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3. The instructor may use his brighter students to help him get before the class work which the others have failed to accomplish.
In general it will probably be agreed that with the exception noted the problems for individual class instruction should not be those assigned for preliminary practice, but should be no more difficult. They should be chosen with the purpose in view of amplification, application, review and unification. In other words, at least the following ideas should be present in the selections made: they should extend the work of the text, should apply the principles involved, if possible, to definite physical problems, should review, clarify and fix portions already gone over and should connect the matter in hand with the material that has preceded it with the purpose of making the subject a logical unit. For such work there are at least four common sources:
1. The exercises at the board may be taken from those in the text that were not assigned for preparation or beyond those assigned.
2. They may be carefully chosen from other texts.
3. They may be invented off hand as the occasion may require.
4. They may be taken from a live, growing and classified private collection.
The first three methods, if used with discretion, are effective but are not sufficient in making a proper preparation at all times for class-room instruction, interest and stimulation.
There can be little dissent from the proposition that a private collection of well chosen or carefully invented, thoroughly classified problems for the blackboard and 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.
1. Relates to Wallis's Coneo-Cuneus or the Ship Carpenter's Wedge.
Ship Carpenters Wedge,
(Students card) l.
Semi-diams.of base a,b;
volume of wedge.
2. Relates to rates as illustrated by the hands of a clock.
5-35-4c0s, where s is distance in inches between the tips and is the angle (in radians) between the hands; between 8:45 and 9:15 9 is increasing uniformly at the rate of śre radians per hour.
(Students Card) 2.
How fast at 2 o'clock ?
Let sedistance in inches between the tips of the hands.
Let 0 = the anglelin radians) between the hands.
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