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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.


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


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.


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).


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

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

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

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

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


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

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

Yes. Did you use your fingers?

You want to know how I did it?


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

Where’d the hundred come from?

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

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

I know that.

You also did it correctly.

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

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

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


Vn81.1 Imitating Machines 9/3/77

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

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


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

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

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

(Blows on pencil and runs in place.)

(Presses his ‘stop’ thumb.)


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

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

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


Vn82.1 Hanging Designs 9/3/77

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

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

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

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

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


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


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

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

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


Knock knock.

Who’s there?


Robin who?

Robbin’ you. Gimme your wallet.

Miriam recalled a second:


Knock knock.

Who’s there?


Ivanitch who?

I’ve an itch I can’t scratch.

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


Knock knock.

Who’s there?


Irish who?

I rish I never said “Knock knock.”

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


Vn86.1 An Unexpected Test 9/8/77

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

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

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

Game 1: Miriam moves first (letters)

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

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

Game 2: Glenn moves first (numbers)

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

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

          A | B | 3
            | 1 |   
            |   | 2 

Game 3: Miriam moves first (letters)

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

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

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

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

Game 4: Glenn moves first (numbers)

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

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

Game 5: Miriam moves first (letters)

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

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

Game 6: Glenn moves first (numbers)

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

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

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

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

Addendum 86-1

Games with Lab Student

Vn 86-1 Games with grad student


Vn88.1 9/8/77

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

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

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

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

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

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


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.


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.


Vn94.1 Miscellany 9/dd/17

At lunch today a variety of topics came up for discussion. Miriam
said she would like to bring her friends over to visit Logo. I thought
of previous visits. “You mean something like the earlier visits of Meg
and Dara?” Miriam added, “I want Michelle and Laurie Ann and Elizabeth.”
I asked if she wanted all her friends to visit at once. She replied,
“No. The whole class.” I am happy that Miriam wants her friends to
visit our lab and get some sense of what she has been doing. After the
final phase of the project ends might be best, since such a visit
would take a lot of preparation.

I discussed with the children our moving into the project’s final
phase. Because there had been some complaints about how specific
experiments were ‘bad,’ e.g. bending rods, I asked if the children had
any good ideas for improving our experiments. Miriam’s response I find
a little bizarre, but worth noting: “We should do more things with clay.
The best thing of all would be if we made things on the computer, the
computer would give you them in clay, like PERSON.” (PERSON is the name
of a procedure Miriam made in Logo Sessions 59 and 60; Miriam uses the
printed images of the procedure’s output for cut-outs and coloring. For
an example see Addendum 94 – 1.)

Later on, a few knock-knock jokes passed across the table. Miriam
noted of them the thing that makes them knock-knock jokes is you have
to say “Who’s there?” We all agreed; then Miriam began what seemed a
divagation. She said to Robby:


Knock knock.

Who’s there?

Will you remember me in 5 years?

I don’t get it. . . . I thought this was a joke.

Will you remember me in 5 years?


Will you remember me in 10 years?

. . . Oh, I guess so.

The conversation generally started drifting another way.


Knock knock (interrupting).

(a little exasperated) Who’s there?

(Looking right at Robby) Don’t you remember me?

I found this a very unusual example of the genre, and asked Miriam if
she had read this in the collection of knock-knock jokes she took out
of the library. Miriam claims to have made it up herself. I find this
an interesting variant because it makes very direct use of the rigid
knock-knock script without having its humor depend on a pun. That is,
the equivocation is at a discourse level, not at a verbal level.

This collection of incidents touches on 3 points: Miriam’s interest
in showing Logo to her class; her imagining computers with a more
physical and less representative output; finally, her articulate knowledge of
the structure of knock-knock jokes.

Addendum 94-1

Using Logo printed images

Vn 94-1 Using Logo printed images


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

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

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

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

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

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

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

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

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

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

Addendum 97-1 Logo Conference Notes

Vn 97-1 notes for IJCAI related Logo Conference


Vn99.1 Puns 9/26/77

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

In school, Miriam invented and told Brian this joke:


What should you do if your toe falls off?

I don’t know.

Call a tow truck.

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


What should you do if your thumb falls off?

I don’t know.

Get a thumb tack.

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

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

Both children enjoyed this joke invented by Andy DiSessa:


What should you do if your finger falls off?

