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32X

HINTS

TEACHING ARITHMETIC

H. s. Maclean,

Assistant Principal Manitoba Normal School ; Author of
The High School Book-keeping.

TORONTO :
THE COPP, CLARK COMPANY, LIMITED.

Entered according; to Act of the Parliament of Canada, in the year one thousand
eight hundred and ninety-flve, by Thb Copp, Clark Ooupant, Limitkd, Toronto,
Ontario, in the Offloe of the Minister of Aj^riculture.

~%oi^\

PREFACE,

As indicated by the title of this maniml, no attempt

lias been made to enter into an exhaustive discussion of

tlie psychological principles underlying the teaching of

Arithmetic. The aim has been, rather, to present in brief

form, and in as practical a manner as possible, what

liave appeared to the author to be the most important

features of the subject, viewed from the teacher's stand-
point.

The early pages are devoted chiefly to the considera-
tion of such fundamental questions as seem to have the
most direct bearing on practical school work. An eflbrt
is made, whether successful or not, to outline what may
be done in the school-room during consecutive periods.
I^astly, suggestions are offered on points where mistakes
are likely to occur in teaching.

This book, containing the substance of talks given at
institutes, and in the Manitoba Normal School, is humbly
submitted for the consideration of teachers generally,
with the hope that it may prove to be of some little
value to, at least, the younger members of the profession.

CONTENTS.

9.
10.
11.
12.
13.
14.
15.

16.
17.

PAOK.

Importance as a Subject of Study 1

Relation to Other Subjects 2

Number a Purely Mental Conception 'S

Development of the Idea of Number 4

What the Idea of Number Includes 5

Units of Mkasurement 6

Numerical Relations 9

Numbering by Ones 15

Numbering by Groups 17

The Use of Objects 18

The Fundamental Rules 21

Arithmetical Processes 23

Problems 26

Educational Value of Probiems 34

Expression , 38

Course of Study 41

Outline of Work for First Period 41

Outline of Work for Sec(md Period 40

Outline of Work for Third Period 56

Plan of Teaching Nos. 1-20 62

Seat Work for Junior Classes 69

[V]

^* CONTENTS.

18. T*RACTICAL SU(;OKSTI()NS ^^^J'.

Reading Numbers q.

Writing Numbers g2

Addition „„

Subtraction or.

Multiplication o .

Division o^

Fractions oq

Decimals q-

Percentage ^^^^

Bank Discount w.q

Compound Interest jq^

' . S<|uare Root ^^-

19. Review Exercises jqh

PAOK.

... 81

. .. 81

... 82

. .. 82

. .. 83

. .. 84

. .. 86

. . . 89

. .. 97

. .. 100

. .. 102

. .. 103

. .. 105

... 107

■J

•.

HINTS ON TEACHING ARITHMETIC.

1. Importance as a Subject of Study.

Arithmetic is one of the few suV)jects whoso rifjht to a place
on the Pu})lic School programme of studies has never l)eeii
seriously questioned. Its adaptability to the different stages
of mental development, it'* suitability as a means of training
in concentration, accuracy, rapidity and logical ariangement
of thought, its tendency to secure clearness, exactness and
conciseness of expression, and its usefulness in the every day
business of life are all so evident as to give it an undisputed
right to a high position among school subjects.

The material world presents objects of endless variety which
invite the attention of man. The human mind reacting on
these ascribes to them qualities of form, size, etc., and classifies
them according to resemblances and differences of such quali-
ties. But these objects manifest diversity of form and size,
and they are acted upon by forces. Hence the necessity
for accurate quantitative measurement arises. A standard of
measurement is determined and the relation between it and
the whole quantity measured is established. This quantity
relation, which is the product of the action of the mind, is
termed number. The study of number is, therefore, necessary
in order that man may be brought into harmony with his
environment. Besides this, a knowledge of number is essential
in all the details of actual life, for there could be no progress
in science, manufactures, or commerce, without exact quantita
tive measurement.

[1]

I I

HINTS ON TKArillNC} AinTMMKTK!.

T]ui conditions of (;xiston(!o sinco tliti vory oarliosl, .'ulvancos
wovo, )n;ui(i in civilization liavo ron(l(M'<'(l it iiniK'mtivc that
inat<M'ial oljjccts, as well as tho forces acting uj)()n them, ,;houi(l
1)0 sul)ject to limitation. Tlio necessity for <!stimatiri<j^, wei«;h-
inn and ni(!asurin«^ sucli ol)j(?cts and forces may 1x5 ret^arded as
th(5 prime motive impelling tlio individual to exercise mental
power in (juantitative determination. An indcjfinitj! whole
«|uantity is presented ; the mind feels the necessity for defining
it. This fact has an important bearing on teaching number,
for it points dircictly to a true basis of interest. The child
passes through stages of growth corresponding, in tho main,
to those through which the race has pass(;d.

2. Relation to Other Subjects.

Arithmetic bears a highly important relation to other subjects
of study. liy means of number we can giv(^ definite si/.o to a
continent, v/ecan compare the heights of its mountain syst(nns,
we can measunnts rivers and river basins, and we can estimate
with a (l(!grec of accuracy the educational, social and commer-
cial progress of its inhabitants. But it is unnecessary to
multiply illustrations ; astronomy, meteorology, chemistry,
botany, physics, etc., will at once suggest many. Indeed,
the natural sciences would forever have remained in a state of
infancy l)ut for the aid of mathematical calculation. Why is
this the case ? Simply for the reason that without number
well-defined notions of size, distance, motion, etc., could never
be attained. As arithmetic is so closely related to other
subjects, would it not be well in the school-room not to
divorce it entirely from them but rather to make it contribute
its due share towards their advancement ? A large proportion
of the exercises in arithmetic should be closely connected
with the daily experiences of the pupil, whether in the school-
room, in the city, or on the farm.

In endeavouring to correlate arithmetic with other subjects.

NUMHKK A I'UUELY MKNTAL CONCEPTION.

, .'ulvancos
ativc tliiit
(Mil, should
ri<,', wcijLfli-
('i,'ai'(l('(l .'IS
is« niontal
iit(; wli()l<5
Dr defining
g number,
The child
the main,

ler subjects

te size to a

in syst(Mns,

m estimate

1 coninier-

cessary to

chemistry.

Indeed,

a state of

Why is

Lit number

)uld never

1 to other

3m not to

contribute

proportion

connected

lie school-

r subjects,

we must guard a<jfainst tlu^ opposite error of making it wholly
subservient to them. Tim arithmetic lesson should b«» a lesson
on arithmetic, and not on g(M)grapliy, botany or j)hysiol()gy.
Arithmetic must be taught in accordance with the *' inh«*rcnt
immoi'tal rationality" of the subject. An attempt to teach
sev(;ral subjects in oiu^ lesson gcnei'ally results in teaching
nothing. The wf^ll-lxnng of tlu; pupil demands "concentra-
tion " of effoi't on the subject in hand, what(;ver that may he.
There n(;ed b<; no contlict of opinion, liow(!ver, betw(>en tlu^
teacher who advocates the con-elation of studies and the teacher
who insists on teaching one thing at a time. J5oth are
riglit ; they are simply viewing the matter from dillerent
standpoints.

3. Number a Purely Mental Conception.

Num))er is not an inh(M'ent (juality of the material ol)jects
or groui)S of obj(>cts, forc(!s, etc., that an; mc'asured numeri
cally. but it is the product of the mind's action in detei'niining
their e.xact limitations as to quantity. This becomes obvious
when we consider that the measure of any definite (piantity
will depend altogether on the unit of measurement which tlu;
mind chooses.

Objects and groups of objects are perceived by the senses,
but not quantity relations. The presence of o))jects is a
necessary condition — but only a condition — ^for occasioning
the mental activity that results in establishing a quantity
relation between a unit of measurement and a whole or aggre-
gate. The sense concepts corresponding to objects presented,
furnish the material upon \vhich the mind operates in deter-
mining numerical relations.

Number is always abstract. There can be no such thing as
concrete number. The unit may Im; particular or limited in its
application, as in 10 pears, 10 yards, etc., :.)r it may be general

HINTS ON TEACHING ARITHMETIC.

I : i

or unlimited, <as in 10. But no matter what the cViaracter of
llie unit may be, an abstraction must be made, otherwise, there
can be no proj)or conception of number. In teaching, this
important fact must not be forgotten.

Development of the Idea of Number.

It has been ah*eady stated that number is a purely mental
conception, and that objects in space merely furnish the
occasion of the mental action which results in establishing
definite quantity relations. To make this clearer, and also to
pave the way for the c<jnsideration of what follows, it nuiy be
well to endeavour to trace approximately the development of
the idea of number. The steps may be roughly outlined as
follows :

(i) Observing objects in space.

(ii) Separating in thought the idea of quantity from other
o])Herved (qualities, as, form, colour, etc., and fixing the atten-
tion on quantity to the exclusion of other characteristics.

(iii) Perceiving differences of quantity. It is on this that
numerical measurement is conditioned. At an early age such
adjectives as much, little, large, small, long, short, many, few,
etc., are used intelligently by children showing that compari-
sons of quantities are made by them.

(iv) Observing that a quaiitity may be separated into like
parts (units), and that those parts when combined make up
the whole.

(v) Giving to each act of attention in combining or separat-
ing the parts its place in the whole series of acts performed.
This is what gives rise to the idea of ordinal number, as first,
second, third, etc.

(vi) Synthesizing the different acts so as to form a complete
whole. By this means the idea of cardinal number, as one,
two, three, etc., is gained.

WHAT THE IDEA OP NUMBER INCLUDES.

iracter of
ase, there
ling, this

ly mental
rnish the
bablishing
nd also to
it may be
>pinent of
utlined as

rom other
bhe atten-
stics.

this that

age such

any, few,

compari-

into like
make up

separat-

t'formed.

as first,

complete
•, as one,

If we accept the foregoing as correct, we are in a position
to mark out the lines along which progress in number nmst be
made. Beginning with an undetermined quantity which is
seen to be capable of being measured, the child separates it —
roughly at first but with sufficient accuracy to satisfy him at
this early stage — into parts, and by counting these he defines
the whole. To his original percept of the whole as a quantity
extended in space he has added the new element of time. The
undetermined whole hns become to him a numbered quantity.
But as he gains strength intellectually greater exactness is
demanded. The indefinite units by which he first measured
the whole must become more and more definite, otherwise
numbering can no longer afford a mental stimulus for him.
Well defined, measured units now become a necessity, for
without them the higher thought processes of number would
be impossible. It is thus seen that the direction of progress
in number is determined by the conditions of mental growth.

There are many other intermediate steps which are more or
less difficult to determine. According to Perez,* M. Houzeau
says : " Children at first can only distinguish l)etween a single
object and plurality. At eigliteen months they can tell the
difference between one, two, and several. At about three, or
a little sooner, they have learnt to understand one, two, and
four — two times two. It is scarcely ever till ca later age that
they can count the regular series, one, two, three, four; and
here they stop for a long time."

4. What the Idea of Number Includes.

In every complete idea of number there are three elements.

(i) A unit of measurement,
(ii) An aggregate or whole quantity measured,
(iii) A quantity relation (number).
Any two of these determine the other. If either the unit

'Perez, The Fimt Three Yearn of Childhood— yt. 186.

;i'::

i !l:^:

6 HINTS ON TKACHINO ARITHMETIC.

or tho quantity relation is unknown or not fully comprehended,
the aggre<(ate cannot be definitely fixed. For example, 20
square feet will not convey its full meaning to a pupil who has
not a clear conception of the unit square foot, even if he knows
the measure (20) of the aggregate 20 square feet. The same
may be said in the case where the unit is well understood, but
the measure of the aggregate (number) is not. From these
considerations we can easily see that in teaching arithmetic,
units themselves, as well as the numbering and relating of
units, demand the careful attention of the teacher.

I''

5. Units of Measurement.

By the term unit, in numbering, we mean one of the like
things of a unity on which the attention is fixed as the basis
of measurement. For example, when we speak of 12 pears the
unit which we have in mind is one pear ; in 12 square feet it
is one square foot, and in $^f it is one twenty-fifth of one
dollar.

Any aggregation of units considered as one whole is termed
a unity. It is evident that if any unity be regarded as a
standard of measurement for like things, it then })ecomes the
unit. Three feet may be taken as the unit of 12 feet, in N/hich
case the measure is 4 instead of 12. Again, if we take | of
a foot as the unit, the measure of 12 feet becomes 48. The
measure, therefore, of any quantity depends altogether 'on
the unit which the mind fixes upon as the basis of measure-
ment. Before an aggr-egate can be measured the unit nmst
be clearly seen. Much of the work in arithmetic consists in
transforming quantities so that they can be measured by par-
ticidar units and rice versa.

Sometimes it is convenient to take one unit and sometimes
another. This will depend on the whole quantity measured
or on the purpose for which the measurement is made. For

UNITS OP MEASUREMENT.

example, 1,000 is more easily thouglit of as 10 liundreds
tlian as 1,000 ones. We can count 120 people in a room
more readily by threes than by ones, and we can measure
off G60 feet much more quickly by using a 16| foot pole, or
G6 foot chain as the unit than we can by means of a foot rule.
Again, it may be necessary to consider a part of a whole thing
which we have in mind as the unit. We may wish to pur-
chase I of an acre of ground. Here the unit is | of an acre.
It is thus seen that the unit may be increased or diminished
to any extent, at will, in order to meet the requirements of
any particular case. The pupil should be given practice in
scilecting different units by which aggregates may be measured.
He should also be led to see the necessity for changing from
one unit to another.

Units may be divided into two kinds, concrete and abstract.
A concrete unit is one which may be used to measure only a
particular kind of quantity, as in 12 horses, 25 yards, etc. An
abstract unit is one that is thought of as applicable to the
measurement of any quantity whatever, as in 12.

Concrete units may be undefined as to quantity, as in C
books, 16 houses, etc., or they may be well defined standards
of measurement, as in 12 yards, 100 ohms, etc.

A fractional unit is one which is dependent for its value on
its relation to some other unit. When we speak of | of one
yard we have in mind the unit ^ of one yard, which is
clearly dependent for its meaning on the unit one yard, to
which it is related. In fact the fractional number 1 does
notliing more or less than express the ratio between the
fractional unit and the prime unit from which it is derived.

What is the purpose of considering units as a factor in
determining a course in number ]

To discover their degree of detiniteness in themselves in

HINTS ON TEACHING ARITHMETIC.

ii :'

order that we may proceed from the vague to the definite in
accordance with the natural progress of thought. The order
then is, (a) Concrete units of single objects which do not repre-
sent any fixed quantity, as block, pin, etc. An aggregate
determined by such units gives rise to counting, and nothing
more definite. (6) Units composed of numbered groups of
such objects. Here the unit, though still somewhat indefinite,
is limited to a degree as it is a numbered unity, (c) Fixed
standards of measurement. These are well defined, and their
use in measuring aggregates leads to the highest conception of
measurement, viz., ratio, by directing the mind not only to
the numerical value of the aggregate, but also to its quantita-
tive value, as compared with that of the fixed unit, (d) Frac-
tional units. Such units are completely defined in so far as
their relation to the unit from which they are derived is con-
cerned. The absolute definiteness of a fractional unit will
depend entirely on the degree of definiteness of the related
unit, as ^ of a line, ^ of 6 feet, etc.

As the simplest fraction expresses a relation based on a
comparison of quantities, it is clear that fractions should not
be introduced until the pupil has made considerable progress
in relating quantities. The fact that many good teachers
advocate the introduction of fractions at the very commence-
ment of a course in number is manifestly the result of a
difference of opinion as to the meaning of the term fraction.

In teaching a pupil to number, none but familiar objects,
unattractive in themselves, should be used. To present objects
as toys, flowers, etc., which possess striking characteristics
may prove interesting to the pupil but the interest will be
centered on the qualities of the objects, and consequently it
will be foreign to the purpose of a lesson on number. As
has already been shown, the pupil must lose sight of the
distinctive qualities of the objects making up a group before

he definite in
t. Tlie order

do not repre-
A.n aggregate
;, and nothing
ed groups of
bat indefinite,
y. (c) Fixed
led, and their

conception of
1 not only to
y its quantita-
lit. (d) Frac-
1 in so far as
lerived is con-
nal unit will
)f the related

n based on a
is should not
rable progress
ood teachers
ry commence-
rei^ult of a
m fraction.

liliar objects,
resent objects
laracteristics
erest will be
tisequently it
number. As
sight of the
group before

■■

KUMEKICAL RELATIONS. 9

they can be regarded as units by which the group may be
numbered. The pupil should be conscious of space limitation
sufficient to distinguish one object from another ami nothing
more, in order that the best condition for numlxM-ing may
be fulfilled. The less definite the unit is, so long as it is
seen to be a unit, the more will the attention be fixed on
numbering as a means of defining the aggregate. Tliat is the
reason why it is easier for the pupil to count ol)jects by ones
than by twos, threes, etc. As soon as the unit becomes
definitely measured the difficulty of numbering is increased,
because greater exactness of thought is demanded. This
being the case, it is easily seen that units which themselves
represent fixed measurements should not be presented until tlie
pupil has gained some power in numbering. After he is once
able to deal with such units, the measurement of aggregates
by them will greatly increase his power, and his knowledge
of number.

In introducing such measured units as inch, foot, yard, pint,
quart, gallon, ounce, pound, etc., the experience of the pupil is
an important factor in determining the order in which these
should be taught. By a judicious selection of such units,
much may be done towards connecting the school life of the
pupil with his home life. Facility of concrete presentation
and simplicity of relation to other units will also be considered
in this connection.

6. Numerical Eelations.

Number is the product of the action of the mind in defining
a quantitative whole. The first step is one of analysis, wheieby
the whole is thought of as being made up of like parts. This
gives rise to the idea of unit. It is a, process of simplification
on which numbering is conditioned. If the mind could go no
farther than this there could be no numbering or numei'ical

HINTS ON TEACHING ARITHMETIO.

liiiii!

I!
Ill 1 1

II i!

measurement. The like pjirts must be related to one another
and to th(^ whole, in some degree at l(?ast, befoi'e any progress
can he made in determining the whole quantity. The first
efforts may, indeed, produce rather indefinite results, but they
are the beginnings of greater things. The mind proceeds from
the vague to the definite. In numbering, this has two appli-
cations, first, as to the units themselves, second, as to the
manner in which the units are related.

It has already lieen hinted that there is a progressive order,
from the vague to the definite, in attitude of the mind with
reference to units as advancement is made in number. The
units first employed by the child are regarded as alike, but
in themselves they are quantitatively undefined by him. The
relating of such units cannot rise higher than mere numbering.
An aggregate composed of such can be defined in so far as
the nnmbe^' of the units is concerned, but it cannot be defined
as a whole quantity. For example, when we speak of 10
apples we think only of the number, the how many, we know
nothing definite about the how much. If we wish to de-
termine the whole quantity we must do so in terms not of
one apple, but of some fixed standard of measurement, as one
pound; We cannot say 3 apples! 15 apples; ; 1 ; 5, but we
can say 3 lbs. of apples ; 1 5 lbs. of apples ; ; 1 ; 5. If we do
make the former statement we have first to regard the units
as quantitatively equal to one another, otherwise the state-
ment is not necessarily true. Undefined units may be
regarded as taken together and separated ; they may also be
enumerated, but a direct comparison of quantities based on
such is impossible. Even in the case of measured standard
units it is not until the exactness of a mathematical concep-
tion of them is gained, that it is possible for the mind to
relate in the highest degree. This indicates not only the
onler in which units should be presented, but also the order
in which they should be related.

NUMERICAL RELATIONS.

Fioin these considoniti()iis wo see tliat tliere are two dis-
tinct stages in relating —

(i) Defining aggregates by numbering. — Number.

(ii) Defining aggregates ])y comparison. — Ratio.

