Miriam recalled a third joke:


Knock knock.

Who’s there?


Irish who?

I rish I never said “Knock knock.”

The first incident contrasts Miriam’s idea of acquiring materials
for the go cart project with Robby’s. The second series of jokes —
the first 2 coming from a book I hadn’t seen and the third from a TV
commercial I did not watch indicate how rapidly Miriam’s perimeter of
experience is expanding beyond the reach of my knowledge. I believe
it is still possible to trace the sources of Miriam’s knowledge but
feel keenly how important it is that she has become accustomed to
discuss her ideas, her thought processes, and their sources.


Vn85.1 9/6/77

When we started playing tic-tac-toe, I asked Miriam how many different ways can you start when you move first. She claimed 9 ways, one for each block in the frame. I pushed the point further by inquiring whether these three frames were really different or the same:

          X |   |   	   |   |   	    |   |      
            |   |   	   |   |   	    |   |      
            |   |  	   |   | X	  X |   | 

              1	             2                3 

She judged the first two to be the same and the third different from them. My intention in today’s play was to work through the range of all game Miriam could see as different responses to the corner opening. We pursued this by my letting her move first in every game with the specific objective of finding those responses which would not lead to my immediate defeat.

Game 1: Miriam moves first (letters)

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

If I go here [the middle of an outside row, not adjacent to A],
can you beat me?


There, in that side place? Or if I go in the corner?
You don’t want me to go in the corner? [opposite diagonal to A]

I want you to go some other place.

How about if I go here. Can you beat me? [the adjacent corner
where move 1 is made]

No. Don’t go there. . . . O. K. You can.

How about if I go over here, in this other corner? [the alternate adjacent corner]

It doesn’t matter [the moves are equivalent].

Oh. If I go there, the moves are the same?


I’ll go in one of these corners here that are the same. . . .
You think you can beat me?

I don’t know [moves B].

You think you beat me already?


No? Do I have a forced move?

Yeah. . . . Actually, I have [beat you]. You have a forced move.

Then what?

I’ll move there [the alternate adjacent corner] and get two ways to win.

So you’ve beat me already.

I know.

Actually, so long as I made that move there (1), you beat me already.
And you told me you didn’t want me to move there. . . . Did you know you could beat me
when i moved there? . . . You did? Did you trick me?

[smiles] Yeah.

Game 2: Miriam moves first (letters)

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

I recapitulate the last game, identify both adjacent corners as responses with which I can get beaten, and recall Miriam’s assertion she can beat me anytime. I respond with a non-adjacent, middle row move.


That means I should either move in this far corner [opposite to the opening]
or in the middle, or here or here [in the two adjacent, middle of row moves]. Let’s suppose
I move here [opposite corner]. Will you beat me?

I don’t know.

I’ll try it [moves 1].

[laughs] I’ll put my B here!

Oh. Oh-oh. Do you have me beat already?

Yep. See. I go there [alternate adjacent corner] and I’ve got two ways to win
[gleeful laughter].

So, as soon as I put my 1 in there, you knew you could beat me,
because you didn’t have a forced move.


Did you know that? Were you just trying to trick me?


You probably didn’t know it really.


Do you know it now?



Game 3: Miriam moves first (letters)

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

This game begins with the moves Miriam originally sought for the execution of her ‘dirty trick.’


If I go here [center], can you beat me?

I think so.

I’ll put my 1 right in the middle. How are you going to beat me now?

[moves B] Whichever side you go [she gestures toward the corners],
I’ll go on the other side [the alternate corner] and get two ways to win.

Ah ha. That’s a good strategy.


But it assumes I make a move in that corner or the opposite corner.

I. . .I know what you’re going to do.

What am I going to do?

You’ll go here [bottom row, middle].

[pointing to the others in turn] Or here or here or here. Does it matter
which of these four I go in?


Will you beat me if I go here? [corner]


I don’t like to lose all the time. I’ll go here [moves 2].

Game 4: Miriam moves first (letters)

          A  | 2  | B   
          1  | C  |      
          D  |    | B1-> 3 

Beginning this game, I review the moves I made and where I’ve been defeated. I cite the adjacent middles of rows as the only locations I haven’t attempted and select them as the next trial.


I’m gonna beat you I think [moves B1 ].

Why do you think you’re going to beat me?

‘Cause. . . . Oh no, I can’t if you go there [in the center].

The move you made is not a winning move. I have a forced move in the center.

I’ll go here [adjacent corner move].

Then I’ll win because you would miss your forced move there.


If you want to take that B out, cross it out and try some other move; maybe you should.

Where else? . . . Here? [move B in adjacent corner] Is that O. K.?

Let’s see. The problem with the other corner [now crossed out]: if I went in the center
you have a forced move in the side. . . but now I must move here [move 2] and you have me beat.


Where are your chances to win?

Here [from A through the center] and here [from B through the center].

If you move where they cross you get two ways to win.

[laughs, moves C]

Oh brother.

At the end of this game, I summarize: “If you start off with a corner opening, you can beat the other guy no matter where he goes — almost — unless he plays in the middle and side as I did in game 3.” Miriam ran off to announce her victories to Gretchen.

These data show Miriam and me working through all the responses (except one: see vignette 71, games 3, 4, and 5) to a corner opening. They provide a good sense of the range of Miriam’s strategic thinking.


Vn86.1 An Unexpected Test 9/8/77

Today, the children’s first day of school, was a tough one for me. The combination of a late arrival at Logo and logistics problems put our work under an unusual time pressure. Miriam was tired (and later said she wished she had taken a nap) and didn’t pursue with enthusiasm her exploration of good numbers for the SEAHORSE (an INSPI procedure). Thus, she yielded up the remainder of her time when I was reluctant to let her have a break. Robby, in his turn started off in what was a normal fashion for him, but soon we ran into a problem, the extent of his reactions to which I still can not fathom. The session with Robby was dreadful, the worst so far since our project began. He was confused, began crying, but refused to stop our session; his allergy caused stuffy nose made his crying dreadful. His reasons for sorrow increased when he began lamenting the time lost which he could have spent making designs,,,, Affairs finally reached such an impasse, we just gave up on the day.

After a few minutes alone, trying to regroup my scattered aims for the day, I carried the video camera into the storage room and saw Glenn (a graduate student) doing paper folding games with the children in the foyer of our lab. Because Glenn enjoys playing with the children and is good at it, seeing them together made me uneasy. Twice through chance, through the availability of materials, and through enjoying games to which I have heightened Miriam’s sensitivity, he has performed before me, in effect, experiments I was developing (confer Vignettes 8 and 77). When I saw peeping out from under a pile for other papers they were folding, the sheet I in Addendum 86-1, I realized my five month long, complete collection of data on Miriam’s development in Tic Tac Toe was in jeopardy.