I don’t know.

Get a fingernail.

Both declared my following joke “not very funny.”


What should you do if you lose your head?

I don’t know.

Call a head hunter.


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

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

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

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

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


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

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

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

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

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

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

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

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

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


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


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

Vignette 106, page 1 Scanned from Original Fair Copy

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

Vignette 106, page 2 Scanned from Original Fair Copy

Vn106-2 scanned; no digital source available


Vn107.1 Self-Understanding 10/22/77

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

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

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

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

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

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

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

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


Vn111.1 Swears 11/30/77

A few days ago I sat at a terminal with Miriam at the Children’s
Learning Lab. In response to the “login” request, Miriam typed “FUCK”
then turned to me and said, “Look, Daddy, I typed a swear.” I responded
non-committally, “Oh yeah. Why don’t you hit new line and see if it
works?” The response came back, “No such user.” I found it amusing to
think back a few months when I overheard Robby making fun of Miriam
because she spelled the word ‘FUKC’. I continued: “You say that’s a
swear. Can you tell me what a swear is?” Miriam didn’t answer.

This evening Miriam demonstrated for me how good she had become at
doing “Miss Lucy.” This is a chanting game for two with partner hand
clapping a la “Patty-cake, patty-cake.” I had earlier seen some third
grade girls playing this game when I rode on the school bus to visit
with the children. With most of her attention focussed on the quite
complex clapping patterns, Miriam began singing:

Miss Lucy had a steamboat, 
   The steamboat had a bell. 
The steamboat went to Heaven, 
   Miss Lucy went to --
Hello, operator, 
   Give me number nine, 
If you disconnect me, 
   I'll cut off your -- 
Behind the 'frigerator 
   There is some broken glass. 
Miss Lucy sat upon it 
   And cut her big fat -- 
Ask me no more questions, 
   I'll tell you no more lies. 
The boys are in the bathroom 
   Pulling down their -- 
Flies are in the meadow, 
   Bees are in the grass. . .

She then called out, “Robby, what comes next?” I was tempted to tell
her myself. The sense of deja vue was very strong, for the tune was one
I knew as a child with these words:

Lulu had a baby, 
   She named him Tiny Tim.
Put him in the piss pot 
   And learned him how to swim. 
He swam to the bottom, 
   Swam to the top. 
Lulu got excited 
   And grabbed him by his -- 
Cocktail, ginger ale, 
   Five cents a glass. 
If you don't like it, 
   Stuff it up your --
Ask me no questions, 
   I'll tell you no lies. 
If you ever get hit with a bucket of shit, 
   Be sure to close your eyes.

When Robby did not respond to her question, Miriam turned to me and said,
“That song sure has a lot of swears in it, doesn’t it, Daddy?” I agreed.
“Michelle taught you the hand clapping, you said. Is she the only one
who knows all the swears?” Miriam confided to me that really everyone
knew them. I admitted I knew many, possibly some she didn’t know.
Miriam’s curiosity rose. I established my claim by running past her some
gutter Italian I had learned in grade school and a few Spanish phrases
I picked up in the Army. Miriam was impressed. I remember being similarly
impressed myself recently when a friend indulged in some exemplary
Afrikaans. I couldn’t understand or mimic his performance, but it
appeared he was mouthing a string of unimaginably vulgar and insulting

Songs such as those of Lulu and Miss Lucy obviously are broadly
dispersed and endure in the child culture we all pass through and no
longer attend to. Beyond the fun implicit in violating the petty taboos
against vulgarity, these rhymes engage the children in memorizing chants,
the crucial humor of which is found in the punning of the terminal rhyme.
Children learn the puns first and realize their double meaning after.
For example, Miriam did not appear to understand the pun on ‘BEHIND’ in
the Miss Lucy song.


