James Pyle Wickersham.

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the study of Mathematics is not well calculated to
increase ability in its use. As Hamilton forcibly
remarks, "Mathematical demonstration is solely
occupied in deducing conclusions; probable reason-
ing, principally concerned in looking out for premises.
All Mathematical reasoning flows from, and, admit-
ting no tributary streams, can be traced back to its
original source : principle and conclusion are con-
vertible. The most eccentric deduction of the
science is only the last ring in a long chain of
reasoning, which descends, with adamantine neces-
sity, link by link, in one simple series, from its
original dependence. In contingent matter, on the
contrary, the reasoning is comparatively short ; and
as the conclusion can seldom be securely established
on a single antecedent, it is necessary, in order to
realize the adequate amount of evidence, to accu-
mulate probabilities by multiplying the media of
inference; and thus to make the same conclusion,
as it were, the apex of many convergent arguments.
In general reasoning, therefore, the capacities
mainly requisite, and mainly cultivated, are the
prompt acuteness which discovers .what materials
are wanted for our premises, and the activity,
knowledge, sagacity, and research, able compe-
tently to supply them. In demonstration, on the
contrary, the one capacity cultivated is that patient
habit of suspending all intrusive thought, and of



810 INSTRUCTION IN THE FORMAL SCIENCES.

continuing an attention to the unvaried evolution
of that perspicuous evidence which it passively
recognizes, but does not actively discover. Of ob-
servation, experiment, induction, analogy, the Ma-
thematician knows nothing."

The above is a true exposition of the nature of
Mathematical reasoning ; but it does not follow that
such reasoning is of no value. It cannot accom-
plish what its nature unfits it for, but it may accom-
plish other ends quite as important.

The habit of rigid demonstration, of close think-
ing, which Mathematics inculcates, must be in itself
very valuable. If no other kind of reasoning be
practiced, it will no doubt lead to a one-sided cul-
ture ; but, pursued with other kinds, any danger of
this sort is avoided, and much is gained by intro-
ducing somewhat of Mathematical exactness and
clearness, both of thought and language, into what
has been called the " common reasoning of life."
!N"eed it be added that the loose forms of reasoning
to which the majority of men are accustomed stand
much in want of pruning ?

Mathematical reasoning is necessary in Mechanics,
Engineering, N^avigation, Geography, Astronomy,
and other arts and sciences ; and when we consider
that the principles of Mathematics are used in all
transactions of buying and selling, the reasoning
peculiar to that branch of study will not be con-
sidered very uncommon.

Hamilton himself admits that the study of Mathe-
matics tends to correct the vice of ^'mental distrac-
tion," and to inculcate the virtue of "continuous
attention." The attainment of this end alone would



MATHEMATICS. 311

justify the study of Mathematics in our schools, for
no one addicted to the vice of "mental distraction'*
can either become a scholar or succeed well in life.

It is easy to see how a mere Mathematician — a
man who knows nothing hut forms and numbers,
might become credulous as to premises, and skeptical
as to conclusions ; but this danger cannot exist when
instruction in Mathematics is combined with instruc-
tion in other departments of learning. Besides^ it
would seem that any one understanding the nature
of Mathematics would scarcely expect to find else-
where self-evident premises or positive conclusions ;
and hence be on his guard against allowing habits
of thought engendered in demonstrative reasoning
to influence him in inductive reasoning. The in-
ductive reasoner, indeed, needs quite as much to
be on his guard against bad mental habits as the
Mathematician.

The sum of all is this : Man and nature correlate.
It takes the wiiole of nature used as means to culti-
vate duly the whole of man. Instruction confined
to one science, or to one class of sciences must be
partial, one-sided, and productive of bad mental
habits. Mathematics may receive more than its
share of attention in some of our institutions of
learning, and bad results may sometimes flow from
it; but that such studies are valuable in them-
selves, in their objective relations, and as a discipline
of the mind, is susceptible of the strongest proof.
The only point that has been seriously questioned is
their value for the purpose of mental discipline;
but until it can be shown that demonstrative reason-



312 INSTRUCTION IN THE FORMAL SCIENCES.

ing is valueless in itself, that the discipline of the
mental faculties it calls into requisition is a super-
fluous work, and that it has no useful application in
the sciences or in the affairs of life — all impossible,
Mathematics will retain a prominent place in our
courses of instruction.

