ROOKS' MENTAL ARITHMETIC,
PHILADELPHIA:
SOWER, BARNES & CO.
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METHODS
TEACHING MENTAL ARITHMETIC,
KEY
THE NORMAL MENTAL ARITHMETIC.
CONTAINING ALSO MANY SUGGESTIONS AND METHODS FOR ARITH-
METICAL CONTRACTIONS, AND A COLLECTION OF PROBLEMS
OF AN INTERESTING AND AMUSING CHARACTER,
FOR CLASS EXERCISE.
BY
EDWARD BROOKS, A. M.
PROFESSOR OF MATHEMATICS IN THE PENNSYLVANIA STATE NORMAL SCHOOL.
SOWER, BARNES & CO.
Entered, according to Act of Congress, in the year 1860, by
EDWARD BROOKS,
In uie Clerk's Office of the District Court of the United States, in and for the
Eastern District of Pennsylvania.
HEARS * DUSENBERY, STEREOTYPED,
PEEFACE.
Tins little volume consists of four distinct parts :
First. An exposition of some of the principles and
methods of successful instruction in the science of Mental
Arithmetic.
Second. A fuller development of the principles pre-
sented in "The Normal Mental Arithmetic/' by many
remarks and suggestions, and by the solution of some of
the more difficult problems.
Third. The presentation of quite a number of
methods of numerical computation by contractions, &c. ;
the object of which is to make pupils ready and accurate
in the mechanical operations of Arithmetic.
Fourth. A collection of a large number of problems
of an amusing and interesting character, under the head
of "Social Arithmetic/' to be used to awaken interest in
a class, or entertain a social circle.
With regard to the second part of the work, in which
it may be considered a " Key/' it will be noticed, that it
differs from Keys, generally, in presenting many remarks
165027. (0
11 PREFACE.
and suggestions, and also in solving only a few of the
more difficult problems, so that even if a class of pupils
should happen to make use of it, in preparing their
lessons, they will find great variety still among the un-
solved problems, for the exercise of their own ingenuity.
The author hopes that the book may be found of value
to the private student, to the teacher of Mental Arith-
metic, and particularly to his numerous friends, who
have shown their appreciation of his former works ; and
thus, in various ways, lend assistance to the great cause
of the age, the cause of Popular Education.
METHODS OF TEACHING
MENTAL ARITHMETIC.
CHAPTER I.
ARITHMETIC is the logic of numbers, and hence its
truths and principles should be derived by logical pro-
cesses. The four logical processes by which these truths
and principles are obtained, are Analysis, Synthesis,
Induction, and Deduction.
ANALYSIS. Analysis is the process of resolving that
which is complex into its elements. A watch is a com-
plex being, consisting of wheels, springs, pins, hands,
&c.. Now, if we take a watch and separate it into these
parts, we are said to analyze it. A house is analyzed
when we separate it into the brick, wood, stone, mortar,
iron, &c., of which it is built. Analysis then meaos
separating, taking to pieces, resolving the complex into
the parts of which it is composed.
SYNTHESIS. Synthesis is the reverse of Analysis. It
is the process by which we form a complex object from
simpler objects. Thus, if, after taking the watch apart
by Analysis, we put the different parts together again,
the process is called Synthesis. The building of a house
is a synthetic process. The putting together of words to
form a sentence, is also a process of Synthesis.
1* (3)
4 METHODS OP TEACHING
INDUCTION. Induction is the logical process by which
we derive general truths from particular ones. Thus,
when we notice that heat expands iron, lead, copper, &c.,
we infer that heat will expand all metallic bodies, and
this process of inferring the general truth is called In-
duction. As simple as this logical process may seem, it
is of vast importance in nearly every department of
science Its utility has been long recognised in the
Natural and Philosophical sciences, but not in the science
of Mathematics ; it is, however, of very great importance
even here, particularly in Arithmetic. In fact, it has
long been used in Arithmetic, although apparently un-
consciously, and also in a slight degree in Geometry and
Algebra. Whenever a general truth or principle is
derived in Arithmetic, without demonstration, it must be
so derived by Induction. We have employed this pro-
cess in Arithmetic as a logical method of procedure, and
one which we claim to be of much importance. Its appli-
cation and utility will be particularly seen in the treat-
ment of Fractions.
DEDUCTION. Deduction is the reverse of Induction.
