### Vn54.1 More Arithmetic 7/31/77

While I was having lunch, Miriam came (in her bathrobe
from the tub) and asked me if I could divide 79. I asked by
what. Miriam didn’t answer but moved away. I pursued the
question: “What do you know that’s a 79?” Miriam replied,
“They said it’s 2 for 79, the quinine water, and I want to
know how much it costs.”

Bob You want to divide by 2. Can you divide 79 by 2?
Miriam No.
Bob Can you divide 80 by 2?
Miriam (long pause) I’ts 40.
Bob That’s very good, sweety. You estimated 79 by 80.
But 79 is a little less than 80; it’s less by 1.
Can you think of a number that’s a little less than
40, but not by 1?
Miriam 30.
Bob That’s less than 40. 79 is 1 less than 80. 39 is 1
less than 40. So your result should be more than 39.
Miriam 48?
Bob No. The correct answer is 39 and a half.
Miriam A half?
Bob Yes.
Miriam That’s silly. How can you cut a penny in half?
Bob You get two halves.
Miriam No. . . . 39 and a half. . . . I guess the rest must be tax.

Later in the day, Miriam complained to me that we hadn’t
done any adding for a long time. She griped further that when
on vacation in Connecticut we hadn’t done any experiments; we
hadn’t done the puzzle I bought; we hadn’t played with the set
of blocks I took with me. I defended myself against the charge
of sloth by arguing that I thought she needed a rest, that I
feared I had been pushing her too hard with experiments. Miriam
went off, still disgruntled. She stopped by a set of addition
exercises left over from Robby’s second grade.

A little later Miriam asked, “Daddy, is 28 plus 48 equal
to 76?” I congratulated her and asked how she had ever fig-
ured it out. Miriam explained: “I know that 2 plus 4 is 6.
So this is like 20 plus 40 and that equals 60. Then I took
the 8 and said 68 and counted (demonstrating on her fingers)
69, 70, 71, 72, 73, 74, 75, 76.”

I told Miriam she had developed a very good procedure and
asked her not to do any more of Robby’s problems until the
evening, at which time I promised to do some adding with her.

Relevance
This vignette shows Miriam extending her way of thinking
about adding to problems she feels she should be able to solve
(given her former successes with much larger numbers). The
first incident also evidences the concreteness of her thinking
about numbers, i.e. division of 1 into halves is impossible
because it’s pennies that are being divided. Further, the cloak
of the mystery of “tax” is thrown over the ambiguous conclusion
of her computation.