John Tilden Prince.

Courses and methods : A handbook for teachers of primary, grammar, and ungraded schools online

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be understood that the recitation will not always take
this precise form or that the combinations, stories, and
illustrations will be limited to what is given above.
See that every child is attending to the work in hand,
whether it is in teaching, telling stories, or drilling, and



116 METHODS OF TEACHING.

when members of the class are inattentive or tired,
change the exercise or stop the recitation.

Objects should not be used for teaching numbers
beyond 20 in the primary course. Knowing the com-
binations to 20, all others to 100 are easily learned. In
addition and subtraction beyond 20, the pupils are made
to see that little work is needed. They have only to
apply tlie knowledge they already have. For example,
in the problems 28 + 6 and 34 — 8, they instantly recog-
nize the results of 8 + 6 and 14 — 8, and have only to
think of the tens in getting the answer. At first the
teacher leads them to see this ; afterwards they do it of
their own accord, until adding and subtracting of all
numbers less than 10 are done at sight.

In multiplication and division, the pupils should be
led to make their own tables from what they know of
addition and subtraction. Adding and subtracting by
twos, threes, etc., will enable them easily to construct
their own tables, and after they have constructed them,
they should learn them so well as to be able to multiply
and divide numbers at sight.

The plan as laid down for the primary grades in-
cludes all combinations to 144, and for the iirst division
of ungraded schools, all combinations to 100. It is not
intended in this plan to add or subtract numbers greater
than 10 or to use a multij)lier or divisor greater than
12. Yet if the pupils become very proficient in
this work, it may be well to have them practise in
other combinations, such as 28 + 64 ; 83 - 25 ; 18 X 4 ;
64^16. To add and subtract large numbers, lead the
pupils to add or subtract first the tens, and afterwards
the units: thus in the problem 28 + 64, first add 60, then



AKITHMETIC. 117

4. The pupil would say 28, 88, 92. In subtracting 25
from 83, he would say 83, 63, 58. After considerable
practice of this kind the computations can be made at
sight.

Primary Drill. — To secure accuracy and rapidity of
work, it will be found necessary, especially in the third
year, to s})end much time in drill. It is not enough to
recite the tables. Pupils may do this, and still not be
able to add, subtract, multiply, and divide at sight.
Practice upon the combinations may be had in prepar-
ing a given lesson, and in the recitation.

In giving work for the pupils " to study " at their
seats, the teacher may give a lesson in a book, or put
the problems on the blackboard. Time may be saved
by putting the problems on pieces of cardboard. These
slips may be used by different pupils and classes. The
following exercises for study Avill suggest what may be
placed upon the board or cards : —



Add





42 + 6 = ?


38-6=?




38 + 9 = ?


40-9=?




4:\-\-'{= 48


35-? = 29




? +26 = 34


? - 6 = 32




8x3=?


42-7=?




9x?-36


36 - ? = 9


6


7




4


8


18 6


9


5


-9 X8



8)18



The same exercises may be used for drill in recita-
tion. In addition to these the following exercises are
suggested.

Add by columns, the teacher or pupil pointing, begin-
ning with very short columns and easy numbers, thus : —



118 METHODS OF TEACHING.



3


5


2


5


7


6


9


2


4


6


4


3


4


8


1


2


3





5


8


3



The pupil will say one, three, six ; two, six, eleven ;
three, nine, eleven ; and so on. Increase the length of
columns gradually, until the pupils can add quickly
twenty or more numbers expressed by a single figure
without mistake.

Another method of drill is to place four columns on
the board, thus : —

+ 6 x4 -5 -3



7


6


16


26


2


9


19


29


9


3


13


23


3


7


17


27


5


4


14


24


8


5


15


25


6


2


12


22


4


8


18


28



As the teacher points to each number, let the pupil
add, subtract, multiply, or divide, as indicated above the
columns. By changing the figures above the columns,
a great amount of work may be dictated in a little time.
The pupils may give the result with or without giving
the formula.

Still another method of drill is to place figures near
the circumference of a circle, and add until a certain
number is reached; or subtract, beginning with a cer-
tain number.

Add by 2's, by 3's, etc., beginning with 2, with 3, etc.
Subtract by 2's, by 3's, etc., beginning with 40, with 50,
etc.



