ARITHMETIC IN THE SCHOOL.
For generations arithmetic has held a prominent place in the school
curriculum. Should you teach any one or all of the first eight years of
school work the teaching of arithmetic will be of practical interest
to you. Even the most modern of school courses has not eliminated
arithmetic nor curtailed to any great extent the proportionate share
of attention given to it. Most teachers will admit that the subject
has been over-estimated, and that it is not of the great practical
value popularly attributed to it. The man in business does not use
the principles of arithmetic half so often as many believe. Outside
of the four fundamental processes and the principles of percentage,
the average merchant and trades people generally do not use much
arithmetic. The use of mechanical means of adding, subtracting,
multiplying, and dividing has become widespread. Few business men now
depend upon adding long columns of figures mentally. The adding machine
is quicker and absolutely correct.
Arithmetic has a practical value, but it is not its so-called practical
value that is its greatest worth. Few subjects in our school course
develop the mind in so many ways. Unless many of our newer subjects
prove their worth even beyond the fondest hopes of many of their
advocates, arithmetic must yet hold a prominent place in the school. It
is the teacher’s duty to teach it so as to make the best out of it for
the child.
I believe most teachers will agree that at present and in the past
much time has been wasted in arithmetic. The results achieved do not
justify the time spent. Whether these teachers will agree with me as
to the reasons for this I am doubtful. To get teachers to question
themselves on this point, and if possible to gain some time in this
subject, else get better results for the time spent, is the purpose of
this little talk.
_1. Our pupils lack intensity in the study._--I believe this is one of
our greatest faults. We allow the pupils to get into bad mental habits.
They snooze over the work. They put in too much time. Their thoughts
are allowed to go wool-gathering. They grow into time-killers instead
of learning clear, sharp business methods.
_2. Much time is lost in dry formalism._--We hold to the form and
neglect the content. We seek the husks rather than the ear. Some
formalism may be helpful, but to hold to it in all things is not only
a waste of time but it weakens the thought power. I knew a graduate of
a township high school, a bright boy and a boy in no way slow to grasp
things, that had been so thoroughly drilled in the formal analyses of
problems, following the dictum, “always teach your pupils to reason
from many to one, and then from one to many,” that after more than a
year in a store he told me he was never certain of an account until he
had written it out in formal fashion:
1. If twelve eggs are worth 18 cents.
2. One egg is worth 1-12 of 18 cents, which is 18-12, or 1 1-2 cent.
3. And 16 eggs is worth 16 times as much as one egg or 16 times 1 1-2
cents, which is 24 cents.
4. If eggs are 18 cents a dozen, 16 eggs are worth 24 cents.
It may have been logical reasoning, some of it may have been useful,
but to grind formalism into a boy for a twelve year’s course of study,
by never allowing anything to pass that was not “analyzed” was a loss
of time and mental discipline.
_3. Pupils are not trained to read and grasp the conditions of
problems._--Fifteen years’ experience with high school pupils leads
me to believe that their ability to understand English causes a loss
of time and results in arithmetic. Practice your pupils on problems
without figures until they look for conditions rather than figures.
Make numerous problems and let pupils tell you how to solve them. For
example, “If you have the length and the breadth of a rectangular
field stated in yards, how will you find the number of acres it will
contain?” If teachers would give all through the grades more attention
to problems without figures, pupils would accomplish more when they
come to using problems with figures because they could grasp the
conditions better.
_4. Teachers neglect the four fundamental processes._--It is not an
uncommon thing for pupils in the advance grades or even in high school
to be unable to add, subtract, multiply or divide with anything like
rapidity and accuracy. I am told the superiority of the German schools
in arithmetic comes very largely in the thorough and accurate manner in
which pupils are held to a mastery of the four fundamental operations
in the first four years. When the advanced work is reached pupils are
not handicapped by inaccurate results.
_5. Pupils are not taught to grasp each new subject and its purpose,
how it is like and how it differs from the subject that precedes
it._--Before beginning the new subject they should take an inventory of
what they already know of the new subject and then the teacher can make
clear to them the new things to learn. When the subject is complete
they have a grasp of it as a whole. Many of our newer and most popular
text-books in arithmetic are weak--unpardonably weak--because of too
close a devotion to the spiral and other methods that try to teach
each process and division of the subject just so far and then return
to it on the next trip around the auger in just such a time. The best
results more frequently come from the teachers and the texts that teach
one thing at a time. Then if the problems are well graded there is a
growing power on the part of the pupil to master the text for himself.
When a boy in my ’teens, I bought a book and solved the problems in
an advanced arithmetic, getting from the effort more growth than any
teacher could have gotten by using a text where the child must be led
all the way or else is lost in a maze of uncertainties. Do not teach
arithmetic in scraps and fragments if you want the pupils to understand
it. Connect each new subject with those that precede, and give them the
feeling of unity of the subject.
_6. Teachers neglect mental arithmetic._--There was once a special time,
set aside in the program for such study, now there is a tendency to
neglect it even in connection with the written work in arithmetic.
