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3V0531.01 COUNTING: beginning of notes. Cookies, hands, and counting (7/7/79)

During interviews at IBM, Moshe Zloof raised the question of whether
or not, in effect, counting is innate. I told him the question was a big
one about which I felt no one could speak with authority but that I had
very strong prejudices. As an example of the kind of experience from
which I felt the knowledge of counting might develop, I cited Peggy’s
reception of cookies. After convincing us to get her a cookie, Peggy
would sometimes open her mouth to receive it directly. More
commonly, she would hold out her hand (usually the right), take the
cookie, and put it in her mouth. Some time ago (we neither can recall
just when), in a situation where a whole stack of cookies was available,
Peggy requested and received a cookie for each hand. In some
circumstances, Peggy ended up transferring two cookies to one hand
and eating a cookie sandwich. The final step, which I witnessed but
can’t date, was Peggy requesting a cookie for each hand, then
transferring the right cookie to the left hand and requesting another.
In this little series of incidents, we see one-to-one correspondence and
a procedure for “getting one more”. These two are enough to base a
counting system on.

Today, Peggy began picking up all the various things on my chair side
table. I gave her three small bean bags to play with. The game of
choice became putting them in my palm and removing them. The
material scraps from which the bean bags were made are all colorful
and quite different from one another. She removed them several ways:
by ones, two first, and two last. When my hand was empty, she twice
scratched my palm after removing the third bag.

Peggy was much engaged with this bean bag play, talking all the while
(the talk is recorded on audio tape #3). I intend to play with these
little bags during our next experiment on videotape. Let’s see if we can
catch the development of Peggy’s knowledge of counting.


3V0585.01 TWO : [two]: counting puddles; spontaneous use: 08/30/79;

The kids and I went down to Bishops “Pick your own” raspberries.
While the older two picked, Peggy and I walked up and down the dirt
road to one side of the bushes. It had rained recently, and there were
puddles. Peggy and I pointed them out to each other. She told me
there was “water”, and I agreed, “Yes, puddles of water”. “Pud-duh”,
repeated Peggy. At a particularly big one, :There’s a big puddle,
Peggy”. After an instant, Peggy said “Doo”. Surely enough, there was a
second small puddle right next to the large one. “That’s right, Peg.
There are TWO puddles of water. One, two.” Gretchen.


3V0594.02 ONE, TWO: [one, two]: note on standardization of Peggy’s counting

You can’t avoid counting, and it’s hard to avoid instructing those who
don’t know what you know — but we’ve been trying to avoid instructing
Peggy. The children are persistent, at odd moments that we can’t
witness. So Peggy’s idiosyncratic counting [one, one, one,…
undecipherable noise] gave way to the more nearly standard
utterance [one two] in contexts of counting as follows: Peggy sees
me drink beer from a can and customarily names that object /kaen/.
She also looks in trash baskets. Today she came upon two in the trash
and said: [can…one…two] where the last had the sound /du(z)/. (The
notation (z) means here that I did not hear the z sound but Gretchen
did). No pointing, unfortunately.


3V0611.01 Counting cookie request [one…two…two ?] 9/25/79

This morning, Peggy asked for a cookie. As I gave it to her, she said
“One,… two…. two ?” She waited until I gave her a second cookie. Gretchen.


3V0674.01 COUNTING (carrying two cookies) [one, two, seven]
ONE, TWO, SEVEN (11/27/79)

Peggy came into the study (living room) with cookies in hand (one
each) and said to me “two”. She continued beyond me, saying, “One,
two, seven”. [FOOTNOTE: Later note on date written up: 12/6 This
evening, I asked Miriam is she had been teaching Peggy to count
(which Miriam denies) after Peggy’s “funny counting”, as “one, four, ten”]

Peggy clearly has learned several number names – perhaps from
watching Sesame street on TV. But her organization of the knowledge
is quite non-standard. Her construction of the number names goes not
much further than “one, two, three and other bigger numbers”.


