I THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT PRIMARY METHODS V A COMPLETE AND METHODICAL PRESENTATION OF THE USE OF KINDERGARTEN MATERIAL IN THE WORK OF THE PRIMARY SCHOOL UNFOLDING A Systematic Course of Manual Training in Connection with Arithmetic, Geometry, Drawing, and other School-Studies BY W. N. HAILMANN, A.M. SUPERINTENDENT OF PUBLIC SCHOOLS, LA PORTE, INDIANA COPYRIGHT, 1887 A. S. BARNES & COMPANY NEW YORK AND CHICAGO Education Library L-B t /&< TO THE TRUSTEES OF THB LA PORTE PUBLIC SCHOOLS AND TO THE GOOD PEOPLE OF LA PORTE, TO WHOSE SYMPATHETIC AID THE AUTHOR OWES SO MUCH, THIS VOLUME IS GRATEFULLY INSCRIBED. 086094 J^ffl&^i&C^Uh^^^ The growing- demand among primary teachers for "busy work," "kindergarten methods," and other means of manual occu- pation, has led to the preparation of this little A r olume. Its suggestions have grown in the school- room, and are the results of careful expe- rience and of a thoughtful study of the children's needs, as well as of the drift and value of manual work as an educational factor. In this work I have "been much aided "by the teachers of the Primary Schools of La Porte, who, with rare intelligence, singleness of purpose, and professional devo- tion; have enabled me to work out the "bearings and possibilities of the work in the various branches of school instruction involved. Whatever credit, therefore, may vi PREFACE. come to tlie book, is largely due to their ready zeal. It is hoped tliat this book will not only supply teachers with the needed means and directions for the methodical and system- atic, the economical and efficient use of the occupations described, but "will also successfully guard them against the evils of random "busy work." W. N. HAILMANN. LA PORTE, IND., July, 1887. CHAPTER I. THINGS BEFORE WORDS IN EXPRESSION PAGE 1 II. THE COURSE OF STUDY 7 III. GIFTS AND OCCUPATIONS . .... 13 IV. THE WORK SELECTED OUTFIT ..... 19 V. SECOND-GIFT BEADS .23 VI. SECOND-GIFT BEADS 33 VII. A NUMERAL FRAME FOR EACH PUPIL ... 45 VIII. COUNTING BLOCKS 53 IX. COUNTING BLOCKS . ....... 65 X. FOLDING SHEET 73 XL FOLDING SHEET . . . 83 XIL PLASTIC CLAY 93 XIII. CUTTING AND MOUNTING 101 XrV. CARD-BOARD 'WORK 113 XV. FRACTION STRIPS 121 XVL SPLINTS OR STICKS 129 XVII. STICKS AND PEAS . . . . . . . .139 XVIII. LENTILS OR DOTS 147 XIX. THE GROUP TABLE . 157 PRIMARY METHODS. CHAPTER I. THOUGHTS BEFORE WORDS IN EXPRESSION. tHE life of man, like all individual life, consists of a series of actions and reac- tions which constitute the experience of the man. They leave a residue in his conscious- ness and in his manner of "being 1 . The former is the furniture of his insight ; the latter, of his conduct. By the former the man is wise ; "by the latter he is virtuous. Thus experience furnishes the material for the wisdom and virtue which hold all worthy objects of education. The tenets of Pestalozzian education con- cede this. " Thing's before words "; "things before ideas " " first the thing-, then its sym- bol " ; " the intellect rests on sense-percep- tions " are utterances of this truth. It has been applied, however, almost exclusively to the in-leading processes in the growth 2 PRIMARY METHODS. of intellect. In the out-leading- processes ^ we have been satisfied -with symbols, with, words ; we have ignored the value of things. Although we had become aware of the in- sufficiency of words in the formation of fundamental ideas, yet their inadequacy in fundamental expression has been overlooked. The organs of speech which express ideas in "words, were to the school the only road for expression of thought, worthy of its notice. The hand which expresses ideas in things has been neglected. Similarly, it has been conceded that in the growth of insight, in the formation of accurate ideas, expression is needed as much as impression, that the intellect owes much to the reflex influence "which comes from efforts to formulate knowledge in words. But it has been overlooked that the more or less plastic expression of ideas by the hands, with the help of suitable material, holds to their formulation in words very much the same relation that things hold to symbols ; that in expression, too, it is necessary to bring things before symbols; things before "words. Thus, in studying the cube, with refer- THOUGHTS BEFORE WORDS. 