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LC1bT06

LC1bT06 Protocol 6

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RAL protocol 6.1

RAL protocol 6.2

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RAL protocol 6-A1

RAL protocol 6-A2

LC1bT07

LC1bT07 Protocol 7

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RAL protocol 7.1

RAL protocol 7.2

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RAL protocol 7-A1

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LC1bT09

LC1bT09 Protocol 9

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RAL protocol 9.1

RAL protocol 9.2

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RAL protocol 9-A1

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RAL protocol 9-A5

LC1bT21

LC1bT21 Protocol 21

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RAL protocol 21.1

RAL protocol 21.2

RAL protocol 21.3

RAL protocol 21.4

RAL protocol 21.5

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RAL protocol 21.8

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RAL protocol 21-A3

Vn00301

Vn003.01

Openness and Reticence

5/10/77

Miriam rarely discusses her activities in kindergarten unless directly asked. I was surprised today when Miriam volunteered that her class had put on a little play. I asked what part she played.

Miriam I didn’t play any, Daddy.
Bob Oh, didn’t you want to?
Miriam I read the story and the others did it.
Bob Was it hard to read?
Miriam Yes, very hard, but I did it good.

It may be that Miriam’s taking her reader Friends Old and New to school had something to do with this selection.

Vn00701

Vn007.01

A Willing Subject

5/18/77

Today was the most difficult experiment of the initial series, separating the variables implicated in the flexibility of bending rods (Cf. The Growth of Logical Thinking from Childhood to Adolescence, Inhelder and Piaget). Of the many experiments through which one may distinguish concrete from formal thinking, this one shows best the distinction between conceiving of things as objects with properties which may occasionally be of interest and conceiving of objects as instantiations of intersecting ranges of properties of which some may be relevant to its use in a specific context.

How could Miriam possibly be interested in distinguishing the contribution of different properties to rigidity? How many adults could, or would care to, distinguish between the relative importance of a rod’s thickness parallel to a force and its cross-sectional shape? How to quantify, or even balance, such factors is not obvious to the untrained intuition at any age. And Miriam is 6, not 16. Need one not subject a child to exorbitant pressure to have her work at such an experiment? If any experiment in this initial series is vulnerable to such a criticism (either through its very nature or through my execution) it is this experiment.

I was hurried into the bending rods experiment by Miriam’s declaration that she would not come to Logo today unless she could do it. Despite feeling imperfectly prepared, I acquiesced in her demand since the obvious problem was that she felt ‘left out’; I had done the experiment earlier with Robby for practice. During the preliminaries wherein I hoped to establish with certainty a common terminology for the experiment (for a description of this work see Miriam at 6: Bending Rods) Miriam was rambunctious. When not interested much in my questions, Miriam began to create chaos by throwing around dice and balls, which were the experimental materials. I stopped the experiment and we had an argument of this general form:

Bob I won’t tolerate your throwing things and causing this confusion; it will ruin the experiment.
Miriam Then I won’t do it. You’re asking me dumb questions. I’ll play SHOOT instead.
Bob That’s not fair. You demanded we do this experiment today and now we have to finish it.
Miriam I never promised I would do all your experiments. I only said I might do them.
Bob That’s not true.
Miriam I wish I was a baby again so I wouldn’t have to do these experiments.
Bob But you’re six now and doing something important.

Miriam reluctantly agreed to continue; she agreed because of the pressure I put on her, because, in effect, I made her do it.

Despite this grim start to the main experiment, Miriam quickly became engaged in working with the materials and finding “which one is the champion” for bendings. She exhibited no interest at all in separating the variables; consequently, the latter part of the experiment bored her silly.

Riding home from Logo, I told Miriam I felt bad because I thought I was pushing her too hard in the experiment. She said she didn’t want to do any more like that, and I replied that we would do no more of that sort for several months. I expanded that we should think of doing other kinds of things. That this thesis project could be not just going to Logo, but going to other places too. Part could be my coming to kindergarten with her; another part could be visiting friends and family and telling them about our project; another part could be trips to places around Boston. I noted that she had been to the Harvard Peabody Museum and I never had; she could take me there. Miriam said we should do that soon.