Vn113.1 Steady State 12/8/77

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

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

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

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

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

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

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

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

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


Vn119.1 Multiplying by Twenty 1/25/78

The children love to get mail and when an envelope comes addressed
to them, to open it. Each has a bank account concerning which they
receive annual earnings statements. The children opened their mail and
puzzled over the contents — a statement of account number, social
security number, and interest earned for the year with no specification
of the current balance. I checked for the latter because, as I explained
to them, I preferred their remaining ignorant because of their
inclination to blat to their friends what capitalists they are. I went
on that they shouldn’t go about bragging how much interest they had
received. “Why not?” I informed them that anyone knowing their interest
could estimate their capital simply by multiplying the dollar amount by

Robby and Miriam realized they could circumvent my not telling them
of their bank balances, and Robby began to do so. Miriam lamented she
didn’t know how to multiply by twenty and received Robby’s promise of
help after he completed his own computation. A few days before he and
I had discussed a good trick for 10 times: just writing down an ‘extra’
zero on the right end of the number. Robby realized he could get the
desired result by doubling the interest (by addition), then adding a
zero. He became confused about manipulating the decimal point during
the 10-fold multiplication, but accepted my procedure for doing do. He
read his balance to Miriam, then went to help her.

Miriam followed Robby’s direction but set up the problem herself.
My role was limited to restraining him from taking over. No problem
with adding 2 plus 2. The carry first arose with 6 plus 6 (see Addendum
119 – 1). Miriam said, “I put down the 2 and carry the 1.” Robby
responded, “Right.” and when she went to mark a carry over the tens
column, he directed her to place it over the hundreds. With some labor
Miriam added 8 plus 8 and 9 plus 9, handling the carries appropriately.
Thus she had doubled $98.62. But what did the answer mean?

Miriam tried to read her answer 19724: “One thousand. . . one thousand
. . . .” She believed her result should be of the same order of magnitude
as his, but was lost because she could not coordinate that correct judgment,
her accurate computation, and the structure of the problem’s solution.
As Robby did at first also, Miriam neglected the 10-fold multiplication;
nor did she understand at all this good trick for 10 times (she had
never been exposed to it before). Comparing Robby’s work to her own
did not help. Rather than protract her frustration, I “showed” her what
to do. (This means I wrote in the decimal point and an arrow and mumbled
a few words). Miriam accepted my answer as correct and sensible.

Both children were able to rejoice once more at having outwitted
their dumb old Dad.

The first incident shows the children applying their arithmetic
skills to a problem too difficult for Miriam. She can effectively execute
complex additions but does not dominate the number representations.
Her writing a carry mark at the top of the tens column shows her sense
that the 1 of 12 still belongs more to the 2 than to the left adjacent
column. I infer that Miriam is working out the problem of what a carry
means. She is very close to understanding. The second incident suggests
I follow up Miriam’s judgment that school arithmetic papers are hard;
why should she find them so?


Adding by Miriam and Robby

Vn 119-1 Sums by Miriam and Robby


Vn121.1 Double Perspectives 2/8/78

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

Miriam has told repeatedly her most recent joke.


What letter of the alphabet do you drink?

I don’t know.

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

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

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

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

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

Addendum 121 – 1

Find the Hidden Picture

Vn 121-1 Hidden Pictures


Vn124.1 Analogical Guidance 2/23/78

This evening at dinner, my family enjoyed a good time at the expense
of Scurry, our Scotch terrier. Earlier in the day, Miriam had played
tug with Scurry, the object of their contention a squeaking toy mouse
she had given the dog. Scurry wrested the toy from Miriam and sat chew-
ing it out of reach. Miriam then rolled Scurry’s small ball across the
floor, and Scurry, the mouse grasped firmly in her teeth, bounded off in
pursuit. She caught the ball and appeared trying to pick it up but
failed because the mouse was still in her teeth and she was definitely
not willing to loosen her grip. After relating this story to Robby,
Miriam rose from the table to demonstrate Scurry’s bind by duplicating
the situation. Scurry would not cooperate; she kept the mouse and
refused to chase the ball. Robby wandered off and Miriam, a little
dejected, draped herself over the back of the couch.

But there, her interest rekindled, for she saw on my toyshelf a
puzzle she has been working at unsuccessfully for a week or more. This
is the Pythagorean puzzle (cf. Vignette 77, Geometric Puzzles) which I
have realized in both 5 and 7 piece forms. This is their form, each
assembled as a single large square:

Miriam failed to assemble the 7-piece puzzle despite her repeated attempts
this past week. It is a vanguard problem for her. When she dumped out
the pieces on the couch, I asked her to bring it to the table and pushed
aside the dishes. Miriam complained, “I can’t do it.” When I asked why
not rhetorically and advised her it was just like the 5-piece puzzle,
she responded, “But I can’t do that either.” Miriam has done the 5-piece
puzzle. Her statement may mean she can not do it at will, without trial
and error, that she does not comprehend the puzzle. Miriam brought both
puzzles to the table but had trouble locating the center square which,
when found behind the couch, I kept.