As the evolution of number is nothing more or less than
tlio evolution of thought in a certain direction, numerical
i('lati<ms must necessarily present different degrees of com-
plexity corresponding to different stages of n)cntal develop-
ment. This can easily Ije shown to be the case. It is plain
tliat units of any kind can be numbered. The best condition
for numbering them, however, is when they are regarded as
not j)ossessing any quantitative value in themselves. This
condition being fulfilled, the mind is impelled in the direction
of numbering, as there is no other means of determining the
aggregate. Further, comparison or ratio is conditioned on
numbering. It would be impossible to say that one yard is a
measure 36 times as great as one inch \inless the one-inch
units composing the aggregate represented by one yard could
be numbered ; but it would be equally impossible to make the
statement without, in addition, taking cognizance of the
measure of the unit itself. This makes the distinction
betwecMi number and ratio perfectly clear, while it shows at
the same time that number is an element of ratio.

There is no psychological necessity for selecting such arbi-
trary standard units as yaixl, second, gallon, etc., as any other
units of definite measurement would do equally w(!ll for pur-
poses of comparison. The point to be noticed is that the
aggregate must be thought of as composed of units which are
thnusf'/ves either absolutely or relatively dejiried as to (piantity^
in order that such aggregate may be determined, not by num-
])ering the units merely, but by relating them quantitatively
to one another and to the whole. This indicates the educa-
biunal value oi practical measurement in a course of study.

HINTS ON TEACTHNO ARITHMETIC.

I !

Tlic preceding statements may ])e illustrated Ihus :

(d) Siip{)()se wo take three different (piantities of 5 peaij
(;acli, tlie unit being undefined (puintitatively. We may adl
tlH!m togethei', 5 p(;ars + 5 pears + 5 peai'S =15 pears, or wl
may go l)ack and count the number of addends and say 5 peaij
X 3 = 15 pears. The latter expression cannot mean anythiiij
more tlian that if we combine 3 groups of pears, having 5 il
each, we get a total of 1 5 pears. This is an example of nuiuf
bering ; tlie idea of ratio cannot 1)0 present as the groups o|
5 pears may bo all different from one another, the units beina
unrelated (juantitativoly. This represents the first and lowes|
form of determining an aggregate.

(b) Suppose we regard the groups to be equal (juantitativelyj
but the individual units in each group to be unrelated. Theij
5 pears + 5 pears -f- 5 pears ^ 15 pears has implied in it tlifl
idea of ratio, altlu)ugh that idea is not fully expressed until
we put it into the form 5 pears X 3 = 15 pears. 3 here indil
catcs the ratio between any one of the ecjual groups and tliel
whole quantity. The thought in mind may be expressed;
either as 5 pears X 3 = 15 pears, or 5 pears =-: ?, of 1 5 pears|
But as the individual pears are unrelated quantitative^ly, w(l
cannot say 1 pear is fV of the whole quantity of 15 pears,!
neither can we say 1 pear is I of each group of 5.

(c) Suppose we regard the pears to be all equal to one
another. Then the groups of five pears nmst be equal. Herci
the whole quantity or any part may ])c delijunl in terms of tlul
unit 1 pear. In this the idea of ratio may be as comj)leteh|
in mind as if a recognized standard of measurement as poun-lJ
peck, etc., were used. We can say 10 pears == I pear x Kf

= 2 pears x 5 = -| of 15 pears, etc.

(d) Suppose that there is only 1 pear. We can think of th
repetition of that one as expressed )jy 1 pear x 15 = 15 pearsj

I'll I

I thus :

ities of 5 peai

Wo may ad

15 pears, or u

and say .5 pear

} mean anythiii

:ars, having 5 i

xample of nun

IS ilie groups o

the units beiii

first and lowes

1 (juantitatively
H'elated. Thoi
nplied in it the!

3 liere ind

^•I'oups and tlie

?{ of 15 pears
ntitatively, \v(
y of 15 pears

NUMERICAL RELATIONS

lore the idea of ratio is nocossarily present, and tliat in the
iii,'liest sense as an abstract conception. It is im{)os8ible to
hink of 1 pear x 15 = 15 pears in this way without having
lie coiresponding thought in mind, 1 pear = yV o^ 1<"* pears,
ndeed, these two are but complementary phases of one thought.

These considerations are of immense importance as a guide

u the preparation of a course of study, as well as in directing

lie thought of the pupil as he progresses from one stage to

iiother. Although there are many gradations between the

iiitial step of determining an aggregate vaguely V)y numbering

nits which are unrelated in every respect, but that they con-

titute the whole, and the Iiighest process of relating, unit to

iiiit or units, and unit or any number of units to the whole,

ireo divisions may serve for all practical purposes in teaching,

^ indicating what should be emphasized during consecutive

leriods of school work. For want of better terms these may

(expressed until »e designated the Nnmberiiuj^ Measuriny and Coinparin(j

)eriods.
During the first or numbering period the chief aim of tiie

r be expressed eacher will be to direct the attention of the pupil to the hoiv

equal to one
)e equal. Here
in terms of tin

as comj)letelv

uaiij/ of the aggregate. It must not be tiiought that a
lerfect knowledge of numbering can be gained at this early
tage. Nothing of the kind is to be attempted, for it would
)e impossible. Both theory and practice have established the
act that not until the how viuc/t is understood can the
■mo mam/ bear its full meaning. The object is to acquire
ucli power in determining aggregates numerically as will
uable the pupil to take, easily and naturally, the next
top along the line of progress. From the beginning.

lient as poun'!.

1 ,,] lowever, the end must be kept in viev/^ if the most direct-

I pear x lU . .^

Pourse is to be taken. The idea of number as the measure

)f quantity must be made proininent. This suggests that

|n think of tlic nore stress shoidd be laid on + 6, 8 + 8, 4 + 4 + 4, 8 + 8 + 8,

1 5 pears, han on 6 + 5, 8 + 7, 9 + 8, etc. These latter are not, of

I
I

HINTS ON TEACHING ARITHMETIC.

course, to be nogloctod, but they are not to roceivo as niu( li
attontioii as tlio formor wIhmi iminheritui is tho main <>l)j«'c'i,
Lessons leading up to nKjasurenuuit and comparison will bt-
given from the beginning. They are necessary, for the reason
that the child has already begun the work of measuring and
comparing. But he has done it in a very indcliniti! mann(!r.
The vague noti<jns he has gained will be made the basis ; also
tlie same means he has already employed will now be used iu
greater advantage under proper direction, togetluu* with sucl
more efficient means as his increasing j)ower may warrant.
This is the time for eye-measuiement, foot-measurem(?nt, etc
At no later date will he so well learn to use the instruments
bestowed l)y nature, for he is now completely dependent on
them. The motto throughout this })art of the course should
be, sPCMre <power in ninnberitKj by mcdim of muhjl/ned tiidt^.
Addition and subtraction will be practiced U) a greater extent
than nmltiplication and division.

The pupil having gained a fair command of number, is
ready at the conunencement of the second or measuring pcM'iod
to give his attention to the (jnantitative vahu; of units, and
thus become prepared for the more difficult task of comparing
quantities on a well-defined basis.

It l^as been shown that the units making up an aggregate
must be quantitatively related to one another, otherwise tho
idea of ratio cannot be present."^ Such being the case the
problem for the teacher is this : How can I best direct the
thought of the pupil so that he will ccm.sider the quantitative
relation of units 1 The answer is plain : Bring before the
notice of the pupil such aggregates as are measured by fixed
standard units as foot, yard, peck, pound, cent, etc., and give

* It may be contender! that as the process of relating: is a purely mental one, it I
cannot matter as to whether the units are actually e(|ual to one another or not. t^uilt j
true, the whole question is one of mental attitude. But how is the proper mental j
attitude to be secured?

NUMUKUING nv ONB8.

iceivc as niucli
e main ohjcri.
[)aris«)ii will Itc
, for tlio reason
moasurin^ and
('finite; mannor,
tlio basis ; also
low })e used to
ther with sucli

may warrant,
isui-cnicnt, etc.
10 instruments

(leiM'ndent on
! course should
tidfjlued units.
greater extent-

of number, is

asuring piM'iod

of units, and

of comparing

an aggregate
lotlierwise the

the case tlie
^st direct the

quantitative
|ig before the

red by fixed

itc, and give

fely mental one, it
• or not. (^uite
le proper mental

Ilim plenty of practice in measuring them. At first the pupil
will measure 10 feet with a foot rule as a mere numbering
proces.s, just in the same way that he would ccuint 10 apples,
but ill measuiing tlie distance 10 fe(!t his atUnition is con-
stantly directed to the leiigth of the foot rule. This work
will delight the pupil as new activities are now called into
play. lie sees more in number than he did before. Jle feels
that he is growing in strength. The work in numbering will
bo continued with great(!r /est than ever. This is th(i time to
fill in what was previously dealt with in outline. The motto
here should he, secure power in numbering by means of measured
units. Multiplicaticm and divisicm will become more and more
prominent during this period, but much practice will still be
giv(»n in addition and subtraction. Factors and nmltiples will
also be dealt with to some extent.

The pupil having gained considerable mastery over the difH-
culties of numbering measured units, is in a position to relate
the units to one another, and thus determine the how much
of the aggregate or of any part of it. In fact, the pupil began
to relate units to one another as soon as liis attention was
directed to their quantity. He is now to be brought into a
full consciousness of the higher thought processes of numeri-
cal ratio. The motto here is, secure power in comparing quanti-
ties. Fractions, reduction of denominate numbers, etc., belong
to this period.

7. Numbering by Ones.

The combining of like objects into small groups and the
separating of groups into their constituents are the external
operations whereby children acquire their first notions of
number."^ The presence of objects is not enough. It is only
when there is consciousness of a whole as being made up of
like parts that progress in number is possible. This conditi(»n
being fulfilled, the mind naturally reacts on the material

* strictly speaking, the order here given should be reversed. See pp. 2, 4 and 5.

IIINTH ON TEACIIINO ARITHMETIC.

r)})joctH })ros(Mit('(l to it, aiul takoH cognizance of its own acts of
attention in passing from the vvljole to the parts (units), and
from the [)arts (units) to the whole. Tliese acts of attention
are rehite<l to one another, and number is the n^sult. Single
o})jectH, as apples, sticks, etc., wliich are (|U«intitatively undeter-
mined, wlien regarded as units, are the h;ast definite possible.
Therefoi-e, counting single objects is the fundamental process
in numbering.

In counting, two distinct mental operations are involved :

(i) Separating and combining units. Although sei)arating
units and combining units an? apparently different, they are in
reality only the analytic and synthetic phases of one mental
act. This is the mode of thought which finds its expression
in the arithmcitical pn)cesscs of subtraction and addition.

(ii) Numbering units. The enumeration of units is the
second mental act in counting. Although it is conditioned
on the separating and combining of units, it is clearly a
dif!ei'(int nuMital proc(\ss. It corisists, essentially, in assigning
to each unit its place in the whole series of units, giving rise
to the use of such terms as first, second, third, etc. Until the
pupil becomes conscious of a first, second, third and fourth act
of attention in counting four objects, say, it is quite in\possil)le
for him to number the objects. As soon as he establishes the
oi'der of the different acts in the series and synthesizes them,
he has found the number, whether he can give that number
a name or not. The arithmetical processes of division and
multiplication give expression to this mode of thought.''

•IN.

* Multipli(^atioii represents two distinct stajres in the evolution of niiinbt r:

(i) The exact riuaiititative vahie of the unit is not considered. Here nuilti plication
implies nndlmj the sum of addends numerically equal to one another and innnhering
them, in this sense nniltiplication is nothing' more than a short fonn of addition.

(ii) The JimltiiiHcand is rejfarded as a unit to which the product bears a fi.\ed ratio,
such ratio hein^ defined hy the nuiltiplier.

Similarly division also represents two distinct stages of thought.

These important aspects of multiplication and division should be kept in mind by
the teacher.

NUMUERINO DY OUOUI'S.

From those oonsidcratioiiH it ir, easily iiifcrro*! tluit counting
forms tlu! hasis of tho four simple rules in arilliiiuitic.*

8. Numbering by Groups.

The child naturally counts at first by ones, but as the labour
(»f counting increases with the number of units, a point is soon
reached when he finds it advantageous to increase tho unit.
As soon as this is the case he counts by twos, thr(M\s, etc.,
with renewed pl(»asure. Why? I'ecauso he has found out a
n(iw and more detinite method of determining tho aggregate.
Why moro definit(!'? Simi)ly because the new unit has a
quantitative value of its own. True, it is not well defined,
but sufficiently so to re(|uir(5 greater effort than before in
numbering tlio whole (piantity by means of it.

During this early stage the pupil is cjuite satisfied if he can
numl)er the objects luifoi-e him, using fust one unit, then
another, etc. Generally speaking, he is not concerned with a
minute analysis of all the possible combinations and separa-
tions of the number, for the reason that such analysis neither
supplies any want which is felt, nor does it give consciousness
of increased power. Every primary teacher knows how weari-
some drill on combinations and separations becomes to the
pu})il, until he attains to a certain degree of familiarity with
number. The cause of the whole difficulty is that an attempt
is made to compel the pupil to memorize facts which are not
mastered. But let a pupil find out that he can number
12 objects by twos, threes or fours inste.id of ones, and he
feels at once the importance of his discovery. He has found
out a new method of doing something he has a desire to do.
How different is the pupil's attitude towards the study of
number when he is required to deal with all the facts of the
number sfiiym before he is allowed even to utter eiyht !

* For an account of the genesis of the fundamental operations, see EducatioiMl
Jteview for January, 1893, pp. 46 and 47.
2

HINTS ON TEACHING ARITHMETIC.

The pupil should be given practice in forming groups and in
recognizing readily groups which he has counted. The pur-
pose of this is to make him so familiar with whole groups
that he will detect them readily as units in the numbering of
larger wholes. The pupil should be put in a position to dis-
cover a unit by which he can number an aggregate, rather
than build up from the unit.* Aggregates which are easily
related to units well known to the pupil should be selected.
For example, if the pupil knows 3, 6, and 9 so well that he can
use them readily as units, he may be asked to measure 18 in
as many ways as he can. It wo; M be a mistake to ask him to
measure 18 by 6's, as naming the unit w^ould deprive the pupil
of the pleasure of gaining a new victory for himself. The
object should be to get the pupil to grasp the idea of number-
ing. The remembering of results at the beginning is of little
importance, except in so far as it carries with it added power.

Counting and grouping is work that is well adapted to
pupils who enter school at the age of 5 or 6 years. It is a
serious error to attempt much work in number with young
children, as they have not the power necessary to deal with it.
Regular, systematically arranged lessons on number may well
be deferred until the pupil is at least seven years old. Up to
that time the subject should be dealt with incidentally. It
must not be forgotten that, no matter how clear the presenta-
tion on the part of the teacher may be, numbering — not the
mere handling of objects — makes no small demand on the
part of the pupil.

9. The Use of Objects.

The extent to which objects should be employed in teaching
number will depend on the mental condition of the pupil.
In the case of young children, speaking generally, objects
should be used freely during the early part of the school course.

*Se9 dsfioition of the tenu unit— page 8.

THE USE OF OBJECTS.

ps and in
The pur-
e groups
bering of
m to dis-
,e, rather
ire easily

selected,
at he can
ure 18 in
sk him to

the pupil
;elf. The
: number-
is of little
3d power.

apted to

5. It is a

th young

il with it.

may well

. Up to

ally. It

presenta-

not the

on the

teaching
[he pupil,
objects
)1 course.

The primary notions of number are occasioned by experi-
ence with material objects. Such early perceptions are very
vague, indeed. For example, a child will be able to dis-
tinguish three objects from two long before he has a true
conception of the numbers two or three. These fundamental,
though imperfect, ideas being based on sense perception,
it is the duty of the taacher to present material of such a
kind and in such a manner as to awaken the higher thought
activities of which number — the relation between a unit and a
whole quantity — is the product.

Again, the power to form direct perceptions is very limited,
indeed ; therefore, any advancement beyond the very lowest
numbers must be made by relating one number to another.
This consideration furnishes another reason why the percep-
tion of relations by tl.\e pupil should be the aim of objective
tea.ching from the beginning.

Objects, as employed in teaching number, serve three pur-
poses :

(i) They are a means of occasioning mental action.*

(ii) They reveal to the pupil what he has to do mentally, or
what his mind is actually doing.

(iii) They furnish a test by which the thought activity of
the pupil may be determined. The pupil may make a state-
ment of a fact which in reality means little or nothing to
him. This will be shown at once l)y his failure to illustrate
it by means of objects or objective representation.

The foregoing applies to all forms of ol)jective teaching,
including representation by lines or diagrams.

* In actual practice it is easy to fall into the error of assuming that the performance
of operations with objects is necennanli/ accompanied by that form of thou;|ht activity
which results in numbering them. A moment's retlccfion will show, however, that the
attention may be fixed to such a dej^ree ujion qualities— form, color, texture, eU?.—
which characterize the object« individually, as to prevent the very kind of lueutal
action which their presence is intended to stimulate.

HINTS ON TEACHING ARITHMETIC.

When the pupil can think readily of number as apart from
particular objects, their use can serve no further purpose in
developing the notion of the how many, and in so far as this
purpose is concerned the use of objects should be discontinued
except as a test of the pupil's work. In the case of young
children, however, it is better to have too much objective
teaching than too little ; in other words, it is better to fail in
reaching the extreme limit of the pupil's power than to appeal
to something which has not yet been developed. Nevertheless,
it must always be regarded as a defect in teaching to allow
the pupil to employ objects if it is possible for him to think
out numerical relations without their aid. Here is a case in
point : A pupil fails to find the sum of 9 and 6 ; he is immedi-
ately sent to the number table to find the answer by counting
blocks. If such a course is necessary, either the teaching is
wofuUy bad or the pupil lacks something which no teaching
can supply. If number is essentially a thought process, then
the pupil should relate 9 + 6 to some known combination, say
10 + 6, 6 + 6, or 9 + 3. Such an analytic and synthetic
operation will develop power to deal with new difficulties,
while the mere handling of the objects can do little more than
inform the pupil of the required result.

A clear distinction must be made between the use of objects
in gaining a proper idea of number — the how many — and
their use in presenting standard units of measurement, illus-
trating numerical relations, etc. In the latter sense, objective
teaching should have a somewhat prominent place throughout
the whole public school course in arithmetic.

The reproductive imagination should be frequently appealed
to in arithmetical work. For example, in the exercise, Find
how often Jf inches is contained in 6 feet, the pupil may form a
mental picture of n. line instead of representing it by mean3 of
a dia<;ram. The value of such exercises does not seem to be
fully appreciated.

THE FUNDAMENTAL RULES.

10. The Fundamental Rules.

All numerical calculation, whether elementary or advanced,
is based on the fundamental processes of addition, subtraction,
multiplication and division. A perfect command of these is
as necessary for the arithmetician as is familiarity with the
keyboard for the pianist. No musical performer could hope
for success in interpretation or expression so long as he is
hampered by lack of acquaintance with the instrument he
uses. In like manner the arithmetician need not expect to
attain to a high degree of perfection in his art until he has
first mastered the fundamental processes. A theoretical know-
ledge of them is not sufficient ; many a one understands them
and can explain them who is not able to calculate either
accurately or rapidly.

There are several reasons why this part of arithmetic should
receive careful attention.

(i) It develops the power of attention. The pupil is required
to concentrate his thought fully on what he is doing in order
to make progress. The work is of such a nature that any
interruption or lack of effort is at once detected.

(ii) It tends to form the habit of being accurate. Arithmetic,
above all other elementary studies, should secure exactness in
thinking and doing.

The teacher who insists on strict accuracy in all school
work does a great deal more for the pupil than to teach him
the subjects indicated by the programme of studies. The
pupil is getting a preparation for actual life, where accuracy
is a matter of the highest im|)ortance.

(iii) It tends to form the habit of thinking quickly. In per-
forming the operations of the simple rules, the attention
shifts rapidly from one object of thought to the next, producing

BINTS ON TEACHING ARITHMETIC.

J« ^1

a continuous mental current, the course of which is modified
hy every new act of attention. - , .

