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plication, or Division of Decimals may be shown to
be true either by reducing the Decimals to Common
Fractions, or from the nature of the Notation itself.
Text-books exhibit both methods, and it is un-
necessary to detail them here.

13. Exercises in Compoujid Numbers. — In the
Compound or Denominate Numbers, the units in-
crease according to varying scales. These scales
are fixed by some authority, and follow no regular
law. Pupils must, therefore, commit them to
memory; but when the tables of Weights and
Measures are well understood, the Addition, Sub-
traction, Multiplication, and Division of Compound



ARITHMETIC. 325

lumbers present little difficulty to the learner that
he has not already encountered in performing the
same operations with abstract numbers.

14. Exercises in Proportioyi., and Involution and
Evolution. — These exercises belong to Pure Arith-
metic, but they are simply modifications of the four
fundamental rules. They present no special difii-
culty in teaching.

15. Exercises in Arithmetical Applications. — A
knowledge of Arithmetic is needed in almost
every kind of business in which men are engaged,
and, therefore, teachers should make its practical
applications a prominent part of their instruction.

In solving practical problems, pupils should be
required to understand the words in which the
problem is expressed, to point out the relation of
the thing required to the thing given, to present a
neat solution, and to explain their work in concise
and appropriate language.

A few additional suggestions will be made.
Problems that involve but a single principle should
be given first, and, afterwards, those which involve
several principles. Text-book or teacher may fur-
nish a form of solution, but the problems should be
so arranged that it cannot be followed mechanically.
Pupils may be required to compose problems involv-
ing certain given principles or answering certain
given conditions. Many miscellaneous problems
add much to the value of an Arithmetic. These
may be classified according to their relations. Im-



326 INSTKUCTION IN THE FORMAL SCIENCES.

portant facts may sometimes be incorporated into
Arithmetical problems.

The preceding series of exercises do not profess
to cover the whole ground of Arithmetic ; but it is
believed that most that is 'essential in teaching it,
has been presented.

Algebra.

Algebra is not a distinct and independent branch
of Mathematics. It is rather a method of repre-
senting quantities and of performing Mathematical
operations, by means of symbols. These symbols
may represent a portion of time, an extent of space,
an amount of matter, value, or force, and, also, the
relations of quantities and the operations which may
be performed on them. These symbols are used in
all the higher investigations of Mathematics, and
they have been productive of results as wonderful
as the}^ are important. They have enabled mathe-
maticians to abridge the processes of calculation, to
overcome difficulties previously considered insur-
mountable, and to express in beautiful language the
truths they elicited. All this should recommend
the study of Algebra to the student.

Owing to the symbolic character of the language
used, the truths arrived at by the process of Algebra
are more general than the truths arrived at by the
processes of Arithmetic and Geometry. Algebra is
sometimes called General Arithmetic; in a larger
sense, it might as appropriately be called General
Geometry. In Arithmetic, particular numbers are
given and particular numbers are required. When
we have demonstrated* a property of a figure in



ALGEBRA. 327

Geometry, we are only sure that it is true of the
class to which that fifi-ure belono:s. But in Alo-ebra,

O O 7

all kinds of quantity may be denoted by symbols,
and the truths arrived at by their means are true of
all quantities whatever when they are subjected to
the same operations. From this it appears evident
that common Arithmetic must be understood before
its operations can be performed Algebraically, and
Synthetical should precede Anal^'tical Geometry in
a course of study. Algebra should be commenced,
however, before Arithmetic and Geometry have
been completed.

In its ordinary signification, Algebra treats of the
relations and properties of numbers by means of
symbols, and it is in this sense that we design to
speak of methods of teaching it. Thus considered,
methods of teaching it must be quite similar to
those of teaching Arithmetic, and a brief discussion
of the subject is all that will be necessary. Any
one who succeeds in teaching Arithmetic will suc-
ceed in teaching Algebra.

In the sense in which Algebra is now considered,
its Fundamental Idea, and its primary Definitions
and Axioms must be substantially the same as those
of Arithmetic. Its Demonstrations difiter only in
being more general ; and its Applications, in being
more extensive. These, therefore, need no discus-
sion here.

The Definitions peculiar to Algebra must be
learned by the pupil, not perhaps, all at once, when
he commences the study, but as he needs them.

