EXERCISE.—Ask a class of children to make up some arithmetic problems about the things that most interest them. Compare these problems with those found in the textbook.

Try to secure statements from your friends of middle age as to the value to them of the mathematics they studied in the elementary or the secondary school. How, if at all, would they change the character or amount of instruction there?

 

The value of mathematics to the world at large.—It would be hard to overstate the importance of mathematics. Our calendar expresses it, historical events are set in order by it, every natural science mounts by it, all industrial arts are wedded to it. Theorist or practitioner, no one can escape it. Nor does our little world bound it. Other worlds may easily have different history, different botany, different psychol­ogy, different literature from ours, but it is next to impossi­ble to conceive of their having any mathematics different from ours.

 

Yet despite this apparently universal truth and utility, there is but slight demand, in the daily lives of most of us, for anything more than the merest rudiments of number and quantity. They are few who do not need a modicum of mathematics, but they also are few who need anything more.

 

Of course, we all profit by the genius of the expert mathematician as we do by the devotion of the expert physician, but that is no reason for becoming either kind of expert unless our "call" takes us that way.

 

Nature of mathematics.—Mathematics is a kind of science. In beginning the study of it, we should make many observations of the world about us, as we do in other sci­ences. But the peculiarity of mathematics is, that we so quickly pass on from the "observing" stage to the "reason­ing" stage. Its generalizations, some of which are called axioms, form the foundation of a great mass of deductions which make up the bulk of the science. It is "the" de­ductive science.

 

Mathematics, then, has few (observed) facts to remember, —no such burden of them as we find in geography, botany, or chemistry. It is a thinker's paradise, for a certain kind of thinker.

 

 

Educational value of mathematics.--In some quarters it is still necessary to dispel the delusion that each branch of study has some peculiar mental power to develop. The truth seems to be that any branch of study can be so taught that it will train any mental power to act on that kind of subject matter, but not, usually, on all kinds. Thus mathematics develops, in one who has the "gift" for it, mathematical observation, mathematical memory, mathe­matical imagination, judgment, reasoning, feeling.

 

But it cannot of itself develop the power to reason well on all subjects. "It is more than doubtful . . . whether the severe study of arithmetic would make any material difference in a man's capacity, as a juryman, to draw sound conclusions from a tangled mass of evidence, or as a citizen, to trace admitted governmental evils to their source. . . Facility in the one kind of reasoning is no more a guaranty of facility in the other than is proficiency in playing golf or proficiency in playing chess."  The general truth of this statement has been borne out by the recent developments of experimental pedagogy.

 

Our chief aim in teaching mathematics to children is to enable them to solve the problems of everyday life, as those problems appear in the household, the market, and the shop. If we accomplish this aim, "discipline" and "culture" will take care of themselves.

This means that it is the art of mathematics, rather than the science, that we are trying to impart. We should not care, until the child himself naturally desires to understand, whether he can "explain" his addition and long division, his fractional and other operations, or not. But we should care constantly that he perform these operations with speed and accuracy. In other words the early lessons in mathe­matics should not be "thought lessons" only, but observa­tion lessons, information lessons, and especially lessons for skill. We are too much afraid of letting our children imi­tate us in mathematical operations, as they do in skating and shooting marbles. Even problem solving is partly habit.

 

In addition to this very practical reason for teaching a little mathematics to everybody, there is a special reason for teaching a great deal of mathematics to the limited num­ber who have strong talent for it; it will serve them in their vocations as engineers, architects, investigators. So far as possible, we should teach mathematics to each child in accordance with what he is likely to do with it. One gains merely the art of reckoning, another glimpses the science of number, a third proves to have been born to further the cause of mathematics in the world. Vocation determines the value.

 

Subject matter.—More and more the principle prevails that subject matter shall be selected according to the needs of the pupils. "What can we leave out?" is the common question, and rightly. We are leaving out of arithmetic such subjects as obsolete or little-used measures and tables, unusually intricate or lengthy problems, progres­sions, series, compound proportion, annuities, cube root, and many other subjects.  At the same time the stress is ever more and more on the applications of such topics as are taught, to practical problems—pupils' problems.

 

From secondary-school algebra we find disappearing the more elaborate method of finding the highest common factor, difficult simultaneous quadratics, all equations beyond the second degree, and other labored and little used topics.

 

  Any principle, having been developed, illustrated, and well drilled in, should be given a wider range of application, represented by circle A.  A pupil who learns how to get the volume of a sphere can apply the formula (roughly) to the earth, sun, or moon. The result is similar with new combi­nations of old things; a boy who knows the circumference of his bicycle wheel can tie a small flag to one of the spokes and measure the distance between home and school.

 

But it is quite wrong to suppose that we can teach much geography or history or any other subject merely by offering information problems in that subject when teaching arith­metic. This is much like wearing skates when learning to dance, so as to master both skating and dancing by a single effort. A few may succeed in spite of the divided attention and effort.

 

General method in mathematics.—Speaking generally, there are but two kinds of work to master in mathematics. They are:

 

1.  Fixed operations, such as addition and multiplica­tion.
2.  Problems to which these operations are applied.

 

As stated before, many of the fixed operations should be regarded as acts of skill, learned chiefly by imitating the teacher, and left unexplained. This method should always be followed when the explanation proves to be a stumbling block, as in the case of division of fractions. Let us teach practical doing, the art of calculating, whether we teach the elegant science of arithmetic or not.