Miriam began with two congruent triangles, thus:

Vn 124-2 Yukky DIamonds

She declared the first a “yukky diamond” and tried, with no confidence,
to suggest the rectangle was a square. I gave her a hint: you have to
use all the pieces to make a square. Miriam then articulated a salient
bit of known knowledge: “This side has to be on the outside.” When
queried, she pointed to the hypotenuse of one of the congruent triangles.
She tried, in order, these intermediate configurations:

Vn 124-3 point to point

As Miriam attempted lining up the corners of the 3rd triangle, she
pushed away the second from the first, saw the configuration below, and
held out her hand to me.

Vn 124-4 pattern of insight

I gave her the center square, and she completed the puzzle. “Now do the
7-piece,” I challenged her.

Miriam laid the two complete triangles of the 7-piece puzzle on top
of the 5-piece assembly, arranged as in V, and noted, “I’m using the
same patterns.” She added the corner-cut-off congruent triangle; first
she put it in backwards, then as below. Her outstretched hand requested
the missing corner.

Vn 124-5 analogy by superposition

I gave her the missing corner and center square. Miriam tried the
smaller, similar triangle abutting the center square, then moved two
vertices to the periphery. She first tried the final piece backwards,
then completed the puzzle correctly.

Vn 124-6 7 piece completed

Miriam was pleased with herself. I removed the 7-piece puzzle and
asked, “Can you make a black-colored square?”

When she had done so, I asked if she could then make a gold one.
Miriam asked, “Do I have to use the square?” I said she should try to
without it.

Vn 124-7 golden square

Arrangement XIII showed Miriam she needed the little square which I
returned to her. The 7-piece puzzle followed. Miriam was stuck for a
while. One hint crystallized her completion of the 7-piece puzzle:
“Find a shape the same as this black part.”

Miriam located the end-cut triangle and fitted the smaller triangle
to it. She proceeded to build a column of these pieces, then added the
rectangle of 2 triangles in the appropriate orientation.

Vn 124-8 3 rectangles


Several themes seem to arise clearly here. Some relate to the
precipitating situation: stumbling into problems accidentally when
the materials are at hand; the existence of this unsolved puzzle as
an item on an internal agenda to be worked at till mastered.

Miriam retained a very specific piece of knowledge as a key element
of the solution: “the long edge goes on the outside.” She was able to
use generally formulated advice when it had specific and obvious appli-
cation (e.g. use all the pieces) as well as very specific direction
(e.g. find a piece with the same shape as the black area). Trial and
error plays a large role.

Finally, for this difficult 7-piece puzzle, the combination of a
few hints and an analogous, simple version about which 1 key point was
known operated as guidance as effectively as do the pictures on the
surface of a picture puzzle.


Vn129.1 Robby Computes a Tax 4/5/78

Robby caught on fire again today. He approached me inquiring,
“How much is half of 423?” Miriam responded to his question from the
other room, “2 hundred and 11 and a half.” I told her to stop butting
in and asked Robby how much was half of 400, then half of 22, then half
of 1. He came to his own conclusion of 2 hundred and 11 and a half.

But why this concern with the specific question? $423. was
the price of a swing set in a catalog the children had been perusing.
They had agreed to go halves on buying this much-desired super-toy.
I opposed their doing so and raised as an objection along the way the
observation that they hadn’t included the amount of tax they would
have to pay.

“Is there a tax on toys?” was the incredulous question. “If
food is taxed,” I responded, “should you not expect toys to be taxed
also?” When he asked how much it was, I explained to Robby that he
could think of the tax as a nickel for every dollar of the purchase
price. Here we got into complicated computations.

Robby tried to figure out how much money is 4 hundred nickels.
His confusion was great, even including such faux pas as “there are 200
nickels in a dollar.” Correcting to 20 to the dollar, he went on to
observe that $100. of the purchase price converted to $5. of tax. Here
he was stymied but began to add $5. and another. I complicated his
computation by suggesting he use the multiplication results he had
learned at school. He looked blankly at me. “How much is 4 times 5?”
I asked, and received an answer: “20.” “How much is 4 times 5 dollars?”
No answer was forthcoming. He came to $20. eventually (I believe by
adding). Robby then computed the tax for 20 dollars more (of the
original $423.), and with Gretchen’s reminder, added another 15¢ for
the last 3 dollars.