Vn00702

Vn007.02

A Willing Subject

5/19/77

As the children left for school this morning, I asked Miriam if she wanted to come to Logo today. She said she would not come. When I asked why not, Miriam replied, “I just need a day off.” After the previous day’s tensions, it seemed a most reasonable request.

I planned at Miriam’s return from school a small shopping trip, with perhaps a detour to a favorite playground. Miriam came bounding in from the school bus. “Hey, Dad, let’s go to Logo.” I objected that I had nothing planned to do for the day because I thought she didn’t want to go. “That doesn’t matter,” she said. “Let’s go anyway; I’ll play with SHOOT or something.” When I asked why she changed her mind, Miriam said, “I don’t know. I just did.”

Vn01201

Vn012

Miriam Collecting Data

5/23/77


While the children were awaiting the school bus this morning, I noticed the little blue notebook in which Miriam was to record her kindergarten activities. I took it out and offered it to her. She said she didn’t want it when I reminded her what it was for and then confided, “I can’t spell all the words of the things I do, but I’ll remember them.”

In a short conference with her teacher, I inquired about Miriam’s role in the skit she mentioned in Vignette 3. I was informed that the skit was an impromptu performance, no adults had been working with the children’s group at the time, and that the other children selected Miriam to be the narrator and tell them what to do.

Miriam and I drove home from school (we stayed a little longer to play in the playground). While Gretchen was making lunch for us, Miriam came to sit beside me, saw the blue notebook, and asked if I would help her write down what she did. Four main events:
– housekeeping corner
– going to the Star Market (a four block walk) with Sue to buy provisions for Wednesday’s farewell party
– making peanut butter
– recess

Relevance

Although this single collection of data is insignificant, such notes for the remaining 6 weeks of school will indicate Miriam’s foci while there. Because I go to the school freely, such data will help me select what parts of her school setting are most relevant to her development and worth detailing.

Vn01301

Vn013

Phonemics

5/23/77


In discussions some weeks ago with a distinguished Genevan psycholinguist, the question of instructing children in phonetics came up. My earlier work on children’s conception of wordhood (‘Pre-Readers’ Concept of the English Word’) and pig-latin (not available) had convinced me that children should learn how to read before delving into phonetics. (Such a point of view is contrary to much current practice in late kindergarten and early first year instruction). My colleague considered it an aberration in American education to emphasize phonetics as an introduction to reading.

Miriam now reads well, at what is generally considered the second grade level (see ‘Miriam at 6: Reading’ for more detailed infor-mation). I speculate confidently that her reading vocabulary is several thousand words. She now knows enough to appreciate the value of knowledge about the correspondence (and lack thereof) between English phonemic and lexical structure. The following observations record the upsurgence of Miriam’s interest in phonemic variation:

In the spring and summer of last year, one of Miriam’s favorite games was bouncing on my knees. We would sing the familiar chant:

Ride a horse to Boston,
Ride a horse to Lynn.
Careful when you get there,
Don’t fall in!

The child rides facing the bouncer, whose knees are the horse. On the last line, the adult separates his knees, and the child tries not to “fall in.” Boston was a place I went to. Lynn was the neighbor who lived across the street from our Connecticut home. (Miriam has yet to discover the existence of Lynn, Massachusetts). The other common variant chant is:

Trot, trot to Boston
To buy a loaf of bread.
Trot, trot home again,
The old horse is dead.

I introduce this history to enforce the idea of how well known are these rhymes to Miriam.

Today at lunch, having finished earlier than Gretchen and me who sat talking and unattentive, Miriam picked up a large rag doll, put it on her lap, and chanted (quietly, to herself):

side a sorse to soston,
side a sorse to synn.
sareful sen sou set sere,
son’t sall sin.