'No Algebraic operation can be performed without
the use of symbols, and a knowledge of such as are



328 INSTRUCTION IN THE FORMAL SCIENCES.

necessary in the solution of simple problems must
be imparted to learners in their first lessons. The
others may be learned when they have made some
progress in the study. All the symbols admit a
neat classification, and a knowledge of them can be
most readily acquired in that form.

In teaching beginners, it is best for the teacher to
illustrate the meaning of the symbols by using them
with respect to numbers. Thus : 4 + 2=6; 8 — 3=5;
4x3=12; 9 - 3=3; 7x4 - 2 — 4 + 2=12; ^/16=4;'
4^=16. He may desire to add 576 to 764; but
instead of performing the operation Arithmetically,
he may say " we w^ill let a represent the first number
and h the second, and the operation can be expressed
by a + 6." I^early all the symbols used in Algebra
can be illustrated in this way, and no one but a
practical teacher can appreciate the value of such
illustrations to the pupil just commencing the study.

The Algebraic symbols which are used to repre-
sent quantity are general in their significance, and
in this respect, difter from numbers. Pupils can
make little progress in the study of Algebra until
they understand this diff"erence. For this purpose
the teacher cannot do better than to make a series
of additions, subtractions, multiplications, and divi-
sions with numbers, and then show that a + 5 + , &c.,
a — h, axh or a h, and a-^b or j are general expres-
sions for all of them in the order named, and for all
others possible. Besides, it is easy to give^ illustra-
tions showing that a, h, c, &c., a:, y, z, &c., can be used
to represent numbers in whatever manner or to
whatever things they may be applied.

The Algebraic idea can perhaps be best commu-



ALGEBEA. 829

nicated by requiring pupils to solve suitable Arith-
metical problems Algebraically. Some of the prob-
lems in our works on Oral Arithmetic can be
selected for this purpose, or, as some authors of
text-books on Algebra have arranged it, they may
be so placed as to be an introduction to the general
subject. Pupils seem to see the practical value of
Algebra more clearly when commenced in this way,
and, consequently to take more interest in the study.

After such an introduction to the subject as is out-
lined in the preceding paragraphs, pupils can be
taught to add, subtract, multiply, and divide Alge-
braic quantities, whether integral or fractional; but
although some elements enter into these operations
that are not found in similar ones in Arithmetic,
they involve no new principle of teaching. The
pupil must be allowed much practice to enable
him to make a ready and intelligent use of the
symbols.

The simplest step in mathematical reasoning may
be expressed in the form of an Equation ; thus, one
added to one equals two may be written 1 + 1=2.
In idea the Equation is constantly before the mind
of the pupil when engaged in the study of Arith-
metic ; and, consequently, the teacher will not find
the task a difficult one to acquaint him with the
Algebraic form of expressing it. A Pair of Scales
can be made to furnish a very good illustration of the
simple form of an Equation. The common weights
can be placed in one scale, and any body or bodies
whose weight is unknown can be placed in the other ;
and, when balanced, the Equation is formed, and
can be represented by letting x, ?/, z, &c., represent

28*



830 INSTRUCTION IN THE FORMAl!- SCIENCES.

the known quantities, and a, b, c, &c., represent tiie
unknown. Having attained the idea of an Alge-
braic Equation, the pupil must next learn to reduce
it to its simplest form. For this purpose, he must
be taught to clear the Equation of fractions, and
to transpose, collect, and reduce its terms. The
method of performing these operations and the
truth of the axioms upon which they depend can
be illustrated by taking the simplest form of an
Equation ; as 4=4, and showing that equals may be
added to or subtracted from equals, multiplied or
divided by equals, and the results will be equal.
In Equations containing two or more unknown
quantities, the various methods of elimination must
be explained and illustrated. The diiferent methods
of solving Quadratic Equations and the forms to
which such Equations can be reduced must undergo
thorough discussion. The theories of all kinds of
Equations should be impressed upon the pupil's
mind by practice in solving numerous, well-graded,
and judiciously-selected examples and problems in-
volving them. These problems may be divided
into two parts : first, that which relates to the for-
mation of the Equation ; and, second, that which
relates to the solution of it. The formation of the
Equation consists in observing the facts given, in
noting their relations, in finding the . equality be-
tween the known and the unknown, and in express-
ing that equality in Algebraic language. Having
attained the elements of a problem, the formation
of an Equation expressing these elements is a syn-
thetic, while the solution of the problem is an
analytic, process. The teacher may require one



GEOMETRY. 331

pupil to form an Equation for a problem, another
to solve it, while still another is engaged in mak-
ing a problem to answer the conditions of a given
Equation.

