custom to insist that each shall know his men individually and thoroughly, shall in difficult cases study their environment. By just that kind of contact it frequently occurs that men are directed and developed who otherwise would be lost to the intellectual world. This is the high missionary work of the true teacher. I am strictly opposed to any method by which this individual contact and influence are destroyed. I say to my men "know your men; I will try to keep the number with whom you are to deal within reasonable bounds, but know your men as individuals and help them as best you can.'

PROFESSOR W. G. RAYMOND: In these papers the question seems to be, how shall we find out whether the students do their work. It appears to me that is not the point. It ought to be, how shall we know that students are doing their work. A good many difficulties have been pointed out and I don't want to bring up any old scores, but possibly I cannot do better than to point out how easy it would be to get around it all if you would adopt that scheme presented by me three or four years ago of individual instruction.

PROFESSOR MANSFIELD MERRIMAN: There is one feature in regard to this work I have not heard spoken of, and that is the importance to the student of first thoroughly understanding what the problem means. After he has done that, he ought to be able to make a mental estimate of the result. For example, on one of these cards there is an area drawn which the student is required to determine by integration. Since this area roughly resembles that of a parabola, a practical man would multiply the base by the altitude and take about

two thirds the product for the area. It is this point of view that the student should also obtain: first, making an estimate by some rough and ready method, and then working out the accurate result and comparing it with the estimate. In this way the engineering judgment of the student is developed, and sound judgment, with ability to make mental estimates with approximate correctness, is one of the elements of success in engineering.

PROFESSOR H. WADE HIBBARD: This subject is interesting to us at Cornell because we are tending towards doing away with examinations at the end of the terms, and I expect that in the course of the next year this result will be attained, which means that more dependence is going to be placed in all our classes upon the understanding of the grade of work the students are doing day by day. I might say we are feeling that the ten days each semester given to examinations is largely time lost.

I did not suppose that this discussion was going to take the form Professor Waldo spoke of, viz., keeping in closer touch with and understanding each student. I think we realize that and now we are endeavoring all the time to keep in close touch with the students. That is very rightly said to be one of the dangers of the large engineering schools. Personally, I have felt that much of that touch with the students may be obtained entirely outside the class-room, and I look to getting my boys over to my house to eat something or play some games or sit on the veranda and talk, and I believe much of the good that we as engineering instructors are going to do for our boys will result from that personal contact with them.

PROFESSOR ALBERT F. GANZ: I do not agree with the last speaker that the ten days devoted to examinations at the end of each term is time lost. On the contrary, I consider it time well spent, first because students in preparing for these examinations must review the work gone over during the term; and secondly because it gives the instructor a chance to find out how much of the work the student has retained.

Regarding the card scheme for handling problem work I can say that for several years I have used the following scheme and have found it to work well. I have a set of problems written out on numbered white cards for each division of the subject which I teach, namely, electrical engineering. I have also sets of yellow cards with the answers and main points of the solution written on them and correspondingly numbered. The white cards are intended to be handed out to students and the yellow cards are for the instructor to facilitate looking over the solutions. I use these cards both for written tests and for work on the blackboard. If, for instance, I am to have a recitation on problems on the blackboards, I will hand out fifteen to twenty problem cards to as many students at the beginning of the hour; as this takes but a few minutes no time is lost as would be the case if the problems had been assigned verbally. The first student to finish is called on to explain his work, the students at their seats following the explanations. By looking at the yellow solution card I can quickly detect, for instance, a numerical error.

I believe that problem work is very important not only in mathematics but in all engineering subjects. These problems should be taken as much as possible

from practice because students will take so much more interest in a real problem than in a made up problem. I add to my stock of problems every year, replace old problems, and so endeavor to keep the stock alive and of interest.

Regarding President Howe's plan for handling large classes in recitations whereby he sends a portion of the class into an adjoining room full of blackboards and lets them work problems on these boards, it seems to me that it would be simpler to hand out blank paper and have students work their problems on the paper and hand these in at the end of the hour. I also have large classes to handle and do it for recitation work partly in that way, and partly by having problems put on the boards in the lecture room and then have these explained before the class.

PROFESSOR H. E. SMITH: I have been much interested in the remarks relative to the card idea of giving out problems in class work, and I may say for some eight or nine years I have been building up this same system and have found that it works admirably. The subjects which I have taught have included applied mechanics, steam engine work, etc. I have now quite a complete file of cards on which are problems, usually numbered under headings, so that I can easily separate them into their different classes. The way I have used them has been like this: The day on which we had a certain subject to consider I have selected a pack of these problems sufficient or more than sufficient to go around the class, and after a discussion of the subject in the class, possibly for the first twenty minutes, I would request the class, if not too large, to go to the board, and on their

way to the board they would come to the desk and I would give out these cards. I have tried to arrange the cards so that each problem as given out would represent some phase of the practical application of the principle that had been discussed during the day or on previous days. Of course, if the class is large, it is a little difficult to get a sufficient number of problems that bear directly on the one or two subjects of that day. These problems are usually numerical, that is, require a numerical result. In most cases I have the results on a separate card or sheet, so that if it is necessary to check in that way I can do so. Outside of this work, usually the class has been given one or more problems to do at their homes, and there again I have tried to make the problems as far as possible practical applications. I think that the result from my experience has been that it is an excellent scheme and the men get hold of ways of applying the principle to actual practical problems. I think in examination it has been clearly shown that the men who fail to hand in these problems are the men who would fall down in examination. I have never used this system for quiz work or examinations, as the first speaker remarked, but I think it might be done.

PROFESSOR C. FRANK ALLEN: There is one point I wish to bring out which I think is in line with what Professor Merriman suggested; this is, that whatever system is used for card problems and the like, the object should not be primarily to get marks to determine the rank of the student. The aim eventually should be for their instruction, to improve the value of the work; but the difference is not always quite so great as might