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Vn00102

Vn001.02

Commutativity

5/8/77

As I sat transcribing the dialogue from recent logo sessions, I heard Robby inquire of Gretchen, at work in the kitchen, how many were 5 twelves. Gretchen simplified the computation by elaborating the problem: 5 twelves is half of 10 twelves. How much is 10 twelves? As Robby worked away on that problem, Miriam, playing at a puzzle within earshot of that conversation, piped up: “the answer is 60.”

Poor Robby! How frustrating when working on a different problem to be prevented by some one else’s interjecting the ‘correct’ answer. And yet, Miriam did have it right. I was quite worried that she had computed the answer by summing twelves (which Robby could have done, albeit with some difficulty and uncertainty) while he wrestled with the transformed problem,

Gretchen had been watching Miriam. She saw Miriam compute 5 twelves by finger-counting thus: 5, 10, 15, 20, 25, / 30, 35, 40, 45, 50, / . . . 60. Thus Miriam’s procedure is more primitive than Robby’s but it is also more sophisticated. She makes use of the commutativity of basic arithmetic operations at every turn. Several weks ago, Miriam gave direct evidence of her use of commutativity in adding. Mimi Sinclair asked her: “How many is 17 plus 6?” ’23’ Miriam responded counting up from 17 on 6 fingers. When the query turned to 6 plus 17, Miriam responded with no hesitation, ’23, because it’s the same problem.

I speculate that she uses commutativity because it permits her to proceed to an answer which costs her little if wrong; Robby, more concerned with the correctness of his results than the unimpeded progress of the computation, is more inclined to ask for advice than to trust to a property, commutativity, which can give him an answer but one about whose correctness he is uncertain. This speculation may demean the actual extent of Miriam’s understanding.

Vn00201

Vn002.01

Productive Cheating

5/9/77

Today was a difficult day. Snow in mid-May for a beginning. Before that problem appeared, Miriam came early with me in to Logo in our joint expectation of going to the Coop to buy a hula hoop. With that option closed by inclement weather, Miriam pushed me early in the afternoon to proceed with the day’s experiment. We proceeded as described in Logo Session 4.

Gretchen and Robby reached the lab later and Robby chose not to engage himself in my work with Miriam, preferring to play with SHOOT by himself in the central portion of the Children’s Learning Lab. Sam Lewis, another child frequently at Logo and a year older than Robby, played with him in the lab at that time. When Miriam declared a break from our work in writing a story, I discussed (with Gretchen) the children’s use of SHOOT and how I was awaiting their discovery of how to cheat. Instead of using the SHOOT : DISTANCE program to project the turtle into the target (which evaluates his location after movement and immediately judges the movement a ‘hit’ or a ‘miss’), one may locate the turtle within the target with a series of forward and turning commands; then, guaranteed of a bull’s-eye, execute SHOOT 0 to register one’s score. Such was my explanation. I noted that the most efficient cheat would be to execute a ‘HOME’ command (which puts the turtle in the target with a single command), then SHOOT 0.

Because of the snow and Miriam’s disinclination to proceed with writing a second story, I suggested Gretchen take the children home while I proceeded with some work they could not be involved in. Robby was most eager to stay and play with SHOOT. After a slow start in the first 3 Logo sessions, Robby was developing skill quickly. He had already, as he noted, scored 5 points that afternoon, and wanted to go on while doing well. I reluctantly agreed. I agreed because I believe the children should be allowed to follow active interests. My agreements was reluctant because I did not want Robby to make further significant advances without my observation. This is precisely what happened. As we discussed the day at supper, Robby noted that he had a good afternoon. His second use of SHOOT garnered him 9 points, giving him a total of 16 (? ). . . this may include in his calculation points from the 3 earlier sessions). Robby then added he had figured out how to score every time. “How?” Robby explained that after drawing the target, the turtle goes ‘Home’ before going somewhere [a setting of his heading and location to random values] and that if one were to key ‘H’ or ‘Home’, then SHOOT 0, he would score every time. To be certain Robby was saying what I thought I posed these questions.

Bob Suppose you key ‘H’, carriage return?
Robby The turtle goes to the center of the target.
Bob Like this?
Robby Yes. Then you say SHOOT 0. illustration:
Bob And what does the turtle say? target and turtle
Robby Ouch. Your score is 1.

