for written tests is as important for the successful mathematical instructor as are stuffed bird skins for the ornithologist. He should be as alert in seeing, recording and arranging his material as the entomologist in noting, capturing and classifying insects new to science. They should come from every possible source, but especially from conversation with his wide-awake professional colleagues. For recording problems, grouping them and making them instantly available a card catalogue with a problem to a card is probably the best that can be done. There will then be at hand a growing and improving collection of problems to clinch principles and to illustrate them in a concrete way. Fully as important for class-room work will be the reflex action upon the instructor himself. His will then be the open, the observing, the alert mind, fresh and juicy. No drying up and fossilizing there. The student will feel that it is an honor to be entrusted with a card problem, that a solution within the hour is a distinct victory usually demonstrating satisfactory mastery of principle and showing commendable progress. In connection with this paper I am exhibiting a few cards taken from the collection of Professor A. M. Kenyon, of Purdue University. The following may be taken as samples of these. 2. Relates to rates as illustrated by the hands of a clock. Rates. Clock Problem. The hands of a clock are 6 in. and 3 in. S= 3√5-4COS, where s is distance in inches 6 Rates. 2. (Students Card) 2. Clock Problem. The collection is classified to suit the subjects taught. The cards are not placed in the hands of students, but in every case such portion of a card is given out orally or on a separate slip as will clearly state the problem, while upon the card itself, for the assistance of the instructor in rapid critical work the salient features of the problem are completely worked out. A complete collection therefore requires that every problem should be in duplicate, one card for the instructor, one for the student. DISCUSSION. PRESIDENT CHARLES S. HOWE: I was asked to write a paper on this subject, but not knowing whether I could be present I said I preferred to take part in the discussion; my part in the discussion will be in regard to what we are doing at Case School of Applied Science. I am sure every teacher in a technical school understands and appreciates the problems which an engineer must solve when he goes out into the world. The engineer is continually solving problems and he must get the right answer; if he does not he is worthless as an engineer. It is therefore important that we teach pupils to solve problems correctly, and to do this we must insist that the problems be right as far as the answer is concerned. We give the man no credit whatever unless the answer is correct. There is no halfway point. The problem is right or it is wrong, and if it is wrong the man should have no credit for it-no more in fact than he will get after he goes out as an engineer. If it is right he should have full credit for it. We do not allow anything for knowledge of theory when a problem is being worked. We get at a man's 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. I can 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 |