et the tapping.

In division four balls were first pushed to the left end of the rod,
thus: 0000. "How many balls are there to the left?" Four taps. They were
now divided into two pairs, thus: 00 00. Pointing to the units of one
group, the teacher asks: "There are always how many in the group?" Two
taps. Three groups were formed, thus: 00 00 00. "There are now how many
balls to the left?" Six taps. "And there are always how many in each
group?", (pointing at them). Two taps. "And how often is two contained
in six?", (pointing to the groups consecutively). Three taps, etc.

The ideas of 'part', of 'whole', and of 'being contained' were
illustrated by means of a chalk line which was interrupted in one or
more places by erasure.

In all these operations Mr. von Osten adhered strictly to the rule, and
required others to do so too, that the number upon which the operation
was performed, must be mentioned first. Thus, one was not to say, "take
3 away from 7", but "from 7 take away 3." Otherwise, he believed, Hans
would become easily confused. Also one was not allowed to say "to
multiply", but to "take" a certain number so many "times". He, himself,
never departed from this practice.

We will not go into the details of the method by which Hans was taught
the meaning of the number signs, of the signs of operation, of the
numbers above 10, or the significance of "digits", "tens", etc. Only
this,--when in problems in addition the sum was greater than 10, the 10
was first tapped and then the remainder of the number added to the 10.
Thus: "You are to add 9 and 5. How much must you add to the 9 to have
10?" One tap. "But now, you were to add not merely 1, but 5; how much
have you still to add to the 10?"--Four taps. In like manner, whenever
the addends were below 20 or 30 and the sum above 20 or 30, Mr. von
Osten would ask for the 20 or 30 taps first. He thought that he was thus
giving his pupil an ever firmer grasp upon the principle of the
structure of our number system, in which all higher numbers are
constituted of tens and digits. For the same reason he used at first,
instead of the words 'eleven' and 'twelve' ('elf' and 'zwoelf' in the
German), expressions which in English might be rendered as 'one-teen'
and 'two-teen' ('einzehn' and 'zweizehn' in the German); and only later,
after the animal had seemingly mastered the meaning in question, did Mr.
von Osten replace them by the usual forms.

All this was beautifully concei