C 342.27 C. A.
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Teaching Arithmetic
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D. C. Hºº CO. *UBLIsº
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M ETH O DS
IN
TEACHING ARITHMETIC
(FO/P CO.)/.]/OM SCHOOL COUA’.S.A.S.)
... ." By
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E. C. BRANSON
Director NorMAI. DepartMENT GeoRGIA NorMA 1. A N D
INDUSTRIAL College
BOSTON, U.S.A.
D. C. HEATH & CO., PUBLISHERS
.*.*
*- 1896
CopyRIGHT, 1895,
By E. C. Branson.
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1 5 34
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C. J. PETERS & SON, TYPOGRAPHERs.
S. J. PARKHILL & CO., PRINTERS.
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METHODS IN TEACHING ARITHMETIC.
CHAPTER I.
A BIRD’S-EYE VIEW.
I. What a Teacher ought to know about Arithmetic.
1. All he possibly can know.
2. Brooks’s “Philosophy of Arithmetic” and a full school
text, say Harper’s “Advanced Arithmetic,” ought to
be on hand for constant reference.
II. What a Teacher ought —
2.
Ö
TO TEACEI.
What is fundamental and neces-
1.
i
sary, say, -
The Fundamental Processes
with whole numbers.
Common Fractions.
Decimals.
Compound Quantities.
Percentage.
Proportion.
Common Measurements.
What every pupil is certain to
have an immediate working
need for every day.
3
TO OMIT OR ADJOURN.
. The subjects put into the book,
merely to make it a complete
scientific cyclopaedia of the
subject, say, -
. The Metric System,
. The Progressions,
Foreign Exchange,
Alligation, and the like.
. What only a few pupils may
chance to have only an occa-
sional need for, — say, Troy
Weight, stocks and bonds,
and the like.
s ºr *
A.
$
t
Y2
4
METHODS IN TEACHING ARITHM ETIC.
II. — (Continued.)
3.
4.
TO TEACBI.
Sensible, practical, work-a-day
problems only. There is al-
ways sufficient occasion for
disciplinary drills in these.
Quick business processes, –
1. Rapid column adding.
2. Quick multiplications and
divisions.
3. Cancellations.
4. Percentage and interest
processes.
5. Common measurements.
. Problems mostly; that is, pro-
cesses should always start in
and centre about problems.
Process and problem should
be wedded, never to be di-
vorced, in the
pupils.
Some further details, as fol-
lows: —
1. Oral work in aliquot parts
of 100 and $1.
2. Oral work with the larger
fractional units, – #’s to
nº's.
3. Specially 10ths, 100ths, and
1000ths in Decimals, — to
be taught mostly with
U. S. Money.
minds of
What a Teacher ought —
TO OMIT OR ADJOURN.
3. Foolish, imaginary, unbusiness-
like problems, like ninety per
cent of the problems in Com-
pound Proportion.
4. The long and confusing analyses
in the special percentage sub-
jects, Compound Proportion,
and the like.
Solutions of naked equations,
the performance of processes
aside from the problems they
concern, – as the long in-
volved solutions of complex
fractions, for example.
Such items or subjects as fol-
lows: —
1. The properties of numbers,
or only as needed in other
work, - L. C. M. with
fractions, say.
2. Work with the very small
fractional units, #'s,
#5's, etc., and most of
the fraction-work unas-
sociated with problems.
3. The smaller decimal units,
as billionths, trillionths,
etc.
3.
METHODS IN TEACHING ARITH M ETIC. 5
II. — (continued) What a Teacher ought —
TO OMIT OR ADJOURN.
TO TEACEſ.
(Reduction of —
Dry and Liquid Measure.
Long Measure (in., ft.,
yd., mi.).
Surveyors’ Measure (omit-
4. ting rods and roods).
\ Square Measure (sq. in.,
sq. ft., sq. yd., sq. A.,
sq. mi.).
Cubic Measure (cu. in.,
cu. ft., cu. yd., cord).
U Time Measure.
4.
\
(English Money; Troy Wt. ;
Apothecaries' Wts.; rods,
perches, poles, furlongs.
Circular Measure (except cir-
cumference and degrees).
Cloth Measure. -
Also omit #6 of the work
laid down in books in ad-
dition, subtraction, mul-
tiplication, and division
of Compound Quanti-
ties.
5. You can safely omit at least
75% of the work ordi-
narily laid down in the
arithmetics in Compound
Quantities.
6. The following special forms
of Percentage: —
Stocks and Bonds, Insur-
5. Practical problems in wood
and lumber, plastering,
papering, carpeting, ma-
sonry. Caution: the pro-
cesses in the books are
usually misleading. Con-
sult business men when
you treat these topics.
6. Common Quantities : £,
shilling, franc, span, pace,
hand, dozen, gross, score,
quire, ream, bbl. flour,
load (earth), 1 hr. = 15°
long., 1 gal. = 231 cu.
in., 1 bbl. = 31, gal.
7. Accounts, bills, and receipts
are a necessary part of a com-
mon-school course in arith-
metic.
8. Percentage as involved in Profit
and Loss, Taxes, Commis-
sion, and Interest.
ance, Exchange, Partial
Payments, Equation of
Payments, Compound In-
terest, Banking, Time Dis-
count, and Partnership. If
the pupil be well grounded
in Percentage in the other
forms, a business situation
or transaction will, later
on, teach the man in five
minutes clearly what you
could not now teach the
boy in five months.
6 METHODS IN TEACHING ARITHM ETIC.
II. — (Continued.) What a Teacher ought —
TO TEACEI. TO OIMIT OIR, ADJOURN.
9. Simple Proportion, and the 7. Most of Compound Proportion
measurement of rectangles, and other forms of work in
triangles, circles, cylinders, Mensuration.
and rectangular solids.
10. Every subject to be prefaced,
reviewed, and finished in
oral arithmetic exercises.
III. Why?
1. Because, with all the time spent on arithmetic in our
schools, the practical results are commonly admitted
to be little short of disgraceful.
2. Because it seems wise to select a brief, practical course
out of all the mass of subjects and matter in the
arithmetics, and to teach that course thoroughly.
3. Because we thus save time for things essential and
necessary, and concentrate attention upon a few
things instead of wasting it upon many.
4. Because the subjects listed “omitted or adjourned ”
can be taught after we guarantee the public that a
brief practical course has been thoroughly mastered.
5. Because, if time be not left for them in the school-life of
the pupil, the business affairs of life will teach them
to mature minds quickly and easily, if the other sub-
jects have been well taught them in school.
IV. Notes. *-º-º-º-º-º-º-º-º-º-º-º-º:
1. There is nothing new in this scheme. General Fran-
cis A. Walker's attack on arithmetic teaching and
results, several years ago, aroused great interest at
the time, – an interest that deserved to accomplish
the reform intended.
2. Publishers are beginning to issue books to suit such
demands. Atwood’s “Complete Graded Arithmetic”
and “The New Arithmetic” are such books.
7
METHODS IN TEACHING ARITHM ETIC.
CHAPTER II.
FIRST TVVO YEARS OF PRIMARY NUMBER. — SYLLABUS.
FIRST YEAR’s WoRK. (First A’eader Pupils.)
I. Extent. — The numbers in order to ten, inclusive.
II. Details. – What the pupils are to learn: –
1. What makes each number (number contents).
a. All the complementary contents (what two num-
bers make the number).
b. The measure contents (how many 1's, 2's, 3's,
etc., make the number). Only the exact meas-
urements at first.
2. The two number operations (actual operations).
a. Taking apart (what can be done with a number).
b. Putting together (what can be done with numbers).
3. The five characteristic story-like problems and draw-
ings for them.
a. The additive story. I had two apples. I bought
two more... I had 4 then ; because 2 and 2 are 4.
Ó C6 and Ó C6 are 4 Cô's.
b. The subtractive story. I had 4 apples. I ate 2.
I then had 2 left ; because 4 less 2 is 2.
ÓCô–é–é– is 2 (665.
c. The multiplicative story. I had two dishes
with 2 apples in each dish. I then had 4
apples; because 2 twos are 4.

