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Peggy Study, Panel PF17, P275

Themes: Counting and Objects, Family Talk, Reading and Moving
Source: (Lawler); date: 4/24/83

Title: Peggy at Five and a Quarter; late session in Paris
Text commentary: number knowledge; thinking about a return to the USA.

PF17A European Coins, 40mb

PF17B Object Geometry, 26

PF17C Family Relations, 20mb

PF17D Reading & Moving, 9mb

PF17E1 Number Knowledge, 22mb

PF17E2 Number Knowledge, 35mb


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


Vn45.1 Going Home 7/15/77

When today’s Logo Session (#39) and errands were finished, we
hurried home to pack up provisions for a 2 week vacation in Connecticut.
The house is empty between tenants, and since we are renting it
unfurnished, it IS empty. What did the children expect of this vacation?
What did they look forward to? And how did they first react to going

Both have looked forward to the trip. The outstanding feature of
our home is lakes and two beaches a few hundred feet away. Learning
how to swim was an activity both talked of with anticipation. Miriam
asked me to commit myself to spending time with her several places:

the playground at the Guilford Lakes School — Miriam said
specifically that she wanted to use the rings where she had learned to
skin-the-cat last year, noting she would be able to do it much better now.
the playground at Jacob’s Beach — this town beach at the Guilford
Harbor on Long Island Sound has swings for babies, tots, and adults,
and small and large sliding boards; from the top of the larger you can
look over the harbor and town dock and watch sailboats and water-skiers
out on the Sound. Miriam remembered as a primary description of the
playground another piece of equipment, a large metal cross with a sit-
upon animal at each end (Elephant [her favorite], Pelican, Turtle, and
Hippo). These four seats are centrally supported by springs which permit
motion vertically with small excursions of rotating and twisting.
Great Hill — this names a specific section of the road from
Miriam’s nursery school behind a hundred foot bluff and down to Lake
Quonnipaug. The road drops and twists quite rapidly and was thrilling to
follow in my old MG.

Notice that Miriam’s focus was on places, whereas Robby’s primary
interest was to play again with his friend Raymond. Miriam has friends
in Guilford (Scott, Toddy, and Sarah are three from nursery school;
Karen and Lisa are girls she liked and played with while at the baby-
sitters’) but her interest did not focus on them. This focus on
places where she had done things may be an artifact of her leaving
Guilford soon after turning 5, before developing the close sort of
attachment Robby shows to his friends.

Upon arrival we unloaded our portable goods into our empty home.
We found in the basement objects of ours and experienced a delight of
repossession. Miriam was obviously as happy to find the mattress from
her crib (which she slept on for 2 weeks) as I was to restore above the
hearth the motto I burned in wood upon first occupying the house —

	I built this house with my own hands 
	And needed helps of friends
	Memento be -- a friendship house --
	Past days when friendships ends.

These notes document some of Miriam’s expectations for
the 2 weeks’ vacation at our house in Guilford.


Vn46.1 Rotten Hints 7/19/77

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

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

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

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

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

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


Vn47.1 Losing a Tooth 7/20/77

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

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

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

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


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

(Surprised and a little disappointed) Oh.

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


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

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

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

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

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


Vn49.1 Finger Counting 7/24/77

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

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

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

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


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

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

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

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

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

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

Addendum 50-1

Addendum 50-2


Vn51.1 Paper Ships 7/25/77

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

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

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

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

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

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

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

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

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

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

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

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

Addendum 51-1

Vn 51-1 Curious George paper folding

Addendum 51-2

Vn 51-2 Curious George Paper Ship procedure


Vn52.1 Fort Griswold 7/26/77

From many places in Boston when you look to the north you can see
the obelisk on Bunker Hill rising above the buildings of Charlestown.
There is a similar but smaller monument in Connecticut, at the site of
the lone major battle of the Revolutionary War that took place in that
state. The dedication of that monument reads:

In memory of the Brave Patriots
who fell in the massacre at Fort Griswold near this spot on the
6th of September A.D. 1781
when the British under command of
burnt the towns of New London and Groton and spread
desolation and woe throughout this region

This monument commemorates the cost of war and its butchery. The
British commander, Major Montgomery, was slain by bayonet in the final
charge over the ramparts. When the British controlled the fort, the
officer in charge asked, “Who commands this Fort?” The rebel commander,
Colonel Ledyard, tendered his sword and said, “I did, Sir, but you do
now.” He was slain on the spot with his own sword. This is a place
where one can not escape reflecting upon Thucydides’ epitaph for the
Athenians defeated at Syracuse:

Having done what men could, they suffered what men must.

