Daniel B[arclay] 1861- Williams.

Science, art, and methods of teaching; containing lectures on the science, art, and methods of education online

. (page 6 of 9)
Online LibraryDaniel B[arclay] 1861- WilliamsScience, art, and methods of teaching; containing lectures on the science, art, and methods of education → online text (page 6 of 9)
Font size
QR-code for this ebook

The different words found in the different readers
of the child should be spelled phonetically as well as
by naming their letters. It is excellent practice to
cause the pupil to first spell a word by naming its
letters and then to spell it by phonics and call his
attention to the real difference between the spellings.
I have had a school of more than fifty children aver-
aging eleven years of age to obtain by this'^method
from 95 to 98 per cent in an oral and written exami-

The importance of teaching phonics may be seen
from the fact that a correct knowledge of the ele-
mentary sounds of the different letters is the basis of
all correct reading and spelling. Deficiency in pro-
nunciation and articulation arises, in great measure,
from an indefinite knowledge of Phonology.



What is Niimher f — - What Cun he Done With it f — How
Must it he Taught f — The First Yearns Worlc — The Second
Yearh Work — The Third Yearns Worlc may Begin with
the First Steps in Addition, Suhtraction, and MuUifplication
— The Numhers from Thirty to Fifty — The Second Steps
in Addition, Subtractio7i, and Multiplication — After Sixty,
Teach Time, Dry Measure, and Long Measure — From Sev-
enty to One Hundred — The Third Step in Multiplication —
The Steps in Division — Other Suhjects of Arithmetic.

"Number is the useful thing,
Which man must know whatever comes ;

For peace and life do often hinge
Upon the way he counts his sums." — The Author.

In teaching Arithmetic, we first aim to impart to
the child a correct idea of number. The question
may be asked, What is number ? Number definitely
limits objects of the same kind to how many. It is
very important to know what can be done with num-
ber. Here are several numbers : the question natu-
rally arises, In what relations can they be seen ? I
can unite them into one number and separate this
number into other numbers. Every operation in
Arithmetic consists of one or both of these simple
processes, — uniting and separating.


Uniting numbers is addition ; uniting equal num-
bers is multiplication. Separating numbers is sub-
traction; separating numbers equally is division.
These are the four fundamental operations in Arith-
metic, and it should be remembered that the applica-
tion of these simple relations enter into the various
operations of it. The instructor should ever hold in
his mind the reasons for teaching number. There
are two important motives for giving instruction in
Arithmetic : first, it trains the intellectual faculties
to calculate with exactitude and rapidity, and develops
the power to reason logically ; secondly, it assists us
in the practical affairs of life.

Let us now consider the question, How must num-
ber be taught? I do not deem it necessary to ex-
plain the old method of teaching beginners Arith-
metic ; for almost every teacher is well acquainted
with it. I shall simply endeavor to present a method
which experience has proved to be superior to it,
and which possesses all its excellencies with the best
features of the "Quincy Methods." As no good
teacher would think of teaching Geography without
maps, Botany without j)lants, or Physiology without
charts or bones, so no skilful instructor would attempt
to teach number without numbers of objects. He
should supply himself with sticks, peas, beans, blocks,
spools, pebbles, shells, buttons, measures, cups, and
other objects.

The first step in teaching number consists in ascer-
taining, by careful examination, just what the pupil
knows. This may be easily done in the following
manner. Hold up two objects and say, "Bring me
so many." If this is done, hold up three objects and


say, "Bring- me three blocks." Then present three
beans and ask, "How many?" Give questions such
as this. I take one stick from two sticks ; how many
sticks are left ? A child may count up to a number,
and not have a clear knowledge of it. Hence, his
ability to count should not be taken as a test of his
knowledge of numbers. From repeated tests given
by myself to a large number of children, I discover-
ed that the average child of five or six years of age
does not know beyond the number two when he en-
ters the school-room for the first time. In most in-
stances, it is best to begin with three in teaching"

There are two important points to be observed in
teaching every number. First, we should teach the
number as a whole, and, secondly, the addition, sub-
traction, multiplication, and division facts of it. I
shall now take the number three to illustrate the best
method of teaching number.