[Ó & J[º]–4 dºs.
8 METHODS IN TRACHING ARITHM ETIC.
d. The division story. I had 4 apples. , I put two
apiece into some dishes. It took two dishes;
because 4 separated into twos is 2.

[33]{63}=2|[ ]s.
e. The partitive story. I had 4 apples, and put them
into two dishes equally. There were two in
each dish ; because # of 4 is 2.
gºl-º-, -, *,
ºne-º-; e.
Here you have the aſphabet of arithmetical relations.
The pupil may know the figures, counting, tables, and
processes, and yet may be helpless in arithmetic in this
fundamental particular. I have seen college students
helpless because they had never mastered these mathe-
matical ideas of relation.
4. The complementary contents of numbers from 2 to
12 inclusive, to be memorized by use and oral
reviews.
III. Tests.
1st Set. To be used for every fact of number studied until
the equation-forms are taken, – say with the number 6.
a. Give operations with objects (for all the contents
with each number in its five relations):
a. Ask for equation-statements in common language.
b. Ask for story-problems.
c. Ask for picture-equations (drawings).
b. Give the story-problems:
a. Ask for answers and statements in common language.
b. Ask for object-solutions.
c. Ask for picture-equations.
6.
4.
METHODS IN TEACHING ARITHMFTIC. ¥ 9
Give picture-equations:
a. Ask for the story-problems (oral).
b. Ask for statements in oral language.
c. Ask for objective solutions.
Discard objects ; ask for complementary and
measure contents, pupils give results. Begin
simple rapid combination drills orally, using
the number facts in all the numbers studied.
To be used after the equation forms are taken.
Give operations with objects as before:
a. Ask for equation-statements (oral).
ð. Ask for written equations.
c. Ask for story-problems.
d. Ask for picture-equations.
Give written equations:
a. Ask for oral reading, and results.
b. Ask for quick answers at sight.
c. Ask for picture-equations.
d. Ask for story-problems.
Give story-problems:
a. Ask for objective solutions.
6. Ask for picture-equations.
c. Ask for written equations.
d'. Ask for quick answers without aids.
Give picture-equations:
a. Ask for oral, and later, written, problem-stories
b. Written equations.
These tests suggest the logical order of the teaching
exercises, and a great variety of busy-work for the pupils.
IV. What Pupils are not to Know.
1. Useless terms, such as Addition, Multiplication, Divis-
ion, etc.; or these are to be learned by use merely.
10 METHODS IN TEACHING ARITH METIC.
*
2. Definitions; not now, later.
3.
Any rules whatsoever.
4. How to count to 100, 1000, and the like, by 1's, 2's,
3's, etc.
5. Roman numbers, or Arabic beyond the use of those in
the readers.
6. The figure-processes; not now, later.
V. Purposes. , -
1. To unlock the meaning of Problems for the child.
(Do not mistake this ultimate meaning in simple
primary numbers.) How?
a. Through sense-experiences with actual numbers,
actual operations with them, and language for
these actual operations.
b. Through suggestions which arise in the child's
mind from these sense-experiences, sugges-
tions of the five fundamental problem-stories,
embodying the five problem-relations of
arithmetic.
2. To develop invention and to train in orderly statement.
How 2
a. By telling the child nothing beyond arbitrary
terms.
b. By leading him into discoveries for himself.
c. By leading him into orderliness of observation.
d. By insisting on his seeing, seeing in order, telling,
and telling in order, exactly what he sees in the
order in which he sees it.
VI. Final Test.
Your pupil must see an equation in any problem-form
he has had, and, on the other hand, a problem in every
equation studied.
METHODS IN TEACHING ARITH M ETIC. 11
If so, you have a basis for a further advance later into
the mysteries of problems, – into problems in which not
one, but a complication, of the five relations occur.
You will note an approach to the problem through all
the necessary details of fact, and, in the second set of
exercises (p. 9), a variety of solutions that thoroughly test
the child’s appreciation of the pictures in the problem.
“Seeing the picture,” “getting the thought,” in a reading-
lesson, has been abundantly stressed; but seeing the pic-
ture and getting the thought in a problem in arithmetic
has been very little emphasized. As in readers, so in
arithmetics, good pupils will not read beyond the limits
of their experiences. What is needed is to supply the
necessary primary experiences and their special forms
of language, – materials for making pictures out of
problems.
Of course a final test of a pupil’s understanding of a
problem is his ability to translate it into its equation (one
or more). He can be trained from the start to see
a problem in every equation and an equation in every
problem. A standing requirement in this department is
to permit no pupil to attempt the figure-processes until he
has stated the problem in equation form.
It will be seen that here we regard problems to be
the staple of work in arithmetic, to be kept in sight from
the very start ; the exercises in number measurements,
operations, equations, tables, figure-processes, and what
not, having to do with problems from first to last.
For instance, no pupil-teacher is permitted to say to her
class, “Divide 675 by 75.” Rather, “$675 buys Shetland
ponies at $75 apiece. How many ponies?” Pupils are
obliged to picture the situation, and see in it the necessary
operation and process. t
12 METHODS IN TEACHING ARITHM ETIC.
CHAPTER III.
SECOND YEAR'S WORK IN NUMBERS. (SECOND READER PUPILS.)
I. Extent. The numbers in order to 20 inclusive.
II. Details.
1. Review. The numbers to 10, as follows: —
a. Regarding them as compound number units.
2, as { 1 quart (2 pints),
1 dime (2 nickels).
3, as 1 yard (3 feet).
1 bushel (4 pecks),
1 gallon (4 quarts).
1 nickel (5 cents),
1 hand (5 fingers).
6, as 1 working-week (6 days).
7, as 1 week (7 days).
8, as { 1 gallon (8 pints),
! 1 peck (8 quarts).
10, as 1 dime (10 cents).
b. Actual objects and measures to be used. Prob-
lems the staple of the work. Problems to be
freely turned into equations, and zice versä, as
busy-work.
2. The numbers in order from 10 to 20, compared in
each case with all the smaller, included numbers.
For example : —
(Z. 12 compared with 1. 12 compared with 2. 12 compared with 3.
5, as
12 – 1 = 11 12 – 2 = 10 12 – 3 = 9
11 + 1 = 12 10 + 2 = 12 9 + 3 = 12
12 + 1 = 12 12 + 2 = 6 12 + 3 = 4
12 × 1 = 12 6 x 2 = 12 4 × 3 = 12
l's of 12 = 1 § of 12 = 2 # of 12 = .
And so on, 12 compared with 4, 5, 6, 7, 8, 9, 10, 11, 12,
METHODS IN TEACHING ARITH M ETIC. 13
b. Problem-stories and picture-equations for all the
facts treated.
3. Complex equations.
You can gradually make the equations more and more
complex; that is, involving two or more relations in
one equation. This is delicate work. Indicate some
of the simpler combinations.
a. The additive and subtractive relations, –
5 + 3 –2 = ?
b. The multiplicative and additive relations, –
4 × 2 + 2 = ?
c. The multiplicative and subtractive relations, –
5 X 2 – 3 = ? -
d. The multiplicative relations doubled, –
3 × 2 + 2 × 2 = ?
e. The partitive and additive relations, –
5 + 3 of 8 = ?
These combinations to be tested thoroughly in all the
ways indicated in the Zests in Chapter II.
4. The pupils by the end of the year ought to know
perfectly, automatically, the following 36 combina-
tions necessary to the figure-processes of addition
and subtraction:—

