Chapter V involve only one discount, the only kind most people ever have to solve. "Finding the rate" and "finding the base" are taught in Chapter VI, with deserved emphasis upon the former. "Finding the interest' on a sum of money by the ordinary general method is the sole subject of instruction in Chapter VII, the six per cent method, a special process, and the problems of finding the rate, the time, or the principal being postponed until grade eight. II. The course must be enriched. By the time pupils reach this grade, a new appeal must be made to them to gain their interest, because they have become somewhat bored with the computational arithmetic. This cannot be done sufficiently by the use of problem material, local or otherwise, grouped about some central theme, as is so often attempted. On the other hand, the beginnings of literal arithmetic and inductive geometry will accomplish this purpose. In this text, the formula is introduced in connection with percentage (pp. 61, 80, 94) and the areas of the plane figures studied in Chapters VII-XIII. This use of the formula is part of a thoroughly considered plan which is developed in subsequent books; it should not be considered lightly, as the use of formulæ is now admitted to be one of the central topics for instruction in algebra. Particular attention, however, is directed to the geometry given in Chapters VIII-XIII. The instruction in this text is centered around the mensuration facts connected with the common plane figures and the rectangular parallelopiped, - these being the practical parts of geometry for most people. More than usual attention has been given to developing the geometrical concepts nvolved by drawing, measuring, and other informal exercises. Experience in teaching this material to children of this grade proves that they can master it in its present form and that they greatly enjoy doing so. III. The course must be thorough. Standardized tests disclose the need of more thorough instruction. In this text, a special effort has been made to instruct in a thorough manner. Observe the separation of the subject matter into small teaching units (e.g. §§ 24, 26-28; 29-30; 31). Observe that the instruction is given in simple language and quite fully (e.g. §§ 7, 34, 35, 36, 37, 43). Observe the large number of examples and problems provided. Separation into groups of oral and written examples has been avoided because pupils should form the habit of doing computations mentally whenever possible. Observe that the instruction is topical, but that review lists of miscellaneous character are given from time to time (e.g. pp. 28-29, 44-47, 63-65, 77-79). IV. The course must begin the process of rationalizing mathematics. In grades I-VI, processes and skills are emphasized, — "how" rather than "why" operations are performed in a certain manner. One of the major aims of secondary mathematics (grades VII-XII) is that of rationalizing mathematical processes. A strong argument may be made for the proposition that such rationalizing in the junior high school shall be chiefly inductive and, in the senior high school, chiefly deductive. In this text, a start is made with the inductive rationalizing of certain processes with which the pupils are already familiar. (§§ 7, 10, 12, 16, 17.) Chiefly and most effectively, however, this form of thinking is developed in connection with the geometry. (See Ex. 5, p. 126; Ex. 11, p. 128; Ex. 1, p. 136; Ex. 3, p. 137; Ex. 5, p. 139; § 65; § 66; Ex. 1, p. 187.) It is urged that these geometry lessons be "tried out" as printed; if necessary, supply more questions or exercises of the same sort, but as far as possible avoid omitting any of them or taking them in other order. The educational value of this geometry depends more upon how it is taught than upon the extent of the instruction. V. The course must prepare for later courses. The mathematics of grades seven and eight have an important rôle in the process of linking together the elementary school and the senior high school. The gap between the mathematics of these two schools has been notorious. This gap, moreover, will not be bridged by merely placing some of the secondary mathematics in these grades seven and eight; to do much of algebra or geometry there is merely jumping the gap and robbing the later courses. The articulation will be satisfactory only if the pupils acquire in the seventh and eighth grades certain correct ideas and mathematical habits and an interest in mathematics which will enable them to study formal algebra and geometry successfully. The articulating process may be started in grade seven with the study of the formula and of inductive geometry. In this text, the formula is taught in §§ 32 and 33, and with it the, for pupils, difficult idea of "substituting" in a formula. This, used repeatedly thereafter, is the only algebraic idea taught in this book. In the chapters on geometry, the definitions and point of view are consistent with the best practices in high school geometry. Besides the more obvious instances of preparation for high school geometry, attention is directed to the psychological preparation for the "superposition" process which is a result of the frequent use of tracing paper. (See Ex. 2, p. 115; Ex. 6, p. 123; Ex. 1, p. 128; Ex. 5, p. 139; Ex. 1, § 66, p. 156.) VI. The course must be flexible. All schools do not wish and cannot use the same material. Enough material has been provided for a full year of work. Certain sections, marked supplementary, are recommended for omission if it becomes necessary to save time somewhere. These sections will be treated adequately in subsequent courses in mathematics. (See §§ 67– 72, and Chapter XII.) Some or all of these may be omitted if more time is required for the rest of the book. Chapters I-VII constitute approximately one half year of work. If they are studied during the first semester, review examples from these chapters and from pages 201–210 should be studied during the second semester, while studying the geometry of Chapters VIII-XIII. In some schools, especially such as have semi-annual promotions, it may be desired to have some of the geometry in the first semester and some of the percentage and interest problems in the second semester. In this case, the following plan is suggested : First semester: Chapters I-V, VIII, IX, and reviews. Second semester: Chapters VI, X, VII, XI-XIII, and reviews. SUGGESTIONS TO THE TEACHER 1. Have all the pupils use the same kind of paper and a No. 2 lead pencil. 2. Obtain a pencil sharpener and attach it to the wall near the door by which the pupils enter the room. 3. Insist upon neat and orderly work from the pupils. 4. When doing the geometry work, the pupils need to be cautioned repeatedly about doing their work neatly and accurately. They cannot do satisfactory work without a sharp pencil and tools. A combination protractor, square, and rule has been furnished with the text. Since it is not needed until the work in geometry begins, it may be advisable to collect all the cards for safe keeping until then. The cards furnished are made of as durable cardboard as can be obtained. If the one furnished is lost or rendered unfit for use, a new one of the same or similar sort should be gotten by the pupil. Pupils of this grade are surprisingly inaccurate in measuring and inexpert in using tools. The only remedy is individual attention. A teacher can give such attention and direction while passing up and down the aisles, when giving directions during a drawing lesson or when supervising a "supervised study" lesson. NOTE. The combination protractor furnished with this text may be bought from D. C. Heath & Co. at the following rates: Less than twenty-five at 5 cents net each. |