In Induction, as has been seen, we pass from the special
to the general; in Deduction we reverse this process,
and pass from the general to the special. Thus, having
derived by Induction our general principle that heat ex-
pands all metallic bodies, we infer that heat will expand
zinc and silver, since zinc and silver are metals. Induc-
tion is an ascending process we go up from the parts
to the whole } Deduction is a descending process we
descend from the whole to the particulars. Induction is
a synthetic process, it puts together; while Deduction is
an analytic process, it takes apart, or separates the few
from the many or the whole.
INDUCTIVE TEACHING. There is a method of teaching
so intimately allied to the logical process of Induction,
that educators have given it the name of the Inductive
Method. It consists in leading the pupil along, step by
step, to the conclusion, announcing such conclusion only
MENTAL ARITHMETIC. 5
after it is clearly seen by the pupil. Thus, in the defi-
nition of Addition, by the Inductive method, we would,
by appropriate questions, lead the learner to a clear idea
of Addition, and then, but not till then, we would give
the name to the process, and also the definition of the
name.
DEDUCTIVE TEACHING. The Deductive method of
instruction is the reverse of the Inductive method, and
is so called because it resembles deductive logic. By
this method a term, and the definition of it, are both
given before the pupil has any idea of the thing defined,
and then he is led to an understanding of it by appro-
priate examples and illustrations.
Without entering into a discussion of the merits of
these two methods, the author would remark that he
prefers the Inductive method for beginners, and the
Deductive for more advanced pupils. In the oral and
mental exercises of the "Primary Arithmetic" the Induc-
tive method has been employed, and also, for the most part,
in the " Normal Mental Arithmetic/' In the more ad-
vanced departments of Mathematics, the Deductive
method is preferred.
CHAPTER II.
WE have, in the previous chapter, given a brief outline
of the general principles of Arithmetical Reasoning and
Instruction. These principles will be found applied in
the two little works of the author THE NORMAL
PRIMARY ARITHMETIC and THE NORMAL MENTAL
ARITHMETIC.
Arithmetic, considered from the stand-point of teaching,
should consist of Oral Arithmetic, Mental Arithmetic,
and Written Arithmetic. It will be noticed that we use
the terms Mental and Written, instead of the more fre-
6 METHODS OF TEACHING
quently used ones, " Intellectual" and " Practical."
Our reason for this is, that the terms are more appro-
priate, as a very little thought will show. The term
"Practical" as applied to Arithmetic, is a misnomer
all Arithmetic should be practical. In one case we solve
questions mentally, and in the other case we employ
written characters to aid the mind in the computation ;
hence the propriety of the division of Arithmetic into
Mental and Written.
ORAL ARITHMETIC. Oral Arithmetic consists of such
instructions in the science and art of numbers as should
precede the use of a text-book by the student. The
learner needs such instructions for several reasons. First,
pupils can learn Arithmetic before they can read, and
hence, of course, before they can use a book. Secondly,
even with pupils who can read, such exercises are a very
valuable preparation to the study of the subject from the
text-book.
The chief instrument to be employed in these oral
exercises, is the "Arithmometer," or Numeral Frame,
although books, pens, pencils, grains of corn, &c., form
a valuable introduction to it.
In the " Normal Primary Arithmetic" we have given
many suggestions for exercises of this kind, which we trust
will be found valuable. The teacher can vary and modify
them to suit the capacity and advancement of the class.
FIRST ARITHMETIC. The author believes that thp
Primary Arithmetic should be based upon the following
principles, some of which have not previously been
recognised in the preparation of such books.
1st. It should consist of both Mental and Written
exercises, and not of either alone.
2d. Addition and Subtraction should be so presented
in the Mental Exercises that they may be taught simul-
taneously, or that an additive process should be immedi-
ately reversed, thus giving rise to a subtractive one.
3d. In the Mental exercises Multiplication and Divi-
sion should be taught simultaneously, Division being
MENTAL ARITHMETIC. /
raade to depend upon Multiplication, as it logically does,
by reversing the multiplicative process.
4*h. The pupils should derive their own " Multipli-
cation Table/' so that they may understand its meaning
and use, being required to commit it to memory after
they have obtained it.
