FIRST STEPS IN NUMBERS; DESIGNED TO LEAD THE PUPIL TO A THOROUGH PRACTICAL ACQUAINTANCE WITH THE ELEMENTARY OPERATIONS ON NUMBERS, AND THE APPLICATION OF THE DECIMAL SYSTEM. BY D. P. COLBURN AND G. A. WALTON, BOSTON: BENJAMIN B. MUSSEY & CO. 1851. GEORGE FR on rettrice Entered according to Act of Congress, in the year 1848, by D. P. COLBURN AND G. A. WALTON, In the Clerk's Office of the District Court of the District of Massachusetts. STEREOTYPED BY GEORGE A. CURTIS; NEW ENGLAND TYPE AND STEREOTYPE FOUNDRY. TO N. TILLINGHAST, ESQ., PRINCIPAL OF THE STATE NORMAL SCHOOL, BRIDGEWATER, MASS., THIS LITTLE WORK IS, WITH HIS PERMISSION, RESPECTFULLY DEDICATED, BY HIS FORMER PUPILS, THE AUTHORS. PREFACE. THIS work embodies a plan of instruction pursued by the authors in their schools; and, thinking that perhaps it may have some merit as an elementary Treatise, they present it to the public. INTRODUCTORY OBSERVATIONS. Ir may appear strange to those who have never thought upon the subject, but we believe that children usually pass over many pages of arithmetic, and often go "through the book," without ever having had any clear idea of the value and use of numbers. The irrelevant answers, or rather wild guesses at answers, which the teacher so often hears, can only be caused by an utter ignorance of the meaning of the words used, and the operations attempted to be performed. The mathematics are called "the exact sciences," and arithmetic is made the first step towards a knowledge of them; yet the pupil is commonly taught to test the exactness of an arithmetical operation, and the truth of its result, only by reference to the "Key." The tendency of such training must not only be to render arithmetic valueless as an exercise of the mind, but, worse than this, to destroy the scholar's confidence in the results of his own reasoning, and to prevent freedom and independence of thought. The difficulties which are the cause of all this uncertainty, may be removed in the first stages of the pupil's progress; but the teacher must remove them by his illustrations and explanations. The best and most perfect book can give the young pupil no idea of number, without the teacher's assistance. To the teacher, then, we suggest the following, as a method of teaching the elementary ideas of number. Having a class of pupils before you, show them a single thing, as a book, and say, "Here is one book;" then, exchanging this book for another, ask, "What have I now?" If none know, tell them, "One book." Show them, in this manner, a great variety of single things, continuing the exercise till they can apply the term one to each. Let them denote actions in the same manner; as, "" one tap," "one motion of the hand," &c. Next write the character 1, and, after explaining that its use is to stand for a single thing, teach the class to write and apply it. Assign them as a lesson, to print the names of as |