I asked Glenn to try to reconstruct the move patterns of the games they had played. His notes are on the 3×5 card shown in Addendum 86-1.

Game 1: Miriam moves first (letters)

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

Glenn remarked on Miriam’s telling him, after her move B, that should he move in either of the other corners, she would win. He did not move there.

Game 2: Glenn moves first (numbers)

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

Miriam and I have not played this game to the best of my recall. Note that had Miriam moved in a space adjacent to A, this diagonal configuration would have permitted the opening to gain two ways to win, thus:

          A | B | 3
            | 1 |   
            |   | 2 

Game 3: Miriam moves first (letters)

          A | O | X
          X | X | O
          O | X | O

Glenn’s only dependable recollection of this game is that Miriam opened at the corner. The tie must have followed one of these patterns of a symmetric variation:

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

The data of Vignettes 71 and 85 argue that game A was most likely the one played (I believe Miriam would have beaten him had Glenn responded to her opening with B or C.

Game 4: Glenn moves first (numbers)

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

Glenn notes that Miriam requested he place his marl at the location of 1. Recall Miriam’s comment (at the time of game 7 in Vignette 61) that she would attempt to get Robby to make such a move so that she could play her newly learned tactic on him. When I asked Miriam, while discussing the game with Glenn, how he had beat her, she was a little apologetic, saying, “Well, gee, Daddy, you can’t win all the time. I guess I must have made a mistake.” She speculated further (at least agreed to my suggestion) that she missed a forced move. As game 5 shows, Miriam learned well how with a corner opening she could defeat an opponent responding with a far, mid-row move. This particular game suggests that she had not yet accommodated her configuration based view of the game to the relative advantage obtaining to the opening player. (Notice her foiling this same opening of Glenn’s in game 6 by abandoning the corner move.)

Game 5: Miriam moves first (letters)

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

Glenn seemed a little surprised at my suggestion that Miriam ‘knew what she was doing’ (i.e. executed a game-length strategy) as she beat him here. When asked her opinion of Glenn as a player, Miriam allowed that he was pretty good. Glenn acknowledged that Miriam did make all forced moves.. . and showed a surprising inclination to adopt the corner opening.

Game 6: Glenn moves first (numbers)

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

This game is notable in showing how quickly Miriam abandons the losing strategy of game 4. I believe this is the third game she has played with a mid-row (non-center) opening.

This vignette raises two issues. First, how does Miriam apply in other situations what she has learned n the structured sessions of this project? Second, how complete can these data really be? It is clear from game 4 that when the knowledge is directly applicable (as in playing Tic Tac Toe with a new opponent), Miriam applies that knowledge directly in a minimally modified form. (She hopers to catch a “naive” opponent with her preceptor’s “dirty trick.”) Learning anew, at her cost, that a significant attribute was not marked in her formulation (its success depending on the corner opening move), Miriam when confronting the same opening a second time retreated to a seize-the-center play (this reduces maximally the opening player’s chances to win).

How complete can these data be? If it be the case that Miriam interacting with one person on the occasions described here and in Vignettes 8 and 77, engages in three significant ventures in learning, must it not also be true that other such incidents occurred which have escaped my notice? I think not. The extent of time I spend with the children and the sensitivity to precisely this sort of influence argue that not much has been missed.

Addendum 86-1

Games with Lab Student

Vn 86-1 Games with grad student


Vn88.1 9/8/77

Over the past few weeks, Miriam has spoken, in the context of
repressing her desire for things she can’t have, of having “an eraser
mind.” When asked to explain what she meant, Miriam conveyed the
image of ideas written on a tablet and subject to erasure.

As supper drew to a close this evening, Miriam cited the existence
of another mind (I believe, but am far from certain, that we were dis-
cussing future meals and Miriam noted her “liver-hating mind”). Remarking
my surprise at her thought of having an ‘eraser mind’ and another kind
as well, I inquired if she thought she had any further “minds.” The
topic lay unheeded for a short while. I made some coffee and sat down
away from the table.

The children picked up the theme as a game between themselves.
Miriam: “I know another mind I have, a “remembering mind”. . . and another,
a “stay-away-from-sharks mind”.” Robby asked if she had a “talking
mind.” Miriam responded that of course she did, it had a voice box in
it. These seemed to exhaust her invention for the moment, so Robby
proceeded: “You must also have a learning mind, or all your other minds
would be empty.” Miriam agreed, going further to claim that her “learn-
ing mind” was the biggest one of all. Robby continued further that he
had an “electric mind” whose function was the manufacture of electricity,
“for that’s what everything else runs on.” In response to Miriam’s
objection that she had no wires inside, Robby pointed to a wall socket
and explained that the electric energy was carried through the bones to
outlets, such as the one in the wall, where it was made available for
local distribution.

At my inquiry of where they had picked up such unusual notions,
Miriam said, “It’s all in your brain.” When pushed further with the
question of whether mind and brain were the same, she clarified thus:
“Actually, it’s all in your everything mind.”

Finally the joking grew stale. On my inquiring, pen in hand, what
was that second mind she had cited, Miriam remarked, “Daddy, if this
shows up in your thesis, I will be mad at you.”

This vignette cites some jocular ways Robby and Miriam discuss
what goes on in their minds. Robby’s relative advancement can be seen
in his concern with a “learning mind” which develops the contents of
others. Though Miriam’s references are not ‘constructive’ they indicate


Vn91.1 Squirming and Thinking 9/14/77

Miriam had a very bad night last night; she had missed a dose of
medicine and played with kittens. Miriam and I were up much of the
night. Still wheezing badly this morning (she had reached the point
where she could not hold down any orally-administered medicine), she
went with Gretchen to the doctor for a shot of adrenalin.

Robby and I were left alone in a quiet house. While I was attempting
to write in the reading alcove, Robby assembled a puzzle on the
living room floor. He left off the puzzle and lay on the floor, bending
his body back and forth at the pelvis. When I told him that was most
distracting, that he should stop squirming, Robby sat up and said:


Daddy? You know all that stuff about 3 hundred and 60? [This is
a back reference to our discussions in Logo Sessions 61 and 62
of the effect of reducing an angle by 360 degrees]


I understand it now.

Wow! How did you figure it out?

