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Vn103.1 Reprise 10/15/77

As we have attempted reducing Miriam’s cortisone dosage, she has
again begun wheezing severely. Her problem was especially unfortunate
today, for she had asked a friend to come visit. Lizzie is an energetic
red-haired colleen of 6. At lunch she disclosed to Miriam and Robby all
the problems they could expect with a younger sibling. (Her sister
Katie, at 14 months, not only keeps her awake but also tackles Lizzie
every time she strolls by). The weather was so bad no one could play
outside. Miriam, hoping for some relief in the better air of the Logo
lab, agreed to Robby’s suggestion that we all go over there to play.

Once at Logo, Miriam showed Lizzie around — through the Learning
Lab, the Music Room; the toilets out here and the coke machine across
the hall. Lizzie said she wanted a picture of a flower and then explained:
earlier in the year Miriam had made copies of the image made
by a FLOWER procedure and took one for each member of her kindergarten
class; the children colored them and some took them home. Miriam was
still feeling low, so I helped Lizzie get what she wanted. First, a
FLOWER picture. Next, looking at Miriam’s work hung on the walls of my
office, she declared she wanted a copy of PF (for PRETTYFLOWER). As we
waited for her picture to come from the printer, Lizzie saw a 6-fold
near triangular polyspiral which Robby had printed from a different
terminal. “Wow! I want one of those. Show me how to do it.”

I started the SHAPES (or MPOLY) procedure. Miriam showed some signs
of improvement and the two girls worked together creating designs which
they would color in later (cf. Addenda 103 – 1 and 103 – 2). Whenever
they got in trouble, I was available to help them get started again.
This play of making computer designs, then printing pictures of them for
later coloring, occupied both girls for about two hours. Both girls
made some very pretty novel designs. For example, the design of Addendum
103 – 2 was one I had never seen before and judge pretty (more so
with the drawing in light on the cathode ray tube). When Robby and I
tried to show SHOOT to Lizzie, she could not get interested in it and
returned to MPOLY.

As the day wore on, Miriam’s apparent improvement faded. We drove
home by way of a playground on St. Paul Street. (Miriam and I stayed
in the car while Robby and Lizzie ran around). As we continued on to
her home, Lizzie revealed that her daddy had a computer too where he
worked, but ours sure was different from his.

These notes record the children’s continuing engagement at Logo
though our project be over, and provide a glimpse of the reaction of
one of Miriam’s peers.

Addendum 103-1

Colored Polyspiral at 155 degreesVn103-1 Colored Polyspiral

Addendum 103-2

Six-fold MPOLY with a person in the middle

Vn 103-2 Six-fold MPOLY with person


Vn104.1 Back to School 10/14/77

During the last days of our project experiments, I promised Miriam
to visit her first grade class as I had visited in kindergarten. I had
the mistaken impression that Miriam had arranged my visit with Ms.
Fieman. The oversight proved to be no problem, for despite my beard
and over-size frame I blended in well with the group of children.

It was “Read me this, read me that. Do you know my name?” David
B. said, “I remember you. Last year you came and we set up that thing
from the ceiling.” His reference was to a 3 string pulley I rigged in
the spring which enable the children to hoist heavy weights, their
desks (!) and each other (!!) a few feet off the floor. One of the
other boys (was it John?) asked if I still had that machine for making
electricity. Curtis brought over a soma cube and the children squabbled
over it. Miriam did not have a chance to work on the puzzle for any time
with 5 classmates each wanting a turn. Meg and Laurie Ann sat with me
and Miriam before the class split into two groups — one headed for the
library, the other for an introduction to the class’ activities for the

The librarian attempted to introduce to the children the distinction
between factual and fictional writing. It is possible my presence, my
sitting on the floor with the children, caused her some unusual confusion.
Nonetheless, it appeared that she neither had articulated for herself
any consistent set of criteria nor had any good language for communicating
her ideas to the children.

Once again in class, Miriam took up the writing activity. Curtis
and I joined her. The task was one of sentence completion: e.g. “With
my eyes I can see ________.” The children’s task is to write a description
and draw a picture of some appropriate object. Miriam chose to spell
and draw flowers. Her other senses led her to taste corn on the cob and
ice cream; to feel fuzzy things (here Scurry was the exemplar); and to
hear a song — which she represented by a person singing the complete
text of “Drive, drive, drive your car, gently down the street” as sung
by Don Music on Sesame Street.

After Miriam’s work was approved, we had a few minutes to play
before I left. She suggested checkers. Lately we have been playing
variations of the standard game. We tried a 4×4 board (played with 2
checkers on each side) and a 6×6 board (played with 6 checkers on each
side). The board fell to the floor while still folded but with squares
showing. I suggested we play ‘half a game’ of checkers. (The board
was thus 4×8 and played with 6 checkers on each side). We played 3
games. Miriam’s friends came crowding around and all wanted their turns.
But I did have to leave and suggested Miriam could play ‘half a game’
with them.

These notes try to capture both the continuity and change of Miriam’s
kindergarten and first grade. There is more structure in that the children
cycle through a set of selected activities (of such a sort that they
could be interesting). The children can get some play time by finishing
their work quickly. Ms. Fieman is good with the children and flexible
enough to let a parent visit with insufficient notice. Miriam seems
comfortable in the situation and enjoys school to the extent that she
chooses to attend even if she feels unwell.


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.


Vn108.1 Miriam Doesn’t Stop 11/1/77

I have ceased collecting data for this project, have focused my
attention on the data reduction problem, but Miriam keeps growing, making
breakthrough after breakthrough. This afternoon, for example, I sat
transcribing videotapes in the reading alcove. Miriam, waiting in place
the ten minutes till Sesame Street should appear on TV (we don’t turn on
the TV till the scheduled time for a chosen program arises), was musing
on the couch. She mentioned something about 10 sixties. I could see
her, in her half-reclining position, lifting fingers up and down. A
short time later she exploded; “Hey, Dad! 10 sixties is 6 hundred 20.”
“Wow! How did you ever get an answer like that?”