3V0700.02 Knives and spoons: learning the word “fork”; called initially a spoon; when I named the object as fork, she called it a “foon”; counting incident. (12/23/79)

When the dishwasher cycle ended, I asked Miriam to put away the
dishes. Helpful Peggy was easily recruited. She started selecting
silverware from the dishwasher and carried it to the appropriate
cabinet. When she was unable to reach high enough to put the
silverware away, I became her assistant. Peggy ran back and forth.
“knife…spoon…spoon.” (The later name applied to forks as well. I
tried correcting her… “That’s a fork, Peg, not a spoon.” Peg brought me
the next fork and said as she gave it to me “foon”)

Peggy began bringing handfuls of silver and said as she handed them to
me, “one, three, four.” on the next trip, (no one speaking between) she
continued “one, three, another”.

Peggy clearly knows some number names, and that they apply to
counting and that a successor name “another” can be used in a
counting series.

Could “two” be left out of her series of well known number names
because of the homonym “too” which is richly meaningful for Peggy as
“me too” a word she uses very assertively ?


3V0733.02 More variations. (1/25/80)

Over the past few days, Peggy has been using the words ‘many’ and
‘more’ in various contexts. Example : sitting on my lap, Peggy looked
up at the picture illustrating Chaucer’s Canterbury pilgrims, and
P : ‘Many (unclear).’
G : What ? What did you say ?
P : Many horses.
G : Oh yes, Many horses.
Another day or so later at dinner, Peg said ‘More potatoes’ (a request
for another helping).


3V0747.03 Number/temporal names (2/8/80)

Miriam tells me she has asked Peggy the time and Peggy responded
“eleven.” The answer was not correct but was significant as a number
name. Peggy may have been imitating a specific response heard from
some one else in response to the same question.

Miriam asked again of Peggy, in my hearing, the time. Peggy responded
“Eleventeen.” This is clearly a made up number name from appropriate
kinds of elements.m


3V0749.01 Words and Numbers; primary roots of discrimination (2/10/80)

Miriam and Peggy play with my yardstick a lot (a free one from a local
hardware store, it has the measure and advertisements on it). Miriam
marches around with it on her shoulder: “hup, two, three, four; hup…”
Peggy marches too, “hup, two, three; hup, two, three.”

Today she LOOKED at the yardstick, then pointing at the symbols as she
clambered along it, said ( in pointing at the numbers) “eleventeen” and
at the words “Peggy Lawler.”

What this means is that she is interpreting alpha-numeric symbols
already — in a very non-standard and idiosyncratic way — but she is
reading the symbol strings as meaningful already.


3V0801.01 Counting (4/2/80)

As I put her floride drops into juice, Peggy counted out
as she has heard me do, “One…two….three…”
Bob notes Peggy counted: one, two three four (checkers)



3V0824.02 Hide and seek (4/25/80)

Miriam and Peggy play “hide and seek” — and Peggy’s imitation is
prominent. The place she picks to hide is always where Miriam hid
immediately before. Counting has resurfaced as an issue in this
context. Peggy hides her head and counts (1, 2, 3, 4…) then runs to
find Miriam. Miriam explained that while she always counts to ten
slowly, Peggy counts as high as she can and (sometimes) waits a bit
before seeking her.


3V0829.01 Counting; conventional now to six (4/30/80)

Robby and I discussed Peggy’s counting and he informs me she counts
now beyond four, to six, quite conventionally. He has waked and heard
her counting in her crib “one, two, three, four, five, six, nine, ten” This
is further evidence of the influence of hide and seek.


3V0865.01 Counting with Mimi: alternate counting game (6/5/80)

Miriam announced a game she and Peggy have been playing – a game of
alternate counting. Miriam and Peg count alternately:
M1, P2, M3, P4, M5, P6, M7, P7, M8, P9, M10, P11, M12, P12

6/8 note: Peg fast count from 4-11 by herself in the other room, as Miriam reports.


