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Tic Tac Toe (2)


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

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

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

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

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

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

 B |   | 2
   | C |
 1 |   | A

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


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

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



Arithmetic Ripples


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

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

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


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


Vignette 21

Miriam’s Room


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

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

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


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


Vignette 23.1

Arithmetic Ripples

6/5 & 11/77

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

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

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

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

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

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

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


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


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

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

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

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


Vn029.01 Making Puzzles 6/18/77


Vn29-2 Addendum1

Vn29-3 Addendum2


Vn031.01 Collecting Tolls 6/19/77

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

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

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

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

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

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

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

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

Addendum 31-1

Toll Collection Records

Vn31 Toll Collection Records


Vn034.01 Candle Fire Crackers 6/23/77

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

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

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

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

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


Vn37.1 Explaining SHOOT 6/26/77

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

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

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

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


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

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

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

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

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

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


Vn41.1 7/5-7/77

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

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

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

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

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

Post Script

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

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

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


Is Daddy a dummy?

Is Daddy a smarty?

What is he?

An idiot!

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


Vn43.1 Binary Counting 7/7/77

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

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

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

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

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

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


Vn44.1 A Boring Session 7/12/77

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

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

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

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

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

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

Post Script

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

Addendum 44-1

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

Addendum 44-2

Hexagonal Maze

Vn 44-2 Hexagonal Maze


Vn46.1 Rotten Hints 7/19/77

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

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

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

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

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

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


Vn47.1 Losing a Tooth 7/20/77

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

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

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

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


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

(Surprised and a little disappointed) Oh.

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


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

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

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

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

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

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

Addendum 50-1

Addendum 50-2


Vn51.1 Paper Ships 7/25/77

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

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

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

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

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

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

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

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

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

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

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

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

Addendum 51-1

Vn 51-1 Curious George paper folding

Addendum 51-2

Vn 51-2 Curious George Paper Ship procedure


Vn53.1 A Birthday Party 7/28/77

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

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

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

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

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

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

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

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


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.


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.