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
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.
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.