1. Three as a Whole. Pick me out two spools, and
put one more with them. How many are there ?
Show me just as many sticks, beans, desks, boys,
girls, trees.

2. The Addition, Subtraction, Multiplication, and
Division Facts of Three. Teach two and one thus.
Show me two sticks. Show me one more stick. How
many sticks have you shown me ? Cause the child
to do likewise with the other objects. Then tell them
such incidents as the following. Nellie had two birds,
and bought one more ; how many did she then have ?
Teach one and two thus. Hand me one pea. Hand
me two more peas. How many peas have you hand-
ed me?


Now teach the subtraction facts. Teach three mi-
nus one thus. Hand me three beans. Take one away.
How many are left ? Do likewise with other objects.
Willie had three apples and ate one ; how many
were left ? Teach three minus two thus. Take three
pieces of chalk. Give me two of them. How many
have you left? Sarah had three dolls, and g-ave away
two ; how many did she have left ? Teach likewise
three minus three.

Now teach the division and multiplication facts as
follows. Teach three divided by one thus. Take
three pencils. How many one-pencils can you find
in them ? Mary had three apples, and g-ave one each
to some girls ; to how many g-irls did she give them ?
Teach three divided by three thus. Take three ap-
ples. Hand them to three boys. How many does
each receive ? In three there is but one multiplica-
tion fact which may be taught thus. Here are three
pens with a pig in each pen. How many pigs are
there? One boy has one nose; how many noses
have three boys ? In teaching this number, a larger
assortment of practical examples should be used than
I have presented. The figure three may be easily
taught after the different facts of the number three
are clearly understood.

When the pupil thoroughly understands the num-
ber three, he should learn the other numbers in reg-
ular order in the same manner. Each number as a
whole and the addition, subtraction, multiplication,
and division facts of it should be systemetically in-
culcated. The different facts of any number from
three to one hundred may be very easily found. In
selecting the addition facts, find what numbers will


give you the number which you are teaching. For
instance, I know that two and two, one and three,
and three and one make four. The subtraction facts
may be readily picked out by remembering that the
number and all lower numbers can be taken from the
number itself. Thus, we observe that the subtraction
facts of six are six minus six, six minus five, six minus
four, six minus three, six minus two, and six minus
one. In presenting the multiplication facts, find
what two numbers multiplied together will produce
the number taught. Thus, we note that the multipli-
cation facts of eight are eight ones, two fours, four
twos ; we sometimes say eight times one, two times
four, and four times two. In teaching the division
facts, find what numbers will exactly divide the num-
ber taught. In each division, the divisor will be one
number, and the quotient will be the other. For in-
stance, we perceive that one, two, and five exactly
divide ten. Hence the division facts of ten are
ten divided by one, ten divided by two, and ten di-
vided by five. There is no necessity to divide the
number by itself.

As soon as the pupil has learned the number ten
the fractions one-half, one-third, and one-fourth may^
be taught. The following method is a good one for
teaching the fraction one-half. Take a stick, and
divide it into two equal parts. How much of the
stick is each part? Some child will probably say,
"One-half." Do likewise with a number of objects.
Show me one-half of two buttons, one-half of four
buttons, one-half of six buttons, one-half of eight but-
tons, one-half of ten buttons. In this manner teach
one-third and one-fourth. Some teachers object to


the teaching- of the fractions one-half, one-third, and
others before the child understands numbers well,
and knows addition, subtraction, multiplication, and
division. They claim that the child is made to at-
l;empt too much at once. On this point, I simply-
state that fractions do not present much difficulty
when they accompany the teaching of whole num-
bers from the beginning*.

In presenting the number eleven, ten sticks should
be tied in a bundle, and the attention of the child
should be called to this fact. With twelve, the signs
of equality -^ , plus I , and minus — should be learn-
ed by the child. The signs may be given by the fol-
lowing method. Who will supply the word that is
necessary to complete this sentence, 5 and 3 . . . 8 ?
Some pupil will probably say, "Equal." Then im-
press him with the fact that the sign = means equal.
Who will give the word that is necessary to complete
this sentence, 8 . . . 2 - 10? Some one will say
"And." Then let him understand that the sign f
means and. Who will tell the word that will fill out
this sentence, 9 ... 7-2? Some one will, in all
probability, say, "Less " Impress the child with the
fact that the sign — means less or minus. Introduce
a large number of exercises, such as the following-,
and cause the pupil to supply the necessary signs and
numbers: 7+3= . . ; 9— 6=- . . . ; 7 . . . 3=4;
10 . . . 7=3; 12—10 ...

