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knowledge of theory in other ways. When a problem is being solved the answer should be obtained and no credit given unless the answer is obtained. speak with more knowledge of mathematical problems because that is the department in which I have worked for many years.

The size of classes is a very important question in the discussion of problem work. I agree with one of the speakers who said that an instructor cannot handle more than twenty-five students at a time. That would be a large number in some institutions but students in institutions grow in numbers very much faster than faculties grow in numbers and hence large classes have become a necessity. Sometimes it is very difficult to arrange a schedule of classes because in some departments a large number of students can be taken care of in the divisions, while in others the divisions should be smaller. We have found it possible to handle very large divisions in mathematics, having frequently from seventy to ninety students in one class. We have one large recitation room having from eighty to one hundred seats. The class assembles in this room. The demonstrations of the day and the problems which ought to be explained are assigned and students are sent to the board for this purpose. All other students are immediately sent to an adjoining room which not only has a number of blackboards around the walls, but has a number of movable boards in the center of the room. In addition to the professor in charge there are two or three instructors as assistants. If the professor is in charge of the main room the instructors go into the blackboard room. In this room the students are

given problems to work and are kept busy until the latter part of the hour. The problems are sometimes those which have been assigned for the day's lesson, but are usually new problems which the students have never seen. They are obliged to work without books and this recitation is in the nature of a test. We have collections of these problems although we have never made as complete and careful a collection as Professor Waldo and Professor Maurer. Near the close of the hour the students all go back into the main room and there listen to the demonstrations and problems which have been assigned from the lesson.

Usually there is a professor at the head of a department and he has a number of assistants. It very frequently happens that these instructors are young and have had very little experience. We have to take young instructors without much experience because we do not pay much salary. It is to be presumed that the professor who has had several years of experience in teaching is a better instructor than a young man who has been teaching but a year or so. By this method all the teaching is in the hands of the head of the department. He handles the class, assigns the work and does the teaching, relying upon his instructors to look after blackboard work, correct problems, etc. The instructors look after details but the professor is responsible for the conduct of the class. I think I have been able to obtain better results with a division of seventy-five students and two instructors than I could with three divisions under three different men, using, perhaps, three different methods of teaching.

The question whether a student is getting the practice in problems he ought to have is a serious one. Many students will not work problems-perhaps, that is not quite true-but many do not hand the problems in on time and so it is presumed that they do not work them. Where they do not hand them in on time we give them other problems entirely outside the text, as far as possible different problems to different men, and every student is obliged to work a number which is practically the same as the number in the text. The problems are examined, and if wrong, go back to the student, who is obliged to solve them again and is kept at them until he gets them right. An accurate list is kept in a note-book and each man is held responsible for all the problems. This method has not been altogether satisfactory because it has been found that when examinations come, the men do not seem to be able to work problems as well as they should. We have recently been trying to find some method whereby we can be sure each man works the problems assigned and works them when he should. The following method has been tried with a good deal of success. A number of problems are assigned each day for examination and study but the men are not obliged to hand them in. During the first part of the recitation hour a written test is given in these problems. Representative problems are picked out, two, three or four, depending upon the length of time necessary to solve them and perhaps fifteen to twenty minutes is given to this test; but it is an absolutely rigid test, every day, day after day, on problem work. The men are marked on this test. If they do not pass they are obliged to take the test again with other problems, so that they are held rigidly to it. We have found the results of regular tests and examinations much more satisfactory. Since this system was inaugurated not over one third as many men have failed as under the old method of assigning problems. This shows that the students feel forced not only to get the problems which are assigned every day, which may mean work with some other students, but shows that they have learned how to work the problems themselves. This is the most satisfactory method we have found of obliging the students to work problems and of actually finding out whether they do work the problems.

PRESIDENT McNair: The subject is now open for discussion.

PROFESSOR MARBURG: In holding a section of seventyfive men concurrently, your periods, I suppose, extend over an hour. Is the half hour devoted to practical work and the other to recitations?

PRESIDENT HOWE: At the beginning of the hour all of the demonstration and text-book work is assigned to some men to be explained and a certain number of problems also assigned to other men; then perhaps for thirty or thirty-five minutes all the rest of the class work problems which they have never seen before. Then all go back into the main room and there listen to recitations and problems which were assigned at the beginning of the hour. Of course, on other days, other men will recite. I may say that with seventy-five students I have divided the class into three sections. When I had charge of the classes in mathematics I would take different sections on different days, so that once in three days a section came under my personal supervision. PROFESSOR EDGAR MARBURG: The large influx of students upon the engineering courses that had become so noticeable throughout the country within recent years was unfortunately seldom or never attended by a proportionate increase in the number of higher-salaried positions on the teaching staff. I think it is generally agreed that most technical subjects can be effectively taught only in small sections, limited preferably to fifteen students. The result is that an increasing proportion of the teaching is entrusted to young and comparatively inexperienced instructors. The speaker believes that under these circumstances the following plan offers distinct advantages, namely, to have the different sections in a given subject report to the various instructors at concurrent hours, and at any of these hours to summon all these sections into the lecture room without previous notice, the lectures being given by the most experienced instructors in that subject. The frequency of these lectures would depend on the nature of the subject and the period of the course. The other instructors should be required to be present at these lectures. In connection with that system, the instructors should change sections from week to week, or in the case of a subject given less frequently than say three times a week, every second week. Thus every student receives the benefit of the average ability of the instructors. Moreover, the system gives rise to a spirit of wholesome emulation among the instructors.

For blackboard drill, assignments should be given out on slips of paper, and, at the end of a half hour, the students should all be sent to their seats to follow the demonstrations during the remaining half hour. In most cases a student who fails to complete his work in

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