(iv) Familiarity with the fundamental rules is necessary for
the business of life. Probably nine-tenths of the arithmetic
practised outside of educational institutions consists of simple
addition and multiplication, and probably ninety-nine-hund-
redths of it does not go beyond easy applications of the four
simple rules. Where higher work is required, as in the meas-
urement of timber, the computation of interest, etc., tables
are used, so that even in such cases a good knowledge of the
fundamental rules will generally suffice.

While these facts do not form an argument in favour of
confining the study of arithmetic within narrow limits, they
show how important it is to teach every pupil to adu,
subtract, multiply and divide numbers accurately and readily.
If the pupils that leave our schools cannot perform well these
every day operations of arithmetic, surely something is wrong.
It is feared that this is too often the case.

How are accuracy and speed in calculation to be secured 1
By careful teaching from tlie beginning, and by constant and
well-directed practice throughout the whole public school
course.

Accuracy should be first aimed at. Why is it that a busi-
ness man succeeds in training a boy to be exact, while the
teacher so often fails 1 Simply because the former impresses
the boy with the importance of strict accuracy, while the
latter does not. This shows the educational value of doing
work under a proper sense of responsibility. Blundering in
school, just as in actual life, should be regarded as a serious
matter, and until such is the case, little improvement need be
looked for.

Accuracy being secured, rapidity will come only through

ARITHMETICAL PROCESSES.

1 modified

saary for
rithmetic
3f simple
ine-hund-
the four
bhe meas-
c, tables
[e of the

avour of
lits, they
to adc!,
I readily,
ell these
s wrong.

secured ?
ant and
school

'< a busi-
lile the
ipresses
lile the
doing
'ing in
serious
eed be

irough

plenty of careful practice. The pupil should be given suffi-
cient time for an exercise, but no more. As he gains in speed,
of course, the time allowed will be correspondingly diminished,
so that he will always be required to make his best effort.

11. Arithmetical Processes.

The question as to when the rationale of arithmetical pro-
cesses should be dealt with is one on which there is much
difference of opinion among the very best teachers. Some
contend that it is a waste of time and a useless expenditure of
mental energy to enter into an explanation of such numerical
processes as adding, subtracting, multiplying, dividing, find
the G.C.M. (formal), extracting the square root, etc., at the
time they are first taught. Others maintain that the pupil,
as an intelligent being, should not be required to perform an
operation that he cannot explain, or at least understand.
Those who take the former position generally defend it on the
ground that arithmetical processes, in their relation to the
whole subject, are merely a means to an end, and as such it is
quite unnecessary to enter into any explanation of them so
long as the desired end can be attained without it.

The main questions to be considered are : (a) At what time
is the pupil able to put forth the mental effort necessary to
investigate the principles underlying a process 1 (b) To what
extent does a knowledge of the principles on which a process
is based affect the performance of it ? (c) What is the exact
place in the development of the whole subject at which the
reasons for a particular process must be understood ?

Some arithmetical processes are much more easily understood
than others. Compare, for example, those of addition and
subtraction. To the pupil who can combine and separate
numbers readily, and who is familiar with the decimal system
of notation, "carrying" in addition will present almost no

1 !

HINTS ON TEACHING ARITHMETIC.

ii^i

" III

ill

difficulty, while " borrowing " in subtraction will give no end
of trouble if the explanation of it ia pushed beyond the very
simplest cases. Is the pupil not to be allovved to use the pro-
cess of subtraction as an instrument of thought simply
because he cannot fully comprehend it ] By all means he
should be taught how to perform the operation so that he
may be able to use it. As thought processes^ addition and sub-
traction are complementary, and they should therefore go
together. This is the controlling fact to which the teaching
of the formal processes must be subordinated.

The following should be carefully distinguished from one
another :

(i) The jyrocess — the instrument by means of which a cer-
tain work is done.

(ii) The use of the process — the work that the instrument
is capable of doing.

(iii) The rationale of the process — the construction of the
instrument.

These may be illustrated as follows : Suppose a pupil to have
such a knowledge of factors and common factors as will enable
him to find for himself the H. C. F. of two or more small num-
bers, as 60, 84, 112. He resolves the numbers into their
prime factors thus :

60=2x2x3x5. ,
84 = 2x2x3x7.
112 = 2x2x2x2x7. - .

After eyarnJ ,.:
covers \^ . t' >i .
underst

li ilie prime factors of the numbers, he dis-
-; . .^ *aie H. C. F. The whole process is clearly
y tie pupil from beginning to end, for he has

five no end
1(1 the very
ise the pro
:ht simply
means he
50 that he
)n and sub-
erefore go
le teaching

from one
hich a cer-
nstrument
ion of the

)il to have

'ill enable

lall num-

Into their

he dis-
Is clearly
he has

ARITHMETICAL PROCESSES.

worked it out himself. He is then asked to find the H. C F.
of 1908 and 2736. He proceeds thus:

1908 = 2x2x3x3x53.
2736 = 2x2x2x2x3x3x19.
2x2x3x3 is found to be the H. C. F. of the given numbers.

The pupil is now required to scrutinize carefully and to state
clearly what he has done in finding the H. C. F. required. He
finds that he has retained all the common factors of the num-
bers, and eliminated all the factors which are not common.
The teacher then shows him a process — places an instrument
into his hands — by means of which the common factors may
be retained, and the factors which are not common may be got
rid of more expeditiously than by the method formerly em-
ployed thus : ■

2)1908
1056

3) 252
216

2736( 1
1908

828(3
756

9/2

72(
72

The pupil understands fully the use of this new process — the
work the instrument is capable of doing. He can, therefore,
employ it intelligently. But he may not be able to compre-
hend the fundamental principles underlying the process, much
less explain them.

This form may be given, but it affords no explanation, it

HINTS 01^ TEACHING ARITHttBTia

merely shows clearly how the formal process of finding the H.
C. F. serves the purpose for which it is intended.

2)2x2x3x3x53
2x2x3x3x46

3)2x2x3x3x7
2x2x2x3x6

2x2x3x3x1

2x2x3x3x76(1
2x2x3x3x53

2x2x3x3x23(3
2x2x3x3x21

2x2x3x3x2 (2
2x2x3x3x2

The points to be noted in teaching arithmetical processes are:

(i) The pupil must feel the need of the process before it is
taught. This means that he has done all he can do conveni-
ently without it, and that he is fully cognizant of its purpose.
Give the pupil plenty of opportunity to reason from first
principles.

(ii) The process is presented by the teacher and the pupil is
immediately required to apply it.

(iii) The process is examined when the pupil is able to deal
with it, so that he may discover the principles on which it is
based. To teach process, and reason for process together, is,
generally speaking, a pedagogical error, because two difficulties
have to be overcome at once, and further, because it is only
after a pupil has become familiar with a process that he can
be in the best position to investigate it.

12. Problems.

During every stage of progress the pupil should be given
ample opportunity of applying his knowledge practicaUy.
Problems should therefore have a prominent place throughout
the whole course in number.

The greatest care should be exercised by the teacher
ill selecting or making problems so that they may be well

PROBLEMS.

adapted to the requirements of the pupil. To give prob-
lems which present no stimulus is to waste time. This surely
finds its application in the thousand and one problems which
are so often given on the number 5. On the other hand, to
ask a pupil to solve a problem requiring for its solution a pro-
cess of thought higher than that to which he has attained, is
to attempt the impossible. There is but one error greater
than this one, and that is for the teacher to give an explana-
tion of the problem and then try to " sense " the pupil into
understanding it. Unfortunately, too many pupils have been
subjected to such unprofessional treatment, and have had to
suffer its evil consequences.

Viewed from the standpoint of discipline, a problem is
worth but little to a pupil unless he solves it himself. As
to whether the solution is the neatest or best matters but
little at first, provided that it is original. After the pupil
lias discovered a path for himself, however, his attention may
be directed to it in order that he may find out whether or not
he might have chosen a better one. Short cuts taken at the
proper time form a valuable means of discipline, as they
necessitate vigorous thought. For example, the solution of
the problem, / sold a Jiorse for $75 gaining 2n %, wimt %
should I have gained if I had sold him for $80 ? may take
several forms as,

(i)

|§§of cost = J
•'• Thrs of co8t = Y^iy of $75.
.-. i§§of co8t=}ogof $75.

Therefore gain in second case is $80 -$60 =$20.

The gain on $60 =$20.
.-. The gain on $1 =^ of $20.
.-. The gain on $100= VjP of $20.

= $33j.

The gain % in second case is 33^.

Hlir 3 ON TEACHING ARITHHETia

(ii) $75=125% of cost.

,\^1 =7?5of 125%ofco8t.
.-. $80=^gof 126 %ofcost.
= 133 J % of cost.

The gain % is therefore in second case 33^,

(iii) $75 = 125% of cost.

.-. $80=f§of 125 %of cost.
= 133 J % of cost.

The gain % therefore in second case is 33^.

(iv) $75=1 of cost.

.-. $80=f§of f of cost.

=f of cost. "

The gain in second case is J of cost =33^ %. v

Suppose that the first form (»f solution is the one which
is obtained. It is roundabout, and therefore indicates a
degree of weakness on the part of the pupil, but if it is his
own, it has developed power. The pupil may be asked to
attempt a shorter form of solution. A question to put him on
the crack may be given, but nothing more. If he gets the
second form, he shows a gain in relating, although there is still
an indication of weakness in that he has not made a direct
comparison between $75 and $80. The third and fourth forms
here given are preferable to the others, in so far as they
indicate a greater degree of ability in establishing a ratio
between quantities. No form should be placed before the
pupil in any case if the purpose is to secure discipline, and
not merely to show how to solve the problem. To give a form
of solution before the pupil has done the thinking correspond-
ing to it is a delusion. To relate all quantities to unity when
more difficult ratios can be established, is another source of
weakness.

PROBLEMS.

What arc the characteristics of an arithmetical problem ?

(i) Certain conditions are given.

(ii) A (Quantity is to be determined.

(iii) Definite relations are expiv^ssed connecting the
undetermined quantity with the given con-
ditions.

The difficulties which the pupil will meet in problem work
are these :

(i) Comprehending the data.

(ii) Analyzing the given conditions and determining
the relations they bear to the unknown quan-
tity.

(iii) Deciding upon the numerical operations that will
correspond to and carry into effect the thought
' process of solving the problem.

(iv) Performing the numerical operations necessary
to evaluate the required result.

These it may be well to consider somewhat in detaiL

(i) The conditions given may not be fully comprehended. If
the data are not understood, the pupil cannot even begin to do
what is required of him. Indeed, the difficulty may not be an
arithmetical one at all. Compare, for example, the problems.
Find the cost of 10 desks at $Jf.87 each, and Find the cost of a
hill of exchange for £10 at J^.87. The former of these is suit-
able for almost a beginner, the latter might put some teachers to
the test, and yet the arithmetical operations required for their
solution are identical. The duty of the teacher is to lead the
pupil by means of questions or illustrations to grasp the con-

HINTS ON TEACHING AltlTHMETIC.

ditions given. If they arc l)eyond his reach, the problem is
unsuitable, and should be withdrawn.

The second of the foregoing problems indicates clearly that
quantity relation is not the only thing to be dealt with in what
is usually termed arithmetic. Often the greatest ditticulty
which the pupil has to contend with arises from the qualitative
elements of the data. He lacks the experience necessary to
enable him to interpret the given conditions and to represent
them to himself in such a manner as to be able to apply either
his mental power or his knowledge of number in relating
them. This shows the disadvantage of attempting to teach
interest, discount, exchange stocks, etc., to pupils, before they
have a proper conception of the business transactions ou which
such divisions of the subject are based.

(ii) The pupil may not be able to analyze the problem
sufficiently to locate the difficulty definitely. In that case he
does not know the exact point at which to concentrate his
energies. For example, a pupil fails to solve, Fiml the num-
ber of cords of wood in a rectangular jnle which is 7 feet hiyh,
and which covers ^ of an acre. He knows what ^ of an acre
means, and he can calculate the number of cords when the
length, breadth and height are given. If asked why he can-
not get the answer, he will probably say that the length and
breadth are not given, but that is not the real difficulty at all.
It is this, he does not see the connection between the linear
measurements of length and breadth in one case, and the sur-
face measurement in the other. The duty of the teacher is to
question the pupil, so as to lead him up to the exact point of
difficulty, and then leave him to grapple with it as best he can.
To solve the problem and then try to get the pupil to see
through it would be a serious pedagogical error.

It is not to be inferred from what has been said that no
yalue whatever attaches to following step by step a neat, logi-

PROBLEMS.

problem is

I early that
th in what
> dirticulty
qualitative
(cesHary to
) represent
iply either

II relating
^ to teach
;fore they
8 ou which

5 problem

it case he

ntrate his

! the num-

feet hiyh,

•f an acre

when the

y lie can-

igth and

ty at all.

he linear

the sur-

ler is to

point of

he can.

il to see

that no
lat, logi-

cally arranged solution worked out by another ; for, under
certain circumstances such an exercise may prove a very effi-
cient means of stimulating independent effort. This is a very
different thing from gi^'ing assistance whenever a difficulty is
met. We must not forget that self-reliance, determination,
and strength are developed by overcoming difficulties and sur-
mounting obstacles, and in no other way. The instructor who
keeps this clearly in view will do much more for the pupil than
teach him how to solve arithmetical problems.

Again the difficulty may be that of determining the connection
between what is given and what ia to be /ound. We may take,
by way of illustration, the following problem : The amount of
$100, bearing compound interest /or two years, payable annually,
is $1^1. Find the rate per cent. The pupil can find the com-
pound interest on a sum of money for a given time, but he fails
to solve the foregoing. What is to be done ? Surely not to
give him a form of solution and ask him to think it out, or to
lead him by a series of questions* — perhaps logically arranged,
but arranged by the teacher, not by the pupil — to arrive at the
answer. Where is the real defect? It lies in the fact that the
pupil did not secure a proper grasp of compound interest. He
looked at the matter from one point of view only. What
should be done is this : He should be asked to perform the
operation of finding the compound interest over again, and
to retrace the steps taken in finding the answer. If he is able
to deduce from his work the relation A=:P (1 -»- r)^, he is in a
position to solve the problem ; if he has not power to do this,
it is too difficult for him.f

(iii) It is unnecessary to illustrate the case where the pupil has
difficulty in determining the operations to be performed. We
frequently, hear such questions as this asked by beginners,

* Proper questioning; at the right time ia a good thing, but the kind of questioning so
often practised, whioh is merely an interrogative form of telling, is pernioioui.

tSee page 104.

HINTS ON TEACHING ARITHMETIC.

whether shall I multiply or divide ? Such is the result of
defective teaching. The important fact that number is the
product of mental action has been lost sight of. The pupil has
been shown how to perform operations without learning how
to make use of them. The remedy consists in getting him
to apply whatever knowledge he may have, and to illustrate
his work objectively when necessary.

(iv) Lack of ability to calculate accurately and readily is
not by any means uncommon. The weak points should be
noted by the teacher, and suitable, well arranged exercises on
the simple rules should be given. This matter is referred to
elsewhere.

One other difficulty may be worthy of consideration. There
may be known facts bearing on the solution of the problem
which the pupil cannot summon to his aid. Sometimes a mere
suggestion may serve to lead him back to a related known
fact or principle. For example, a pupil is asked to find the
area of the right-angled triangle whose sides are 6 feet, 8 feet
and 10 feet. He draws a diagram of it, and he attempts to
find the required result, but he cannot succeed. He then
informs the teacher that he does not know how to find the
area of the triangle. The teacher asks him to say what he
knows about finding areas. Probably this may be quite suffi-
cient to enable the pupil to relate the area of the right-angled
triangle to that of the corresponding rectangle.

The recitation may take various forms, according to the
object in view :

1. Stating in simple language the conditions of the ]}roblem.
This is often a very necessary preliminary exercise. It
awakens interest and it assures the pupil that he is in posses-
sion of the facta on which the solution is to be based.

tji

PROBLEMS.

Confidence will thus be gained, and the pupil will engage in
the work with a determination to succeed.

2. Analyzing the conditions carefnUy and relating them, to
the required answer. This is what may be termed solving the
problem. In the case of advanced classes it is generally better
to make no use of figures at all, except in so far as they may
be required to express relations. The object is to deal with
the maiu thought process of the problem.

3. Stating the operations to be 'performed in arriving at the
answer. This will be found to be an excellent exercise after
the pupil has had sufficient time to think over the problem. It
directs attention to the uses of the fundamental rules as instru-
ments for serving the purposes of thought. In addition to this,
it is a time-saving exercise. Probably ten problems can be
dealt with in such a manner in the time taken by two or three
worked out from beginning to end. At the close of the
recitation, or some time afterwards, the pupils should be
required to perform the operations so as to find the required
result, as a quantity in its simplest form. The latter should
often be an oral exercise.

4. Thinking out the solution^ finding the answer, and stating
the ivhole process in written form. This is an excellent exer-
cise, especially when large numbers are involved. It gives
the pupil an opportunity of doing the whole work and of
expressing it clearly and neatly. Tlie time given must be
limited, however, so that the pupil will not fall into sluggish
habits. This exercise, though valuable, should not be allowed
to usurp the place properly belonging to other forms.

5. Thinking out the solution, finding the answer and stating
it orally. This invaluable exercise should have a prominent
place in every schoolroom in which number is taught. Mental
arithmetic is highly advantageous for several reasons : (i) It

J-' I

HINTS ON TEACHING ARITHMETIC.

requires the pupil to hold all the conditions and relations of the
problem in mind at one time, thus tending to give him a strong
mental grasp, (ii) It leads to accurate and rapid calculation,
(iii) It is the most practical in everyday life, (iv) It stimulates
the pupil to put forth his best effort — an education in itself.

There is far too much use made of the pencil in our schools
at the present time. Twenty minutes devoted to vigorous
mental work is worth more than an hour spent in leisurely
performing written exercises.

New principles should always be developed by means of
oral exercises, using the very simplest numbers. In fact, the
greater part of the whole time devoted to arithmetic should
be taken up with mental work. Improvement in this direc-
tion is greatly needed.

13. Educational Value of Problems.

The value of problems as a means of intellectual training is
so generally recognized that it is unnecessary to discuss it here
at any great length. But the consideration of two or three
points which are likely to escape notice may not be out of
place.

Number, in common with other school subjects, has its
mechanical side. Fundamental facts must be remembered,
processes must be learned, or there cannot be progress. In this
there is always a danger that the pupil may become so confirmed
in a habit of thought belonging to a lower stage of mental
activity, as to prevent him from passing easily and naturally
to a higher. Herein consist chiefly the evil effects of memoriz-
ing facts before they are comprehended, of learning tables
which are not understood, hi dealing with fractions which are
not fractions at all (to the pupil), of becoming familiar with
forms of expression which do not correspond to thought

EDUCATIONAL VALUE OP PROBLEMS.

ns of the
a strong
culation.
imulates
in itself.

p schools
vigorous
leisurely

leans of
fact, the
) should
is direc-

l-inmg IS
it here

•r three
out of

las its

ibered,

|In this

^firmed

lental
rurally
kmoriz-
I tables

3h are
with
lought

activity, etc. The solving of problems is one of the best safe-
guards against this, as the pupil is constantly required to
relate new ideas to those already in his possession. By this
means he is enabled to rise by easy gradations from one stage
of advancement to the next, always making use of what he
already knows. Two conditions are here assumed :

(i) That the problems are suitably arranged in a progressive
series.

(ii) That in solving them the pupil does the whole work for
himself.

But if these conditions be not fulfilled, problems may be a
source of injury rather than benefit. By way of illustration
take the problem, If 5 lbs. of sugar cost 30c., what is the cost
of 10 lbs.? Supposing the pupil not to have grasped the idea
of ratio, the problem is clearly beyond him. Although it
appears simple, the knowledge which he has cannot be applied
in solving it. But he may be shown how to get the answer
thus :

The cost of 5 lbs. is 30c.

1 lb. " } of 30c.
10 lbs. " 10 times ^ of 30c. = 60c.

