Twenty Talks to Teachers

by Thomas E. Sanders

Chapter 17: Arithmetic in the School

Additional Information
  • Year Published: 1908
  • Language: English
  • Country of Origin: United States of America
  • Source: Sanders, T. E. (1908). Twenty Talks to Teachers. The Teachers Co–Operative Company.
  • Readability:
    • Flesch–Kincaid Level: 10.4
  • Word Count: 3,474
  • Genre: Informational
  • Keywords: education, educational, learning, teaching
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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 men tally. 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 under stand 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 746,234 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 reason ing. 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 great est 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 con fuse 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 teach ers 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 auto mobile 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 auto mobile 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 re view 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 an other 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.