To this historic place I brought 3 children: Robby and his friend
Raymond and Miriam. After climbing the 166 steps of the monument, we
could see up and down the coast line, and in the Thames River (/thæmz/
not /temz/) below a Nautilus class nuclear submarine. I pointed south
to the green hangar of Electric Boat, wherein are produced most of
America’s nuclear submarines, where 2 years ago we had witnessed the
launching of the submarine Philadelphia. Back at ground level we found
in the museum the essential toilets, interesting uniforms, muskets,
dolls, and a model of the Fort itself as it was on the date of the

What remain now of the fort are earthworks, scattered stones, and
an ammunition dump close by the old cannon emplacements overlooking the
harbor. The children raced over the ravelin and in the main gate, past
the plaque commemorating Colonel Ledyard, to the dungeon and the secret
passage through the south rampart. Raymond and Robby scaled the ramparts
and ran about in the moat. Miriam complained at being left behind
as they refought the battle in their imaginations.

Another submarine came gliding down the Thames. Miriam asked me
about the funny smoke stacks. Robby explained what I did not know:
they were underwater launch tubes for Polaris missiles. In the middle
of their morning’s play Robby asked, “Dad, who won the battle of Fort
Griswold? The British or the Americans?” I answered that the British
had defeated the Americans. Robby turned to Raymond. “We’ll be the
Americans the British didn’t defeat.” And they resumed their play.
Recall the rebels’ song in which Ireland is a land

. . . where the past has been lost and the future is yet to be won.
Recall not only the songs, but also the bombs and deaths of innocents.

We bought sandwiches from a shop I knew and divided them in Ledyard
Park. The children played after on one of the best equipped playgrounds
we’ve visited. Miriam ran to ride on the spring mounted toy elephant
and cried, “Daddy, now I know where they put them.” (Cf. Vignette 45)
Miriam assumed her favorite playthings, missing from the town beach in
Guilford, were transported to the park in Groton. In this playground
on can see the impact of EB’s Triton submarine contract. Not only has
Groton been immune to the 2 year real estate depression which plagues
south central Connecticut, but even the parks and playgrounds show the
presence of armament money. As Aeschylus observed, Plutus is the
ultimate god of war, who changes men for gold.

Raymond decided we should visit the Mystic Aquarium instead of the
beach at Rocky Neck. Aside from the sharks in the central tank, only
the trained animal show held interest for the children. Miriam liked
best the high jump of Sassy the dolphin (20′ above the water surface)
and recalled seeing a similar show the summer before. We went home by
way of an ice cream parlor and finished the day with a swim at the
beach of our lake. Raymond, aged 9 and taking regular lessons, swims a
little. Robby appeared to be inspired by his example. Miriam continues
to imitate the splashy surface appearances of swimming but can’t
penetrate the appearances to the activities unseen.

This vignette recounts a day of our vacation presented more from
my viewpoint than most others in this series. It may indicate how the
children and I can simultaneously enjoy different aspects of one situation.

Fort Griswold – 1

Vn 52-1 Fort Griswold

Fort Griswold – 2

Vn52-2 Fort Griswold

Fort Griswold – 3

Vn 52-3 Fort Griswold


Vn53.1 A Birthday Party 7/28/77

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

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

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

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

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

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

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

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


Vn54.1 More Arithmetic 7/31/77

While I was having lunch, Miriam came (in her bathrobe
from the tub) and asked me if I could divide 79. I asked by
what. Miriam didn’t answer but moved away. I pursued the
question: “What do you know that’s a 79?” Miriam replied,
“They said it’s 2 for 79, the quinine water, and I want to
know how much it costs.”

Bob You want to divide by 2. Can you divide 79 by 2?
Miriam No.
Bob Can you divide 80 by 2?
Miriam (long pause) I’ts 40.
Bob That’s very good, sweety. You estimated 79 by 80.
But 79 is a little less than 80; it’s less by 1.
Can you think of a number that’s a little less than
40, but not by 1?
Miriam 30.
Bob That’s less than 40. 79 is 1 less than 80. 39 is 1
less than 40. So your result should be more than 39.
Miriam 48?
Bob No. The correct answer is 39 and a half.
Miriam A half?
Bob Yes.
Miriam That’s silly. How can you cut a penny in half?
Bob You get two halves.
Miriam No. . . . 39 and a half. . . . I guess the rest must be tax.

Later in the day, Miriam complained to me that we hadn’t
done any adding for a long time. She griped further that when
on vacation in Connecticut we hadn’t done any experiments; we
hadn’t done the puzzle I bought; we hadn’t played with the set
of blocks I took with me. I defended myself against the charge
of sloth by arguing that I thought she needed a rest, that I
feared I had been pushing her too hard with experiments. Miriam
went off, still disgruntled. She stopped by a set of addition
exercises left over from Robby’s second grade.

A little later Miriam asked, “Daddy, is 28 plus 48 equal
to 76?” I congratulated her and asked how she had ever fig-
ured it out. Miriam explained: “I know that 2 plus 4 is 6.
So this is like 20 plus 40 and that equals 60. Then I took
the 8 and said 68 and counted (demonstrating on her fingers)
69, 70, 71, 72, 73, 74, 75, 76.”

I told Miriam she had developed a very good procedure and
asked her not to do any more of Robby’s problems until the
evening, at which time I promised to do some adding with her.