I asked Robby if Sam had showed him that and received a negative answer and the claim that he had figured it out himself. I recall informing Robby, before his second terminal session of the day, that because of his squabble with Miriam in Logo Session 3, I changed the SHOOT program so that if the turtle were within the target after execution of GO-SOMEWHERE, he would be made to GO-SOMEWHERE-ELSE, i.e. land at a different location.

Miriam then confided to Robby in her most conspiratorial stage whisper: “Robby, you shouldn’t have told me; I’m going to do that every time.”

I pursued this question, asking Robby whether he had used this new idea to score all his points during the afternoon. Robby denied it, saying the trick didn’t work. I was surprised (it should work perfectly) and asked why not. Robby said the computer would respond ‘You didn’t tell me how to H or Home.’ I asked if he knew it wouldn’t work and how. The answer was that he hadn’t tried it, thus he couldn’t say why he knew it didn’t work.

Interpreting this incident depends on how open Robby is with me, generally, and on the extent to which his final comments were an attempt to delude Miriam by convincing her that his discovery isn’t worth attempting. Robby is usually quite open with me. Nonetheless, given the intellectual rivalry between the children, I would not be surprised at Robby’s attempting to throw Miriam off the track of a discovery he made which his revealing to her had made useless to him. An alternative explanation for Robby’s not trying the “Home SHOOT 0” cheat (and perhaps the impetus for it’s coming to his mind) is my explaining that I had modified SHOOT to forbid those lucky landings of the turtle within the target. He may have believed any time the turtle were found in the the circle at the beginning of executing the SHOOT procedure he would GO-SOMEWHERE-ELSE before being shot at the target. [Indeed, such is possible and is the way one would prohibit the ‘forward and turn commands/SHOOT 0’ cheat if one were so inclined.]

This incident promises further interest in that part of my intention is to guide Miriam’s concerns from getting a correct answer to attending to the process and operations by which one can achieve an answer. Her obvious engagement with the desire to succeed immediately will lead her to pursue Robby’s discovery. I expect and intend to have her succeed thus. My following countermove (which will be to relocate the target off center screen) may show how too simple “an answer” is inadequate and must give way to deeper comprehension of process by which “an answer” is developed. When, later, both children realize they can still succeed by deferring execution of SHOOT until the turtle has been relocated within the target circle through forward and turn commands, they will have extracted all the value they can get from the use of this introductory game.

Vn00302

Vn003.02

Journals

5/11/77

When collecting the data reported in ‘Pre-Readers’ Concept of the English Word,’ I was shown by one of the children, Tina, a 3×5 notebook her mother gave her for writing down any words she wanted to learn to read. I bought Miriam such a notebook (a green one) some weeks ago for such a purpose. She asked me how to spell ‘floccinaucinihilipilification’ and proceeded to print that in her notebook (today she refused to give me that datum because she still wants to learn to spell that word). Her uses of the notebook are more various than my plan. I find: a list of models she would like to have; an upper and lower case alphabet page (the first set done by Robby as a set of patterns, the second Miriam’s copy); the address and telephone number of her friend Maria. The most frequent use of that notebook has been to make greeting cards for her friends: typically Miriam draws a picture, then prints “For ___” and “Love Miriam”. (It also has many blank pages.) Clearly, Miriam’s use of this notebook has been richer than my conception

I recently purchased another such small notebook (a blue one) for jotting down notes about what Miriam does during the day. Miriam saw that notebook and asked if she could have it for making cards. A conjunction of things occurred. Miriam wanted the notebook; the time she spends in school is one wherein I can’t observe what she is doing; an observation several years ago by Sarene Boocock at an AAAS seminar that children themselves had access to data most researchers could not get at and that one should consider enlisting them as agents of data collection. I told Miriam she could have the blue notebook if she would write in it the things she did in kindergarten. Miriam refused, saying she wanted the notebook for drawing pictures. When I said she could not have the notebook for drawing pictures, Miriam responded, “If you don’t give me the notebook, I won’t even tell you what I do.

At the current round of negotiations, Miriam agreed to use the notebook as I wanted when I agreed to buy a replacement for her green notebook after she uses up all the pages. Her intentions are suspect though because of the final inquiry she made: “If I do a rotten job can I keep the notebook and not write down what I do?