5th. The pictures of objects, marks, stars, &c., should
be omitted, since the objects themselves are better than
their pictures or the marks, &c., and also, since the pupils
should not leave the Oral Exercises until they can operate
without the aid of objects or pictures.
6th. Principles and definitions should be taught In-
ductively, and the methods in fractions should be derived
by inductive inferences from analytic processes.
MENTAL ARITHMETIC. Although the first book, The
Primary Arithmetic, should consist of a combination of
Mental and Written exercises, yet it has been found
most advantageous in teaching to separate them after the
tirst book, having a complete course of Mental Arithmetic
in one book, and a complete course of Written Arithmetic
in another book.
The Mental Arithmetic, it is believed, should be based
upon the following principles :
1st. It should be Analytical and Synthetical. Num-
bers are formed by Synthesis; hence, in the solution of
problems, this synthetic process must often be reversed
by the process of Analysis. Almost every solution, if
properly given, involves the two processes of Analysis
and Synthesis. Since the synthetic process is so easy
after the analysis has been made, the two processes have
been denoted by the term Analysis.
These two processes are included in the more general
process of Comparison. Comparison is more properly
the reasoning process, Analysis and Synthesis the me-
chanical processes.
The simplest process of Comparison which involves
Analysis, is where we compare a number with one, a col-
lection with the unit, since the relation is intuitivelj'
8 METHODS OF TEACHING
apprehended; and the simplest process of Comparison
which involves Synthesis, is when we compare the unit
with the collection. After the pupil becomes familiar
with these elementary processes of Comparison, he begins
to perceive relations existing between different collections
of units numbers and he should be taught to apply
those relations in the solutions of problems. This may
be seen illustrated on pages 42 and 43 of the " Normal
Mental Arithmetic/'
2d. Fractions should be treated by Analysis and Syn-
thesis and Induction, or more briefly by Comparison and
Induction. In the treatment of fractions it is necessary
to obtain methods for operating upon them, such as re-
ducing to lowest terms, to common denominator, &c.
These methods may be derived by Induction from the
results obtained by the analytic process. This feature,
which has not previously been introduced into Arithmetic,
we deem a very valuable one, viewed either from the
stand-point of science or teaching.
3d. The problems should be so varied that pupils will
be forced to think for themselves. If all the problems
under any class of problems, are like the one which is
solved in the text-book, the pupils can take the solution
given and apply it directly to them all, and thus no more
thought is required than by the old method of working
by Rule. To prevent this mechanical way of operating,
we have taken much pains to give great variety to the
problems.
The development of these features, and also of others
which we deem of much value, will be seen by an exami-
nation of the Key, and also of the work itself.
OBJECT OF THE KEY.
This Key is designed to show the manner in which
Mental Arithmetic may be successfully taught. In pur-
suing this object, many remarks and suggestions have
been given, quite a number of solutions presented aa
MENTAL ARITHMETIC. 9
models, and remarks given for the solution of some of
the more difficult problems. The object has not been,
as in Keys generally, to solve the difficult problems
merely, and give results, but to suggest to teachers
Methods of Teaching, which will be of use to them in
the school-room.
When a solution is given as a model for the recitation-
room, we head it with " SOL. ;" but when suggestions
merely are given, we use the heading " Suo." Remarks
are indicated by the abbreviation " REM."
CHAPTER III.
METHODS OP RECITATION.
THE attention of teachers is respectfully solicited to
the following Methods of Recitation. Some of them are
preferable to others, but all may occasionally be used
with advantage.
COMMON METHOD. By this method the problems are
read by the teacher and assigned promiscuously, the
pupils not being permitted to use the book during reci-
tation, nor retain the conditions of the problems by
means of pencil and paper, as is sometimes done. The
pupil selected by the teacher arises, repeats the problem,
and gives the solution, at the close of which the mistakes
that may have been made should be corrected by the
class or teacher.
SILENT METHOD. By this method the teacher reads
a problem to the class, and then the pupils silently solve
it, indicating the completion of the solution by the up-
raised hand. After the whole class, or nearly the whole
class, have finished the solution, the teacher calls upon
some member, who arises, repeats the problem, and gives
the solution as in the former method.
10 METHODS OF TEACHING
By this method the whole class must be exercised
upon every proolem, thus securing more discipline than
by the preceding method. It, however, requires more
time than the first ; hence, not so many problems can be
solved at a recitation. We prefer the first method for
advanced pupils, and the second, at least a portion of the
time, with younger pupils.