5 || 5 || 6
5 || 6 || 6
4 || 4 || 4 || 4 || 5 || 6 || 7
4 || 5 || 6 || 7 || 7 || 7 || 7
3 || 3 || 3 || 3 || 3 || 3 || 4 || 5 || 6 || 7 || 8
3 || 4 || 5 || 6 || 7 || 8 || 8 || 8 || 8 || 8 || S
2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9
2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 9 || 9 || 9 || 9 || 9 || 9 || 9
4 || 5 || 6 || 7 || 8 || 9 |10|11||12||18 ||14|15|10|17|1s
14 METHODS IN TEACHING ARITH METIC.
5. Begin short single-column additions.
6. Teach numeration and notation of numbers to 100.
Use numeration-box.
7. Teach carrying of units in short additions.
8. Teach subtraction, — numbers under 100.
a. Objectively, with analysis story.
6. Written work, with analysis story.
c. Written work, without analysis story.
9. Teach multiplication tables through 5's.
a. Objectively in table buildings.
b. Without aids.
These subjects are all treated in detail later.
III. Notes.
1. No equation and no figure-process to be divorced
in sense from the problem. Thus, it is as easy to
say to a class at the board, “I bought a cow for $17,
another for $35, and a calf for $12. How much in
all 2 ” as to say, “Now, class, put down 17, 35, and
12. Add them.”
2. No problem to be worked until it has been turned by
the pupil into its equation form.
3. Details of these first two years of number teaching—
outlines of lessons, purposes, principles, methods,
and busy-work — are given fully in
Branson’s “Grube Number Text for Teachers,”
D. C. Heath & Co., Boston, Mass.
4. For busy-work in abundance, see
White's “First Two Years with Number " and
“Junior Arithmetic,” D. C. Heath & Co., Boston.
Bacon’s “First Four Years in Number,” Ginn &
Co., Boston.
Atwood's “Graded Arithmetics” (a complete
course), D. C. Heath & Co., Boston.
METHODS IN TEACHING ARITHM ETIC. 15
CHAPTER IV.
METHODS IN THE FUNDAMENTAL RULEs.
NUMERATION AND NOTATION.
I. Materials.
1. Toothpicks or straws bundled by the pupils into 10's
and 100's (of 10, 10's). Held by small rubber bands
or cotton thread.
2. Also a numeration-box, made of three shallow cigar-
boxes held together end to end by eight matting
tacks (two-pronged).
N
Nilºllſ
Nº-Nºss N .
The numbers to be read are to be taken out of
the boxes, and arranged along the lower part of the
lids, the 100's, 10's, and 1’s, each on their own lids.
Ull
—ws---
II. Order of Teaching.
1st. Numbers of one period.
Kinds of numbers to be treated specially, -356,
306, 300, 360, 060, and 006.
2d. Numbers of two periods (then three periods, and
so on).
Teach the second period with drawings instead of
the objects, thus, –











16
METHODS 1 N TEACHING ARITH M ETIC.
The teacher
draws the two
boxes in the
board, and writes
in the numbers,
one in the boxes,
the others below.
Require the deci-
mal point and the
separatrix from
the start.
THOUSANDS.
100’s. 10's.
1’s.
\
100’s. 10's. 1’s.
| 3 || 7 || 6 || 5 || 8 || 9 |
3 8 5 , 6.
3 0
3 5
7
III. Steps in Teaching.
O
:
4 7
0.
S.
5.
, 0
, 0
, 0
etc.
6
()
6
The pupils nu-
merate 1's, 10's,
100’s ; 1's, 10’s,
100's of thou-
sands, and read
the numbers in
periods, as the
device naturally
inclines them to
do.
1. Build the numbers on the box-lids; ask pupils to read
them.
2. Call out numbers; ask pupils to build on box-lids.
3. Draw the box, write in a number; ask pupils to dupli-
cate it on the box-lids.
4. Build numbers on the box-lids; ask pupils to write the
numbers on the board.
5. Write numbers on the board; ask pupils to read them.
6. Call out numbers; ask pupils to write them.
ADDITION.
I. What the Pupils Must Know.
The thirty-six combinations listed in Chapter III.
II. How Taught.
*
1. By discovery (in course of the work already planned).
2. By drill.
a. Quiz rapidly (8 + 7= ? or, what +7 = 152 and
the like, for all the 36 combinations).
b. Sight additions.
a. Use combination-cards, containing: (1) The 36 com-
binations; (2) all the addends, 3 in number, that
METHODS IN TEACHING ARITHMETIC. 17
make 10; (3), all the addends, 3 in number, for
15 ; (4) all the addends, 3 in number, that make
20; and (5) all the addends, 4 in number, that
make 20, Make these cards yourself. Expose to
sight an instant; call for sums.
b. Use blackboards; write columns, erase quickly, call
for sums.
c. Call out addends rapidly ; ask for sums.
c. Cat-and-rat games may well preface these sight
additions.
Put on the blackboard any device, say a star, – one num-
ber in the centre, the others outside. Pupils conduct the
game; the one at the blackboard the cat, the others the
rats. The cat points to the out-
side numbers, and a rat calls the
sum made with the inside number
in each case. If the cat accepts
a wrong answer, and some rat
detects her, then the rat has
caught the cat.
Some teachers easily get a great
deal out of this simple game.
Vary the sign of relations before
the inside number (make it —, X,
8
12 A. 9
4X + 3 3
10 2 5 7
-ll

--, or make the inside number 4, , etc.), and the same
device and game serve to drill upon all the fundamental
tables.
d. Write columns of 2's, 3's, 4's, etc.; write different
numbers below; pupils add orally, quickly.
e. Graduated columns.
These are columns of figures in which the
36 combinations are involved gradually accord-
ing to difficulty, the hardest introduced later,
and being involved oftenest. Sold by Kidd &
Co., Milledgeville, 5 cents each, $4.00 per 100.
18 METHODS IN TEACHING ARITHMETIC.
III. How to Add a column.
1. Stupidly, as follows: 5 and 3 are 8; and 8 and 5 5
4 are 12; and 12 and 3 are 15; and so on, 3
slowly and drawlingly to the top. 4
2. Poorly, as follows: 8, 16, 20, 23, 30, etc., to the 5 20
8
top.
This may be called spelling up a column. 7 10
3. Expertly, as follows: 20, 30, 50, 55. 3
This may be called reading up a column. 4
The column was put down at random. The 8 20
pupil ought to be drilled to see that the first
four figures spell 20, just as S-h-i-p, spells
“ship; ” the next two spell 10; the next four,
20; and so on. In short, he ought to be drilled to
pick out the 10's, 15's, and 20's at a glance in adding
columns.
5
IV, Notes.
1. I would very rarely give pupils sums to add at their
seats. They do so slowly, and usually learn to add
on their fingers, – which is not adding, but only
counting by ones.
2. Ginn's “Addition Manual" (10 cents) is the best
thing on column adding I know. Ginn & Co.,
Boston.
SUBTRACTION.
I. Steps in Teaching.
1. Objective subtraction. (Use the numeration-box.
Shut the box, and arrange the minuend on top of
the lids.)
2. Written forms with analysis.
3. Written forms without analysis.
METHODS IN TEACHING ARITH M ETIC. 19
II. Ideas and Terms to be kept in mind as the teacher draws
from the child by questions the subtraction story
(analysis).
1. Visiting. 10
2. Borrowing. 2 4 10
3. Changing. 3 # 2. W. The written form.
4. Placing. 7 8.
5. Subtraction. 2 7 4.
III. The Subtraction Story.
1. “I want to take 8 ones from 2 ones. I can't do it, so
I go "isiting to the tens place. I borrow 1 ten from
the 5 tens, which leaves 4 tens. I change the 1 ten
to 10 ones, and place them with the 2 ones, making
12 ones. I then subtract 8 ones from 12 ones, which
leaves 4 ones.” And so on for the next step.
This story is to be developed by questions. The
ideas being clear, then the story may settle finally
into some such form as that given above.
2. The last step — written work without analysis. “8
from 12 leaves 4; 7 from 14 (not 8 from 15) leaves
7; nothing from 2 (not 1 from 3) leaves 2. Answer,
27.4.”
IV. Notes.
1. The child that has thoroughly learned the 36 combina-
tions, as given in Chapter III., - has learned them
as complementary contents, – has in mind already all
the number-facts necessary to subtraction.
2. This method to be used with beginners only. If
the pupil has already learned the old way of sub-
traction, do not make it distraction by having him
learn the new way also. Both methods are mathe-
matical and right. This, however, is the simpler for
20 METHODS IN TEACHING ARITHMETIC,
beginners, and easily grounds memory upon under-
standing.
3. Give the subtraction example in form of a problem
always. Or, if not, have the pupil to make a
problem for the example, orally or in writing.
4. Practise pupils in making change, by addition, as
all clerks do.
MULTIPLICATION.
I. Busy-work. — The tables from 2 to 12 inclusive.
FIRST FORM. $ECOND FORMI.
17 || 2 = 2 2 || 1 × 2 = 2
1 / 11 || 2 + 2 = 4 2 + 2 || 2 × 2 = 4
11 11 11 || 2 + 2 + 2 = 6 2 + 2 + 2 || 3 × 2 = 6
11 11 11 11 || 2 + 2 + 2 + 2 = 8 2 + 2 + 2 + 2 || 4 × 2 = 8
Pupils finish. Pupils finish.
THIRD FORMI.
2 || 3 || 4 || 5 || 0 || 7 || 8 || 0 ||10||1|12
2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
2 || 2 || 2 || 2 || 2 || 2 | 2
2 || 2 || 2 || 2 || 2 || 2
2 || 2 || 2 || 2 || 2
2 || 2 || 2 | 2
2 || 2 || 2
2 2
2
2 || 4 || 6 || 8 |10|12 ||14 |10|18 |20 |22 |24