Nothing paves the way better for good work in written arithmetic than
mental arithmetic. The teacher states the problem orally, the pupil
states the conditions carefully, gives a good logical solution, and
the result or conclusion. The long problems solved without the use
of figures is not so important as the clear statement of conditions.
To multiply 746234 by 278 without making any figures is not half so
valuable in mental arithmetic as to tell how to solve a problem, giving
the reason for each step, without giving the numerical result.
These, it seems to me, are the chief reasons why our results in the
study of arithmetic are not richer. Now there should be three chief
aims in teaching arithmetic. If these are accomplished, the pupil has
not only the highest practical value of arithmetic, but the best and
richest of the cultural value also.
_The first aim should be accuracy._--This accuracy should include not
only accuracy in result, numerical result, but accuracy in reasoning
process, and accuracy in expression also. Mathematics is an exact
science. Its great cultural value lies in its training in exact
reasoning. The arithmetic that does not give exact numerical results is
poor--extremely poor. In fact, much of our work in arithmetic drills
to poor standards. The boy who gets nine problems out of ten--even
in routine drill work--is graded high. Suppose the boy goes into the
store or the bank and makes one mistake in ten computations--how long
would he hold his job? He would pass, but it would be off the pay roll.
Do we not make a mistake unless we hold up to pupils in all drill
work a standard of absolute accuracy? After the pupil has mastered
the mechanical processes until he is accurate, make sure that he is
accurate in reasoning process. Teachers often err here by confusing
numerical results with proper thought results. If a field is 40 rods
wide and 80 rods long, how many acres does it contain? After the pupil
is accurate in the fundamental processes, the real food in this problem
for the pupil is what is given, what is required, and how the results
required may be obtained from what is given. It is the steps in the
process, why and how they may be obtained--and not the numerical result
that is wanted. After these steps are fully in mind, as a drill in the
fundamental processes he may solve the problems and give you the answer
in figures.
_The second aim of arithmetic is rapidity._--This is the age of
electricity. Speed counts. Time is money. We have no time for frills.
The clearest, quickest, most direct processes are the best. Get the
correct results and get them quickly. The person who can do this is at
a premium, the person who cannot do it is at a discount. Teach pupils
to concentrate the attention upon the problem, shutting out all thought
of irrelevant things. This is discipline as well as training for good
results. Strive to keep your pupils from slothful habits in written
or mental arithmetic. The habit grows and lessens results as well as
wastes time.
_The third aim of arithmetic is neatness._--This is less important than
the other two, but it is important and should not be neglected. It
was made a shibboleth by teachers in the past. The form and neatness
counted for more than the thought. Manuscripts were carefully bound to
show the beauty and neatness of the work. Not long ago in a private
school I saw exhibited with pride the work of pupils in the class in
arithmetic, the chief merit of which was the faultless neatness of
which it was placed on the paper. Ignorant people and those who are
untrained in judging good work in school give credit for the form.
Those who judge intelligently look for the content as well as the
form. Teach pupils to orderly arrangement of work. The visitor in the
school-room, if familiar with the processes of arithmetic, should have
no trouble in following a solution on the blackboard or paper.
I cannot in the limits of this chapter go into detail of how particular
subjects are taught. Your teaching should center about these three
aims. Any method or device that helps any one of these is helpful.
Center your energies upon the essential things. Train pupils to
careful, accurate, rapid work. See that principles and processes are
made clear, but do not waste time trying to develop the processes down
in the grades where the children are and should be interested in the
memory work. Do not fool away time on the process of developing the
multiplication table and let pupils leave the subject without being
able to give the multiplication table. It may be useful to the child to
know how to find how many six times seven are by using chalk marks or
counters. The one thing that must not be neglected is to make sure that
the child knows that six times seven are forty-two. The results are the
practical things.
The first four years in school should make the pupils able to add,
subtract, multiply and divide, _and it should enable them to do this
quickly and accurately_. If I could get a class proficient in these--up
to my standard of proficiency--at the end of the fourth year, I should
give them a thorough knowledge of the practical arithmetic by the end
of the seventh year and they would have the eighth year for algebra or
commercial arithmetic.
A principal of one of the most thorough and reliable business colleges
of which I know told me that his greatest trouble in a commercial
course was to hold the students down to a regular systematic drill in
the fundamental processes of arithmetic. The trouble was that they
could not add accurately and quickly. Their results could not be relied
upon. Many of them he kept two periods of thirty minutes each daily on
rapid calculations in fundamental processes.
See that your pupils get correct ideas of number. Children think
figures instead of number. Two dollars to many is a figure two with
a dollar mark to the left. Two feet is a figure two with ft. to the
right. See that they think number rather than figure. This comes often
by having them make figures before they can use numbers.
Teach ideas instead of words. One-half does not mean a fraction to
many pupils--it means the figure one above the figure two with a
line between. Do not confuse a fraction with the manner in which a
fraction is expressed. Principles should precede rules and pupils
should comprehend these principles. I am even old-fashioned enough to
believe that often, very often, pupils should be asked to commit the
definitions and principles and rules as given in the book, but the
language and purpose of these same definitions, principles and rules
should be understood. The fault of the old plan was not that pupils
were to commit definitions but that the pupils failed to comprehend the
language and meaning of the definitions.
