to appear which were organized upon what has been known as the "spiral plan."

The Werner Arithmetics, by Frank H. Hall a three-book series for graded schools, and The Hall Arithmetics by the same author, a two-book series for graded or ungraded schools, were pioneers in the exploitation of the spiral plan of organization. A copy of the Werner Arithmetics, Book I, which I have bears the date 1896. to the plan of this text, Mr. Hall says in the preface:

As

The first five lines of this book present problems in addition, subtraction, multiplication, division noting the number of groups, and division noting the number in each group. Then, by a kind of spiral advancement, the pupils move around this circle and upward through all the intricacies of combination, separation, and comparison of numbers.

The arrangement of topics is unique and convenient. In this book measurement problems appear on pages 43, 53, 63, 73, etc.; a certain class of fraction problems on pages 45, 55, 65, 75, etc.; facts of addition, subtraction, multiplication, and division on pages 41, 51, 61, 71, etc. This decimal arrangement of subjects makes the books almost as convenient for reference as are the books that are made on the strict classification plan, while the frequent recurrence of similar matter insures thorough review. This spiral arrangement, which was followed in the other texts of the series, found favor very quickly. Within the next few years a number of texts were published which were organized upon the spiral plan. A few were as extreme as The Hall Arithmetics, but in general the spiral plan was modified in part. The spirals were less numerous, and the "decimal arrangement" was not followed. Within the last few years there has been a pronounced reaction. The spiral plan has been severely criticized, and authors of some of the spiral texts have found it necessary to revise them, eliminating some of the spirals. At present the consensus of opinions seems to be in favor of a moderate spiral for grades one to four, a topical plan for grades seven and eight, and a transition from the one plan to the other in grades five and six. Some authorities, while agreeing with the above general opinion so far as the actual instruction is concerned, contend that the best results can be obtained from using a topical text above the primary grades. The teacher can then adopt a spiral which will meet more nearly the needs of the community and the particular class.

The Grube method.-Grube (1816-1884) was a German whose "claim to rank as an educator lies largely in his power of judicious selection from the writings of others." The features of Grube's writing which stand out most clearly are objective teaching, the measuring of each number with fixed units, the spiral or concentric circle plan of organization, thoroughness and complete mastery, making of each arithmetic lesson a language drill, and the simultaneous teaching of the four fundamental operations for each number.

1 D. E. Smith: The Teaching of Elementary Mathematics, p. 89.

Grube presented his method in "Leitfaden für das Rechnen in der Elementarschule, nach den Grundsätzen einer heuristischen Methode" (Guide for Reckoning in the Elementary School, according to the Principles of a Heuristic Method). This was published in 1842. The beginning of the method in this country dates from 1870, when F. Louis Soldan presented to the teachers' association of St. Louis an account of Grube's plan for teaching the numbers 1 to 10. The plan was tried in the St. Louis schools and later elsewhere. In 1876 Soldan presented the remainder of Grube's plan, which includes the numbers 10 to 100 and above, and common fractions. This was intended to cover the work of the first four years. In 1888 Levi Seeley wrote Grube's Method of Teaching Arithmetic. This is really a complete text for the first four years.

The method rapidly became popular in many sections of the country. One writer suggests that the reason for the popularity of the method in this country was due to Grube's original treatise being brief and written so as to be easily translated and to the fact that it was a "German" method. Furthermore, it seems that the friends of the method, or at least those who first used it, saw most clearly the good features and emphasized them to the partial or entire exclusion of the less desirable features. Doubtless they, in their enthusiasm, secured commendable results. But as is often the case, as the method was passed on to other teachers, the attention was fixed primarily on the most obvious phase of the method, which said that the four fundamental operations should be taught for each number before the next was taken up. Thus within recent years this single feature has come to stand for the Grube method.