Keeping in mind the assumption made, it is at once seen
that the statement, the cost of 1 lb. is ^ of 30c. cannot be the
expression of the pupil's thought. He may admit the truth of
the statement, but that is neither here nor there, because he
has not thought it out. For all this, experience proves that a
few repetitions will enable him to make use of the form in the
case of other similar problems. But if the problem be varied
thus : If 5 men can do a ivork in 30 days, in what time can 10
men do it ? he is at as great a loss as ever. Why 1 Simj)]y
becauiie the solution given means nothing more to him than a

HINTS OX TEACHING ARITHMETIC.

I I

.:$

I I

matter of form. It is indeed bad enough to repeat over and
over again hundreds of times, such combinations 3 + 4, 5 + 6,
etc., before they are understood, but it is infinitely worse to
deliberately furnish the means for causing what ought to lie a
highly valuable intellectual exercise to degenerate into a mere
mechanical operation, and thus arrest mental growth perhaps
for many a day to come.

Next, let us suppose that the problem is presented at the
time the pupil is prepared for it, and that he is left to his own
resources for its solution. The chances are ten to one that he
will not adopt the standard school form of three lines. He
will probably compare the quantities of sugar, and discover
that there is twice as much in the second case as in the first.
He will connect this idea with that of the given price, and
immediately arrive at the answer. But it matters not as to
the form of solution, whether it should be the best or other-
wise. The point is this : the pupil has soloed the problem in-
dependently/, and he has gained strength in doing so. His power
to grapple with new difficulties has been increased. He is
stimulated to greater effort, having experienced the pleasure of
gaining a victory.

Another advantage of problem work is that it reveals defects
in teaching. Take the following problem : J/ .3 of a farm cost
$24.00, what will ^ofit cost?

Here is the solution actually given by a student tJoing some-
what advanced work :

3^ of the farm cost $2400.

((

• (

\ of $2400.
10 times \ of $2400.
10 times \ of $2400.

\ of 10 times \ of $2400.

3 times \ of 10 times \ of $2400=

EDUCATIONAL VALUE OP P'tOPLEMS.

The whole solution is made up of six statements, the second,
third, fourth and fifth of which are unnecessary, or should be
so at least. This form of solution may be attributed to one or
more of the following :

(i) The relation between common and decimal fractions had
not been properly taught, otherwise .3 and § would have been
related directly as .3 and .6.

(ii) The work of comparing quantities had not received due
attention.

(iii) Attention had not been sufficiently directed to the
advantage of taking short cuts. (The pupil should be en-
couraged in finding these out for himself.)

The information given by such a problem as this w^ill greatly
assist the teacher in determining the nee(^ of the pupil.

A third advantage of problems consists in the fact that
they afford the best opportunity, not only for applying new
knowledge as it is acquired, but also for reviewing the old in
relation to the new. Much time may be saved by giving this
matter the attention it deserves. Tables of weights and
measures, reduction, the compound rules, fractions, etc., may
be kept fresh in the mind by means of problems which are
carefully prepared with that end in view.

Problem work may be made an important means of con-
necting school exercises with home experience. A boy brought
up on a farm will take great pleasure in solving such prol>
lems as : (i) A farmer tests his seed wheat and finds that 85
out of every 100 grains will grow. How much seed must he
provide for 100 acres, allowing 1| bushels of good wheat per
acre 1 (ii) A team travels at the rate of 1| mile per hour in
ploughing a field of 26 acres. How many days of 10 hours
each will be required for the work, supposing the width of the
furrow to be 15 inches? (iii) Find the cost of 20 boards 16

i' ;

HINTS ON TEACHING ARITHMETIC.

f<;et long, ,14 inches wide and 1 inch thick at $2*2.50 per
thousand, (iv) A macliine saws 40 cords of wood per day.
How many days' work does a rectangular pile 80 feet long,
20 feet wide and 6^ feet high represent?

Problems on marketing grain, purchasing goods, laying out
and fencing fields and gardens, building barns, houses, etc.,
will be found particularly suitable for pupils of rural schools.

Such problems as the foregoing say be easily defended on
the ground of their practical utility. But this is not the only,
or the strongest argument in their favour. The data of these
problems, being closely associated with the everyday life of the
boy, direct his attention to his surrorouljngs, and lead him to
take a deeper interest in them, lie) i-''u^ '-linp a truer concep-
tion of the world he lives in, and coi -eqiuiDtlj, of his duty in
relation to it. •

Much of the material employed in problems ciiOi ' ■ Ix, drawn
from the other subjects of study, as geography, botany, physics,
chemistry, etc. By this means the pupil's knowledge will
gradually tend to become unified into a complete whole.

Problems should be thoroughly practical. Power to think
out relations is only one of the ends to be kept in view.
Another important end is to train the pupil to do something use-
ful. It must not be forgotten that the value of power depends
altogether on how it is applied. Hundreds fail in life from a
lack of training in doing for every one who fails merely for
want of mental power. The aim should be to secure discipline
as the result of doing what is in itself useful.

14. Expression.

Language is the expression of thought. Number is the
product of mental action. Numbering requires, in com-
mon with all other forms of thought, language which appro-

:||

EXPRESSION.

priately expresses it. In order to adapt the moans of expres-
sion to the thought to be expressed and to its purposes, tlie
introduction of numerical symbols became necessary. This
gave rise to the use of characters — letters and figures. Several
systems of notation have been devised but only two of them
are now generally known, viz., the Koman and Arabic sys-
tems. In the expression of numbers, four things have to be
taught :

(i) Spoken words,
(ii) Figures — Common notation,
(iii) Letters — Roman notation,
(iv) Symbols of operation.

Owing to the interdependence and interaction of thought
and language, it is imperative that the pupil, in the lowest as
well as in the highest stage of progress, should be in a position
to express his thought clearly and fully. Applying this
general truth to the case of number, it is seen that the pupil
must be able, from the beginning, to use either words, figures
or letters in order to make advancement in the subject. Now
the practical questions to be considered are, at what time and
in what order should these different forms of expression be
taught 1

Spoken words should be used from the beginning of the
course. As tlie pupil becomes able to distinguish groups of
two, three, four, etc., objects from one another he should be
given the symbols one, two, three, four, etc., as spoken sounds.
These will serve for him every purpose of thought and ex-
pression for some time. Indeed, it is quite possible to make
considerable progress in numbering without a knowledge of
any other symbols. '

Figures may be taught as soon as the pupil has reached
a true conception of number, that is, when he can think of a

HINTS ON TEACHING ARITHMETIC.

-;i

whole quantity in relation to a unit. It must be kept in
mind, however, that many vague determinations of quantities
will necessarily precede the condition of thought here in-
dicated.

When a new symbol is introduced the pupil should be
required to use it frequently until he can associate it readily
with the idea for which it stands. In such exercises as these
the teacher will place the figure carefully each time on the
blackboard, using it instead of the spoken word. Take up 6
blocks. Bring me 6, the teacher pointing to the objects.
How many ones in 6 ? How many more than 5 is 6 ? How
many threes in 6 *? How many twos in 6 ? etc. It is highly
important that the symbol be of as accurate form as possible.
The primary teacher, above all others, should write accurately
and neatly on the blackboard.

The signs of operation should be presented one at a time as
they are required for written exercises. As in the case of
figures, they should take the place of words which are well
understood by the pupil.

Roman characters may be introduced gradually after figures
and symbols of operation are well known.

The decimal system of notation will present no difficulty
until the number 10 is reached. It is best to show the pupil
how to express 10, 11, 12, 13, 14, without any reference to
the notation for these reasons :

(i) No difficulty is experienced in doing so. The pupil is
perfectly satisfied to know the how, the why does not concern
him.

(ii) By the time that, say, 14 is reached the pupil has gained
some experience in relating these numbers to 1 0. He has a
few facts at his command, and he is, therefore, in a position to
make comparisons, 11=10+1, 12=10+2, 13=10+3, etc.,

I I.

COURSE OF STUDY.

from tlieso he iniiiiediatoly gets the idea of 1 ton -{-one, 1 ten-f-
2 ones, 1 ten-|-3 ones, etc. The main thincr is, of course, to
make sure that the pupil has the thought giving rise to thes<^
forms of expression.

Plenty of practice should be given in both reading numbers
and writing them after the system of notation is understood.
The pupil should become thoroughly familiar with it as a pre-
paration for performing numerical operations by means of
figures.

Much attention should be given to expression in arithmetic.
In fact the sul)ject cannot be well taiight without this. It
demands logical arrangement and continuity of thought,
exactness in the use of terms and clearness of statement. This
is too often lost sight of in actual practice.

15. Course of Study.

Outline of Work for First Period.

Numbering.

A — Counting — from 1 to 20.

B — Grouping —

(i) Objects — from 1 to 10.

(ii) Squares of equal size systematically arranged —
leading up to comparison, as :

D > DD

DDDDDDD DDD
DD DD DD , DD

ODD
DD-

C — Separating and combining — from 1 to 20.*
(i) E(jual groups of objects,
(ii) Unequal groups of objects.

m
5

See outline of plan of teochini;, Nos. 1 to 20, p. 62.

HINTS ON TEACHING ARITHMETIC.

\k:

■« '

D — Addition.

(i) Numbers from 1 to 20 — Addition Table,
(ii) Numbers from 20 to 100 — Types of Exercises.

(«)

6 7

6 6

~j ~")

-, etc.

('>)

10 20 30 30 40

10 20 30 20 30

10 20 30 10 20

~~")

-, — , etc.

(^•)

— , etc.

(d)

' /

— , etc.

(e)

— , etc.

;/)

s*^ /

)

— , etc.

(y)

15 .

)

— , etc.

(h)

-, — , etc.

COURSE OF STUDY.

While facility in addiug may be regarded as a highly important end in
teaching addition, it is not the only thing to be aimed at. The pupil
must be trained to apply his knowledge, he must comprehend, and be-
come familiar with, the coniinou system of notation as a means of
thought expression, and he must gain such mental jjower as will enable
him to pass on to the next higher phase of relating (quantities.

The question to be considered is, How are these results to be
attained ? Although this question may be viewed from several stand-
points, we are here concerned with only one aspect of it, viz., the
character of the exercises which the pupil is required to perform. They
may be indicated as follows :

(i) Exercises requiring the pupil to use what he knows in discovering
new facts. In («), as given above, the pupil is required to relate the
known, C + 6, and C + 6 + 6, to 6 + 6 + 7, 6 + 7 + 7, and 7 + 7 + 7.
After the pupil has performed the work, the teacher will question him
in such a manner as to make him fully conscious of what he has done.

(ii) Exercises specially intended to familiarize the pupil with the ten-
unit. Every primary teacher is aware of the rapid progress that a pupil
is capable of making after he is once able to relate by tens. Why?
Certainly not because the ten-unit possesses any magic power in itself.
The reason is simply this : the decimal system of notation is an arbitrary
one, which cannot, therefore, be employed by the pupil as a natural
means of expression until he has become familiar with the unit which
forms its basis. But just so soon as the pupil can think in tens, the
resulting unification of thought and expression manifests itself, not
only in the increased power which it gives, but also in the pleasure
derived from the consciousness of such power. Much practice should
therefore be given in finding the sum of such addends as will direct
attention to the decimal system of notation. See exercises given above.

(iii) Exercises which pave the way for multiplication. Finding the
sum of equal addends forms the most direct preparation for this.

(iv) Exercises which are designed to emphasize particular combina-
tions. This is referred to elsewhere.

(v) General review exercises.

Nothing should be said about "carrying" until the pupil has actually
practiced it. If the exercises are properly graded and related to one
another, the pupil will almost unconsciously do the work for himself.

•'>'

HINTS ON TPJACIIINO ARITHMETIC.

if!

At tli« propnr tiiiio, liowever, lio Hliould bo made fully aware of how
iliu opcrutioiiH lit! has already purforinud are related to the notatioD.

Before the formal process is taught the pupil should have a good
knowledge of notation.

E — Subtraction.

(i) Numbers from 1 to 20.
(ii) Numbers from 20 to 100.

. — Multiplication.

(i) Numbers from 1 to 20.

(ii) Numbers from 20 to 100 — Types of Exercises.

(a) The multiplicand a multiple of 10, as 10 x 2,
20 X 3, 20 X 4, 30 x 2, 30 x 3; etc.

(6) The multiplier a multiple of 10, as 2 x 10,
2 X 20, 3 X 10, 3 X 20, 3 X 30, 4 X 10, 4 X 20.
etc.

(d) 15x2, 15x3, 25x4, etc.

(e) 2x 15, 3x 15, 4x 25, etc.

(/) 24 X 2, 24 X 4, etc.

The exercises should be related in such a way that the pupil will dis-
cover the law of commutation. Objects should be used for this purpose ;
a blocks X 5 = 5 blocks x 3.

G — Division.

(i) Numbers from 1 to 20.

(ii) Numbers from 20 ^.o 100.

Multiplication and division should be taught as related to each other.
For example, 10 blocks x 3 = 30 blocks may be read as 10 blocks taken

Pi
bel

8h^

.; :ii J

COUHSK OF STIDY.

.? times = 30 bloekH, nml it may he further interpretod a8 implying that
.'{0 blocks contaiiiH 10 bloc-kH '.\ tiinoH, ^Multiplication an^l (livision aro
complementary to each other, and they should from tlie beu:innin^ bo
taught together. Kven in giving practice in the more advanceil work
of a later stage it is well not to lose sight of this completely. . When a
pupil has found that 1728 -f 12 - 144, he will understand it all the
better for proving his answer.

H— Multiplication Table— from 1 to 100.

The pupil I make his own table as ho proceeds. Each number

should be traced back to addition, at least for a time ; thus, 6 + 6 + ti
+ 6 + 6 = 30 = 5 sixes.

Mkasurino and Comparino.

A — Volume.

(i) Comparative size of objoct.s — Estimating; and
testing by means of displacement of water
when possible.*

(ii) Compai'ative size of rectanf^ular piles of cubical
blocks, leading up to definite measurement.
The blocks should be of uniform^ size.

(ill) Relation of size to weight- Estimating ,nd
B— Weight.

testing.

(i) Comparative weight of objects — Estimating and
testing by means of balance when possible.

(ii) Relation of weight to size — Estimating and
testing.

C — Surface.

(i) Comparative area of plane surfaces of objects,
as tables, desks, cubes, sheets of paper,
pieces of cardboard, etc — Estimating and
testing by superposition when possible.

•I-
'I

""We are not certain as to the practicalDe&i of this. It is merely suggested.

i •

i ■

46 BINTS ON TEACHING ARITHMETIC.

(ii) Comparative area of coloured circles, squares
drawn on V)lackboard. (The whole figure
must be chalked over, not merely the out-
line. Nothing should be said about linear
dimensions here.) Estimating and testing by
superimposing pieces of cardboard.

D — Length.

(i) Comparative length, breadth, height, etc., of
objects — Estimating and testing.

(ii) Comparison of length with breadth, etc. —
Estimating and testing.

(iii) Distance between objects — Estimating and test-
ing by pacing.

E— Value.

(i) Measurement of sums of money as lOc, 15c.,
20c., etc., using one-cent coins (afterwards
five-cent coins, ten-cent coins, etc.).

(ii) Comparative value of small coins.

(iii) "Making change," etc.

(iv) Easy problems.

F— Time.

(i) Comparative length of intervkls between suc-
cessive acts, as ringing the bell, raising the
hand, etc.

(ii) Comparative length of time taken by different
objects in falling, as hail and snowflakes,
pebbles aiid feathers, etc.

The main purposes of these lessons are (a) to give the pupil a clear
conception of what is meant by voluit^e, surface, etc., so that he may be

COURSE OP STUDY.

prepared to measure them ; (6) to direct attention to the idea of quanti-
tative measurement ; (c) to train the pupil to observe closely, to
measure with the eye, etc.

Expression.
A— Oral.

(i) Names of numbers from 1 to 1 00.

(ii) Language of number used in explaining or illus-
trating operations performed.

B.— Written.

(i) Figures.

(ii) Notation and numeration — Numbers from 1 to
100.

(iii) Symbols of operation.

(iv) Roman notation from 1 to 20.

(v) Exercises.

Problems.

A — Addition. *

(i) One-step — John has 4 marbles and James has 5 ;
how many have both together.

(ii) One-step — John has 4 marbles and James has 5
more than John ; how many has James ?

(iii) Two-step — John has 4 marbles and James has
5 more than John; how many have both
together?

B — Subtraction.

. , (i) One-step — John had 6 marbles and he lost 2;

how many had he left ?

hi '

HINTS ON TEACHING ARITHMETIC.

l' >r

I
ft

' i

(ii) One-step — John has 8 marbles and James has
15; how many more has James than John ?

(iii) Two-step — John won 12 marbles and lost 6. He
then had 18 ; how manv had he at first.

The problems given at this stage should be chiefly based on addition
and subtraction. Until the pupil has grasped the idea of ratio he has
but little power in the appUcation of the processes of multiplication
and division to the solution of problems.

Drill Exercises — Generally without objects.

A — Adding to or subtracting from a given initial number
by twos, threes, fours, etc. This forms an excellent
exercise on the different combinations.

B — Adding rapidly such as J ^ ^^ ^J ^J

C — Adding columns. The columns should be arranged,
generally speaking, so that difficult combinations would
be repeated frequently.

T> — Finding the value of a series of numbers connected by
the signs of operation, as, 64 - 32 + 1 7 - 8 + 3 - 1 2 = 1

E — Adding columns purposely design-
ed to give practice in all possible
combinations within certain limits.
For example, this will furnish 16
different exercises in addition, pre-
senting nearly a.U the combina-
tions of the first 6 numbers.

F — Performing operations indicated as follows :

(i)

5
8

-f-14

(ii)

X S

■28

24J

20^
12
(iii) 16 !> -=- 4

(iv) 24 =

ni-r-

48 -i-

40^

(V) 4 = J

16-^

28-=-

12 X

32 -f-

ll2-

imes has
m John ?

it 6. He

rst.

d addition
itio he has
tiplication

number
excellent

rranged,
IS would

Bcted by
2= ?

/'24-f-

48-r

40-=-

28-=-
32 -f^
12-^

COURSE OF STUDY.

iSuch exercises as the foregoing may be varied to any extent to suit
the requirements of the pupil. They shouhl l»e performed accurately,
and as rapidly as possiltle. In so far as the pupil is ahle, he should be
required to relate results. Why is 24 e((ual to four tiuies 6, but cit/ht
times 3? Why are 24 divided by six and 4S divided by twelve equal to
each other ?

It is an excellent plan for the teacher to prepare a set of number
charts for the purpose of drill. Exercises carefully thought out are
likely to bo much more effective than those hurriedly placed on the
blackboard. The charts may be made of ordinary manilla paper, and if
properly mounted they will last for a long time.

Outline of Work for Second Period.
Numbering — I to 1000.

A — Addition, Subtraction, Multiplication and Division.

Plenty of practice should ])e given in addition and subtraction ; but
multiplication and division should become more and more prominent as
the pujjil advances.

B— Multiplication Table— From 1 to 144.

Measuring and Comparing.

A — Volume.

(i) Measurement of quantities of sand, water, etc.,
using pint, quart, and gallon measures — one
measure at a time. Comparison of whole
quantity with the unit. Comparisons of quan-
tities measured. Problems.

(ii) Measurement of one standard by means of
anotlier, as finding the number of pints in a
gallon, quarts in a gallon, etc. Compai'isons
based on such as 2 gallons = ? quarts, '2
gallons = ? pints. llelation of pint and
quart without actual measurement, etc.
Other standards. Problems. ^.

(iii) Measurement of rectangular solids. ^ '<-^'^^

HINTS ON TEACHING ARITHMETIC.

li, '

^li;

■ ■♦*■■■

Kectangular blocks of uniform size should be first used. These can
be arranged together so as to represent rectangular solids. The aggre-
gates may be measured, taking one block, two blocks, etc., as the unit,
as the pupil may choose. * Comparisons will then be made. The great
difficulty which is usually experienced with cubic measure arises from
the fact that standard units of solidity are derived from correspond-
ing units of length. For that reason nothing should be said about
linear measurement, at least in this connection, until the pupil has
grasped the idea of solidity, and until he is able to select units which
are applicable to the measurement of solids.

B— Weight.

(i) Weighing quantities, using ounce and' pound
weights. Comparison. Problems.