Arithmetic.

Arithmetic may be defined as the science of dum-
ber. The idea of number, probably, has its origin
in a consciousness of successive mental states con-
stituting periods, and is therefore involved in the
more fundamental idea of time. But whether this
is a correct account of its origin or otherwise, it is
certain that external objects furnish the occasion of
its formation, and that children possess it at a very
early age.

Arithmetic has its Definitions and Axioms, its De-
ductions and Demonstrations, and its Applications.

Among Arithmetical definitions, there must be
those of number^ a U7iit, a fraction^ ratio, &c. ; and
among Arithmetical axioms, there must be the fol-
lowing : " Two magnitudes are equal when they can
be divided into parts which are equal each to each;"
"The whole is greater than any of its parts;" "The
whole is equal to the sum of all its parts ;" "If with
the same means the same operations be performed
upon equal quantities,* the results will be equal."

It is maintained by good authority that "Pure
Arithmetic contains no demonstration," but while
the operations of adding, subtracting, multiplying,
and dividing may, perhaps, be resolved into pro-
cesses of simple intuition, there seem to be other



ARITHMETIC. 813

Arithmetical operations which cannot be so re-
solved. For example, that the product of the two
means of a proportion is equal to the two extremes,
or that if the numerator and denominator of a frac-
tion be multiplied or divided by the same number
its value will remain the same. Arithmetic may
re.quire fewer steps of reasoning than Geometry, but
its methods of operation are substantially the same.
All the reasonings of Arithmetic are properly
deductive or demonstrative. Some writers upon
Arithmetic use the term induction with reference
to certain methods of operation ; but in all cases
the truth sought is capable of being demonstrated
without the series of facts from which it is inferred
by induction, and, besides, universal truths which
it is the special province of deductive science to
attain, can never be arrived at hy an inductive
method.

The greater part of our treatises on Arithmetic is
taken up with the Applications of the science. Its
practical importance renders this desirable.

From what has been stated above, it will be seen
that the general remarks made upon Methods of
Instruction in the Formal Sciences, apply to Arith-
metic; but as already intimated, it is my purpose
to enter upon a more detailed discussion of Methods
of teaching this subject.

Before proceeding to describe these methods,, it
may be well to state the principal ends for which
Arithmetic is studied, and the most necessary con-
ditions of their attainment. These ends are : 1st, To
obtain a knowledge of the properties of numbers ; 2d,
27



814 INSTRUCTION IN THE FORMAL SCIENCES.

To give practice in mathematical reasoning ; 3d, To
attain precision in the use of language ; and 4th, To
secure skill in the application of numbers to the concerns
of life. There are several secondary ends which
must not be overlooked. Among them, the follow-
ing: Ist, Rapidity and accuracy in the solution of
problems ; 2d, Skill in the use of abbreviating artificer;
3d, An acquaintance with methods of proof. The
following may be named as the most necessary con-
ditions for the attainment of these ends : 1st, Tlie
object-matter of the s-cieytce should be distributed in a
logical order; 2d, Pupils should commence tvith the
simplest Arithinetical operation, and be thoroughly
grounded in each step of their progress before taking
another ; 3d, Arithmetical definitions and rules should
be understood- by pupils before they are required to use
them; 4th, Pupils should be taught to explain their
work in clear, concise, and appropriate language ; 5th,
Numerous, well-graded, skilfully varied problems, em-
bodying every principle learned, should furnish ample
opportunity to pupils for making a practical application
of their theoretical knowledge.

Arithmetic is usually divided into two parts. Oral
Arithmetic and Written Arithmetic. These names
are derived from the manner in which the operation
is performed. All Arithmetic is "Mental," "Intel-
lectual," and "Practical" in its character. Written
Arithmetic may embrace all Aritlmietical topics.
In preparing their work, pupils write it out on slates
or blackboards ; and in reciting, they are expected
to explain what they have done. Oral Arithmetic
embraces only such topics as admit of a convenient
oral discussion, and such problems as do not con-



ARITHMETIC. 315

tain large numbers or require complicated fractional
reductions. Pupils are expected to prepare their
lessons in Oral Arithmetic without writing down
their work, and to repeat the problems and solve
them orally, upon hearing the teacher read them.
Instruction in both Oral and Written Arithmetic
should be given at the same time, and some
advantage may be gained by making the lessons
correspond. The peculiar advantage of the Oral
method is that it enables a teacher to accomplish
more disciplinary work in the same time than the
Written method, and gives more exercise to the
powers of conception and memory. Being unaided
by written symbols it tends more to cultivate con-
tinuitv of thouo'ht.