Grube's method has been severely criticized by several recent writers on the teaching of arithmetic. As a plan of teaching it has been discredited. Much of Grube's method was not new to the United States. In fact all the features are to be found in texts published prior to 1870, though some were not given quite as extreme form. Objective teaching began with Colburn. Davies held that the unit was the basis of all numbers and treated each number "as a collection of units." Emphasis had already been placed upon thoroughness and drill in language. In the Child's Book of Arithmetic, 1859, D. P. Colburn approximates the concentric circle plan and the simultaneous teaching of the four fundamental operations. However, this does not alter the fact that Soldan introduced the Grube method directly from the writings of Grube.

The relation of the Grube method to the spiral plan.—The opinion is prevalent that the spiral plan is simply an outgrowth of the Grube method. However, the writer has failed to find evidence to show

1 See D. E. Smith: The Teaching of Elementary Mathematics; McMurry: Special Method in Arithmetic; McLellan and Dewey: Psychology of Number. It is also true that, in its entirety, the method did not gain any considerable prestige.

this. In fact, there is evidence to indicate that the spiral plan was the result of attempting to fit the organization of arithmetic to the child and to secure thoroughness.

The texts of the previous period were topical, but the order pursued by the pupil was spiral. Not only this, but there were frequent review exercises. Now, when the slogan was "adapt arithmetic to the child," what would be more natural than to put the spiral into the text rather than leave it to the pleasure of the teacher. Thoroughness was the cry, and psychologists were saying that only by repetition is thoroughness secured. Then, the plan of the text make repetition certain. Frank H. Hall suggests this conclusion when he says, "Proper sequence with reference to the pupil has been constantly in the thought of the author in his selection and arrangement of matter," and later, "the frequent recurrence of similar matter insures thorough review."

A careful comparison of the Grube method and the spiral plan reveals many essential differences and few points of contact. Grube did not go beyond the work of the fourth grade. Within the year his spirals were all of the same size; no new matter was admitted in the successive revolutions. The spiral plan usually provided for some preliminary number work in which the pupils learned to count (Hall says up to 100). Also they learned some of the number facts. Grube did not provide for this. The spiral plan did not make the magnitude of the numbers the basis of the spirals. Furthermore, the work of Grube had been severely criticized by Dewey in 1895.

Rationalizing the teaching of arithmetic.—The changes in the aim, subject matter, and organization of arithmetic, together with other factors, have combined to change the method of teaching arithmetic. The present period has been one of transition. Perhaps less has been accomplished in modifying the method of teaching than in the other aspects of arithmetic. Certain it is that school practice has fallen far short of realizing the ideals of method proposed by leaders in arithmetical reform.

The most important factor in this transition has been the child, and progress has been made in the direction of adapting the method of teaching to the nature of the child as revealed by modern psychology. But this progress has been attended by unfortunate wanderings after "single idea methods" and devices. However, the period has been marked by progress. Methods which in themselves are open to serious criticism have rendered service by making obvious defects of the dogmatic, memoriter, disciplinary methods of the past.

Development of topics.-It follows immediately from Dewey's first thesis that the pupil's understanding of number and operations with numbers must result from his own psychical activity. This implies that it is the function of the teacher to provide situations

which will exercise the pupil's mind and to simply guide the pupil in this activity. The method of such teaching would consist of a plan for providing situations which call for the use of number and number relations, for moving the pupil to work upon them, and for guiding the activity of the pupil. The plan for guiding is to be based upon the normal way in which the child's mind works in "making a vague whole definite."

The Herbartian plan.—The leaders in the Herbartian movement in the United States emphasized inductive thinking, and their concept of inductive teaching became quite popular and was applied to arithmetic along with other school subjects.

Charles A. McMurry says:

"The study of arithmetical processes furnishes one of the best opportunities to apply inductive methods. And nearly every topic in arithmetic has these two phases: First, to derive these general processes; second, to apply them variously to important practical and theoretic affairs that need arithmetical clarification.1

The derivation of "these general processes" consisted of the steps (1) preparation, (2) presentation, (3) comparison and abstraction, and (4) generalization.

