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LC1bT01

LC1bT01 Protocol 1

Included Text Pages(14)

RAL protocol 01.1

RAL protocol 01.2

RAL protocol 01.3

RAL protocol 01.4

RAL protocol 01.5

RAL protocol 01.6

RAL protocol 01.7

RAL protocol 1.8

RAL protocol 1.9

RAL protocol 1.10

RAL protocol 1.11

RAL protocol 1.12

RAL protocol 1.13

RAL protocol 1.14

Included Materials(8)

Addendum 1
RAL protocol 01-A1

Addendum 2, BGB, BigBuilding
RAL protocol 01-A2

Terminal Log Pages (6)
RAL protocol 01-A3

RAL protocol 01-A4

RAL protocol 01-A5

RAL protocol 01-A6

RAL protocol 01-A7

RAL protocol 01-A8

LC1bT02

LC1bT02 Protocol 2

Included Text Pages (2)

Discussion
RAL Discussion before Protocol 2

Protocol 2.1
RAL Protocol 2.1

Included Materials (3)

RAL 2-A1 Terminal Log

RAL 2-A2 Terminal Log with Notes

RAL 2-A3 Terminal Log

LC1bT03

LC1bT03 Protocol 3

Included Text Pages

RAL protocol 3.1

RAL protocol 3.2

RAL protocol 3.3

RAL protocol 3.4

Included Materials

RAL protocol 3-A1

RAL protocol 3-A2

LC1bT04

LC1bT04 Protocol 4

Included Text Pages (2)

RAL protocol 4.1

RAL protocol 4.2

Included Materials (3)

RAL protocol 4.1 Add1

RAL protocol 4.1 Add2

RAL protocol 4.1 Add3

LC1bT05

LC1bT05 Protocol 5

Drawing a Fox (cf. discussion in Development of Objectives)
n.b. hand-written date at top of first page in error by 2 years.

Included Text Pages

Included Materials

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

PG11

Peggy Study, Panel PG11

Themes: Reading “Hop on Pop,” Adding with Fingers
Source: (Lawler); date: 1/5/84

Title: Peggy at Six Years
Texst commentary: documentation of observations (right or wrong); Symbols, symbol manipulation



PG11A1 Reading 1, 3mb


PG11A2 Reading 2, 22mb


PG11A3 Reading 3, 21mb


PG11A4 Reading 4, 22mb


PG11A5 Reading 5, 23mb


PG11B1 Addition with Fingers, 19mb


PG11B2 Even More Fingers 20mb


PG11B3 New Method, 18mb


PG11C Large Number Names, 9.5mb

PG15

Peggy Study, Panel PG15

Themes: Reading, Number, Drawing
Source: (Lawler); date: 3/8/84

Title: Peggy at Six and a Quarter
Text commentary: ; the penultimate video in Guilford; the last has poor video quality.



PG15A Introduction, 6mb


PG15B1 Frontier Reading (1), 31mb


PG15B2 Frontier Reading (2), 32mb


PG15C1 Numbers, Adding (1), 29mb


PG15C2 Numbers, Adding (2), N 32b


PG15D Drawings, Notes, 25mb

Vn13201

Vn132.1 BIG-SUMS Extension 4/21 & 22/78

4/21 “Oog. I’ve got to add all these big numbers.” So I called to
Miriam’s attention a situation in which one could make use of the kinds
of skills she had developed in adding multi-digit addends: I was summing
the number of calories in my day’s diet. Miriam asked if she could add
the numbers for me. I protested that there were four big numbers to add
together and asked if she could do that. Taking a piece of note paper,
she replied, “I think I can do it.”

Miriam worked the problem of Addendum 132 – 1 in the notebook
I held on my lap. I expected her to perform 3 separate additions.
Instead, after I had read the first two addends (235 and 560) and Miriam
wrote them down in the vertical form, she asked me what was the next one,
and said she was going to do them all together. With the four addends in
place, she put in columnar division lines and plus signs, then underlined
each addend. I asked Miriam if she had been doing any such work in school
and why she underlined the addends. She answered “No” to the first
then question and “I don’t know. I just like to.” to the second. She declared
that this was the first time she ever tried “anything like this”, never in
school, and she had never seen Robby do it; she wanted to try just because
she thought she could do it.

The units sum was done first. As she summed the tens column
digits, Miriam asked for confirmation. “You put down the 4 and carry the
1, right?” I agreed. In the hundreds column, Miriam said, “You put down
the 7 and carry — no.” She wrote 17 in the hundreds column and continued,
“It’s 17 hundred and 45.” Then Miriam went on to draw her star and
added a ‘happy face’, which is her teacher’s symbol put on work well done.

4/22 As I recorded my supper calorie count, Miriam, who had a head
ache and felt too ill to eat supper, eagerly asked if she could add my
big numbers. (See Addendum 132 – 3). One addend I read as 12 hundred
85. As Miriam started to write 12 in the hundreds column, she asked if
it were correct. I told her another way to say the number was “1 thousand
2 hundred 85.”

She began adding at the left but immediately thereafter realized
there would be a carry-in and declared her first result an error. After
restarting on the right, at the sum of the tens column Miriam asked, “Do
you put down the zero and carry the 2?” I responded, “Do you think that’s
right?” She answered confidently, “Yes” and proceeded. When I inquired
whether her carry mark were a 1 or a 2 (it was initially illegible),
Miriam said “2” and rewrote it.

