demonstrate. Each pupil put on the blackboard the figure to be used in his demonstration, with a few of the more important statements in the demonstration. This usually took from five to ten minutes. The remainder of the period was spent by each section listening to and discussing the demonstrations by its members. Usually these were demonstrations. that had been studied out at home, but if the teacher were wise enough to suggest it, the period was partially used for group study and discussion of new propositions or exercises. Inasmuch as no rowdy behavior was tolerated in the school, and the students spoke in low tones but sufficiently loud to be heard by a small, compact group, the three recitations proceeding simultaneously produced no confusion. The teacher divided his time between the sections as he thought best. Sometimes he simply looked on; sometimes he made suggestions; sometimes he took full charge of a section for as long as he saw fit.

Stimulated intense study of difficulties by groups. This scheme had many educational advantages apart from its variation of the pace to suit the capacities of the three types of students. The chief of the advantages was the intense geometrical thinking done by the members of each section when it was working out puzzling points in propositions or exercises. The sections being fairly homogeneous, a serious difficulty for one member of a given section would be likely to be a difficulty for all its members. Hence it would be worth while for the group to work it out. This condition contrasts very strongly with that presented in recitations conducted for the whole class. In the latter case, difficulties which would puzzle the slow students for fifteen minutes might not be difficulties at all for the bright pupils. The teacher either has to let the bright pupils tell the solution, thus depriving the slow ones of participation in the activity of finding it, or he has to call on the slow ones to work it out, allowing the fast ones to mark time meanwhile.

Another advantage of these sectional recitations is the fact that each student gets an opportunity each day to recite at some length to an audience that is usually attentive and critical.

Only skilled teacher can use method successfully.-Needless to say, an unskilled teacher in a school where there was no well-defined spirit of work and serious behavior would have difficulty in administering this monitorial sectional method in geometry, but a skilled teacher in whose classroom the spirit of serious work and application always prevailed could probably secure better educational results by some modification of this system than by any other. It could easily be adapted to use in other mathematics classes, such as those in algebra and trigonometry. It could probably be used to advantage in any subject where there is a large amount of fairly routinized practice in studying and mastering material in textbooks for example, in Latin translation. In all such cases the teacher must be more interested in the process of thoughtful study and mastery of the material by the students than in covering a given amount of ground.

3. Recitations only for students who need them; seat work for others. Both of the methods described so far (namely, the Pueblo scheme of individual instruction or progress and the monitorial sectional plan) provide for different rates of advancement by different students throughout the course. The scheme now to be described holds the class together as far as progress through the essential steps of the subject is concerned, but enables the brighter pupils to do more intensive work at each step and saves their time by excusing them from paying attention during some of the recitations.

Time wasted by bright students in algebra recitations.— The desirability of such a method is illustrated from one period in my own experience, when I taught algebra very unskillfully in high school, using the ordinary method of

conducting recitations upon material that had been studied at home the night before. My younger brother was in one class, and, being a bright, conscientious student, he mastered at home every evening, without assistance, the lesson for the next day. Consequently, when he came to class, there was practically nothing for him to do but sit as a spectator, except when he was given an opportunity to contribute. I recall two other students in the same class that were the only ones who caused me any concern about discipline. They also were bright pupils who had got all the meat out of the set of problems under discussion, but instead of sitting as polite spectators they insisted on devising ingenious pranks that amused me so much that I was always at a loss to know what to do. Unfortunately it did n't occur to me to excuse them from participation in the recitation and to devise assignments of extra problems that would interest them and serve as an outlet for their ingenious energy.

Three factors in scheme for special seat work. Such a scheme for saving time and for proportioning to their capacities the amount of work expected of individual pupils is often used by skilled teachers who have mastered the ordinary routine of handling classes and are interested in devising and organizing varied assignments. When skillfully administered, the scheme involves at least three features; namely, (1) the minimum essentials of the subject to be covered by the slow students; (2) supplementary assignments for the bright pupils; (3) arrangements to conduct recitations only for those who need them.

1. Standardized minimum essentials for the slow. - The idea of standardizing quite definitely the minimum essentials in elementary and high-school studies has only recently been generally applied. The importance of the practice can readily be appreciated when one considers, for example, what a very small part of grammar-grade arithmetic is necessary for the student who is lacking in mathematical capacity and who, as

a consequence, would not secure a position in life as an accountant. For such a student, only the bare essentials of the subject should suffice to give him a passing grade. The same principle applies to most high-school subjects, and its application is readily illustrated by the topic of factoring in algebra, which has already been discussed from this point of view on page 82.

2. Standardized, mimeographed, supplementary assignments for bright pupils.-Supplementary assignments for the brighter students should be just as definitely routinized and standardized as the minimum essentials for the slow. Hence, the teacher should have, very definitely outlined on paper, statements of supplementary topics or problems, with exact references to material to be used in studying, means of securing the material, etc. Sometimes the supplementary material could be in the form of another textbook to be bought by the student and brought to class regularly. More commonly, the teacher will have in the room a small working collection of books for supplementary assignments. Such supplementary collections are provided by the school authorities for every room in well-equipped elementary schools and should also be provided in high schools. Sometimes the student could be sent to the high-school library. The main point of this paragraph is that all the supplementary assignments should be definitely planned, and, if possible, mimeographed outlines prepared so that a student's assignment could be given to him in a moment by brief reference to the outline. Much of this assigning could be done while the class is assembling. In some cases supplementary assignments for a week could be outlined in a five-minute conference with the student.

3. Excuse individuals to work at desks or in library or laboratory. If the classroom is of ordinary fair size, students who are excused from participation in the recitation can work at the back of the room without any serious loss of time. If

the teacher establishes the right general spirit of work, all students will form such habits of application and of disregard of distracting conditions as were described above on page 359. In some cases students who are excused from participation in the recitation could be permitted to work in the library or in a laboratory with perfect assurance that they would use their time seriously and profitably.

Examples of successful use of the method.-A description of a method of conducting algebra classes along lines somewhat similar to those described above is found in a pamphlet entitled "The Laboratory Method in Teaching Mathematics," by Hornbrook (3). I have seen the method used in classes in German where the class work consisted of reading or translating selections that had been studied at home. In each class two or three of the brighter students were excused two periods a week on the average and given long reading assignments in other German books. These they read and prepared in the form of epitomes or selected narratives, which they told in German to the class from time to time. The ordinary class in foreign language, when conducted by the translation method, furnishes perhaps the best argument for such variations as we have been describing, since the class time is very commonly wasted for the brighter pupils who have prepared their translations conscientiously at home or during study periods.

4. Required supervised study periods supplementary to recitation periods: the Batavia scheme. Up to this point in the chapter we have considered three types of experiments in varying class instruction so as to provide for individual differences in capacity; namely, the Pueblo plan of individual instruction, the monitorial sectional plan, and the plan of providing supplementary assignments for the brighter students instead of requiring them to participate in all recitations. The fourth scheme for securing individual instruction does so by having regular supervised study periods for all