2. The work will appeal to more faculties of the mind, and hence afford opportunity for alternate use.

Greenwood says of the book:

The information composing the problems is drawn from at least a hundred sources. It is highly instructive as well as eminently practical. A good title to the book would be "Useful and Scientific Information Treated Arithmetically." A revised edition ought to be in the hands of every teacher. It has never been properly appreciated, and few copies are in existence; even the publishers do not have a copy.1

Organization. In the internal arrangement many of the texts in the active period explicitly professed to be upon the inductive plan, and we have shown that the authors did build upon this plan with some degree of understanding of the mental process involved in induction. But it appears that about the middle of the century this understanding of the inductive plan faded or became overshadowed by philosophical considerations.

In the Practical Arithmetic of 1857 Ray says that the "inductive and analytic methods" are adopted. Two paragraphs later he states that "the arrangement is strictly philosophical; no principle is anticipated; the pupil is never required to perform any operation until the principle on which it is founded has first been explained.” This is a contradiction, because induction and what he defines as the "philosophical arrangement" are fundamentally opposed. The text itself exhibits both plans of organization, but the spirit of the text is more in accord with the latter. In 1877 there is no reference to the "inductive and analytic methods." Ray's text does not represent an extreme in respect to this trait. Davies, Greenleaf, Brooks, Robinson, and others are more pronouncedly "philosophical."

Higher arithmetic. -The higher arithmetics were simply an advanced treatise on the general plan of the practical arithmetics. All topics were included from notation and numeration and the fundamental operations for integers to involution and evolution and series. The topics were not introduced by a few questions to be answered orally. The organization was strictly logical, first such definitions as were necessary, then the rule followed by abstract exercises, and finally practical problems. The subject was looked upon throughout the text primarily as a science. Emphasis was placed upon "clearer definitions, more rigid analyses, and briefer and more accurate rules." These features represent the prime merit of not only the higher arithmetics, but of the practical arithmetics as well. This is undoubtedly one of the main reasons for the long and extensive use of such texts in our schools.

Colburn's influence upon the textbooks of this period.-Primary texts, "mental arithmetic," the use of objective materials, and the inductive method were the most significant features of the arithmetics of

1 Op. cit., p. 854.

H

the period. The authors of some of the texts were acquainted with Pestalozzi's system of arithmetic and his educational principles, but it is probable that all were acquainted with Colburn's texts, particularly the First Lessons, and the interest in Pestalozzi's system of arithmetic was due in a large measure to the popularity of this text. The primary texts were patterned after the First Lessons; the "mental arithmetics" followed it very closely; and the inductive method, before it was formalized, was very similar to that in Colburn's texts. The objective materials were changed only by the omission of the Pestalozzian tables and by adding pictures. Thus, much is due to Warren Colburn for stimulating and directing the development of American textbooks on arithmetic during this period.

Chapter IX.

TEACHING ARITHMETIC BY DEVICES AND DRILL.

The introduction of mental arithmetic, the concept of mental discipline as the function of arithmetic, the teaching of arithmetic to young children, the ideal of skill and thoroughness, the very great increase in the number of pupils studying arithmetic, and other changes necessitated modifications in the methods of teaching arithmetic. These modifications are described under the following heads. Class instruction.-In view of the fact that in the ciphering book method, the individual contact between the teacher and pupil was for examining the pupil's work and telling him whether it was right or wrong, and not for instructing the pupil, much time was wasted. Before 1821, the monitorial system of instruction had been applied to arithmetic, and after this date class instruction in arithmetic was the rule. This was probably because it was more economical, but some teachers believed that in a group superior instruction was possible. For example, Ray says: "Pupils study best in classes; it is almost as easy for a teacher to instruct 15 pupils in a class, as 1 alone."

For a teacher to handle a group of pupils successfully, some technique was necessary, and much attention was given to this phase of teaching during this period. What was written on the teaching of arithmetic was confined almost wholly to the elaboration of the technique of class instruction. The following report of the Boston Monitorial School gives a good description of one type of class in

struction.

The next exercise is arithmetic. I have already said that even the youngest is taught to count and perform simple operations with beans, her fingers, and such aids. Soon a little mental arithmetic is introduced; but, as the excellent little work of Colburn is too difficult for such small children, manuscript questions prepared by the instructor are used. Next, Colburn's First Lessons are studied; and about the same time, written arithmetic is gradually introduced. This, however, is for the present completely subordinate to the intellectual. The monitors of arithmetic recite to the master, and then disperse to their stations to act as monitors. Their classes form around them; and the lesson which has been previously set is recited. If any explanations are necessary, the monitor, who has gone over the ground before, explains; but if she is at a loss, she applies directly to the master. In this way, the little classes get a great deal of practice, and the monitor reviews her studies. For the sake of variety, they then take slates and cipher. The monitor dictates sums verbally, and

Key to Ray's Practical Arithmetic, p. 177.