She explained and demonstrated so quickly I had trouble keeping
pace while I wrote down her computation. She used her finger counting
to control her decadal arithmetic addition procedures thus:

first result -- 620
recapitulation --

Finger count   intermediate computation
       1               60 + 60          120
       2              120 + 60          180
       3              180 + 60          240
       4              240 + 60          300
       5              300 + 60          360
       6              360 + 60          420
       7              420 + 60          480
       8              480 + 60          520     [an error]
       9              520 + 60          580
      10              580 + 60          6. . . .

At this point Miriam hesitated. . . . “Wait. . . . 560 plus 60 is. . . no. . . 580
plus 60 is 640. 640 is the answer.”

This performance of Miriam’s is noteworthy several ways. Contrast
this “product” with 4 x 90 of Vignette 106. Notice that the computation
of Vignette 106 is assembled AD HOC. The intermediate results, as numbers,
are manipulated with legitimate and varied operations (addition
and subtraction) to give other especially good intermediate results
which simplify the computation, e.g. 20 is taken from 80 and added to
280 to produce the ‘better quality’ intermediate sum 300; 60 is tacked
on as a simply addable residuum (confer here Seymour Papert’s article
on ‘The Mathematical Unconscious’). This computation is different. The
addition procedure itself is manipulated by controlling its iterations
through counting; the control is independent of the intermediate results.
Counting by non-unary increments (Miriam’s method of multiplying single-
digit numbers) has been replaced in a hierarchical control structure by
the general addition operation. (Confer here the data of Protocol 21,
Multiplying, from the series on Robby’s development.)

This is a further incident giving evidence that “you can’t schedule
learning” (cf. Vignette 91, Squirming and Thinking). Although Miriam
lives in a micro-culture wherein computational issues easily surface,
this particular problem of 10 times 60 is one she posed herself, one
clearly at the expanding periphery of her competence. I claim that, for
whatever reasons (including but not limited to sibling competition),
multiplication has become a vanguard issue of Miriam’s concerns; that
one sees the natural surfacing of such concerns and real intellectual
growth occurring in the interstices of other activity. This claim does
not argue Miriam learns no other way — but this incident shows how
engaging and powerful such learning is. It also argues again that to
study learning, you have to go where the person does it and be there
when it happens.


Vn109.1 Tic Tac Toe 10/4/77

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

Game 1: Miriam moves first (letters)

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

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


[placing 1] Can you beat me?

Think so.

Go ahead.

[moves B]

Do you have me beat already?


Can you show me how?

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

Can I go anywhere to stop you from getting those?

I don’t know.

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



[places her index finger on 2]

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

[moves C] C. Go!

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

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


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

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

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

Game 2: Bob moves first (numbers)

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

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

Game 3: Miriam moves first in the center

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

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

It will be sort of like the same game.

Same game as what?

The last game. Go!

You think it will?



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

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

I just pick a space [moves C].

Why is that a good space?

I don’t know.

You have no idea?

I just pick a space.

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

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

Is there any reason?


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

[quickly moves D1 between 2 and 3]

Did you block me?


‘Cause you thought I had a way to win?

Yeah [it’s obvious] 3 and 2.

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


Do you think you have a way to win?


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

[moves D1 to D2]

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

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

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

Game 4: Bob moves first

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

         1 |    |     
         b | a2 |     
         2 | a1 |   

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

One place.

Is there a safe place? Where is it?

[moves a1]

You believe that’s a safe place?


Well. . . shall I prove you wrong?


[moves 2] What now?

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

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

No [replaces a2 in the center].

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

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

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

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

[removes a1 to b1]

Miriam! That’s just not fair.

[reluctantly replaces a1]

Now, where can you move?

I know [moves b1].

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

Good idea! [moves b1 to b2]

O. K.? You want to do that?

What? [moves b2 back to b1]

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

[moves b1 to b2] Win?

Take a look at my chances to win.


Do my chances to win come together?


[gesturing to the fourth corner] No?

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

You think that’s a good defense?

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

Yes, they do.

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

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

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


Vn110.1 Tic TacToe 10/30/77 & 11/12/77

10/30 When Robby and Miriam agreed to play tic-tac-toe together (intending to use Miriam’s ‘magic slate’ which would have left no record of their play), I suggested they play on the chalkboard in the reading alcove. Miriam was granted first move (letters).

Game 1

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

Game 2: When Robby moved first (numbers), he chose the corner opening

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

At his move 2 Miriam said, “Oh oh,” apparently sensing the fork’s distant threat, and attempted to circumvent it by moving twice. Her attempt was met by Robby’s loud and justifiable complaint. The game proceeded to Miriam’s defeat.

Game 3: Miriam moves first

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

In her turn as aggressor, Miriam first complained “Wah!” when Robby moved 1, then went on to defeat him with an expert win. Her conversation highlights her sense of the situation. While Robby, still confident that he had blocked the corner offensive, walked away, Miriam said somewhat gleefully, “Robby, you’re gonna kill me for this,” and then moved B. She then went on to promise him a million dollars in reparations if she should move between A and B. Robby then moved 2; with his move in place, she added, “I wasn’t going there anyway.” When I asked Miriam if she could beat Robby now, she replied, “Yeah, I think so,” and then she did. Robby was not accustomed to being beaten fairly by Miriam, and he was angry. Miriam offered to let him turn the table on her.

