CHARACTERISTICS of the IDEAL NUMERICAL PROBLEM* ALEX. C. BURR Massachusetts Institute of Technology, Cambridge, Massachusetts
When used, the numerical problem should constitute an integral part of the structure of a course. Each problem should have definite functions to perform and should add its own increment to the educational process. Certain desirable characteristics oj the ideal problem are here explicitly stated and briefly presented.
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HE instructor in charge of a course in chemistry, be it general, analytical, organic, physical, or industrial, is frequently hard-pressed to find or prepare suitable problems for class work. A critical study of the problems available in the text used or in special problem books will usually reveal constitutional defects which greatly reduce their educational effectiveness. Further, by the very nature of the collection, such ready-made problems will seldom fit precisely into the structure of the course as designed by the instructor. The alternative is to prepare problems for each given situation. This requires a thorough appreciation of the characteristics of the ideal problem. It will be assumed that numerical problems are used in instructional work because they can accomplish the following results: 1. To illustrate, iix in mind, and make quantitative, principles and methods treated in lecture and text. 2. To test effectively and conveniently a student's knowledge of the subject and grasp of essential points. 3. To encourage logical thinking and methodical procedure on the part of the student. 4. To give the student practice in handling and criticizing experimental data and in extracting a maximum amount of information from a minimum amount of original data. The author does not wish to imply that all problem work, or that even most problem work, will lead to these results. He is of the opinion that a great deal of the problem work is a t best an additional burden on the student which is not justified by the results obtained in the time involved. In many cases the individual problems used have not been formulated with definite and unique purposes in mind, are not carefully stated, involve non-essential points, and are not closely and clearly connected with the lecture and text work of the course. The ideal problem possesses a number of character-
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*Based upon a paper read before the Division of Chemical Education at the 85th meeting of the A. C. S., Washingon, D. C., March 26-31, 1933.
istics which, taken together, distinguish i t from the common or garden variety of problem. The ideal problem should be definitely and closely related to the immediate subject matter under consideration in the classroom. Any lack of adjustment in this respect will prove a t least disconcerting, if not actually confusing, to the student. Also, with such maladjustment, the problem loses a great deal of its value as an illustration and does not as effectively fix in mind the major principles involved. Each problem should be formulated with specific and unique purposes in mind. The problem is an instrument of education and as such should have very definite aims. It is not in the nature of penance for taking the course but should deliberately attempt to accomplish one or more of the four results, previously mentioned, in immediate connection with contemporary class work. The problem should be so framed that the student must contribute both t h o u ~ h and t information in oroceeding to its solution. Problems entailing only substitutions in formulas, or recasting into set forms, and turning the crank may yield very interesting and even valuable information, but they constitute part of the drudgery of technical work and soon lose all educational value. The statement of the problem should be clear, concise, and accurate. Such forms of expression, desirable and valuable when adopted by the student, are best taught by example. The student should be able to tell a t a glance what it is desired that he do and what he has with which to work. A sure knowledge of what he is attempting will give him added confidence. Time spent in deciphering the statement of the problem and in exploring the mysteries of the instructor's mind might be better spent in formulating an attack and in perfecting a solution. To further avoid confusion, not more than one new major idea should be presented in any one problem. There are a limited number of new ideas to be presented within the limits of any one course. These can be distributed throughout the assignments so that each in turn receives its share of concentrated attention. Each new idea thus presented should be presented more than once. This is the principle of reiteration and should be rigorously observed. The 6rst presentation of a new idea frequently results in a poor performance on the part of the student. A second presentation will not only result in emphasis but will allow the student to redeem and perfect himself.
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In presenting data to the student, the number of significant figures used should be determined by the precision of the measurements involved. For the mas> part students are unfamiliar with the concept of significant figures and acquire familiarity by example and practice. I t should be emphasized that throughout the solution the student deal only with terms containing the proper number of significant figures. The results should be expressed in the proper number of significant figures; no less and certainly no more. The individual problems making up a set should be of approximately the same degree of difficulty. This does not mean that at the beginning of a course a student should necessarily be able to work the first and last problems of a set with equal facility. It does mean that toward the end of a course the last problem of a set should give the student the same concern and require the same relative effort as did the first problem. Further, a set of problems should present the new fundamental principle involved in various guises and in
different combinations. The major difficulty is not so much solving the problem, once the necessary principles are explicitly noted, but in recognizing and isolating these principles. Practice in doing this is most effective when accompanied by variation in presentation. Finally, a numerical problem is more than a mere exercise in computation. Its purposes are not primarily arithmetical. Each problem should be so designed as to reduce the mathematical work to a minimum. This is necessary not only as a means of saving time and effort, but that the student may concentrate on the technical points involved and not dissipate his effortson the detail of calculation. Problems designed with these points in view are few and far between. They are not found ready-made in books but are tailored to fit the precise situation. This requires added effort on the part of the instructor and limits the number of problems which can be used. The increased educational effectiveness of such problems will justify the extra labor involved.
