And it is not too bold an assertion to say that no man ever actually learned mathematics in any other method than by analytic induction; that is, by learning the principles by the examples he performs, and not by learning principles first, and then discovering by them how the examples are to be performed.

The full significance of this feature of Colburn's method appears only when we compare it with the practice of his time. It marks, as do other features of his work, an absolute break with the past. The principle is fundamental with him, and its effect is clearly evident throughout both texts as well as in his method of teaching.

Objective method. In the First Lessons the pupil is not told the "combinations," but he is expected to discover them by using objective materials, the Pestalozzian tables, or beans, peas, etc., in performing the operations which the "practical questions" called for. The advantage of asking the child to think in terms of concrete objects is mentioned in the above quotation. It should be noted that Colburn recommends the use of objective material only when a pupil has need of it. It is not his purpose to introduce objective material for the purpose of amusing pupils, and he intends that they shall transcend the use of it. The objective method, next to the oral instruction, is the most conspicuous feature of Colburn's method of teaching.

Assisting the pupil.-It has already been indicated that Colburn had a definite and accurate conception of the working of the human mind. He also knew the appropriate manner in which to assist this working. This he discusses in the preface to the Sequel.

When the pupil is to learn the use of figures for the first time, it is best to explain to him the nature of them to about three or four places, and then require him to write some numbers. Then give him some of the first examples without telling him what to do. He will discover what is to be done, and invent a way to do it. Let him perform several in his own way, and then suggest some method a little different from his, and nearer the common method. If he readily comprehends it, he will be pleased with it, and adopt it. If he does not, his mind is not yet prepared for it, and should be allowed to continue his own way longer, and then it should be suggested again. After he is familiar with that, suggest another method somewhat nearer the common method, and so on, until he learns the best method. Never urge him to adopt any method until he understands it and is pleased with it. In some of the articles it may perhaps be necessary for young pupils to perform more examples than are given in the book.

One general maxim to be observed with pupils of every age is never to tell them directly how to perform any example. If a pupil is unable to perform an example, it is generally because he does not fully comprehend the object of it. The object should be explained, and some questions asked which will have a tendency to recall the principles necessary. If this does not succeed, his mind is not prepared for it, and he must be required to examine it more by himself, and to review some of the principles which it involves. It is useless for him to perform it before his mind is prepared for it. After he has been told, he is satisfied, and will not be willing to examine the principle, and he will be no better prepared for another case of the same kind than he was before. When the pupil knows that he is not to be told, he learns

to depend upon himself; and when he once contracts the habit of understanding what he does, he will not easily be prevailed on to do anything which he does not understand.

Also in his address he speaks at length upon how the teacher should assist the pupil:

If the learner meets with a difficulty, the teacher, instead of telling him directly how to go on, should examine him and endeavor to discover in what the difficulty consists; and then, if possible, remove it. Perhaps he does not fully understand the question. Then it should be explained to him. Perhaps it depends upon some former principle which he has learned, but does not readily call to mind. Then he should be put in mind of it. Perhaps it is a little too difficult. Then it should be simplified. This may be done by substituting smaller numbers, or by separating it into parts and making a distinct question of each of the parts. Suppose the question were this: If 8 men can do a piece of work in 12 days, how long would it take 15 men to do it? It might be simplified by putting in smaller numbers, thus: If 2 men can do a piece of work in 3 days, how long would it take 5 men to do it? If this should still be found too difficult, say, If 2 men can do a piece of work in 3 days, how long will it take 1 man to do it? This being answered, say, If 1 man will do it in 6 days, how long will it take 3 men to do it? In what time would 4 men do it? In what time would 5 men do it? By degrees, in some such way as this, lead him to the original question. Some mode of this kind should always be practiced; and by no means should the learner be told directly how to do it, for then the question is lost to him. For when the question is thus solved for him, he is perfectly satisfied with it, and he will give himself no further trouble about the mode in which it is done.