ARITHMETIC. 119

Place in the centre of the circle desciibed above
a figure or figures. Point to any figure in tlie
circumference, and requhe the pupil to multiply
or divide. By occasionally changing the figures in
the circle, a large number of problems may be given.
Keep the attention of every pupil to the recitation
of every other pupil, and require answers only as you
point.

Practical Problems. — Mention has been made of
the advisability of making stories in connection with the
learning of combinations. When the combinations are
known, they should be applied as soon as possible to
practical problems. These problems should contain the
common weights and measures, and should be of such a
nature as to induce pupils to think. The following
problems will suggest what may be given daily during
the latter part of the third year in school. Some of the
problems will have to be carefully taught with objects
before they can be understood by the pupils. Encour-
age the pupils to give one another original problems of
a similar kind.

What will three pints of milk cost at eight cents a quart ?

How many cupfuls of milk in a quart, if each cup holds half a
gill?

A piece of tape six inches long costs three cents. What will a
yard cost at the same rate ?

How many apples, at the rate of t^Yo for a cent, can I buy for
twenty cents ?

• What will six apples cost, at the rate of two for three cents?
what at the rate of three for two cents ?

I buy 12 two-cent stamps at the post-office, and give a half-
dollar. What change will be given me ?

Eighteen eggs are worth what at twenty cents a dozen ?



120 METHODS OF TEACHING.

What will one pound and four ounces of meat cost at twelve
cents a pound ?

If you should walk six rods north, and then turn and walk eight
rods south, how many rods would you be from the place where you
first started? how many yards? how many feet?

If a peck of potatoes will last a family one week, how many
weeks will two bushels last them ?

At one dollar a yard, what will be the cost of a piece of carpet-
ing twenty-four feet long ?

Notation and Numeration. — The writing and read-
ing of numbers should be begun in the fourth year,
but it is not well to give numbers of more than seven
or eiglit places at this time. Sticks, and bundles of
tens and hundreds should be used to teach notation.
The sticks may be counted, and the number expressed
by figures on the blackboard. Eighteen sticks should
be counted as one ten and eight, the ten being bound
into a bundle. When two more sticks are placed with
them, they will make two bundles of ten sticks each, and
should be called two tens, or twenty. When ten bun-
dles of ten sticks each are counted, they should all be
bound together into one bundle and called one hundred.
As the numbers are thus taught, the expression should
be placed upon the board in figures and read. It may
not be necessary to teach notation in this way beyond
thousands. When this is done, the pupil will see that
ten of one denomination will make one of the next
higher, and can write and read numbers readily to
millions. In the fifth year billions, trillions, and quad-
rillions should be taught.

Numbers of two places of decimals should be taught
and used in the fourth year. Pupils should learn to
write decimals at first through a knowledge of writing



APaTHMETlC. 121

dollars and cents. When tliey can write and read num-
bers in dollars and cents readily, lead them to see that,
as there are one hundred cents in a dolhir, one cent is
one-hundredth part of a dollar, and that one-hundredth
is expressed precisely as one cent is when written with
a decimal point. Six dollars and one cent is expressed
thus, 86.01, and may be read six dollars and one cent,
or six and one-hundredth dollars. From this it may
be readily seen that any number of cents represents so
many hundredths, and may be read as hundredths.
The first figure at the right of the decimal point repre-
sents the number of dimes, and may be called tenths of
a dollar. This may be taught in the same way as hun-
dredths is taught. Numbers of three places of decimals
can be taught in the fifth year. As one thousand mills
make a dollar, one mill is one-thousandth of a dollar ;
and therefore one mill, when expressed by figures, may
be read one-thousandth.

Fundamental Processes. — At the beginning of the
fourth 3'ear in graded schools, and of the second period
in ungraded schools, the pupils are supposed to have a
thorough knowledge of the four fundamental processes
to 144 or to 100. They can, without hesitation, add
and subtract twelve and all numbers below twelve.
They can with equal facility multiply and divide, when
the multiplier and divisor do not exceed twelve. They
have learned to express in figures the numbers to 144,
and can count to one thousand. They have learned to
use and to write the fractions J, J, J, J, J, in connection
with the combinations, and have had much practice in
the application of their knowledge of numbers to prac-
tical problems.



122 METHODS OF TEACHING.

All of this work should be constantly reviewed in
the fourth year, while other things are being taught.
As soon as pupils have acquired a knowledge of units
and tens in writing numbers, addition of numbers of
two figures should be- begun. Tins should be taught
with sticks, beginning with the addition of two num-
bers that will not require "carrying." We may, for
example, wish to add 22 and 21.