This incident required a surprising amount of time, as much
as 5 minutes, to develop.

This was a very exciting incident for Robby — his first
computation of a sales tax. He brought the idea of “a tax” under
control as a comprehensible percentage, thus eliminating that
mysteriousness which has troubled his world of money since
at least last summer (cf. Vignette 54).


Vn131.1 Miriam’s 7th Birthday 4/8 & 9/78

4/8/ Miriam began planning her birthday party several weeks ago.
On the 3 x 5 cards of Addendum 131 – 1, she listed the friends to bo
invited, the candy, and her selection of party games. The children were
all from her class at school. The games are all familiar, the first
being a party standard, the second played at Meg’s party, and the third
one of Miriam’s favorites from gym. (She also spoke of playing Red Rover
outside and was much concerned that Brian and Miceal should be on opposing
teams.) Miriam thought of getting cards for invitations, but did not.
Thus at the last minute we had to make our own. Miriam liked the idea
of preparing invitations at Logo, so we made a special trip there and
used the letter-writing procedures and her pretty flower to create her
unique invitations (cf. Addendum 131 – 2). Yesterday morning was dedi-
cated to preparing the house. We pushed the furniture out of the living
are of the loft to make a big play area, nonetheless praying for sun
shine so that we would not have 12 active kids confined in our small
apartment on a rainy afternoon.

A week of allergy-driven fitful sleep left Miriam physically
depressed but cheerful on her party day. She donned the party dress made
by her great-grandmother and played in the courtyard waiting for guests.
As they arrived, Robby helped first by carrying in presents and then by
playing soccer with the boys. At the one point where all the guests had
arrived and were inside, I spoke above the pandemonium to announce that
we would have an ice cream cake about 3 o’clock but that otherwise they
should enjoy themselves in whatever way they chose. The children gathered
about while presents were being opened. . . and then began a problem. An
early-opened gift was a set of face paints, which appealed to everyone, and
some children went off to the bathroom and decorated themselves. Somehow
two girls ended up fighting in the hallway, pulling hair and crying. At
this pass, Robby led the boys off to the tree fort and either Dara or
Lizzie suggested playing on the space trolley out back. I joined that
group of girls for fear they might get too close to our land lord’s
horses. Miriam and two friends stayed inside with Gretchen. From the
space trolley landing, the girls could see the boys across the lawn and
made an assault on the tree fort. That short-lived battle was ended by
my recalling the children for the party meal.

The children sat in a circle on the floor, and Miriam asked if
Peggy could join them. Everyone sang ‘Happy Birthday’, had his soda and
goodies, and after a quick clean up played in the courtyard until parents
started arriving.

Miriam was disgruntled, mainly because her face-paint crayons
had been used against her will and some got broken. She was also disap-
pointed that no one played the games she had picked out. (We discussed
later whether I should have directed them to, and Miriam opined that I
was right in not taking over). Miriam cheered up a little when I gave
her my present, a small string art design in the shape of a heart (which
she had requested) with a large letter ‘M’ in the middle, and when she
chose the evening’s dinner (pizza).

4/9/ This birthday began on a cheerful note. Since I had neglected
to give Miriam her weekly allowance on Saturday, the normal day, and
because it is calculated as a dime for each year of her age, on this day her
allowance was 70¢, where yesterday it would have been 60¢. This joke,
heightened by my feigned aggravation, delighted Miriam.

After a good night’s sleep, Miriam was considerably more chipper
than yesterday and eagerly accepted my suggestion that we should go riding
the trolley cars of Boston. She, Robby, and I took the Riverside line to
the terminus. The conductor, finding the kids and I were out for a ride,
would take no money, so we enjoyed a free ride both ways as we headed
down into the city. At Park Street we took the red line to Quincy,
stayed on board when the train reversed directions, and emerged at
Harvard Square. After a late lunch at Brigham’s we returned home by
the red line and the Commonwealth Avenue trolley for a quiet afternoon
and a small party this evening for the 5 of our family.

Addendum 131-1

Party Planning

Vn 131-1 Party Planning

Addendum 131-2

Party Invitations Made at Logo

Vn 131-2 Party Invitations