Contrast now the relative openness of the two children, Robby and Miriam. For reference, consider the protocols and 21 (on adding units of English length and on multiplication, respectively) from the series of Robby’s arithmetic development. These protocols are exemplary of a child’s uninhibited exposition of his thought processes; at 7, Robby is a ‘loud thinker’. Contrariwise, Miriam’s characteristic behavior shows her inclination to display a perfect result, a reluctance to exhibit pre-competence confusion

One may laud such a stance as reflecting essential good taste (for who wants to overwhelm the possibility of communication with the mess inside every mind). However, for an ignorant child (is anyone at six years anything else) such a performance criterion is unrealistic and counterproductive. One of my objectives is to render Miriam less sensitive to revealing her imperfect comprehensions in the expectation tha her doing so will provide guidance to those who are willing to help her learn.

Vn00401

Vn004.01

The Clever Hack (2)

5/12/77

At dinner this evening we talked over some of the incidents of the day. It had been one of novelty for Robby. His friend John (the son of a former naval officer with whom Robby shares an interest in naval battles and model building) came to Logo with Robby as a stopping point on a visit to the Hart Nautical Museum (a museum of models) in MIT building 5. Miriam stayed at Logo with me (the events are recorded in Logo session 7) while Gretchen took the boys to the museum. By the time the boys had returned, Miriam and I had finished our work for the day. While we played otherwheres with a hula hoop and some tennis balls, Gretchen and the boys went into the music room where Robby introduced John to SHOOT. When Miriam heard of this at dinner, she said, “I should have shown John the clever hack.”

The immediate surfacing of this suggestion to Miriam’s mind made me curious about what role, if any, it had played in Robby’s introducing John to SHOOT. These selections, from the transcription made by the chance of the tape recorder’s having been left running, are extracted from Logo session 7.

GretchenWhy don’t you play SHOOT?
Robby That’s a good idea. (To John) Let’s play SHOOT.
John What is this SHOOT?
Robby [Robby logs into Logo, reads the file “SHOOT from secondary storage] You’ll want to see SETUP. Are you looking at the screen?
John Yeah.
Robby [executes the procedure SETUP. The procedure clears the terminal display, then puts a message on the logging portion of the display and creates a ‘screen’ (a movement domain for the turtle); the procedure increases the screen size in steps until it reaches the standard size, thereby pretending to simulate the distant first appearance of something coming into view. While the screen size increases, messaged are printed on the logging portion of the display. Robby reads them.]
Robby Look in the sky!
It’s a bird — it’s a plane.
No, it’s super turtle.
(The SETUP procedure then draws a bull’s eye on the screen and sets the turtle at random location and heading.)
Gretchen I never saw that before. Why don’t you explain to John what you are doing? (When Robby fails to respond, Gretchen continues), The object. . . this is the turtle, and the line in the middle shows which way he is pointing. The object is to get the turtle pointing towards the target. And then say SHOOT-
Robby Something.
Gretchen A certain distance. . . and see if he stops, if that gets him to the target.
Robby [keying]
GretchenRobby has just made a turn of 90 degrees. You try to get it lined up. Now he has to figure out how far to tell it to go.
Robby Yes! (meaning his shot hit the target)
Gretchen Just made it. If it’s a hit, the turtle says ‘ouch’. [exits]
John(laughs) . . . Can I try?
RobbyNo. I want to show you something. [keying] H . . . (then realizing he has omitted the carriage return) Oh . . . Now SHOOT O. It worked! Isn’t that a great trick?
JohnYeah.
RobbyIt’s sort of easy.
JohnHow do you do it? Show me how to do it. I want a turn.
Robby(after shooting successfully at the target, but in no way describing what he did). Here, John. You do everything now.
John Well, you have to help me.
Robby O.K. Right or left?
John Right. I want to do right . . . What do I do?
Robby I want to go home.
John What button do I push? This one? To aim it at this?
Robby Oh, . . . just do Home, SHOOT O.
John No, I don’t want to.
Robby [keying in Home, carriage return, despite John’s preferences] Just say SHOOT O. He’s already at Home . . . I want to go to my house, do you?
John What would we do there, Robby?
Robby Play soccer . . . play in the tree fort.
John SHOOT O. Oh, come on. (shortly after this point, the tape ran out)

Out of three cycles of SHOOT (from setting the target through hitting it), the clever hack was used for two, the basic turtle commands being used for the first cycle. In the second cycle, Robby clearly paraded for John a solution to a problem John could not appreciate. The second use of the clever hack was to short-circut John’s interest in finding out “how to do it”; by appealing to John’s interest in achieving the objective, Robby attempted to circumvent the more time consuming process of showing how to play the game. The mark of the clever hack in both uses is its salience; whenever little time is given to the problem solving process, either through motives of setting a dramatic effect or simply to reach a quick solution, the clever hack comes into its own.