This vignette shows Miriam extending her way of thinking
about adding to problems she feels she should be able to solve
(given her former successes with much larger numbers). The
first incident also evidences the concreteness of her thinking
about numbers, i.e. division of 1 into halves is impossible
because it’s pennies that are being divided. Further, the cloak
of the mystery of “tax” is thrown over the ambiguous conclusion
of her computation.


Vn55.1 About Taxes 8/1/77

Since the beginning of the High School Studies Program,
the children and I have come to Logo to use the system from
8 to 10 am. One consequence is that we occasionally skip
breakfast. Even when we do not, the children have become
accustomed to mid-morning snacks. The favorite: apple pie
and milk.

At their young age, Robby and Miriam get money from me,
and we talk about how they spend it. A piece of pie costs
59¢. A half pint of milk is 32¢. So Miriam told me this
morning, and these figures are familiar. After her adding
(cf. Vignette 54) 28 and 48, as we got her snack I asked her
how much we would have to pay the cashier. After a few mis-
calculations, she came to a sum of 91¢ and seemed confident
it was correct. I congratulated her on a correct sum and
asked the cashier to ring up our tab. 92 cents!

92 cents? I asked the cashier to explain. She said the
pie is 55¢ and the milk, 30¢ thus 85¢ and the tax 7¢. “See.
Look at the table.”

I am at a complete loss as to how to explain this to
Miriam. Not only is the 8 per cent food tax dreadful in it-
self, it is rendering incomprehensible a primary domain of
arithmetic that children regularly confront — paying small
amounts of money for junk food.

This incident clarifies Miriam’s comment in Vignette 54
wherein “the tax” appears to be the difference between what is
a reasonable computation and what you actually have to pay
somebody to buy something.


Vn56.1 TicTacToe 7/19/77

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

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

After Miriam’s move C:

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

Game 2: Bob first

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

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

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

Game 3: Miriam first

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

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

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

Game 4: Bob first

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

After Miriam moves A:

B You have frustrated my tactic.

M [laughs]

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

M I always like to frustrate your plans.

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

M [moves B]

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

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

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

Game 5: Miriam first

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

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

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

M [moves B]

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

M Yeah.

B I will frustrate your tactic.

M How?

B I will put my 2 here.

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

Game 6: Bob first

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

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

M [moves A]

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

M What?

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

M [moves B]

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

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

B You tell me where to move.

M Here.

B Shouldn’t I make a forced move?

M Unh-uh.

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

M Yeah.

B O. K. [moves 4]

M [moves D]

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

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


Vn57.1 Desserts 8/3/77

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

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

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

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


Vn58.1 Owning an Angle 8/4/77

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

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

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


Vn59.1 Air Conditioning 8/6-11/77

8/6 Logo came into our conversations twice this day at lunch. When
asked if she knew what a palindrome was (cf. Logo Session 39, 7/15/77),
Miriam offered two examples: ‘mom’ and ‘I’ (I checked that she did not
mean the word ‘eye.’) Miriam later said she would like to sleep at
Logo. I recall having told her that I once slept at Logo (when last
winter the city suffered 22″ of snow and I lived atop Corey Hill).
Miriam’s request was justified by her hope to sleep better there than
at home. She explained that she had slept well in Connecticut and had
slept very ill since returning to Massachusetts. During the heat wave
of mid-July, we had run the air-conditioner regularly. Miriam believed
she would sleep better in the air-conditioned computer room.

8/7 Miriam woke me at 4 am (a fairly regular occurrence) with her
coughing. Despite having had her standard dosage of medicine she was
wheezing. Because I believe it is important she not conclude that her
malady is hopeless, beyond remedy, I asked if she would like to go to
Logo. We left home at 5 with her pillow and medicine, and tape recor-
dings from my transcription backlog. We drove through a deserted city
to sign in at 5:30. Both wide awake, we walked through the lab. Tom
Knight was using the terminal at the mainframe, so we assumed Logo was
unavailable and found other entertainment. Miriam first set up her
pillow in a chair (the king size pillow barely left room for her) and
showed me a peculiar book she had found in the Children’s Learning Lab
(Ça Ne Va Pas, Charlie Brown). Miriam asked me to read it. I read her
a few frames from the first cartoon. A better resting place was needed.
We brought a bean bag chair to my office (Miriam preferred that option
to sleeping on Seymour’s or Hal’s couch). She curled up with her pillow
and the cartoons in the corner. Her last words before dropping off to
sleep: “Daddy! I can read ‘The Doctor is in’.” Miriam slept from 6
until 11. Her nap of 5 hours was the longest uninterrupted sleep she
has had since our return to Boston.

8/9 Miriam told me this morning she had had a good night’s sleep, her
first in a week. When I mentioned this to Gretchen in her hearing,
Miriam qualified the statement by “besides sleep at Logo.”
After her bath this evening Miriam stood at the balcony over the
court yard and said, “Hey, I see the first star:

Star light, star bright,
First star I see tonight,
I wish I may, I wish I might,
Have the wish I wish tonight. . . .