the children are taught to write amounts from dictation. They are never allowed to copy sums, and consequently must acquire a knowledge of numeration, as useful as it is uncommon. In addition the highest adds the first column aloud and tells the next what to set down and what to carry; the next takes the second column, and does the same. Anyone who corrects another goes above her, as in spelling or reading; and, as all must aid in doing the sum, the attention of all is secured. It is so with subtraction, and all the other rules. The highest scholars cipher in Colburn's Sequel, and record their operations in a manuscript.1

The monitorial plan of group instruction was not generally adopted. Ray describes the practice of about 1840 as follows:

When practicable, the pupils should be arranged in classes, due regard being had to their ages, acquirements, etc. After this, the proceeding in the best schools, is somewhat as follows:

A certain number of examples is arranged as a lesson; it will, also, frequently be necessary that a part, or even the whole, of the lesson shall consist of the illustration of principles, or the memorizing of definitions or rules. When the class meets for recitation, each pupil passes his slate into the hands of the pupil next above him, except the pupil at the head, who passes his to the foot scholar. The teacher then reads the answer to the first question, while each pupil examines the slate he holds, to see if the answer is correct and properly obtained.

In addition to reading the answer, the teacher, in many cases, such, for example, as proportion, should state the general method of working the question. The pupils mark the answers that are wrong, or obtained improperly. In the same manner, each question is examined and marked. Instead of the teacher reading the answers, the pupils in succession may read them.

When there is a blackboard (and there should be one in every schoolroom, 4 or 5 feet wide, and as long as the room will permit), each pupil should be required to work out one or more of the examples, and give the reasons for performing the operation. The time required to examine the questions is generally short, while the habit of closely scrutinizing each other's work, improves the perceptive faculties of the pupils.2

William B. Fowle was the author of an arithmetic and for a number of years was an instructor in teacher's institutes in Massachusetts and New York. In the Teacher's Institute he discusses the teaching of the common branches. The following "methods" of teaching addition are interesting as well as typical:

When the children are ciphering on the blackboard, there are various ways of keeping them at work. I will try to describe a few of them. Suppose the class consists of six, and the exercise to be in addition. I first dictate one line of a sum to each pupil, as follows:

3, 746, 389, 467

7, 680, 895, 089

8, 070, 688, 496

9, 009, 900, 090

7, 508, 785, 687

8, 687, 768, 686

515
253

The pupils stand in a semicircle around the board, the teacher or monitor standing on the left, the head of the class being always on the right.

1 Amer. Jour. of Educ., 1826, 1:35.

2 Key to Ray's Practical Arithmetic, p. 6.

First method.

Let the first child begin, and say aloud, “6 and 7 are 13." Let the next child say, "and 6 are 19;" and the next, "and 9 are 28;" and the next, "and 7 are 35." The next sets down 5, and if the children are very young, he sets a small 3 under the 5, as a guide to the next, who says, "3 tens carried to 8 tens make 11." Then head begins again, and says, "11 and 8 are 19;" the next says, "and 9 are 28;" the next, "and 9 are 37;" the next, "and 8 are 45;" the next, "and 6 are 51;" the next sets down 1 in the tens place, and puts a 5 under it. The next says, "5 hundreds carried to 6 hundreds make 11 hundreds;" the next says, "and 6 are 17," and so on until the sum is finished.1

This "method" is given several variations. Pupils may be called upon "promiscuously." Or each pupil may add a column silently and place the result upon the board. The next is held responsible when the sum is not correct.

These "methods" are simply types of technique for effectively focusing the attention of the class and arousing interest in the work. No fundamental principles of teaching are stated, but these specific rules for carrying on the classroom work are typical of the method of teaching during this period. Objective and examinable results. were desired, and devices which would give these, and would secure attention, were accordingly exalted as methods of teaching.

In teaching mental arithmetic a procedure was adopted for the purpose of forcing the continuous attention of the class. As in the case of written arithmetic, the plan was an artificial device. Stoddard gives in his Methods of Teaching the following "methods:"

First method.

The teacher reads the problems and calls upon the different members of the class promiscuously. Each pupil named arises, repeats, and analyzes the problem. Members of the class who have discovered mistakes, or who take exception to the method of analysis, raise their hands, and the teacher designates some one of them to make the necessary correction, or he makes it himself.

Modifications of the above method:

1. Call upon different pupils to solve different parts of the same problem, each as he is named being required to proceed with the analysis where the pupil who has just taken his seat left it. This method furnishes an opportunity for "stirring up," or jogging the memory of the inattentive.

2. The pupil designated to analyze a problem arises, repeats it, and names another to solve it.

Second method.

The teacher reads a problem, the class solves it in silence, and as rapidiy as possible; each raises the hand on obtaining the result.

After giving sufficient time the teacher, if he wishes a simultaneous answer, says "Class," and all who can pronounce the result together. Or he names a pupil, who arises, gives the result, and solves the problem.

Third method.

The teacher reads and assigns a problem to each member, or a part of the members, of the class without waiting for a solution. He then calls upon pupils promiscuously

1 P. 55.