3 ence to its sliape, the cliild will probably at first see the cube, handle it, use it in his games, and thus gain many impressions concerning its shape. These may be ex- pressed in "words or, plastically, in clay. Both modes of expression will react favor- ably upon the child's idea of the shape; yet there is no doubt that the efforts at plastic representation will be the more ef- fectual mode to clear his idea of inaccura- cies and insufficiencies. At every step the child has opportunities to compare his rep- resentation of his idea with the idea and with the original, to correct faults and to supply omissions. Again, it may be possible to give the child a fair idea of a square by showing him square pieces of paper, square figures, square objects, and by accompanying this with judicious instruction and skillful ques- tioning on the details of the form ; yet all this is but the glare of gas-light compared with the noonday light that is poured into the pupil's mind by the actual hand- ling of squares, by using them in the con- struction of other forms, by drawing squares singly or in symmetrical groups, by cut- 4 PBIMARY METHODS. ting such forms from paper, pasteboard, or wood, "by fashioning them from clay or wax. It will "be readily seen that this hand- training has wider aims than industrial training in the various arts. Industrial training supplies some particular or tran- sient need of self-preservation ; -whereas the aims of the hand-training here proposed lie in general and permanent self-expansion. Hand-training in this sense is as much a need of the professional and literary man, of the merchant and clerk, of the capitalist and land-owner, as it is of the artist and artisan, of the laborer and farmer ; as much a need of woman as it is of man ; its need rests on the immanent being of man, not on transient industrial circumstances. Industrial work selects its material pri- marily in accordance with the use to -which its products are to be put, "whereas the hand-training here proposed looks prima- rily, in the selection of materials, to the capacities and needs of the little workers. Here it is of the first importance that the material should yield readily to the work- er's limited skill. It should be of such a THOUGHTS BEFORE WORDS. S shape and character that it -will adapt itself without worry to the worker's aim, so that he may reach automatism in manual ex- pression as readily as he does in speech ; it should, therefore, he so prepared that the arranging- and transforming activities of the hand may receive ready answers, as is done to a large extent in the materials suggested by Froebel. The business of life is adaptation to sur- roundings, to nature, to the universe. This implies knowledge and control of self and surroundings. The business of education is to lead the young human, being on the surest and shortest road to this adaptation. Education should see to it that the income of the senses be properly interpreted and appreciated by the mind, and that the tongue and the hands properly represent the mind and execute its behests. The mind should learn to rely implicitly upon its powers to see, say, and do. In seeing, saying, and doing, it should acquire the habit of success, a calm sense of power, a firm conviction of mastership. This is pos- sible only, if head and hand are trained simultaneously and in unison with each 6 PRIMARY METHODS. other ; and for this purpose the hand- training 1 here proposed is needed in the school. The advocates of industrial training are met with the objection that the school is already loaded down with work, and that it will be ruinous to the child to add fresh burdens. The hand-training 1 here proposed is not open to this objection ; it removes 'bur- dens. It enables the child to gain the knowledge which the current subjects of school instruction represent, in a manner more suited to his tastes and powers ; in a complete, all-sided, active, ideal child-life in which he is upheld and strengthened by the constant joy of success, the steady glow of growing power. The chief object of this book is to show how this may be accomplished, to lay be- fore the teachers the possibilities of avail- able material, to indicate its many-sided applicability to primary school work, and to suggest in a number of model lessons modes of directing their life-giving sun- shine into the school-room. CHAPTER II. THE COURSE OF STUDY. (5^, tN tlie work indicated I shall have be- fore me a quasi-ideal average Course of Study, the essentials of which appear in the following 1 considerations : " In framing a course of study for Pri- mary and Grammar Schools, it should he constantly borne in mind that the period involved corresponds chiefly to the earlier portion of the psychological period of con- ception. When the child enters school he is still gathering perceptions, though upon some things he has quite clear and compre- hensive conceptions ; and when he leaves the Grammar School, his intellect should have grown into a fair