The question is often asked. How much should the
child attempt to learn during the first year ? Accord-
ing to the testimony of those who are best acquainted
with this method, the pupil, during the first school
year, can really learn between ten and fifteen. I am


convinced that the teacher can, by careful training,
enable the child to master at least twelve in that time.
Some think that the attempt to teach four operations
at once confuses the pupil ; but experience shows that
the child readily grasps them, when they are proper-'
ly and systematically taught. As I have before inti-
mated, the figure which represents a number may be
taught as soon as the nuruber itself is thoroughly mas-
tered. As far as my experience is concerned, 1 see
no serious objection to teaching figures along with
numbers. Hence the child may, during the first year,
commence slate and blackboard work.

The second year's work may be begun with thir-
teen, if the first year's teaching ended with twelve.
During the first year, the teacher should not attempt
to teach beyond the number fifteen, and the second
year's instruction in number should not commence
with any number beyond sixteen. The signs of
multiplication and division may be given with fifteen
in the same manner as the others. After the child
learns nineteen, tens should be inculcated as prepara-
tory to the teaching of twenty . One hundred sticks
should be put into bundles of ten each. The pupil
should be so thoroughly drilled that he can readily
tell how many there are in two tens, three tens, four
tens, five tens, six tens, seven tens, eight tens, nine
tens, and ten tens. Practical examples and slate
work should be given. After the number thirty has
been learned, the fractions one- fifth, and one -sixth,
may be taught as one-half was. Liquid Measure may
be also taught at this point. To do this with efficien-
cy, the instructor should supply himself with gill,


pint, half -pint, quart, and gallon measures. Dry
Measnre may be then taught.

The first step in addition, when the sum of the
units does not exceed nine, may be taught after thirty
thus. Three boys had some marbles; one had thir-
teen, another, twenty-four, and the other, thirty-two.
Let some pupil show by the sticks or some other ob-
jects what the ^um is. He picks out the bundle of
ten sticks and three separate sticks for thirteen, two
bundles of ten sticks and four separate ones for twen-
ty-four, and three bundles of ten sticks and two sepa-
rate ones for thirty -two. He then counts nine single
ones and six tens, which make sixty -nine. Give a num-
ber of examples such as the foregoing, and cause the
child to show their sum by the objects. When he
can readily do this, the examples may be transfer-
red to the board and slate, and added in the usual

The first step in subtraction, when the number in
each order of the minuend is greater than the corre-
sponding number in the subtrahend, should be taught
in the following manner. I have fifty -seven chickens,
and sell twenty -four. Let some child show from the
objects how many are left. He removes from the five
tens and seven single sticks two tens and four single
sticks, and he at once sees three tens and three single
sticks remaining. Give several like examples, and
when the pupil is familiar with their solution by
means of the sticks, the same or like examples may be
transferred to the board and slate, and subtracted
in the usual way.

The first step in multiplication, when the multi-


plier is less than ten, and the product of any order in
the miiltiplieand is hiSs thanten, may be well j^^iven by
the following method. If it takes twenty-two cents
to buy one melon, how many cents will it take to buy
four? The child should take the sticks or some ob-
jects, and show how many twos are needed. He then
shows how many tens are wanted. After he has
worked mentally a number of examples in this man-
ner, he is prepared to solv^e them on the board and
slate. After these steps are learned, we teach from
thirty to fifty. I shall say at this point that the sec-
ond year's work shonkl end with thirty, and that the
pupil should be^in the third year with the first steps
in addition, subtraction, and multiplication. Some
teachers, however, pn^fcjr to include these steps in the
second year's teaching.