Then:

ide a orse to oston,
ide a orse to ynn.
areful en ou et ere,
on’t all in.

And then:

fide a forse fo foston,
fide a forse fo fynn.
fareful fen fou fet fere,
fon’t fall fin.

It is clear that phonemic separation is an issue that engages Miriam now.

Relevance

Although she knows that letter represent sounds, it is not clear that Miriam understands the way in which phonetic knowledge will help her with reading. She may. Do note, however, that Miriam was introduced to phoneme separation two weeks ago (confer Logo session 5) through using my pig-latin system to drive the voice box. Five days ago, Miriam selected that activity as one she wanted to engage in (confer Logo session 10) even though I attempted to dissuade her from it by arguing that the programs I had available were limited and the output boring. Today I asked Miriam what I should plan for tomorrow: story writing, drawing, voice box. . . we didn’t always have to play SHOOT or READY, AIM, FIRE. “Voice box,” she exclaimd. “That’s what I want for tomorrow.”

Vn01401

Vn014.1

Housekeeping Corner

5/24 & 27/77


Since Miriam started recounting where she spends her time in kindergarten (see Vignette 12), it has become clear that she spends most of her time in the ‘housekeeping corner.’ When I’ve been in the kindergarten class, I’ve usually found myself playing with blocks, or making designs and elaborate towers from Cuisenaire rods. Miriam has frequently played otherwheres. I have seen her there, curled up in the baby carriage, but I’ve had no good idea of what games she and her friends invented for that location.

That question received a major clarification today while Miriam was stringing beads for necklaces. I had given her a Bic pen cap to poke the string through the beads. After she learned the cap was from an exhausted pen, Miriam began this conversation:

Miriam Can I keep the cap?
Bob Sure.
Miriam Thank you. Tomorrow I’m going to take it to school.
Bob The cap?
Miriam Yeah.
Bob What for?
Miriam Because we usually play Doctor, there in the housekeeping corner.
Bob Un-huh.
Miriam And we give each other pretend shots.
Bob Oh my goodness. You’r not really going to poke anybody with that, are you?
Miriam No (you silly Daddy).
Bob But that will be your needle?
Miriam Yes. Sometimes we use pencils.
Bob I hope nobody ever gets hurt.
Miriam We don’t.
Bob Good. . . I think that’s kind of funny. That you have a house-keeping corner and you play Doctor. Is that because everybody likes giving shots?
Miriam Yeah.
Bob Does anybody like getting shots?
Miriam No. We always run away from the Doctors.
Bob Well, who’s the Doctors. . .or does it change?
Miriam It changes. We run away ’cause we don’t want our shots.
Bob Yeah.
Miriam We always have it in the summer. We run away because we don’t want it and the door’s always open in the summer.
Bob You mean the door to the kindergarten? Or the housekeeping
corner, a pretend door?
Miriam A pretend door to the housekeeping corner. It’s always open so we run right out.

One fact, of possible relevance in initially suggesting the game to the children, though not at all accounting for its continued interest, is that Dara’s mother is a nurse. Dara and Maria are the two girls Miriam most plays with in kindergarten.

After we focussed a while on the beads, I resumed the theme of the housekeeping corner by attacking the game of ‘Doctor.’

Bob I still think it’s kind of silly that you play Doctor in the housekeeping corner. Do Doctors come to houses or something?
Miriam Sometimes they do.
Bob Don’t you ever play anything else? Or is it always shot giving?
Miriam We like the Doctor but sometimes we play House of the Wicked Witch.
Bob Wicked Witch? How’s that go? I never heard of that. Is that
like something from the Wizard of Oz? Or a different wicked
witch?
Miriam From the Wizard of Oz.
Bob Does anybody know the song or what?
Miriam I and Dara know the song.
Bob You and Dara?
Miriam And Maria
Bob How’s it go? “La la the wicked, la la the wicked witch, la la the wicked witch is dead”? No? That’s not the song?
Miriam No. It’s about the Wizard.
Bob Oh. We’re off to see the Wizard?
Miriam Yeah.
Bob The wonderful Wizard of Oz?
Miriam Yeah.

