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in doubt for a long time as to the meaning of a great many things and the uses that he is to make of them when the two parts are taken up separately.
PROFESSOR CLARENCE A. WALDO: We have had considerable experience along these lines and have settled down at present to this method of procedure: During the freshman year, in the subject of college algebra, we bring out as clearly as we can the meaning of the differential calculus so far as the infinitesimal rating of the triangle is concerned as spoken of by Professor Cajori. That gives a basis for the analytical geometry, so we are not compelled to use the secant method there but can use the calculus method at once in our analytical geometry. So, therefore, we keep the calculus before our students, so far as its simple ideas are concerned, right from the beginning, until we take up calculus itself as a formal subject. We do that the second semester of the sophomore year, and a thing we have in mind when we take that up is that our calculus is to be used in the first semester of the junior year, especially for theoretical mechanics. So in that second semester we go over completely a first course in calculus. In starting in upon that it is my custom, and I think it is the custom of my colleagues largely, to begin at once with a sort of lecture upon the scope of the calculus showing its method of operation in differentiation and what it will lead to in integration, etc., giving them right at the beginning some concrete examples which enable them to understand what it is and what the thing is good for to begin with. In that way we are not compelled to mix up the differential and integral calculus too much. I think if that is done constantly there is a feeling in the student's mind of a lack of system and completeness of the work. If the man sees clearly first what he can do with his differentiation and his attention is being called all the time to the fact that there is the reverse of that to be taken care of, then this first course is taken rapidly with many of the difficult parts omitted in order that the essentials of the subject shall be completed. I think the student in that way gets a firm grasp of the fundamentals of the calculus so he can use them easier.
Then, we take up the second course, going over the whole subject again, starting in the junior year, taking it all through the junior year, introducing lecture and problem work and running the higher work along three nearly parallel lines, having in mind the application of analysis to three different classes of questions, those that arise in civil, mechanical and electrical engineering. In that way we hope to give our boys before they get through with it some real knowledge and experience of the value of the calculus in their work.
PROFESSOR EDGAR MARBURG: Recently the speaker had occasion to examine the curricula of a number of engineering schools, especially with a view to the time devoted to pure mathematics. That inquiry seemed to show a growing tendency to complete the teaching of mathematics at as early a period in the course as possible; doubtless in order to make more time available for the more advanced technical subjects in which a knowledge of calculus is indispensable. At the University of Pennsylvania the engineering students now complete calculus by the end of the sophomore year.
The speaker thinks it very important that the teaching of mathematics, especially calculus, to engineering students should be in the hands of engineers, or men who have had at least an engineering training, so that wherever possible the problem work may be given a proper trend in the direction of engineering. This would serve not only to stimulate the interest of the students, but would give them valuable training along right lines. The teaching of mathematics to prospective engineers should not be conducted from the purescience standpoint. It is encouraging to note that a number of text-books in mathematics have been written within recent years especially with an eye to the needs of engineering students. There would seem to be ample room for more and better books of that nature.
SYMPOSIUM: METHODS OF HANDLING PROBLEM
WORK IN LARGE CLASSES.
BY EDWARD R. MAURER,
I have assumed that the object of this symposium is preëminently pedagogical—to secure an exchange of information on our practices in a certain line of teaching work. Instead of describing different schemes that I use in the conduct of problem work, I shall limit myself to one, explaining it in detail. It may be called a card system for assigning problems in written quizzes.
I have used such a system only during the past half year. By the end of the year, the card collection will include at least one problem on each important subject in the ordinary engineering course in mechanics, thus one each on composition of forces, center of gravity, bending moment and shear, stresses in a simple truss, moment of inertia, friction, etc. These problems are numbered consecutively in the order in which they are gotten out, and not with reference to their occurrence in the course. In the next year, a second set of problems will doubtless be gotten out, so that no problem shall be used in consecutive years. The numbering of this second series will begin where that of the first ends.
Each problem is prepared in four forms, differing in data or in requirement but all involving the same principles. The forms are distinguished by the letters a, b, c and d, thus la, 1b, 1c, 1d, 2a, 2b, 2c, 2d, etc. All a forms are printed on white cards, b on blue, c on yellow and d on buff, colors being used to facilitate identification of the form of problem on any given card.
The problems are printed on 3 x 5 cards for the use of students and on 10 X 12 sheets for the use of instructors. Each card bears one form of a problem, while each sheet bears the four forms of one problem and their solutions.
To explain the use of the problem cards and the solution sheets, let me suppose that the class has been studying a subject for some time, and is nearly ready for a written quiz on it. The instructors prepare for the quiz together as follows: First each form of the quiz problem is worked out by at least two instructors, the time required is noted, and the solutions are checked. The average time multiplied by a suitable factor of safety is fixed as the time allowance for the quiz. Then each instructor copies on a solution sheet as much of the solutions just made as will be useful to him in correcting the quiz papers. (I am now assuming that the problem is being used for the first time. For the second time, the solution sheets are all ready, and the proper time allowance for the quiz is known from the first use.) Next the problem cards are arranged in a certain way. Thus it being problem 7, say, Ta and 7b are placed alternating into one pile, likewise 7c and 7d. Then each instructor takes his solution sheet and enough of the cards (a, b, c and d) to go around his section of the class.
On the day appointed for the quiz, the instructor