characteristics of both of Colburn's texts. Colburn's treatment of percentage, interest, etc., is perhaps most typical of this feature and of his attitude toward the applications of arithmetic. On page 21 of the Sequel, in the section on multiplication, this paragraph is given immediately preceding the first problem on interest:

Interest is a reward allowed by a debtor to a creditor for the use of money. It is reckoned by the hundred, hence the rate is called so much per cent or per centum. Per centum is Latin, signifying by the hundred. 6 per cent signifies 6 dollars on a hundred dollars, 6 cents on a hundred cents, £6 on £100, etc., so 5 per cent signifies 5 dollars on 100 dollars, etc. Insurance, commission, and premiums of every kind are reckoned in this way. Discount is so much per cent to be taken out of the principal.

Colburn evidently considers this sufficient explanation for such problems as the following, for he gives nothing additional either here or in Part II:

What is the interest of $43.00 for 1 year at 6 per cent?

What is the interest of $247.00 for 3 years at 7 per cent?

Imported some books from England, for which I paid $150.00 there. The duties in Boston were 15 per cent, the freight $5.00. What did the books cost me?

A merchant bought a quantity of goods for 243 dollars, and sold them so as to gain 15 per cent; how much did he gain, and how much did he sell them for?

The next mention of percentage is on page 83. This problem is given:

A merchant sold a quantity of goods for $273.00, by which he gained 10 per cent on the first cost. How much did they cost?

Following the problem is this note:

10 per cent is 10 dollars on a 100 dollars, that is 10/100. 10 per cent of the first cost therefore is 10/100 of the first cost. Consequently $273.00 must be 110/100 of the first cost.

A little farther on in the list we find the following problems and

notes:

A merchant sold a quantity of goods for $983.00, by which he lost 12 per cent. How much did the goods cost and how much did he lose?

Note. If he lost 12 per cent, that is 12/100, he must have sold for 88/100 of what it cost him.

A merchant sold a quantity of goods for $87.00 more than he gave for them, by which he gained 13 per cent of the first cost. How much did the goods cost him, and how much did he sell them for?

Note. Since 13 per cent is 13/100, $87.00 must be 13/100 of the first cost.

A man having put a sum of money at interest at 6 per cent, at the end of 1 year received 13 dollars for interest. What was the principal?

Note. Since 6 per cent is 6/100 of the whole, 13 dollars must be 6/100 of the principal.

A man put a sum of money at interest for 1 year at 6 per cent, and at the end of the year he received for the principal and interest 237 dollars. What was the principal? Note. Since 6 per cent is 6/100, if this be added to the principal it will make 106/100, therefore $237 must be 106/100 of the principal. When interest is added to the principal, the whole is called the amount.

What sum of money put at interest at 6 per cent will gain $53 in 2 years? Note.-6 per cent for 1 year will be 12 per cent for 2 years, 3 per cent for 6 months, 1 per cent for 2 months, etc.

Suppose I owe a man $287, to be paid in one year without interest, and I wish to pay it now; how much ought I to pay him when the usual rate is 6 per cent?

Note. It is evident that I ought to pay him such a sum as put at interest for 1 year will amount to $287. The question therefore is like those above. This is sometimes called discount.

Later in the sections on decimal fractions special methods for interest are given in the same way, i. e., by means of a note following a problem which calls for a special method.

Chapter VI.

COLBURN ON THE TEACHING OF ARITHMETIC.

In the preface to his texts and in his address on the teaching of arithmetic Colburn has given a good presentation of his method of teaching arithmetic. These accounts are supplemented by the texts. In this chapter we shall present the most significant features of his method.

The pupil is introduced to a topic by means of practical problems.— Colburn's introduction of the pupil to arithmetic is in striking contrast to that in the texts used prior to 1821. (See p..) For example, the first page of the First Lessons is as follows:

1. How many thumbs have you on your right hand? How many on your left? How many on both together?

2. How many hands have you?

3. If you have two nuts in one hand and one in the other, how many have you in both?

4. How many fingers have you on one hand?

5. If you count the thumb with the fingers, how many will it make?

6. If you shut your thumb and one finger and leave the rest open, how many will be open?

7. If you have two cents in one hand, and two in the other, how many have you in both?

8. James has two apples, and William has three; if James gives his apples to William, how many will William have?

9. If you count all the fingers on one hand, and two on the other, how many will there be?

10. George has three cents, and Joseph has four; how many have they both together?

These problems are followed by 22 of similar nature, and these in turn are followed by 163 drill questions on the combinations. This plan is continued through the book.