It’s clear that my wicked witch was she of the west upon whom did fall Dorothy’s house. In retrospect, I’m sure the children think more of the Witch of the East, she commander of flying apes and profoundly allergic to water. No dancing Munchkins for them.

6/27 Miriam arranged for Dara to come play at our house today. Because Miriam expected to come to Logo, I asked her if she intended to bring Dara with her and wondered whether Dara would want to come. Miriam responded that she could get Dara to come to Logo by telling her it was a good place to play Wicked Witch. I had no idea why this was so.

Dara and Miriam at lunch told me a little more of Wicked Witch, not clearly perhaps but enough to reveal what sorts of sides and tensions exist. They mentioned that the boys build spaceships in the kindergarten and should they be left unattended, the girls play Wicked Witch, swoop down on the spaceships, and keep the boys away. This, I saw too late, was relevant to Logo’s being a good place to play Wicked Witch. Robby and Sam have been playing war games in the Learning Lab, building barricades or trenches from unoccupied chairs. When Miriam and Dara seized the momentarily unoccupied trenches, I realized from the commotion how Wicked Witch was being applied at Logo.

Gretchen informs me that while the children played at home, most of their time was spent playing Doctor in the tree fort.

Relevance

Both Doctor and Wicked Witch are highly mobile fantasies which appear to be role centered with improvised skits focused on dramatic actions: giving and getting needles; seizing somebody else’s place. From outside the kindergarten, the setting dependence of the games I speculate to be primarily in the nature of a space allocation. The girls play in the housekeeping corner. They use it as their home base for whatever fantasies they can construct with a sufficiency of roles for themselves.

Vn01701

Vn017A

Arithmetic Ripples

5/28/77


After the session in which I introduced Miriam to adding large numbers (see Home Session 4, 5/28), passing Miriam’s room I noticed in her open loose-leaf book a page of computation. Miriam later gave it to me and I include it as Addendum 17 – 1.

Note that the written form of the equations mimics the horizontal form used in our introduction (see addendum 1 in Home Session 4). Additionally, Miriam attempted here a subtraction with large numbers (i.e. 80 – 7 = 73), her suggestion which I turned down during Home Session 4. Place value, as a topic of interest to Miriam, appears not only in her large numbers, but also in the directly contrasting sums: 11 + 1 = 12 and 1 + 1 = 2.

When she gave me the page, Miriam explained her attempt to subtract 7 from 1; how 1 minus 1 was zero and 1 minus 7 was zero. I expect she will conceive of the negative integers soon.

Relevance

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

Addendum 17-1

ADDENDUM 17-1

Comments Off on Vn01701

Vn01801

 

Vn018.01

Housekeeping Corner (3)

6/2/77

 
In her race to the school bus this morning, Miriam left behind a present for her friend Maria and scripts for the play rehearsal (see Vignette 19). When I entered the kindergarten about 9 (which means the children had finished their quiet reading time and the general class meeting), I saw Miriam and several friends playing in the “math” area. This area is defined by a table and a cupboard which is laden with toys such as cuisenaire rods, blocks, geometric puzzles, et cetera. The children were playing with Willy Fangel’s “Connector Set,” a sort of oversized tinker toy collection of rods and 4″ wooden disks. All the disks were in use — as lollipops. Three or four children pretended to eat these wooden lollipops and were squabbling over the distribution. As the newest entrant to their group I was given one by Elizabeth, a friend I knew from Miriam’s birthday party, visits to our house, and my previous visits to the kindergarten. I took another disk and then claimed my lollipop was great because it had candy on both ends. A few imitations later the children were using their dumbbells to show how strong they were.

 

When I gave to Miriam the present she had made for Maria, that interrupted the play. Miriam explained it was a going away present she had made for Maria (her closest friend and a frequent visitor at our house) who will soon be returning with her family to Spain. The present was, of course, to be opened immediately. After Maria saw her U-Bake-It Owl, she and Miriam discussed what presents Maria would give her in return and when.