Perceiving no necessity for pursuing the subject
further, it may be well to remark in conclusion,
that the ends for which Als^ebra is studied are
similar to those for which Arithmetic is studied,
that the general conditions which must be observed
in their attainment are the same, and that the sug-
gestions mentioned in reference to conducting reci-
tations in Arithmetic or arranging its object-matter
for study apply equally well to like questions in
teaching Algebra.

Geometry.

The Etymology of the word would lead us to
suppose that Geometry has reference to measuring
the earthy and no doubt it had this reference in
early times ; for the necessities of the race would
compel them to adopt some means of measurement
long before abstract truths like those now composing
the science of Geometry could be appreciated, much
less reduced to a system.

Geometry as now understood, may be defined as
the science of form. Its Fundamental Idea is space.
There are two kinds of form, pure and real. Pure
form is a portion of space limited in idea but not in
fact. Real form is a portion of space limited in fact.
Geometry proper treats only of pure forms, but it
may be applied to real forms.

Geometry furnishes the most perfect model of a
deductive science. It may be considered a type of



832 INSTRUCTION IN THE FORMAL SCIENCES.

all the rest, l^o Mathematician douhts that its
basis rests upon the Idea of space. Its Definitions
and Axioms are better understood than those of
any other of the same class of sciences. The De-
monstrations which form the body of it, comprise a
beautifuil system of applied logic, each admitting
an easy reduction to the syllogistic form. And its
Applications are among the most important in the
practical affairs of life.

The two most common divisions of Geometry are
Elementary Geometry, and Higher or Transcendental
Geometry. Elementary Geometry treats of the line
and the circle. Higher Geometry embraces the
consideration of all curves except the circle. A
brief discussion of methods of teaching Elementary
Geometry is all that is contemplated in this con-
nection.

Elementary Geometry as we find it in books like
those of Euclid and Legendre, is not a study for
children. Its abstract conceptions and long pro-
cesses of reasoning require for their full compre-
hension, minds of some maturity and some discip-
line. The idea of form, however, must be one of
the earliest which springs up in the mind of a child ;
and it would seem to follow that he can receive
instruction in Geometry at as early an age as in
Arithmetic. It may be shown that this theoretical
conclusion can be verified in practice.

Young children can learn to distinguish a great
many Geometrical forms ; as a line, a square, a circle,
a triangle, a rectangle, a cone, a pyramid, a cylinder,
a prism, kc, &c. For this purpose, they can be
taught to draw them on their slates or on the black-



. GEOMETEY. 833

board, and tliey can be shown blocks which represent
them as wholes, .or are cut into sections of which
they can be engaged in making them.

Young children can also be taught the meaning
of many Geometrical terms. It is not meant that
abstract definitions should be given ; but certain
Geometrical terms can be so illustrated as to render
them comprehensible to children. The following
are examples : a plane, an atigle and its different
Jcirids, the different kinds of triangles, sl perpendicular,
a diagonal, parallel lines, the parts of a circle, chords,
polygons, the hinds of prisms, kc, &c.

IVIany Geometrical truths can be made known to
children as matters of fact. They can perceive these
truths without being able to demonstrate them, that
is, they can perceive the particular truth, but cannot
make it general. It is not a difficult thing, with
blocks suitably made, or pieces of pasteboard suitably
prepared, to shotv children that " If one straight line
meet another straight line, the sum of the adjacent
angles will be equal to two right- an gles ;" "When
if two straight lines intersect each other, the opposite
or vertical angles, which the}^ form, are equal ;" "In
every triangle the sum of the three angles is equal
to two right angles;" " Every triangle is half the
parallelogram which has the same base and the
same altitude ;" " The square described on the
hypothenuse of a right-angled triangle is equivalent
to the sum of the squares described on the other
two sides;" &c., &c. A well-graded course of in-
Btruction of this kind, if judiciously given, would
furnish very valuable discipline to children of the
age of ten or twelve years, and greatly diminish for



334 iNSTEucTiojsr in the formal sciences.

them tlie labor of Geometrical demonstration when
their minds become sufficiently mature to enter upon
it. Besides, it seems to be the natural method.
Solid objects first meet the eye, not points, and
lines, and angles ; and here, as elsewhere, the method
of proceeding should be from the concrete to the
abstract — from the particular to the general.