CHANCE ASSIGNMENT. This method differs from the
first only in the assignment of the problems. The
teacher marks the number of the lesson and the number
of the problem, upon small pieces of paper, which the
pupils may take out of a box passed around by the
teacher or some member of the class. The teacher then,
after reading a problem, instead of calling upon a pupil,
merely gives the number of the problem, the person
having the number arising, repeating, and solving it.
By this method the teacher is relieved of all responsibility
with reference to the hard and easy problems, and it is
also believed that better attention is secured with it. It
is particularly adapted to reviews and public exami-
nations.
DOUBLE ASSIGNMENT. By this method the pupil
who receives the problem from the teacher, arises, repeats
it, and then assigns it to some one else to solve. It may
be combined with either the first or second methods.
The objects of this method are variety and interest.
METHOD BY PARTS. By this method different parts
of the same problem are solved by different pupils. The
teacher reads the problem and assigns it to a pupil, and
after he has given a portion of the solution, another is
called upon, who takes up the solution at the point where
the first stops; the second is succeeded in like manner by
a third ; and so on, until the solution is completed. The
object of this method is to secure the attention of the
whole class, which it does very effectually. It is parti-
cularly suited to a large class consisting of young pupils.
UNNAMED METHOD. By this method the teacher
reads and assigns several problems to different members
MENTAL ARITHMETIC. 11
of the class, before requiring any solutions, after which
those who have received problems are called upon in the
order of assignment for their solutions. The advantages
of this method are, first, the pupil, having some time to
think of the problem, is enabled to give the solution with
more promptness and accuracy, and secondly, the neces-
sity of retaining the numbers and their relations in the
mind for several minutes, affords a good discipline to the
memory.
CHOOSING SIDES. This is a modification of the old
spelling class method, and is one calculated to elicit a
very great degree of interest. By it two pupils, appointed
by the teacher, select the others, thus forming two parties
for a trial of skill, as in a game of cricket or base ball.
The problems may be assigned alternately to the sides,
by the teacher, by chance, by the leaders of the sides, or
in any other way that may be agreed upon by the teacher
and class.
In regard to these methods, the first, second, and third
are probably the best for the usual recitations, but the
other methods can very profitably be employed with
younger classes, or, in fact, with any class, to relieve
monotony and awaken interest. With advanced pupils
we prefer the first method, or the first combined with the
third.
ERRORS TO BE AVOIDED.
There is a large number of errors to which pupils in
every section of the country are liable, a few of which
\ve will mention. We classify them as errors of Position
and errors of Expression.
ERRORS OP POSITION. Pupils are exceedingly liable
to assume improper positions and awkward attitudes
during recitation, such as leaning on the desk or against
the wall, putting thefoot upon a seat, jamming the hands
in the pockets, particularly when the problem is hard,
playing with a button, watch chain, &c. All of these
2
12 METHODS OF TEACHING
faults should be carefully guarded against, for reasons so
obvious that they need not be mentioned. An erect and
graceful carriage, aside from its relation to health, is of
advantage to every lady and gentleman.
ERRORS OF EXPRESSION. Under this head we include
errors of Articulation, Pronunciation, Grammar, &c.
There is quite a large number of words which pupils in
their haste mispronounce, and also quite a large number
of combinations, which by a careless enunciation make
ridiculous sense, or nonsense. We will call the attention
to a few of them, suggesting to the teacher to correct
these and others he may notice.
"And" is often called "an;" "for" is called "fur;"
"of" is pronounced as if the o was omitted; words com-
mencing with wh, as when, which, where, &c., are pro-
nounced as if spelled "wen" "wich" "were" &c.
"Gave him" is called "aavim^" "did lie" is called
"diddy;" "had he" is called "haddy;" "give him" is
called "givim;" "give her" is called "giver;" "which
is" is often changed into "witches;" and "how many"
is frequently transformed into "hominy." " How many
did each earn" is often rendered " hominy did e churn"
A very common error, and one exceedingly difficult to
correct, is involved in the following solution : " If 2
apples cost 6 cents, one apple will cost the. J of 6 cents,
which are 3 cents." Here "the" is superfluous, and
"are" is ungrammatical.