As soon as the child has learned fifty, he may pro-
ceed to master the next steps in addition, subtraction,
and multiplication. The second step in addition,
when the sum of the ones exceeds nine, should also be
taught by objects in the following manner. Three
boys were counting their marbles. One had twenty-
six, another had thirty-five, and the other had twen-
ty-four. Let us see how many marbles they all had.
Let the pupil show each numl)er by sticks. Let him
find the numb(;r of on(;s, which are fifteen. He
knows that, in fifteen, th(;ie are one ten and five sin-
gle oijcs. I*ut a string or rubber band around the
ten. Now let him count the tens, whi(;h are seven,
and add the one ten to them. He will then have eight
tens and five single ones, which make eighty-five. Do
other examples in the sann; way, and then put them


on the board, tiud have liiiri work them. Now he
may be readily iaii^ht to add aliriowt any numbers.

The Keeond step in Hubstraetiori, when th(i ntnnber
in the right hand order oC the minuend Ib leHH than
the number in the corresponding order of the subtra-
hend, may bf^ inculcated by objects in the following
manner. .John had lil'ty-three marbles, and sold twen-
ty seven. We wish to know how many he had left.
We must subtract th(5 ones iirst. When the teacher
asks, 'Mlow many ones must we subtract?" some one
wi 1 1 say , ' SSeven . ' ' l'>u t we cannot take seven ones from
three ones. The instructor says, ^^ Who knows what
to do?" Some one will probably say, ^' Break a bun-
dle of tens." We now break a bundle of tens, and
put them with the ones. Now we may take away
seven. The child then sees that two tens must be
taken from the remaining four tens. He then knows
that John had twenty -six marl)le8 left. When the child
can readily solve such examples by means of the ob-
jects, he is prepared to work them on the ])oard and
slate. When he understands how to work this kind
of exami)les on the slate, he may be easily taught how
to subtract any number from a larger one.

The second st(q) in multiplication, when the multi
plier consists of one figure, and the product of the
number in the lowest order of the multiplicand by the
multiplier exceeds nine, may be well taught thus. A
farmer had seven pc^ns and forty six pigs in each.
How many pigs did he have in alH Jjead the pupil
to find seven sixes first. He knows that there are
four tens and two single ones in forty-two. He puts
a string or rul)ber band about each ten, and lays the


two single sticks on one side. He next finds seven
four-tens which are twenty-eight tens. Lead him to
see that the four tens must he added to the twenty -
eight tens, making thirty -two tens. The whole num-
ber of pigs was three hundred and twenty -two. The
child can very readily learn that the three in this an-
swer means three hundred. Commence board work
as soon as the child can easily separate the tens from
the ones in the product of units by units, and can
add these tens to the product of tens. When these
different steps in addition, subtraction, and multipli-
cation are learned by the pupil, he should proceed to
master the numbers from fifty to one hundred inclu-
sive according to the method which is explained in
the preceding part of the lecture.

After sixty is presented, the pupil may learn Time
and the Time Table. This should be inculcated by
the following method. Let the teacher suspend in
front of the class a string with a weight attached to it.
The string should be thirty -nine inches long. He
must impress the child with the fact that whenever
the weight moves under the hook, one second of time
passes. He must now teach the child that sixty sec-
onds make a minute. He can make the child easily
comprehend this by showing him that the smallest
hand of a watch moves once around the little circle
while the weight swings sixty times . The pupil learns
from this that it takes the little hand sixty seconds,
or one minute to go entirely round the circle. Now
see if some child cannot discover that it takes the
long hand one hour, or sixty minutes to move round
the face of the clock. Cause him to know how many


minutes it takes the long hand to move from twelve
to one, from one to two, and to the other figures on
the face of the clock. When the scholar thoroughly
understands seconds, minutes, and hours, he is pre-
pared to learn, with little effort, the number of hours
in a day, the number of days in a week, the number
of weeks in a month, and the number of months in a
year. Kow he can easily write the Time Table.

After seventy is learned, he may be taught Long
Measure. The teacher should supply himself with a
foot-rule and a yard-measure. Let him cut a piece
of paper one inch in length, so that the pupil will be
enabled to tell an inch whenever he sees it. He then
easily learns that twelve of these little measures make
a foot, and that three foot-measures make a yard.
Kow let the pupil learn the length of a rod by seeing
a string five and a half yards long. He may then be
impressed with the fact that there are three hundred
and twenty rods in a mile . Cause the pupil to take the
measures and find out lengths for himself. Proceed
to teach the remaining numbers from sixty to one
hundred according to the method illustrated in teach-
ing the number three. Before one hundred is taught,
Eoman I^umbers may be given. Almost any Arith-
metic gives a simple method of teaching them.