Vn00501

Vn005.01

Tic Tac Toe

5/13/77

Years ago I bought a tic-tac-toe game for playing with Robby. The board is 12 x 12; the X’s and O’s are large yellow pieces. At that time I taught Robby a single strategy for playing the game: look for two ways to win. Robby quickly became quite good at the game. Miriam, at age four, learned that one took turns and that winning was getting your three pieces in a row. She did not take the strategy instruction so readily. The children played with the game with different competences: Robby winning with his strategy; Miriam winning frequently enough when he made errors to be satisfied. As Robby came to make fewer errors, Miriam played less, the pieces falling to other uses: the dog chewed one of the X’s, the play group used an O as a hockey puck.

Tic-tac-toe came forward again as a game at a recent visit to the Children’s Museum. Miriam lost to the computer when it, moving first, chose the center square. When she moved first in the center square she never did better than a draw. Robby beat the computer with a first move corner choice (let Robby’s sequence be 1, 2, 3, 4, and the computer’s A, B, C, D):

 1  |     |  3
--------------
 |  A  |
--------------
 B  |     |  2

By move 3, Robby had forked the computer, had “two ways to win.” Since neither Miriam nor his friend John had done better than a draw, he gladly showed them the gambit on his next turn.

Tonight, after dinner, Miriam asked me to play tic-tac-toe with her. She was quite familiar with the terminology of two ways to win and implicit victory whenever she achieved a forking pattern. She understood and accepted the terminology of a forced move.

Whenever Miriam had the first move and chose the center square, and I chose a non-corner, she consistently won. This was true regardless of orientation of the board. Let these two games represent the rest as well (Miriam’s moves are the digits, mine the letters):

    |     |  2       2 |  A  |
 --------------    --------------
    |  1  |            |  1  |
 --------------    --------------
  B |  A  |  3       3 |     | B

In both, her “two ways to win” victories were not noted as being ‘tricks’ of any sort.

I gradually altered my responses to her first move center choice until I chose a corner square in responsse to every center square first move. Making no mistakes, we come inevitably to a draw, thus:

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

Miriam began then a new gambit, the first move corner choice. When I responded with a center square choice (as did the computer at the Children’s Muuseum), Miriam had her two ways to win by the third move. Miriam described this new gambit as her ‘dirty trick.’ I was quick to tell her I thought it was not a ‘dirty trick’ but a ‘good trick,’ and we both later referred to it that way.

After being beaten several times by her new gambit, I blocked its effectiveness by refusing to make the center square response to her opening move. When I responded thus

  1 |      |
 -------------- 
    |      |
 -------------- 
    |      |  A   

Miriam complained vociferously. It was clear she did not know what to do. She eventually proceeded by placing her piece anywhere. In another variation, she was quite surprised at my victory:

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

She was so intent on her ‘good trick’ she failed to see the simple victory she might have achieved on her third move. This was the only ‘mistake’ I recall Miriam making.

I conclude from these observations that Miriam’s strategies are very specific in nature. When consolidated, they may be orientation insensitive, but not even orientation insensitivity is immediate. Further, Miriam made her ‘mistake’ because she had not yet integrated her ‘good trick’ with the primary rule of tic-tac-toe: make all forced moves. One might better conceive of her “two ways to win” less as a strategy than as a more complex and various, more immediate objective, more immediate than the victory criterion of three pieces in a row.

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.

Vn00901

Vn009.01

Tic Tac Toe (2)

5/22/77


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

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

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

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

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

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

 B |   | 2
----------
   | C |
----------
 1 |   | A

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

Relevamce

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

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

Vn01702

Vn017B

Arithmetic Ripples

5/30/77


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

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

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

Relevance

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

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.