Let this form stay on the blackboard. Erase the
numbers, and substitute 3's, 4's, etc., as the drill
progresses. -
Pupils also construct this table on their slates.
METHODS IN TEACHING ARITHMETIC. 21
FOURTH FORM.

5 || 6 || 7 || 8 || 9 || 10 || 11 || 12
\
10 | 12 || 14 | 16 || 18 20 22 || 24
15 18 21 24 || 27 || 30
3 6 No.
4
8
12 33
N
4 || 8 || 2 | tº 20 |24 |28 |32 || 36|40 || 4 || 48
30
24
5
1
0
1
5
in 20 35 | 40 45 50 55 60
6 || 12 | 18
56
8 16 || 24 32 40 || 48 & Nº 12 80 88 96
2
9 | 18 || 27 | 36 || 45 54 || 63 7 90 99 |108
N
10 20 30 40 50 | 60 70 108110 120
99 |110 N 132
12 24 || 36 || 48 || 60 | 72 | 84 || 96 108 |113"|132 |1
º
90
11 22 33 44 55 | 66 || 77
1. Teacher finally constructs the table on the board, and
drills upon it in a great variety of ways, until the
pupils discover all its properties, and at last what
the diagonal means. Let them discover it. Don't
tell them.
2. Make them pick out each table entire on the left-hand
side of the diagonal, say. -
3. There are 77 products in all. The 2's, 5's, 9's, 10's,
and 11's are easy; 20 are hard for children (they are
dotted). Drill upon these most, and be sure they
are known finally.
4. Let the children construct this table.
II. Drills.
1. Cat-and-rat games.
2. Rapid oral quiz.
22 METHODS IN TEACHING ARITHM ETIC.
3. Put the 77 sets of factors on cards. Give each
member of the class one. Give each pupil another
as soon as he can answer the one he has. The
number in each pupil’s hand sets a value upon
his recitation. The last cards obtained by each
pupil indicate the weak points.
4. Practise upon each table as learned, in slate-work,
with long multiplicands.
5. Grade your multipliers as follows: 2, 12; 3, 13, 23,
123; 4, 14, 24, 34, 134, 1234, etc. Thus you con-
stantly review all the old tables to date.
6. Matches upon the tables (like spelling-matches).
III. The Multiplication Idea.
Remember that the pupil is not completely in pos-
session of multiplication till the following ideas are
lastingly associated in his mind : —
The actual operation, *N
The oral statement for it,
The equation form,
The problem,
The problem-picture,
The figure-processes. J
Constantly test this asso-
ciation of ideas in the
pupil’s mind.
i
IV. Business Processes.
1. When the multiplier ends in a naught.
2. When the multiplier is an aliquot part of 100 or $1.00.
3. By grouping units as you multiply.
a. This last covers all the special multiplications
noted in the business arithmetics, and renders
such classifications useless.
b. It means simply that you perform mentally all
the multiplications that produce units, all that
produce tens, all that produce 100's, and write
METHODS IN TEACHING ARITHMETIC. 23
down the results, so as to make one product.
Thus you avoid writing down all the partial
products, and taking their sum as usual.
Thus : —
2 × 3 gives the 6 units.
32. 3 times 3 tens and 2 times 4 tens give
X 43. 17 tens.
1376. 4 tens × 3 tens gives 12 hundreds; plus
the 1 hundred to carry=13 hundreds.
The multiplications may be pictured thus : —
3d 2d 1st The eye quickly gets accustomed to
1, X, 1 adding these products as it goes.
The principle applies to any number of figures in
either factor. It is specially easy and service-
able when the multiplier contains 2 or 3 figures.
c. Note that the drill prepares the way for double-
column adding.
DIVISION.
I. The Division Idea. — Associate firmly and lastingly as
before : —
The actual operation,
The oral statement for it,
The equation form for it,
The problem,
The picture-problem,
The figure-process. 2
TEST.
Given any one of these
* ideas, call upon the
pupil for any of the
others.
i
II. Outline of Work.
1. Teach short division first.
a. Short division for all divisors from 2 to 12,
inclusive.
b. Division by factors of a divisor. Illustrate,
prove, and derive rule for the whole remainder.
24 METHODS IN TEACHING ARITH METIC.
c. Division by aliquot parts of 100 or $1.00. Oral
arithmetic freely.
2. Long Division.
a. Divisors.
a. Grade them (13, 14, 15, 16, and so on, in order).
b. Construct multiplication tables for each divisor in
order to 20, inclusive.
Leave these on the board for reference by pupils
in using these divisors.
b. Dividends. Grade them in difficulty.
a. One period, then two, etc.
b. Multiples, at first.
c. Then dividends requiring remainders.
1. Write remainders as whole numbers.
2. Have them written as part of the quotient.
c. Quotients.
a. Write them above the dividend, the first figure over
the last figure of the partial dividend.
1. It brings the quotient nearer the divisor in
multiplying, and
2. It keeps the pupil out of confusion when
there are naughts freely in the dividend.
1446.
26)37570.
26.
115
104
117
104
130
130
III. Note. —Long Division is distinctly hard for the average
child. It can be rendered easy only by carefully
grading the steps in teaching. The notes above have
this special purpose in view.
c):
METHOIDS IN TEACH ING ARITHM ETIC. **)
CHAPTER V.
FRACTIONS.
I. Objective and Oral Treatment.
1. Course of study.
(2.
.
%
J.
#’s with a 2-pane window (or drawing).
#'s with a 3-pane window (or drawing).
#’s and 4's with a 4-pane window (or drawing).
}'s with a 5-pane window (or drawing).
#’s, A's, and 's with a 6-pane window (or
drawing). §
}’s with a 7-pane window (or drawing).
#'s, 4's, and A's with an 8-pane window (or
drawing).
A's and #’s with a 9-pane window (or drawing).
#'s, l's, and l'o's with a 10-pane window (or
drawing).
#’s, A's, 4's, l's, and P3's with a 12-pane window
(or drawing).
2. Use strips of paper, drawings of pies, etc., for a
variation.
3. This course can easily be given — ought to be given —
in the work of the first two years, with first and
second reader pupils.
4. Aids.
(2.
b.
Davis's “Suggestions for teaching Fractions,”
Bardeen & Co., Syracuse, N. Y.
Sloane’s “Lesscns in Fractions,” D. C. Heath &
Co., Boston.
26 METHODS IN TEACH ING ARITH METIC.
c. For oral exercises, see Colburn’s “Mental Arith-
metic,” Houghton, Mifflin, & Co., Boston.
ſº You cannot have too much simple oral work in
this course.
II. Illustrations. Some details of work in 3's, 's, and 's
with a six-pane window.
1. Reduction.

