The description of the process of solving an arithmetical problem as "detecting and formulating the question concerning a functional relationship, plus answering this question," suggests that the more important functional relationships should be explicitly taught as principles. This will probably require the use of such questions as: "What calculations must be performed to find the area of a rectangle when the base and altitude (or two adjacent sides) are given?" "How does one find the rate of profit, given the cost of goods, and the expenses and losses?" "What calculation must be made to find the time required to travel a specified distance, given the distance and rate of travel?" What problems should be used as learning exercises. It is generally agreed that problems differ in their merits as learning exercises. Formerly authors of arithmetic texts included many problems that did not occur in the activities of adult life. Some of these were verbal puzzles designed to "sharpen the wit" of the pupils; others were derived from obsolete activities; and a third group, although identified with an adult activity, asked questions that would never arise because the answer would always be known in the practical situation. These kinds of problems appear to be less effective as learning exercises than real problems, that is, problems similar to those that arise in the normal activities of children and adults. "Similar" means not only that the problem involves the same functional relations but also that the quantities (prices, amounts, and so forth) be in substantial agreement with those of real life. Problems may be real and still differ in ways that affect their value as learning exercises. The analysis of ten texts described in the preceding chapter showed the variety of vocabulary used and the degree of complexity of many of the problems appearing in current texts. The desirability of an extensive technical vocabulary is largely a matter of objectives. If the objectives of arithmetic specify one thousand technical terms to be mastered by the end of the eighth grade, these terms should be used in the stating of the verbal problems used as learning exercises. If a small number of technical terms is considered satisfactory, there should be less variety in the language of the verbal problems. The objectives of arithmetic not yet determined in detail. The objectives of arithmetic were described at some length in Chapter I. Reference was made to investigations by Osburn and others (see page 9) which indicate that the number of combinations to be learned is much greater than commonly supposed. Certain functional relationships in problems were listed (see page 41) and the vocabulary used in stating problems partially analyzed. It was observed that "ability to solve problems" means ability to solve a problem that has not been solved before. Doubtless the reader of the preceding pages probably has wished for a more complete and definite exposition of the objectives of arithmetic. With the exception of the skills that function in calculating, the description of objectives was in very general terms, especially in the case of those relating to problem solving. Even in the case of the calculation skills used with integers, the objectives have not been completely determined, and in the case of fractions and decimals less is known concerning what a pupil should learn. Hence, in attempting to determine the number and kind of learning exercises needed in the field of arithmetic, the assistance to be derived from formulations of objectives is limited. However, the exposition of objectives in Chapter I, supplemented by the analysis of examples and problems presented in Chapter IV, should lead the reader to formulate a concept of the objectives of arithmetic which will be very helpful in determining the number and kind of learning exercises needed. An estimate of the teacher's responsibility for devising and selecting learning exercises in arithmetic. The kind and number of learning exercises needed in arithmetic depend upon several factors including the objectives, pupil's capacity to learn arithmetic, and his previous school experience. The preceding discussion should make it clear that our present state of ignorance about the various factors makes it impossible to do more than estimate the learning exercises needed. The estimate given here is in general terms and represents only the writer's judgment. Furthermore, the reader should bear in mind that the details. of the teacher's responsibility for devising learning exercises will be affected by the text adopted for use. Usually it will be necessary for the teacher of arithmetic to devise some exercises to provide perceptual experiences. In the primary grades opportunities for counting, measuring, and estimating should be provided. In the intermediate and upper grades the teacher may require pupils to observe adult activities that produce arithmetical problems and to give a description of these experiences but frequently it will be advisable to ask the pupils to listen to, or read, accounts of the experiences of others. An important phase of the devising of exer The norms established for standardized tests constitute definite objectives but they are available for relatively few types of examples (see page 35). For a summary of types of examples for which norms are available see: HERRIOTT, M. E. "How to make a course of study in arithmetic." University of Illinois Bulletin. Vol. 23, No. 6, Bureau of Educational Research Circular No. 37. Urbana: University of Illinois, 1925. 