supremacy, and the dawn of insight into the deeper relations of "being should he full upon his mind. During the first years of school life, the subjects of study should be of a character to facilitate the formation of perceptions 8 PEIMARY METHODS. and their transition into comprehensive conceptions ; they should lie on the side of the concrete, the actual, the outer ; they should deal with facts, with space, with objects. They should, then, gradually merge into forms that lie on the side of the ab- stract, the possible, the inner; that deal with principles, with laws, with time, steadily leading the child out of the com- plexity of things into the simplicity of thought. * * * The essentials of the outer world that interest man most nearly in his efforts to obtain intellectual control of his surroundings are centered in space "which in its limits involves form, position, size, direction, and number. Of these, number and size have a special interest, inasmuch as they consti- tute the chief bridges in the transition of the mind from outer space to inner time. To these may be added color as an impor- tant element, depending on certain relations of material surfaces to light. Lying nearer the emotional side of sensation, it has much power to arouse interest in related elements of space, hence its educational value is very great. The school will, then, find the first subjects for instruction with reference to THE COURSE OP STUDY. 9 the pupil's individual development, in the provinces of Geometry, Drawing, Coloring, and Arithmetic. " Almost simultaneously, however, the phenomena of motion and life to which the changes of position, direction, size, form, and number among- surrounding objects are referred, and which intensely affect the child's comfort and welfare, point to studies connected with the provinces of Physics, Chemistry, Natural History, Geography, and Ura- nography. "Long before the child's entrance into school, too, the helpful presence of others aroused in his heart feelings of gratitude, of affection, and good-will. These may or may not have been brought more clearly to the child's consciousness, and more fully within his control in the social games and group-work of the kindergarten. Howsoever this may be, the school should afford con- stant opportunity for social enterprises, involving common interests, common pur- poses, and common effort, leading to an in- terest in the occupations of men, and in the relationships among men. This leads to studies connected with Sociology and History, 1O PEIMARY METHODS. through which man connects himself con- sciously with the past, and bases his life on the experience of earlier days. " In all that relates to motion and life and, consequently, to the social phases of being 1 , sound lying also nearer the emotional side of sensation plays a part similar to that of color in the realms of space. Con- nected with rhythm in the harmonious combinations and melodious successions of 'music, it has wonderful power in freeing the mind from the material, and leading it to the spiritual, and is, therefore, of incal- culable value in lifting man to the highest planes of mental life. " The chief medium of the work is lan- guage. At the moment when the child is awakened to self-consciousness, language appears as the chief outward reaction of growing self-consciousness in the inter- course with others. Language binds man to man, makes the past an ingredient of the present, and holds this fast for a future. In the development of the intellect and of reason, it is the medium of thought, the indispensable condition of their growth. Hence language, with all that pertains to THE COURSE OF STUDY. 11 it, -will furnisli subjects of instruction dur- ing the entire school-life."* In the details of the course the school should "be guided largely by local and individual circumstances, needs, and -wants. In no case, it is true, can the school afford to follow the child through the maze of facts and phenomena, as they occur in nature and in immediate practical experi- ence. Yet, in all cases, it should create around the child a "world of objects and events, more or less idealized, and more or less systematized, where the child may attain a fair understanding of the essentials of life with comparatively little friction; and, in all cases, it should strive to place this world as fully as possible within the child's control. "While these remarks apply equally to the Primary and Grammar Schools, it is evident that what I may have to say con- cerning the use of things in the school- room for purposes of expression will apply chiefly to the "work of the Primary Grades. Nevertheless, even a superficial survey of * Prize Essay: "Application of the Principles of Psychology to the Work of Teaching." By W. N. Hailmann. 