Skip to content
Archive with last of tag-string Nbr

3V0531.1

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.

3V0594.2

3V0594.02 ONE, TWO: [one, two]: note on standardization of Peggy’s counting
09/08/79;

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.

3V0611.1

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.

3V0674.1

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”.

3V0700.2

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 ?

3V0747.3

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

3V0749.1

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.

3V0801.1

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)

Gretchen

3V0824.2

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.

3V0829.1

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.

3V0865.1

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.

3V0882.2

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).

3V0931.1

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.

3V0996.1

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.

3V1010.1

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.

3V1025.1

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.”
Gretchen.

3V1043.2

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’.”

3V1049.1

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.”
Gretchen

3V1049.6

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

no further content.

3V1056.1

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.

3V1058.1

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? )
.

3V1063.2

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.

3V1070.1

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.”
Gretchen

3V1074.1

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.”
Gretchen

3V1084.1

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.

3V1148.1

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.’
Gretchen

3V1155.1

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…”
Gretchen

3V1179.1

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.

3V1267.1

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”

3V1275.1

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.

3V1416.1

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.)
.

LC1bT06

LC1bT06 Protocol 6

Included Text Pages (2)

RAL protocol 6.1

RAL protocol 6.2

Included Materials (2)

RAL protocol 6-A1

RAL protocol 6-A2

LC1bT07

LC1bT07 Protocol 7

Included Text Pages (2)

RAL protocol 7.1

RAL protocol 7.2

Included Materials (2)

RAL protocol 7-A1

RAL protocol 7-A2

LC1bT13

LC1bT13 Protocol 13

Included Text Pages (7)

RAL protocol 13.1

RAL protocol 13.2

RAL protocol 13.3

RAL protocol 13.4

RAL protocol 13.5

RAL protocol 13.6

RAL protocol 13.7

Included Materials (6)

Figure 1
RAL protocol 13 Figure 1

Addendum 1
RAL protocol 13-A1

Addendum 2
RAL protocol 13-A2

Addendum 3
RAL protocol 13-A3

Addendum 4
RAL protocol 13-A4

Addendum 5
RAL protocol 13-A5

LC1bT17

LC1bT17 Protocol 17

Included Text Pages

RAL protocol 17.1

RAL protocol 17.2

RAL protocol 17.3

RAL protocol 17.4

Included Materials

RAL protocol 17-A1

LC1bT19

LC1bT19 Protocol 19

Included Text Pages (7)

RAL protocol 19.1

RAL protocol 19.2

RAL protocol 19.3

RAL protocol 19.4
RAL protocol 19.5

RAL protocol 19.6

RAL protocol 19.7

Included Materials (2)

Addendum 19-A1
RAL protocol 19-A1

Addendum 19-A2
RAL protocol 19-A2

LC1bT20

LC1bT20 Protocol 20

Included Text Pages

RAL protocol 20

Included Materials

None

Vn01701

Vn017A

Arithmetic Ripples

5/28/77


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

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

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

Relevance

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

Addendum 17-1

ADDENDUM 17-1

Comments Off on Vn01701

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.

Vn01703

Vn01703

Arithmetic Ripples

6/1/77


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.

Relevance

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

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.

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

Vn04901

Vn49.1 Finger Counting 7/24/77

At lunch, I inquired of Miriam how she used to add on her fingers
numbers like 2 plus 7. After saying ‘9’ and my refusing that answer,
she counted up, i.e. Miriam said ‘7’ then lifting her pinky and fourth
finger on the right hand, ‘8, 9.’

I again rejected the answer: “Try hard to remember when you couldn’t
do any sums greater than 10, how did you add 2 plus 7 then?” Miriam
counted from 1 to 7 using her right hand and the pinky and fourth finger
of her left hand; she then raised her thumb and index finger, saying
‘1, 2’ thus leaving her middle finger unused.

Miriam complained that she no longer enjoyed doing such easy sums,
so I asked her to add 37 and 12. She looked shocked — then said ’49.’
When asked, she explained: “I knew it had to be more than 40 ’cause it’s
like 30 plus 10, so I said ’47,’ then ‘8, 9’ because of the 2 left over.”
(Miriam counted upon her hand for the last 2).

Relevance
This vignette confirms a speculation (cf. How Miriam Learned to
Add) that Miriam’s early use of commutativity is an artifact of her
finger counting procedure in that selecting the larger of two addends
to first represent is less confusing where the sum approaches 10.

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.

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.

Vn06501

Vn65.1 Arithmetic Ripples 8/13/77

1. Miriam brought me this morning a dime she had found in the laundry.
My speculation is that it had been left in some pocket, fell out in the
wash, and was pinned in and nearly cut in half by the washing machine.
How it got on the floor where she found it I don’t know. When she
showed it to me, Miriam said, “See, Dad, somebody tried to cut this dime
in half. I bet they thought they would get two things. . . two nickels.
What a silly idea.”

2. Miriam’s most common purchase is chewing gum. She knows the going
rate for Care Free and Trident packs is 15 cents. When Gretchen went
shopping recently, Miriam gave her a quarter and a dime, placing an
order for two packs of gum.
Expecting a nickel change, Miriam asked Gretchen for it on her
return. There was no change. Gretchen explained that where she had
purchased the gum the price had been 20 cents per pack, 40 cents for
both. Miriam looked worried: “Do I owe you a nickel?” Gretchen told
her not to worry about it. Miriam muttered to herself, “20 cents, that’s
5 cents tax on each pack.”