See that each new subject is properly related in the child’s mind with
the subject that preceded it. It will then become one step of the
ladder, something specific and definite in the child’s mind. When they
study decimals they should be able to recall and use any principles
learned when studying common fractions. When they learn percentage
there is little new to learn if they apply their knowledge of decimals.
If percentage is well learned there is little else to learn in interest.
These principles applied carefully will make your teaching of
arithmetic more fruitful. It will save you time to do the more
important work. Many teachers tell me it is hard for them to keep
children interested in the drill and abstract work. They find it hard
to keep pupils interested in addition until they become accurate and
rapid in results. I have never found it so. In fact, I have always
found the abstract drill problems to be the most fascinating work.
The only thing necessary is to call into play the spirit of contest
or rivalry. Take an excellent test for reviewing and drilling on the
tables.
Suppose pupils have learned all the combinations up to 100. Suppose
they have been drilled on adding by endings and the thing the class
needs is practice. Some such device as this will make splendid
practice. Draw a circle on the blackboard. On the circumference of
the circle write the numbers 6, 9, 5, 7, 8, 11, 9, 4, 5. Then in the
center of the circle write some number, let us say 7. To show what is
to be done with the numbers write a plus mark before the seven. That
will show that seven is to be added to each of the numbers on the
circumference in rapid succession, the pupil calling the result. John
is given a pointer to pass to the board and call the result. He will
point to each of the numbers written above calling the result after
adding 7 to it, as 13, 16, 12, 14, 15, 18, 16, 11, 12. Time John in
giving the answers. Then send Mary to the board and see if she cannot
beat him. The rivalry will be intense. Interest will be at fever
heat and drill will become a game and a pleasure. Place a sign of
multiplication before the seven and you have the whole drill changed
in a moment from addition to multiplication. In the same way it can be
changed to subtraction or to division. Change your numbers as soon as
the pupils begin to repeat the results from memory of position.
Baseball terms and other things may add spice and awaken some of the
sleepy boys that have never taken an interest. If he makes a mistake it
may be called a foul, if he does not get far without a mistake he is
out on first, etc. In the same way you may have an automobile race. The
circles may well be thought of as the automobile and try which can beat
giving the results. Should some one fail to give the correct results
his automobile breaks down. A little ingenuity of the teacher will keep
the drill work on abstract number and the learning of the tables a
fascinating game instead of a continuous grind.
The spirit of the contest can be profitably used in all subjects of
arithmetic by means of the ciphering match. Perhaps all teachers have
used the ciphering match as a stimulus to two of the great aims in
teaching arithmetic, accuracy and rapidity. Nothing will excel it in
this. To those who have not tried a ciphering match the following
directions will make it clear as to method. The more you use it and
study the results on the arithmetic work of your school the more you
will come to value it.
The ciphering match may be between members of the same class or
between different classes, or the whole school may be divided into
two classes by choosing up, or the teacher may call on pupils on any
order he chooses. You may state in advance what subjects may be used
in the match at a given time. To begin with it is well to limit to
addition. Later it may be any one of the four fundamental processes.
Then it makes a splendid review test when the class has completed any
particular subject or group of subjects. Then to encourage them to keep
well up on all subjects completed it is a good plan to allow them to
choose from any subject over which the class have passed during the
year. Short problems make the best drill for the ciphering match.
Send the two “captains,” if the school has chosen up, to the board.
Read them a problem. The first to _call_ the answer turns the other
one down. Make calling of the answer the test, as it relieves you of
watching them so closely. It is best to have another arithmetic or a
list of problems so that you will not lack for problems. As the next
pupil passes to the board he has the right of choice of subject and
calls this as he passes to the board. He may choose a subject in which
the first pupil is known to be weak or one perhaps in which he himself
is considered strong. The choosing of subjects is a fine stimulus to
keeping up in review all subjects previously gone over.
Do not give long, involved, complex problems. The purpose of the
ciphering match is drill on fundamentals and accuracy and quickness in
getting results. Long problems kill interest. If one class is pitted
against another or if the school is divided into two parts it is easy
to determine the victor. The individual who turns the most down is the
victor in one sense and then keeping a score of which side turns the
most down is often a good plan.
The best results of the ciphering match is the voluntary work and drill
you get from the pupils in preparation. They will time themselves.
They will practice to see who can solve the most problems in a given
time, they will solve problems at home or meet other pupils at a
neighbor’s house and get as much enjoyment out of it as they would at
card-playing and more profit. I have seen scores of boys quickened and
interested permanently in arithmetic from the ciphering match. Pupils
quit snoozing over their work and try to go direct for results. Try a
ciphering match a few times this year--often enough to get the pupils
thoroughly acquainted with the plan and see if you and they do not both
feel that it is profitable.