12 PRIMARY METHODS. the subjects of instruction 'will show that so far, at least, as Geometry, Drawing, Col- oring-,- Physics, Chemistry, Natural History, Geography, and Uranography are concerned, I might with profit follow the pupil through the highest grade in the grammar school, aiding him in expressing thought in things as well as words. However, this -would ren- der my task too cumbersome ; and I shall, therefore, not go "beyond the pale of the primary school, save in an. occasional hint concerning the extension of the work to higher grades. CHAPTER III. SYNOPTICAL TABLE OF GIFTS AND OCCUPATIONS- GIFTS AND OCCUPATIONS CONTRASTED. <2o tN" order to enable teachers to choose in- telligently the material for the needs of their pupils, I shall present in this chap- ter a survey of the so-called gifts and occu- pations selected and proposed "by Froebel on the basis of considerations essentially in ac- cordance with the two preceding chapters. Inasmuch as the play-and-work with these things is to lead the children to the study and control of an external world, their formal key lies in shape. There is promi- nent in them, as a whole, first a mathe- matical analysis descending from the "body, through the surface and line, to the point. This is followed among the gifts by a syn- thetical gift in which from the point and line the child ascends to less material rep- resentations of the surface and body. In the occupations, the synthetic elements are 14 PRIMARY METHODS. so intimately and prominently interwoven with, their very e'ssence, that one is almost tempted to treat them as the specifically synthetic side of Froebel's scheme. The following- synoptical presentation, al- though quite different from those generally accepted, is in strict accordance with Froe- bel's spirit. I have no doubt that he him- self would have given us an arrangement not unlike this one, had he found time to look more calmly upon the revelations that came to us through him. I. GIFTS. A. BODIES. i. Things, objects (color) : Six colored soft worsted balls. First Gift. ii. Shape : Wooden ball, cylinder and cube. Second Gift, in. Number: 1. Two (2x2x2): a. Divisibility: Eight one-inch cubes, forming to- gether a two-inch cube. Third Gift. b. Dimensions : Eight bricks (2 x 1 X in.), form- ing together a two-inch cube. Fourth Gift. 2. Three (3 x 3 x 3) : a. Direction (Beauty) : Twenty-seven one-inch cubes, forming together a three-inch cube. Three of the cubes are cut diagonally once into halves ; and three are cut diagonally twice into quarters. Fifth Gift. SYNOPTICAL TABLE. 13 b. Proportionality and Position (Law) : Twenty- seven bricks (2 x 1 X \ in.), forming together a three-inch cube. Three of the bricks are cut lengthwise once into square prisms (2 x \ X \ in.), and six bricks are cut crosswise once into flat square prisms (1 x 1 X \ in.). Sixth Gift. B. SURFACES. i. Squares derived from third gift, cut from veneer or card-board. 1. Entire squares (one square inch). Seventh Gift A. 2. Half squares, right isosceles triangles. Seventh Gift B. n. Equilateral Triangles. 1. Entire triangle, each side one inch long. Seventh Gift C. 2. Half triangles, right scalene triangles. Seventh Gift D. [These were formerly derived, less logically, from a double square or oblong rectangle, cut diagonally.] 3. Thirds of triangles, obtuse isosceles triangles. Seventh Gift E. in. Circles. [The scheme calls for these, but as yet manu- facturers do not furnish them, nor do manuals give directions concerning their use. With a one-inch gun-wad cutter, which may be obtained of any gun- smith, they may be cut cheaply from stiff paper or card-board.] C LINES. i. Straight Line: Sticks or splints of various lengths. Eighth Gift A. n. Curved Line: Rings and half-rings of various sizes. Eighth Gift B. 16 PRIMARY METHODS. D. POINTS. Lentil-seeds, pebbles, beans, etc. Ninth Gift. E. RECONSTRUCTION. (Synthesis.) Softened peas and sticks or straws. Tenth Gift. II. OCCUPATIONS. A. BODIES. 1. Plastic clay (or substitutes, such as wax, putty, etc.). 2. Card-board. 3. Sand. B. SURFACES. 1. Folding sheets. (To the square and oblong sheets, sug- gested by Froebel, I have added the circle and equi- lateral triangle.) 2. Cutting and Pasting. (Paper, card-board, wood.) 3. Painting. (With brush or crayon, or with the lead or slate pencil in " shading.") C. LINES. 1. Interlacing slats. 2. Jointed slats. 3. Intertwining strips. 4. Weaving mats. 5. Thread-games. 6. Embroidery. 7. Drawing. D. POINTS. 1. Stringing beads. (Mrs. Hailmann's Second-Gift beads.) 