I wish I had ten more wishes.” Thus well provided with wishes and still
talking to herself, she made her first real wish: “I wish I had no
allergies at all.” Then her friend Scurry should get a new collar and leash.
I told Miriam those 2 wishes could happen, but the first could not, that
she would continue to suffer from her allergies into her teens, at the
end of which they might become less severe.

8/10-11/77 After calling those who advertised air conditioners in Tech Talk
and waiting to find out none would fit in the windows of Miriam’s room,
we purchased and installed an air conditioner in Miriam’s room. It is
not at all clear that air conditioning Miriam’s room will help her in
any physical way. It is most important, however, that she not feel
alone in confronting her problems and that we will attempt whatever
reasonable means are available to ameliorate her discomfort.

These notes may indicate how profoundly burdensome to Miriam are
her allergies to dust, trees, and mold. August and early September are
the worst times.


Vn60.1 Surprise Party 8/8/77

Spoiled by living in the air-conditioned comfort of our Connecticut
home during the mid-July heat wave, when the next spell of hot weather
found us in the hot air heated loft of our Boston carriage house little
persuading was needed to induce Gretchen to join Miriam and me at Logo
yesterday. With the hot weather continuing and both children expecting
to do an experiment this morning, it was a natural consequence that
Gretchen should join us at her later convenience, bringing lunch if she
so chose, and plan to spend the afternoon at the lab.

We three gave Gretchen birthday presents, wished her happy birth-
day, and sped off to our morning’s work at Logo. As we drove across
town in the MG, I broached the idea of a surprise party with the chil-
dren. They were as enthusiastic as I was and far more certain that it
would work out.

We completed our morning’s experiment, enjoyed together the lunch
Gretchen brought a little later, and settled down each to his afternoon’s
occupation: the children browbeat Margaret Minsky to carry them around
and played at frisbee with the students of the HSSP; I worked at data
transcription; and Gretchen read a book newly selected from the library.
I had alerted a few friends and hoped others would drop by the lab in
the afternoon. Since the children and I planned to get an ice-cream
birthday cake, we had to concoct some plausible excuse for the three of
us to ride off leaving Gretchen behind at Logo. My script’s argument
called for moving the MG from a block away to the Tech Square lot to
render easier carrying down to the car the remains of lunch, my recor-
ding equipment, and so forth. The children were to set up a cry in
their normal fashion that they wanted to go for a ride with me.

Our little ruse worked a little bit, for Gretchen surely knew it
was her birthday and the children kept approaching me to whisper, “Is
it time to go get the cake?” The circumstance that gave away the secret
was unforseeable. We moved the MG at 3 o’clock, thereby escaping the
earlier ban on cars without the appropriate parking stickers. Gretchen
said her car was parked on the street right in front of mine and she
should walk along to move hers also. I tendered some completely inade-
quate reason for not doing so, and Gretchen was sufficiently insightful
not to push the argument.

We picked out a cake at Baskin-Robbins. Robby held the cake on
the way back (the privileged function) and Miriam rode in the boot (the
seat of choice). We gathered a collection of dishes, forks, and friends
and sprung our surprise on Gretchen. She was pleased.

As is the case with most Logo parties, as many people were absent
as present; the place seems sometimes a crossroads in the paths of
over-committed people, but Andy, Donna, Margaret, Marvin, José, and the
children and I met the challenge of consuming Gretchen’s birthday cake.

This vignette shows the children in preparing a surprise birthday
party. This informal party was more or less typical of those at Logo
in that the summer dispersion and other commitments kept the size
small and made the guest list a nearly random selection of people from
the lab.


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

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

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

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

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

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

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

Game 2: Bob moves first (numbers)

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

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

Game 3: Miriam moves first (letters)

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

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

Game 4: Miriam moves first (letters)

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

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

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

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

Game 5: Bob moves first (numbers)

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

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

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

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

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

Game 6: Miriam moves first (letters)

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

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

Game 7: Bob moves first (numbers)

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

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

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

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

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

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

Addendum 61-1

from Home Session 15

Vn 61-1 Addendum from Home Session 15


Vn62.1 Multiplication 8/7 & 11/77

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

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

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

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

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

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

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

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

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

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


Vn63.1 Another Birthday Party 8/12/77

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

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

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

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


Vn64.1 Jumping Rope 8/13/77

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

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

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

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


Vn65.1 Arithmetic Ripples 8/13/77

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

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

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


Vn66.1 Pre-History 8/16/77

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

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

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

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

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


Vn67.1 Think and Do 8/17/77

I asked Miriam today if she had read all the stories in the book
Friends, Old and New (the book used in initially assessing Miriam’s
reading ability in Miriam at 6). She thumbed through the book, said
she hadn’t read them all, but found the stories just too easy.