Game 4: Robby first

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

After this game, Miriam was surprised that Robby didn’t know how to turn the tables. I agreed that he did not have so specific an idea as we did of what we meant by “turning the tables.” I interpret Miriam’s failure to block row 3 – 1 – 4 as a gift to Robby, so he wouldn’t “kill her” for her prior victory. Although she also achieved her preferred three-corner configuration, Miriam appeared not very interested in winning the game.

At this point, I was called outside the house on peripheral matters. I told the children I would return shortly and asked them not to play with each other while I was gone. Miriam, true to her word, played a game against herself on her magic slate (game 5).

	 A | D | C    
	   | a | g    
	 b |   | B  

Miriam explained on my return that she had been playing tic-tac-toe against herself, making “smart moves” for both herself and “the other guy.”

11/12 This evening, when I stopped videotape transcription, a quiz show override signal from channel 4 displayed a tic-tac-toe board with this configuration:

	 X |   |      
	   | O |      
	   |   | X 

I shut off theTV, then called Miriam and asked if she had figured out yet how to block a corner opening. She said, “Give me some chalk,” thus volunteering to show me. I reached to the chalk supply, drew a game frame, and placed the corner opening. Once she controlled the chalk, Miriam made moves for both players.

Game 6

	 1 | D | C    
	 b | g |      
	 B |   | a 

Miriam claimed she could block the corner opening thus: “Go in the opposite corner.” After response a, Miriam knew the next move would be B. She made that move, the forced b, and moved C — at which point she realized she was forked! Miriam then claimed “he” would not go there. I replied, “Yes, he would.” Miriam responded, “Yes, he would try hard to win. So I block him there [moving g] and he wins there [moving D].

Two points stand out in these data. As aggressor, Miriam is unquestionably able to defeat an opponent with an opening game (first two moves) as in game 3. The last two games show a major new capability as the culmination of Miriam’s development: she can now play both sides of the game simultaneously. I consider such an accomplishment the ultimate decentration in any domain, which, when achieved, renders competition-engaged analysis possible.


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.


Vn112.1 How Her Teacher Sees Miriam 12/7/77

Miriam’s teacher, Sue, sees her as a special child in several ways.
Her surprise at Miriam’s easy solution of class inclusion problems (cf.
Vignette 90, Meeting Miriam’s Teacher) shows she had reason outside of
anything I told her in our first meeting. She learned of Miriam’s continuing
work at the Logo project and was favorably impressed by our links
with the now-respectable scientist Piaget. Thus Miriam appears special
by developmental progress for her age and by the experience of her ongoing
engagement in a serious study.

As The Intimate Study concluded, the children asked if they could
bring their classmates over to visit Logo. I agreed to help them work
that out if they wanted to, on condition that a few children came at one
time and that Robby and Miriam be the ones who ran the show. Both accepted
this scenario as the best one. Robby suggested that their teachers
be first to visit (I don’t know why). Miriam was not keen on the idea
but didn’t argue enough to undermine Robby’s support of the plan. About
the middle of November, the two teachers spent approximately 2 hours at
Logo. The children showed off their computer pictures and their desks,
then explained their work to the teachers. I stayed in the background
as much as possible. Both wanted to play Wumpus, but because this was
confusing to their teachers, they showed them SHOOT and its variations,
explaining the primitives and exhibiting the arithmetic tasks the game
involved them in. Otherwork included the use of POLYSPI and INSPI,
drawings, and a text manipulation work. I believe the teachers were
impressed by the work and the children’s command of it. Sue’s note (see
Addendum 112 – 1) witnesses her response.

Yesterday Gretchen met with Sue for an evaluation conference. (The
report is attached as Addendum 112 – 2, 3, and 4). I was unable to attend
the meeting, but Gretchen recalls these comments:

- Miriam gets a great deal of pleasure from seeing and playing with 
     her school friends.
- Miriam always did her work with a great deal of attention to detail, even
     if she was merely drawing to fill in time between organizeed activities.
- Miriam didn't copy from other people, either to get directions 
     for what she should be doing or to get an idea.
- Miriam cooperated and worked well with her classmates, but not 
     merely that. She tried to help them and was able to do so.
- Miriam seemed to enjoy solving problems. Her focus was not on getting 
     the answer; she seemed to enjoy the process of working out problems, 
     to take pleasure in the process more than in the result.

These notes record a view of Miriam independent from mine.

Addendum 112-1

Note from Miriam’s Teacher

Vn 112-1 Teacher note

Addendum 112-2

Conference Report, page 1

Vn 112-2 Conference report, pg 1

Addendum 112-3

Conference Report, page 2

Vn 112-3 Conference report, pg 2

Addendum 112-4

Conference Report, page 3

Vn 112-4 Conference report, pg 3


Vn113.1 Steady State 12/8/77

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

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

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

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

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

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

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

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

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


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


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

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

Vignette 114, page 1, scanned from Original Fair Copy

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

Vignette 114, page 2, scanned from Original Fair Copy

Vn 114-2 Scanned Original Fair Copy


Vn116.1 Transferring a Good Trick 1/3/78

Miriam, not imagining yet that she will one way or another make
a living, sees her best hope of getting a lot of money as inheriting my
money. Thus my impending demise is a subject on her mind and one that
involves her in computations. At lunch today:


Daddy, if you die in another 37 years — no, if you die in another 30 years you will be 67.

Right. But suppose I live those 37 years. How old will I be then?

(After a short pause, wherein she raised and lowered a few fingers) 74.

That’s absolutely correct. How did you ever figure that out?

I know a good trick. See, you have the 67. And you know the other 7?

(Nodding assent here)

Well, that’s like a 3 and a 4. So I took the 3 with the 67 and that’s 70 and then the 4 left over made 74.

That’s beautiful, sweetheart.