2. Mrs. Hailmann's lentils or dots. 3. Buttons, papers, and straws, etc. 4. Perforating. I am aware that tlie distinction "between gifts and occupations is post-Fro~belian ; "but GIFTS AND OCCUPATIONS. 17 it is not on that account less real. The gifts are intended to give the child from time to tim'e new universal aspects of the external world, suited to the child's powers of comprehension. In the schedule the es- sential features of these are indicated for the first six gifts, "by the italicized words introducing the gift. The occupations, on the other hand, furnish the child with mate- rials on which to exercise certain phases of skill. Any thing will do for an occupa- tion, provided it is sufficiently plastic and within the child's powers of control ; hut the gift, in form and material, is largely determined "by the cosmic phase to be brought to the child's apprehension, and by the condition of the child's mind. Nothing but the first gift can so effectively arouse in the child's mind the feeling and conscious- ness of a world of individual things and of its own dawning individuality ; but there are numberless occupations, in addition to those enumerated in the schedule, that -will enable the child to become skillful in the manipulation of surfaces. The gift gives the child a new cosmos ; the occupation fixes the impressions made 18 PRIMARY METHODS. by the gift. The gift invites only arrang- ing activities ; the occupation invites chiefly controlling, modifying, transforming, crea- ting activities. The gift leans toward the in-leading processes, toward instruction ; the occupation toward out-leading process, toward expression. The gift leads to dis- covery ; the occupation, to invention. The gift gives insight; the occupation, power. CHAPTER IV. THE WORK SELECTED OUTFIT OF SCHOOLS. 'ROM tlie mass of material presented in the preceding chapter, I shall select a few gifts and occupations, and show in a series of somewhat detailed directions and "model" lessons how they may be used in the school-room in such a way as to aid in rendering the children's growth com- pact, sound, all-sided, and rounded ; and, at the same time, in giving* them full and ready knowledge and mastership in the di- rections of ordinary subjects of instruction. In this, while I shall aim to regard the types best suited to give an insight into the scheme as a whole, I shall be guided largely by considerations of expediency. Thus I shall give preference to those that seem to stand nearest the school as it is, and that will prove most serviceable in the ordinary school studies. Again, I shall con- sider the cost of the material, and put 2O PRIMARY METHODS. more stress upon occupations that may "be procured with comparatively little expense, so that teachers may "be less exposed to an- noying opposition on the part of conserva- tive friends of education. Probably each teacher, according 1 to the circumstances in which she may he placed, will have to select again from my selection. Possibly, too, some may be able to use ma- terial not treated in this volume. In this case I hope my presentations may be suffi- ciently lucid and comprehensive to enable her to make the new applications independ- ently. Most of all should I rejoice, if -what I may say will enable and induce teachers to reach out in their surroundings for new and simpler material, to seek simpler "ways of applying the principles that have guided me, and to discover new avenues of useful- ness for the materials I may present. In the order of the schedule of the pre- ceding chapter I shall consider : (1) The counting-blocks, a modification of the third gift ; (2) the square and half-square tablets ; (3) the sticks or splints ; (4) the lentil- seeds ; (5) sticks and pease ; (6) clay ; (7) card-board work ; (8) folding ; (9) cutting THE WORK SELECTED. 21 and pasting; (10) intertwining strips; (11) Mrs. Hailmann's second-gift, beads ; (12) Mrs. H.'s dots. For various reasons, however, these will "be presented in a different order, more in accordance wi;th the needs of the school, as it is, Thus : (1) Mrs. Hailmann's second- gift heads, "with special reference to lessons in number; (2) the counting-blocks, with special reference to number lessons' ; (3) the folding-sheets, with special reference to drawing and geometry ; (4) clay, with spe- cial reference to drawing, geometry, and coloring ; (5) cutting and pasting and, sub- sequently, card-board -work, -with special reference to arithmetic, geometry, and draw- ing ; (6) intertwining strips, with special reference to arithmetic (fractions), geometry, and drawing; (7) the sticks or splints and, subsequently, sticks and pease, with