Vn02101

Vignette 21

Miriam’s Room

6/5/77


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

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

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

Relevance

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

Vn02301

Vignette 23.1

Arithmetic Ripples

6/5 & 11/77


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

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

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

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

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

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

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

Relevance

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

Vn02401

Vn024.01

Writing Stories


Robby called me from Miriam’s bedroom: “Dad, come see the puppet show.” They have played with, even made, hand puppets for a while and enjoy giving shows — whose typical script has been, “Hello, my name is Owl. Goodbye.” I was in high spirits after a very successful arithmetic session with Miriam (see Home Session 7). Sucker that I am, I came along. Walking through the door, I found the children were playing “Ambush” — both lying under covers on the top bunk and crying “BAM BAM” as I walked through the door. Riotous laughter.
Suffering only flesh wounds, I managed to return their fire, then said I thought it was a dirty trick for them to call me to see a puppet show, then shoot me. Miriam responded, “This was our puppet show —

Once upon a time,
There were two guns.
Bang. Bang.
The end.

Her joke was a spontaneous expression of the WRITER program’s story format. Her use of it in this explanatory way shows her recognition that it is a shared model of what a story structure is.

The puppet-show play continued; the children agreed to put on a real puppet show. I had to leave the bedroom, and when ready, they summoned the audience with toots on a toy horn.

Act I The fierce Owl (Robby) swooped down and pounced upon the dancing ballerina (Miriam’s doll).,
Act II The fierce Owl, joined by Eeyore, climbed a tree, and both pounced upon whom? The ballerina had become a teacher and had to be pounced upon because the owl and donkey didn’t want to go to school.
Act III The owl and donkey climbed a tree whence better to pounce upon the teacher, now riding upon two toy horses because of injuries received in act II. (Miriam plays such an action herself because she is so big and her favorite toy horses so small). Miriam argued that it was O.K. for Robby to pounce upon the teacher, but he shouldn’t bite “little horsey.”
Act IVThe horns again called me into the room. “BAM BAM BAM” — ambushed again.

When I had resettled in the reading alcove of my bedroom, the children rushed upon me with cried of “Act V.” We engaged in a general rough house and pillow fight.

Vn02601

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

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

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

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

Vn02901

Vn029.01 Making Puzzles 6/18/77


Vn29-1

Vn29-2 Addendum1

Vn29-3 Addendum2

Vn03101

Vn031.01 Collecting Tolls 6/19/77

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

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

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

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

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

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

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

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

Addendum 31-1

Toll Collection Records

Vn31 Toll Collection Records

Vn03401

Vn034.01 Candle Fire Crackers 6/23/77

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

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

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

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

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

Vn03701

Vn37.1 Explaining SHOOT 6/26/77

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

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

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

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

Vn03801

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

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

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

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

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

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

Vn03901

Vn39.01 Good News 7/1/77

At lunch today we told the children that Gretchen is pregnant.
When the children and I finished our morning’s experiment (Logo Session
33), we joined Gretchen at home, picked up a stack of their favorite
books, and all went to the doctor’s office.

We all suffered a delay (the doctor was late in returning from a
delivery). Each child read his own books, then the other’s, and finally
whatever children’s books they could find in the office. They seemed
to take no special notice of most women being about 7 months pregnant.

After our interview with the doctor ( and our listening to the
fetal heart beat!) Gretchen remained for some laboratory tests while
I took the children outside. They were rambunctious from their hour’s
wait and argued with me that we should have left them home. When I
raised the surface objection that we couldn’t find a baby sitter, they
claimed to be big enough to stay home by themselves and that a compro-
mise solution would be for them to work in the garden with Steve (our
landlord’s handyman).

When we sat down to lunch at home, I asked both children if they
had noticed anything unusual about the patients in the doctor’s office.
Robby said several of the women had big bellies. When I asked why,
Miriam said that meant they had babies inside. Neither guessed why
we had gone to that doctor’s office. “Kids, we have some good news.
Your Mommy has a baby in her belly!” The children were thrilled. To
their first question, “How do you know?” we could say we had heard its
heart beat. “Is it a boy or a girl?” Both children were looking for
an ally. Robby: “I hope it’s a boy.” Miriam: “I hope it’s a girl.
What do you hope, Dad?” I replied that I hope the baby is healthy,
and was content to wait, as I must do, until the baby is born to find
out if it’s a boy or girl. They started to plan:

Robby If it’s a boy, he can sleep in my room and I’ll give him the crib.
Miriam It’s my crib.
Robby No. I just loaned it to you.
Miriam Well, I’ll give the baby my little blanket. What toys should we get for it?