1 = } 1} = } # = ?
# = } 1} = } # = ?
# = } 1} = { # = ?
# = } 23 = } | * = ?
Pupils draw windows for these equations. Or show with
strips of paper, folded and torn. Always keep one strip untorn,
to show the unit.
2. Addition.


# -- * = ? J | } + 1 = ?
# + 3 = ? 13 + \ = ? 1% + 3 = ? etc.
Pupils illustrate as in the first two equations. Call for
problems from the children.
3. Subtraction.
| | TFE— | | | TFJ
[T]
1 — 3 = ? 1} – } = }
§ – A = ? § – 4 = ? § – 3 = ? 14 – 3 = ?
||| REGſ, Illustrates § — $. Vary, using
strips of paper or drawings. Pupils make problems for each
equation.
METHODS IN TEACHING ARITH METIC. 27
4. Multiplication.


sº-, sº
(; T • R {-º-º-º:
(2 times #) º 5 × 1 = ?
2 × 1 = ? 3 × 1 = ? 2 × 1 = ? 2 × 13 = ? etc.
You can easily make multiplication too difficult at the start.
Multipliers now to be whole numbers. After Partitioning, they
may be fractions and mixed numbers.
5. Division.
4 #
DSTSs.RS] # H | H || ||

1} + k = 4;
1 + 1 = ? § -- = ? A + i = ? § -- # = ? 13 + 4 = ?
The same caution here as before. Go from easy to difficult
gradually. Call for problems, especially for these forms in
multiplication and division.
6. Partitioning.
L-T-s


| | | | | | | | TTº
# of 1 = } # of # = ?
# of # = ? § of 13 = ? § of 13 = ? etc.
Note the form of the Partition Story in Chapter II., and how
it differs from the Division story.
III. Busy-work.
1. Give the drawings; children give the equations.
2. Give the equations ; children illustrate them with
drawings.
28 METHODS IN TEACHING ARITHM ETIC.
3. Give the equations; children write problems for them.
4. Give a variety of equations; children complete them.
Illustrations : —
* + 2 = | | 14 -i- ? = 4; # of 2 = #| ?-- 3 = 3
1. – ? = { | ? of 4 = }| ? × 13 = 23 || 5 × 2 = 14
? × 1 = 2\| 3 -i- ? = 2 || 3 – ? = }| ? – 4 = }
etc.
IV. Purposes.
1. To ground the understandings of pupils safely in the
facts of fractions.
2. To preface the formal course in definitions, rules, and
figure processes.
3. To drill them into an easy and certain mastery of the
larger fractional units, the fractions they will most
need in every-day life.
W. What Next 2
1. The book-course in fraction forms in any good text.
2. Special attention to cancellations.
3. Fractions to be worked in perpendicular arrangements,
wherever possible.
43
* for i º f :
× 3} or instance, instead O
2% 43 x 34 = * x , = # = 164.
14 3 2 3
16,
METHODS IN TEACHING ARITHM ETIC.
CHAPTER VI.
1. Illustrations.
a. A silver dollar, Io dimes, and Io cents.
b. A drawing of a large cube containing 1000 small
cubes.
DECIMALS.
. Objective and Oral Treatment.
29
Outline the top layer in colored chalk; the left-hand
upper row front in another color of chalk; and the upper
cube in the nearest corner in another color.

II. Development Questions.
1.
2.
3.
How many layers ?
1 layer = ? part of
the cube.
How many rows in
each layer?
How many rows in
the cube 2
1 row = ? part of
the cube.
How many little
cubes in 1 row Ż
in 1 layer? in the big cube 2
Sºś
S
S
N
<>
3
&
&
3
3
>
3
&
3.
<>
Sºś
*
SRS
<>
*Sº
<>
>
<>
<>
S-S-
Sºc
<>
<
3
3:
*Sº
<>
S
S
s
1 Wittle cube = ? part of the
big cube.
Sam may go to the board, and fill out these statements:
<>
S&S
<>
>
3.
3.
1
1 layer
1 row
= 3 of the cube.
= i. of the cube.
1 little cube = + of the cube.
ſ
L’
<>
2
30 MFTHODS IN TEACHING ARITH METIC.
9. I will show you another way of writing l'o's, rºo's, 10% o’s.
Thus,
1 cube = 1. 1, 6 cube = .01
To cube = .1. rooo cube = .001.
10. Class may go to the board. The decimal forms are
taught as follows: —
(4 big cubes. 4. 11. Test. ( 1.
3 layers. .3 .8
1 big cube and 2 .02
layers. 1.2 Show me | 1.35
8 layers. .8 these in the l .003
Write - 3 rows. .03 cube. | 032
gº 1 big cube, 2 layers, How many j .253
and 3 rows. 1.23 layers ? rows 2 | 1.876
2 little cubes. .002 | little cubes 2
3 layers and 2 rows. .32
1 big cube, 2 layers, 5
\ rows, Slittle cubes. 1.258 \
12. You write one dollar thus, $1. Show the coins in the
following. O
13. This (1 dime) is what part of a dollar? Write it.
These (3 dimes) are what part of a dollar 2 Write.
This (1 cent) is what part of a dollar 2 Write.
These (9 cents) are what part of a dollar 2 Write.
Write these (3 dimes and 5 cents). Read.
Write these ($1, 8 dimes, and 3 cents). Read.
14. Explain that the last decimal unit to the right gives
the name to the whole decimal.
Read these as money; as decimals.
$3.85 || $1.10 $ .056 $ .005
$ .03 || $ .05 $3.083 || $1.853.
METHODS IN TEACHING ARITH METIC. 31
III. Tests. – The exercises in any good arithmetic involving
10ths, 100ths, 1000ths; say Cook & Cropsey's,
pp. 252, 253, §'s 8, 9, 10, 11.
IV. What next 2
1. Take up the book-course in decimals in any good
text-book.
2. Stress 10ths, 100ths, and 1000ths. Teach all the diffi-
culties of decimals — reading and figure processes —
with these. Later you can treat the smaller decimal
units.
3. Teach decimals very largely with U. S. money problems.
32 METHODS IN TEACHING ARITHM ETIC.
CHAPTER VII.
PERCENTAGE.
I. Objective and Oral Treatment.
1. Course of study. The following %'s pictured and
treated in the order indicated in abundant and
varied oral exercises : —


tºº



5. 6. o
b. Beginning with 100%, and at last ending with 1%.
c. Vary illustrations. Use squares occasionally.
Draw square with 100 little squares in it for
the 1%.
II. Development Questions and Drills.
Tell that % means per cent. Write per
cent on the board. Explain nothing else
now. The meaning of the term is to be
taught, not defined or explained.

1. Here is a blackberry pie.
:
10.
11.
METHODS IN TEACHING ARITH METIC. 33
How have I marked it 2
How much of that pie is 100% of it?
All of the pie is what % of it?
All of my hat is what % of it?
All of these desks are what % of them 2 all of
anything?
100% of 4 = ? Of 6 = ? Of 1 = ? Of 100 = ?
Of 200 = ?
Bought shoes at $1.50, and sold for $3.00. 9% of gain :
1. Cost what?
a. An invariable | 2. Sold for what?
order of 3. Made how much 3
4
O
quiz. . Made what part of the cost?
. Made what %
Bought readers at 25%; sold for 50%. 9% of gain
Quiz as above; and so on till the facts are easily
translated by pupils into terms of 100%.
Sold candy at 20%, and made 10¢ ; cost 7%
1. Sold for what 2
& 2. Made what 2
a. Quiz 3 hat 2
form. . Cost what :
4. Made what part of the cost 2
5. Made what %
My cloth cost 16% per yard; want to make 100% :
selling price 2
Want to make how much
Sell at what then 2
1. Cost what ?
;, 2. Want to make what %
a. Quiz
f 3. Want to make what part of the cost 2
orm. |4
5.
and so on, with a great variety of examples.
METHODS IN TEACHING ARITHMETIC.
:
10.
11.
12.
13.
14.
15.
16.
What is it? A blackberry pie.
How many equal slices 2
The whole pie is what %
Each slice is what %
5. How many 50 %’s in a pie 2
:

50% is what part of the pie 2
# of the pie is what % of it?
# of this class is what per cent of it 2
# of anything is what % of it 2
50 % of anything is what part of it 2
50 % of 100 = ? of 25 = ? of 4 = ? of 2 = ?
Then go right into the three cases, and employ the
three forms of quiz, always asking the questions
of the quiz in the same order.
[ºf Don’t memorize the questions; master the order
of them.
Then treat the other 9%’s given in the order in which
they are given, ending with 1 %.
Quiz rapidly; call for a show of hands; watch the
weak pupils; address most of your teaching to
them.
Treat the four forms of Percentage noted as funda-
mentally important in Chapter I. Then the others,
if you have time.
III. What Next.
1. The book-course in Profit and Loss, Commission-
Taxes, Interest, in any good text. Terms, defini-
tions, processes, and rules in Wentworth’s “Grammar
School Arithmetic” are specially good. The prob-
lems in “The New Arithmetic” (D. C. Heath & Co.,
Boston) are much better than usual.
METHODS IN TEACHING ARITHMETIC. 35
IV. Two Methods in Interest.
1st Method. (By courtesy of D. L. Earnest, Georgia
Normal and Industrial College.)
a. Explanation. Interest $1., 12 mo., at 6% = .06;
for 2 mos. = .01. Pointing off two decimal
places in the principal gives interest for 2 mos.
Take half of that, and multiply by the time in
mos. (Reduce days to decimal of a mo. Point
off 1 decimal, and divide by 3.)
Example: Int, on $840. for 2 yrs., 6 mos.,
18 da., at 6% -
$8.40 Int. for 2 mos. For 5%, subtract #.
$4.20 “ “ 1 “ For 7%, add #, etc.
x 30.6 The time in mos.
2 5 20
_1 260
$ 1 28.520
2d Method. State the process in full; employ cancellation.
Illustration : Int, on $400., 63 da., 5%.
50 .01 7
* * x 63 = $3.50.
#2 *
ź -
36
METHODS IN TEACHING ARITHMETIC,
NOTES.
An indicated division of one
number by another of the
same kind.
CHAPTER VIII.
4. Properties } :
1.
5. Reduction of {:
3.
PROPORTION.
SOE (EMI.E.
I. The Ratio.
1. What? 1.
2. Written? 2
3. Read? 3
4. Terms 1. Numerator, 4.
2. Denominator.
5. Reduction of É. 5.
II. The Proportion.
1. What? 1.
2. Read? 2.
1. Numerators and
denominators.
3. Terms 2. Means and ex-
tremes,
\
. 8 : 2; 3: 12.
. 8 divided by 2.
3 divided
by 12.
8 is numerator; 2, denomi-
Inator.
Same as any fraction.
An equality of ratios.
2 : 6 = 4: 12. 2 divided by
6 = 4 divided by 12.
º
. Equality of ratios.
2. Equality of products of
extremes and means.
. By reduction of the ratios.
2. By multiplying or divid-
ing both numerators by
the same number.
3. By multiplying or divid-
ing both denominators
by the same number.
METHODS IN TEACHING ARITH METIC 37
III. Statement of the Problem.
1. General statement.
2. Special statement.
3. Test.
IV. Solution.
1. Reduction of proportion.
2. Reduction to an equation.
3. Reduction of the equation.
V. Additional Notes.
1, 1st cause : 2d cause = 1st
effect : 2d effect.
2. Of the particular problems.
3. (1) Drill pupils thoroughly
in statements first. (2) In-
dicate the unknown quan-
tity by (?). (3) Name
each quantity in the pro-
portion.
1. Apply the three laws of can-
cellation in the proportion
just as it stands.
2. Write product of extremes =
to product of the means.
3. Divide one product by the fac-
tor associated with the (?).
1. Here are the necessary facts stripped of unessentials.
2. Ideas, terms, and signs are related to what the pupil
already knows. Note the changes.
3. Have pupils to put down as first cause the very first
quantity in the problem, always.
4. Cause and effect is not arithmetical proportion per se;
but it is the very simpiest approach to the subject,
and serves all practical purposes.
5. Illustration of the only kind of puzzle to children.
If 5 men can do a job of work in 5 days, how long
will it take 3 men to do it
?
1st cause : 2d cause = 1st effect : 2d effect.
5 men . 3 men
5 da. • ? da. -(tº rº)
3 P = 25
? = **
=x 8} da.
38 METHODS IN TEACHING ARITHM ETIC.
CHAPTER IX.
SQUARE ROOT.
I. Ideas and Terms.
1. A square root is the root (= side) of a square.
2. 8649 is a perfect square. It may be pictured thus : —
3. If the area of the square is 8649,

how find its root; that is, its
side 2
II. An Objective Solution. * 8649
8100 Wanted the line A.C.
8649 =
— 8100 A B C
–549 = or 2 's +
Now, 1 side =90 :
of the *
which is the line AB. The
line BC is the
end of the 9
Therefore : and the side of the little
2 × 90 × 2 + 2* By substitution.
–549 = 180 º’s + 2* 549 = about 180 °’s, or,





180 × 3 + 3* {1} = 3. Substitute 3 for
540 -- 9 = 549 the 2 and test equality.
METHODS IN TEACHING ARITHM ETIC. 39
III. Process.
1. Take out the largest known square.