Well, you know if you have an angle that’s 3 hundred and 61?


And you take away 360?

Uh huh.

It’s 1, and that’s like it’s starting all over again.

That’s really great, Rob. When did you figure that out?


Just now? When you were squirming around there on the floor?

Yeah. Squirming around helps me think.

Robby returned to his puzzle. Shortly thereafter, Miriam came bounding
into the loft, so full of energy that she pushed me into leaving early
for our Logo session today.

This particular incident, though it occurred with Robby and not
with Miriam, highlights what I see as the central methodological
problem in the study of learning: being able to observe the
manifestation of a centrally-determined mental process, being there
when it happens.


Vn92.1 Company for Dinner 9/14/77

This has been a week for company at our house. Fernando Curado and
José Valente first, then Bertrand Schwartz and Antoinette together with
Laurie Miller, and this evening Seymour and the Minskys. My intention
in asking Marvin and Gloria here at this time was to provide a sense of
setting for the variety of descriptions of our lives that Marvin, as a
member of my thesis committee, will encounter in my data; and further,
through a short exposure to one evening in my family’s life, to provide
a sense of the relations and qualities of interaction from which the
observations in these data arise.

Unfortunately for my purposes this evening’s guests arrived too late
to tour the grounds of our landlord’s mansion, those places where the
children have played this summer when not under my eye (and under foot);
yet they did have a chance to participate in a more or less typical
evening at home. If the evening was atypical, it was so in two respects
mainly: Robby was tired and went to bed directly after our late dinner;
Miriam (could she possibly have been still energized by the adrenalin
shot in the morning?) was lively and stayed up much later than usual.
Since Miriam was expected to go to school the next day, I told her
several times to go to bed. She took my instructions as reminders
merely, and chose to ignore them. Further, it was appropriate that
Marvin should see as much of her as she wished to show him.

We talked some of Miriam’s work (I showed Marvin one of Miriam’s
“Seahorses” [an INSPI with an angular increment of 13]; Marvin allowed
that he did recognize it — indeed, he noted he was the first person in
the world ever to see that particular design) and of some of the unusual
turns of mind that Miriam now exhibits (the data of Vignette 76, Where
Do Ideas Come From, were then much in my mind). Gloria gave us her
appreciation of the Brookline schools, from the perspective of her special
knowledge and from the experiences of Margaret, Henry, and Julie. When
Gretchen and Seymour brought dinner to the table, talk turned more
intellectual for a short while. Miriam redirected that tendency after
dinner by engaging Marvin’s help in her weaving of a potholder. Eventually
both Miriam and the evening wound down and our guests departed.

This evening, representing a for us natural mixture of social,
intellectual, and family concerns and activities, provided a more or
less typical experience of an evening in our family for two members
of my thesis committee.


Vn96.1 Tic Tac Toe 9/19/77

After the final arithmetic session of this date, Miriam wanted to play some tic-tac-toe. (Her purpose was to get the material over with so she could have a friend come play with her on the morrow; her assumption that such was necessary was an error.) These games began with a discussion of Miriam’s play with Glenn Iba (cf. vignette 86).


I remember a funny thing happened the last time we were at Logo.


We were there and you ended up playing tic-tac-toe with Glenn.


Was he surprised when you beat him that time?


What did he say to you? Did he say, “How did that happen?”

I told him how I was going to beat him in the first time and he spoiled it.

In the very first game?

I went here [a corner response].

Yeah. You had a corner opening.

And he went here [center]. I went here [opposite corner], and said if you go here [adjacent corner],
I’ll go here and get two ways to win. And he went here [side move] and I had to go here [forced move].

He spoiled your game.


That’s why you didn’t beat him in the first game. Do you remember that other game you did beat him in though?…

[no response]

After Miriam selects a pen (by applying the chant “Engine, engine number 9, going down Chicago line. . .”), we try to discuss the possible responses to Miriam’s opening corner move. Miriam shows no inclination to reduce her count of possible responses based on symmetry arguments.

Game 1: Miriam moves first (letters)

	 A | 1 | B1->D   
	 2 | C |     
	 B |   | 3 

Miriam’s initial counter-move (B1) to my opening response was not optimal (for a game of the form classed as Game VII in Learning: Tic-tac-toe ). Together we worked through the recorded game above.


Let’s see if you can beat me when I move right close here to you. Do you know whether you can beat me or not?

Unh-uh. Rats. This red pen [moves on opposite side of my markers].

That’s where you want to go? Miriam, I’m really surprised. Why do you want to go there?

‘Cause then if I go here [opposite adjacent corner], I can beat you.

Now hold on. You’re trying to move there. So then you’ll get two ways to win? … Let me show you’ve got a bug.
Miriam, I get to go next, and I’m planning on going there [center].

Go here.

Oh no. The good trick to beat me there is you have to force me to go someplace else.


Like, if you crossed out that move and moved in a different corner, like down there.


Would it work then?

Maybe. Would it?

What do you think? You force me to move over there. Then the center will be free. You think that’ll make it?
O. K. Go ahead.

I did.

Great. Well, Miriam, I am forced to move here, in the side place. . . . Your chance to win do what?

Come together, this way and this way [moves C].

I will go down there with my 3 then.

I will make this. I win, I win.

Why don’t you put a big M over the top for Miriam. . . . But if you fail to force my move, the next time around,
my number 2 — I could have put my number 2 right in the middle and that would have screwed your strategy all up.

Games 2 and 3:

After a replaying of the game situation in which Glenn beat Miriam (cf. vignette 86) — at her request — in which I beat her (the opening game, of form X, is determinate), Miriam spontaneously turns the tables on me in game 3.

	numbers first	letters first 
	 A  |    | 2 	 1  |    | B    
	 C  | 3 | 1 

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

I remember when you were playing with Glenn, you did a lot of playing on the side. He started on the side a lot.
Is that because you told him to every time? Or just because he wanted to after the first time?

Will you go over here [the far side from upper left corner].

Over here? What were you going to do?

Here [upper left corner move].

Who do you think will win?


Let’s see. Ah. You’re the letters, I’m the numbers.


Now you think you’re going to beat me by going up in the corner?

[moves A]

That’s a bug, Miriam. Shall I show you why? . . . Do you think you’re going to win, or do you think I’m going to win?

I think you’re going to win.

How come?

You just told me.

You want to see how I do it?


I put my 2 right up here, over top of the 1. Now you have a forced move…. My good trick is that your second move
(B) comes right under my 1. Now, tell me, what I’ve got [3 in center square], where my chances to win intersect.