When pupils are prepared to understand Geomet-
rical Demonstrations, they should be supplied with
a suitable text-book. The iirst pages of such a book
will present to them certain Axioms and Definitions
relating to Geometry which must be carefully studied.
If the author of the book has done his duty, its sub-
ject-matter will be arranged in a rigidly logical
order, starting with the simplest and most inde-
pendent propositions, and containing no missing,
imperfect, or superfluous link in the chain.

Geometrical propositions admit of two kinds of
demonstration ; the first, with axioms, definitions,
or previously proven propositions as premises, seeks
to show that the proposition to be demonstrated is
included in these premises, and is therefore true ;
while the second consists in forming hypotheses
which contradict the proposition, and in reasoning
upon these hypotheses until conclusions are reached
which contradict truths before known, and thus
prove the proposition by demonstrating that the
hypotheses which contradict it are false. The
former of these methods of demonstration is called
direct, and the latter, indirect, or reduetio ad ahsurdum.
Both are equally philosophical ; but where a choice
is optional between them, the first as the more
simple is generally preferred to the second. Some



GEOMETEY. 835

propositions admit both kinds of demonstration,
and many can be demonstrated by different methods
of the same kind. With snch propositions, when a
pupil has followed the text-book in one method of
demonstration, he might be greatly benefited by an
effort to find others. It would be an admirable
feature in a text-book to present here and there
undemonstrated propositions, because pupils ought
not only to be trained to follow the reasoning of
others ; but to invent processes of reasoning for
themselves. The connection between certain propo-
sitions is so obvious, that ^ pupil, after having
demonstrated one, ought to be able to infer the next
w^ithout being helped to it by the book or teacher.
Original thinking is always much more valuable
than that w^hicli is second-hand. If the teacher
desire fully to impress upon the minds of his pupils
the truths they demonstrate, he should teach them
to make an application of them at once in the solu-
tion of well-selected problems. Mensuration might
be very profitably taught in connection with Geom-
etry. It might be well also to require the pupil
sometimes to give Algebraic demonstrations of
Geometrical propositions, and to solve Algebraic
problems by Geometrical methods.

In conducting a recitation in Geometry, the prop-
osition should be stated, and the diagram drawn,
from memory; and the demonstration should be
given clearly and precisely, in the pupil's own lan-
guage. In placing letters or numbers to the dia-
gram, it is best to use them in a different order
from the text-book, or the practice of demonstrating
without a diagram may be productive of benefit,



336 INSTRUCTION IN THE FORMAL SCIENCES.

especially in reviews. In addition to this, the pupil
should be taught to give a complete analysis of each
demonstration. He should be able to tell —

1st. The kiyid of quantity under consideration.

2d. The relation of the demonstrated proposition
to those which have preceded it.

8d. The kind of demonstration used.

4th. The axioms, definitions, or previously de-
monstrated truths used as premises.

5th. The relation of the conclusion to the premises.

6th. The relation of Corrolaries, Scholiums, and
Lemmas, to the principal proposition.

m. Logic.

The aim of this book does not require that Logic
should undergo a lengthy discussion. Much has to
be omitted, and the vast majority of teachers will
miss a discussion on Logic less than one on most
other branches taught in our schools. Still some-
thing must be said, and it is proposed to say it
under two heads : 1. The utility of Logic as a study ;
2. Tlic methods of teaching Logic,

1. The Utility of Logic as a Study. — Some extra-
vagant claims have been made by Logicians in res-
pect to the utility of their favorite study. It has
been called the Art of Arts, the Science of Sciences,
Catharticon Lntellectus, Caput et Apex Philosophice, kc. ;
and these names indicate the estimation in which
it was held by the authors who used them. But
while these claims should be moderated, it will ap-
pear from what is to be said that the utility of Logic



LOGIC. 837

is such as to demand for it a prominent place in
every liberal course of study.