Upon her completing the sum, I asked what was the result.
Miriam answered, “2 thousand twenty five [for 2205]. Is that right?”
I said “No.” After a little while, Miriam said, “2 thousand 2 hundred
and 5.”
Later in the evening Miriam explained, “When I said that answer
had twenty five, I thought it was like the 2, 0 was twenty and then the
5, but I know that’s wrong.”

Relevance
I consider the incident of 4/21 an extraordinarily rich one.
Consider the precursors of the achievement. Miriam’s use of the standard
algorithm for addition in the cyclic notation is well-developed but
shaky, because of the difficulty of verifying the results of such a
computation. The second source is one Miriam denied (cf. Addendum 132 – 2)
as being like this problem. The triple single-digit sums of her school
work from last week clearly suggested to her the possibility of adding
more than two addends at a time. Her sense of the differentness of the
kinds of problems is evidence that she sees them as belonging to dif-
ferent worlds of thought and thus potentially connectible to different
cognitive structures.

The second significant aspect is that this problem shows as [an?]
internally directed extension of integrated knowledge as the elaboration
of a new procedure (handling triple single-digit sums) and its insertion
as a subprocedure in the controlling superprocedure Miriam uses for
multi-digit addition.

Finally, in drawing the teacher’s ‘smiley face’ on her own
math paper, Miriam shows herself playing both roles of ‘learner’ and
approving authority that is reminiscent of her playing both sides of
games at Tic Tac Toe.

In the problem of Addendum 132 – 3 Miriam confronted again
the double naming of ten-value hundreds as thousands and met her first
carry of value not 1. Her reading of the result suggested that with a
single, well-presented contrast of numbers to read, she should be able
to lock in the distinctions necessary to read any 4-digit results cor-
rectly. A day or so later, Miriam was able to read numbers from this
parallel list, and when shown the number names first, predict where the
zeroes and fives would appear.

             5     FIVE
            50     FIFTY
            55     FIFTY FIVE
           500     FIVE HUNDRED 
           505     FIVE HUNDRED FIVE
           550     FIVE HUNDRED FIFTY 
           555     FIVE HUNDRED FIFTY FIVE
          5000     FIVE THOUSAND 
          5005     FIVE THOUSAND FIVE
          5050     FIVE THOUSAND FIFTY
          5500     FIVE THOUSAND FIVE HUNDRED
          5505     FIVE THOUSAND FIVE HUNDRED FIVE
          5550     FIVE THOUSAND FIVE HUNDRED FIFTY 
          5555     FIVE THOUSAND FIVE HUNDRED FIFTY FIVE

Addendum 132-1

Multiple Addends -1

Vn 132-1 Multiple Addends -1

Addendum 132-2

Schoolwork Sample

Vn 132-2 Schoolwork Sample

Addendum 132-3

Multiple Addends -2

Vn 132-3 Multiple Addends-2

Vn13301

Vn133.1 4/28/78-5/8 & 31/78

4/28 The calorie counting continues for me, and the adding for
Miriam. The sum of Addendum 133 – 1 is interesting because of its error:
the left justification of 3 and 4 digit addends in the same vertical grid.
Despite this mal-arrangement, with its implication that the place-value
basis of column alignment has not been mastered, Miriam’s carrying
procedure was standard.

5/8 When the children started squabbling over whose turn it was to
add up my daily calorie count, I refused either the privilege. But today
Miriam took up the task with no complaint from Robby so I let her proceed.
She wrote the addends and result thus:

Vn 133-2 3T like grid for Addition

Most interesting was her procedure. Beginning at the left, Miriam cal-
culated 14 for the column and carried a 10 to the next right column. I
intervened, “Miriam, don’t you remember your good trick for when you
have carries?” She restarted on the right and reached a standard result.

5/31 After staying up late last night at a dinner with Mimi Sinclair
and Marvin and Gloria (whereat Miriam showed off a little by working with
the soma cubes and the 7-piece pythagorean puzzle), Miriam stayed home
from school today. During the course of the day, I posed the following
problem, designed to contrast for her results of mental addition and
vertical form addition to point up the question of place value in the
vertical form.

Vn 133-3 Place Value Alignment Test

When Miriam saw the horizontal sum, a sheet of paper covered the question
and sums below the dotted line. She knew the result was 20 and after she
had written down the answer I took away the covering sheet. Miriam read
the question, then mused, “For both, the numbers are eleven.” Puzzled,
I asked what she meant, but Miriam continued without responding to add the
left sum from the left to 65. She stepped back, then with a “No. That’s
not it.” returned to the chalk board to erase her result. I stopped her
erasure and directed her to the second sum, which she added to 20 (from
the right, with only mental marking of the carry), then encircled. When
I asked how she knew the right sum was correct, Miriam argued that the
answer was 20.

Relevance
These three incidents all focus on the imperfections of
Miriam’s place value comprehension. The last I expect to be seminal,
leading Miriam to recognize that the results of addition, in whatever
form, should be consonant, and that she can herself judge the correctness
of small vertical sums by checking through mental computation.

Addendum 133-1

Column Alignment Error

Vn 133-1 Column Alignment Error