 

When it seems likely that a class can get some understanding of the process to be taught, it may be approached as a problem to be solved. The following is suggested as a good type of general procedure.

 

(a) Let the new truth appear in the guise of a problem. For instance, when the class is ready to work out a rule for finding the area of a rectangle, we may propose the ques­tion, How can we find the number of squares in a checker board without counting them all?

 

(b) Let the pupils solve the problem under the guidance and leadership of the teacher. The work should be concrete and mainly oral. When a rule or principle is developed, a brief statement of it should be formulated and recorded. For example:

 

Area of rectangle = W x H.

 

(c) This terse statement should be fixed by repetition and drilled upon orally until it can be applied accurately and readily. Let it circulate all round the class.

 

(d) Fix the most desirable written form by practice on board or paper, the teacher carefully supervising.

 

(e) Let the pupils, as independently as may be, apply the knowledge gained to situations which to them are real and interesting problems.

 

2. The solution of particular problems may proceed somewhat as follows:

 

(a) Image vividly the conditions of the problem, drawing a figure or picture if necessary, to aid.

 

(b) Discover just where or how the "answer" must fit in with the rest, and consequently how it can be obtained from the facts given.

 

(c) Translate the language of the problem into figures, and solve, writing no explanations except such as are neces­sary to aid the solver himself.

 

(d) When the answer is obtained in figures, state what it means in the concrete terms of the problem.

 

Concerning analyses and explanations, so often abused, Dr. Smith is eminently quotable:

 To require that every applied problem should be solved in steps is to encourage arithmetical dawdling.

1. To split hairs on such ques­tions of form as 9 x r5c or r5c x 9 is to get away from the essential point
2. To require no analyses of the ap­plied problems is an extreme that is about as bad as to require them for all, and perhaps worse.

 

To require some particular form of analysis, only to meet the idiosyn­crasy of the teacher, is also a danger against which we need to be on our guard. . . . In general, therefore, the teacher should see to it that there is a reasonable amount of rapid, accurate solution, the answers being the paramount object. He should see also that there is a reasonable amount of written analysis, preferably in the convenient form of steps, but not limited in any notional way that would destroy originality or make a solution unnecessarily long."

 

Formal, detailed analysis should not be required too early—say before grade seven—but problems should be talked over and reasoned out informally from the very be-ginning of school work.

 

The psychology of arithmetic.—Fundamentally, arith­metic is counting up and down the number scale, with the invention of such short cuts as addition, division, etc., to quicken the process. Whatever the adult may make of his concept of number when he analyzes it, there is little doubt as to how the child gets hold of the idea. Out of the "bloom­ing, buzzing confusion" that surrounds the young child, there comes home to him very early the consciousness of changes, and particularly those rhythmical changes that readily form series. This series idea is built up from many sources, breathing, running, the clock tick, the drum, the accented notes in music.

 

The series idea is the basis of the number concept.

 

The series need not be named at first. The essential thing is that it should be abstracted, that is, separated from any particular concrete objects or events that have helped to build it up. This abstract series idea can then be applied in a manner that seems very much like counting, as in the case of the child who reproduces the strokes of the clock without using number names, saying, for example, "Boom! boom! boom!" when the clock strikes three.

Next comes the learning of the number names. The first "counting" should not be applied to things, any more than one would point to objects when repeating a Mother Goose rime. It is purely a memory drill on a series of names. It is important, then, that the numbers be taught in their natural order. Having now gained his abstract series idea named with the number names, he is ready to apply it. Care is necessary that the number names be not applied at random, or regarded as the names of certain individuals. Phillips gives a case of a boy who counted his neighbor's four dogs as follows: "Tip is naught, Bob is one, Nero is two, and Dandie is three."

 

The pupil may now proceed to count all sorts of things in which he is interested, using his fingers, the original basis of the "tens" system, as freely as he chooses. Objects also find a large place in the development of addition tables, etc., but the transition from things to pictures of things and then to symbols purely, may follow rapidly.

 

FOR FURTHER STUDY

• It is sometimes urged that pupils should write their problems, because they must certainly think of the problem while they are writing it. Is this argument sound? Why? 
• "Mathematics teaches us to reason." Discuss this pro and con.
• State the psychological reason for not using, in the very beginning of arithmetic, such general symbols as those employed in algebra.
• State, in terms of habit, the reason why children should not long continue the use of objects in computation.
• Show that all arithmetic can be reduced to a matter of counting up and down the number scale.
• Visit some classes in mathematics, and try to work out the psychological reasons for the errors you find.
• What use can be made of arithmetic in the teaching of algebra?
• "Mathematical study begets accuracy, the prime requisite of the truth teller, and so has a high moral value." Comment on this.
  
• Make a list of those topics in arithmetic for which you have found frequent actual use since you left school.
• Discuss "The use of imagery in mathematical study."
• Should models be used in the study of geometry?  Why?
• What proportion of the work in arithmetic should be oral? On what do you base your answer?
• How should you proceed with the boy who counted, "Tip is naught, Bob is one," etc.?