Two ways to win.

How did I do that to you? . . .

I’m going to go here.

So you block my 1 – 3 and I go there. O. K. One for Bob. . . . Where do you want to start?

[moves A where my 1 had been in last frame]

You’re going to go where I went last time?


Oh. . . . Is this “turning the tables”?

[laughing] Yeah.

Well, that means I have to go up where you went last time, right?


O. K. So you’re turning the tables on me. You’re going to be able to beat me now? . . .


I will put my 2 down here.

Here? [i. e. is move C in right location?]



Oh. You put it there because you’ve got one chance from the B through the center and one chance from the A across.

[unclear comment]

Well, I’ll go here. 3. So you turned the tables on me all right. You like turning the tables on me?
Is that a good way to do tic-tac-toe?


Here. I go first again? Shall we do another side one?

Let’s stop. I’ll do some more after a while.

We leave tic-tac-toe for a game of frisbee in the courtyard and do not return to it.

Miriam’s play in game 1 shows the residual dominance of the three-corner configuration and an imperfect integration of the idea of a forcing chain (we have not explicitly discussed this idea). Miriam requests we replay the game in which she was defeated in vignette 86, the re-executes it of her own volition in ‘table-turning’ mode. I cite this event as evidence that Miriam has adopted this procedure as a powerful learning tool.


Vn97.1 Retrospective: Logo Conference 10/1/77

The week of IJCAI, August 22-26, concluding as it did for some of
us with attendance at the Logo conference, was an especially busy one
for me. I can see now this was the point at which I lost control of
processing the data of the project “on line” (keeping the transcription/
vignette backlog to about a week or so). To maintain the density of my
interactions with the children, 3 Logo sessions preceded the conference
and two Home sessions occurred during it. At the same time, I attempted
to attend most of the IJCAI conference session on “Knowledge Acquisition”.
The antihistamine dosage I required to suppress my hay fever kept me
drowsy. The final complication was Hal’s asking me to speak at the Logo

It seemed appropriate for me to speak for two reasons: first, for
general communication’s sake in a community that suffers from too much
intense self-absorption; second, because my work with Miriam is one of
the best answers available to the external criticism that Logo is all
talk and no demonstration. However appropriate, this talk worried me —
it represented my first public appearance in a community within which
I could best hope for my work’s friendly reception, however controversial
might be some elements of methodology. I thought a lot about what
I should say, was much troubled and perplexed.

Miriam asked if she could attend the conference. She knew Danny
Hillis was expected back from Texas on August 25. She knew the conference
was a Logo conference and expected Danny to be there. Miriam
recalls with delight attending Seymour’s seminar at DSRE in the spring,
has asked to attend Marvin’s class expecting to sit in Danny’s lap as
she did the semester before. Here, for me, was a central problem. To
the extent that Miriam is my colleague, to the extent that this project
is our joint construct, I believe her role in it must be manifest.
I warned her that she would be bored, that people talk and talk and talk.

Miriam decided to attend the conference. She spent most of the
morning playing with Claudette Bradley’s son under a table at the back
of the room. With the pressure of her arising visit to the doctor, I was
given a short time to speak before lunch. Miriam chose to stand with
me and be introduced to the community. This was her decision, which
I felt bound to respect. . . . One might ask why.

A behaviorist I once knew, who had since worked his way through to
a richer perspective, one of Zen mastery, advised me: “Don’t worry about
who you are; there is only one valid description; let it emerge from the
process of your being.” The day preceding this conference, Marvin had
been willing to publish fragments of a comprehensive theory of the mind
about parts of which he was still unclear himself. Compared with the
polish of Feigenbaum and his protegé, or Simon’s didactic revelations,
Marvin appeared to be a confused amateur. What is the lesson one may
infer from that performance?

Shakespeare has the villain Iago ask with ironic deprecation,
“Shall I wear my heart upon my sleeve, for daws to peck at?” Othello,
Shakespeare’s hero of greatest heart, by his action answers “Yes.” Of course,
he suffers for it, does stupid things, and is generally considered a fool.

So Miriam chose to stand with me, be introduced as my colleague,
in a study I consider more serious than the others discussed that day.
Because we were in the middle of things, I did not discuss too specifically
what we were doing. (I recall Lee Gregg complained of that, as
did Ira Goldstein.) Howard Peele advised me to attend to the role of
establishing a vocabulary in my work with Miriam, the tracking of that.
Cynthia Solomon thought much of the audience was freaked out by the
prospect of my doing an experiment “on” a subject with whom I was
obviously so intimate.

At first Miriam dogged my steps, then asked me to carry her. I did.
While I was responding to questions she kept asking me to relate to the
audience the joke she invented the night before — while cutting her
food at dinner, her pork chop went flying; she described the incident
as showing her pork chop had a “jump on the floor” bug. I could have
spun a story from that incident, as she wanted me to, and I did not.
Since then, Miriam has invented better jokes. Perhaps, I will one day
give a more polished description of our work.

These notes try to capture some few aspects of my introducing our
intimate study to the Logo conference.

Addendum 97-1 Logo Conference Notes

Vn 97-1 notes for IJCAI related Logo Conference


Vn99.1 Puns 9/26/77

At a recent visit to the library, Robby borrowed two books which
both he andMiriam have read and re-read since then (cf. Vignettes
84 and 94). The first book is one of Knock-knock jokes, the second is
one of riddles. I believe this second book broadened Miriam’s view of
making jokes and led subsequently to today’s little story.

In school, Miriam invented and told Brian this joke:


What should you do if your toe falls off?

I don’t know.

Call a tow truck.

Robby and I though it was very funny. I congratulated her and
told Miriam it was a great advance over her ‘fart bomb’ joke (cf.
Vignette 35). When they started squabbling over who could retell the
joke to Greg Gargarian at our next visit to Logo, I told both children
that Miriam should have the inventor’s privilege. Robby was quite put
out until I saved the day by asking:


What should you do if your thumb falls off?

I don’t know.

Get a thumb tack.

I gave this joke to Robby for his use at Logo.

This joke of Miriam’s shows her first invention of a pun. I consider
this proof positive of her effective mastery of context-based word-idiom
disambiguation. (For a discussion of this issue, see Pre-Readers’ Concept
of the English Word
.) Note that she now reads at third grade level.

Both children enjoyed this joke invented by Andy DiSessa:


What should you do if your finger falls off?

I don’t know.

Get a fingernail.

Both declared my following joke “not very funny.”