Relevance
These data are further examples of Miriam’s assigning any non-
explainable variation in price to the category of tax and her puzzle-
ment over the idea of dividing coins into fractional parts. (She does
not know about the obsolete piece-of-eight, a Spanish peso or dollar
designed to be cut into eight reals or ‘bits,’ whence our expression
‘two bits’ designating a quarter.)

Vn07901

Vn79.1 Sums Over a Hundred 8/29/77-9/1/77

8/29 While we sat at lunch today, Miriam introduced the topic of adding
with this claim: “Daddy, if you live for another hundred years, I know
how old you’ll be.” When I expressed surprise Miriam demonstrated:
“A hundred 37.” Two complications derailed this discussion. Robby
introduced my birth on February 29th with its implication of quadrennial
birthdays. Before we entered more complicated computations on this
basis, I noted that I would be dead before a hundred more years and
that one stops counting a person’s birthdays when he dies. Both children
looked at me blankly, and we proceeded to a discussion of what death is like.
(If curious, confer the note appended at the end of this vignette.)

9/1 This evening, I read aloud to Gretchen an excerpt from a draft-
section of Seymour Papert’s Logo book, a sardonic description of the
class structure of the mathematics education world:

Mathematicians create mathematical knowledge, math education
researchers package the material for children, teachers deliver
the packaged stuff, evaluators measure how badly the whole
process worked.

When Gretchen laughed, Miriam, out of sight in the adjacent area of the
loft, commented, “I don’t get it. I don’t think that’s funny.” Although
in one sense this is not at all funny, in another way it is, and so I
told Miriam. She replied, “What do you mean?”

Bob

How much is a hundred 70 plus 27? [original has a hundred 7]
Miriam

97. . . a hundred 97. Did I do it right?
Bob

Yes. Did you use your fingers?
Miriam

You want to know how I did it?
Bob

Sure.
Miriam

I said 70 plus 20. That’s 90, so I have the 97.
Bob

Where’d the hundred come from?
Miriam

It was a hundred 70. . . . Did I do it right?
Bob

You did it beautifully. . . and that’s more important than doing it right.
Miriam

I know that.
Bob

You also did it correctly.

Miriam went back to playing at what had occupied her before the dis-
traction of my reading aloud, so I did not explain why this problem she
solved, documenting as it does her ongoing progress in constructing her
own algorithms for addition, shows how ‘funny’ in another sense are the
best efforts, even the well-intentioned efforts, of the mathematics
education establishment.

Relevance
Since Miriam’s forgetting how to add multi-digit addends and her
subsequent reconstruction of adding procedures on a different basis,
I have let her curiosity guide our discussion of the algorithms she
employs for computation. This vignette records Miriam’s recrossing
of the hundred barrier with her own method of adding.

* For the curious: when I elaborated somewhat further, I said,
“You don’t count birthdays ’cause you can’t think at all when you’re
dead. You don’t eat or breathe either, but that doesn’t matter because
you can’t feel anything at all.” Robby came back: “Oh, I get it now.
Being dead is like you blew a fuse.” I agreed: “And each of the major
organs in your body — your heart, your lungs, your liver — each of
those is like a fuse and when one of ’em goes, you die.” Robby has
spent time since building two models, the Invisible Man and Invisible
Woman, attempted over a year ago and judged too complicated then.

Vn08201

Vn82.1 Hanging Designs 9/3/77

After today’s session was complete, I asked Miriam why she had not
pinned on the wall — as she had said she intended — those designs made
in yesterday’s session (Logo 58, 9/2/77). She explained that she had
started to do so earlier but needed help.

I separated the designs from the interleaved blank pages in the
pile on her desk, then asked where to hang (“Up there.”) and how.
Miriam’s directions: “In alphabetical order, by the numbers.” When I
found this opaque, Miriam explained, “Like the way Robby did it.”

At Miriam’s direction, we set up a display of poly spirals varying
from the base of 60 degrees (we had originally called such a shape a
‘maze’) in order by the turtle’s angle of turning up to 67 degrees.
Miriam had created this complete set of designs with considerable direction
from me (cf. Session 58), and she used Robby’s arrangement of designs
as a model. Nonetheless, the creation of this family of shapes was her
idea.

We came to a last design. All the others had been made with an
increment (‘delta’ we call it) of 2 turtle steps. At the angle of 67
degrees, we made a design with delta = 1. (This was done because I had
been too directive earlier in the session, requiring Miriam to hold
delta constant.) I asked: “Where do we want to put this one? We have
a 67 degree design already, but this one’s got a different delta; should
we just put it under like the others?” Miriam instructed me (by placing
the design in this place) to tack the design on the wall at the side of
the other 67 degree design and “we may want to make another family later
like the other one.”

Relevance
In the directions Miriam provides for how her poly spiral designs
should be hung on the wall, one can see her beginning to organize them
into groups defined by the changing of one variable while the others
are held constant.

Vn08301

apparently, this file needs to be recreated, from earlier sources.
The tags attached to the source suggest it is important.