We shall now endeavor to present a series of
Arithmetical exercises which will conform to the
principles already indicated.

1. Exercises in Counting. — A child will be found
to possess the idea of number at a very early age.
He undoubtedly obtains it through the medium of
objects. It is the teacher's duty to exj^and this idea
in the way nature indicates. If a child can count
ten when he enters school, the teacher must begin
his instruction at that point and teach him to count
twenty, fifty, and a hundred in the same way he
learned to count ten. Convenient objects riiay be
found for this purpose in beans, grains of corn,
pebbles, strokes on a blackboard, or balls on a
frame. The pupils should be taught to count back-
wards as well as forwards, and without objects as
well as with them.



316 INSTKUCTION IN" THE FORMAL SCIENCES.

2. Exercises in adding, subtracting, multiplying, and
dividing orally. — These exercises must first be taught
with objects; but the pupil must be gradually
accustomed to do without them. Small numbers
must be used until the pupil is prepared for larger
ones. The manner of conducting such exercises is
so obvious that no description of it here, is deemed
necessary. Besides, any teacher who may need aid
can obtain it from works on Oral Arithmetic.

3. Exercises in combining these Processes. — These
exercises are of the same nature as the preceding
and can be conducted in the same way. The teacher
will do well to introduce into the lessons the names
of the pupils in the class, the objects about the
school-room, trees, flowers, sheep, horses, cows,
dogs, &c.

4. Exercises in learning the written Symbols for
Numbers. — Pupils have now the idea of number.
They can readily count, and it is a task of no diffi-
culty to make them acquainted with the nine digits.
It is only necessary for them to make an arbitrary
association between the number and the character
which is used to represent it. The pupils may
count while the teacher forms the characters, or the
teacher may name the numbers, and the pupils
either" point them out or name them. The meaning
of the cypher must likewise be taught.

5. Exercises in Numeration and Notation. — For the
purpose of teaching ISTotation and I^umeration, I






ARITHMETIC. 317

would arrange columns of figures upon cards or
blackboards thus:

1 10 100 1000 10,000 100,000

2 20 200 2000 20,000 200,000

3 30 300 3000 30,000 300,000

4 40 400 4000 40,000 400,000

5 50 500 5000 50,000 500,000

6 60 600 6000 60,000 600,000

7 70 700 7000 70,000 700,000

8 80 800 8000 80,000 800,000

9 90 900 9000 90,000 900,000

This done, I would use the first two columns in
giving the first lesson. One may be called the units
column, and the other the tens column. We now
suppose that the class have leai-ned to read and
write the numbers in the column of units, and we
use it only to assist us in the task of teaching them
to read and write the numbers in the column of tens.
The teacher should call attention to the fact that
there are single figures to represent any number of
objects up to nine; but that ten cannot be repre-
sented by a single character. He may then arrange
objects in collections of ten, and have his pupils
count one ten, two tens, three tens, four tens, &c.
If now he tell them that one ten is designated by
the figure one with a cypher placed to the right of
it, as in the column of tens, they will be prepared
to understand that two tens are designated by the
figure two with a c^q^her placed to the right of it,
and so on to nine tens. The pupils should be
exercised in pointing out two tens or twenty, five
tens or fifty, seven tens or seventy, &c. ; and after-
wards, in writing them.

27*



318 INSTRUCTION IN THE FORMAL SCIENCES.

The second lesson should consist in teaching the
class to read and write numbers between ten and
tw^enty, twenty and thirty, kc, to ninety-nine. The
teacher may write the number 10 upon the black-
board and ask how many added to ten will make
eleven, twelve, thirteen, &c. lie may then ask how
these numbers are written, and if no one can tell,
he may erase the cypher and put 1 in its place, and
say the 1 on the left hand signifies one ten, and the
1 on the right hand one unit, and one ten and
one unit are eleven. If when 1 is put in place of
the cypher, the number becomes eleven, pupils will
readily understand that when 2 is put in its place
the number will become twelve ; 3, thirteen, and so
on to nineteen. The numbers between twenty and
thirty can be taught in the same way, and so on to
ninety-nine. Pupils must not only read the num-
bers but write them. Questions like the following
will also be very useful: What number is that
which it composed of two tens and seven units?
four tens and three units ? eight tens and &ve units ?
&c. ; how many tens and units in twenty-four? in
thirty-seven ? in seventy-six ? &c.