What does one believe? I asked Miriam if she had done all the
exercises in the Think and Do book (a set of companion exercises to
accompany the text). She had not, but picked up the book, announcing
that she would do some then, while I continued with my work. I asked
Miriam to write the date (8/17) on those exercises she elected to do.
Miriam did write the date on page 3 (in red pen) and did some of the
exercises there as well as those on pages 2, 3, 4, 6, and 13 (these are
integrated here as Addenda 67 – 1, 2, 3, 4 and 5).

Page 2 apparently tests the extent to which a child can inflect
verbal forms and recognize the significant lexical morphemic correlates.
Miriam did the exercise without error.

The exercises of page 3 ask the child to identify the initial or
final consonant sounds of items pictured. Miriam responds generally
with the initial and final letters in the spelling of the word, e.g.
‘w’ for wheel and ‘h’ for the th of mouth. Two other anomalies appear.
Miriam does not attempt (perhaps can not recognize) the peach and
appears to spell chain with a terminal e (‘chane’).

Page 4 tests reading comprehension (and possibly, recall). Miriam’s
conclusions are error-free.

Page 6 examines a child’s ability to select appropriate modifiers
in a context. Miriam made no errors.

The questions of page 13 make an interesting contrast because
Miriam got them all wrong. The question posed is, in which [picture
word] do you hear the vowel of it? Miriam apparently identifies the
sound of the ‘vowel of it’ as the sound of the letter name i rather
than the phoneme /I/, thus choosing ‘lion’ in preference to ‘pig’ and
‘fire’ in preference to ‘bridge’.

That speculation is not confirmed by the final 4 questions wherein
Miriam does not select ‘mice,’ ‘pie,’ ‘slide,’ or ‘knife.’ Her decisions
may be both spelling-based and based on imperfect spelling.

These data sample the language skills expected in school of some
one who can read at Miriam’s level. They show, not surprisingly, that
Miriam is largely ignorant of phonetics as it is taught in school.

Addendum 67-1

Vn 67-1 Think and Do pg.2

Addendum 67-2

Vn 67-2 Think and Do pg 3

Addendum 67-3

Vn 67-3 Think and Do pg 4

Addendum 67-4

Vn 67-4 Think and Do pg 6

Addendum 67-5

Vn 67-5 Think and Do pg13


Vn68.1 Continuous Quantity 8/18/77

As is the case with many who have a few fine things, we hardly
ever use them. Our silver and china are in some dark corner, our
Venetian glassware sits empty, hardly touched. At dinner our common
wine goes into common wine glasses. Through accidents at table and
sink, the usable collection has become one made of odds and ends.

This evening a guest joined us at a picnic supper on the patio
behind the house. Unloading supplies from the basket, I found (beside
the dinner meats) one jug of wine and three glasses of roughly these

the figures are on Addendum 68-1, original text of the vignette.

I placed the empty glasses on the picnic table in the order shown and
posed a problem abstractly to the children: “How can I be sure nobody
gets gypped when I pour the wine?” No response was forthcoming.

As I poured wine into the first glass, Robby cried out: “I got it.
Pour the same amount at both ends. Empty one into the center glass,
then refill the one you just emptied.” It was clear he meant refilling
the glass would result in its matching the first. Miriam concurred in
this solution.

Because the middle glass had a non-standard shape, as I followed
Robby’s procedure I arrived at a wine distribution whose appearance was
deceptive. There appeared to be a greater volume in the center glass
because its top circumference was greater than that of the two matching
glasses and its height was greater than both (since its cross-section
was more nearly conical than cylindrical).

the figures are on Addendum 68-1, original text of the vignette.

When finished pouring, I exclaimed, “You’re wrong, Rob. Look. The
center one’s got more in it. I’ll take that one.” (My overacting was
supported by a few gleeful chortles). When I then disbursed the
matched glasses to Gretchen and our guest, Miriam censured me: “Daddy,
you’re just being silly.”

Consider this anecdote as an informal post-test of Miriam’s
conservation of quantity. I do. I intend to introduce such ‘experiments’
into our everyday life as this project draws to a close. My purpose
is to reduce the testing burden Miriam will face by performing informally
those post-tests whose conclusions should be beyond question, without
rendering the evaluation sequence subject to the criticism of

Addendum 68-1

Vn 68-1 Original Fair Text of Vignette 68


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.


Vn74.1 The Light Brigade 8/27/77

Miriam, Robby, and I watched on TV this evening a movie about ‘The
Charge of the Light Brigade.’ The movie was an important one for several
reasons. Whenever the opportunity presents itself, I try to lead the
children in extending their sense of connection to other places and
other times. Since my grandfather’s grandfather rode with the Light
Brigade in their famous charge at Balaklava, this movie presented a
unique opportunity for a sense of personal involvement in remote events.