All illustrations should be given by practical examples, having reference to sensible objects. Most people use the reverse of this principle and think to simplify practical examples by means of abstract ones. For instance, if you propose to a child this simple question: George had 5 cents, and his father gave him 3 more, how many had he then? I have found that most persons think to simplify such practical examples by putting them into an abstract form and saying, How many are 5 and 3. But this question is already in the simplest form that it can be. The only way that it can be made easier is to put it into smaller numbers. If the child can count, this will hardly be necessary. No explanation more simple than the question itself can be given, and none is required. The reference to sensible objects, and to the action of giving, assists the mind of the child in thinking of it, and suggests immediately what operation he must perform; and he sets himself to calculate it. He has not yet learned what the sum of those two numbers is. He is therefore obliged to calculate it in order to answer the question, and he will require some little time to do it. Most persons, when such a question is proposed, do not observe the process going on in the child's mind; but because he does not answer immediately, they think that he does not understand it, and they begin to assist him, as they suppose, and say, How many are 5 and 3? Can not you tell how many 5 and 3 are? Now this latter question is very much more difficult for the child than the original one. Besides, the child would not probably perceive any connection between them. He can very easily understand, and the question itself suggests it to him better than any explanation, that the 5 cents and 3 cents are to be counted together; but he does not easily perceive what the abstract numbers 5 and 3 have to do with it. This is a process of generalization which it takes children some time to learn.

In all cases, especially in the early stages, it will be perplexing and rather injurious to refer the learner from a practical to an abstract question for the purpose of explanation. And it is still worse to tell him the result, and not make him find it himself. If the question is sufficiently simple, he will solve it. And he should be allowed time to do it and not be perplexed with questions or interruptions until he has done it.

But if he does not solve the question, it will be because he does not fully comprehend it. And if he can not be made to comprehend it, the question should be varied, either by varying the numbers, or the objects, or both, until a question is made that he can answer. One being found that he can answer, another should be made a little varied and then another, and so on till he is brought back to the one first proposed. It will be better that the question remain unanswered than that the child be told the answer, or assisted in the operation any further than may be necessary to make him fully understand the question.

It is clear that Colburn understood that a dfficulty initiates reflective thought. The pupil is at first to meet a difficulty, feel a need, have a problem. This is the first step. Second, the pupil is to make his own hypothesis; the teacher is to keep hands off. Unless the problem is one for which the pupil is not prepared, he will "invent" a way to solve the problem. It may be a crude one, but nevertheless a method which will control the value. The thought process involved here is that of making hypotheses and verifying them. The instructor is in the background. Colburn would have his function to be that of explaining to the pupil the meaning of the problem and its demands, and to see that the pupil was finally made acquainted with the best method of solving the problem.

Inductive instruction.-In the titles of both of his arithmetics, Colburn explicitly states that the method of presentation is inductive rather than deductive. His inductive development is not formal and mechanical, but here as elsewhere he has grasped the manner of the working of the human mind. The complete texts must be studied to appreciate fully the quality of his inductive development of a topic, but the development of division in the Sequel will give an idea of the charm of Colburn's inductive treatment of a topic. (See P. 74.)

This is as near real induction as it is possible to get in a textbook. The pupil is given problems which he can understand and appreciate; the first he may solve in a crude fashion, more difficult problems force him to make hypotheses, and the rule is delayed so that the pupil has had an opportunity to test his hypothesis empirically. As a consequence, the pupil probably has discovered the appropriate rule before he reaches the statement of it in the text.

Class instruction.-During the ciphering-book period, the instruction of necessity was individual. Before 1821 the need was being keenly felt for a more expeditious manner of teaching arithmetic. The attendance was increasing very rapidly, and arithmetic was beginning to be taught quite generally to all pupils. This condition made it necessary to instruct the pupils in groups. Colburn not only advocated class instruction, but gives suggestions as to the technique.

It is chiefly at recitation that one scholar can compare himself with another; consequently they furnish the most effectual means of promoting emulation. They

are an excellent exercise for the scholar, for forming the habit of expressing his ideas properly and readily. The scholar will be likely to learn his lesson more thoroughly when he knows he shall be called upon to explain it. They give him an opportunity to discover whether he understands his subject fully or not, which he will not always be sure of, until he is called upon to give an account of it. Recitations in arithmetic, when properly conducted, produce a habit of quick and ready reckoning on the spur of the occasion, which can be produced in no other way except in the business of life, and then only when the business is of a kind to require constant practice. They are therefore a great help in preparing scholars for business.

Directions concerning recitations must be general. Each teacher must manage the detail of them in his own way.

In the first place, the scholar should be thoroughly prepared before he attempts to recite. No lessons should be received by the teacher that are not well learned. If this is not insisted on, the scholar will soon become careless and inattentive.

