faction I at first expected; i. e., where there are several boys in a class, some one or other must wait till the boy who first has the book finishes the writing out of those rules and questions he wants, which detains the others from making that progress they otherwise might, had they a proper book of rules and examples for each; to remedy which I was prompted to compile one, in order to have it printed, that it might not only be of use to my own school but to such others as would have their scholars make a quick progress.'

Daniel Adams says in the preface to his Scholar's Arithmetic:

We have now the testimony of many respectable teachers to believe that this work, where it has been introduced into schools, has proved a very kind assistant toward a more speedy and thorough improvement of scholars in numbers and at the same time has relieved masters of a heavy burden of writing out rules and questions under which they have so long labored to the manifest neglect of other parts of their schools.

The Pupil's Guide, by Benjamin Dearborn (1782), is simply a collection of the "most useful rules in arithmetic." The book contains no examples or problems. Its purpose was "to lessen the labor of the master."

A few authors attempted to facilitate this plan of instruction even more than by simply providing a source for problems and rules. Isaac Greenwood, the first American to write an arithmetic which was published, says in his preface:

The Reader will observe, that the author has inserted under all those rules, where it was proper, Examples with Blanks for his practice. This was a principal End to the Undertaking; that such persons as were desirous thereof might have a comprehensive Collection of all the best Rules in the Art of Numbring, with examples wrought by themselves. And that nothing might be wanting to favour this design, the Impression is made upon several of the best Sorts of Paper. This method is entirely new,

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Daniel Adams embodies this same feature in his Scholar's Arithmetic in 1801. He says in his preface:

To answer the several intentions of this work it will be necessary that it should be put into the hands of every arithmetician; the blank after each example is designed for the operation by the scholar, which being first wrought upon a slate or waste paper he may afterwards transcribe into his book.

This text apparently attained some popularity, for by 1815 it had gone through nine editions. I have seen a copy printed as late as 1824. I have a copy printed in 1820 which bears the imprint of the hand of some pupil who doubtless labored long over the involved and obscure exercises.

When Dudley Leavitt edited an abridged edition of Pike's System of Arithmetic in 1826, there was also published

A New Ciphering Book, adapted to Pike's Arithmetic abridged; containing illustrative notes, a variety of useful Mathematical Tables, etc., with blank pages of fine paper, sufficient for writing down all the more interesting operations.2

1 Francis Walkingame's Tutors' Assistant, preface. This is a text by an English author, but was reprinted in this country.

2 Amer. Jour. of Educ., 1826, p. 511.

Texts in the hands of the pupils grew in favor, and thus the masters were relieved from the burden of setting sums and dictating rules. The pupil had both in his text. But in doing this another need was created. Many of the instructors possessed practically no arithmetical ability. So little, that the pupils believed that the masters could not do many of the sums, and without their own ciphering books would be helpless. Thus when a new or different problem appeared in the texts, master and pupil alike were perplexed if they could not locate it under a known rule. The following is typical of what one may imagine happened frequently:

A law had just been passed requiring that teachers' examinations should be conducted by three county commissioners instead of the township trustees, as had been the practice before. "I shall not forget," says Hobbs, "my first experience under the new system. The only question asked me at my first examination was, What is the product of 25 cents by 25 cents? We were not as exact then as people are now. We had only Pike's arithmetic, which gave the sums and the rules. These were considered enough at that day. How could I tell the product of 25 cents by 25 cents, when such a problem could not be found in the book? The examiner thought it was 64 cents, but was not sure. I thought just as he did, but this looked too small to both of us. We discussed its merits for an hour or more, when he decided that he was sure I was qualified to teach school, and a first-class certificate was given me." 1

As will be shown a little later, new types of texts began to appear in 1821. To make possible their use, it was necessary to provide a key for the use of the teacher. A key, bound either with the text or separately, became an essential part of a series of arithmetics.

The use of the ciphering book is so conspicuous in the plan of instruction and in the purpose of the texts of this period that it seems appropriate to call the period in the development of arithmetic up to 1821 the "Cyphering Book Period."

Although the ciphering book represents the most conspicuous feature of the teaching of arithmetic during this period, a careful analysis of the method of teaching reveals other factors of fundamental importance.

From the abstract to the concrete.-We have shown that the texts were organized upon this principle. It represents also the order of the instruction. The experience of the boy who was started on a "sum in simple addition-five columns of figures, and six figures in each column"-seems to be typical.

In his Scholar's Arithmetic Adams presents abstractly the four operations for integers and addition and subtraction for "compound numbers." Following the completion of this section of the text, he says:

The scholar has now surveyed the ground work of arithmetic. It has before been intimated that the only way in which numbers can be affected is by the operations of addition, subtraction, multiplication, and division. These rules have now been

1 Quoted by F. Cajori, The Teaching and History of Mathematics in the United States, p. 16.

taught him, and the exercises in a supplement to each suggest their use and application to the purposes and concerns of life. Further, the thing needful, and that which distinguishes the arithmetician, is to know how to proceed by application of these four rules to the solution of any arithmetical question. To afford the scholar this knowledge is the object of all the succeeding rules.'

Not only was the above dictum literally followed, but it manifested itself in the form of proceeding from the general to the particular. It appears that educators of those days really thought this the proper order.