What should you do if you lose your head?

I don’t know.

Call a head hunter.


Vn101.1 The Death of Robin Hood 9/29/77

“Did Robin Hood really die in the charge of the Light Brigade?”
This peculiar question of Miriam’s, rising from no external cue that I
noticed, recurrently perplexes me. Its background is this. Months ago,
Robby, Miriam, and I watched the movie “Robin Hood” wherein Errol Flynn
performed at his swashbuckling best. Both children stayed awake to the
end. Miriam saw Flynn again on film in “The Charge of the Light Brigade.”
It appears that Miriam identified Geoffrey Vickers, the character Flynn
played in the latter movie, with Robin Hood, the character he played in
the former. Further, she identifies Flynn (if one can even say she
accords him independent existence) with Robin Hood.

The perplexity this question raises is whether or not this exemplifies
a sense in which children’s thinking is concrete. An alternate interpretation
is that the child is not sufficiently knowledgeable in encoding symbolic
descriptions — consequently, he just gets it all wrong, but in such a way as
to make it appear that his encoding is “object-fixed.”

Miriam answered her Robin Hood question after a short pause. “Oh
no, that’s silly. . . .” but she still left me puzzled. When later I raised
the issue again obliquely, she asked whether Robin Hood was still alive
or not after averring she did not believe he died in the charge of the
Light Brigade. The ensuing discussion between Robby and Miriam indicated
both are confused. Miriam believed we had seen a film of the battle it-
self. Robby disbelieved the immediate reality of the filmed battle but
stated he felt the actors in the film were the people who had really been
in the battle (he later changed his mind).

This example of a small confusion suggests one way through which
children, while living in a constructed, representational, mental reality
of the same sort as an adult’s, may appear to think very “concretely” in
contrast to the more “abstract” thought of an adult. Lacking the effective
knowledge that a story character may be represented by an actor,
Miriam has bound too tightly to the character of Robin Hood and Geoffrey
Vickers in her story frames the impression made by the actor Errol Flynn.
To the extent that a structure of frames is insufficiently rich, with the
consequence that the ranges of terminal values are restricted, the
representations of the frame will appear concrete; the complexity and
abstractness of adult thought derive from enriching intermediate levels
of frame-like structures.


Vn105.1 Hotel Magee; Two Microworlds; Decadal Computation 10/20 & 27/77

10/20 With Robby’s introduction of WUMPUS to Miriam yesterday, the
mechanically recorded sessions at Logo cease. Vignettes continue to
round out and close off at natural stops various themes of the project.
The sense of closing off the mechanical recording is that the project
has REALLY ended. Thus our trip to witness my cousin’s wedding in
western Pennsylvania is both a vacation, an obligation, and a celebration.

After 7 and more hours of driving, nightfall found us in Bloomsburg,
on the east fork of the Susquehanna. We passed motel after motel with
NO VACANCY signs. After dark, we came to the Hotel Magee. (Their bill
board advertisements along the road declared ‘children stay free’; I
thought staying in a hotel (their first time) would offer them an interesting
contrast with the motel room we knew awaited us the next night at
our journey’s end.) We piled into the hotel, and while Gretchen and the
children freshened up after a day on the road, I sought a table at the

A grandmotherly hostess first informed me there was no room now and
no empty tables were expected till 8 in the evening. When I asked for
recommendations to other dining places about town — for my two hungry
children would not peacefully wait another hour for service — the woman
scratched a reservation from her list, making room for us.

Soon we were at table; the food was good and the variety quite
surprising. So even though Miriam was tired and refused to eat, the
meal had a festive sense for all of us for our various reasons. During
the evening we talked about the children’s sense of the project and some
of the amazing things they had done. I told Miriam how her addition of
96 plus 96 impressed me (cf. Vignette 100) and contrasted that with her
attempt to sum 89 plus 41 by counting hash marks 5 months earlier (cf.
Miriam at 6: Arithmetic). When I recalled that detail, Robby convulsed
with laughter. How could anyone attempt so absurd a procedure? I
asked Robby to think back, reminding him of the night he showed the same
response when I asked him to add 75 and 26 (Robby recalled having a late
pizza at the European Restaurant with our friend Howard Austin — Cf.
ADDVISOR, Logo W.P. #4). This reflection sobered him some. Miriam
piped up: “That’s a hundred and one.” “And how did you get that result?”
Miriam replied (to my surprise), “It’s like 70 [sic] and 20 is 95 and then you
add 6. 75 and 20 is 95 plus 6.” I was surprised because with those
particular numbers I thought Miriam might compute the result using a
money analogy. After assuring her of the correctness of her result, I
posed a different problem. “Miriam, suppose you had 75 cents and I gave
you 26 cents — say a quarter and a penny — how many cents would you
have?” When she responded “A dollar ten,” I asked where the extra 9
cents came from. Miriam computed for me in explanatory mode: “75 cents
is like 3 quarters and another quarter is a dollar. That’s a hundred
cents and one more is a hundred and one.” She denied her first answer
was a hundred and ten cents.

Note first that Miriam did not carry the result from one computation
to the second. Note further that although she applied directly her
decadal then unary algorithm for the numbers (75 plus 26), the same
numbers applied to money engage with a most minor variation the
well-known result that 4 quarters make a dollar. I can not confidently
explain the penny-dime confounding. I speculate that when not central,
they are not well distinguished. A dime won’t buy a 5-pack of bubble
gum and you can’t use pennies for anything but paying food taxes (cf.
Vignettes 54 and 55).

10/27 While waiting for the school bus this morning, I asked Robby if he
were doing anything interesting. He was enthusiastic about certain games
and said he liked especially the play time when the first graders come to
play with his class (3rd grade). I asked if they ever did any academic
stuff — TIMES problems and so forth.

Miriam informed us both she knew how to do TIMES. She argued her
point concretely: “Four twenties are eighty.” I laughed and reminded
her that I was driving the car yesterday while she and Robby discussed
that sum in the back seat. She protested, “I can do it.” “You can do
4 times 20. Can you do 4 times 90?” I challenged her. Robby knew and
said the answer. Miriam complained to him and walked down the driveway
kicking leaves. She returned. “The answer’s 3 hundred and 60.” Robby
claimed credit: “I told you first.” I argued that having the first
result was not so important, that what matters most is having an answer
you can understand yourself. Miriam said, “Can I tell you how I figured
it out?” and proceeded to do so: “I had a hundred eighty and a hundred
eighty. I took the two hundreds and one of the eighties. That’s 2 hundred
eighty. Then I took away 20 from the other 80 and I had 300
with 60 left over. 3 hundred 60.” I congratulated Miriam on good execution
of a very complicated computation and wished both children a good
day as the school bus came to rest where we waited.