The following is a frequent error : "If one apple cost
3 cents, for 12 cents you can buy as many apples as 3 is
contained in 12, which are 4 times" The objections are,
first, 3 is not contained any apples in 12 ; secondly, the
result obtained is times, when it should be apples, or a
number which applies to both times and apples. The
solution should be, "You can buy as many apples for 12
cents as 3 is contained times in 12, which are 4."
With regard to is and are, it is not easy to determine
which should be used in some cases in Arithmetic. I
am rather inclined to think that it would be better to use
MENTAL ARITHMETIC. 13
the singular form always when the subject is either an
abstract or a concrete number ; thus, 8 is 2 times 4, or 8
apples is 2 times 4 apples. But, since custom sanctions
the use of "are" with a concrete number as a subject, we
have adhered to that form. There is some authority for
using "is" in the " Multiplication Table/' and I think it
would be better if the singular form was universally
adopted. We have adopted the following rule in deter-
mining the form of the verb when it has a numerical
subject: If the idea is plural use "are;" if the idea is
that of a whole singular use "is."
Pupils have some difficulty in knowing how to read
such expressions as |. They object to saying "| dol-
lars," since there are not enough to make dollars, and
they also object to saying "| of a dollar," since there
are only 3 thirds in a dollar. The correct reading is
undoubtedly the second, remembering that | is an im-
proper fraction.
The following error is almost universal : 2f apples is
read "2 and 3 fourth apples" instead of "2 and 3
fourths apples" The expression " J times" is sanctioned
by custom, although it is not in accordance with gram-
matical principles. It is rather more convenient than
the expression f of a time, although evidently a violation
of the rules of language.
But it is unnecessary for us to swell this list larger.
A little care on the part of the teacher will detect a large
number of errors, similar to those we have noticed, and
we suggest that the attention given in this direction will
be time profitably employed.
KEY.
SECTION I.
LESSON I.
A FEW of the more difficult problems in this section
may be omitted the first time in going over by a class of
young pupils. They are introduced for the purpose of
interesting older pupils, that they may be induced to
study " Simple Addition," in a Mental Arithmetic.
REMARK. This lesson is so easy that it is not neces-
sary to give the solutions of any of the problems. The
15t*h problem is designed as a puzzle, as will readily be
seen.
LESSON II.
41. SOL. Since, when he sold 10 none remained,
before selling these 10 he had 10, but 6 of these he
bought, hence before buying them, he had 10 minus 6,
or 4.
54. SOL. If Hiram gave Oliver 10 cents, and Oliver
gave Hiram 6 cents, Oliver had 10 6, or 4 cents more
than at first, and Hiram had 4 less than at first ; hence
Oliver had 26 + 4 or 30 and Hiram had 26 4 or 22.
2* (15)
16 KEY. [SECTION I.
57. SOL. If A sold 30 and then bought 12, he had
30 12, or 18 less than at first, and if by selling 30
more, none remained, he had at first 18 -f- 30, or 48.
SUGGESTION. The 57th may also be solved as follows :
SOL. In all he sold 30 + 30, or 60, but 12 of these
he bought, hence he had at first 60 12, or 48.
58. SUG. After A gave B 10 and B gave A 6, A had
4 less and B 4 more than at first ; hence A had 26 and B
34. Then after B lost a certain number, 26 12, or
14, equals B's number; hence B lost 34 14, or 20.
LESSON III,
38. SUG. Six times what they both have is 90;
hence C has 90 10, or 80.
40. SUG. After A gave B 10 and B gave A 20, A
had 10 more and B had 10 less than at first; hence A
had 50 and B had 30. Then after B lost a certain
number he had twice 10, or 20 ; hence he lost 30 20,
or 10.
-
LESSON IV.
NOTE. The problems after the 36th may be assigned
to the class and recited as the other problems, or to give
variety to the exercise the pupils need not repeat the
problem, but name the result as soon as the problem is
announced by the teacher. It will be.well for the teacher
to give quite a number of such problems, as the exercise
will be found very valuable.
LESSON V.] KEY 17
LESSON V.
81. SOL. Twice a number, plus 8 times the number
equals 5 times the number, minus 4 times the number
equals once the number, plus twice the number equals 3
times the number.
37 & 38. REM. These problems and others like them
may be made very interesting to the pupils by the teacher's
stating to the class that he can tell the result which they
obtain, no matter what number they begin with. This