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time-saver for it economizes much time for instructors inasmuch as they are furnished quiz problems already for use and solutions, too, once the collection is stocked, and it economizes time for the student in the class-room, for the quizzes can be started very quickly. Then, too, his work is facilitated by his having the problem immediately before him and better stated than a blackboard problem is apt to be. Last, and very important, the system furnishes the occasion for frequent meetings of the instructors of the department for profitable and fruitful discussions of their own teaching and the work of their students as disclosed by the examination of the quiz papers.

SYMPOSIUM: METHODS OF HANDLING PROBLEM

WORK IN LARGE CLASSES.

SECOND PAPER.

BY CLARENCE A. WALDO,
Professor of Mathematics, Purdue University.

It is the purpose of this paper to speak briefly of problem work in large mathematical classes and to exhibit samples of a collection of problems.

To stimulate the student to do his severest work, to set him thinking upon some combination of affairs which taxes skill and strength to their utmost, but which at the same time is fairly within his powers, this is the instructor's problem. At the same time that something should be a link in a chain connecting a man in a rational way with the purposes of his education. It is a well-recognized principle in sound class-room work that at every class-room hour there should be complete mental contact between student and teacher, a contact which reveals the true mental status of the pupil, at least in its relation to the lesson material.

The accomplishment of such contact places certain limitations upon the organization of class-room work and one of these is found in the number of pupils in one class. Too few students in a class tend to make the teacher lazy; too many make him superficial. Unless endued with extraordinary ability the teacher will find twenty-five near the maximum number of students that can be properly handled in any one class—this, at least, seems to be the conclusion of good teachers-and such a class is large.

In the hour and a half or two hours spent at the home in preparation, what work has been done and how? This the teacher must know if he is to deal with his students as individuals. But how? Perhaps the most common method of the diligent teacher is to insist upon the student's handing in on paper a certain number of selected and carefully solved representative problems. It is certain that this method frequently leads to excellent results. But if the purpose of the written work is to insure originality it is a failure. If the student tries to shirk and deceive it is an easy matter for him to copy from others. Knowing this to be true, is it necessary, is it even wise, for an instructor to burden himself with such a vast amount of critical reading as this method when well done of necessity implies? Is it not often and largely a waste of energy-a useless and unappreciated piece of drudgery?

A second method in frequent use for ascertaining the amount and kind of preparation is to pick problems at random from work assigned for the student to do at the board. In small classes of five to ten, good results may be reached in this way, but in classes from twenty to thirty the lazy, indifferent or bashful student may for weeks escape wholesome criticism and admonition.

In our technical schools which are increasing so rapidly in numbers, the provision for instruction does not keep pace with growth. Congested classes are the result, thorough individual examination impossible. Why not put students at once on their honor! At the opening of every lesson period ask every student directly concerning work done and record his answer or ask him to report on a blank furnished for that purpose. After such inquiry you are justified in assuming that the students have told you the truth and have a right to modify the work planned for the hour according to reports received. Students will sometimes make false claims, but the alert teacher will soon discover that fact and prevaricators may be easily made to see that honesty is the best policy. But the instructor with a flexible program will know how to divide his time in explanation, in development and in criticism. Certain members of the class are kept at their seats and given such council and assistance as they need singly or together while the better students are honored and stimulated with fascinating original work.

One of the easy, obvious ways of conducting a mathematical recitation is to send students to the board for the hour to work problems proposed in the text, and it is a way too often used to kill time. Three good reasons may be given for giving problems at the board from the exercises in the day's assignment and it would be difficult to enlarge the number:

1. If the student has orally claimed at the beginning of the hour that he has actually solved certain exercises, he should occasionally be called upon to verify his report.

2. The instructor should make sure that a student is grasping and using the simplest and best method of solution. For this purpose the whole class can be profitably engaged at times upon the same problem or upon two problems so that those standing next to each other will not be doing just the same work.

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

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