These notes close off my informal observations on Miriam’s computational
development. Miriam shows herself clearly in command of com-
plicated procedures for mental arithmetic, as witness her computation
of 4 times 90 with her decadal additive procedures and their integration
with unary adding. The contrast of computation performed on numbers and
money document the interaction of computation and microworld well-known


Vn106.1 Tic Tac Toe and Nim 10/22/77


Miriam’s Tic Tac Toe play shows an opening game played only with Glenn before and some surprising rigidity. When we play a subtraction arithmetic form of Nim, Miriam adduces “going second” as the efficient cause of her winning game 2. This appears to be as a consequence of our playing with hexapawn; this idea — I call it a vanguard issue — appears to be one Miriam has become sensitized to and is trying to fit into other microworlds.

Vignette 106, page 1 Scanned from Original Fair Copy

(click to enlarge scanned image; back-arrow to return here)
Vn106-1 scanned; no digital source available

Vignette 106, page 2 Scanned from Original Fair Copy

Vn106-2 scanned; no digital source available


Vn107.1 Self-Understanding 10/22/77

My cousin’s wedding has been a day of reconciliations, of growing
closer to family from whom I had been long and much estranged. After a
late breakfast, we attended the wedding. I felt proud of Robby later
when he told me the nicest part of the wedding was a piano-organ duet
(‘Jesu, Joy of Man’s Desiring’) even though my engagement was other.
As I later told my cousin, the groom, in a scene reminiscent of the end
of The Madwoman of Chaillot wherein I stuttered several times
then spoke clearly, I came to bear witness that marriage and paternity were
the two great blessings of my life.

At the reception, as we arrived early I took a table for 8 and then
asked my brother, his family, and my father to join us four. There, and
at a later party for the immediate family, we spoke much with Dave (my
brother) and his wife. As their daughter has gone through school they
have become appalled at the quality of the “education” to which she has
been subject and indignant at the pretense of knowledge ignorant
teachers make. (We spoke freely because I told them my difficulty in
foreseeing an academic future was that I could not endure the pretense
of knowledge with its implicit deceit and manipulation of other people
that the professorial stance systematically demands.) I explained to
them parts of our newly completed project: one of our goals was to render
a child more articulate, to give a child better control of his own
mental procedures and knowledges.

Miriam was playing chase outside with Robby and Peter (a second
cousin, her junior by nine months). When Peter last tagged her, he hit
her in the back of the neck and pulled her hair (thus her story goes).
I found Miriam outside, sobbing and very much out of breath. I would
have judged she needed a dose of her wheeze-suppression medicine at
that time. I loaned Miriam my handkerchief and speculated that his
unkindness had been an accident, or perhaps a thoughtless act, but
surely not a mean one directed at her as a person. Inside, my brother
sat down with Miriam, who was still wheezing heavily, in an out-of-the-
way place. As he subsequently related their conversation to me, Dave
told her of his severe childhood asthma, a difficulty he suffered when
the practice was less sophisticated and medications fewer than today’s:
he had found that through conscious effort, he could stop an impending
asthma attack, bring his breathing and his emotions under sufficient
control that his bronchi could recover from the particular assault they
suffered in a given incident. Miriam tells me they made friends. Dave
said if Miriam comes to visit him, she can play in the large playhouse
he made for his daughter (almost 7 years Miriam’s senior) and could
watch for the deer which visit at his four apple trees.

Later in the evening I accosted Miriam outside. She was again
breathing heavily, engaged once more in a game of chase with the two
boys. “Come walk around slowly with me.” When Miriam refused, I
pointed out how she was breathing so heavily and that I didn’t want her
to end up wheezing. She explained to me, “Daddy, I have a very good
trick, to stop it when I have trouble breathing.” “How’s that?” I asked.
“I just think about it [pointing to her head], and after 5 minutes, or
maybe even 15, I won’t be breathing so hard.” I left Miriam playing tag.

I reported Miriam’s reply to my brother, who said this was
substantially the advice he had given her and filled in the information
I noted previously. Dave remarked further that he didn’t really under-
stand my description of our project’s work at Logo but volunteered the
judgment that he had never met so young a child so well able to under-
stand the idea of controlling her own processes.

This incident reports one example of how Miriam’s work on this
project has developed a perspective on self-control which may be
profoundly valuable for her in an entirely separate area of her life —
controlling her allergic reactions.

Some more detailed notes. My brother is an engineer, not an
educator nor a psychologist, so his exposure to young children is limited
to his daughter and her friends. His daughter is in her school’s pro-
gram for ‘gifted’ children, which fact I cite as witness that he is used
to having a bright girl child around. Further, he is a design engineer
for microcomputer-based milling machine control systems; by this I imply
that he is used to thinking in terms of procedures and control.

I would not claim that Miriam understands herself in the profound
sense of placing herself coherently in her world. It is clear she can
talk with and comprehend the ideas of a mechanistically-oriented but
sophisticated 40-year-old engineer in his attempt to explain what he
views as a milestone of self-understanding. It is very likely that her
ideas of herself in this respect are influenced by our work at Logo (cf.
Vignettes 87, Turtle Tactics, and 88, One or Many Minds). It might be
more direct to say that Miriam can establish a theory of herself as an
object. (For a discussion of whether that is a good thing, see Vignette 81,
Imitating Machines.) If one criticizes a culture or subculture for
leading people to think mechanistically about themselves, one criticizes
an approximation to the actual human condition — and are not approximate,
wrong theories a first step toward the truth? Contrast a theory I might
impute to Miriam, wherein she sees herself partially as a coughing robot
who can be commanded to stop (by another agent’s insistent
will), with an alternative conception — that of a small creature wakened
in the dark of her bedroom at midnight by coughings which fall her way
through ill luck, whom nothing can help. The wrong, mechanical theory
may be the lesser evil.


Vn109.1 Tic Tac Toe 10/4/77

These 5 games are revealing of Miriam’s knowledge and ignorance both. Game 2 reveals more of my failings than I am happy to admit, but its contrast with game 3 permits a central revelation of her thinking about tic-tac-toe. These two together show by how much good fortune (when it occurs) is preferable to good planning. Throughout this session I prompted Miriam to think out loud and make predictions, hoping that she would thereby illuminate her representation of the game. The consequence is evidence how well articulated is her knowledge of what she does in specific cases.