(ii) Comparison of ounce, pound, hundred weight.
Problems.

(iii) Relation of weight to volume. A pint measure
of water weiglis so much, what will a gallon
weigh] etc. Problems.

C — Surface.

(i) Measurement of rectangular plane surfaces,
using one square foot as the unit. Compari-
son. Problems. j

(ii) Measurement of a square yard of surface by
means of the unit one square foot. Also the
measurement of a square foot by the units, 1
square inch, 2 square inches, 36 square inches,
72 square inches, etc. Comparison. Problems.

(iii) Relation of different standards to one another.
Problems.

A piece of cardboard exactly 1 foot square may be cut out neatly to re-
present the unit. By this means the attention of the pupil will be fixed
on the idea of surface measurement. The main thing is to lead the pupil
to see that a surface can be measured only by some unit of surface. The

*The Uaoher will observe ihat th^ operation Atai performed is one of aiialysis.
Why?

COURSE OF STUDY

measure

form of the unit should be changed ; for instance, the square may be cut
into two e(jual parts, and the parts placed so as to form an oblong 4 times
as long as wide, etc. If one unit is thoroughly taught, and if the pupil
once sees that it is applicable to the measurement of surface, the main
difficulty of square measure is overcome. What has been said regarding
cubic measuro applies equally to square measure.

D — Length.

(i) Measurement of linear dimensions of objects, as
table, desk, floor, wall, blackboard, etc. Com-
parison. Problems.

(ii) Measurement of distances between objects, as
two pickets placed in the ground 30 feet
apart. Comparison. Problems.

(iii) Relations of the standards, inch, foot, yard, rod,
to one another. Other standards. Problems.

E— Value.

(i) Measurement of amounts of money by means of
standards, as cent, 5-cent piece, 10-cent piece,
25-cent piece, etc. Comparison. Problems.

(ii) Comparison of values of different coins. Prob-
lems.

(iii) " Making change." Problems.

F— Time.

(i) Measurement of time taken for the pacing of
distances, the oscillations of a pendulum, etc.
Comparison. Problems.

A weight suspended by a thread will be found very serviceable for this
work, as the time of each oscillation may be varied to suit the circum-
stances, by shortening or lengthening the distance from the point of sus-
pension to the centre of gravity of the weight. A clock or watch will
be used to indicate the standard units of measurement, as second,
minute, etc.

UNIAKlU CULLtiit Uf ^UUCAII0I»

IS '

HINTS ON TEACHING ARITHMETIC.

(ii) Relation of second, minute, hour, day, etc.
Comparison. Problems.

Lessons on the sun dial and face of the clock will be found profitable.

Factors.

A — Factors of numbers — From 1 to 144.

B — Common factors of two or more numbers.

Multiples.

A — Easy multiples of numbers.

B — Common multiples of two or more numbers.

Fractions.

A — Exercises on division into equal parts.

B — Comparison of the whole and one of the parts, taking
the part as the standard.

C — Comparison of the whole and one of the parts, taking
the whole as the standard.

D — Easy fractional relations, as halves, fourths, eighths,
thirds, sixths, ninths. i

This work should not be begun until the pupil is able to form a pro-
per conception of the word times; that is, as implying the ratio of one
quantity to another. It is to be noted that the pupil will use this term
long before it conveys to him its full meaning. He may say 3 times 6
feet = 18 feet, Just as he would say 3 times 6 apples = 18 apples, almost
at the commencement of the course. There is a long step between 6
feet X 3 =- IS feet, as meaning 6 feet + 6 feet + 6 feet = 18 feet, and
6 feet X 3 = IS feet, as implying that IS feet is a quantity 3 times as
great as 6 feet. In the former case the unit may be regarded merely
as one of the like things which make up the whole, while in the latter it
must be regarded as a measured quantity. The teacher should satisfy
himself that the pupil has actually made a comparison of the quantita-
tive values of G feet and 18 feet. When the pupil has done this, how-
ever, he has the thought in mind which finds one form of its expression
by moans of a fractional number.

COURSE OF STUDY. 1^

Expression.

A — Notation and Numeration — 100 to 1000.
B— Roman Notation— 20 to 100.

Problems — Types.

A — Counting by fives and tens. (Introductory.)

(i) I bought 20 yards of cloth at one store and 30
at another. How many yards did I buy?
How much more is there in one piece than in
the other? If I sell lx)th pieces at $2 per
yard, how much shall I get for them 1 I
divide the money equally among 20 men, how
much does each get 1 If there had been
twice as many men how much woiild each
have got 1

(ii) John sold 20 quarts of milk in the forenoon and
30 pints in the afternoon. How many pints
did he sell altogether 1 How much more did
he sell in the forenoon than in the afternoon 1
At 5c. a quart, how much money did he get 1
If John's money is made up of 5-cent pieces,
how many has he 1 How much cloth can he
buy with it at 25c. per yard 1

B — Counting by sixes and twelves. (Introductory.)

(i) In 1 2 yards how many feet ?

(ii) In 48 feet how many yards ?

(iii) Measure the length of the table, using a foot
rule. What is its length in inches ]

(iv) In 48 yards how many 6 feet measures ?

(v) What is the value of 18 lbs. of rice at Gc. 1
\ y How much butter at 12c. could be bought for

the same money ?

HINTS ON TEACHING ARITHMETIC.

lis

! T

C — Goricnil Exercises.

(i) How many quarts in 3 gallons ? In 3 gallons
and 3 quarts 1 Compare 3 gallons and 3
quarts. At 5c. per quart find the value of 3
quarts. Find the value of 3 gallons. Com-
pare 15c. and 60c. with 5c. How many
times is 15c. contained in 60c. 1 How much
greater is 60c. than 15c. ? How many times
greater? At 15c. per hour, how long would
it take to earn 60c. 1

(ii) How many yards in 18 feet? In 19| feet? In
72 inches?

(iii) I had 15c. and I got 45c. more. How mfiny
pencils at 5c. each did I buy with one-half of
my money? I spent the other half in buying
oranges at 3 for lOc, how many did I get?
I sell the oranges at 5c. each. How nmch do
I get for them ? How much must I add to
what I have in order to make it $1 ? How
many oranges at 3 for 10c. can I buy for .^1 ?
How many at 3 for 5c. ? Why are there
twice as many oranges in the latter case ?

In giving problems, the following points should be noted :

1. The problems should usually be related to the exercise
preceding them.

2. They sho:ild be related to one another. It is generally
profitable to give a series of problems bearing on one another
and developing one main idea.

3. They should call forth the pupil's best effort. Problems
or other exercises which are too easy are useless. Of course
the opposite error must be guarded agamst also.

COURSE OP STUDY.

4. They shouM be carefully graded according to degree of
difficulty.

In the foregoing, the primary object is to give the pupil a
clear conception of a measured unit. This is done in two ways,
(a) by presenting the unit so that attention may be directed
to it as a measured quantity, (h) by requiring the pupil to mea-
sure suitable aggregates by means of it. All measurements
should be performed as accurately as possible. It is not the
amount of work done that counts most for either the purposes
of number or the formation of right habits, it is the character
of the work. Whenever a measurement is made, it should
become the basis of a series of questions, leading the pupil to
make comparisons in so far as he is aVjle to do so.

The pupil should be allowed to use one unit until he has a
good working knowledge of it, before another is introduced.
New units should be related to those already known. Sup-
pose, for example, that the pupil has become familiar with 1
foot as a unit of measurement by using it in determining such
lengths as 12 feet, 16 feet, etc. Let him measure off say 12
feet, place in his hand a yard stick, saying nothing about its
length, and ask him to measure the distance of 12 feet with
it. He finds that the yard measure is contained 4 times in 12
feet. Question him so as to lead him to compare the length of
the new unit with that of the old one. This is not by any
means the easiest way to get the pupil to see the relation
between the units, but it may be the best way, provided he
has gained sufficient power to make the necessary comparison.
But if he has not the power, it would be a serious mistake to
try to force him to a conclusion. The stimulus should be as
great as the mental condition of the pupil will warrant, but no
greater.

56 HINTS ON TEACHING AlllTIIMPlTIC.

OuTiJNK OF Work for Tniiin Pekiod.
Fundamental JIules.

A — Addition, Subtraction, Multiplication and Division.
Practice to secure accura(;y and rapidity.

B— Multiplication Table. From 1 to 400.

C — Factors and Multiples — oral exercises.

Weights and Measures. ,

A — Tables as required for exercises.
B — Transformation of denominate units.

(i) Simple — one denomination.

(a) Reduction descending )

. Relate to each other.

(6) Reduction ascentling )

(«) Problems — oral and written.

(ii) Compound — two or more denominations.
(o) Reduction descending
(A) Reduction ascending
(c) Problems — oral and written.
Weights and measures ii<tt in cummon use should be omitted.

C — Compound Rules,
(i) Addition
(ii) Sul)traction

(iii) Multiplication
(iv) Division

(a) Finding the value of
each part

(b) Finding the number

Relate to each other.

'Relate to each other.

of times
(v) Pi'oblems — oral and written

COURSE OF STUDY.

nsion.

other.

D — Coiiiinercial transactions.

(i) Evaluation, Finding values in connection with
everyday business transactions.

(ii) Bills and accounts. Making out and receipting
})ills of goods as groceries, dry goods, hard-
ware, farm produce, etc., in proper business
form.

In all the foregoing exerciaea care will be taken not to introduce too
many large numbers, as they make the work Ijurdensome and serve no
educational or practical purpose. In teaching principles small numbers
should always be employed,

E — Measurements.

(i) Length.

(a) Practical measuring of distances — finding
the length of lines, the dimensions of rect-
angular surfaces and solids.

(h) Coiui>arison of lengths t>f lines, etc.

(r) Problems — oral and written.

(ii) Suiface.

(a) Area of rectangles — land measurement.

(b) Comparison of are«is of rectangidar sur-
faces.

(iii) Volume.

(tt) Measurement of rectangular solids.

(b) Comparison of volumes of different solids.

(iv) Practical applications.

(a) Fencing, ditching, tree-planting, etc.

Ill

0S HINTS ON TRACHINO ARITlIMETIfJ.

(h) Carpt^tiiig, pHixM'iiig, painting, phwtoring, otc.

(d) Measurement of lumber.

(e) Measurement of cordwood, stone, stonework,
brickwork.

(/) Capacity of rectangular tanks, bins, etc.

(g) Problems.

In all of this practical work the pupil should find his own data by
actual uieasuremeut as far as possible.

F — General review exercises and problems.

Measures.

A — Integral factors as measuroft of quantities.

(i) Denominate units (siini)l(>). Exercises,
(ii) Abstract units.* Exercises.

B — Common measures of two or more quantities,
(i) Denominate units (simj)le). Exercises,
(ii) Abstract units. Exercises.

C — Prime factors.

D — Greatest common measure of two or more quantities.

(i) By means of resolving numbers into prime
factors.

(a) Exercises.

(6) Easy problems.

(ii) By means of the formal process. '

(a) Exercises.
(6) Easy problems.

• See imge 7.

COURSE OP STUDY.

E — Gonoral roviow exerdaes and proV)lomH.

Multiples. •

A — Quantities which can ho niea.surod exactly by other
({uantities.

(i) Denominate units (simple). Exercises on
divisors.

(ii) Abstract units. Exercises on divisors.

B — Common nmltiples of two or more quantities,
(i) Denominate units (simple). Exercises,
(ii) Abstract units. Exercises.

C — Prime factoi's. Tleview.

D — Least Common Multiple of two or more quantities.

(i) By means of i-esolving numbers into prime fac-
tors. Exercises.

(ii) By ni(\'ins of formal process. See Hamblin
Smitli's Arithmetic, p. 48. Exercises. Easy
problems.

E — General review exercises and problems.
Fractions — Common.

A — Exercises on the comparison of quantities. (Review.)

B — Division of units into equal parts. (Review.)
C — ProliminMty exercises — no formal process.

^ 'omparison of the fractional unit with tlie prime

unit, as j j j is 4 times also

is \ of i j j ; 12 inches rr 4

times 3 inches, also 3 inches = | of 12 inches;
5 feet 5 times 1 foot, also 1 foot = J of 5
feet, <

HINTS ON TEACH INO ARITHMETIC.

t'.

t:
;»

Such complementary thoughts must be kept before the pupil in order
thut thu true idea of a fraction may be developed. It is one thing to
learn to manipulate fractions, but a quite different thing to think of the
relations which fractions express.

(ii) Comparison of different fractional units with a
common prime unit and then with one
another, as 6 inches = | of 12 inches = 3
inches x 2 =: f of 12 inches = 2 inches x
3 == g of 12 inches. Therefore, J = f = g.
Elxercises.

(iii) Comparison of multiples of the fractional unit
with the p;'ime unit, as 12 inches = t>vice 3
inches x 2, also twice 3 Inches = ^ of 12
inches.* Therefore, twice ^ = h. Exercises.

(iv) Comparison of equal parts of the fractional unit

with the prime unit, as 12 inches =12 times

J of 3 inches, also ^ of 3 inches = ^^ of 12

inches. Therefore, l^ of | = j^g- Exercises.

The pupil should be given many exercises such as chose indicated
above before the formal teaching of the subject begins.

D — Formal study. ^

(i) Notation. I

(ii) Changing from one form to another.

(a) Mixed numbers to improper fractions.

(b) Lnpropcr fractions to mixed numbers.

(iii) Changing from one denomination to another.

(a) Equivalent fractions having different denom

inators.

(b) Equivalent fractions in lowest terms.

denominator. Comparison.

*It is here nssumed, of course, that 3 inches = J of 12 inches is well known. Tnu
CMmes under (i).

COURSE OF STUDY.

Tnu

(iv) Addition, subtraction, multiplication and di-
vision.

(v) Complex fractions. Easy exercises.

(vi) Practical problems.

E — General review exercises and problems.
Fraciions — Decimal.

A — Notation. Relation to the Arabic (decimal) system.
Reading decimals. Exercises.

B — Changing from decimal to vulgar fractions and ince
versa. Exercises.

C — Addition, subtraction, nmltiplication and division of
decimals.

D — Practical exercises. Problems.

E — General review exercises and problems.

Fractions — Percentage.

A — A particular case of fractions. ^

(i) Meaning of the term. Illustrations.

(ii) Relation of percentage to decimal and common
fractions.

(iii) Exercises and problems.

B — General application — Exercises and Problems.
C — Particular applict^tions.

(i) Trade discount. ♦

(ii) Insurance, taxes, etc.

(iii) Commission.

(iv) Profit and loss.

HINTS ON TEACHING ARITHMETIC.

y <

(v) Simple interest
(vi) Bank discount,
(vii) Partial payments,
(viii) Compound interest,
(ix) Stocks,
(x) Exchange.
D — General review exercises and problems.

Proportion.

(i) Simple, Exercises and problems,
(ii) Compound. Exercises and Problems.

Involution.

Evolution.

(i) By inspection.
(ii) Formal process,
(iii) Application — Easy exercises.

General Review.

16. Plan of Teaching Nos. 1 to 20.* ,

Two.
(a) 2=1 + 1; 2-1-1=0; 1x2=2; 2-^1=2.

Three.
P 3=1 + 1 + 1; 3-1-1-1=0; 3=1x3; 3+1=3.

Four.
(c) (i) Counting — forwards and backwards ; 4^=1x4;
4 + 1=4.

(ii) Forming groups of two.
(iii) 4=2 + 2; 4-2-2=0; 4=2x2; 4+2=2.

* In onler to save H\vwii mich combinations iw 3 f 4, r>+2, 7 — 2, 11 — 6, et«., are hero
oniittcti. TheHe are to be fully dealt witii lus explained elsewhere.

It will be understood, of uounte, that concrete uniUi are to be dealt with at the
beginning. See page 18.

PLAN OP TEACHING.

I, are here
lith ut the

Five.

(d) (i) Counting — forwards and backwards.

(ii) 4+1; 2 + 2 + 1.

Six.

(e) (i) Counting — forwards and backwards ; 6=1x6; 6+1 = 6.
(ii) Forming groups of three.

(iii) 6=3 + 3; 6-3-3=0; 6=3x2; 6+3=2.
(iv) Review (c) iii.

(v) 6=2 + 2+2, 6-2-2-2=0; 6=2x3; 6+2=3.

Seven,
if) (i) Counting — forwards and backwards.

(ii) 6+1; 3+3+1.

Eight.

(g) (i) Counting — forwards and backwards.

(ii) Forming groups of four,
(iii) 8=4 + 4; 8-4-4=0; 8=4x2; 8+4=2.
(iv) Review (c) iii and (e) v.

(v) 8=2 + 2 + 2+2; 8-2-2-2-2=0; 8=2x4; 8+2=4.

^ine.
(h) (i) Counting — forwards and backwards,
(ii) Review (b) and (e) ii.
(iii) 9=3 + 3+3; 9-3-3-3=0; 9=3x3; 9+3=a

Ten.
(i) (i) Counting — forwards and backwards.

(ii) Forming groups oi five. '

(iii) 10—5 + 5; 10-5-5=0; 10rrr5x2; 10-t-6==2.

(iv) Review (c) iii, (c) v and (j/) v.

(v) 10--2 + 2 + 2 + 2 + 2; 10-2-2-2-2- 2-0; 10 2x5;
10+2=5.

;[

1^9

tr;

HINTS ON TEACHING AlUTUMETIO.

' Eleven.

(j) (i) Counting — forwards and backwards,
(ii) 10 + 1.

(iii) 5+5 + 1.

Twelve.

(k) (i) Counting — forwards and backwards.

(i1) Review (c) i.

(iii) 12:^0 + 0: 12-0-6=0; 12=6x2; 12+6=2.

(iv) Review (c) i and (y) ii.

(v) 12=4 + 4 + 4; 12-4-4-4=0; 12=4x3; 12 + 4=3.

(vi) Review (/>), (c) iii and (h) iii.

(vii) 12=3 + 3 + 3 + :i; 12-3-3-3-3=0; 12=3x4; 12 + 3=4.

(viii) Review {<j) v and (/) v.

(ix) 12=2 + 2 + 2 + 2 + 2 + 2; 12-2-2-2-2-2-2=0;
12 = 2x6; 12 + 2 = 6.

It is unnecessary to give a detailed analysis any further, as
the general plan is easily seen. It may be briefly stated thus :

(i) The chief aim at the beginning is to give the pupil
power to number'. With that end in view the idea t)f determ-
ining the hoiv many of an aggregate by means of a known
unit is constantly kept l>efore him. Beginning with counting
(by ones) as tiie fundamental process in numbering, the pupil
is led step by step to detine aggregates by means of numbered
units, thus pro«<>o<ling from the indefinite to the definite, in
accordance with the natural order of thought development.
This accounts for the fact that composite numbers, which are
etusily me.asured by difterent units, receive mo» attentictn at
first than pritoe numbers.

(ii) Numbers such .*Vs 7, 11, 13, etc., are not analyzed to any
great extent at the commencement of the course, because to

PLAN OF TEACHINO.

(leal with them in detail wouhl only lead to the memoriz-
ing of facts Ijofore they are mastered --an error to be guanled
against as carefully as that of conmiitting algebraic or trig-
onometrical formuhe before they have been de<luced from fiivst
principles. ]t must not be forgotten that numbering is essen-
tially a form of mental activity.

But although prime numbers are not to be analyzed minutely,
care must l)e taken not to overlook them entirely. Their
position (ordinal) in the regular series must be brought clearly
before the pupil. The exercises in counting objects, etc.,
suggested in the foregoing outline, if properly performed, can-
not fail to secure this.

(iii) Such con\binations as 7 + 5, 9 + 7, etc., are for the
most part to be omitted at the very beginning f(jr twt) reasons,
in addition to those which may be inferred from the foregoing.

(a) They present difficulties which the pupil is not prepared
to meet.