A migration then began to the housekeeping corner. The children asked me to join their play to be the Daddy. I agreed on condition that I not be made to eat any more wooden lollipops (which the girls carried over). I found myself seated on a small chair at a small table. The housekeeping corner has a set of toy sinks across the wall, a refrigerator, a cupboard wherein is kept the food, and shelves. A large (big enough for two 6 year olds to crowd into) baby carriage and a pile of clothes were the other main items in the area. The children immediately began arguing over roles. “You be the mother. I’m the baby.” To reduce the conflict I noted we might have twins. That was acceptable and Michelle, who didn’t seem to mind too much, was constrained to be the mother. She began struggling with a short nightgown– long enough for her– while Miriam and Elizabeth climbed into the baby carriage. Since there were two blankets, this seemed a nicely balanced situation.

For a moment only. The children, still carrying their collection of wooden lollipops, began playing GOO-GOO-GA-GA. This game has three main features: the children say nothing but GOO-GOO-GA-GA; they fight with each other over the blankets; they throw to the floor whatever they can reach. The additional element they explained to me is that the parents get to pick everything up. I refused to play any such parental role. Michelle then acted her part. Going into the cupboard, she took out all the plastic fruit and boxes of cereal and dumped them over the babies — who proceeded to throw them to the floor.

Miriam, pulled out of the baby carriage by Michelle, crawled under the table to maintain possession of the blanket. Enter Meg, the largest child in her class. She too joined GOO-GOO-GA-GA. The noise level and contention increased. When Michelle rid herself of the nightgown there were four girls playing tug-of-war with two blankets amidst unspeakable noise. The one English word I was to hear come up: “share.” Elizabeth declared she would take a bath, hopped on the sink with her end of one blanket, and invited Michelle to join her in the bath. Then Meg and Miriam joined the bath. Four girls in the bath jabbering GOO-GOO-GA-GA and laughing like crazy.

I picked up some of the plastic fruit and walked away juggling. While some of the other children took an interest in juggling (and Maria boasted that her father can juggle 10 eggs at once), the game in the housekeeping corner came to an end and the girls decided to rehearse a play (as described in Vignette 19).

Relevance

This incident exemplifies a game’s functioning as an excuse to engage in a small repertoire of satisfying actions. It appears to be the case that some peripheral factor (such as having a type-cast Daddy available) may evoke and not at all constrain these shared actions.

Vn01901

Vn019

Rehearsal

6/2/77


Last night Miriam asked me to come to kindergarten this morning to help out with a rehearsal of ‘Goldilocks.’ (see Logo Session 17, 5/1). When Miriam began showing an interest in plays (cf. Vignette 3), I mentioned to her that those years I spent at Yale were in the Drama School, that I had also done the kind of thing she was describing to me. I can not justify it by having impressed Miriam. She told me she believed I could be of help to her because I had a good book about plays for little people [A Dozen Little Plays, Parents’ Magazine Press 1965]. (She has read that book and finds its costuming of major interest; she asked if I would make for her Fox and Duck costumes. That book is a piece of flotsam remaining from a Master’s thesis project on ‘role rotation’ I once proposed and dropped.)

When I retreated from the housekeeping corner (as described in Vignette 18), Miriam’s intention to stage a rehearsal came to the fore. She asked her teacher if the children could go rehearse in the gym but that was not possible so her cast gathered near the clothes closet. Miriam brought out the scripts of ‘Goldilocks’ we had created the previous day at Logo. The potential cast was 6 in number: Miriam, Maria, Michelle, Elizabeth, Meg, and me. Given the dramatis personae of the 3 bears and Goldilocks, though I was immediately type cast as Daddy Bear I declined and elected myself to the audience. Then the squabbling began. Counting remaining noses, I was quick to suggest that we needed an “announcer.” Miriam declared she would do that and assumed directorship: “Meg, you be Daddy Bear.” (Meg is the largest child in the class). “No. I was Daddy Bear last time. I want to be Baby Bear.” “Michelle, you be Mommy Bear.” “No. I was the Mommy already.” (confer Vignette 18). To Elizabeth: “Here.” (as she gave her a script). When I asked Elizabeth what she thought of the script, she said it was nice but gave it to me because she can’t read. Maria made a cone-shaped hat of hers: “I’ll be the witch.” (a probable reference to the “Wicked Witch” game). Miriam tried to cast Maria as Goldilocks — Maria quit. Elizabeth and Michelle also refused the role. We were at this impasse in casting:

