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8 CHAPTER 1
not even need to know how to count! In a book about Sicily in the
1950s (Dolci, 1959), a young shepherd boy was interviewed who did
not know how to count:
I can’t count, but even when I was a long way away, I could
see if one of my goats was missing. I knew every goat in my
herd—it was a big herd, but I could tell every one of them
apart. I could tell what kid belonged to what mother....The
master used to count them to see if they were all there, but I
knew they were all there without counting them.
You can see that for the shepherd boy, counting was not neces-
sary. Nor is it required if we want to sell our ﬂock for one dollar a
sheep. We just pair up the dollars and the sheep. And in the case of
two sacks of potatoes, we can take one potato out of each sack and
throw the pair of potatoes over our shoulders. We repeat until one
sack is empty. If the other sack is also empty, we have conﬁrmed
that there were the same number of potatoes in each sack to begin
with.
Counting Viewed as a Representation
But if there are thousands of potatoes, or if we want to keep
a record, or tell someone far away how many sheep we have,
something else needs to be done, involving language—in this case,
mathematical language. The ﬂock of sheep is our “thing, our source
object. For a target, we need a standard object that we know how
to count in a standard wa y. This is the series of counting words,
for example, in English, “one,two, three,...” As each sheep enters
the fold, we count it with the next word in the series, and the last
counting word that we utter is the number of sheep.
Again we have made a one-to-one correspondence, but this time
with a standard object, so we have something to write home about.
The folks at home have the same standard, so they will know how
to interpret our report. (If we report our result to people who do not
know the English counting words, they will not know how many
sheep we have.)

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