2. The remainder equals 2 ’s +
3. Multiply the long side of the . by 2,
and use it for a trial divisor.
4. Substitute the quotient for the (?), and test the equality.
If it satisfy the equality, -
5. Add the two lines thus found, and the result is the side,
or root, wanted.
IV. Notes.
1. Practise the pupils in this form of solution till they are
completely in possession of all the ideas involved.
2. Or till they can extract roots of two places by inspec-
tion readily.
3. Then take up the subject in the books, and have them
master the definitions, and explain every step in the
rule there.
28 MA. THEMA 7'/CS.
The Pupils' Series of Arithmetics.
By W. S. SUTTON, Supt. of the Houston Public Schools, and W. H. KIM-
BROUGH, Principal Oak Grove School, Dallas.
Lower Book. 212 pages. Cloth. Introduction price, 35 cents. By mail, 40
cents. Higher Book. For Intermediate and Grammar Grades. 280 pages.
Half leather. Introduction price, 60 cents. By mail, 70 cents. The Primary
Section and the Intermediate Section of the Lower. Book are bound separately to
meet special cases. Primary, 18 cents. By mail, 22 cents. Intermediate, 20
cents. By mail, 24 cents.
H E Primary Book embraces the four fundamental operations, the
whole book is devoted to practical work. While this is true,
sufficient theory, without consumption of space, is introduced by a
system of lessons in ORAL WORK and ORAL DRILLs to develop in the
pupil's mind and almost unaided perception of the truths necessary to
to his advancement.
The Intermediate Book embraces practical work through the four
operations cancellation, factoring and properties of numbers, simple
and decimal fractions, percentage and simule interest, — the pupil
being led to the comprehension of truths before formal definitions and
rules are thrust upon him.
The Higher Book embodies within its 275 pages all that is essential
to a thorough teaching of the principles of numbers. It will be wel-
comed by those progressive teachers who sympathize with the demand
for shortening and properly adjusting the distorted course of secondary
studies.
C. C. Cody, Prof. Math., S. W.
Univ., Georgetown, Tex.: You have
eliminated the old, worn-out relics of the
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thing practical, thereby actually bridging
the chasm between the dreary work in the
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Oscar H. Cooper, Supt. Galveston
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I find much to commend in the book.
R. H. Buck, County Supt., Fort
of practical problems, it commends itself
to the eye and the mind. (Sept. 13, 1892.)
Palmetto Teacher, Columbia, S.
C.: The authors take it for granted that
children are not idiots, and they do not
propose to subject them to the intermin-
able drivel quite too common in these
days.
H. C. Pritchett, Supt. Sam Hous-
ton Wormal Institute: The work indi-
cates careful preparation on the part of
its authors. You will permit me to add
that I am glad to see so excellent a work
prepared by two of our Texas teachers.
Popular Education, Boston : We
Worth, Tex.: In typographical appear. are glad to see so much space in this
ance, suitable arrangement, and its legion | volume given to oral exercises.
A/4 7//EM/A 7T/CS. 29
Mathematics for Common Schools.
A graded course in arithmetic, with simple problems in algebra and geometry.
By John H. WALSH, Associate Superintendent of Public Instruction, Brooklyn,
N.Y. Part I. Cloth. , 198 pages. Introduction price, 30 cents. By mail, 35
cents. Part II. Half leather. 411 pages. Introduction price, 65 cents. By
mail, 75 cents.
ATHEMATICS for Common Schools is a one-book arithmetic
in two parts. Part I, the PRIMARY ARITHMETIC, is designed
to cover the work of the first four years, and contains those portions of
the subject needed by all pupils of the common schools: addition,
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fractions; and the most commonly used denominations of compound
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Part II, the GRAMMAR School ARITHMETIC, completes the ordi-
nary course in this subject, and contains, besides, two chapters on
algebraic equations and one on elementary constructive geometry,
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The special features of this work are its division of the arithmetical
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the great number and variety of the examples; the use of the equation
in the solution of arithmetical problems, especially in percentage and
interest; and the introduction of the elements of algebra and geometry.
Believing that there is some foundation for the complaints frequently
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graduates are too often slow and inaccurate in ordinary computations,
the author has furnished throughout the entire work systematic drills
and reviews in the addition, substraction, multiplication, and division
of ordinary numbers and of common and decimal fractions.
Teacher's Manual to Mathematics for Common Schools.
By Jº H. WALSH, Associate Superintendent of Public Instruction, Brooklyn,
N. Y. Cloth. ooo pages. Price ooo.
HE manual contains suggestions as to methods of teaching the
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3O * MMA 7/7/2MA 7'/CS.
Flementary Arithmetic.
&Joº H. WALSH, Associate Superintendent of Public Instruction, Brooklyn,
N. Cloth. 218 pages. Introduction price, 30 cents. By mail, 35 cents.
HIS book, intended for pupils below the fifth school year, com-
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Infermediate Arithmetic.
§Jº H. Walsh, Associate Superintendent of Public Instruction, Brooklyn,
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HIS book contains the first five chapters of the Grammar School
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intended for pupils of the fifth and sixth school years.
Higher Arithmetic.
By John H. WALSH, Associate Superintendent of Public Instruction, Brooklyn,
N.Y. Half Leather. 365 pages. Introduction price, 65 cents. By mail, 75
CentS.
HIS book contains the last six chapters of the Grammar School
Arithmetic previously described, with many sets of examination
papers for supplementary and review work. It is intended for pupils
above the sixth school year.
From the hundreds of highly commendatory letters concerning these
books that have come to us within the last few months, wany of which
are printed in our complete circular, we select the following represen-
tative opinions :
Lee R. Knight, Prin. Kent School, G. T. Fletcher, Agent Mass. State
Akron, Ohio: I have taken much pains Board of Education: The author keeps
to make inquiries among the teachers con- the right end in view, -development of
cerning the merits, and especially the de- thought-power, accuracy, and readiness of
merits, of the Walsh books, and am ex- expression. He begins and proceeds
ceedingly well pleased to be able to report upon the supposition that the child knows
almost universal satisfaction. In fact no some things and can find out other
criticisms worthy a passing notice have things. The exercises for thought and
been offered. All seem delighted with the | practice are commendable for quality and
plan of the books so far. (Wov. 10, 1894.) quantity.
MA 7'HEMA 77CS. 31
7 wo Years with Numbers.
A book for Second and Third Year Pupils. By Charles E. White, Principal
Franklin School, Syracuse, N. Y. Cloth. 207 pages. Introduction price, 35
cents. By mail, 4o cents.
HIS book is intended to be put into the hands of pupils at the
beginning of the second year (or when they have mastered all
combinations to Io) and used for two years. It deals with numbers
thoroughly and progressively, using easy fractions and Roman num-
erals from the beginning. Each lesson takes up a single page and
presents a variety of matter made up from work that has been long
tried and approved by the very best primary teachers. A notable
feature is the simplicity of the language and the means by which full
comprehension of its meaning is secured. Original problem pictures
and problem making are required.
A. B. Blodgett, Supt. of Schools, Syr- Miss E. Harding, Buffalo, N.Y.:
acuse, M. Y.: I know this book will meet | It seems to be just what we have been
a great need in primary number teaching in looking for, and we hope to be able to in-
our public schools. It will not only do |troduce it into our schools. (jan. 14,'92.)
away with just criticisms on exclusive Wnn. E. Buck, Sze
tº ge e = ** | * p pt. of Schools,
blackboard work,-straining of the eyes, Manchester, N. H.: The material and
tº: but it. will .Ct i. : arrangement of the lessons are admirable
breviated questions and hastily, blindly and and unusually well adapted to the purposes
incorrectly worded problems. It comes as for which they were designed
a boon to primary pupils and teachers and y
will lead to more progressive thinking and (jan. 13, '92.)
doing. A. I. Branham, Supt. of Schools,
L. McCartney, Supt. of Schools, Glynn Co., Ga.; I am greatly pleased
Hannibal, Mo. : The plan of the book is with it and shall introduce it in our schools
9 y
excellent. No teacher can follow the lººt * (jan. 12, '92.)
spirit of it without reaping rich results and L. R. Fowler, Supt. of Schcols,
none will be misled by its letter. Dunmore, Pa...: I like the book. It
(jan. 13, '92.) |ought to take well. It is just the book in
W. H. Hatch, Supt. of Schools, oak the primary department. (7am. 14, '92 )
Park, Ill; I am pleased with what I see | S. S. Kemble, Supt. of Schools,
upon looking hastily through the book. Rock Island, Ill. The book carries an
I want to give it a trial. air that attracts and interests. It pleases
I. N. Mitchell, State Normal, Mil. me very much. (jan. 22, '92.)
waukee, Wis. . I find it is 3. good book. EI. M. Sawyer, Council Bluffs, 'a. ."
