In conclusion, we find that the general notion is a pivotal centre of discussion not only in elementary and higher studies of all sorts, but also in the great fields of psychology and philosophy. It is not claimed that the method by which general notions have been worked out in our text-books is uniformly correct and valid. This is a question that we are not called upon to settle at this point. Whether or not an inductive or deductive approach to general truths is the correct one, we can leave for further consideration. But one leading aim of instruction in every important study is a mastery, in the full sense, of its general truths. Without this basis no method of instruction has any validity. It may be that the method by which this aim can be best realized has been so thoroughly misinterpreted and misapplied that we have approached a uniformity of error in our methods of teaching. It may be that definitions and abstract formulæ have been set too much in the forefront of every lesson, and also that systematically formulated knowledge has been forced prematurely into lower grades. Yet it is a great step in the right direction to have fixed clearly the aim of instruction, to have determined the goal toward which all proper mental movement tends. Assuming that our conclusions thus far are justified, we may move on to a discussion of the essential steps in a correct method of instruction. CHAPTER II ILLUSTRATIVE LESSONS SHOWING THE PROCESSES OF REACHING GENERAL TRUTHS mount. IN the first chapter, having located the goal of instruction in general notions and in their proper use, the question, how to reach them, now becomes paraIn the present chapter a number of lessons is worked out to illustrate the different processes in vogue for mastering general truths. In each example two different methods are presented: first, that common to many of our text-books and to the usual practice of teachers; and second, the fuller inductive and developing method now followed in some schools. The purpose of this chapter is not only to show the two ways of reaching a comprehension of such truths, but also to suggest other important phases of recitation work. In the discussions of recitation method which follow, these lessons may be kept in mind as illustrating the principles under treatment. The lessons are taken from different studies,arithmetic, geography, literature, natural science, and history. They recognize generalizations as the goal of instruction, but leave open the question as to whether or not any further principles of method may be found in the treatment of these various materials. The Addition of Fractions In first learning to add fractions, one method of the arithmetics is fairly illustrated by the following: What is the sum of 2, 18, and 1? 2 Process:++ } } = 2 + 18 + 10 = }} = 1}}. Process: Since unlike fractional units cannot be added, reduce the fractions §, 12, and , to a common denominator and then add the resulting fractions. After ten or a dozen problems the following rule is given : "To add fractions, reduce the fractions to a common denominator, add the numerators of the new fractions, and under the sum write the common denominator." The following more detailed process is suggested for consideration : How shall we add fractions whose denominators are unlike? What fractions have you already learned to add? Try these, and . and . and . Can you add and? What change is necessary before adding them? Why not add them in the same way as the others? How can you add one bushel and one peck? Change the bushel to pecks. Add two yards and one foot. What change was necessary in both examples? Add and . }=}. 1+1=}. Illustrate this with a square divided into fourths and eighths. Add and. Add and . What was done in all these cases before adding? How shall we add and ? How can you change the two fractions so that they will be alike, that is, have the same fractional unit? Change them to twelfths. One-third equals how many twelfths? One-fourth equals how many twelfths? +=+= Illustrate this with a sheet of paper folded into thirds, fourths, and twelfths. Add and . What is the common fractional unit? 12 Notice, now, what was done in each of these problems: ++, and +. The fractions in the first were changed to twelfths, in the second to tenths, and in the third to twentieths. Was the value of the fractions changed? But in each example the fractions were changed to a common fractional unit, or a common denominator. What was done to the numerators? In each fraction they were changed to correspond with the change in the denominator. Then, in adding, the numerators were added. Make a rule for adding fractions that will cover all the cases so far worked: "To add these fractions, change the fractions to equivalent fractions having a common denominator. Add the numerators for a new numerator and use the common denominator for the new denominator." To acquire skill and accuracy in this kind of addition: 1. Add oral problems as follows:— {+}. J+1. 12+ §. 1+12. 2. For written work add such as these: Trade Centre in the Northwest-Minneapolis as a Type This topic may be treated in two ways, briefly, as in the geographies, or in a fuller inductive manner. One of our grammar school geographies says: "Minneapolis, which adjoins St. Paul, so that the two are called the 'Twin Cities,' manufactures more flour than any other city in the world, its capacity being 40,000 barrels a day. The two cities have had a remarkably rapid growth." Tilden's "Commercial Geography," which is very much fuller on this topic than the regular geographies, says: Minneapolis, on the Mississippi at the Falls of St. Anthony, is the greatest flour-milling city in the |
