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

! The following questions require the
same process, which, as will be perceived,
! recognizes the principle of Fractions.
What will l-fourth of barrel of -four
cost at 8 dollars per barrel.
If 4sevenths of yard of cloth cost .
cents, what will 1-seventh of yard cost?
! The processes thus hastily sketched are
all which can occur in Multiplication and
Division, except when we come into the
province of Al zebra and Geometry They
will not always assume precisely the forms
which have been given, but in spirit and
essence they will be the same. And they
are the key to all operations in Multipli-
cation and Division. Equally simple and
general are the processes required in Ad-
ditionand Subtraction. We would not be
understood to say that no problem re-
quiries the application of more than one
of these processes; far from it. A prob
lem may require several of them, or that
the same process shall be many times re-
peated; but each process shall of itself be
simple, and in al such cases the original
problem can be resolved into a series Of
simple ones, each as simple and easy of
solution as those We have given. We
say, then, that these processes, are the
key to all arithmetical operations, and
submit the question,-Is it not better, is
it not more philosophical to require our
pupils perfectly to master these, and to
base their work upon them, and learn ev-
cry where to apply them, than to burden
their memory with SO many arbitrary rules
and useless distinctions? In the one case
We are teaching principles, developing
the reasoning powers, and cultivating the
whole mind; while in the other we are
teaching forms and cultivatiDg the memo
ry only.
In Mental Arithmetic We take such
course as has been recommended. We
do teach principles, We do require our pu-
pills to follow out rigid reasoning process.
es. What teacher in using Warren Col
burns First Lessons ever thought of Civ
ing his pupils rule! Yet every one
praises that book as the best ever written;
every one who ever studied it speaks of
it as the one from which be derived his
most valued arithmetical knowledge and
discipline. Why is this? Simply, I fan
cy, because it has no rulers, because it
throws the pupils so much upon their own
resources, compelling them to learn prin-
ciples, to follow out rigid reasoning pro-
cesses, and connected trains of thought,
to examine and know for themselves the
necessity and reason fthe steps they take
and the operations they perform.
When the scholar has been through
Mental Arithmetic and take up Written,
he seems to have entered ou an entirely
different field, where all that he has FORM
erly learned is to be thrown away. At
the very outset he is required to learn an
arbitrary rule, then another, then anoth-
er, &c, &c, learning each as a new and
distinct thing, having nothing to dc with
any other principle or process, or with
anything previously learned; when per-
haps precisely the same operations may
be required, and the same principles in
volved in all of them. Is this philosophi-
cal? What difference is there between
Mental and Written Arithmetic, to 1e-
quire so wide difference in our methods
Of teaching them? The only real differ
cnce in their nature is that in Mental A
rithmetic We must retain in the mind the
numbers we use and the results we obtain,
while in Written Arithmetic We write
them, and thus relieve the memory.
Another point which | would suggest
is, that scholars ought always to prove
their work for themselves, instead Of YER
flying it by comparison with the work or
answer Of another | believe that the
practice of placing the answers to arith
metical problems within reach of the pu-
pill, either in the text boob or key, is al
ways injurious.
In the first place such tests arc unprac-
teal, for they can never be resorted to in
the problems of real life. What mer-
chant ever thinks of looking in a text
book or key, or of relying on his neigh
Dor to learn whether he has added coll
umn correctly, drawn correct balance
between the debit and credit sides of an
account, or made o mistake in finding the
amount of bill?
When pupil, having left the school
room, performs problem of real life, how
anxious he is to know whether bis result
be correct! Neither textbook nor key
can aid him now, and he is forced to rely
on himself and his own Investigations to
determine the truth or falls ty OF his work
If he must always do this in real life, and
if his school course is to be preparation
for the duties of real life, ought lie not to
do it as learner in school! Is it right
to lead him to rely on such false tests!
But the labor of proving an operation
is usually as valuable arithmctical work
as was the labor of performing it; and it
will oftentimes make process or solution
appear perfectly simple and clear, when
it would otherwise have seemed obscure
and complicated.
Again, the science of Mathematics, OF
which Arithmetic is a branch, is an exact
science; it dwells in no uncertainties, ITS
reasonings are always accurate, ana /,
based on true premises, must always ieau
to true results. In Arithmetic, the pupil
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