I like its plan and its contents. * For years I have felt the need of a simple
J. S. McClung, Supt. of Schools, practical book in numbers for second and
Pueblo, Colo.: It appears to me to be an third grades. This is just the book, and
excellent book. }I cordially give it my endorsement.
32 MA 7THEMA 7TWCS.
Complete Graded Arithmetic.
By GeoRGE E. Atwood, Principal of Grammar School, Tarrytown, N.Y. Part
First. For Fourth and Fifth Grades. Cloth. 200 pages. Introduction price,
30 cents. Price by mail, 35 cents. Part Second. For Sixth, Seventh, and
Eighth Grades. Half leather. 382 pages. Introduction price, 65 cents. Price
by mail, 75 cents.
This books present a carefully graded course in arithmetic to
begin with the fourth grade, or fourth year, and continue through
the eighth grade. They are a departure from the stereotyped text-
books in this subject. The departure consists in an entirely new and
different plan and arrangement of the work and in the omission of much
commonplace theory usually contained in arithmetics.
All needed work for a course in arithmetic extending over five years
is planned, prepared, and arranged. This statement implies: —
1. That the work is planned for the teacher in every respect. It is
first divided into grades, the work of a grade being the work for one
year. The division of the work by topics does not appear. The work
of each grade is divided into 150 lessons, each lesson being intended
for a day's work. The order and time of presenting new work are
indicated by notes to teachers. These notes are given at the end of
\essons, and they indicate that new work will begin in the next lesson.
2. That there is abundance of work to develop skill in the funda-
mental processes of arithmetic, and that there is also a sufficient
amount and variety of work to secure the greatest possible development
of intellectual power.
3. That each lesson has its review as well as new work.
4. That, in view of the unusual amount of all kinds of needed work,
no supplementary work will be needed, either for the purpose of
reviews or for the purpose of giving additional work. Teachers are thus
relieved of the burden of providing supplementary work.
5. That these books can be used with the minimum amount of labor
on the part of the teacher and with the best possible results.
This book forms with the White's “Two Years in Numbers'
(designed for second and third grades) a complete course in arith.
metic for graded and common schools.
Geo. Griffith, Suet. Schools, Utica, J. O. Briggs, Co. Suet. Schools, M.
M. Y.: They are decidedly and refresh- Sterling, Ills.: I like them better than any
ingly out of the usual rut. (Dec. 21, 1893.) | Arithmetic I have seen. (49ec. 25, 1893.)
MA 7THEMA 7'/CS. 33
7%e New Azithmetic.
By 3oo authors. Edited by SEYMOUR EATON, with Preface by TRUMAN
HENRY SAFFORD, Professor of Astronomy, Williams College, Mass. Fifteenth
Edition. Cloth. 230 pages. Price by mail, 75 cents. Introduction price, 65
Cents.
HE exercises in this book were selected from more than three
hundred contributions sent to the editor by teachers in all parts
of the world. Explanations of theory were purposely omitted—the
endeavor having been to prepare a pupil's hand-book of exercises.
The practical character of the book will commend it to all who have
at heart the mental development and the business success of their
pupils. All problems requiring much labor and affording very little
practical or intellectual benefit have been avoided. It is so graded
that the elementary departments prepare the pupil for the study of
algebra, geometry, and the higher mathematics, while the more ad-
vanced work prepares him for the active duties of every-day life, and
at the same time gives him such mathematical training as he may not
have time to secure in any other way. There is a complete set of
answers, which may also be had bound separately.
H. F. Fisk, Prin. Prep'y School of
Northwestern Univ., Ill.: I very cordial-
ly admire it. It is superior to any other
I have seen in the adaptation of its exer-
cises to develop in the pupil the habit of
rapid and accurate work.
Albert G. Boyden, Prin. State
Mormal School, Bridgewater, Mass.: It
is a valuable book for examples. They
are well selected and in good variety. We
have found it very useful for drill work.
H. Conover, St. Paul’s School, Con-
cord, W. H. : I am greatly pleased with
it. All good teachers, I think, will admit
that for the average student methods are
best taught and explained viva voce, and
what one needs for his pupils is just what
this book gives, – plenty of good ex-
amples and a few hints.
C. L. Hunt, Supt. of Schools, Brain-
tree, Mass. : It is giving much pleasure
and satisfaction
W. H. McFarland, Prin. Peary St.
Schools, Springfield, Ohio : I have used
it ever since it came out. I never saw an
arithmetic worth one-tenth to me that this
one is.
G. I. Hopkins, Instructor in Mathe.
matics and Sciences, High School, Man-
chester, M. A. : I have examined it with
an interest that gradually grew to enthus-
iasm. It is a book after my own idea.
C. S. Campbell, Prin. McCollome
Inst., Mt. Vernon, W. H. : I like the
book for our use better than any other
that I have seen. I expect to continue its
use. I like it for its onission of artificial
methods, set rules and tables, and obsolete
or useless subjects.
J. H. Lee, Supt. of Schools, Riley
Co., Kans.: I think it the completest
work of the kind I have ever seen. It
gives the facts and principles of arithmetic
in a nutsk- and there is apparently not a
Superfluous word in the book,
MUMBER.
White's Two Years with Numbers. Number Lessons for second and third year
pupils. 4o cts.
Atwood's Complete Graded Arithmetic. Present a carefully graded course in
arithmetic, to begin with the fourth year and continue through the eighth year. Part I.
200 pages. Cloth. 4o cts. Part II. 382 pages. Half leather. 75 cts.
Walsh’s Mathematics for Common Schools. Special features of this work are
its division into half-yearly chapters instead of the arrangement by topics; the omission,
as far as possible, of rules and definitions; the great number and variety of the problems;
the use of the equation in solution of arithmetical problems; and the introduction of the
elements of algebra and geometry. Part 1. 218 pages. 35 cts. Part. II. 252 pages. 4o cts.
Part III. 365 pages. Half leather. 75 cts.
Sutton and Kimbrough's Pupils’ Series of Arithmethics.
PRIMARY Book. Embraces the four fundamental operations in all their simple relations.
80 pages. Boards. 22 cts.
INTERMEDIATE Book. Embraces practical work through the four operations cancellation,
factoring and properties of numbers, simple and decimal fractions, percentage and simple
interest. 128 pages. Boards. 25 cts.
Lower Book. "Combines in one volume the Primary and Intermediate Books. 208
pages. Boards, 3o cts. Cloth, 45 cts.
High ER Book. A compact volume for efficient work which makes clear all necessary
theory. 275 pages. Half leather. 7o cts.
Safford's Mathematical Teaching. Presents the best methods of teaching, from
primary arithmetic to the calculus. Paper. 25 cts.
Badlam's Aids to Number. For Teachers. First Series. Consists of 25 cards for
sight-work with objects from one to ten. 4o cts.
Badlam's Aids to Number. For Pupils. First Series. Supplements the above
with material for slate work. Leatherette. 3o cts.
Badlam's Aids to Number. For Teachers. Second Series. Teachers' sight-work
with objects above ten. 4o cts.
Badlam's Aids to Number. For Pupils. Second Series. Supplements above with
material for slate work from 10 to 20. Leatherette. 3o cts.
Badlam's Number Chart. 11 x 14 inches. Designed to aid in teaching the four
fundamental rules in lowest primary grades. 5 cts, each; per hundred $4.o.o.
Luddington's Picture Problems. 70 cards, 3 x 5 inches, in colors, to teach by pic-
tures combinations from one to ten. 65 cts.
Pierce's Review Number Cards. Two cards, 7 x 9, for rapid work for second and
third year pupils. 3 cts. each ; per hundred $2.40.
Howland’s Drill Card. For rapid practice work in middle grades. 3 cts. each; per
hundred $2.40.
For advanced work see our list of books in Mathematics.
D. C. HEATH & Co., PUBLISHERS,
BOSTON. NEW YORK. CHICAGO,
|liili
ITY OF MICHIGAN
|
3 9015 08028 O764
Bowser's Academic Algebra. A complete treatise through the progressions, includ-
ing Permutations, Combinations, and the Binomial Theorem. Half leather. $1.25.
Bowser's College Algebra. A complete treatise for colleges and scientific schools.
Half leather. $1.65.
Bowser's Plane and Solid Geometry. Combines the excellences of Euclid with
those of the best modern writers. Half leather. $1.35.
Bowser's Plane Geometry. Half leather. 85 cts.
ry
Bowser's Elements of Plane and Spherical Trigonometry. A brief course
prepared especially for High Schools and Academies. Half leather. $1.o.o.
Bowser's Treatise on Plane and Spherical Trigonometry. An advanced
work which covers the entire course in higher institutions. Half leather. $1.65.
Hanus's Geometry in the Grammar Schools. An essay, together with illustrative
class exercises and an outline of the work for the last three years of the grammar school.
52 pages. 25 cts. *
Hopkin’s Plane Geometry. On the heuristic plan. Half leather. 85 cts.
ry
Hunt’s Concrete Geometry for Grammar Schools. The definitions and ele-
mentary concepts are to be taught concretely, by much measuring, by the making of
models and diagrams by the pupil, as suggested by the text or by his own invention.
1oo pages. Boards. 3o cts.
Waldo's Descriptive Geometry. A large number of problems systematically ar-
ranged and with suggestions. 9o cts.
The New Arithmetic. By 300 teachers. Little theory and much practice. Also an
excellent review book. 230 pages. 75 cts.
For Arithmetics and other elementary work see our list of books in Mumber.
D. C. HEATH & CO., PUBLISHERS,
BOSTON. NEW YORK. CHICAGO.