The Jeffersonian. [volume] (Stroudsburg, Pa.) 1853-1911, 1854-12-07

plies the scholar Mary.
But this is not enough The schola,
should trace clearly and state the connec
ton between the problem and the result
a times 3; but he should first be led tc
see the deficiency of his former answer
To show him this, the teacher may reply
Yes, L know that times cents are II
cents, and so times 4 cents are ! 6 cents
Why do you not say l6 cents them Be
cause the apples cost 3 cents piece; not
a. Then why not say 15 cents, because
S times cents are Is centsl' Because
there were only apples, and they cost
3 cents apiece The pupil will now sec
that to make his reasoning perfect, he
must take into account the number of ap-
Piles and the price of each, and will after
little effort be able to give perfectly
rigid demonstration, similar TO the follow
ing. IF one apple cost 3 cents, apples
will cost times cents, which are 10
cents. Therefore apples at 8 cents a
piece cost II cents
1t is much better that the scholar should
thus discover this process for himself, than
that the teacher should give him an ar
bitrary form for it, for he will better un-
derstand and appreciate his nature
Moreover he will be thrown more fully
upon his own resources, and will do more
of the work for himself 1t should be al
ways borne in mind that it is work which
the scholar does for himself which edu-
Gates him. The work done by the teach
er cannot do it. He is the best teacher
who throws the most work on his pupils,
and does the least direct work for them.
Indeed, were it possible for teacher to
stand before his school and do nothing
himself, yet keep every scholar proftably
and constantly employed in performing
the appropriate work of the school-room,
he should do it; and he who could do it,
would best deserve the title of Model
Teacher.
The reasoning process now given, SIM
PIE as it is, is the key to all processes in
Multiplication which are required in A
rithmetic, even those which depend di-
rectly on Algebraical or Geometrical prin-
ciples There is not in Arithmetic, from
beginning to end, question requiring a
multiplication not depending directly on
Algebra or Geometry, which does not re-
quire essentially this process. By fully
mastering it, then, in its simplest form,
We are preparing to refer to the same
simple principles, questions apparently
entirely unlike.
Thus yards equal how many feet
This question is, in works on Written A
rithmetic, classed with questions in Re
duction Descending, and a special rule is
given for their solution. But it requires
(the Tables being lcarned) no new prin-
ciple, reasoning process, or operation.--
Thus since one yard equals 8 feet, yards
must equal times 3 feet, which are II
Reduce 4 to thirds. This is classed
with questions in the Reduction of Whole
Numbers to improper Fractions and is
honored with a new rule The simple
solution, however, is Since one equals
three-thirds, 4 must equal 4 times three
thirds, which are 10 thirds
What will yards of cloth cost at g.
twentieths of dollar per yardl' This
again is thrown into a new class, viz -
TO multiply Fraction by Whole Num-
ber,' and has its peculiar rules. The So
luton is however as before. IF one yard
costs 3-twentieths of dollar, yards will
cost times 3-twentieths of dollar, which
are l2-twentieths of dollar The list
might be extended indefinitely, but cases
enough have been given to show the abb
surdity of the common classification, or
rather the absurdity of requiring scholars
to burden their memories with formal ar
btrary rules. | have given four ques-
tons, all, as We have seen, alike in prin-
ciP!e, and al! involving the same reason
ing process; yet by the system of rules.
the pupil is required to learn them as
though they had nothing to do with each
other. He first learns his rule for Sim
pie Multiplication, then, after turning o.
yer a few pages, he comes to Reduction
Descending, when he must learn a new
rule, and how to work by it; little furth-
er on is Reduction of Whole Numbers to
Fractions, with o new rule to be learned,
and process presented as new to be
mastered; as so on again to Multiplication
of Fraction by Whole Number, when
the same process is to be repeated. Now
is not this unphilosophical! Does it not
render the subject altog ether too compli-
Gated, and impose great amount of need
less labor on the pupil!
In Division, there are two forms ofreas-
onng process corresponding to two dis
tinct classes Of questions. One of them
will be required in the solution of the
question. How many apples at cents
apiece can be bought for II centsl' The
reasoning process required is in spirit as
follows :-'If for cents one apple can
be bought, for II cents as many apples
can be bought as there are times cents
in 2 cents, which are 4 times. There
fore 4 apples at 3 cents apiece can be
bought for II cents
The following questions require this
process
IL yards equal how many feet!
II thirds equal how many ones?
How many yards of cloth at 3-twen-
t;aths of dollor nor ya~d ooD bo bouob >