(6) They will serve for a most important purpose after the
pupil has gained some conception of what numbering
really is. He will be given the opportunity of relating
6 + 7 say to the known combinati<m (5 + 6 or 7 + 7.
5 + 7 may be related to 5 + 5, 6 + 6, or 7 + 7, all of which
are well known. These exercises will not only aiTonl
pleasure by occasioning a{)p?'opriate mental activity, but
they will also gnuitly increase the pupil's power to
analyze and compare quantities. N«) attempt to get the
pupil to memorize these should be made at first. The
remembering of facts may be allowed to take care of
itself for a time at least, provided the right kind of
thuikhnj is done. There is little difficulty in cf>m-
mitting to memory facts that are thoroughly under-
stood. One of the weaknesses of primary number
6

HINTS ON TEACHING ARITHMETIC.

I-Sf'i

teaching consists in drilling mechanically on facts which,
in so far as the pupil is concerned, have never passed
through the thought stage. The proper order is, first
comprehending the idea, second, mastering the idea hy
making use of it frequently, third, drilling on it to
secure perfect familiarity with it — the mechanical phase.*
To reverse this order is to put the cart before the horse.

(iv) By the tinie 20 or 30 is reached the pupil will have
acquired some grasp of the subject. He will then Ihj in a
position to make a fuller analysis of the lower numbers. This
work should be carried on in no haphazard manner, but
according to a well designed plan, leading the pupil to connect
each new idea with others already in his possession. For
example, such exercises as these taken in the order given are
comparatively useless for the purposes of teaching : 6-1-7,
5 + 4, 14 + 8, 9 + 11, 23-8, etc. The pupil cannot see any
connection which one part has with another, therefore the
exercise can have little value except as a test of what is
known. How different is the effect when a lesson is arranged
so as to develop one leading thought, as for example :

6+7, 17 + 6, 23-7, 16 + 7, 33-6, 37 + 6, 16 + 17, 23-17, 33-16,
43 - 17, 2(i + 27, 43 - 26, and 37 + 26.

This exercise is valuable because by performing it

(a) The pupil has been led to relate numbers to one another
as 6 + 7, to 16 + 7, etc., and he has consequently gained
power.

(6) The combination 7 + 6 means more to him than it did
before, for he has made a wider application of it.

*Stre.s8 should belaid on the first and s<*cond of these rather than on the third.
Drill exercnaes, although quite iiecesHury, if carried beyond a proper limit have a dead-
ening effect.

fLAN OP TKACHINO.

hat is

it did

third,
bdead-

(fl) llo lias don^ at least something towards forming the
habit or relating one thing to another.

The numbers given above extend beyond the stated limit of
20 or '^0 in order to show how the primary lessons may be
connected with those more advanced.

The pupil should l)e encouraged to discover different units in
numbering an aggregate. When 16 blocks, say, are presentwl
for the first time it is best to simply ask how many there
are. Altliough an order for the selection of units is laid down
in the foregoing outline, it is nijt intended that such order
should l)e forced upon the pupil. One-half of the value of the
lesson will be lost if he is not allowed to determine units for
himself. In fact very much of the difficulty of arithmetic
consists in recognizing units as being applicable to the
measiu'ement of quantities.

(v) At the same time that the analysis of the lower numbers
is being carried on there will be progress ma«le in connection
with the higher, proceeding as before, from outline to detail.
The l>upil will pass gradually from 20 to 100, counting by tens,
then by fives. The relations established between numbers
under 20 should be inunediately applied to those above it.
Whenever the pupil learns that 7 -|- 3 = 10 he should be re-
(juired to use this knowledge in so far as he can. He will have
a better grasp of 7-\-'.] when he has related it thus, 17-J-3,
27 -f 3, etc., than he could possibly have had Ix^fore doing so.
Every new idea must be used in order to be fully mastered.

The exercises in multiplication and division indicated
above should be based on correspontling ones in addition and
subtraction. The latter processes will therefore preccnle the
former. It is not the intention here to advocate a return to
the old metluxl of teaching tlu^ fundamental rules, either inde-
pendently of one another, or in consecutive order ; but it is

HINTS ON TEACHINO ARITHMETIC.

the intention to advocate their presentation in accordance
with the logical sequence of their development. The pupil
should be able to add 3 pears, 3 pears and 3 pears readily
before he is asked to perform even the additional operation of
counting the three addends. If this is true as to the lower,
what must be said with reference to the higher phase of multi-
plication? The same principle applies in the case of sub-
traction and division. The prevailing practice of teaching
addition, subtraction, multiplication, division and fractions (?)
simultaneously, from the very commencement of the course,
is based on the fundamental error of assuming that the pro-
cesses of thought which these involve are co-ordinate rather
than correlative.

The course here outlined is, for the most part, suitable for
the average pupil of 7 years of age. Any work in number
df)nq with pupils below this limit should be very elementary
imleed, such as counting and forming small groups of objects,
stick-laying, drawing, etc. No regular number wt)rk should be
prescribed for pupils below the age referred to for two reasons :

(i) They have not, generally speaking, attained to the matur-
ity of thought necessary to enable them to deal with the
subject intelligently or profitably.

(ii) There is plenty of other school work which is well
suited to them — reading, drawing, nature study, oral composi-
tion. Little children delight in these, because they can be
adapted to their mental condition.

17. Seat Work for Junior Glasses.

A — Objects.

(i) Arranging shoe-pegs, sticks, etc., into groups of
two, three, four, five, etc.

SEAT WORK FOR JUNIOR CLASSES.

(ii) Laying sticks or pegs so as to roproscnt t'orms
placed on })lackboard, arranging IkjIow each
form the number of objects used, thus :

AD HI OD H O^^^

III nil mil mill iiiiiii iiiiiiii iiiiiiiii inmiiii

(iii) Arranging sticks, pegs, etc., by threes, fours,
fives, etc., so as to represent as many different
forms as possible, thus :

(iv) Arranging sticks, pegs, etc., into groups of ones,
twos, threes, fours, etc., numbering the groups
in consecutive order, thus :

I I

I I I

II I I

II 1 I I

A A

AA AA

AAA AAA

AAAA AAAA

AAAAA AAAAA

(v) Forming bundles of toothpicks, etc., by tens,
and afterwards making up such numbers as
16, 26, 36, 46, etc., by adding, and also by
taking away.

HINTS ON TKACHING ARITHMETIC.

(vi) Illustrating by means of pegs, trjotlipicks, etc;.,
operations -indicated on blackboard, ilius ;
(See No. iv.)

1+0 =?
1 + 1 =?
1+1+1 =?
1+1+1+1 =?

1+1+1+1+1-?

1x1 = ?
1x2 = ?
1x3=?
1x4 = ?
1x5 = ?

2+0

^__ (

f 3+0

-if

2 + 2

»'

1 3+3

-r

2 + 2 + 2

1 3+3+3

2+2+2+2

f 3+3+3+3

2+2+2+2+2

= 5

f 3+3+3+3+

3 = ?

2x1 = ?

3x1 = ?

2x2=?

3x2 = ?

2x3=?

3x3=?

2x4 = ?

3x4=?

2x5 = ?

3x5=?

(vii) Illustrating by means of blocks operations
already performed on slate, thus :

5x4—4x5

6x4 — 3x8

5 X 4 = 10 X 2

nnnnn
nnnnn
nnnnn
nnnnn

DDD

DDD
DDD

nan

DDD
DDD
DDD

nnnnn
nnnnn

(viii) Laying one-inch, two-inch, three-inch, etc., sticks
end to end, so as to represent lines of given
length as 6 inches, 1 foot, 1 foot 6 inches, etc.
Comparison.

(ix) Finding the measurement of sticks numbered
1, 2, 3, 4, etc., by means of one-inch, two-inch,
etc., measures and writing results on slates.
Thus stick No. 4 contains the two-inch
measure 8 times, therefore it is 2 in. X 8 == 16
inches long. Comparison.

SEAT WORK FOR JUNIOR CLASSES.

(x) Defcerniinint» tlio Icnj^ths of sticks l»y placing
small measures end to end heside them.
Thus stick No. 7 requires 6 ft»ur-irich measures
to make up its whole l(ui<(th,it is therefore 4 in.
X 6=24 inches long. Questions as, how
many 8 inches contained in the length of
stick No. 7 1 How often must a threcNinch
measure be rep(5ated to make up its length, etc.
Comparison.

(xi) Laying sticks of measured length so as to en-
close 2 sq. inches, .3 s«|. inches, 6 sq. inches,
1 sq. foot, etc. Corresponding oral or written
statements. Comparison of one unit, 2 units,
3 units, etc., with the whole area enclosed.

The unit of surface must be made clear to the pupil.

(xii) Laying sticks of measured length so as to enclose
a stated area, giving it different forms.

(xiii) Forming cubes and other rectangular solids of
given size by means of cubical blocks. Com-
parison.

In all of the preceding either the idea of counting or mea-
suring should be prominent.

B — Representation of objects.

(i) Representing on slates, pegs, sticks, blocks, etc.,
by twos, threes, fours, etc., as :

D DD DDO DD DD DDD

HINTS OX TEACHING ARITHMETIC.

(ii) Copyin;^ from lilacklxMinl dniwiu'^s, ilhist rating'
o{M;niti<)iis, tlie pupil (l(>U'i'iiiiiiiii<{ tin; li^'lil-
hund side of tlie equation, as .

AA-^AA=? AAA+AAA=?
TTTT-^TT=? HHHH-HH=?

(iii) Using figures in addition to that given in No.
(ii), as :

nnn + iiiin = ?

3 + 3 =

(iv) Illustrating exercises, problems, etc. (first per-
formed with figures), by means of circles,
• dots, squares, etc., as :

5 + 4=9 4+4+1=9 4x3=3x?

• I =M00TS

9» • • •
DOTS ! • • !
• • • •

5+5+4= 5+5+5+5+5= 6x4=4x?

•:l

(v) Measuring lines drawn on slate by means of
sticks, etc. Comparison.

flFAT WOIIK FOR .TtTNtOn OfiAflSKft.

[ns of

(vi) |)r.iAviiii,' oil slut(? s(|U)in's and rcctan^losof ^ivcn
livvn. CoiiipariHuii of arttiiH.

(vii) KoproH(Mitin^ rectanfflos of given area in different
fonuH, etc. Comparison.

C — No objects.*

(i) Addition.

(a) Adding l)y ones, twos, threes, fours, etc., the
results being expressed by figures, as :

2, 4, fi, 8, 10, 12, 14, K;, 18, 20, 22, 24, etc.

4, 8, 12, Ifi, 20, 24, 28, 32, 36, 40, 44, 48, etc.
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, etc.
16 ft., 25 ft., 35 ft., 45 ft., 55 ft, 65 ft, 75 ft, 85 ft, etc.
13 pts. , 33 pts. , 53 pts. , 73 pts. , 93 pts. , etc.
25 ds., 50 ds., 75 ds., 100 da., etc.

(h) Adding nund)ers arranged in columns, as:

4
»

4
>

4
— , etc.

8
— , etc.

— , etc.

__ __ __. ftf/>

,,,,,, V\Aj.

* Standunl uiiiUs of iiieaiiurc'iuent, utt ft., .yd., )>t., wk., etc;., should Ite kept proiui-
nently before the pupiL

>■■.

HINTS ON TEACHING ARlTHJTETIC,

f
i

2+2=? 4+4=]

8 + 8 = ? etc.

2+3=? 3+4=?

8 + 9 = ? etc.

!l:

G+4 = ? 16+4 = ?

2vS+4 = ?

10+9+1 =? 20+ 9+ 1

= ? 30+ 9+ 1

7+7+8 =? 7-f- 7+ 8 +

= ? 7+ 7+ 7+ 7 +

7 + 8 + 9 + 10=? 10 + 11-12+13 = ? 14 + 15 + 16 + 17 + 18-?
{d) Problems based on preceding exercises.

(ii) Subtraction.

(a) Subtracting })y ones, twos, threes, fours
etc., from a given initial number, as:

12, 10, 8, 6, 4, 2.

30, 28, 26, 24, 22, 20, 18, 16, 14, 12, etc.

100, 96, 92, 88, 84, 80, 76, 72, 68, 64, etc.

150, 140, 130, 120, 110, 100, 90, 80, 70, 60, etc.

165 ft., 160 ft., 155 ft., 150 ft., 145 ft., 140 ft., etc. ^

(b) Finding the diflterence between numbers
arranged in columns, as :

25.

—

"~"i

•

172

44,

etc.

, etc.

___^d£^^

SRAT WORK FOR JUXTOR OLASffES.

(fi) Perff)nning operations indicated by signs, as:
9- 6 = ? 19- 6 = ? 29- 6 = ? 39- 6 = ? etc.
13-7 = ? 23-7 = ? 33-17 = ? 43-17 = ? etc.

75 = 15 + ? 75 = 25 + ? 75 = 45 + ? 75 = G5 + ?

30 = ?
34=?
32 = ?
20 = ?
24 = ?
22 = ?
10=?
14 = ?
12 = ?

125-

5 = ?

15 = ?

35 = ?

06 = ?

75 = ?
105 = ?
115 = ?
125 = ?

. ((l) Problems based on preceding exercises.

(iii) Aiidition and Subtraction.

(a) Adding to and subtracting from a given
numlmr, by twos, threes, fours, fives, etc., as:

50
90

55, ()0, 05, 70, 75, 80, 85, 90, 95, 100.
45, 40, 35, 30, 25, 20, 15, 10, 5, 0.

99, 108, 117, 12G, 135, 144, 153, 162, 171, 180.
81, 72, 63, 54, 45, 36, 27, 18, 9, 0.

When pupils are sutticiently advanced they should be required to
account for difierences, as, 99-81, 108 - 72, 117 - 63, etc.

(b) Performi ng operations iixdicate<l by signs, .'is:

G + G-J= ? 0-3 + 6= ? 6 + 3 = 6-? 6 + 6 = 3-? 9 + 12-6-4=?
9 + 12 = 5 + 4 + ? 9-5 = 12 + 4-? 25 + 30-15 + 10=?
25-15 + 10+30-? 25-15 = 30-10-? 25-15 = 30-10-?

These exercises are of little value as a means of securing facility in
additiiin aiid subtraction. Their pur|M)8C8 arc mainly, (1) to stiuiuhit*-
thought aotivit>f ; (2) to familiarise the pupil with the signs of operation.

HINTS OS TEACHING ARITHMETIC

The former of ihoRe is too often lost Hij^ht of. A proper time-limit should
be set for the performance of all such uxerciscH.

(a) Making tables, as :

2x 1= 2
2x 2= 4
2x 3= 6
2x 4= 8
2x 5 = 10
2x 6=12
2x 7 = 14
2x 8 = 16
2x 9 = 18
2x10 = 20

3x1= 3
3x2= ()
3x3= 9
3x4 = 12
3x5 = 15
3x6 = 18

4x1= 4
4x2= 8
4x3 = 12
4x4 = 16
4x5 = 20

5x1= 5
5x2=10
5x3=15
5x4 = 20

(/>) Multiplying numbers, as :

2
3

2
4

2
5

2
6

2 2
8 9

6 6 6 6 6

6 6

2 3 4

1
4

8 9 10

6x1= 6
6x2 = 12
6x3=18

10 10 10 10 10 10 10 10 10
234^^6789 10

SEAT WOIIK FOR JUNIOR CLASSES.

imit Bhoukl

16 16 16 16 16 16 16 16 16
23456789 10

20 20 20 20 20 20 20 20 20

23466789 10

26 26 26 26 26 26 26 2() 26

23456789 10

666666 6 (> 6

20 30 40 50 (W 70 80 90 100

In such exercises the true idea of multiplication should be kept in
mind, viz., the ripetition of the multiplicand so many times. For that
reason one nn'^ipiicand (unit) should be kept before the pupil for a time.
The exercit»> .. . be varied by changing the mu/dplvr. Tlie pupil
should be rt. * >.>•<'■ to compare products in so far as he is able, and ac-
count for their relation to one another. When the viultiplicaml is
changed the new one should bear to the old a relation which the pupil
can appreciate. For example, 2 might be followed by ^ or ti, 7 by 1 4
or 21, etc.

5x4-?
4x8=2x?

1 = ?

2 = ?

3 = ?

4 = ?

25x 5 = ?

6 = ?

7 = ?

8 = ?

9 = ?

{(i) Problems based on preceding exercises.

4x3 = ?

3x4 =

-l

8x3=? X 6 3x8=

^ 6 X ? 8 X

3 = ?

5 = ?

6 = ?

8 = ?

9--?

9 = ?

10 = ?

11 = ?

HINTS OS TEACHING ARITHMETIC.

(v) Division

(a) Making tables as

60-

-10= 6

1 =

= 72

GO-

- 6 = 10

= 36

60-

- 5 = 12

= 24

- 4 = 16

*?o •

= 18

60-

- 3 = 20

<«i^-

iy.

= 12

60-

- 2 = 30

= 9

60-

- 1 = 60

= 8

12 =

= 6

(b) Dividing

nunil)ci>

I, as :

•

2)12

2)16

2)20

2)24

2)32

4)12

4)16

4)20

4)24

4)32

2)48

2)72

2)1)6

2)144

2)i88

4)48

4)72

4)96

4)144

4)288

6)48

6)72

6;96

6)114

6)288

8)48

8)72

8)96

8U44

8)288

2)108

3)108

4)108

6)108

12)108

(vi) Multiplica.tion and Divisioc.

(i) Exercises as follows :

12 ft. X 3= 18 ft. X? 12 ft. X 4 = 4 ft. X? 12 ft. x 6=3 it. x ?

12x6 = 144-r? 12x6-r3 = 12x? 12-i-6x8 = ? x 2.

100x2-=-50=16x2-f? 100-^2x8=? x2x4.

(6) Problems involving multiplication and di-
vision.

(vii) Iteview exercises, involving the four hiutple
rules. Problems.

SEAT WORK FOn JUNIOR CLASSES.

Seat work may l)e excecMlina^ly profitable, or it may be just
the opposito. This will (h'peiid on the circumstanoes umlrr
which it is carried oti. The following are the principal con-
ditions for successful work :

(i) The exercises should Ix^ properly graded.

(ii) They should be related to one another, so that what is
taught U>-day may be iiscul to-morrow.

(iii) Each exercise should, as a rule, develop <me main i<i^^•l
— a particular combination or relation.

(iv) The exercises should stimulate thought activity. Gener-
ally .speaking, tln^y do not at all compare with class exerrisrs
for securing facility in calculating ; they should, therefore, lie
constructed so as to make the pujiil think.

(v) A proper time-limit nmst be .set for the p(;rformance of
an exerci.se. No more than a reasonable tiuie should ever Ik?
allowed, otherwise sluggi.sh thinking and acting will be the
result.

(vi) Accuracy and nea^^ncss must be insisted upon. On no
account should these points be overloo'-.cd. Hal)its which
will affe<;t the whole life of the i)upil are being formed.

(vii) The pupil's work nui.st be can^fully sup<»rvised. Ex-
perience has j)roved this to be the case. It is better to
allow the pupil to enjoy a gam(5 on the play ground, than to
have him doing a school exercise to which thn teaclu^r cannot
give proper attention, either during the time of its perform-
ance, or shortly afterwards.

In the foregoing, no attempt has been made to exhaust the
different forms of ex(;rcises that will Im? found useful in actual
practice. An efVnrt has been made, iiow(!V«fr, to present
only such types hs an^ nigardrd Hie most vahiabic. Mu<'li of
the stjat work done by primary pupils under the designation of

Ill

HINTS ON TKArillNCJ AKITIIMETIC.

" numb(«r l)usy work," such as ropresenting or copying draw-
ings from tlio hlacklxKinl, into which the element of counting
scarcely cnlcjrs, is comparatively us(^l(?ss except as a means of
keeping pupils employt^d. Now, while the ability to provide
a variety of attractive exercisj's for young pupils is one of the
essential characteristics of the successful primary teacher, we
cannot consider such ability a mark of the highest degree of
excellence unl(?ss it be accompanied ])y such professioiuil know-
ledge as is necessary to determine the educational value of the
exercises given. To furnish " busy woik " for the sole purpose
of preventing pupils from getting ijito mischief, no matter how
necessary it may be, is a tacit confession of weakness. All
sch(X)l exercises should be more o!ian merely interesting or
attractive, they sliould be educativf?. True interest, let it l>e
remembered, is that which grows out of the study of a subject
for its own sake.