This is the first time I have witnessed Miriam doing a sum with
a decade crossing without the use of a counting-up procedure to sum the
secondary units addend with the intermediate result.

What I find most striking in this decomposition of a single
units digit is that the trick (though similar to her reduction to nines
technique for carrying) was first explicitly applied as a procedure for
mental addition in decadal arithmetic (cf. Vignette 105, Decadal Compu-
tation). Thus here we witness a procedure developed for summing large
numbers being retrofitted to addition of small numbers, and in that
microworld supplanting (in this case) the serial counting-up procedure.


Vn117.1 Playing Both Sides 1/14/78


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

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

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

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

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

Vn 117, page 2, scanned from Original Fair Copy

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

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


Vn118.1 Introducing Peggy 1/26/78

The calculated arrival date for Peggy, our new daughter, was
January 24th. Gretchen, because of her past experience with Robby and
Miriam who were both late, did not expect the birth until the very end
of January. This expectation was a source of some comfort over the
past weekend (Jan. 20-22) during which Boston was subject to a storm
which dumped 26 inches of snow in our area. This was the most snow
from a single storm in the city’s history. Had the baby come Friday
the 20th, a police escort to the hospital would have been our only
hope of getting there. I discussed with our landlord in more or less
serious jest a home delivery. (A psychiatrist, he offered to help as
much as he could but warned me he would not be especially useful.)

Two days passed; the roads were again usable though their
sides were piled high with snow. Gretchen woke me at 4:30 a.m. on
the 23rd, two hours after entering labor, and we proceeded to the
hospital with cautious haste. Arriving at 6 a.m., the obstetrician
predicted an 8:30 delivery. After a short time, he predicted an imminent
delivery. Peggy was born 10 minutes later at 6:46 a.m. This 4
hour labor was very short in contrast to 14 hours with Robby and 8 with
Miriam. Ninety minutes after delivery, with Peggy in her arms, Gretchen
was able to talk to Robby on the phone and tell him she and the baby
were well and feeling pretty good.

With the Monday morning arrival, our plan to take care of
Robby and Miriam had been straightforward. Our landlady would wake
the children and be available to help as they got dressed in preparation
for school. Should they return from school before I returned from
the hospital, she would be available then also, but the children were
to amuse themselves in our house. (The rare permission to watch after-
noon cartoons I expected to keep them out of mischief.) School was
canceled because of Friday’s snow. Robby and Miriam took care of
themselves quite well. They escaped any major mishaps during the day,
though infringing a few rules, i.e. they bounced on my bed as if it
were a trampoline. I met them at home after noon. Subsequently I
prepared an early supper and left them with permission to watch more
TV (a Charlie Brown special and “Rikki-Tikki-Tavi”) while I returned
to the hospital.

The next day each child took a picture of Gretchen and Peggy
(made with Robby’s new Polaroid One-Step) and the good news to share
with their classmates. They visited the hospital late in the afternoon.
As Peggy was wheeled away from the viewing window, she flipped her arm
about. The children claimed she had waved good-bye and began squabbling
over whom Peggy had waved at.

I expected the children to be in school Thursday as I brought
Gretchen and Peggy home. School in Brookline was canceled again that
day, today. The children preferred being on their own this morning to
an indefinite wait in the hospital lobby. We are now 5 at home.


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


Vn120.1 Designing a Box 1/26/78

Miriam approached me holding up a flat, cross-shaped piece of card-
board about 13 inches long, marked as below, with this challenge: “Daddy,
you’ll never guess what this is.”

Vn 120-1 Fig 1.

I gambled: “An airplane?” “No,” Miriam chuckled, “it’s a box, for a
Valentine’s day present for you and Mommy.” Miriam showed me how to
bend it to create a cubical cardboard box. When I praised her new creation,
she explained the way it came about. Then and later, in response to
questions of mine she told this story.

Miriam needed a box for a Valentine’s day present. This requirement
came first. Then, in her words:


I saw the Tic Tac Toe board (a lined 12″ x 12″ cardboard) and I said to myself, “If I bend it along one of those lines and another one, then if I turn it around and bend it the other way, I’ll make a box.” But that doesn’t work because of the corners, so I did it different: I had to cut it and put an excess piece on one end.

Did you really say that to yourself? Or did you see it all at once.

I saw it all at once.

In the flotsam of the play area, I found a piece of cardboard with a figure on each side. One side shows the figure above Miriam describes as her plan. She explicitly denies that she ever intended to make so small a box. The other side appears as Addendum 120 – 1. This was her first attempt to draw the outline for a potential box. The box she finally made is of larger size, the 13″ length first shown.
I reconstruct Miriam’s procedure for making the box as follows. She first concluded she needed a box, then, guided by the Tic Tac Toe grid, she imagined a solution to her problem. Miriam worked out the plan shown in the top figure and produced the unsatisfactory pattern of Addendum 120 – 1. She then selected a piece of material of size comparable to the Tic Tac Toe board, the inner half of a box she had been saving for another purpose (it had been tentatively assigned as the body of a cardboard elephant to be made when Miriam should collect enough cylindrical rolls) 10″ square and 3″ high. Miriam ripped down one side of the box and drew her pattern on the box bottom.

Vn 120-2 Fig 2.

Miriam cut out her pattern, showed it to me, and demonstrated how she had made a cubical box by folding along the dotted lines and the bottom-side crease. The size of the cube sides, of a magnitude comparable to the squares of her Tic Tac Toe frame, was determined by the piece of material Miriam found conformable to her imagined objective.