Use of symbols delayed.-One phase of the organization of the subject matter is Colburn's treatment of the symbols of notation which seems to exemplify one of his fundamental notions of arithmetic. For example, he wishes the pupil to learn that two objects and one object make a total of three objects; that five plums and four plums are nine plums, and not that the symbols 2+1 equal the

1 This address was delivered before the American Institute of Instruction in Boston, August, 1830. It was published in the proceedings of that society and was reprinted in the Elementary School Teacher for June, 1912.

symbol 3, or the symbols 5+4 equal the symbol 9. As a means to this end, in the First Lessons, the characters 1, 2, 3, etc., are not given until page 50, and the system of notation and numeration is not given beyond 10 until page 69. Before these symbols and the system of notation and numeration are given, the pupil has learned the four fundamental operations for integers. The symbols are introduced by saying, "Instead of writing the names of numbers, it is usual to express them by particular characters called figures." Thus before the pupil is asked to learn number symbols, he doubtless has felt the need for them.

In giving his reason for these two features Colburn says, referring to the contemporary practice:

The following are some of the principal difficulties which a child has to encounter in learning arithmetic, in the usual way, and which are seldom overcome. First, the examples are so large that the pupil can form nó conception of the numbers themselves; therefore it is impossible for him to comprehend the reasoning upon them. Secondly, the first examples are usually abstract numbers. This increases the difficulty very much, for even if the numbers were so small that the pupil could comprehend them, he would discover but very little connection between them and practical examples. Abstract numbers, and the operations upon them, must be learned from practical examples; there is no such thing as deriving practical examples from those which are abstract, unless the abstract have been first derived from those which are practical. Thirdly, the numbers are expressed by figures, which, if they were used only as a contracted way of writing numbers, would be much more difficult to be understood at first than the numbers written at length in words. But they are not used merely as words, they require operations peculiar to themselves. They are, in fact, a new language, which the pupil has to learn. The pupil, therefore, when he commences arithmetic is presented with a set of abstract numbers, written with figures, and so large that he has not the least conception of them even when expressed in words. From these he is expected to learn what the figures signify, and what is meant by addition, substraction, multiplication, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of all these things, and that is, how to perform these operations on figures. He can perhaps translate the figures into words, but this is useless since he does not understand the words themselves. Of the effect produced by the four fundamental operations he has not the least conception.

After the abstract examples a few practical examples are usually given, but these again are so large that the pupil can not reason upon them, and consequently he could not tell whether he must add, substract, multiply, or divide, even if he had an adequate idea of what these operations are.

The common method, therefore, entirely reverses the natural process; for the pupil is expected to learn general principles before he has obtained the particular ideas of which they are composed.'

Oral instruction.-Just as the most conspicuous feature of the method of teaching arithmetic during the ciphering-book period was the absence of a textbook in the hands of the pupil, and the consequent exclusively written arithmetic, so the most conspicuous feature

1 First Lessons, preface.

of Colburn's method is oral instruction, or the solving of exercises in the mind. Colburn does not provide for written computations in the First Lessons. In fact, as we have mentioned, he does not introduce the number symbols at all in the first third of the book. The quantities of the problems throughout the book are small enough to bring the numbers within the comprehension of the pupil and also so small that he may solve the problems mentally. It is therefore probable that pupils solved the problems of the First Lessons without recourse to written calculations. When there were no "sums" to be done on paper or slate and submitted to the teacher for inspection, it became necessary for the teacher to hear the pupils give an oral solution of the problem. Thus, at least in the case of the younger pupils, instruction in arithmetic was largely oral after the appearance of the First Lessons. The Sequel was a "written arithmetic," but in it close connection is made between "operations performed in the mind" and the "application of figures to these operations."

From concrete to abstract.-Colburn invariably introduces a topic or a new combination by a "practical question." In the case of a new combination the "practical question" is followed by the same combination in abstract form. For example, the multiplication of an integer by a fraction is begun as follows:

If a yard of cloth costs 3 dollars, what will 1 half of a yard cost?
What is 1 half of 3?

If a barrel of beer costs 5 dollars, what will 1 half of a barrel cost?
What is 1 half of 5?

In the preface to the First Lessons the necessity of this order in teaching children is emphasized:

The idea of number is first acquired by observing sensible objects. Having observed that this quality is common to all things with which we are acquainted, we obtain an abstract idea of number. We first make calculations about sensible objects; and we soon observe that the same calculations will apply to things very dissimilar; and finally, that they may be made without reference to any particular things. Hence from particulars we establish general principles, which serve as the basis of our reasonings and enable us to proceed step by step, from the most simple to the most complex operations. It appears, therefore, that mathematical reasoning proceeds as much upon the principle of analytic induction as that of any other science.

Examples of any kind upon abstract numbers are of very little use until the learner has discovered the principle from practical examples. They are more difficult in themselves, for the learner does not see their use, and therefore does not so readily understand the question. But questions of a practical kind, if judiciously chosen, show at once what the combination is, and what is to be effected by it. Hence the pupil will much more readily discover the means by which the result is to be obtained. The mind is also greatly assisted in the operations by reference to sensible objects. When the pupil learns a new combination by means of abstract examples, it very seldom happens that he understands practical examples more easily for it, because he does not discover the connection until he has performed several practical examples and begins to generalize them.