Pupils have now learned to read and write all
numbers up to ninety-nine. The next lesson should
make them acquainted with the third column, or
the column of hundreds. To do this, the teacher
-will take the ten collections of objects of ten each,
place them all together and ask the number. It is
one hundred. He points to the number, has the
pupils notice how it is written, and then they readily
read and write the other numbers up to nine hun-
dred. Any number may now be placed in the units



AKITHMETIC. 819

column by erasing the cypher and inserting the
nnmber, and so with the tens column, or both
columns at the same time.

It is unnecessary to describe further, as the same
method applies to the column of thousands, tens of
thousands, hundreds of thousands, &c.

6. Exercises m Addition, Subtraction, Mudtiplication,
and Division. — A pupil who can read and write
numbers is prepared to understand the operations
of Addition, Subtraction, Multiplication, and Divi-
sion ; and, therefore, he should not only be taught
how to perform these operations, but why they are
so performed.

For the pupil to understand the process of Addi-
tion, it will be necessary for him to know that those
numbers only which represent things of the same
denomination can be added together. This he can
l)e taught readily with objects. He will see at once
that live grains of corn and .three beans neither
make eight grains of corn nor eight beans, and,
hence, that units must be added to units, tens to
tens, &c. He must know how to convert lower
denominations into higher ones, that is units into
tens, tens into hundreds, &c. This, however, more
properly belongs to l^otation and I^umeration.
Finally, he must be made to see that to render such
reductions more convenient he must commence in
adding at the right-hand column of figures.

To perform the operation of Subtraction nothing
more is necessary than for that of Addition, except
the converting of higher denominations to lower
ones, and that is as easily done as its reverse.



320 INSTKUCTION IX THE FORMAL SCIENCES.

There is no principle in Multiplication tliat is not
found in Addition ; and Division is but a different
kind of Subtraction.

The iirst examples in Addition should consist of
such numbers that the sum of those under each
denomination can not exceed nine. The first ex-
amples in Subtraction should consist of such num-
bers that each number of a certain denomination in
the minuend should exceed the number of the same
denomination in the subtrahend. The first exam-
ples in Multiplication should consist of such num-
bers that none of the products of numbers in the
multiplicand by the multiplier can exceed nine.
The first examples in Division should consist of
such numbers that the divisor can be contained in
each number of the dividend without a remainder.
The first divisors used in what is called Long Divi-
sion should be less than ten. In all cases the pro-
gress of the pupils should be gradual ; but one point
of difiiculty should be presented at a time. Much
practice should be allowed them in order to secure
rapidity and accuracy in the performance of their
work. Solutions should be neatly written upon
blackboards and properly explained. Forms of ex-
planation may be obtained from text-books ; but
teachers should be careful to have their pupils un-
derstand them and not merely' commit them to
memory. Teachers will find the construction of
Addition, Subtraction, Multiplication, and Division
tables, by their younger pupils, a very valuable
auxiliary in familiarizing them with the processes
involved. The terms applied to the numbers used
in Subtraction are Minuend, Subtrahend, and Dif-



ARITHMETIC. 821

ference. Any two of these being given, a third
can be found. The same is true in Multiplication
with reference to the Multiplicand, Multiplier, and
Product; and in Division with reference to the
Dividend, Divisor, and Quotient. I mention these
facts here, in order to say that such problems pre-
sent w^ork of much value to learners.

7. Exercises in the Solution of practical Examples
involving the four fundamental Rules. — Pupils not
only need to know hoiv to perform simple Arith-
metical operations, but when they are required to be
performed. Por this purpose numerous practical
problems must be presented. All text-books con-
tain some such problems ; but none of them within
my knowledge contain one-fourth as many as are
needed. The teacher must supply this deficiency.
They are so well calculated to give interest to the
study and to make pupils think, that I am disposed
to consider them almost indispensable.