This vignette is a cameo of Miriam’s problem solving in her own world of objectives and materials. She was happy with her product and the act of imagination implicit in seeing its pattern two ways, as a “raw material” and as an object creatable from the material. (Her challenge to me was that I could only see it one way, or only as a nexus of pretense, i.e. as representing some other object.)
Also noteworthy is her comfortable use of “interior dialogue” as a convention for communicating her thought processes and her admission that it was a fabrication. Her final use of the box two days later to hold a wedding present shows her commitment of the material to her
original objective was slight. I infer that the objective’s import is as an occasion for the working out of an idea.

Addendum 120-1

Unsatisfactory Pattern noted in text

vn 120-3 Miriam'splan


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


Vn122.1 Carrying Bugs 2/5/78

Invited to play at a friend’s house, Miriam waited for Gretchen to
drive her there. During this vacuum of activity, I asked her if she
remembered how to add with carries (cf. Home Session 23). Miriam
reacted impatiently, as though it were foregone that she did. She
agreed to solve a problem I posed on my chalk board and showed
sufficient interest that she tried to peer over my shoulder as I wrote
the sum in vertical form.

                1000   100   10
         |  4  |  7  |  3  |  4  |  5  |
       + |  2  |  2  |  8  |  5  |  7  |	
         |  7  |  0  |  1  |  9  |  2  |

After drawing the columnar division lines, Miriam first said, “5 plus 7
is 2 carry the 1.” “Carry the what?” I asked. “Ten,” she replied and
wrote her marks above the tens column (these marks of hers are hand-
written in the sum above [italics]). She then proceeded: “5 and 4 are 9 plus
10 is 19; put down the 9 and carry.” Miriam did carry a hundred but
failed to add it to her sum of 8 plus 3. Adding the carry from that
11 into the thousands column sum (7 plus 2), Miriam wrote the carry
from that 10 above the identical column with four zeroes (see above)
and added the carry of 1 into the ten thousands column sum (4 plus 2).
Satisfied with her result, Miriam asked me to indicate any columns she
should check.

When I drew an arrow under the tens column and asked whether the
4 was a 4 or a forty, Miriam crossed over the 9 with a zero. Upon my
pointing to her dropping the carry into the hundreds column, Miriam
(who knew the 3 and 8 were 3 and 8 hundreds and that a hundred had been
carried) quit and refused to do more arithmetic before going to her

Even though Miriam appears to have gained a sensible way of
thinking about carries and representing them for herself, her command
is still imperfect, as these two mis-steps of hers indicate. Can she
make such errors and still be judged as understanding carrying?
I believe so. One test would be to see whether on a similar sum
she exhibits these same errors or shows confusion.


Vn123.1 Computation Finale 2/12 & 14/78

2/12 Since completing Vignette 121 (Double Perspectives) I have tried
to engage Miriam in executing a difficult addition. My purpose was to
introduce the idea of a simultaneous, double perspective as what one
needs to appreciate carries by challenging her with a puzzle — “What
number is 10 when you take it away but 1 when you add it in?” Thus,
days ago, I wrote on my chalk board the problem: 22857 plus 47345.
(N.B.: this is the sum of Vignette 122 with addends inverted). Miriam
has refused to look at the problem because, as she explained at lunch
today, I had told her before that she had done so much arithmetic for
me she wouldn’t have to do any more.

She is quite correct, and I tried to make it clear she should feel
no pressure to do any more experiments with me. We continued talking
about how great her skill in computation has become. I speculated that
playing SHOOT at Logo was most important in her learning how to add.
Miriam disagreed and averred finger counting was most important; she
specifically identified her counting up procedure as the most useful.
I objected. Such a procedure was fine for small numbers but not for
big ones, such as 20 plus 30, because one does not have so many fingers.
Miriam demonstrated base-10 finger counting. . . and then generalized her
procedure for my confounding: 20, 40, 60, 80; 40, 80, 120, 160, 200.
I asked if she could count by 12’s. Miriam did so easily up to 60, then
continued on her second hand: “72, 84, 98 — no, 96. . . (a fairly long
pause), 1 hundred 8. She stopped at 9 twelves but answered “120” when
I asked her what the next number would be.

We discussed multiplication in passing. Miriam volunteered her
knowledge of 4 times 90 and when asked, said 2 times 90 was 180. She
was at first non-plussed when I inquired how many were 3 times 90. She
produced her result through counting up in decades from 180.

2/14 What an afternoon! The children and I returned late from shopping
(this was our first auto trip since the Blizzard of ’78 left us snow-
bound). We had gone out for staples, but on this Valentine’s Day
Miriam would have been heart-broken did I not stop to buy her some
heart-shaped candies (she was very explicit). During the course of
lunch, I promised the children we could play with the Logo Cuisenaire
rods afterwards. They ate quickly and began pestering, but I demanded
the right to finish at a relaxed pace the bottle of ale I enjoyed with
my lunch.

While I talked with Robby in the reading alcove, Miriam entered
that area and executed “the next experiment” before I was ready (as she
put it later, “on purpose, to trick you.”)

      10000  1000  100    10 
     |  2  |  2  |  8  |  5  |  7  |
  +  |  4  |  7  |  3  |  4  |  5  |
     |  7  |  0  |  2  |  0  |  2  |

Miriam executed the sum perfectly, writing in the carries as I have
copied them above. When I asked how she could do this sum perfectly but
had manifested bugs on a similar sum days before, she replied, “I remem-
bered how to do the carries.” When Miriam had completed the sum and was
confident that it was correct, I recalled for her her jokes about “what letter
do you drink?” (cf. Vignette 121) and asked if she would like to try a
puzzle of mine. She agreed but was utterly unable to guess “what number
is 10 when you take it away and 1 when you add it back?” Miriam did
understand when I told her the answer was “a carry.”