The Herbartian plan applied to arithmetic was received with enthusiasm by many teachers and was by them unquestioned. The attempts to use it in actual teaching became and are to-day widespread. But the Herbartian plan of inductive development has been severely criticised, and it has been pointed out that it is a special case of reflective thought with the steps of problem, data, hypothesis, and verification. In addition, the practice of developing or rationalizing every topic in arithmetic has been criticized recently by some educators. They point out that some parts of arithmetic, such as the fundamental operations, must be reduced to habit if they are to function efficiently. They contend that to attempt to explain the "why" in such processes as "carrying" in addition and "borrowing" in subtraction "is merely to stir up unnecessary trouble, trouble unprompted by any demands of actual efficiency." This position with respect to rationalization is summarized by Suzzalo in his book, The Teaching of Primary Arithmetic, 1912.

He says:

(1) Any fact or process which always recurs in an identical manner, and occurs with sufficient frequency to be remembered, ought not to be “rationalized" for the pupil, but "habituated." * (2) If a process does recur in the same manner, but is

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so little used in after life that any formal method of solution would be forgotten, then the teacher should "rationalize" it. * * *(3) If the process always does occur in the same manner, but with the frequency of its recurrence in doubt, the teacher should both "habituate" and "rationalize." * * * (4) When a process or relation is likely to be expressed in a variable form, then the child must be taught to think through the relations involved, and should not be permitted to treat it mechanically, through a mere act of habit or memory.

1 Special Method in Arithmetic, p. 60. See pp. 165–167 for some lessons planned according to this method. 2 See S. C. Parker: The History of Modern Elementary Education, p. 425 ff. for a summary of these criticisms.

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This reactionary movement should not be interpreted to mean a return to the former memoriter plan of instruction and drill. It indicates rather that mental processes are being carefully examined and modes of instruction are chosen with reference to the subject matter involved and the end sought.

In the Herbartian plan, deduction came in the step of application and was treated as an incident in the total cycle of inductive development. But it has been pointed out that in life we make many deductions for every induction, and that in the rationalizing of arithmetic deduction has an important place. It is also a special form of reflective thinking, the distinction being that the general principle is a part of the data, and the hypothesis consists in subsuming the particular case under the appropriate rule.1

Motivation. Interest as a motive.-Along with the efforts to adapt the mode of instruction to the child, there have been endeavors to work out plans for securing incentives for the mental activity of the child. In reacting from the plan of securing motive by rivalry, emulation, fear of punishment, etc., interest was conceived of as a motive, and the plans for securing motive were plans for arousing interest. Interest and its attendant conditions were very imperfectly understood by the great majority of teachers, and blunders were made in attempts to arouse interest.

For instance, it was proposed that children like easy things, that difficulties were uninteresting. Hence to make arithmetic interesting, make it easy. So difficulties were divided and subdivided or removed. The pupil was "prepared" by the teacher for each topic. And this anemic subject matter was to be interesting and attractive to the pupil because it was easy. Or, the uninteresting became interesting when "associated" with the interesting. Hence to make an uninteresting topic in arithmetic interesting, "associate" it with some activity which the pupil has already found interesting. For example, children are interested in games; they are not interested in the multiplication table. Thus to secure interest in the multiplication table, associate it with some game. This has been done by devising a game with which the multiplication facts could be "associated."

The efficacy of this plan depends upon the interpretation of the word "associate." If it is taken to mean that the game is to be so arranged that the pupils will need, or find useful, the number facts to be taught, the number facts will become interesting. They are then a means to a valuable end. But this was not the way it sometimes worked out in practice. As recently as 1911, an author gives a lesson plan of teaching the multiplication table of fours in which the game of bean bag is to be utilized. The pupils have already played it in

1 For deduction applied to arithmetic, see Strayer: A Brief Course in the Teaching Process, p. 182.