3V0882.02 Counting: pauses at places where sequence goes wrong (6/22/80)

Peggy was up late last night. Around eleven o’clock, while Robby
played with Miriam, I heard Peggy counting to herself: “one, two, three,
four, five, six, seven, eight, nine, ten, eleven,…eight,…twelve,…nine… ”
(where the dots represent short pauses).


3V0931.01 Generalization; logical thinking accidentally wrong: pennies and
quarters. (8/10/80)

Peggy came running around the table. “Somebody left these pennies
and quarters on the table,” she exclaimed as she handed them to me.
There were two pennies and, folded up, two dollar bills. So Peggy
knows two coin names and knows that both coins and specie can be
money. she has (as frequently witnessed in reference to coins) applied
“penny” as a label for any coin. she has chosen to apply “quarter” as a
different money name to another kind of money, i.e. currency.
Beautiful thinking, accidentally wrong.

The interesting problem this highlights is that the processes of
generalization and specification are much more complex than attaching
labels at the right description level and then extending them. There are
problems of shifting labels as classification refinements are developed.


3V0996.01 Using incomprehensible numbers: “Eighty” (10/12/80)

Miriam reported that Peggy was counting with such high numbers.
I recall Miriam saying that Peggy said things like “85, 86” and so forth
but have little confidence in that. See note of 10/27/80 on Counting
Jumping Jacks.


3V1010.01 Playing with coins: progressive discrimination (10/26/80)

After P143 (where we played with many coins) Peggy found the pile of
change and asked me to join her in playing with them on the floor. As
we did so, Peggy separated them and said, “I’m picking the big pennies
out and putting them on the floor.” This is significant as showing
Peggy’s primary classification of the coins is based on size — further
that the discrimination proceeds by qualification of the THING before
discriminating different kinds of things.


3V1025.01 85 dollars (1/12/80)

For several months Peggy has been coming out at random times with
odd numbers. She will look at a supermarket tag and say with a
decisive air “This costs 86 dollars.” to which my usual reply is “I hope not.”


3V1043.02 Shooting Monsters (22/30/80)

Miriam was in the basement watching King Kong on TV. Peggy came
into the living room and told Robby there was a monster. He drew out
his gun and undertook shooting all the monsters. Peggy was not
content with this form of their game. She wheedled the gun from
Robby and went after them herself. Since they had run away, Peggy
mounted her bouncing horse and took off in hot pursuit, “Bang bang”
and so forth. As she kept it up, I asked how many bullets she had left.
“Four” was the answer. She shot them all. “Which is more Peggy, four
bullets or a ‘whole lotta bunch’?” She answered “‘A whole lotta bunch’.”


3V1049.01 Finger counting: [I want fifteen childs] (12/6/80)

Asked if she thought it would be nice to have a baby, Peggy held up her
hand and said, “I want a baby. a boy, and a girl.” holding up a finger for
each. We tried again, “Peggy, the baby will be a little boy or a little
girl.” “I want a baby, a boy, and a girl.” By this time, she was running
out of fingers and had to bring up the other hand. Finally, Miriam
asked her how many boys and girls she wanted, and Peggy responded, “I
want 15 childs.”


3V1049.06 Finger Counting: 1-1 correspondence, up to 2 (12/6/80)

no further content.


3V1056.01 Counting objects for herself (12/6/80)

What I remember as significant about this episode was Peggy’s putting
her fingers and the number names into 1-1 correspondence. Now she
can “count” (as documented below) but the limits of her correspondence
appears to be TWO. But that fact that she attempts correspondence at all
shows a preliminary grasp of the relation and some sensitization to its applicability.