Cards containing written exercises will ])e found convenient
— especially wIumi there is lack of blackbo;ird space. They
may be |)repared by the teacher alter school hours, thus saving
time during the day. Kach c;ird should emphasize one main
thought, /.fter the cards are prepare«i they should be num-
bered, indexed, and filed, so as to l>e easily n'tVii'.'d to.

ProV)lems should be related to oth»M- exercises. A few types
are here j)resented. They may suggest one ov two Uiiefui
lines of work.

1. John has S apples an<i James has IS, how many have
both ?

2. 16 pencils is how numy more than 41

3. ]\iary Iniught o |>enciKs, lost 2 an<l gave away 1, how
many has she left f

4. .lohn bouv;ht 10 lbs, of sugar, and Willinrn iMiught 5
niore than John. H<»w mu<'h did ImhIi buv f*

* Fixeti MtaiHlanlH uf ineiuiiin'Uiuiit

ho prominent in problem work.

PRACTICAL SUGGEHTI0N8.

5. AVluit must I add to 20 wiits to havo 3') cents?

G. A l)oy oarns 10 cents every day, liow much will he earn
during the week ?

7. How many gallons in 5 times 12 gallons]

H. How many 3 inches in 12 inches?

9. 3 times $15 contains how many $5 bills?

10. How many dollars in 400 cents?

11. A man earns 75 cents every fon'n<M)n and 50 cents
every afternoon, how much will he tarn during the week?

12. How many 2 square inches in 6 s(|uare inches?

13. ]' many inches in 2 feet?

14. How many (juarts in 8 pints?

15. How many pints in 1 quart 4 pints?

16. 3 tinws 2 pints make how nuiny quarts?

17. H»»w many 3 pints in 3 quarts 3 pints?

18. How much is 2 feet greater than 15 inches?

19. Mark off 1 foot 6 inches into 3-inch lengths.

20. How many 4-inch measures are there in twice 1 f(K)t 6
inches ?

18. Practical Suggestions.

Kkadino Numrers.

(i) l{ead 44, as 4 tens, and 4 onrs; 444 as I hundreds, 4
tens, and 4 ones. C^uestiuns on relation of j)osili()n to value.
Excrcist s.

(ii) Kead 459, as 4 humlretls, 5 tens, and 9 ones. Ques-
tions. Exercises.

fiii) Read 459 in other ways as 4 hundreds, 4 tens and 19
ones; 3 hundreds, 15 tens, and ones, etc. Exercises.

HINTS ON TEACHING ARITHMBTIO.

Writing Numbers.

(i) Write 300, 30, and 3. Add together.
3 hundreds, 3 tens, and 3 unes. Explain,
ercises.

Read the sum as
Questions. £x-

(ii) Write 5 hundreds and 16 ones. Read in different
ways. Exercises.

Addition.

(i) Add hy twos, threes, fours, etc., from 1, 2, 3, 4, T), 6, 7,
8, I).

(ii) Rea<l sums rapidly.

8 8 8
5 2

8 8 8 8

9 10 12 24

(iii) Add (intro<lucing carrying).

m u

m m

■mm *>«•

44 88

110

ti6

.88

1(»5

1(;5

418

(iv) Two column addition. Add the numbers 15, 65, 47,
84, thus : 15, 75, 80, 120, 127, 207, 211. This is an excellent
exercise for senior pupils.

e sum as
18. Ex-

different

. ^ 6, 7,

i5, 47,
iellent

PRACTICAL SUGGESTIONS.

(v) Testing results. '

28+ 20+ :)0+ :u

32+ 3:{+ 34+ 35

3(>+ 37+ :W+ 39

40+ 41+ 42+ 43

44+ 45+ 40+ 47

48+ 4S)+ 60+ 51

228 + 234 + 240 + 240

118
134
160
1(>6
182
198

948

342

986
735
429
834
927

4253

20(>3

2190
4263

(vi) Prove by casting out tlio 9's. This is the inetluxl
generally enjployi;d by accountants.

Much practice should he given in adding columns. L<?t the
pupil add short columns over and over 'lin until he can a«ld
them rapidly. As progress is made tin- length of the columns
should he increase<l. Kach exercise should, g(Mierally, em-
phasize some combination, as 7 + 6, 27 + 0, 47 + 6, etc.

The fundamental combinations of addition should be mas-
tered. They are quite as important as the multiplication
table. Charts may be used with advantage for rapid drill
work.

SUBTKACTION.

(i) Count backwards, commencing at HO, by twos, threes,
fours, etc.

(ii) Read remainders only.

10 18 20 26 28

8 8 8 8 8

;{0 ;)0 48 50

8 8 8 8

(iii) Find the difference by adding to the less between 1000
and 120, 320, 840, 847, 850, 920, 940, 947, 950.

HINTS ON TEACillNO ARITHMETIC.

(iv) Making cliange. Much practice should ho given hy the
ordinary IjusinesH nietliod.

It nmy Ije applied thus :

$10 - HC.Sn = 13. 15. (5 + 5= 10, 9 + 1 = 10, 7 +3= 10 ; or hetter,
85 + 15- im), 7 + 3 =10.)

(;497- 3988 -2501). (8 + 9 = 17, 9 + 0=9, 9 + 5=14, 4 + 2 = 6.)

(v) Steps leading up to formal process.

8 18 18 28 12 22 22 32 52
4 4 14 14 9 9 19 19 39

(vi) Proofs of correctness of work, (a) Adding remainder
and subtrahend. (/>) Casting out the 9's.

It is not \v(;ll to spend much time in trying to g(^t the pupil
to comprehend the principles underlying the formal process
until after he has acijuired a knowledge of the relations of
denominate units. The uae of the process must be thoroughly
understood.

Subtraction should be taught in its relation to addition.

Multiplication.

(i) Exercises leading up to formal process.

600x8 = 4800

60x8= 480

6x8= 48

666
8

480

4800

666
8

4800

480

666
8

6328

5328

PRACTICAL SUGGESTIONS.

nhy the

•r better,
2=6.)

siuaimler

lie pupil

pi'()C«'SS

ktions of
Di-oughly

bion.

|66
8

(ii) III tliu following, coiiqNiro the multiplicuid Hiid the
(liH'ereut pro<luct8. Account for relations of products.

120

300

30x

3= 90

5 = 160

6 = 180
10 = 300
12 = 360
15 = 450
18 = 540

1440

7200

43200

(iii) The parts of the niultiplicution table* up to 144 which
will ie«iuire special drill are 18, 21, 24, 27, 28, 32 36, 42, 48,
54, 50, C3, 72, 84, 9(5, 108, 132.

(Iv) Rejwl the products only. Drill.

250 252 2520 2522
It 12 12 12 12

Drill exercises should not only secure facility in i>erf()rming operations
hut they should also give increased power to relate quantities. Oliarts
will b« found valuable.

<v) Practical Methods. Compare the following:

(a)

765
16

4530
765

12080

43(i5
623

(6)

13095
8730
21825

2282895

755
16

756

12080

4365
523

13095
10039

2282895

* After the multiplication tabic up to 12 x 12 \« learned, nnich pnuHice will be driven
ill finding products up to 20x20.

IMAGE EVALUATION
TEST TARGET {MT-3)

1.0

I.I

1.25

2.8

IIS ""'~

«^ IIIIM

2.0

111=
M. 116

/}.

'9j^/ %^''

Photographic

Sciences
Corporation

23 WEST MAIN STREET
"^ WEBSTER, N.Y. 14580
(716) 872-4503

■n>^

:^^

t<*/

t/u

Hints on teaching ARiTHMETrc.

(c)

{d)

^2.25

448

1800
900
900

$1008.00

^11.28
75

5610
789«

$840.00

^896
$112

$1008

$1128

$ 282

$840

Short methods should not be dealt with until the pupil hcas compre-
hended the ordinary proce.socs fuUj'. At the proper time, however, such
a problem as this will furnish an excellent means of discipline. Find
the cost of 1728 cords of wood at $1.()'2^ per cord. (Short method.)
The pupil must, of course, devise the method himself. The importauce
of the law of commutation \\-\\\ be readily se-.n in this connection.

Division.
(i) Exercises leading up to short division.
0-7-2= 3
00-=-2= 30
600 -r- 2 =300 3)60 3)006 3)078

Find the greatest nurabrr of hundreds, tens, and ones,
exactly divisible bj 4 in 80, 84, 85, 92, 95, 440, 560, 564, 507,
648, 650, etc. Questions and exercises.

(ii) Exercises leading up to long division. (To be performed
by inspection.)

18-M5 = ? 22-f-21 = ?

36h-15=' 25~21 = ?

45 -f 15 = 2 42-r-21 = ?

tllACTICAL StGGEStlONS.

compre-
ver, such
5. Find
method. )
portauce
a.

[ ones,
4, 5G7,

formed

50-M5 = ?

60-M5 = ?

6C-M5-?

90-M5-?
100^15 = ?
J50-^J5=?
ICO H- 15 = ?

484-21 = ?

50-f-21 = ?

84-h21 = ?

90-: 21 = ?
100-^21 = ?
210-:- 21 = ?
220^21 = ?
425-=-21 = ?

Find the greatest number of tens exactly divisible by 25 in
250, 255, 422, 525, 7G0, 840, etc.

Practical methods.

39r))G5923(l(;G
3<>G

(a)

2G32
2373

25G3
237G

39G)G5923(166

2G32

2563

187

(b) Such exercises as these will be found useful :
(o) Divide 1342 by 90 (factors).
(^) " " 21 (factors).

(d) " - 226. '

(e) " " 33J.

Such as the foregoing are suitable for pupils who have made
considerable ad vancenient in the subject.

HINTS ON TEACHING ARITHMETIC.

(1) (2)

1434.227 1434227

1436

434
1

662

1437

808

101

This method cannot be regarded as very practical. It is applicable
only in the case of divisors a little less than a power of 10.

Exercises in factoring and finding multiples should accom-
pany work in multiplication and division. These exercises
will greatly increase the pupil's knowledge of number. They
may take such forms as tiie following :

(a) Find by inspection three factors of 64, 72, 96, 108, 210,
450, etc.

(b) Find three multiples of 15, 20, 25, 75, etc.

If the exercises are well chosen the pupil will soon discover
that the work may be shortened by cancellation.

In teaching the simple rules, care must be taken not to
allow the exercises to degenerate into performing merely me-
chanical operations. For example, the pupil must understand
when he multiplies $10 by 12 he has either combined 12
addends of $10 each or he has found an amount 12 times as
great as $10. The latter conception is the higher, in that it
involves a comparison of the unit $10 with the amount found.
The aim should be to present the simple rules so that the
pupil will gain a true notion of number as expressing the
ratio between the unit and the whole quantity. Until this
stage of thought is reached, multiplication and division cannot

)plicable

accom-

cercises

They

)8, 210,

3x8x

iscover

I not to
ily me-
[rstand
led 12
lues as
that it
Ifound.

it the
lig the
111 this

cannot

PRACTICAL SrGGKSnONS.

moan any tiling more than sliort proviesses of addition and
subtraction respectively — a meaning which carries with it but
little power.

Fractions.*

The term fraction seems to convey different meanings. It
is use 1

(i) To ex})ress a ratio between two quantities, as 5 feet is ^ of
6 feet. Here a direct comparison is made between the quanti-
ties 5 feet and 6 feet, and f expresses the relation between
them, 6 feet being taken as the standard of comparison. It is
this mode of thought which is employed in the solution of the
problem : If 6 pencils cost 12c., what will 5 pen^'ils cost J
when the solution assumes either of the forms

6 pencils cost 12c.,
.•. 1 pencil costs ^ of 12c.,
.•. 5 pencils cost 5 times ^ of 12c.;

or, 6 pencils cost 12q.,

.-, 5 pencils cost g of 12c.

We see this the moment we answer the (juestion, why does
5 pencils cost f of 12c.? Because 5 pencils is f of 6 pencils.

The first form of solution differs from the second only in one
respect — an intermediate step is taken in order that the compari-
son may be more easily made.. It is less dilHcult to establish
a comparison between 1 and 6 and then between 5 and 1, than
to establish it directly between 5 and 6. The point to be
noticed is that in both cases the fractions express ratio. .

(ii) To denote an unperformed process of division and
multiplication. According to this idea 5 feet = | of 6 feet,

* For a complete treatment of fractions see Psychology of Number, by McLellan
and Dewey — an admirable book just published.

1 1 'I'

■4 '

ii-'v

;•■!!

HINTS ON TEACHING ARITHMETIC.

means that if we divide 6 feet into six equal parts and take
five of them, the result will be five feet. The mode of thought
in this case corresponds to that of finding the value of ^ of 1 2c.
in the preceding problem, after the ratio has heen determined
and stated. To find the value of | of 12c., we divide 12c. into
6 equal parts and take 5 of them. This view does not eiiable
one to use a fraction in the solution of a problem, but merely
as a process in evaluating the result.

It is clear that when the pupil can perform the operations of
multiplication and division intelligently he has but little to
learn in order to deal with fractions used in this sense, If
asked to divide 12c. into 6 equal parts and take 5 of them he
will immediately do so, provided that his attention has been
carefully directed to definite units of measurement, this beii g
the necessary condition for the development of the true idea of
fraction. But, it must be remembered, the difficulty of a
fraction does not so much consist in finding the equal parts —
the initial step — as in relating the parts to the whole. The
latter implies establishing a ratio between a unit, or a number
of units, and a whole quantity taken as the standard of com-
parison.

(iii) To denote a kind of concrete unit.* Suppose one j'^ard
to be divided into three equal parts, then we may give
the name foot to each part. 1 part would be designated

1 foot, 2 parts 2 feet, etc. In the same way, if we allow our-
selves to lose sight of the relational idea, we may call eacli
part by the name third. One part would be called 1 -third,

2 parts, 2-thirds, etc. Similarly, if we divide any other
quantity into three equal parts we may give each part the
name third. What is supposed to be a fraction is not a
fraction at all, but a kind of concrete unit which has no fixed
value. Unfortunately, much valuable time is wasted in drilling

* See The Public School Journil, vol. xiv., No. 10.

PRACTICAL SUGOKSTIONS.

d take
tiought
of 12c.
rmined
2c. into
eiiable
merely

bions of
ittle to
ise, If
hern he
as l)eeii
is bell g
5 idea of
by of a
ts—
The
umber
f com-

16 yard

pan

give
ignated
Iw our-
lU each
i -third,
other
trt the
not a
fixed

irilling

small children unable to number 10 objects on fractions of
which they cannot possibly have any proper idea.

It is exceedingly important for the teacher t6 distinguish
from one another the mental processes corresponding to the
different cases given above. It is easy to fall into the error of
assuming that because a pupil can manipulate fractional forms,
which mr.y mean to him, in reality, very little, he is able to
make use of fractions in the solution of problems, or in other
cases where they represent not merely arithmetical operations,
but definite quantity relations.

As there is much difference of opinion regarding the time at
which the teaching of fractions should begin, it may be well to
indicate the ground on which such difference rests. The
real question at issue is, how does the idea of fraction grow-
up in the mind ? Does it arise from merely nwrnhnrhig acts
of attention occasioned by objects, sounds, events, ideas, etc.
(tliere can scaicely be any doubt that the idea of number — the
liow 'ma7i}j — may be developed by any means that will occasion
successive acts of attention), or from numbeiing definite quan-
titative units ] In other words, is the fundamental idea that
of time only, or does it involve both space and time ? If num-
bering units, which are not quantitatively defined, can give
rise to the idea of fraction, why should it be considered im-
portant in teaching fractions to deal with units which are
quantitatively e(i[ual to one another 1 On the other hand, if
both number and definite quantity are included, why should
fractions be inti-oduced until some prOj^ress is made in number-
ing measured units ?

It is sometimes urged by those who favour teaching frac-
tions from the beginning of the course in number that young
children use the language of fractions intelligently in ordinary
conversation. For example, a child may say that 50 cents is
half a dollar. Certainly, there is nothing to prevent a five

HINTS ON TEACHING ARITHMETIC.

year old boy from rememl)eriiig two names for one thing. Ho
will also remember that a dime is one-tenth of a dollar or one-
fifth of 50 cents if t(ild frecjuently enough. He may learn
further that one-fifth is equal to two-tenths! H this is taught
by means of objects he may even "catch the trick" of illus-
trating it after a fashion. Surely it is perfectly plain that
all of this may be done and much more without involving a
true conception of a fractional unit as related to a prime unit.

The teaching of fractions usually presents considerable diffi-
culty. The cause of this may generally be traced back to one
or more of the following :

(i) Attempting to deal with fractions at too early a stage.

(ii) Regarding fractions as indicating merely processes of
multiplication and division.

(iii) Losing sight of the prime unit from which the frac-
tional unit derives its meaning. This is closely allied to the last.

(iv) Introducing symbols before their meaning is fully de-
veloped.

(v) Showing the pupil how to perf mn operations with frac-
tions instead of allowing him to discover rules and methods
for himself.

Preliminary Work.

In teaching fractions the first step is to develop a proper
notion of what a fraction is. This may be done by dividing
strips of paper, lines, etc., into equal parts, and by dealing
with denominate units. Compaie a line 1 foot long with
another 2 feet long ; a line 2 feet long with another 4 feet
long. Compare 5c. and 25c., 10c. and lOOc, etc. It makes
no difference whether the pupil answers that 25c. is 5 times
5c. or that 5c. is | of 2oc., provided that a comparison of
the quantities has actually been made. The idea of fraction
is implied in both, although expressed only by the latter.

PRACTICAL SUOCIKSTIONS.

with
feet
akes
mes
of
tion

Tho order of steps indicated by tlie foregoing may be more fully
statrd thus: (i) Separating the whole into parts (roughly) and /n/m/jfr-
iiKj the parts, (ii) Dividing the whole into equal parts and numbering
the parts, (iii) Comparing one or more of the efjual parts with the
whole ; also, the whole with one or more of the ((jual parts. When the
third stage is reached whole numbers are found insutHcicnt to define all
of the ratios which may be established ; hence the necessity for frac-
tional numbers. It will be seen that the very first mental effort made
in numbering a quantity involves the idea of fraction as existing, not
actually, but polentuUly. As the mind develops it demands a greater
degree of accuracy for the satisfaction of its wants. The unit of mea-
surement is made definite, and the necessary condition for the realization
of the idea which at first existed only in possibility is afforded. As
soon as fractions are dealt with in relation to the whole subject of
number, many of the difficulties which are now experienced in teaching
tiiem will vanish. Work in fractions will not be prematurely forced
upon the pupil, but it will be deferred until such time as the require-
ments of mental growth necessitate the comparison of (iuantitiea on the
basis of the well-defined unit.

The second step is to give such exercises as will dcn^elop in the
mind of the pupil the principles underlying all fractional work.

These principles may be stated as follows :

(i) Multiplying or dividing the numerator of a fraction by a
number multiplies or divides the fraction by the number.

(ii) Multiplying or dividing the denominator of a fraction
by a number divides or multiplies the fraction by the number.

(iii) Multiplying or dividing both terms of a fraction by a
number does not alter the value of the fraction.

The method of dealing with principle i. may be illustrated
thus : Compare \ oi a, foot and | of a foot ; Jj of 80 and ^ of
$0 ; ^ of $14 and ^ of $14 ; 5 of a pint and ^ of a pint, etc.

Compare f and | ; | and | ; f and |, etc.*

There should be no haste in getting the pupil to make a
formal statement of the principle. The aim should ])e to give

* At first the pupil should be required to supply concrete units where no unit is given.

HINTS ON TEACHING ARITHMETIC.

him power to make use of fractions, not simply to teach
him how to perform operations. Tlie power gained will be
proportionate to the amount of well-directed, independent
thinking which the pupil does in connection with exercises in
comparing fractional quantities.

Formal Study.

The formal study of the subject may begin by giving
the pupil practice in changing the form of a fraction without
altering its value. No new principle is involved in this. To
change fourths to twelfths is quite as easy as to change yards
to feet, provided the relation between one-fourth and one-
twelfth is as weir known as that between one yard and one
foot. Here, again, we see the necessity for exercises in com-
parison.