Miriam Self-declared narrator and director
Meg grudgingly willing to be Daddy Bear (and practicing growls)
Elizabeth both bears but neither willing to be the Mommy Bear
Michelle both bears but neither willing to be the Mommy Bear
Goldilocks persona non grata

The problem was solved by ignoring it. “We need costumes,” said Meg. I suggested shopping bags with head and arm-hole cut-outs as making good ‘bear suits.’ “No. Masks is what we need.” Seizing on this suggestion of Meg’s, the children got paper plates to make masks. My contribution was to mark the position of their fingers when each held a plate to her face and located her nose, eyes, and mouth. The children had finished cutting out the necessary holes and made the faces those of bears with brown crayon. Miriam was told there was no time for putting on the play before gym, but they could do it after. So the children closed off their activities and got ready for gym.

After gym, it was clean up time. Put away the wooden lollipops and those fruits Michelle had dumped out in Vignette 18. The children did not go further with their rehearsal that day and I left.

Relevance

This incident is important as an example of the cross fertilization of ideas from different domains of Miriam’s life: kindergarten and Logo. It also hints at some of the constraints: what good does it do a 6 year old to make a script if none of her friends can read it? This last question is obviously rhetorical — one may distinguish between a project’s being immediately effective in achieving a goal and its value in a person’s development.

Vn02001

Vn020A

In the Gymnasium

6/2/77


Twenty four children filing into the gym, a space about 20 by 40 with a wall-wide, wooden climbing rack at one end. During the half hour, when any children did not want to play some game or other he might climb around on that climbing rack, come sit on the sawhorse where I was standing, or sit against the opposite wall. It was common for several children not to join each game.

During the period, I recall 5 games being played:
ICICLES — a few children are ‘it’ in the center of the gym; the others at command run lengthwise to the gym’s other end; anyone tagged should ‘freeze.’

COWBOYS AND INDIANS — a chase game with two teams; one team hides its eyes against the wall; as the second team sneaks up on the first, the teacher intones, “Oh, you sleepy cowboys, you’d better wake up. Here come the INDIANS!!!”; with this cry the cowboys chase the indians across the gym.

BATTLESTATIONS — a general exercise game in which the children leap to stereotyped postures (e.g. at attention, saluting for “here comes the captain”) or actions (e.g. swimming gestures for “here comes a shark”) when the teacher gives commands. These were usually given in pairs, the exercise value in going from one state to the other — port/starboard here comes the captain/all hands on deck naptime/chowtime

FIRE CHIEF — one child is the chief, getting to select which of three teams responds to the teacher’s alarm: “Fire, fire, fire: alarm at station. . .” When the chief yells his number, the selected team members race to the climbing rack, climb far enough to tag the top, then race to the opposite end of the gym. The winner of the race gets to be the new chief.

ANIMAL FARM — this final game of the period is the decrescendo of the hectic excitement of gym. The children sit in a circle. The teacher, in the center, covers the eyes of one child who must guess the identity of another, selected from the several who volunteer to make animal sounds.

After gym, the children spilled back into kindergarten, where
they began, relatively slowly, to pick up the variety of materials they
had so liberally scattered before gym.

Relevance

The observations focus on an important part of the world of Miriam’s peers, and her standing apart from that. I infer that, because of her limitations and specific experiences, she has a different framework for thinking about gym activities from her peers.