22 sticks and 21 sticks are placed in position, properly
separated into tens and units. Putting the one unit
with two units, we have three units. Putting the two
bundles of tens with the other two tens, we have four
tens. Answer, four tens and three units, or forty-three.
The figures should be written out in proper order, and
each result indicated as we go on. Several problems
equally easy should be wrought in the same way.

When the pupils have had sufficient practice of this
kind, take an easy problem in which the sum of the
units is more than nine. Thus, in the problem 24 + 38,
take the sticks as before, and put the 4 sticks with the
8 sticks, making 12 sticks equal to 1 ten and 2 units.
Put together the 10 sticks in one bundle, and add the
tens thus, 1 + 2+3 = 6 tens. Answer, 6 tens and 2
units, or 62.

Subtraction can be tauglit by taking from a given
number of sticks a part. Thus, to teach 34 — 22, we
would take 2 sticks from the 4 sticks and 2 tens from
the 3 tens, leaving 1 ten and 2 units, or 12.

To subtract 17 from 35, we should put before the
pupils 35 sticks, consisting of 3 bundles of tens and 5
mdts. Asking them to first take 7 units from the 5
units, they will see at once that they will have to untie



AKITHMETIC. 123

one of the bundles and put the 10 sticks with the 5
sticks, making 15 sticks. Now they take 7 sticks from
15 sticks, and have remaining 8 sticks. 1 ten from 2
tens leaves 1 ten. Answer, 1 ten and 8 units, or 18.
^ Multiplication and division should also be taught
with objects, each operation being expressed in figures.
It will not be necessary to carry the objective teaching
beyond hundreds, but it will be found useful to have
considerable practice with smaller numbers before num-
bers of tlie higher denominations are taken. No num-
ber higher than ten thousand should be used duiing
the fourth year, so as to allow time for a sufficient
amount of drill and for work upon practical problems.

Fractious. — According to the prescribed course,
fractions are taught during the sixth year in the graded
school, or the latter part of the second period in the
ungraded school. Before this, simple fractions have
been taught objectively, and used to some extent in the
various operations. Circles of pasteboard will be found
to be the most convenient means of teaching fractions.
The idea of a fraction should be first taught b}^ present-
ing tlie circles cut into halves, fourths, eighths, thirds,
and sixths. The expression may follow, first oral and
then written. Three-fourths will be seen to be three
of the four equal parts into which the circle is divided,
and is expressed by placing one figure above another,
and a line between them. The lower figure will be
seen to express the size of the parts, and the upper fig-
ui'e to express the number of parts taken. Considerable
practice of this kind, with the fractional circles and
expressions, may be followed by giving the terms denom-
inator and numerator^ and havijig them defined by the



12-4 METHODS OF TEACHING.

pupils. The same objects may be used in teaching
redaction of mixed numbers to improper fractions, of
improper fractions to mixed numbers, and of fractions
of one denomination to those of another. Care should
be taken to occasion the idea before the expression is
given. Thus the reduction of 5\ to fourths, 4| to
eighths, 3 1 to sixths ; of f, |, f, to whole or mixed num-
bers ; of J, f, f, y%, to lower terms, should be known by
means of objects before the operation is expressed in
writing. When these facts have been presented many
times to the pupils, they may be expressed in figures,
and the pupils may be led to see the process by which
the answers are obtairjed. For example, in the state-
ment 5f = ^-^'^ the pupils should be led to see, after the
fact has been taught by objects, that the answer could
be obtained by the following course of reasoning. In 1
there are 4 fourths, in 5 there are 5 times 4 fourths, or
20 fourths ; add 3 fourths, and the answer is -\^-. If it
is thought advisable, the rule could be deduced in the
same way.

In teaching addition, subtraction, multiplication, and
division of fractions, the same method should be pur-
sued. First use the objects, and afterwards express the
operations b}^ performing tliem in figures on the board.
Practise much in this Avay with small numbers before
the book is used, and from the problems performed lead
the pupils to deduce their own explanation or rule.

To illustrate the method of teaching fractions the
following examples are given, one for teaching addition,
and one for division. It will be understood that reduc-
tion of fractions has been taught before these subjects
are reached.



