I have ceased collecting data for this project, have focused my
attention on the data reduction problem, but Miriam keeps growing, making
breakthrough after breakthrough. This afternoon, for example, I sat
transcribing videotapes in the reading alcove. Miriam, waiting in place
the ten minutes till Sesame Street should appear on TV (we don’t turn on
the TV till the scheduled time for a chosen program arises), was musing
on the couch. She mentioned something about 10 sixties. I could see
her, in her half-reclining position, lifting fingers up and down. A
short time later she exploded; “Hey, Dad! 10 sixties is 6 hundred 20.”
“Wow! How did you ever get an answer like that?”

She explained and demonstrated so quickly I had trouble keeping
pace while I wrote down her computation. She used her finger counting
to control her decadal arithmetic addition procedures thus:

first result -- 620
recapitulation --

Finger count   intermediate computation
       1               60 + 60          120
       2              120 + 60          180
       3              180 + 60          240
       4              240 + 60          300
       5              300 + 60          360
       6              360 + 60          420
       7              420 + 60          480
       8              480 + 60          520     [an error]
       9              520 + 60          580
      10              580 + 60          6. . . .

At this point Miriam hesitated. . . . “Wait. . . . 560 plus 60 is. . . no. . . 580
plus 60 is 640. 640 is the answer.”

Relevance
This performance of Miriam’s is noteworthy several ways. Contrast
this “product” with 4 x 90 of Vignette 106. Notice that the computation
of Vignette 106 is assembled AD HOC. The intermediate results, as numbers,
are manipulated with legitimate and varied operations (addition
and subtraction) to give other especially good intermediate results
which simplify the computation, e.g. 20 is taken from 80 and added to
280 to produce the ‘better quality’ intermediate sum 300; 60 is tacked
on as a simply addable residuum (confer here Seymour Papert’s article
on ‘The Mathematical Unconscious’). This computation is different. The
addition procedure itself is manipulated by controlling its iterations
through counting; the control is independent of the intermediate results.
Counting by non-unary increments (Miriam’s method of multiplying single-
digit numbers) has been replaced in a hierarchical control structure by
the general addition operation. (Confer here the data of Protocol 21,
Multiplying, from the series on Robby’s development.)

This is a further incident giving evidence that “you can’t schedule
learning” (cf. Vignette 91, Squirming and Thinking). Although Miriam
lives in a micro-culture wherein computational issues easily surface,
this particular problem of 10 times 60 is one she posed herself, one
clearly at the expanding periphery of her competence. I claim that, for
whatever reasons (including but not limited to sibling competition),
multiplication has become a vanguard issue of Miriam’s concerns; that
one sees the natural surfacing of such concerns and real intellectual
growth occurring in the interstices of other activity. This claim does
not argue Miriam learns no other way — but this incident shows how
engaging and powerful such learning is. It also argues again that to
study learning, you have to go where the person does it and be there
when it happens.