50 p. cises of the latter type is the preparation or selection of the descriptions that the pupils are asked to listen to or read. The authors of some texts suggest construction exercises, games, and the dramatization of certain adult activities, but it is usually necessary for the teacher to plan at least the details of these types of exercises. In some cases the textbook will give little or no assistance. However, it may be noted that construction exercises, games, and the dramatization of adult activities may not be highly efficient learning exercises, especially after the primary grades are passed. Arithmetic texts provide for practice in reading and copying numbers, supplying the missing number from specific quantitative relationships, and calculating, but analyses of the example content of texts show that there are "gaps" in the practice on the combinations, both basic and secondary. The teacher is responsible for discovering the "gaps" in the adopted text and for devising the exercises necessary to round out the practice. Usually the practice to be provided is on the more difficult basic combinations and certain of the secondary combinations. Several sets of practice exercises have been devised to relieve the teacher of a portion of this responsibility but investigation has revealed that some of the sets are not satisfactory. In some cases it will be necessary for the teacher to provide additional exercises in reading and especially in copying numbers. The "gaps" in the verbal problems provided by a text vary but the analysis of the problem content reported in the preceding chapter indicates that the teacher will need to provide some additional problems. However, it appears likely that the number needed will not be large and pupils may be requested to collect some of the needed problems. On the other hand, a number of omissions usually will be justified. There is no justification for the very high frequency of certain problem types (see page 51) and some of the problem types listed in the Appendix are not sufficiently valuable to be included in our objectives. The teacher will need to supply exercises that will lead to the mastery of functional relationships (see page 19), rules, definitions, and the like. Among these are requests to explain, explanations to be listened to, and thought questions. Other types of learning exercises for which the teacher must assume at least some responsibility are requests to check calculations, to inspect and verify solutions of problems, to collect quantitative information, and to generalize experience. Another very important responsibility of the teacher of arithmetic is to provide exercises that will lead to the mastery of the abstract and general terms in this field, especially those that are used in stating verbal problems. APPENDIX A Types of Functional Relations Found in the Problems of Ten Series of Arithmetics. The first line of numbers gives the frequencies of the simple occurrences of the relation, the second line the frequencies of the total occurrences of the relation. For a description of the method of analysis see page 48. A. OPERATION PROBLEMS A1 To find totals by addition, given two or more items, values, etc. A82 B112 C62 .D66 E132 F43 G49 H82 146 J75 749 A2 To find the difference, given two items, values, etc. A96 B134 C60 D68 E123 F30 G72 H168 131 J50 837 A395 B648 C370 D445 E429 F186 G345 H532 1314 J410 4074 A3 To find the amount, or number needed, by multiplication, given a magnitude and the number of times it is to be taken. A53 B96 C20 D36 E75 F47 G107 H161 145 J38 678 A4 To find the size of a part of a magnitude, given the magnitude and the number of parts into which it is to be divided. (Averages which are the result of division only are included here.) A28 B36 C39 D34 E30 F44 G86 H53 122 J15 387 G164 H104 156 J47 859 A5 To find how many times a stated quantity is contained in a given magnitude, given the quantity and the magnitude. A16 B59 C11 D15 E26 F8 G37 H82 129 J26 A6 To find how many when reduction ascending is required, given 309 184 J144 1101 104 1556 45 322 A6 B9 C1 D2 E20 F6 G27 H10 I16 J7 A163 B125 C118 D220 E115 F151 G191 H125 1167 J181 b. a magnitude expressed in terms of two or more denominations. All B3 C1 D5 F4 G2 H5 113 J1 A37 B5 C46 D49 E13 F78 G32 H16 145 J1 A7 To find how many when reduction descending is required, given a. a magnitude expressed in terms of a single denomination. A6 B20 C2 D17 E16 A97 B152 C72 D148 E90 F63 G116 H101 1145 J164 1148 b. a magnitude expressed in terms of two or more denominations. G14 H5 127 J13 A6 B4 C2 D3 E8 F5 G6 H4 110 J2 J13 110 310 50 217 D2 E14 C2 D8 F6 G5 H3 J7 14 J15 E1 G3 D1 E2 F2 G6 114 J4 32 24 9 f. the cubic contents of a parallelopiped and two dimensions. F2 E2 H1 17 J5 17 A4 B1 C5 D5 E2 F5 G13 H4 110 J9 58 78 D1 j. the cubic contents of cylinder, and the diameter. k. the cubic contents and area of the base of a prism or cylinder. A9 To find a diameter equal to two or more smaller diameters, given the smaller diameters. 5 A10 To find the area, given a. dimensions of a square, rectangle or parallelogram. HI 1 2 149 A11 B10 C4 D15 E28 F3 G8 H47 19 J14 b. the base and altitude of a triangle. (Includes right triangle with two legs given.) e. the perimeter of a cone or pyramid, and its slant height. f. the altitude and two bases of a trapezoid. C1 D2 E4 F1 G12 g. the altitude of a cylinder and radius of the base. F1 13 H2 16 31 20 G1 15 23 12 10 2 |