Logic is a useful study in itself. Thought, as
thought, presents a noble object for investigation.
It is man who thinks and thinking is his highest
attribute. A thought is greater than a thing.
Things pass away, thoughts are immortal. If
science as science is worthy of study anywhere, it
is surely worthy of it when it treats of the laAvs of
thought. "And is it nothing," says a writer, "to
watch the secret workshop in which nature fabri-
cates cognitions and thpughts, and to penetrate into
the sanctuary of self-consciousness, to the end that,
having learnt to know ourselves, we may be qua-
lified rightly to understand all else?"

Logic is a useful study on account of its objective
relations. Men can do nothing well unless they
think well. All science and all art .are the fruit of
right thinking. Wrong thinking is at the root of
all error. In this sense. Logic would almost be
entitled to be called Ars Artium or Seientia Scienti-
arum. It is only in theory, however, that Logic
holds this place, for the best Logicians are far from
linding all truth or escaping all error. All that
can be claimed is that as reasoning takes place in
every thing we do, the study of the laws of thought
must aid us in reasoning correctly. Besides, nature
in all its departments fills with matter certain logical
forms, and cannot be well understood in itself or
well arranged into systems of science without a
knowledge of these forms. Logic is an indispen-
sable instrument in scientific investigation.

Logic is a useful study because it disciplines the
29



338 INSTRUCTION IN THE FORMAL SCIENCES.

Understanding. Tlie Understanding is the faculty
by which we reason. The end of Logic is to reason
welL Hence it follows that the study of Logic dis-
ciplines the Understanding. It not only imparts
skill but power, for reasoning about reasoning must
be at least as capable of strengthening and develop-
ing the Understanding as reasoning about some-
thing else.

2. MetJiods of teaching Logic. — If the nature of
Logic is as we have stated it to be, its subject-matter
will be composed of Definitions and Axioms, Deduc-
tions and Demo7istrations, and Aj^plications.

Every one who has the least idea of Logic is
aware the first step in teaching the science must
consist in making pupils acquainted with the defi-
nitions of concept, judgment, reasoning ; term, prop-
osition, syllogism, induction, deduction, &c. Indeed,
Logic consists in much greater part than Mathe-
matics in definitions and explications of the pro-
ducts of the intuitions of the Reason. The axioms
of Logic, too, admit as clear a statement as those
of Mathematics and bear the same relation to the
science. Hamilton speaks of Fundamental Laws
of Thought, and slates them as follows : 1. The Law
of Identity ; 2. TJie Law of Co7itradiction ; 3. The
Law of Excluded Middle ; 4. The Law of Reason and
Consequent. Other Logicians give substantially the
same laws. But all of these laws admit of state-
ment in the form of axioms, and many Logicians
have so stated them.

The Body of Pure Logic is arranged by Hamilton,
and substantially so by many others, into two great



LOGIC. 339

classes wliicli may be expressed as follows : 1. The
Means of Thinking ; 2. The Methods of Thhiking.
The Means of Thinking include Concepts, Judg-
ments, and Reasonings. Concepts are the products
of conception. Judgments are the arrangement
of concepts as subjects and predicates. Eeasonings
are processes by which one judgment is deduced
from another, by means of a third which is inter-
mediate. Reasonings, when fully stated, assume
the form of syllogisms, of which concepts and judg-
ments are the elements. The Methods of Thinking
include the doctrine of Definition, the doctrine of
Division, and the doctrine of Proof. Logical defi-
nition is the complete development of a concept.
Logical division is the separation of a whole into its
parts according to their relations. Proof consists
in deducing on& judgment from another known to
be true.

This whole Body of Pure Logic is made up in the
main of definitions and judgments which are known
to be true only by intuition. A pupil who does not
realize in his own mind the thing spoken of will not
be profited in the least bj^the words of the Logician.
A teacher of Logic must be constant in his efforts
to induce his pupils to investigate the products of
thought as they lie in their own minds. The study
of Psychology should precede that of Formal Logic,
both because the habit of introspection into one's



Online LibraryJames Pyle WickershamMethods of instruction .. → online text (page 21 of 31)