3V1058.01 Counting objects: near standard sequence with omissions (12/13/80)

Miriam and Peggy went to visit Mrs. Smith. She keeps toys in her house
for children she takes care of. Peggy selected a ring pyramid and
Miriam (as she later tells the story) inverted the rings. Peggy began
re-stacking the ring and spontaneously reciting number names: “one,
two, three, four, five, six, (seven omitted), eight, nine, ten, eleven,
twelve.” (cf. P150 ? P151? )


3V1063.02 Reciting number names: varied responses to correction (12/20/80)

I drove to New Haven. Miriam and Peggy came along for the ride. On
the return trip, Peggy stood behind and between the two front seats of
the Saab, holding on and exclaiming amazedly at nearly everything
seen. As I drove from I-91 down onto route 80, Peggy noted “There’s a
whole lotta tell-poles for people to count.” and began reciting number
names, “One, two, three, four, five, six, eight, nine, ten, eleven,
twelve… eighteen, nineteen, sixteen.” Let that list represent her basic
recitation. Miriam criticized the omission of “seven” — so Peggy added
it to the list by displacing and omitting “six”. Apparently she knew
there were more “teen” numbers, because at one point she repeated
several times “eighteen, nineteen sixteen, eighteen, nineteen sixteen,
eighteen, nineteen sixteen.” Finally, after “eighteen, nineteen, ” on one
occasion she concluded, “one, two, three, go.”

Was Peggy reciting merely ? She wasn’t counting. “tell-poles,” at least
gave no evidence of doing so. We can’t tell if she was counting objects
in her mind separate from her name list — but I doubt she was.


3V1070.01 Counting: scrambled eggs super ? (12/27/80)

Reading “Scrambled Eggs Super (Dr. Seuss) one page has a line of birds
winding back into the distance over a mountain. Peggy spontaneously
started counting, at the beginning of the line, ” 1…2…3…4…5…6…[here
the line turned and became less detailed]… many birds.”


3V1074.01 Counting: for hide and seek (12/31/80)

Peggy was playing hide and seek with Robby. He was “it” and after
finding her [she didn’t really hide, but stood in another room ready to
laugh when he appeared] told her to go into the end of the kitchen (by
the basement) and hide her face while he went to hide. She obligingly
leaned against the wall and said (not too fast) “1, 2, 3, 4, 5…Here
I come, ready or not.”


3V1084.01 Counting Irregularities (1/10/81)

Peggy “counts,” ie. recites the number names in a quasi-standard
fashion. (omitting “seven” more often than including it.) Although she
has put objects in one to one correspondence, she has not done so
successfully in the standard sense. She counted on her fingers today
showing no non-standard variations. First she counted on her fingers,
at some point reciting several number names before going on. She
stopped (was there here a global criticism that she didn’t have twelve
fingers on one hand), she started again at the number three.

I believe (1/25/81) she is very close to being able to apply the number
names to objects in the standard fashion. Today, P157 (i.e. 3;0;2) we
want to try finger counting.
I believe (1/25/81) she is very close to being able to apply the number
names to objects in the standard fashion. Today, P157 (i.e. 3;0;2) we
want to try finger counting.


3V1148.01 “Tendy” (3/15/81)

While working on dinner in the kitchen amidst a circus of children, I suddenly heard out of the chaos Peggy counting, ‘eighty, ninety, tendy, eleven…’. She trailed off there, perhaps having said ‘eleventy.’


3V1155.01 Cuisenaire rods: playing with them after experiments (3/22/81)

Guessing games
Peggy had used Cuisenaire rods in the immediately previous videotape. Somehow she got hold of them again and I became aware that she was laying them out [Bob had, I think, tried to see if she would build a “stair.”] As she did so, she ran her finger along and chanted, “They get smaller,,, they get tinier…they get bigger…”


3V1179.01 Counting Plates with numbers in various ranges (4/15/81)

Peggy loves to help empty the dishwasher. After stacking the small plates on the shelves, she began counting: 1, 2, 3, 4, 5, 6, 9, etc…18… She continued from stack to stack, using ‘big’ numbers as well, “eighty, ninety, tendy” repeating them as well as smaller numbers and in no obvious order. After getting up, she announced to the world, “I counted all the plates.” — as she had done in her terms by assigning a number name to each item (mauger the lack of order and repetition of tokens).
What does Peggy have to learn about number?
1. use each number name once only.
2. use the number names in a fixed order.
Let’s observe closely how she picks up these ideas — not pushing them on her… but focusing on her natural learning of them, probing some in videotape experiments but not too much otherwheres.