The movie was a pre-war (1936) romance with Errol Flynn cast as
Geoffrey Vickers (I suggested to the children that the character was
the grandfather of the Sir Geoffrey we had met at DSRE). The first 90
minutes of the movie was unalloyed “mush.” One incident stood out: the
massacre of the garrison at Chokoti by the Suranis Indians. Miriam had
been content and a little smug when noises of clashes sent men to the
border and left the women behind. Both children were terribly shocked
that ‘the bad guys’ killed everyone when they captured the fort. They
were especially appalled to see the children and their mothers shot and
stabbed. When I recalled the many documentaries on World Wars I and II
which both have watched, even some with the charnel houses of the
concentration camps, when Robby said he thought this was dreadful, I said:
“I understood you liked wars, Rob.” He replied, “They’re all right to
read about in books, but they must really be terrible.” It seems to me
that Robby went a long way towards growing up in those few minutes of
the movies. He went away.

Miriam and I continued watching, waiting to see the cavalry charge.
I recalled Robby when the charge was under way, and he returned, with
only a little enthusiasm. As one could expect, when the horses leapt
over the cannon emplacements, Errol Flynn (or Geoffrey Vickers) drove
his lance through the chest of the chief ‘bad guy.’ Miriam was happy
again as she asked me, “Daddy, everything’s all right again, isn’t it,
now that they killed the bad guys?” I couldn’t give Miriam the
reassurance she wanted, but I said those bad guys were dead, that our
ancestor survived the charge. She took those two positives as adequate,
though not meeting her hope.

This incident shows the children becoming engaged and shocked by
the portrayal of a battle in which one of our ancestors took part.


Vn75.1 Logo Seahorses 8/29/77

At the end of the day’s work (Logo Session 56), when the recording
equipment was packed away for the trip home, I was preparing material
for tomorrow’s session. Robby had been using procedures where DELTA
named a variable increment applied to a linear distance and today was
introduced to a use of DELTA as an angle increment. I had written a
POLYSPI analogue procedure (call it “A”; examine its listing on Addendum
75 – 1) and was showing it to Gretchen. (An execution of “A” creates
1 of the s-shaped curves in the picture of the addendum.)

Miriam entered my office and asked, “What’s that, Daddy?” I told
her it was a SEAHORSE and tried to distract her attention. This is work
I intend to pursue with Miriam in the near future, and did not want to
expose it to her early. Miriam was most insistent; she wanted to do
SEAHORSE. I would not tell her how to spell it. She spelled ‘SEA’
then got Robby’s help in spelling ‘HORSE.’ When Logo complained that
no one had told it how to SEAHORSE, Miriam complained to me. I relented
and wrote this procedure:

        1  A  10  60  2

Miriam cleared the screen and was delighted when SEAHORSE executed
to create the figure she expected. She called Robby, executed it a
second time. He remarked, “It looks like you’re making something.”
“I am,” declared Miriam. “A flower.” She proceeded through another 7
executions and happily printed her flower in triplicate — with copies
for Robby and me.

This vignette documents Miriam’s engagement in a small project
(which won’t appear in the Logo Session recordings), her attraction by
something not-quite-familiar, her elaboration of the artifact of the
procedure through repetition, and her fitting in the developing design
to a class of objects she is accustomed to. (To Miriam, the ‘flowers’
of drawing or design include any shape of manifest circular symmetry).

Post Script:

Miriam was sufficiently pleased with her SEAHORSE/FLOWER to send
copies to her great-grandmother (G.G.) and to her friend Maria (who has
moved to Spain). To the latter’s copy she appended a hand-written note:
‘I made this on the computer’.

Addendum 75-1


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).


Vn78.1 Allergies Controlled 8/26/77-9/2/77

8/26 Today we took Miriam to her doctor, a pediatric allergist, and
explained both her symptoms (waking in the middle of the night, wheezing,
coughing, and vomiting) and our own sense of their severity (it has been
happening regularly; Miriam has had only 2 uninterrupted nights of sleep
since our return from vacation). The doctor’s response was threefold.
He gave her a shot of adrenalin, which rapidly cleared up her wheezing
(it was not so bad as it had been anyway). He prescribed for use tonight
a sustained release form of her regular medicine and offered to proceed
with a cortisone analog medication if the other prescriptions were not
effective. Gretchen and I are reluctant to proceed to cortisone treatments
unless absolutely necessary because we believe titrating the hormonal balance
is a much more profound intrusion than is the bronchodilator medication.

8/27 Miriam slept ill last night. She waked again in the middle of the
night coughing and threw up accumulated mucus. The change resulting
from the sustained release drug is that Miriam waked at 4 am instead of
2 am. Once she reaches such a state, she is unable to take a new dose
of medicine without risk of regurgitating it. This will not do.

8/29 Since entering the cortisone regimen, Miriam’s condition has
improved radically. She has had 2 successive nights of uninterrupted
sleep and is no longer physically depressed. I judge this regimen, to
be continued for about 4 weeks, worth the risk of the two most prominent
side effects: fluid retention and increased appetite (Miriam has lost
4 pounds since the beginning of the summer).