Gretchen cautioned the children that the wait would be a long one. I
added that the baby would be a late Christmas present for everybody.

Robby asked to be excused from the table so he could tell our
neighbors. I warned both children I wanted no racing or fighting
about the good news. They could tell whomever they wanted but they
had to do it together. Miriam left her lunch and both went across the
court. We could hear their cries: “Steve, Carol, Annette, Jim — we’re
going to have a baby!”

Relevance
This vignette witnesses the children’s excitement over an event
which will complicate and enrich our common life.

Vn04001

Vn40.1 Logo After Hours 7/4/77

During the bicentennial year Miriam was too young to enjoy the
fireworks. She was frightened by the noise of amateurs’ exploding
firecrackers and so sleepy by 9 o’clock that we abandoned a half-
hearted attempt to watch the display from the vantage of Corey Hill
in Brookline. Radio forecasts promised this year a smaller crowd
and a more impressive exhibition than last year. Uncertain that we
would be near Boston in the future, I decided the children should
seize this opportunity to see the biggest fireworks display on the
east coast.

Having heard of how impossible is parking near the Charles, I
brought my family to Logo early in the evening. We all casually enter-
tained ourselves while waiting for nightfall. Gretchen and Robby occu-
pied themselves with reading, Miriam with drawing (Robby did that too)
and making letters. I reviewed material in various workspaces on the
Logo system, to refresh my memory with possibilities for future work
with the childlren. I showed Robby (but not Miriam) Danny Hillis’
“STRING” design procedure and an elaboration I had made thereon for
developing Lissajous figures. He was impressed, but drawn away by
witnessing the Cambridge police respond to an apparent mugging on the
corner of Main and Vassar. Miriam wanted to use the Slot Machine but
it did not work (as we had discovered earlier in the day: cf. Logo
Session 34A). We all watched the traffic build to an impenetrable
mass as dark approached.

We walked to Memorial Drive near the foot of Longfellow Bridge and
beheld that crowd of evening picnickers who had come prepared with
incredible paraphernalia and seized all the choice locations early on.
The air was acrid and pulsating from the frequent but irregular ex-
plosions by amateur incendiaries. The children’s chronic impatience
was only relieved by the distraction of fudge popsicles and the dis-
tressingly late beginning of the fireworks. Very few seemed to care
that hearing the 1812 Overture was impossible until the cannon fire
declared the beginning of the long-awaited fireworks. The display was
worth the waiting. Even though they were quite tired, both children
were excited and delighted.

At the end of the show, we repaired to Logo to await there the
subsidence of the traffic. We were all glad to find the lab occupied
by friends. Miriam perched herself on Margaret Minsky’s lap and
announced that we’re going to have a baby. Upon hearing that Danny
Hillis was back from Texas on a visit, we all trooped up to Marvin’s
office and interjected ourselves into their conversation. Miriam seized
Danny’s lap as her own property, and I shanghaied him to repair the
Slot Machine so that the two-terminal experiment could be executed the
next day (this was essential because of rearranging the lab for the
summer high school program). After Danny did a little magic to make
the Slot Machine work, we sat talking til midnight with Margaret, Bruce
Edwards, and Ellen Hildreth.

Relevance
These notes record the casual use of Logo as a place to pass the
time and meet friends.

Vn04101

Vn41.1 7/5-7/77

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

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

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

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

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

Post Script

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

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

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

Leader

Respondent
Is Daddy a dummy?

No!
Is Daddy a smarty?

No!
What is he?

An idiot!

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

Vn04201

Vn42.1 7/6/77

Because the High School Studies Program begins next week and the
lab will be filled with teenagers all day every day, we moved into my
office the equipment not to be used by the high school students: the
slot machine, the floor turtle, and the music box.
Since moving things around brings change and sometimes adventure,
I asked the children to come to Logo though no computer sessions was
planned. In a day full of disorganization, pushing, pulling, and helping
out, the greatest excitement for the children was in re-routing
data lines from the terminals to the computer. This involved lifting
up floor panels. The floor panel lifter goes in place with a loud
bang as it’s slammed down. The panels are heavy — a challenge Robby
can barely meet and Miriam feigns attempting. They greeted the under-
floor space, a dark maze of tangled wires, as a new, mysterious world
and began prospecting in the openings for souvenirs. As Hal Abelson
and I traced wires, the children invented impromptu games — being
stranded on islands or trapped by moats with escape possible only by
the fine balance that permitted them to walk on the floor panel holding
frame.