Memoriter method.-As must necessarily be the case when the above order is followed, the memory was emphasized almost exclusively. This was certainly true in this early period. Adams probably only expresses the consensus of opinion when he says: "Directions to the Scholars: Each rule is first to be committed to memory; afterwards, the examples in illustration, and every remark is to be perused with care." 2

A very vivid description of the method of teaching arithmetic iś given by another author:

The boy, advanced perhaps some way in his teens, is sent to a winter school for two or three months to complete his education; for he can not attend in any other season, nor then indeed but quite unsteadily. But as he is almost a man he must go to school to cypher; and as he has but a short time for the business he must cypher fast. He goes to school, vulgarly speaking, raw, perhaps scarcely able to form an arithmetical figure. His master sets him a sum in addition, and it may be tells him he must carry one for every ten; but why, is a mystery which neither master nor scholar gives himself any trouble about; however, with a deal of pains, he at length gets his sum done, without ever being asked, or knowing how to read the sum total, or any number expressed in the statement.3

But it is cyphering, and that is sufficient. If he is taught to commit any of the rules to memory, he learns them like a parrot, without any knowledge of their reason, or application. After this manner he gropes along from rule to rule, till he ends his blind career with the rule of three; and in the end, the only and truest account he can give of the whole is, that he has been over it. But he has completed his school education, and is well qualified to teach a school himself the next winter after.*

The idea of assisting the pupil not a part of the teacher's creed.-In these early days the function of the teacher was to maintain order and hear lessons. The doctrine of interest had not yet been promulgated in this country, nor was it considered necessary to motivate the work of the school, except, perhaps, by punishment in case the lessons were not satisfactorily prepared. In the accounts of the teaching in these early days we find no mention of attempts to assist the pupil to appreciate arithmetic. The study of arithmetic was not compulsory during the greater part of this period. The pupil did not undertake the study until he desired to do so, and pre

1 P. 50.

2 Scholar's Arithmetic, p. 7.

As a striking example of this method of instruction, I have actually known a lad of 18, who, after having in this way, gone over all the first rules of arithmetic at a common school, was utterly unable to read or enumerate any number consisting of four places of figures.

Chauncey Lee: The American Accomptant, preface.

sumably studied it only so long as it pleased him, or possibly his parents. Certain incentives combined to excite interest in the subject. For example, ciphering was a relatively rare accomplishment, particularly until after 1800. Because of the few who could claim the title of "arithmeticker," the title carried with it considerable distinction and honor as well as some practical advantages. Also, the subject was relatively new and considered difficult. These conditions combined to cause rivalry, and one can easily imagine the zeal with which the children to whom such a thing appealed attacked sums which had a reputation of being difficult. Under these conditions the need of motivating the work in arithmetic would not be felt as keenly as under our present conditions. But when interest in the work flagged, as it must have at times, or when a pupil failed to become interested, we have no evidence that it was considered the teacher's function to stimulate interest except by one means, i. e., punishment. In some cases the pupil was allowed to drop the subject.

The master set or dictated sums and rules and examined the pupil's work, or where the pupil possessed a text he had only to examine the work. These were the essential features of the instruction. In the case of some teachers they represented the total of instruction; other teachers were "more communicative." Just what this quality was we are not told, but we may draw some conclusions from the texts of the period which were presumably written by some of the "more communicative" teachers. In the text the assistance given to the pupil is limited to an explanation, or demonstration, of an example which has been selected to illustrate a rule. The explanation is little more than an elaboration of the rule for this special case. The following explanation of the first example in division in Adams's Scholar's Arithmetic is typical. The example is, Divide 127 by 5.

Proceed in this operation thus: It being evident that the divisor (5) can not be contained in the first figure (1) of the dividend, therefore assume the first two figures (12) and inquire how often 5 is contained in 12; finding it to be 2 times, place 2 in the quotient, and multiply the divisor by it, saying 2 times 5 is 10, and place the sum (10) directly under 12 in the dividend. Subtract 10 from 12 and to the remainder (2) bring the next figure (7) ́at the right hand, making the remainder 27. Again, inquire how many times 5 in 27; 5 times; place 5 in the quotient, multiply the divisor (5) by the quotient figure (5), saying 5 times 5 is 25, place the sum (25) under 27, subtract, and the work is done. Hence it appears that 127 contains 5, 25 times with a remainder of 2, which was left after the last subtraction.1

Such assistance is "telling" with no attempt at development. The attitude is: The rule is difficult because it is not finely enough divided. Hence we will state it for a particular case in more detail. No attempt to develop the topic or to teach the pupil to think is indicated.

IP, 27.

Deductive instruction.-It has just been shown that there was practically no attempt to instruct pupils, but in so far as there was any plan for guiding the pupil in his learning, it was deductive, i. e., from rule to problem. The textbooks were arranged on this plan, and the evidence indicates that school practice followed this order exclusively.

No drill.-As has been seen, the texts made practically no provision for drill, and ciphering books of the period indicate that no drill was given. Speed contests and rapid drills were wanting. A partial reason is that blackboards and slates were unknown until near the close of this period. Paper was expensive, and the manufacture and repair of quill pens took much time. There appears to have been no drill, even upon the multiplication tables, except in the dame schools. At this stage of his education the pupil could not write, and hence his work must be oral, and drill upon the number facts was a feature of it.

No oral arithmetic. The work was all written except the very elementary instruction in number given in the dame schools. The beginning of oral arithmetic, or as it is more generally called, mental arithmetic, belongs to the next period.

Individual instruction.-The very nature of the ciphering book method made impracticable class or group instruction. Though little instruction was given, the pupils had individual contact with the master. The practice of having examples explained to the class by either the teacher or a pupil comes in a later period.

An objective result.-The accumulating collection of examples and their solutions furnished tangible evidence of the pupil's progress. By both master and pupil this objective result was consciously striven for. Incidentally, neatness both in the making of characters and in the arrangement of the work was insisted upon and usually secured. Ciphering books which are still preserved are models of neatness and skill in writing.