Vn02002

Vn20.2

In the Gymnasium (2)

6/3/77


This evening I asked Miriam whether she usually stood aside from the games in gym as she did yesterday (except for Animal Farm). She replied that she doesn’t like running around (her allergy to dust and its chronic incipient wheezing make her feelings quite understandable). But Miriam justified her alternative further: “I was on the balance beam.” She had been but I had not recognized it.

In the summer of 1975, Robby was enrolled in a gymnastics class. Miriam and I took him there and either waited inside, watching the older children exercise, or walked about outside. Miriam felt left out (she was too young for that class). When we arrived early one morning, I held her hand as she walked along the elevated balance beam. Back at our house she found a ‘balance beam’ (a 14′ 4×10 under the porch) and had me set up the timber so she could use it. The next summer brought the Olympics when those young girls from eastern Europe were so impressive in tumbling and on the balance beam. Miriam asked me to set up her balance beam again. Of the activities available in the after school program, Miriam chose gymnastics as the one she most wanted to pursue.

Relevance

The observations focus an important part of the world of Miriam’s peers, and her standing apart from that. I infer that, because of her limitations and specific experiences, she has a different framework for thinking about gym activities from her peers.

Vn09501

Vn95.1 Why the Project is Ending Now 9/18 & 27/77

9/18 Miriam has expressed her desire to do the same work as her class-
mates, her preference for doing the whatever ‘math’ they will do at
school to learning the kind of math we do together. I take this prefer-
ence as a strong commitment on her part to be one with her peers and
not as a rejection of the arithmetic we have done together. (My doing
so is justified by her initiating most of our math sessions and her
enjoying them.) Further, Miriam’s social needs begin to conflict with
our engagement at Logo. She likes to play with her friends; seeing
more of them now that school has begun, she will demand playing with
them more often. With her friends back from vacation and more readily
available for play, she will have less time available to play with me
at Logo.

For the time being at least, the focus of Miriam’s interests has
shifted out of the home toward the social world of her peers. I con-
sider this a natural change, my struggling against which could be bad
for Miriam and counterproductive for me. I believe Miriam will become
bored with school relatively quickly (perhaps by November or December)
and will rebound with a newer interest in our learning together at home
and Logo.

I discussed this situation tonight with the children. I further
explained my sense of exhaustion — that as much as Miriam has had
allergy problems since our return from vacation, I have also had them.
The antihistamines I have taken to suppress hay fever symptoms have
made me often drowsy and have undermined my ability to stay on top of
the data I have collected.

We three agreed that now is the time for the project to close. We
begin our final series of evaluations tomorrow.

9/27 When attempts to circumvent my allergy/medication based drowsiness
by changing medication failed, I arranged for skin tests to specify
precisely my allergies in the hope of controlling them some other way.
I respond, as Miriam does, to a broad range of substances: I am most
sensitive to house dust, mold, and cat dander; I am slightly less sensi-
tive to ragweed and various grass pollens; at a lower but still signifi-
cant level I am allergic to varieties of trees. This allergy profile
is the same as Miriam’s with minor variations (she is more sensitive to
oak, I to maple). It helps explain the common difficulties we have
experienced these last two months. (This has been an especially bad
year for ragweed.)

Relevance
These notes document the ways in which two factors — the children’s
return to school and allergic reactions — lead us into the final project
phase two weeks earlier than I had anticipated.

Vn09801

Vn98.1 Miriam’s New Reader 9/24/77-10/3/77

9/24 Since I was uncertain how much Miriam’s reading skill had developed
over the summer (focused as much of it was on reading Peanuts and Pogo
cartoons), I could not easily judge what would best test Miriam’s capacity.
We discussed the problem. Miriam characterized the book used in
Miriam at 6, a reader for the first half of second grade, as being
“easy-bezy.” “Was it so back in April?” I inquired. Miriam answered
that the book was pretty hard for her to read earlier. We agreed the
solution to my problem was for her to select the book for her final
reading evaluation. When we went to Hammett’s supply store, Miriam
checked out the fifth and sixth grade readers, declaring them too hard;
examining the third and fourth grade readers, she selected More Roads
to Follow
(a third year spring semester book) as having a level of
difficulty comparable to the book we used in April. Back at Logo, our
experiment of the day was a reading evaluation. Her judgment was proved
correct: she was able to read the book but exhibited some difficulty.