Days later, Miriam told me she had enjoyed surprising me, doing
“the next experiment” before I was ready, because she likes to trick me.
But more, she said she would not have done it except for one thing: the
day was Valentine’s Day and her effort was a kind of present for me.

On this day, Valentine’s Day, the children and I spent the
afternoon playing with Cuisenaire rods, building the Logo-style right
rectangular polygonal spiral as described in Home Session 24.

Miriam exhibits fairly clearly her grasp of carrying and distributed
addition is sufficiently strong that she will remember it. She may
produce occasional errors and may even suffer minor confusions, but
I believe she now understands distributed addition. By this I mean her
understanding of the parts and wholes of numbers in vertical form
addition will permit her to reconstruct the addition procedures she
needs however many times she forgets them.


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.


Vn126.1 Turtle on the Bed 3/14/78

This Saturday morning I sat in the reading alcove working away, and
Miriam came to join me. Robby was downstairs and Gretchen out of the
house. Miriam offered to sit in my lap, but I protested to being busy
and turned her down.

Miriam moped a little, then crawled on my bed and into the center.
She began to move and spin in a most puzzling and distracting fashion.
“What are you doing? You’re driving me batty!” My gripe inspired
Miriam to explain. Requesting a pen and a 3×5 card, she drew the picture
below of what she was doing in her “crawling on the bed game.”

Vn 126-1 Turtle on the Bed

Miriam’s verbal description was that she was “making one of those maze
things.” (Cf. Home Session 23, 2/14/780

I value this incident as an example of Miriam’s exporting into her
play world the kinds of knowledge and activities The Intimate Study
involved her with at Logo.


Vn127.1 Moo Shu 3/19/78

There is an old joke of this simple script:


Do you know how to read Chinese?

I don’t know. I’ve never tried.

At lunch today I described to Miriam my lunch of yesterday at a
Chinese restaurant — showing her then the take-out menu. As we looked
at the menu, I mentioned that Seymour had talked about learning to read
Chinese — and I asked Miriam if she knew how to read Chinese. . . so she

She was able to read in English “Moo Shu Shrimp.” Since we had
brought home dinner from another restaurant a few days before, Miriam
still remembered the “Moo Shu” as indicating the thin pancakes, and she
recognized “Shrimp.” She announced her discovery. “Hey, Dad. There
are 3 of these (ideographs) and 3 words. That must be Moo Shu Shrimp
(indicating the translation by one to one correspondence).”

“But how do you know which thing means which word?” With this
challenge, Miriam turned back in the menu and located “Moo Shu Pork”,
then flipping from one leaf to another, “See. Look here. The first
and second ones are the same. Moo Shu! I can read Chinese!”

Recall Vignette 17, wherein Miriam claimed to be able to add
big numbers and divide (she knew one division result: 8 ÷ 8 = 1, and
a single addition of big numbers, 1035 + 2000 = 3035). With a little
reflection, looking at the 60 or more characters on a leaf of the
menu, Miriam knows she can’t read Chinese and considers her claim a
joke. This incident marks by contrast how seriously Miriam took her
earlier claims of Vignette 17, when solving a single problem represented
to Miriam an example of a general capability.


Vn128.1 Robby’s Topological Game 4/2/78

About a month ago, Robby was shown a paper-cutting game by a
classmate’s parent. The procedure to follow was this:

1. Cut two paper strips of equal length (8″ will do)
2. Draw a line down the middle of each (using lined paper makes
this unnecessary)
3. Bend each strip of paper into a circle and tape the juncture
4. Join the circles perpendicularly and tape the juncture
5. Cut around the mid-line of each circle.

When two strips of equal length are so connected and cut, the surprising
result is that, though having passed through a circular phase, the strip
halves end up taped together as a square.

Squaring two circles

Robby enjoyed this game when shown it. Yesterday, I removed
a paper form he had made in the past (an 8 x 11 sheet divided into 11
strips 8″ long) from my clipboard and gave it to him. When I inter-
rupted his reading to give him this sheet of paper, Robby recalled the
game and quietly took it up on his own. He was very happy when the
procedure produced a square and showed it to Gretchen and me. We neither
paid much attention.

Going on to three circles, Robby cut two of the three along
their mid-lines. He judged (in error) that he had finished by finding
a square with a bar (a double strip) across the center. It lay flat.
Still no one paid attention. Robby went on to four circles, and he
cut all the mid-lines. What he got was a confusion of floppy paper.
I advised him to try to get it lying flat. Robby again borrowed my
clipboard, clipping and taping the product to it. He was delighted
when he succeeded in flattening the strip-figure and subsequently
taped it to a large piece of cardboard. The resulting shape is this:

But why stop at 4? Robby went on to connect and cut 5 circles. Here
he met another surprise. When cut, the 5 circles separated into
identical, non-planar shapes. Robby likewise taped these to another
piece of cardboard. When he made a cutting of 6 circles, controlling
the floppy strip-figures became a big problem. Robby succeeded at
taping it to the box from which he had been cutting cardboard backing
pieces, but in doing so went over an edge. He decided the problem
was getting too complicated to be fun and quit.

This morning I told him I had been thinking about his paper
cutting game and asked Robby to find the figure made from three circles.
When he returned, I asked him if he had cut all three circles. Robby
thought so, but when I pointed out the middle bar in his square was
double thick, he agreed he had only cut two. Robby saw immediately
that his square would divide into two rectangles. He cut the center
strip. “The 5’s made 2 too. Hey! I’ve got a new theory: the odd-
numbered circles make 2 and the evens all stay together.” I agreed
that this was an interesting speculation and that I could believe it
might be true, but that I couldn’t see immediately why it should be.