3V1267.01 Computer-based cuisenaire rods (7/12/81)

Peggy enjoyed playing with the Cuisenaire rods during out experiment P181. Either in that one or the next P182, Peggy first accomplished a set of “stairs.”

After the end of the experiment, she continued playing with rods and I heard her mention (at a point where she omitted the 3-length green rod from a series) “Oops. I left out the poor little green one.” After knocking them over and restarting, she went on to omit the 4-length and said something similar – perhaps “left out the purple-y”


3V1275.01 Computer “rods” (7/20/81)

Seeing the trouble she had with the rods always falling over, I asked is a Rods microworld would be easier to manipulate and thus intellectually more accessible to her. So I proceeded to make one, substituting (a later idea) the blinking of numbers in place of partial blanks — that is the active rod is so indicated by its number name flashing at the center (end unit) of rotation.

After introducing this system (P182) later the same day, Peggy;s spontaneously adopted the objective of building a set of stairs on the table and achieved that objective. Since then, she has usually made such a construct whenever she plays with it.

This is not entirely true — for Peggy has used the active rod (usually the white one) driving it over the other rods to make them disappear. I left this feature in the system as a child-correctable bug — ie when a rod has holes in it, it can be repaired by rekeying it’s number name. when I saw Peggy had made all the rods disappear, I asked her where they were. Miriam responded that Peggy had made the white one “eat” them . I don’t know if the idea and word were Miriam’s or Peggy’s.


3V1416.01 A Big Penny and a Little One (12/8/81)

We went to Boston this day for a pre-Christmas visit. Rob hung around LCSI with me. Miriam took Peggy over to the Childrens Museum. Late in the day, the kids were going out with Greg to buy sodas and Peggy — of course — wanted to bring some Regal Crowns. She was distressed because everyone was dressed to leave and she had no money. She came crying to me asking for two pennies to buy Regal Crowns. “I need two pennies to buy my Regal Crowns, a big penny (by which she meant a quarter) and a little penny (by which she meant a dime.)


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Vn 001.01

Everyday Calculation


Miriam suffered a queasy stomach today, so she didn’t join us at our evening meal. She lay on the loveseat near our dining area. The speakers for our radio are directly behind the loveseat, thus more enforcing Miriam’s attention than ours.

During a radio commercial we others chose to ignore, Miriam burst out in disbelief, “That would be 60 dollars.” When I asked what her reference was, she explained that if the four of us attended a certain fixed-price dinner (one we had no interest in) the total cost would be 60 dollars. “How did you figure that out?” I asked. “Did you learn how to multiply already?”

Miriam coupled a disclaimer of any knowledge about multiplication with one of her ‘dumb-Daddy’ looks and explained that each meal cost 15 dollars and she knew that 15 plus 15 were 30 (which, as she added in parentheses, accounted for 2) and another 15 and 15 make another 30. The two 30’s making 60 dollars gave her a conclusion and accounted for the four meals she knew we would want.





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.



Arithmetic Ripples


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

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

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


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

Addendum 17-1


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Arithmetic Ripples


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.”


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



Arithmetic Ripples


Miriam was playing in the kitchen with Scurry this morning. Gretchen and I were discussing some topic, and I mentioned a division problem. Miriam piped up, “I can divide, Daddy. . . . 8 divided by 8 is 1.”

I congratulated her on her prowess. For Miriam the formula she recited constitutes division. The division problem is the one I executed in playing Dr. World’s computer game (in Home Session 5, 5/30). Despite her ability to divide sets concretely (see Miriam at 6: Arithmetic), Miriam does not appear to associate dividing with “division,” a process for which she has, I believe, this one example.


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