begins a recitation as usual and conducts it until the time for the quiz arrives. Then the recitation is promptly stopped, "next lesson" is assigned, the quiz is announced and a problem card is given each student. Cards 7a and 7b are distributed to students in the oldnumbered rows of seats, and 7c and 7d to those in the even, and so that 7a and 7b alternate in the rows, likewise 7c and 7d. Near the close of the quiz, the instructor stations himself at the door of the recitation room and collects the problem cards as the students pass out. This insures collection of all the cards, a necessity to preserve the full value of the system.

At Wisconsin, all sections of the mechanics class do not meet at the same hour but always within two consecutive hours; therefore, it has been practicable to use the same problem for all sections, the short intermission affording insufficient time for consultation between students.

As before stated, I have used this system only one semester. In that limited experience I have noted but one objection (not serious) and more merits than I anticipated. The objection lies in the occasional necessity of stopping a recitation for the quiz in the midst of an important discussion, but this can be avoided usually by careful management on the part of the instructor.

The following advantages of the system may be mentioned: The problems cards are of permanent value, as each problem may be safely used at least once in two years. Once prepared, each problem is ready for instant use. (I regard my collection just as useful as any good laboratory equipment.) The system is a great

time-saver for it economizes much time for instructors inasmuch as they are furnished quiz problems already for use and solutions, too, once the collection is stocked, and it economizes time for the student in the class-room, for the quizzes can be started very quickly. Then, too, his work is facilitated by his having the problem immediately before him and better stated than a blackboard problem is apt to be. Last, and very important, the system furnishes the occasion for frequent meetings of the instructors of the department for profitable and fruitful discussions of their own teaching and the work of their students as disclosed by the examination of the quiz papers.

SYMPOSIUM: METHODS OF HANDLING PROBLEM

WORK IN LARGE CLASSES.

SECOND PAPER.

BY CLARENCE A. WALDO,

Professor of Mathematics, Purdue University.

It is the purpose of this paper to speak briefly of problem work in large mathematical classes and to exhibit samples of a collection of problems.

To stimulate the student to do his severest work, to set him thinking upon some combination of affairs which taxes skill and strength to their utmost, but which at the same time is fairly within his powers, this is the instructor's problem. At the same time that something should be a link in a chain connecting a man in a rational way with the purposes of his education. It is a well-recognized principle in sound class-room work that at every class-room hour there should be complete mental contact between student and teacher, a contact which reveals the true mental status of the pupil, at least in its relation to the lesson material.

The accomplishment of such contact places certain limitations upon the organization of class-room work and one of these is found in the number of pupils in one class. Too few students in a class tend to make the teacher lazy; too many make him superficial. Unless endued with extraordinary ability the teacher will find twenty-five near the maximum number of students that can be properly handled in any one class-this, at least,

seems to be the conclusion of good teachers—and such a class is large.

In the hour and a half or two hours spent at the home in preparation, what work has been done and how? This the teacher must know if he is to deal with his students as individuals. But how? Perhaps the most common method of the diligent teacher is to insist upon the student's handing in on paper a certain number of selected and carefully solved representative problems. It is certain that this method frequently leads to excellent results. But if the purpose of the written work is to insure originality it is a failure. If the student tries to shirk and deceive it is an easy matter for him to copy from others. Knowing this to be true, is it necessary, is it even wise, for an instructor to burden himself with such a vast amount of critical reading as this method when well done of necessity implies? Is it not often and largely a waste of energy-a useless and unappreciated piece of drudgery?

A second method in frequent use for ascertaining the amount and kind of preparation is to pick problems at random from work assigned for the student to do at the board. In small classes of five to ten, good results may be reached in this way, but in classes from twenty to thirty the lazy, indifferent or bashful student may for weeks escape wholesome criticism and admonition.

In our technical schools which are increasing so rapidly in numbers, the provision for instruction does not keep pace with growth. Congested classes are the result, thorough individual examination impossible. Why not put students at once on their honor? At the opening of every lesson period ask every student directly

concerning work done and record his answer or ask him to report on a blank furnished for that purpose. After such inquiry you are justified in assuming that the students have told you the truth and have a right to modify the work planned for the hour according to reports received. Students will sometimes make false claims, but the alert teacher will soon discover that fact and prevaricators may be easily made to see that honesty is the best policy. But the instructor with a flexible program will know how to divide his time in explanation, in development and in criticism. Certain members of the class are kept at their seats and given such council and assistance as they need singly or together while the better students are honored and stimulated with fascinating original work.

One of the easy, obvious ways of conducting a mathematical recitation is to send students to the board for the hour to work problems proposed in the text, and it is a way too often used to kill time. Three good reasons may be given for giving problems at the board from the exercises in the day's assignment and it would be difficult to enlarge the number:

1. If the student has orally claimed at the beginning of the hour that he has actually solved certain exercises, he should occasionally be called upon to verify his report.

2. The instructor should make sure that a student is grasping and using the simplest and best method of solution. For this purpose the whole class can be profitably engaged at times upon the same problem or upon two problems so that those standing next to each other will not be doing just the same work.