9/25 Miriam has been praising her new “Dick and Jane” book to Robby.
She explained that not only have Dick and Jane been left out (a great
advance in Robby’s eyes), but that it also contained a chapter from the
Pooh stories with much prettier illustrations than their paper-back
versions. Miriam asked if she might take her book to school. I agreed
it was a good idea, since Ms. Fieman wanted to know where should she
start Miriam.

10/3 Miriam has been reading More Roads to Follow quite regularly.
This evening she recommended to my attention “The Gingham Dog and the
Calico Cat”, remarking on its absurd good humor of the fight ending
with each eating up the other.

Relevance
Miriam’s continuing reading of her evaluation book confirms her
original judgment in selecting it and my conclusion that it represents the
right level of challenge to her skill. Without tutelage, her reading
level has advanced a year in the six months since the beginning of the
project.

Vn10001

Vn100.1 Arithmetic Ripples 9/26/77-10/2&7/77

9/26 Miriam tells me today was her first day of doing math at school.
“But we did it differently there from what we [she and I] did here.”
She explained that school math was playing with cuisiniere rods. I
told her I thought that was great, and asked what she did with them.
Miriam said she used them for building. As this project comes to an
end, I will ask Miriam to build, out of cuisiniere rods, a POLYGONAL
SPIRAL. . . or perhaps ask her to describe my procedure for doing so in
the Logo language.

10/2 Robby and Miriam have lately been making home-made clay. They
mix flour, salt, and a little water, knead thoroughly, and thereby develop
a clay which they later fix by baking. Robby began counting the layers
of material he made by folding the material over and into itself. After
reaching a count of 96 (by what path I am not certain), Robby cut his
clay ply in two pieces, and superposing one on the other, declared he
had 96 plus 96 layers.
Miriam said, “That’s a hundred 92.” Robby asked me if she were
right. Miriam responded, “90 plus 90 is one eighty (looking at him for
concurrence); so it’s one eighty six, seven, eight, nine, one ninety,
one ninety one, one ninety two.” thus completing her proof.

10/7 To inquire whether Miriam’s 90 plus 90 sum might derive now from
the sum 9 plus 9, I asked this morning (after warning her I wanted her
first answer, not one thought about too much). “How much is 9 plus 9?”
After a shosrt pause, Miriam responded, “18.” “How did you get that
result?” Miriam answered, ” ‘Cause 8 plus 8 is 16; so it’s 16 plus 2.”

Relevance
These notes document Miriam’s beginning of math at school and
suggest one simple way to begin binding her experience at Logo to her
future school work. The second observation documents the ease with
which Miriam has incorporated well-known sums from turtle geometry’s
decadal arithmetic into her procedures for mental computation. This is
an indication of their permanence as members of her repertoire.

Vn10401

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
day.

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.

Relevance
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.

Vn11201

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.

Relevance
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

Vn11301

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
school?”

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
thing.

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.

Relevance
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.

Vn11801

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.

Vn12201

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.

                      0	
                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
friend’s.

Relevance
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.

Vn12301

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.

Relevance
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.

Vn12901

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.

Relevance
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).

Vn13001

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:

             1 
             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.”

Relevance
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

Vn13101

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

Vn13201

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.”

Relevance
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
          5500     FIVE THOUSAND FIVE HUNDRED
          5505     FIVE THOUSAND FIVE HUNDRED FIVE
          5550     FIVE THOUSAND FIVE HUNDRED FIFTY 
          5555     FIVE THOUSAND FIVE HUNDRED FIFTY FIVE

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