Skip to content
Archive with last of tag-string Q24

Vn11701

Vn117.1 Playing Both Sides 1/14/78

Relevance

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

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

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

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

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

Vn 117, page 2, scanned from Original Fair Copy

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

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

Vn11801

Vn118.1 Introducing Peggy 1/26/78

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

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

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

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

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

Vn11901

Vn119.1 Multiplying by Twenty 1/25/78

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

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

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

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

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

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

Addendum

Adding by Miriam and Robby

Vn 119-1 Sums by Miriam and Robby

Vn12001

Vn120.1 Designing a Box 1/26/78

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

Vn 120-1 Fig 1.

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

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

Miriam

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

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

I saw it all at once.

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

Vn 120-2 Fig 2.

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

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

Addendum 120-1

Unsatisfactory Pattern noted in text

vn 120-3 Miriam'splan

Vn12101

Vn121.1 Double Perspectives 2/8/78

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

Miriam has told repeatedly her most recent joke.

Miriam

What letter of the alphabet do you drink?
Victim

I don’t know.
Miriam

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

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

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

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

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

Addendum 121 – 1

Find the Hidden Picture

Vn 121-1 Hidden Pictures

Vn12201

Vn122.1 Carrying Bugs 2/5/78

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

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

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

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

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

Vn12301

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

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

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

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

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

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


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

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

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

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

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

Vn12401

Vn124.1 Analogical Guidance 2/23/78

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

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

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

Miriam began with two congruent triangles, thus:

Vn 124-2 Yukky DIamonds

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

Vn 124-3 point to point

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

Vn 124-4 pattern of insight

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

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

Vn 124-5 analogy by superposition

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

Vn 124-6 7 piece completed

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

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

Vn 124-7 golden square

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

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

Vn 124-8 3 rectangles

Voila!

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

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

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

Vn12601

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

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

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

Vn 126-1 Turtle on the Bed

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

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

Vn12701

Vn127.1 Moo Shu 3/19/78

There is an old joke of this simple script:

Adult

Do you know how to read Chinese?
Child

I don’t know. I’ve never tried.

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

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

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

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

Vn12901

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

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

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

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

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

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

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

Vn13001

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

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

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

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

3000 700 —

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

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

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

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

POST SCRIPT: 4/11/78

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

Addendum 130-1

Chalk Board Sums (home session 7)

Vn 130-1 Miriam solves an old problem

Chalk Board Sums (home session 8)

Vn 130-2 A second old problem

Vn13101

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

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

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

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

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

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

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

Addendum 131-1

Party Planning

Vn 131-1 Party Planning

Addendum 131-2

Party Invitations Made at Logo

Vn 131-2 Party Invitations