After one hundred is learned, the third step in mul-
tiplication, when the multiplier consists of two fig-
ures and the left hand figure is one, may be presented
thus. A man had eighteen coops and thirty -six
chickens in each . How many did he have in all ?
Icead the scholar to find by the sticks how many
chickens there were in eight coops. Then let him


find how many there were in ten coops, and add both
nnmbers together. Give a number of practical ex-
amples of this kind, and then let the child work
them on the board and slate. When he has master-
ed this step, he can very easily be taught to work ex-
amples of this kind : Thirty-six times forty-eight are
how many? He can then be led to multiply any
number by any number of two figures . After this he
very readily learns to multiply, in the usual manner,
any number by any number of three or four figures.

The first step in division, when each order in the
dividend is exactly divisible by the divisor, may be
presented thus. A farmer has forty -four sheep, and
wishes to put two in a fold. How many folds does
he need ! Lead the child to pick out all the twos pos-
sible from forty-four. Give other examples like these :
Divide sixty-six by three ; ninety-six by four ; one
hundred and five by five 5 and ninety by six. Let the

pupil solve all these by sticks, and then put the work
on the board and slate.

The second step in division, when the number of
tens in the dividend is not exactly divisible by the
divisor, may be taught thus. A farmer has ninety-
six sheep, and puts them in six folds. How many are
there in each fold "? How many sixes are in nine tens,
and how many tens are left ? We break the three
bundles of tens, and put them with the six, and make
thirty -six ones. How many sixes are in thirty-six
ones "? Then take an example such as this : How
many eights are there in one hundred and thirty -
eight? Do this likewise, and show that two are left.
As soon as the pupil comprehends the first and sec-
ond steps of division, he is prepared to begin the di-
vision of any number by any number of three or four


I have presented the best approved methods of
teaching Primary Arithmetic to children. I do not
deem it necessary to give methods for teaching prop-
erties of numbers, fractions, percentage with its ap-
plications, and other subjects of Higher Arithmetic ;
for good methods of inculcating these subjects are
found in the Arithmetics of Quackenbos, White,
Greenleaf, and Wentworth. The teacher must lead
the pupil from the simple to the complex, from ob-
jects to ideas, from the known to the unknown. When
a new subject is begun, let the pupil discover facts
concerning it for himself. When he commences frac-
tions, he should be led to discover what fractions are
by means of sticks or some other objects. Let him
look at the fractions and handle them. Then pursue
the usual course in teaching them. By this method
the child will discover for himself mixed numbers^
proper, and improper fractions. In teaching reduc-
tion, addition, subtraction, multiplication, and divi-
sion of fractions, good use can be made of the objects.
The question is often asked, When should objects
be dispensed with in the teaching of number ? Wheth-
er this should be done at ten, fifteen, or some other
number, I am not prepared to say ; however, the in-
structor may safely follow this rule. When the child
can comprehend a number without the presence of ob-
jects, their use may cease. I would exhort the teach-
er to seek to know the subject of number in its length,
breadth, and depth? Do not imagine that you fully
comprehend number, because you are well acquainted
with figures ; for this is a common mistake. Lead
the pupil to discover step by step the various rela-
tions of each number. Definitions, processes, and
rules are excellent agencies in mental growth, if they
are discovered by the pupil for himself. Eemember
the old maxim, '^Is^ever do anything for a pupil that
he can be led to do for himself."



A Definition of Geography, and its Relation to other
Sciences — The Aim in Teaching it, and How it Should be
Taught — The Teachmg of the Forms of Land and Water —
Moulding the Continents — Elementary Geography — An
Easy Method of Teaching States and Countries — Teach the
Cities, Rivers, Mountains, Boundaries, Occupation, His-

1 2 3 4 6 8 9

Online LibraryDaniel B[arclay] 1861- WilliamsScience, art, and methods of teaching; containing lectures on the science, art, and methods of education → online text (page 6 of 9)