It is best that the recitations, both in intellectual and written arithmetic, should be in classes when practicable. It is best that they should be without the book, and that the scholar should perform the examples from hearing them read by the teacher. Questions that are put out to be solved at the recitation should be solved at the recitation, and not answered from memory. The scholars should frequently be required to explain fully and clearly the steps by which they solve a question and the reasons for them. Recitations should be conducted briskly and not suffered to lag and become dull. The attention of every scholar should be kept upon the subject, if possible, so that all shall hear every thing that is said. For this it is necessary that the questions pass around quickly, and that no scholar be allowed a longer time to think than is absolutely necessary. If the lesson is prepared as it should be, it will not take the scholar long to give his answer. It is not well to ask one scholar too many questions at a time, for by that there is danger of losing the attention of the rest. It is a good plan, when practicable, so to manage the recitations that every scholar shall endeavor to solve each question that is proposed for solution at the time of the recitation. This may be done by proposing the question without letting it be known who is to answer it until all have had time to solve it, and then calling upon someone for the answer. No further time should be allowed for the solution; but if the scholar so called on is not ready, the question should be immediately put to another in the same manner.1

He also shows a trace of the monitorial system when he says:

It will often be well to let the elder pupils hear the younger. This will be a useful exercise for them, and an assistance to the instructor.

Teaching pupils to study.-Colburn recognized the value of teaching pupils how to study. He says:

There is one more point which I shall urge, and it is one which I consider the most important of all. It is to make the scholars study. I can give no directions how to do it. Each teacher must do it in his own way, if he does it at all. He who succeeds in making his scholars study will succeed in making them learn, whether he does it by punishing, or hiring, or persuading, or by exciting emulation, or by making the studies so interesting that they do it for the love of it. It is useless for me to say which will produce the best effects upon the scholars; each of you may judge of that for yourselves. But this I say, that the one who makes his scholars study will make them learn; and he who does not will not make them learn much or well. There

1 Address, "The Teaching of Arithmetic."

never has been found a royal road to learning of any kind, and I presume there never will be. Or if there should be, I may venture to say that learning so obtained will not be worth the having. It is a law of our nature, and a wise one too, that nothing truly valuable can be obtained without labor.1

In another place he suggests some necessary conditions:

This subject also suggests a hint with regard to making books, and especially those for children. The author should endeavor to instruct by furnishing the learner with occasions for thinking and exercising his own reasoning powers, and he should not endeavor to think and reason for him. It is often very well that there should be a regular course of reasoning in the book on the subject taught; but the learner ought not to be compelled to pursue it, if it can possibly be avoided, until he has examined the subject and come to a conclusion in his own way. Then it is well for him to follow the reasoning of others, and see how they think of it.1

Motivation. Although Colburn recognized that there were several ways for making arithmetic interesting, he selected the problems which especially appeal to children and caused them to feel a need for a process or definition before it is given. The types of problems are well illustrated by those already given. A feeling of need for the process is created by introducing each topic by problems. The very plan of dividing the texts into two parts, and thus separating the problems from the development of the principles, operates to create motive for the study of the principles. Even in the development of the principles, the rules are not stated until after the explanation of the operation which is itself based upon a problem. Whatever drill seems necessary is not given until after a considerable number of practical problems have been solved by the pupils.

But even these devices do not represent all that Colburn has done to motivate the arithmetic work. His style of writing and his ability to see things from the child's point of view assist materially in this respect, and the way he guides the learner in the development of the principles adds a touch of genius to the whole work. The following is from the Sequel, p. 193:

A boy wishes to divide of an orange equally between two other boys; how much must he give them apiece?

If he had three oranges to divide, he might give them one apiece and then divide the other into two equal parts, and give one part to each, and each would have 1 orange. Or he might cut them all into two equal parts each, which would make six parts, and give three parts to each, that is, =14, as before. But according to the question, he has or 3 pieces, consequently he may give 1 piece to each, and then cut the other into two equal parts, and give 1 part to each, then each will have 1 and of 1. But if a thing be cut into four equal parts and then each part into two equal parts, the whole will be cut into 8 equal parts or eights; consequently of is 1. Each will have and of an orange. Or he may cut each of the three parts into two equal parts, and give of each part to each boy, and then each will have 3 parts, that is. Therefore of

is.

Ans. 3.

1 Address, "The Teaching of Arithmetic."