I see this incident as one exceptionally valuable for
characterizing how significant learning occurs very naturally in a
mildly supportive milieu. First note that the initial exposure to the
“phenomenon” was quite memorable and puzzling. (How can you make a
square from two circles?) Robby clearly marked this phenomenon in his
mind as one which he would explore later. This pending explorarion was
invoked by the accident of his seeing a piece of paper approximately
meeting the material requirements for use in the game. The circum-
stance was one of no pressure. (He had been reading all of Gretchen’s
collection of Oz books and was probably a little bored.) He had no
outside direction or motivation at all. Once Robby succeeded at
making a square, he continued executing the procedure with stepwise
complications all focussed on one variable — the number of circles.
(He might have chosen to make the strips of different lengths — a
possibility he mentioned.) With the 3 circles, Robby stopped prema-
turely because he had produced a result (a square with a bar) only a
little different from the next simpler case (2 circles make a square).
With 4 circles, the outcomes of cutting were apparently sufficiently
confusing that completion could not be judged from the product but
depended on verifying that individual steps of the procedure were
completed. With the figure of 4 circles he was excited and delighted
to have succeeded in imposing some sort of order on the tangle — and
that the final product showed a family resemblance to the earlier
products. Finally, Robby was quick to jump to conclusions (his new
theory) in explaining why some figures were connected and others were

Post Script — 4/3/78

After writing the preceding, I spoke to Robby again of his
game and his theory, inquiring whether or not he could prove it correct.
His method of choice was to test the case of 7 circles (which, as he
later found, splits into two planar figures of overlapping near-squares).
I tried to introduce the idea of a proof in place of another case study,
suggesting he take all possible cuttings of 3 connected circles and
figure out which one cuts the strip-figure in half. He said he had cut
the center first one time and at another had cut from one end.

Robby then drew the two pictures below on my chalk board:

Vn128-2 intermediate state squaring circle

He argued that it is always the last cut that severs the strip-figure
in two, representing the situation as at the above left. By cutting
along the dotted line, one joins the two small circles (here he made
motions of pulling apart the strip pieces) into the one large one.
Note well that this argument is merely a restatement of how he
appreciates the deformation, but it contented him.


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).


Vn130.1 4/3 & 10, 11/78

4/3 Miriam noticed a sum in Home Session 7 as I worked on a paper
and asked if she could do it. When I wrote the sum on the black board,
Miriam added right to left with carries, thus:

             3     7     4     1
       +     2     5     3     0  
             6     2     7     1

However, before I wrote down the problem, I had asked Miriam to do so,
saying the first addend 3 thousand 7 hundred 41. Miriam wrote:

3000 700 —

then complained that she had run out of room on the black board.

After Miriam had a snack, I called her back to a cleared
chalkboard and asked her to write this number — 7443, and then 2322.
Miriam wrote both in the standard form. When I asked why she had earlier
written 3741 differently, she replied, “To get you confused.”

Miriam’s peculiar notation for 3741 shows the upsurgence of an
obsolete representation. It is not surprising that it surfaces in a
task where Miriam must produce the representation rather than merely
manipulate it (cf. Vignette 29, Making Puzzles). I interpret her final
remark as a sign that she is becoming increasingly defensive about her
thinking. (It is also, of course, an excuse for her embarrassing confusion.)

This second problem confirms as robust, both in execution and
against challenge, Miriam’s application of the standard algorithm for
vertical form addition in the cyclic notation. Miriam has “learned to
add” in the common sense, as well as in her own, less common ways.

POST SCRIPT: 4/11/78

To verify that Miriam would not be confused by cascading carries
as she was in the past (Cf. Home Session 8), I left upon my chalkboard the
sum 248,443,575 plus 531,576,428 (Cf. problem 2 of Addendum 130 – 1).
Miriam expected me to bargain with her over doing the problem. Whenever
she inquired, I told her not to do it — rather she should play outside on
this sunny afternoon. Later, she came determinedly up to my chalkboard:
“I’m going to do that problem.” She proceeded right to left, taking the
cascading carries in stride. Miriam did not recall the result of 4 plus 7, [but]
achieved the correct column result through finger counting. When she
finished, I asked her if she could read the result. She could not. Her
best attempt was 7 million 8 hundred 2 thousand and 3 (Had there not been
so many zeroes in the result she would have given up completely). Miriam
knew that her reading of the result was not standard.

Addendum 130-1

Chalk Board Sums (home session 7)

Vn 130-1 Miriam solves an old problem

Chalk Board Sums (home session 8)

Vn 130-2 A second old problem


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


Vn132.1 BIG-SUMS Extension 4/21 & 22/78

4/21 “Oog. I’ve got to add all these big numbers.” So I called to
Miriam’s attention a situation in which one could make use of the kinds
of skills she had developed in adding multi-digit addends: I was summing
the number of calories in my day’s diet. Miriam asked if she could add
the numbers for me. I protested that there were four big numbers to add
together and asked if she could do that. Taking a piece of note paper,
she replied, “I think I can do it.”

Miriam worked the problem of Addendum 132 – 1 in the notebook
I held on my lap. I expected her to perform 3 separate additions.
Instead, after I had read the first two addends (235 and 560) and Miriam
wrote them down in the vertical form, she asked me what was the next one,
and said she was going to do them all together. With the four addends in
place, she put in columnar division lines and plus signs, then underlined
each addend. I asked Miriam if she had been doing any such work in school
and why she underlined the addends. She answered “No” to the first
then question and “I don’t know. I just like to.” to the second. She declared
that this was the first time she ever tried “anything like this”, never in
school, and she had never seen Robby do it; she wanted to try just because
she thought she could do it.

The units sum was done first. As she summed the tens column
digits, Miriam asked for confirmation. “You put down the 4 and carry the
1, right?” I agreed. In the hundreds column, Miriam said, “You put down
the 7 and carry — no.” She wrote 17 in the hundreds column and continued,
“It’s 17 hundred and 45.” Then Miriam went on to draw her star and
added a ‘happy face’, which is her teacher’s symbol put on work well done.