ARITHMETIC. 125

Look at these circles and fractions as I hold them before you.
How much is J of a circle and ^ of a circle 1^ + ^11 + ^1 i + f?

i + f? i + f? i + f? Let us now express in figures the answers
you give me : ^ + i = 1 ; f + i = 1 ; i + i = | ; etc.

Do you see how we added the halves and fourths ? How did
we add the fourths and eighths ? Give an example adding halves
and fourths, fourths and eighths. Your lesson to-morrow will be
these fifty problems on the board (or chart).

The problems given for study are of course similar
to those which they have had with the circles in the
class.

The first part of the following exercise is designed to
show how to teach the division of a fraction by a whole
number. The second part illustrates a method of
teaching the division of a whole number by a fraction.
The directions and questions should be many more than
are here given, and each exercise may be enough for
two recitations.

Divide this circle into two equal parts ; how much in each part?
Divide this half-circle into two equal parts ; how much in each
part? Divide these two circles into four equal parts; how much
in each part? Divide these two circles into eight equal parts;
how much in each part? Divide this half -circle into four equal
parts ; how much in each part? Divide one-fourth of a circle into
two gqual parts; how much in each part? Divide three-fourths
into two equal parts ; how much in each part ? Let us now see
what you have done (writing on the board) : —

l-2==i; J-2 = i; 2-4=A; 2-8 = i; *-^4 = i; i-2 = i;

1-^2 = 1-^

"Who will divide any of these fractions into equal parts and
place the result on the board?

Practise in this wa}^ with halves, fourths, and eighths,
and then with thirds and sixths. When a large num-



126 METHODS OF TEACHING.

ber of problems and answers is placed upon the board,
lead the pupils to see and express for themselves the
fact that we may divide by a whole number b}^ dividing
the numerator or multiplying the denominator. When
they have done this, give out a large number of simple
j)roblems for them to perform before the book is taken.
To divide an integer by a fraction.

Call these circles pies. I have eight pies, and give them to the
persons in the room ; each person receives four pies ; how many
persons in the room ? Put down on your slate each operation as
you find it. I have eight pies, and give to each person in the
room two pies; how many persons in the room? I have one pie,
and give to each person at the table one-half a pie ; how many
persons at the table? I have one pie, and give to each person at
the table one-fourth of a pie ; how many persons at the table ? I
have two pies, and give to each person at the table one-half a pie ;
how many persons at the table? etc.

Xow let us see what you have on your slate. Yes, —

8^4 = 2; 8-2 = 4; 1-^ = 2; 1-1 = 4; 2-J=4; 2-1 = 8;
2-i = G; 2^1=12.

Keep these upon your slates, and do as many more as you can
before to-morrow.

Now call tlie circles cents. I have four cents (holding up four
circles) ; if apples are one cent apiece, how many apples can I
buy? how many at h cent apiece? how many at } of a cent apiece?
If the apples were three times as much apiece, how many could I
get; more or less? what part as much? If these apples were f of
a cent apiece, how many apples could I get? Now let us'take
eight cents. Who will give ns the same kind of a problem?
What are the expressions on your slate ? Let us put them on the
board : —

4_^1^4; 4-§=8; 4 - } = 16 ; 4 - f = -i/ = 5] ; 8-J = 16;
8 -^ I = -1^6 ^ 51 . 8 -^ 1- = 82 ; 8 - f = -»/ = lOf .

From this w^^rk both the explanation and rule may
be deduced by the pupils.



ARITHMETIC. 127

Applications. — From the time the child enters
school lie is led to apply his knowledge of numbers in
making and doing practical problems of various kinds.
With a knoAvledge of common and decimal fractions his
work of this kind can only be limited by his immatu-
rity. As the pupil matures, his field of stud}^ and prac-
tice widens. Some part, at least, of all kinds of business
he may know, and the teacher should gather from every
source material with which to work; several reference
books, both written and mental, should be upon the
desk to suggest ways in which the pupils' knowledge
of numbers may be applied.

Weights and Measures. — These subjects, according
to the prescribed course, are taken in the seventh year.
During all the preceding years the pupils have per-
formed problems which involve nearly all of the weights
and measures commonly used in every-day life. They
will not, therefore, have to spend much time in learning
the tables. The metric system, and some parts of the
tables of square and cubic measures, and of Avoirdupois
and Troy weights, will have to be learned: also miscel-
laneous facts, such as the weight of different commodi-
ties, the number of units in a gross, and score; number
of sheets in a quire and ream, and the value of common
foreign coins. All of these tables which will be of use
to the pupils should be made and learned by them.
When they are learned, they should be reviewed and
applied so frequently that the pupils will not have to
go to the book for information.