Game 1: Miriam moves first (letters)

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

[after Miriam’s opening] I’m going to ask you some questions. Will you answer them?


[placing 1] Can you beat me?

Think so.

Go ahead.

[moves B]

Do you have me beat already?


Can you show me how?

If I put one here [at ‘C’], I’ll get two ways to win. . . 3 ways to win. One [B – C], two [A – C], three [A – B].

Can I go anywhere to stop you from getting those?

I don’t know.

Suppose I go up here [at’C’]; could you still beat me?



[places her index finger on 2]

Ah, yes. The way things are [gesturing from A to B], do I have a forced move? . . . So I have to go here [moves 2], and you still get two ways to win.

[moves C] C. Go!

Go, huh? Hum. All right [moving 3], all right.

[moving D] D! [pointing to C] You know why I went there?


If I went here [pointing to 3], you would put yours down there [pointing to C].

That’s right. I guess you had a forced move too.

Yeah [agreeing that such was her reason]. Yay! I win!

Game 2: Bob moves first (numbers)

After Miriam’s center response, I realized I was myself so unfamiliar with games of this opening I didn’t have any specific plan to follow. I was confused and not wanting to keep Miriam waiting, moved aimlessly at 2. The game thus becomes pointless but does exhibit Miriam’s defensive play without confusion by any aggressive objective (hers or mine).

         c  | 2 | b    
         1  | a | 5    
         3  | d | 4  

Game 3: Miriam moves first in the center

         3 | D1 | 2    
         1 | A  |
         B | D2 | C  

If I move here (1), can you beat me?

It will be sort of like the same game.

Same game as what?

The last game. Go!

You think it will?



[moves B — after hesitating and moving her hand between corners B and 2; laughs]

Let’s see. I have a forced move now [moves 2]. How do you figure out where to go next?

I just pick a space [moves C].

Why is that a good space?

I don’t know.

You have no idea?

I just pick a space.

Why don’t you move here [pointing to side opposite 1]. I think that would be a good place.

Nahh. I want to move there [pointing to C].

Is there any reason?


You just don’t want to tell me. Here. . . . I’ll stop you [places 3] along your way to win there.

[quickly moves D1 between 2 and 3]

Did you block me?


‘Cause you thought I had a way to win?

Yeah [it’s obvious] 3 and 2.

That’s right. I had a way to win. Do you think it’s better to block somebody who’s got a way to win or do you think it’s better to win yourself?


Do you think you have a way to win?


May I call your attention, Miriam, to a way to win you could have had? [points to D2]

[moves D1 to D2]

That’s why I asked so many questions. I wanted to know if you knew you had two ways to win.

No, I didn’t. . . . Tic-tac-toe, three in a row.

Miriam became angry when I argued her victory ‘didn’t count’ since I had to show it to her.

Game 4: Bob moves first

This game shows Miriam’s confusion of move 2 in a game of form VII-B (the only safe response to a corner opening) with move 1 of game form IV (cf. Learning: Tic-tac-toe ). This is an explicit example of configuration dominating to the exclusion of serial information.

         1 |    |     
         b | a2 |     
         2 | a1 |   

Can you go any place at all so I won’t beat you? If I move in the corner [moves 1].

One place.

Is there a safe place? Where is it?

[moves a1]

You believe that’s a safe place?


Well. . . shall I prove you wrong?


[moves 2] What now?

[making forced move b] Hold it. I want to have him. [cheats: she moves a1 to a2]

That’s not fair. You moved here [removes a2 to a1].

No [replaces a2 in the center].

Let’s back off, then, if you don’t want to play that game.

Game 5: Bob moves first (restarting game 4 with his opening marker at 1 and Miriam’s at a1)
RESET the figure in a sensible fashion

         1 |   | b3          1 |   | b3   
           | a1|               | 3 |  
         b2| b1| 2           a2|   | 2 

Let’s say you moved there [a1] right off. If I move here [2] what do you do? Can you move any place?

[removes a1 to b1]

Miriam! That’s just not fair.

[reluctantly replaces a1]

Now, where can you move?

I know [moves b1].

[pointing to b2] Why didn’t you move there?

Good idea! [moves b1 to b2]

O. K.? You want to do that?

What? [moves b2 back to b1]

Go ahead and move here. I’ll show you what I’ll do.

[moves b1 to b2] Win?

Take a look at my chances to win.


Do my chances to win come together?


[gesturing to the fourth corner] No?

[grabbing Bob’s hand] No! [moves b2 to b3]

You think that’s a good defense?

[laughing, points to the empty space b2] Here.

Yes, they do.

[moves a1 to a2 as in the second frame] No. I didn’t want you to go there.

When I move 3 in the space just vacated, Miriam sulks and we give up tic-tac-toe for another game.

Miriam exhibits her extensive and flexible command of games of the form of game 1. Her comment, after the opening two moves of game 3, that it will probably be the same as game 2 renders explicit the absence from her thinking of the concept of move order variations as significant in tic-tac-toe. I consider it staggering that anyone could play so well as Miriam does and yet not have a well formulated idea of opening advantage. Game 3 also appears to show a game whose play has (may be interpreted as having) degenerated to a serial procedure with loss of an original, configuration-oriented forking objective. Game 4 shows strong confusion between the 2 move of game VII-B and 1 of game IV. These games permitted no show of table-turning because Bob never clearly won any games.


Vn113.1 Steady State 12/8/77

A few nights ago, Miriam approached me: “Dad, why do we have to
spend 6 hours in school every day?” “Why do you ask?” I countered.
Miriam continued, “It sure is a long time.” When I first asked what
was the problem, the answer came back that the work was too hard, there
were so many math papers to do, and so forth (but note that Miriam’s
work of choice at school is doing math papers; Cf. Addenda 112 – 2, 3).
Finally Miriam said, “It’s just boring.” And then, “Do I have to go to

Two years back, I recall Robby asking if he could quit school at
the end of 3 months in the first grade. He argued that he knew how to
add and had learned how to read and that there was little more the schools
could teach him. Miriam’s position is the same. I told her she can stay
home from school any time she wants except on certain days when Gretchen
and I might both have to be out of the house — and that this would be
the case especially when the baby is due. Beyond giving that permission,
I offered a little advice of this sort. “School may be boring, but you
will have friends to play with there. It can be boring at home as well;
while I’m working I won’t be able to play with you as much as you might
like, nor will I be going over to Logo too frequently.” I offered to
take Miriam to Logo whenever I go there, either going over after school
or telling her in the morning of my plans.