The reduction of fractions may take several forms of which
the following may be regarded as types : —

(i) Find the number of fourths of a dollar in $1|.

(ii) Find the number of dollars in 1 fourths of a dollar.

(iii) Find the number of fourths of a dollar in $|%.

(iv) Express $f§ in its lowest terms. '

(v) Express %\ and %% as fractions having a common de-
nominator.

The exercises should be performed chiefly by the method of
inspection. Viewed from either the standpoint of discipline oi'
practical utility, it is well to give ample practice in easy frac-
tions before proceeding to those involving large numbers.

The next step is to teach the application of the fundamental
rules to fractions.

In introducing addition and subtraction, exercises should
be given as : Find the value of %\ -f ij, IJ + Ii? i V^^^
4- I pint, %\ - $iV, i yd. - i yd., etc.

PRACTICAL SUGGESTIONS.

teaeli
all be
Mulent
ises in

giving
rithout
is. To
! yards
id one-
,nd one
in coni-

liicli

ollar.

kon

de-

thod of
lline or
ly frac-

[nental

should
pint

The pupil should V)e requirechto point out the resemblance
which adding and subtracting fractional units bears to cor-
responding operations which ho has aln^ady performed with
other units. He should frecjuently illustrate his work by means
of objects. The representation of lines on the blackboard will
be found both convenient and useful.

Multiplication and division present to the pupil greater
dilUculty than addition and subtraction. This is chietly owing
to the fact that the former involve a two-fold ratio.

(i) The ratio expressed by the fractional unit.

(ii) The ratio implied in the operation to be performed.

It is true that $^ x 3 may mean nothing more to the pupil
than $1 -j- $^ + $1 = $|, which expresses directly but one ratio,
viz. : that which the fractional unit $^ bears to the prime unit
$1. But with oidy the thought of a single ratio in mind S;^ x |
must be meaningless. As soon, however, as the pupil grasps
the idea that in S| x 3 = $|, 3 expresses the ratio between $\
and $|, he is in a position to comprehend the meaning of

$i X J - $A.

In teaching multiplication and division of fractions much
practice should be given in comparing quantities. Compare
$J and $2, $J and $f, J gal. and 5 gals., J lb. and 2 J lbs, etc.

By what must we multiply J?| to get $2 1 How is this
expressed 1 What part of 5 gals, is ^ gal. ? Express this on
blackboard.

The steps in teaching multiplication of fractions may be
indicated as follows :

(i) Multiply a fraction by a whole number, as : $ j x 2,
pt. X 4, ^ X 2, j§ X 5, etc.

HINTS ON TEACIIINO AIUTIIMETIC.

(ii) Multiply a w1h»1(3 number by a fraction, as : $6 x ^, J?()
X I, 8 gals. X i, 10 X g, etc.

(iii) Multiply a fraction by a fraction^ as : SJ* x \, S'f x J,

The steps in teaching division are as follows :

(i) Divide a whole number by a fraction, as : $2 -r $h, ') gals,
-r 'i gal., 5 -f 3, etc.

(ii) Divide a fraction by a whole number, as : .M -^ ^, ^$1 -^ 1 6,

5 JL •)r) »>t,o

(iii) Divide a fraction by a fraction, as : '^\ -f $|, $^ -r g,

A full interpretation of such an expi-ession as $2-f8^ = 4,
generally pi'esents difficulty to the beginner. What are the
lucanings which it conveys 1 (The pupil's answer to this will
depend altogether on his power of interpretation.)

(a) $^ may be taken from $2 f.)ur times, or to express the
same idea in other words, $2 contains 4 half-dollars.

{b) $2 is made up of 4 equal parts of $^ each.

These represent different stages in relating. In (a) the parts
(units) are numbered^ in (6) the equality of the parts (units)
is considered, and in (c) the ratio between the unit, $^, and
the whole, $2, is stated.

Both {a) and (6) precede (c) in the order of thought. With-
out the idea of equal parts the comparison implied in (<?) could
not be made, and without taking cognizance of the number of
parts, the ratio could not be defined. Hence (c) includes (a)
and {b) as elements. Compare what has already been said
with reference to the development of ratio.

PRACTICAL SUOGEBTIONS.

parts

[units)

\j and

With-

I could

)er of

js(a)

said

It will 1)0 found valuable to write occasionally an expression
on the l)laiklK)Hr(l, and get the pupils to interpret it as fully as
they can. What d(M's $2 represent in the foregoing ? What
does $^ represent? What dors 4 show? What moaning is
implied by not definitely expressed ? Express clearly this im-
plied meaning, etc.

Important Points.

(i) Lay stress on the efpiality of parts. Objects should be
used freely at first.

(ii) Oive many exercises in comparing denominate numbers.
There should be much practical measurement.

(iii) Requii-e the pupil to use each new fact as it is learnt.
By this means it will become to him a source of power.

(iv) Let thought always precede expression. Figures should
not be introduced too early.

(v) Require clear statements, both oral and written.

(vi) Get the pupil to illustrate his statements by means of
objects, lines on blackboard, etc.

(vii) Ask the pupil frequently to interpret the written state-
ments made by him.

(viii) Require the pupil to perform exercises in the shortest
and most practical manner. For example, in adding 3^, 18|,
42|, let the whole numbers and the fractions be added
separately.

(ix) Give plenty of mental work in the form of practical
problems.

Decimals.

(i) Introduce decimals by getting pupils to point out the
effect of change of position on tlui value of digits, as 6, 66, 666,
6666, 8, 88, 8888, etc. Apply to 66/6, 66/66, 88/88, etc. (The

HINTS OM TEACHING ARITHMETIC.

advantage of repeating the same digit will be seen at once.)
These numbers should l>e expressed as 6 tens, 6 ones, 6 tenths,
6 hundredths, etc. Change to usual form as 66.66, etc.

(ii) Oral exercises.

(a) Read 3.5, 33.25, 423.625, etc.

(6) Express by figures 3 tens, 4 ones, and 5 tenths, etc.

(d) Compare .2 and .4, .2 and .6, .2 and .8, .4 and .8,
etc.

(iii) Reduction of decimal to vulgar fractions and vice versa.

(a) Represent a line 5 feet long on the blackboard.
Divide it into foot lengths and bisect each of them.
Give the class such work as : Point out ^ of the line.
Express this as a decimal. Point out .25 of it, .6 of
it, etc. Express these as vulgar fractions.

(b) Divide the line into inches. How many inches make

^ of the line ? Express as a decimal, etc. Com-
parison. A great vruiety of valuable exercises may
be given in this way. Tlieir difficulty, too, may be
increased to any degree by taking lines differing in
length. In this work the idea of comparison should
be made prominent.

(c) Represent a line 2^ feet long. On it describe a
square. Divide the square into 25 equal squares.
Give work similar to the foregoing. Divide the
square into equal parts in other ways and proceed as
before.

PRACTICAL SUGGESTIONS.

once.)
.enths,

etc.
E .25 of

and .8,

ze versa.

jkboard.
if them,
he line,
it, .6 of

js make
Com-
ses may
I may be
jring in
should

Icribe a
squares,
tde the
Lceed as

{d) Write oa blackboard such fractions as f , f, f, etc.,
and have pupils give their equivalents as decimals,
etc. In case there should be doubt as to whether or
not the pupil has thought out any relation which he
has expressed, have him illustrate it objectively.

Addition and subtraction of decimal fractions are eatdly
understood. Multiplication and division usually present diffi-
culty in one particular, viz., finding the position of the decimal
point in the product or the quotient as the case may be.
Such exercises as these will direct attention to the main
thing to be taught.

(a) Write on blackboard the expressions 5.5, 55.55, 12.335,
183.932, etc. Change the position of the decimal point.
How is the value of the number affected, etc. %

(b) Multiply 642.937 by 10; by 100; by 1000. Divide
the same by 10, etc. Give a sufficient number of exercises to
familiarize the pupil with multiplying and dividing by powers
of 10.

(d) Multiply 125 by 4, .4", .04, .004, 4.4, 4.44, etc. Com-
pare products, and account for their relations to one another.

(e) Multiply 12.5 by 4, .4, .04, .004, 4.4, etc. Exercises.

(f) Divide 44 by 20, 2, .2, .02, .002. Compare, and account
for relations of quotients.

(g) Divide 4.4 by 2, 20, 200, 22, 2.2, .22, .022, etc. Com-
parison of quotients.

A large number of easy exercises related to one another
should be given. The teacher must not depend entirely on

100

HINTS ON TKACHINO ARITHMKTIC.

the textbook for these, because a textbook, no mfttter how
much merit it may possess, cannot, in some cases at least,
meet the wants of the individual.

Tt is important that the pupil should at first read deci-
mals so as to bring out their full meaning. For in stance,
72.3 15 should be read as seven tens, two ones, three tenths,
four hundredths and five thousandths. After decimals are
well known a briefer form of expression will be adopted, as
seventy-two and three hundred and forty-five thousandths.

A common error in the school-room is to read such an ex-
pression as 7.125 as seven decimal one hundred and twenty-
five. This must be guarded against.

Percentage.

It is often considered necessary to give a series of lessons
leading up to percentage as if it were an entirely new division
of arithmetic. Such is a waste of time, for the pupil who has
a proper knowledge of fractions has really nothing new to
learn except the meaning of the term; 3% means y§^ or .03,
both of which are well known to the pupil. In fact no such
phase of the subject would appear in our textbooks but for the
fact that it is found convenient to adopt 100 as a standard on
which to base certain arithmetical calculations. Percentage
should therefore be taught in its relation to fractions, of which
it forms a particular ease.

The steps in teaching percentage may be taken as follows :

(i) Meaning of the term, per cent.

(a) Statement.

(6) Easy exercises to develop meaning of term, as what
is 3% of $100] Of $2001 Of $400^ Of 1000
bushels, etc.

PRACTICAL SUGGESTIONS.

how
least,

deci-
tance,
enths,
Is are
ed, as
lis.

an ex-
wenty-

lessons
ivision
ho has
ew to
[or .03,
lo such
'or the
,rd on
intage
which

lows:

what
1000

(ii) Relation to decimals.

(«) Express 5%, 10%, 15%, 20%, 25%, 50%, etc., as deci-
nials. Problems.

(b) Express .5, .55, .525, etc., as percentages. Problems.

(iii) Relation to vulgar fractions.

(a) Express 10%, 20%, 25%, 30%, 50%, etc., as vulgar
fractions in their lowest terms. Problems.

(6) Express yV. ^o, zuj h h h Ih etc., as percentages.
Problems.

The pupil should become thoroughly familiar with relations commonly
used, as ^ = .5 = 50%. By means of oral exercises, such as find tlie value
of I of $88, find . 125 of $648, find 33i% of $900, etc., the practical ad-
vantage of what is liere stated may be shown. The fractions most im-
portant in this respect are i, ^, J, i, J, J, ^V. uV» tsV. »"<! tV-

.(iv) General application. Types of problems.

(a) How much is 5% of $200? Of $50? Of $40'? Of
, 90 gals.? Of 110? Of 420?

(h) Of what is each of the following 6%, $G0, $72, 96
feet, 75 cords, 1200 bushels?

(d) What sum increased by 5% of itself amounts to
$105? $420? $1050?

(e) What sum diminished by 6% of itself equals $94?
$470 ? $658 ? •

(v) Particular applications.

The chief difficulty that will be experienced in dealing
with the applications of percentage to commercial transactions
will be in comprehending the transactions themselves. For

102

HINTS OU TEACHING ARITHMETIC.

instanco, if the pupil hsis not a clear idea of what Bank Dis-
count is he may not be able to solve inteUigently the simplest
problem on it. Lessons should, therefore, be given on the
business side of such subjects as interest, insurance, taxes,
commission, banking, exchange, etc., in so far as such can be
given in the school-room. The main thing, however, is for the
pupil to get a firm grasp of the general principles underlying
all of these. This being the case, he will be in a good position
to deal with such subjects as the occasion arises in actual life.

Simple Interest and Bank Discount, on account of their
practical importance, should receive special attention. Before
arithmetical lessons are given on these the pupil must under-
stand the nature of the transactions involved. To attempt
to te?ach Bank Discount to pupils who have never seen a note,
or who have no idea of what discounting a note means, is sheer
folly. The pupils are simply working in the dark. But when
the proper conditions are fulfilled the subjects here men-
tioned present almost no difficulty to anyone who has a good
knowledge of the general application of percentage. The only
new element in^'olved in them is that of time.

Bank Discount. (

The steps in teaching Bank Discount may be somewhat as
follows :

(i) Let teacher and pupils have a short talk about lending
money. As interest has already been taught, this will be
familiar.

(ii) Discuss conditions undter which money is lent.

(iii) Suggest a transaction and request pupils to write a
promise to pay according to the given conditions.

(iv) Show them a promissory note in proper form. Ques-
tion pupils on its chief conditions until they are understood.

PRACTICAL SUGGESTIONS.

103

only

ill be

(v) Have a talk about buying and selling notes. Let the
teacher present a note drawn in his favour, say, for $100 due 6
months hence. What will it sell for if money is worth 6%?
Exercises.

(vi) Introduce notes bearing interest in a similar manner.
Exercises.

(vii) Find the face of a note when the selling price is given.
The pupil should be able to work this out for himself. It is
well, at first, to ask him to represent, by means of diagrams,
the face of the note, the amount, the selling price, and the
bank discount taken off by the purchaser.

The regular business method should be followed.

Compound Interest.

To the pupil who understands simple interest compound inter-
est presents only one new idea, viz., that interest instead of
being paid over to the lender when it becomes due may be retain-
ed by the borrower from perio^l to period for a specified time at
the same rate of interest as the original principal. As soon as
the pupil understands this condition he should be given exer-
cises to work out. No form of solution should be presented,
neither should any explanations be offered, until the pupil has
done all he can independently. The following will illustrate a
method of dealing with the subject.

Find the compound interest on $500 for 2 years at 6%
payable annually.

Solution :

Principal for 1st year =$500

.06

Interest

=$ 30.00
$500

$530

104

HINTS ON TEACHING ARITHMETIC.

Principal for 2nd year , =$530

.06

Interest

=$ 31.80
$530

Amount at. the end of 2 year.s=$561.80

$500

Interest for two years

=$ 61.80

No objection can be taken to this form of solution as a means
of giving the beginner a clear notion of what compound interest
means and of how it may be calculated ; but it is evident that
after a few exercises are worked out the pupil cannot gain
anything more except, perhaps, facility in additicm and multi-
plication. But a great deal is still to be done if he is ever to
master compound interest fully. He must consider such ques-
tions as these : What is the relation between $500 and $30 ?
Account for this relation. Compare $530 and $500. Explain
this. Compare$530 and $31.80. Compare $561.80 and $530.
"Why is this relation the same as that between $530 and $500'?
What must $500 be multiplied by to give $561.80? What
relation does this bear to that between $500 and $530 1 etc.
By this means the pupil will not only get excellent practice
in comparing quantities, but he will arrive at a short method
or finding compound interest, thus :

The amount = $500 x (1.06)2 = $561.80.
The interest = $561.80- $500 = $ 61.80.

To be sure these statements might have been formed directly,
but in that case they could not possibly me.an as much to the
pupil as they do after he has deduced them in the manner here
indicated.

After the pupil has done this work he ought to be in a
position to solve arithmetically such a problem as : The prin-

PRACTICAL SUGGESTIONS.

105

cipal is $2000, the compound interest for 2 years is $420; find
the rate.

Solution :
$2420 1.21

lectly,

to the

here

in a
Iprin-

Ratio when time is 2 years.

$2000 1 .

.'. Ratio of amount for 1 ye.ir to I*rincii>iil^- j/ |— ^1.1

. • . Rate per unit = . 1

. '. Rate per cent = 10.

Of course it will be seen that these exercises are suitable
only for advanced pupils, for they demand considerable thijuj^ht
power.

Square Root.

During the early part of the course .there should be given
in connection with the work on factors and multiples, exercises
on squaring small numbers, also on finding the square root of
such numbers as 25, 36, 49, 81, 121, 144, etc.

Before the formal process is taught the pupil should have a
knowledge of terms and symbols ; he should also know some-
thing of the application of square root in connection with
such problems as these :

(i) The area of a square is 1 6 square feet ; find its side.

(ii) The area of a rectangle whose length is twice its breadth
is 288 square yards ; find its length and breadth.

(iii) The length of a rectangular piece of ground is to its
breadth as 5 : 3, its area is 2160 square rods; find its length
and breadth.

106

HINTS ON TKACHING ARITHMETIC.

Constructive exercises will make clear how the square of a
numl)er is made up. These may be carried on as follows :
Describe on the blackboard a square whose side is 24 iiiciies,
thus : .

24 in.

80 sq. in.
20 in.

400 Bq. in.

16 sq. in.
4 in.

80 sq. in.

Area of square AC = area of square EH

4-area of rect. AK /

-j-area of rect. CK
+ area of square GF '

= 400 sq. in. + 80 sq. in. x 2-f- 16 sq. in.
4 = 576 sq. in. '

After the pupil has worked out a few problems of this kind
his attention will be directed to the relation between the
number of units of length and the number of units of surface.
He will thus perceive that 24^ = 202+20x4x2+42. Great
care must be taken here, or there may be confusion in the
mind of the pupil with regard to the different kinds of units.

PRACTICAL SUOOESTION8.

107

of a
ows :
clies,

sq. in.

kind
the

rface.

Ijlreat
the

dts.

Simple exercises requiring the application of what lias heen
learnt will follciw, as : find the square of 15, 16, 18, 21, 2.'i, 25,
27, 31, 32, 40, 41, 42, etc.

The next step is to present such numbers as can easily he
separated into squares by inspection, as :

121 = 10-i + 10 X 1 X 2 + 1'^ /. |/121 = 11

144 = 102 4-10x2x2 + 22 .-.1/144 = 12
225 = 102 + 10x5x2 + 62 .-.1/226 = 15
400 = 202 .-.]/ 400 = 20

484 = 202 + 20x2x2 + 2- .-. i/ 484 = 22

The formal process should then be taught by applying it to
numbers whicli have ali-eady htton dealt with by the method
of inspection.

102 = 100

10x0x2 = 120

62= 36

256

102

20x6-

6- =

256(10+6
100

166
120

36
36

256(16
1

156
156

Review Exercises.

Among all the words found in the teacher's vocabulary none
is more important than the word Review. Reviewing a sub-
ject does not mean merely repeating exercises previously
performed. It includes far more than this, and serves a much
iiigher purpose. When a particular section or division of a
subject is first brought under consideration, attention must
necessarily be confined to somewhat narrow limits, otherwise
little progress could be made. The new, in order to be ap-
perceived at all, must be related to the old, it is true, but we

108

IIFNTS ON TEACH INO ARITHMETIC.

must remomb«;r tlwit tlu; points of contact will be few, and the
bonds of union will be w(!ak. To pass on to the next division
of the subject, and then to the next, and so forth in rapid suc-
cession would, under such conditions, result in the ac(|uisition
of a mass of uni'(;lated or imperfectly related knowledge which
would be of almost no value to its possessor. Knowledge gives
power only when it is well organized.

The chief value of review work in arithmetic consists in
givin,:^ tlu; pupil a connected view of the subject in so far as he
has stadied it. Suppose, for example, that he has finished a
series of exercises on percentage, he is now in a better position
than ever before to comprehend the whole subject of fractions.
He can see the advantage of having different fractional forms
and, conseciuently, he can avail himself of this in their appli-
cation. The pupil will take the greatest pleasure in such
review lessons, for they are the means of representing to him
old ideas in new relations. -

Review exercises should consist chiefly of problems involving
principles, and also facts which should be remembered, as
tables, etc. Even in the regular daily exercises much time may
be gained by requiring the pupil to recall incidentally such facts
of arithmetic as are of practical importance in everyday life.

and the
division
pid suc-
luisition
e which
ie gives

}ists in
ir as he
ished a
)osition
ictions.
forms
appli-
n such
to him

' /

olving
ed, as
le may
I facts
y hfe.