8. Exercises in imparting the Idea of a Fraction. —
The basis of all Arithmetical operations is the unit.
The unit may be multiplied or divided, and these
processes really constitute the whole of Pure Arith-
metic. All Integers may be called multiplied units,
and all Fractions, divided units. Particular whole
numbers denote the extent of the multiplication,
and particular fractions denote the nature of the
division.

The idea of a fraction is formed upon seeing things
broken up or divided.. Pupils have the idea when
they enter school, but the teacher must expand it by



322 INSTEUCTION IN THE FORMAL SCIENCES.

exhibiting and naming the parts of objects. For
this purpose, an apple may be cut into parts, a stick
may be broken into pieces, or a line, a square, or a
circle, drawn on a blackboard, may be divided into
sections. Such instruction should be continued
until the pupils can readily name the fraction upon
seeing the object, or find an object which is repre-
sented by the fraction ; or, in other words, until
they learn to count fractionally.

9. Exercises in adding, suhtr acting, multiplying, arid
dividing fractions orally. — At this stage of their pro-
gress, pupils may perform orally with much advan-
tage some of the simpler problems in Addition,
Subtraction, Multiplication, and Division of Frac-
tions. Such questions as the following may be
asked: In Addition: What is the sum of one-half
and one-half? one-third and. one-third? one-fourth
and two-fourths ? one-half and one-fourth? one-half
and one-third ? &c. ; in Subtraction : What ip the
difference between one and one-half? three-fourths
and one-fourth? one-third and one-sixth ? one-half
and one-third? &c. ; iji Multiplication: What is the
product of two times one-half? three times one-third ?
four times one-sixth? one-half times two? one-half
times one-half? &c. ; in Division: how many halves in
one? in two? in five? how many times is two contained
in one-half? in one-third? in two-fourths? how
many times is one-fourth contained in one-fourth ?
in one-half? in one-eighth? &c. All this can be
beautifully illustrated with squares drawn upon the
blackboard and divided into the requisite number
of parts. As soon as possible, however, pupils



AEITHMETIC. 823

should be taught to solve such problems without
depending upon objects.

10. Exercises in teaching Fractional Expressions. —
When pupils have attained a clear idea of a fraction,
it will not be difficult to teach them to express it.
The simplest fractions are those in which the
numerator is unity, and, therefore, pupils should
first be taught to write J, |, i, 4, J^, &c. ; and after-
wards fractions in which the numerator is greater
than unity ; as |, |, |, j^, &c. Pupils maybe required
to write fractions representing the given parts of
squares or circles drawn upon the blackboard, or
they may divide such figures so that certain given
fractions will represent them.

11. Exercises in the Addition^ Subtraction^ Multipli-
catio7i, and Division of Fractions, and their Ajyplications.
—Pupils are now prepared to enter upon the work
of adding, subtracting, multiplying, and dividing
fractional numbers, and of making an application
of them in the solution of practical problems. The
work may be done orally or by writing. The sim-
pler operations of fractions can be understood by
inspection ; but when pupils are prepared for it, the
rules for finding the Greatest Common Divisor, the
Least Common Multiple, and all other rules relating
to Fractions must be rigidly demonstrated.

12. Exercises in Decimal Fractions. — With a know-
ledge of the Decimal ^Notation and of Common
Fractions, it will be no difficult task for a pupil to
learn Decimal Fractions, for there is no new prin-



324 INSTRUCTION IN THE FOEMAL SCIENCES.

ciple involved. A Decimal Fraction is a fraction
whose denominator is always 10 or some product
of 10. Such fractions are written by placing a point,
called the Decimal Point, before the numerator.
This point indicates that the number of figures in
the numerator to the right of it is equal to the
number of cyphers in the denominator, and hence
does away with the necessity of writing the de-
nominator.

Instruction in Decimals must begin by making
pupils thoroughly acquainted with the Decimal
E"otation. They must be taught both to read and
to write Decimals with facility. The Decimal Nota-
tion may be taught in the same manner as the
notation of integers ; but this trouble need scarcely
be taken, as pupils can almost as easily read or
write tenths, hundredths, thousmidths, as tens, hun-
dreds, thousands.

All the rules in the Addition, Subtraction, Multi-



Online LibraryJames Pyle WickershamMethods of instruction .. → online text (page 20 of 31)