In choosing work for the pupils, give only that which
is practical. Omit all parts of compound addition, sub-
traction, and division, which are rarely or never used.



128 METHODS OF TEACHING.

Do not give impossible areas or volumes to measure or
absurd puzzles to solve, but let the work be such as
occurs, or may occur, in practical life.

The time may be well spent in the reduction of com-
pound numbers, both ascending and descending, the
computation of longitude and time, and the mensura-
tion of surfaces and solids, such as papering and carpet-
ing rooms, measuriiig boards, wood, bins, etc. Select
from two or three books placed upon your table for the
purpose such problems as you think most practical and
best adapted to your pupils. If the reasoning of any
problem is complex, give the same problem first with
small numbers, and always encourage the pupil to
use blocks and diagrams to illustrate the problems.
The practice of illustrating problems by diagrams
cannot be too early begun or too constantly insisted
upon.

Very much time need not be spent upon the metric
system. It is enough to teach by objects the different
measures, and to lead the pupils to work enough upon
the various applications to see the great saving of time
which would follow the introduction of the sj^stem. As
the denominations are rarely used in practice at the
present time, they would be soon forgotten if learned
ever so well. Therefore,' not so thorough work in the
application of metric measures should be attempted as
in those measures which are in common use.

Percentage. — The kind of work to be done in per-
centage is indicated in the best text-books upon the
subject. The amount to be done is limited only by the'
time of the pupil, for very much drill is needed to
distinguish readily the various conditions of problems



ARITHMETIC. 129

wliich are classed under the head of percentage. In no
part of arithmetic is the necessity greater of passing
slowly from the known to the unknown than in per-
centage. Teach each part of the subject with great
care, using familiar illustrations and small numbers.
Avoid, so far as possible, all work by rule, but lead the
pupil by slow degrees to understand the principle in-
volved in each problem as it is presented. Review fre-
quently, and arrange the problems in such a way as to
encourage the pupils to think. Sometimes pupils are
directed to look over a " model solution," and to perform
all the problems of a given lesson by it — a course which
is likely to discourage independent thinking. To indi-
cate how the subjects may be taught, a few illustrative
examples are here given. The process of each problem
should be indicated upon the blackboard as the answers
are found, and when the principle is understood,
2"


i


i


i


i


t


¥


2


H


1


1(>


2


3
4


i


i


1


2


i


f


-V-


i


3|


IV


13


3


f


f


i

4


f


4


i


1


¥


3


n


If


19


4


f


1


i


i


i


f


f


¥


8


5|


2|


17


5


1


5

8


i


1


t


*


f


¥


6


n


3|


U



The table may be used as follows ;



A + B, B + C, A + C,


B + E,


B + D,


A + E, etc.


A-B, A-C, A-B,


D - E,


D-F,


I — A, etc.


A X B, B X C, A X I,


Ax J,


AxF,


A X E, etc.


I ^ A, I - C, A - I,


J-^I,


A^B,


A-C, etc.


Reduce 1 pounds to ounces.








Reduce E feet to inches.








What will A jDounds of meat cost at L


cents a


pound? etc.



Besides the oral work Avhich is done in connection
with written arithmetic, there should be a few minutes
set apart each day for miscellaneous mental practice.
The problems given should be of a varied character,
sometimes consisting of operations with abstract num-
bers, in which accuracy and rapidity are mainly sought;



ARITHMETIC. 135

at other times the problems should be of such a nature
as will call into active exercise the reflective faculties.
As no one book would furnish a sufficient variety of
problems, there should be upon the teacher's table sev-
eral difterent mental arithmetics, from wdiich to gather
and give problems of a proper kind. It will not be
found best for the teacher or pupils to read the prob-
lems from a book. Let the teacher glance over two or
three pages of a book, and select such problems as will
induce the pupils to think, giving them in language of
his own. Sometimes the problems may be analyzed and
explained, and sometimes, especially in examination,
answ^ers only may be required. One good method of


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Online LibraryJohn Tilden PrinceCourses and methods : A handbook for teachers of primary, grammar, and ungraded schools → online text (page 8 of 22)