### Vn119.1 **Multiplying by Twenty** 1/25/78

The children love to get mail and when an envelope comes addressed

to them, to open it. Each has a bank account concerning which they

receive annual earnings statements. The children opened their mail and

puzzled over the contents — a statement of account number, social

security number, and interest earned for the year with no specification

of the current balance. I checked for the latter because, as I explained

to them, I preferred their remaining ignorant because of their

inclination to blat to their friends what capitalists they are. I went

on that they shouldn’t go about bragging how much interest they had

received. “Why not?” I informed them that anyone knowing their interest

could estimate their capital simply by multiplying the dollar amount by

twenty.

Robby and Miriam realized they could circumvent my not telling them

of their bank balances, and Robby began to do so. Miriam lamented she

didn’t know how to multiply by twenty and received Robby’s promise of

help after he completed his own computation. A few days before he and

I had discussed a good trick for 10 times: just writing down an ‘extra’

zero on the right end of the number. Robby realized he could get the

desired result by doubling the interest (by addition), then adding a

zero. He became confused about manipulating the decimal point during

the 10-fold multiplication, but accepted my procedure for doing do. He

read his balance to Miriam, then went to help her.

Miriam followed Robby’s direction but set up the problem herself.

My role was limited to restraining him from taking over. No problem

with adding 2 plus 2. The carry first arose with 6 plus 6 (see Addendum

119 – 1). Miriam said, “I put down the 2 and carry the 1.” Robby

responded, “Right.” and when she went to mark a carry over the tens

column, he directed her to place it over the hundreds. With some labor

Miriam added 8 plus 8 and 9 plus 9, handling the carries appropriately.

Thus she had doubled $98.62. But what did the answer mean?

Miriam tried to read her answer 19724: “One thousand. . . one thousand

. . . .” She believed her result should be of the same order of magnitude

as his, but was lost because she could not coordinate that correct judgment,

her accurate computation, and the structure of the problem’s solution.

As Robby did at first also, Miriam neglected the 10-fold multiplication;

nor did she understand at all this good trick for 10 times (she had

never been exposed to it before). Comparing Robby’s work to her own

did not help. Rather than protract her frustration, I “showed” her what

to do. (This means I wrote in the decimal point and an arrow and mumbled

a few words). Miriam accepted my answer as correct and sensible.

Both children were able to rejoice once more at having outwitted

their dumb old Dad.

**Relevance**

The first incident shows the children applying their arithmetic

skills to a problem too difficult for Miriam. She can effectively execute

complex additions but does not dominate the number representations.

Her writing a carry mark at the top of the tens column shows her sense

that the 1 of 12 still belongs more to the 2 than to the left adjacent

column. I infer that Miriam is working out the problem of what a carry

means. She is very close to understanding. The second incident suggests

I follow up Miriam’s judgment that school arithmetic papers are hard;

why should she find them so?

**Addendum**

**Adding by Miriam and Robby**