8/30 Information relayed by Hal Abelson from his wife supports the
doctor’s judgment that the addition of cortisone medication is now the
treatment of choice. In her work Lynn has seen many children whose
conditions have been radically improved by this treatment with no ill
effects. The synthetic analogs of cortisone engender fewer side effects
than the drug form of which my mother had a very bad experience a decade
or more ago.

9/2 Last night was Miriam’s first following a day wherein her cortisone
intake was nil (she has begun a schedule of alternate day doses).
Miriam slept well till 8 am. The doctor now advises reducing her
medication further to 2 cortisone tablets every other day.

These notes recount Miriam’s rapid improvement from the intensified
allergic reaction experienced this August (the first time) at the beginning of
the ragweed season. Skin tests last winter indicated her vulnerability, but we
had not before seen such a manifestation of her allergic sensitivity.


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.


Vn80.1 Planning for School 9/2/77

Miriam, showing her unprompted concern, began the following
dialogue. I transcribed it from memory (not tape) about 2 hours after the
fact. The content is accurate, though the sequence of points may be a
bit muddled.


What do you think the teacher will say when she finds out I can add?

What do you think?

I think she’ll be mad at me.

Are you worried about that?


Don’t worry, sweety. I’m going to have a meeting with your teacher next week. She knows you’ve been working with me at Logo and wants to know what she should try to teach you.

What do you think?

I don’t know. What do you want me to tell her?

I guess I should just do the regular stuff.

You mean like 2 plus 3 is 5?


For a whole year? When you already learned to add big numbers at Logo?

I didn’t learn that at Logo. You taught me.

Oh. I don’t mean the really big ones. I mean numbers, say, that you use in
playing SHOOT. Like 90 plus 90 is a hundred 80.

I didn’t learn that [I didn’t figure it out]. You told me.

But I don’t have to tell you any more, do I?

No. . . . When do they usually do numbers like that in school?

At the end of second grade, maybe third grade.

You mean I can skip a grade?

You can read well and do computations. I guess you could skip a grade if you wanted.
Do you want to?

Do I have to?

No. You said before you wanted to stay with your friends. I think that’s a good idea
and you shouldn’t skip a grade. But how will you feel about school?

Art should be a lot of fun. And so should gym.

I bet they’ll let you read whatever books you want. That should be good.


About the arithmetic: maybe I should worry about that, make the work for you to do.
Maybe I could get some good advice from Dan Watt. How would that be?

Well, I don’t know. Maybe it would be O.K.

At this point, Miriam terminated our conversation, drifting out into the
court yard to watch people moving furniture.

In this dialogue, Miriam and I discuss what she should do when she
starts school. She expresses fear that her teacher will be mad at her
because she already knows how to add. I inform her of an impending
conference with her teacher and ask her advice.


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


Vn87.1 Turtle Tactics 9/7/77

This night is the last of summer, so defined by the children’s
beginning school on the morrow. Over this summer they have gradually
become accustomed to going to bed late, and now, in order to rise early,
they should go to bed early. No one found this argument convincing.
We negotiated a compromise that the children get into pyjamas, return
for dessert (delayed by conversation with dinner guests, José and Fernando),
and then go off to bed. Robby lived up to the agreement. Miriam did not.

When given a direct order to go to bed, she went to my bed instead
of hers. I had mentioned during dinner the children’s inclination to
play turtle. Fernando tried to help. “Miriam, forward.” She did nothing.
I advised him that he had left out the carriage return. Upon his “carriage return”
Miriam complained, “You haven’t told me how far to go,”
chuckled, and popped back onto my bed. Gretchen attempted, “Forward 100,
carriage return.” With gripe “You haven’t told me how to FD100” still
in the air, I described the bug as the well-known space omission between
operation and input. Fernando was then precise: “Miriam — forward,
space, 100, carriage return.” Miriam played fair and proceeded stepwise
(and counting each step) down the length of the loft. At first we
expected 100 steps to be too few. Miriam counted 70 in the kitchen, and
at 88 gleefully announced, “Out of bounds!” as she walked into the wall
in the hallway. While so close to her bed room, she picked up her
‘security blanket’ (the air was a little chilly) and came skipping
back into the living room.

The game wore on (hide turtle under the blanket and so forth), after
a while becoming wearing, and I directed her to bed with the threat of
physical force. Miriam replied, as she has for some months now, with
the counter-threat of “I’m quitting your thesis, Daddy, I really am.”
Having thus preserved her dignity, she acquiesced to the demand that
she go to bed.

This vignette describes the way Miriam employs her knowledge of
Logo as a delaying tactic to avoid going to bed early. The ease with
which she adopts the turtle’s role in a command/execution script (using
it, of course, for her ends) shows how directly that script can represent
the actual ‘authoritarian’ portion of relations in a family. Lest
this seem mechanization obtruding into a human relationship in an un-
healthy way, remind yourself that this was to her a useful game which
permitted her to survive for a while by her wits in a situation where
her other obvious options were to submit to authority, to rebel (make
a fuss), or to wheedle from me some relaxation of the order.