Margaret Minsky agreed to move to make room for the equipment in
our office. The children decided this was now their office, which
required getting nameplates for the door. They further dubbed the room
‘The Little Learning Lab’ since they were little (to distinguish it
from the Children’s Learning Lab which the high schoolers would be
taking over). Pope’s couplet

A little learning is a dang’rous thing:
Drink deep or taste not the Pierian spring.

and the obvious joke that this was a lab where little learning takes
place caused me pause but no inhibition so severe as to halt their
momentum.

Each child was allotted one of my 3 bookshelves, which they
provisioned as best they could. We walked to the Coop to buy each child
a large notebook for keeping the pictures they made with the Logo
printer. As we three trekked across the campus, the children fell
into ‘Follow the Leader’ and an immediate argument over who should be
leader. My turn-taking suggestion (one to the Coop, the other on the
way back) was no solution: it left the problem of who should be leader
first. Robby went first despite objections. Miriam undercut him by
giving Robby turtle commands to follow the obvious path whenever that
path was clear. To the Coop and back this game gradually was elaborated
as Robby raised syntactic quibbles to avoid doing what Miriam
commanded. For example, “You haven’t told me how to forward 30” (by
which he indicated that Miriam had not verbally specified that a space
separated the word ‘forward’ from the word ’30’). The most puzzling
impediment Robby introduced occurred while we were returning, skirting
the side of building 26. Miriam tried to make Robby walk into the wall
by commanding left 90 (to be followed by a forward). He stopped and
said nothing. After several commands and repetitions, Robby burst out
laughing. “You haven’t done a carriage return!” Miriam said, “New
line!” and Robby obliged her by walking into the wall.

Relevance
This vignette recounts the excitement of a moving day at Logo and
an example of how Playing Turtle arose as a game outside the lab.

Post script

This game of ‘Follow the Turtle’ has become a common game the
children engage in whenever we three walk together where there is no
crowd.

Vn04301

Vn43.1 Binary Counting 7/7/77

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

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

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

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

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

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

Vn04401

Vn44.1 A Boring Session 7/12/77

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

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

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

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

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

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

Post Script

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

Addendum 44-1

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

Addendum 44-2

Hexagonal Maze

Vn 44-2 Hexagonal Maze

Vn04501

Vn45.1 Going Home 7/15/77

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

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

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

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

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

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

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

Vn04601

Vn46.1 Rotten Hints 7/19/77

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

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

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

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

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

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

Vn04701

Vn47.1 Losing a Tooth 7/20/77

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

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

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

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

Bob

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

(Surprised and a little disappointed) Oh.

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

Vn05001

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

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

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

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

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

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

Addendum 50-1

Addendum 50-2

Vn05101

Vn51.1 Paper Ships 7/25/77

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

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

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

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

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

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

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

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

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

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

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

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

Addendum 51-1

Vn 51-1 Curious George paper folding

Addendum 51-2

Vn 51-2 Curious George Paper Ship procedure

Vn05201

Vn52.1 Fort Griswold 7/26/77

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

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

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

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

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

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

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

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

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

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

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

Fort Griswold – 1

Vn 52-1 Fort Griswold

Fort Griswold – 2

Vn52-2 Fort Griswold

Fort Griswold – 3

Vn 52-3 Fort Griswold

Vn05301

Vn53.1 A Birthday Party 7/28/77

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

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

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

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

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

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

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

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

Vn05501

Vn55.1 About Taxes 8/1/77

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

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

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

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

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

Vn05701

Vn57.1 Desserts 8/3/77

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

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

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

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

Vn05801

Vn58.1 Owning an Angle 8/4/77

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

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

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

Vn06001

Vn60.1 Surprise Party 8/8/77

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

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

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

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

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

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

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

Vn06201

Vn62.1 Multiplication 8/7 & 11/77

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

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

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

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

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

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

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

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

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

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

Vn06301

Vn63.1 Another Birthday Party 8/12/77

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

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

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

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