Since that conversation, Miriam has several times declared she was
not going to school. She stayed in bed, and I didn’t argue or disapprove
at all. All those times she subsequently changed her mind, got dressed
in a rush, and hurried out to await the school bus.

Recently Miriam has learned two things at school she values. The
‘academic’ learning is that there are 2 sounds for the A vowel. She
knows one is long A and the other short A and that the first is
distinguished by its spelling with a terminal silent E. Her example of the
distinction was the couple HAT/HATE. She was not too interested when
I suggested we play with the voice box at the lab to make it talk with
long and short vowels. Miriam comments that she can’t remember learning
anything else besides the spelling of a few words — and one important

The student teacher of her class taught Miriam how to twirl a baton.
Baton twirling first engaged Miriam’s interest in kindergarten when her
friend Michelle brought hers to school. At Miriam’s request, I bought
her one which she has played with discontentedly since then. After her
one day’s instruction, Miriam has marched, posed, and practiced before
the glass doors of our china closet, declaring herself a “batonist” (a
word she is conscious of having made up.)

At Logo, too, Miriam’s current interests are primarily physical
skills. She plays with the computer (Wumpus, and lately some new facil-
ities I’ve shown her) but her first choices are the hula hoop or jump
rope. An incident occurring last night gives evidence of what may be
the outstanding consequence of her learning during The Intimate Study —
what I refer to is her sensitivity to instruction and advice couched in
procedure-oriented terms:

Miriam had convinced Margaret Minsky to turn a long rope for
Miriam’s jumping (the other end being tied to doorknob). Miriam tried
hard and long to jump into an already turning rope. She attended
carefully to the rope and at the right time moved toward the center —
but only a short distance in that direction. In consequence, she got
her head inside the space, but the turning rope regularly caught on her
arm. Miriam had no good answer when I asked if she could recognize the
specific problem. I asked if she could take some advice and said she
should jump onto a line between Margaret and the doorknob. Miriam could
not. I put a paper napkin on that line — but the turning rope picked
it up and away. José Valente drew a chalk line. Miriam took the chalk
and drew a box to jump into. Now she was ready.

Miriam’s first attempt failed because she jumped into her box with-
out attending to the rope. Then she regressed to watching the rope and
moving only a little. Finally, “Miriam,” I said, “you’ve got a bug in
your SETUP procedure. You’re doing only one thing at a time. You have
to do both things at once.” On her next try, Miriam jumped into the
turning rope successfully. I did not see her thereafter exhibit either
of her two earlier bugs (too little movement or not watching the rope).
This incident occupied about 3 minutes.

Miriam finds school boring, but not depressing. Though allowed to
stay home, she goes to play with her friends. Of most immediate and
spontaneous interest to her are physical skills. She shows herself
very capable of using advice formulated in concrete terms focused on
separate procedures.


Vn114.1 The Game Goes Ever On 12/28 & 29/77


In the first incident, Miriam invents the idea of opening advantage for “Tic Tac Toe two in a row.” I believe this is connected to her introduction to Hexapawn (a pawn capture gain played on a 3×3 board) as a reduced form of chess, and my invention of “half a game” of checkers as a reduced form. This invention of Miriam’s is a significant advance whose development I will follow in its application to Tic Tac Toe Three in a Row (cf. Home Session 20, Tic Tac Toe Finale).

Miriam’s defeating the Children’s Museum computer brings her back as master to her point of engagement with the game.

Vignette 114, page 1, scanned from Original Fair Copy

(click on the image to englarge it; back arrow to return here.)
Vn 114-1 Scanned Original Fair Copy

Vignette 114, page 2, scanned from Original Fair Copy

Vn 114-2 Scanned Original Fair Copy


Vn117.1 Playing Both Sides 1/14/78


These data establish that when Miriam plays against herself she does not use the games merely for easy satisfaction of a certain win. Game 3 is especially striking in that it shows the unusual side opening Miriam rarely used before the table-turning sequence for the game of form X she initiated in Vignette 115, Tic Tac Toe Finale.

I consider it a reasonable speculation that during Game 1, she formulated for herself a perspective on gamaes of form VII that permitted her articulate description of the distinction between games of VII-A and VII-B she exhibited in Game 5.

Vignette 117, page 1, scanned from the Original Fair Copy

(click on image to enlarge it; back arrow to return here.)
Vn 117, page 1, scanned from Original Fair Copy

Vignette 117, page 2, scanned from the Original Fair Copy

Vn 117, page 2, scanned from Original Fair Copy

Vignette 117, page 3, scanned “game-side” of 3x5card

Vn 117, page 3, Miriam's games against herself


Vn121.1 Double Perspectives 2/8/78

While school has been canceled this week due to the Blizzard of ’78,
the children have spent a lot of time outside, playing on the snow
mountains the plows and people have piled up. Inside much of the time,
they have followed their own inclinations, playing the card game War,
reading Gretchen’s collection of Pogo and Peanuts books, drawing and

Miriam has told repeatedly her most recent joke.


What letter of the alphabet do you drink?

I don’t know.

T. . . . T, E, A, get it? Tea.

In her turn, she has had to suffer our variations of her joke. A second
group of similar jokes is expressed in drawings Miriam made for me and
Robby. They are like puns in that the gift is coupled with a request
that you “find the hidden picture.” (Confer Addendum 121 – 1).

In the first picture, “the hidden picture” is a whale, underneath
the house, whose eye is formed by the ‘O’ of ‘TO’. When I asked how she
ever came to make such a picture, Miriam replied, “After I drew the hill,
I looked at it and saw it looked like a whale.” I surmise that the
whale’s mouth and tail fluke were later additions.

Subsequently Miriam made a gift for Robby, swearing me to secrecy.
(Confer Addendum 121 – 1). “The hidden picture” is once again a whale,
but rendered less incongruous by his rising under the boat. The whale’s
mouth says ‘TO ROBBY’ and his eye, pencilled in, has been covered over
by blue coloring both ocean and whale. The theme of sea warfare is a
direct catering to Robby’s taste.

The seeing of some entity from two different perspectives is an
activity that is forward, a vanguard issue, in different areas of
Miriam’s concern, as documented here and otherwheres. It strikes me
I might help foster her understanding of carrying by posing for her
the problem, “What number is ten when you take it away and one when
you add it in?”

Addendum 121 – 1

Find the Hidden Picture

Vn 121-1 Hidden Pictures