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


Vn89.1 The Ten in Fourteen 9/7-10/77

9/7 After considerable confusion at the beginning of yesterday’s arith-
metic work (Home Session 18, 9/6/77), in a reprise after games of Tic
Tac Toe, I was able to explain ‘carrying’ to Miriam in a manner access-
ible to her. I cited a recent comment of hers while doing mental arith-
metic that “there’s a ten in the number 14.” This point of connection
permitted the only explication of carrying so far that has been able to
compete with Miriam’s “reduction-to-9’s” procedure.

Where did this reference “there’s a 10 in the 14” come from? I
examined recent vignettes and found no reference to it. Since I could
recall no more detail, this morning I put the question to Miriam. I
noted it might have come up the last time we rode to MIT in the Audi
(I vaguely remember a discussion in such a setting about the sum of
170 and 87). Miriam said, “I remember. It was at dinner a day or two
ago. Robby asked how much was 30 and 14, so I said it was 44, ’cause
there was a 10 in the 14; that made it 40, plus 4.”

9/8 Before Miriam went off to school this morning, I asked her if she
could still see the 10 in 14 and the 20 in 27. She apparently under-
stood and said yes. I reminded her that reducing to 9’s was a buggy
procedure for carrying.

9/10 While typing a fair copy of the work in Home Session 14 (July 31),
Gretchen found the reference I sought to Miriam’s explanation of there
being a 10 in 14: Episode I, page 2.

These notes mark the reappearance of the idea of being able to see
a 10 in 14. When I, attempting to find the specific reference of
Miriam’s first using the phrase, ask her about it, she reconstructs for
me an incident which seems plausible enough but is probably entirely
a fabrication.


Vn90.1 Meeting Miriam’s Teacher 9/11-12/77

9/11 This Sunday morning, Miriam inquired of me if I knew what kind of
a teacher she had. When I admitted I did not, she continued:


Nice. She says if the math problems are too easy, I won’t have to do them.

What will you do instead?

Play with other games, I guess.

Why did she talk to you about this? . . . Did you have math on Friday?

No. I went and asked her about it.

9/12 Early in April before this project began, I discussed with Bob
Gracia, guidance counselor for the Heath/Baldwin schools, the possible
impact on Miriam’s school life of her work on my thesis project. I
raised such questions as these: if Miriam’s level of knowledge and
capacity for rapid development placed her markedly ahead of her peers,
would this be a problem for her? The senses in which I imagined possible
problems were these: her teacher, coming under an additional burden to
provide separate guidance for Miriam, might come to dislike her; to find
continued intellectual challenge, might Miriam be forced to skip into a
higher grade with other children who would be more mature in other ways?
Bob assured me that neither of these possible problems would arise, that
where special needs were clearly shown, the school assumed the respon-
sibility to provide individualized curricula. He proposed a meeting with
Miriam’s first grade teacher at the beginning of the academic year.

Today Gretchen and I met with Bob Gracia and Sue Fieman, Miriam’s
new teacher. Our concerns were three — that Sue know of Miriam’s
allergic vulnerabilities; that she know that our project is still con-
tinuing, that during the next month after school Miriam will be coming
to work at Logo with me; finally, that she not get mad at Miriam. The
first two points are of information, and not too difficult to address.
Both Bob and Sue were interested in the work of our project as I des-
cribed it to them. Sue was excited at the opportunity to see first hand
how Miriam’s experiences at Logo would interact with her standard school
work; she mentioned in passing having done some preliminary assessments
of her students and how surprised she had been that in a class with a
number of ‘non-conservers’ she found Miriam solving class inclusion
problems without difficulty.

Given Sue’s reaction to my description of Miriam’s work on this
project, I believe Miriam has little to fear of teacher antipathy. Nonetheless,
I told Sue the little story of Miriam’s discussion with her from
yesterday to indicate Miriam’s hope to avoid boring work. I also
mentioned Miriam’s earlier fear that the teacher wouldn’t like her because
she knew too much already. On the contrary, Sue seems quite eager to
work with Miriam and wants to know what sort of instruction would be
best for Miriam — “where should she start her out?” I avoided answer-
ing that question in a school-oriented way. I did tell her not to worry
about special instruction: Miriam had previously expressed a wish to
do what everyone else did; and that, with the exception of adding, I had
not been working on school-like material. Miriam would have some special
knowledge, for example about geometry, but such special knowledge would
be outside the normal curriculum. The greatest differences between
Miriam and her peers I expect to be in her tendency to focus on mental
processes and her ability to discuss thinking articulately. I concluded
that the experiments from now on will best provide answers about her
question of where to start Miriam out. Sue remarked that was no problem,
that first grade never began any academic work until October so that each
child had a month to get used to the people and surroundings.

This report of a meeting with Miriam’s new teacher leads me to
conclude that Miriam will have no difficulty with her. She is sympathetic,
open-minded, and considers it an opportunity that a child with Miriam’s
unique experience is in her class.


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.