4/22 As I recorded my supper calorie count, Miriam, who had a head
ache and felt too ill to eat supper, eagerly asked if she could add my
big numbers. (See Addendum 132 – 3). One addend I read as 12 hundred
85. As Miriam started to write 12 in the hundreds column, she asked if
it were correct. I told her another way to say the number was “1 thousand
2 hundred 85.”

She began adding at the left but immediately thereafter realized
there would be a carry-in and declared her first result an error. After
restarting on the right, at the sum of the tens column Miriam asked, “Do
you put down the zero and carry the 2?” I responded, “Do you think that’s
right?” She answered confidently, “Yes” and proceeded. When I inquired
whether her carry mark were a 1 or a 2 (it was initially illegible),
Miriam said “2” and rewrote it.

Upon her completing the sum, I asked what was the result.
Miriam answered, “2 thousand twenty five [for 2205]. Is that right?”
I said “No.” After a little while, Miriam said, “2 thousand 2 hundred
and 5.”
Later in the evening Miriam explained, “When I said that answer
had twenty five, I thought it was like the 2, 0 was twenty and then the
5, but I know that’s wrong.”

I consider the incident of 4/21 an extraordinarily rich one.
Consider the precursors of the achievement. Miriam’s use of the standard
algorithm for addition in the cyclic notation is well-developed but
shaky, because of the difficulty of verifying the results of such a
computation. The second source is one Miriam denied (cf. Addendum 132 – 2)
as being like this problem. The triple single-digit sums of her school
work from last week clearly suggested to her the possibility of adding
more than two addends at a time. Her sense of the differentness of the
kinds of problems is evidence that she sees them as belonging to dif-
ferent worlds of thought and thus potentially connectible to different
cognitive structures.

The second significant aspect is that this problem shows as [an?]
internally directed extension of integrated knowledge as the elaboration
of a new procedure (handling triple single-digit sums) and its insertion
as a subprocedure in the controlling superprocedure Miriam uses for
multi-digit addition.

Finally, in drawing the teacher’s ‘smiley face’ on her own
math paper, Miriam shows herself playing both roles of ‘learner’ and
approving authority that is reminiscent of her playing both sides of
games at Tic Tac Toe.

In the problem of Addendum 132 – 3 Miriam confronted again
the double naming of ten-value hundreds as thousands and met her first
carry of value not 1. Her reading of the result suggested that with a
single, well-presented contrast of numbers to read, she should be able
to lock in the distinctions necessary to read any 4-digit results cor-
rectly. A day or so later, Miriam was able to read numbers from this
parallel list, and when shown the number names first, predict where the
zeroes and fives would appear.

             5     FIVE
            50     FIFTY
            55     FIFTY FIVE
           500     FIVE HUNDRED 
           505     FIVE HUNDRED FIVE
           550     FIVE HUNDRED FIFTY 
           555     FIVE HUNDRED FIFTY FIVE
          5000     FIVE THOUSAND 
          5005     FIVE THOUSAND FIVE
          5050     FIVE THOUSAND FIFTY

Addendum 132-1

Multiple Addends -1

Vn 132-1 Multiple Addends -1

Addendum 132-2

Schoolwork Sample

Vn 132-2 Schoolwork Sample

Addendum 132-3

Multiple Addends -2

Vn 132-3 Multiple Addends-2


Vn133.1 4/28/78-5/8 & 31/78

4/28 The calorie counting continues for me, and the adding for
Miriam. The sum of Addendum 133 – 1 is interesting because of its error:
the left justification of 3 and 4 digit addends in the same vertical grid.
Despite this mal-arrangement, with its implication that the place-value
basis of column alignment has not been mastered, Miriam’s carrying
procedure was standard.

5/8 When the children started squabbling over whose turn it was to
add up my daily calorie count, I refused either the privilege. But today
Miriam took up the task with no complaint from Robby so I let her proceed.
She wrote the addends and result thus:

Vn 133-2 3T like grid for Addition

Most interesting was her procedure. Beginning at the left, Miriam cal-
culated 14 for the column and carried a 10 to the next right column. I
intervened, “Miriam, don’t you remember your good trick for when you
have carries?” She restarted on the right and reached a standard result.

5/31 After staying up late last night at a dinner with Mimi Sinclair
and Marvin and Gloria (whereat Miriam showed off a little by working with
the soma cubes and the 7-piece pythagorean puzzle), Miriam stayed home
from school today. During the course of the day, I posed the following
problem, designed to contrast for her results of mental addition and
vertical form addition to point up the question of place value in the
vertical form.

Vn 133-3 Place Value Alignment Test

When Miriam saw the horizontal sum, a sheet of paper covered the question
and sums below the dotted line. She knew the result was 20 and after she
had written down the answer I took away the covering sheet. Miriam read
the question, then mused, “For both, the numbers are eleven.” Puzzled,
I asked what she meant, but Miriam continued without responding to add the
left sum from the left to 65. She stepped back, then with a “No. That’s
not it.” returned to the chalk board to erase her result. I stopped her
erasure and directed her to the second sum, which she added to 20 (from
the right, with only mental marking of the carry), then encircled. When
I asked how she knew the right sum was correct, Miriam argued that the
answer was 20.

These three incidents all focus on the imperfections of
Miriam’s place value comprehension. The last I expect to be seminal,
leading Miriam to recognize that the results of addition, in whatever
form, should be consonant, and that she can herself judge the correctness
of small vertical sums by checking through mental computation.

Addendum 133-1

Column Alignment Error

Vn 133-1 Column Alignment Error