Pupil takes 1⁄2 of a half of something and answers "," writing 1⁄2 of 1⁄2 = 4. Repeat this type of question and answer with several combinations, as: = In this way the child will discover that the numerators are multiplied together for a new numerator and the denominators for a new denominator. All reductions of answers to simpler forms, as 4 of 2 = 2/12 = 1/6 and all short cuts as 4 of 3 1/6. Thus, dividing mentally before multiplying should follow, and should be discovered by the child, guided by the skilful questioning of the teacher. After the principle of multiplication of fractions has been discovered by the pupil, much practical work and drill should follow immediately. Following are some suggestions for practical work. 1. Taking 1⁄2 of a recipe. (As: Efficiency Arith., Intermediate, P. 76.) CHOCOLATE NUT FUDGE. 11⁄2 cups granulated sugar butter, size of walnut 1⁄2 cup chopped nut meats vanilla For drill, follow with other recipes and take 34, or 11⁄2, or 2 times the recipes. Also give as abstract drill all the fractions involved in the problems, and other similar fractions, as 1⁄2 of 1⁄2, 1⁄2 of 34, 2X34, 2X12, 2X21⁄4, 1⁄2 of 24, etc. 2. Playing store and making bills, as: The multiplication of a mixed number by a mixed number does not involve anything new. The term "cancelling" should not be used in the reduction of fractions, the multiplication of fractions, or the division of fractions. It is wholly unnecessary and often confusing to the child. Use the expression "Divide both numerator and denominator by the same number." A splendid form of checking the result in multiplication of fractions is to begin with zero and count by the fraction. Example Check 7X34 = 21/4 = 54 0,34,6/4, 9/4, 12/4, 15/4, 18/4, 21/4, 5 or 34, 12/4, 24, 3, 33, 4 2/4, 5. ILLINOIS STATE ACADEMY OF SCIENCE. APRIL 27-29, 1922. Section Chairmen for the Rockford meeting: 1. Section of Biology and Agriculture-Geo. W. Hunter, Knox College, Galesburg. 2. Section of Chemistry and Physics-R. C. Hartsough, Illinois Wesleyan University, Bloomington. 3. Section of Geology and Geography-W. S. Bayley, University of Illinois, Urbana. 4. Section of Mathematics and Astronomy-C. E. Comstock, Bradley Polytechnic Institute, Peoria (Chairman), Program Com. of Ill. Sec. of Math. Assoc. of America; Sec. E. B. Lytle, University of Illinois, Urbana. 5. Section of Medicine and Public Health-W. G. Bain, M. D., Springfield. 6. Section of Psychology and Education-E. S. Ames, University of Chicago. Academy Committees for 1922: Program Committee for the Rockford Meeting: H. C. Cowles, University of Chicago; C. Frank Phipps, State Teachers' College, De Kalb; Chas. T. Kinpp, Urbana (Chairman). Rockford Local Committee on Arrangements: Ruth Marshall, Prof. of Zoology, Rockford College (Chairman); R. D. Mullinex, Prof. of Chemistry, Rockford College; Agnes Brown, Instructor in Botany. Rockford High School; E. E. Lewis, Supt. of Schools, Rockford; J. O. Marberry, Prin. of the Rockford High School; Dr. Edward Weld, the Rockford Clinic; Seth Atwood, Sec. of the Rotary Club; Mrs. Maud Cormack, Pres. of the Rockford Nature Study Society. Membership Committee: C. F. Hottes, University of Illinois, Urbana (Chairman); W. H. Haas, Northwestern University, Evanston; W. H. Packard, Bradley Polytechnic Institute, Peoria; Stuart Weller, University of Chicago, Chicago. Committee on High School Science and Clubs: J. S. Hessler, Knox College, Galesburg (Chairman); Frank H. Coyler, Carbondale; C. M. Turton, 2059 E. 72nd Street, Chicago; Harriett Strong, 72 Loomis Street, Naperville. Publication Committee: The President; The Secretary; Geo. D. Fuller, University of Chicago, Chicago. THE FREQUENCY OF CERTAIN PROBLEM SOLVING SITUATIONS IN THE HIGH SCHOOL CURRICULUM AND A SUGGESTED GENERAL METHOD OF SOLUTION.1 BY FRANK C. TOUTON, Department of Education, University of California, Berkeley, Calif. PROBLEM. Teachers generally agree that many secondary school pupils experience much difficulty and little success in their attempts to solve independently those verbally stated problems which require in their solutions both reasoning and operations with the fundamental processes. That such problems do occur with great frequency in the mathematical and natural science subjects of the high school curriculum is a matter of common knowledge. It has been called to my attention that the mathematical abilities of an entire family are sometimes exercised and even taxed in an attempt to do for the pupil that which the teacher had evidently intended the pupil to do for himself. In problem solving as in the formal work of mathematics, the two real tests of good teaching are successful achievement and evidence of increasingly independent activity on the part of pupils. PURPOSE. It is the purpose of the writer at this time: 1st. To make a brief report on the place taken by problem solving situations of the quantitative sort in the secondary school curriculum. 2nd. To consider the need for a special technique in training pupils to solve those problems which involve both reasoning and operations with fundamentals. 3d. To suggest a method of solution, the several steps of which correspond to a psychological analysis of the thought processes utilized in solving problems of a quantitative type. TEXTBOOK ANALYSIS. Having accepted the thesis that "It is the duty of the school to teach the pupil to do better those desirable activities which he is to do anyway," it occurred to me that it would be both interesting and perhaps profitable to know the extent to which pupils encounter problem solving situations of the type here considered while engaged in the study of certain subjects in the Paper read before the Mathematics Section of the California State Teachers Association at San Francisco, October, 1921. secondary school curriculum. In the securing of data on this point I have had the assistance of seniors and graduate students enrolled in my classes in Education at the University of California. The study made, in so far as it is here reported, shows for each text examined (a) the number of problems completely solved for the guidance of the pupil (b) the number of hints or partial solutions given with the evident purpose of teaching pupils to solve, with a minimum of help, such problems as require in their solutions both reasoning and computation of an arithmetic or algebraic sort and (c) the number of problems to be solved by the pupil. The following table shows for each text examined, (a) the number of problems solved in full in the text, (b) then number solved in part in the text, and (c) the number to be solved by the pupil. These items, only, are selected out from those listed in the more extended analysis made by those students who examined the several texts. Text. (b) Text prob- Subject. First Year Algebra. A B C D E F G H I J K 33 30 105 13 17 29 15 40 17 32 10 17 33 60 17 16 10 28 17 14 0 38 540 550 439 359 482 262 310 354 441 225 520 Subject. Second Year Algebra. (b) Text problems solved in part......... 22 |