special reference to geometry, drawing, and arith- metic; (8) Mrs. Hailmann's dots and lentils; (9) the square and half-square tablets. For the majority of the exercises, it is desirable that the child should work upon a surface laid off in square inches, similar to the surface of a kindergarten table. In 22 PRIMARY METHODS. my own schools I have found it quite sat- isfactory to rule sucli a net-work with, a sharp scratch-awl on the ordinary desk- fronts. For many exercises, particularly in group-work though these are not indispen- sable it is desirable to have small tables, thirty inches square and of suitable height. The surface of these, too, is ruled with a net-work of square-inches, and each table accommodates four children. For my own schools I have been able to procure such tables at a trifling cost of $1.50 a piece, and suitable low stools at $2.50 per dozen. Other matters of outfit will be mentioned in the proper places. CHAPTER V. THE SECOND-GIFT BEADS.* (HE Second-Gift Beads consist of wooden cubes, cylinders, and balls (the shapes of Froebel's Second Gift), one half inch in diameter, colored in the hues of the rain- bow red, orange, yellow, green, "blue, violet, and perforated for stringing. For the school, they are particularly useful in number lessons. The contrasts of color and form afford effective means for presenting analyses, aiding the mind through the "sub-conscious" in the formation of clear notions and ready mastership. This will become evident if we compare the follow- ing three typical ways of presenting the number five: (1) Five cubes (five cylinders, or five balls) of the same color. (2) Two pairs of cubes separated by one ball, all of the same color. * Mrs. Hailmann's Second-Gift Beads are manufactured by the Milton Bradley Company, of Sprinarfleld, Mass. 24 PRIMARY METHODS. (3) Two pairs of red "balls separated by a green ball. In the first mode (1) it will be found that the little child can not arrive at the unit : five balls (or beads), without laboriously counting : one, two, three, four, five. This process constantly tempts him to look upon these counting numbers as names, and to call the first ball one; the second, two; the third, three, etc. Thus the very unity in the outer fact hinders the growth of the desired corresponding mind unit. In the second mode (2) the contrast in form breaks up the number five into the familiar constituents, 2 + 1 + 2 ; and these are readily united by the common color into five BEADS. Similarly, in the third mode (3), the separation into 2 + 1 + 2 is made by color, and the common form unites the beads into five BALLS. Another important feature of the work lies in the possibility of arranging 1 the beads in rhythmic waves of form, color, and num- ber. This is illustrated in the following lesson of threes, in "which the square ([ [) stands for cube ; the circle (O)> ^ or "ball '> the semicircle (Q), for cylinder; and the let- ters for the colors (r for red, o for orange, THE SECOND-GIFT BEADS. 25 y for yellow, g for green, 6 for blue, and v for violet) : In tnis exercise two waves are repre- sented, separated from eacli other "by a green bead at the point S. The color wave on each side of this bead is as follows : (r-o-y) (g-b-b) (v-v-v) (b-b-g) (y-o-r). The cor- responding form Avave will be easily read from the diagram. The number wave reads, (1 + 1 + 1) (1 + 2) (3) (2 + 1) (1 + 1 + 1). It will be seen that the second half of the wave is in arrangement the reverse of the first half, descending where the latter ascends. The outfit for the class is simple. Each child is furnished with a box of 50-100 as- sorted beads, and a shoe-string, two or three feet long. This outfit is ample for all ex- ercises within the limits of 1 and 10, and admits of many exercises beyond these limits. For a class of twenty children the cost will not exceed $2.00. One end of the shoe-string is tied to some convenient part 26 PRIMARY METHODS. of the desk or table, and the other end is used for stringing the beads. The beads may be used for all funda- mental operations within the limits of one and ten, and one and twenty. In proof of this I suggest below several series of exer- cises in (1) Counting by ones, twos, threes, fours, and fives. (2) Analyses and syntheses of the num- bers two to ten. (3) The making of addition and subtrac- tion tables. (4) The making of multiplication and di- vision tables. In the exercises the arrangement of the beads will be indicated with the help of the symbols already mentioned, and the respective number lesson, indicated in fig- ures, will accompany each exercise. I. COUNTING EXERCISES. l.\g[r}g[r]glr]g (r^HYr}&c^ d, THE SECOND-GIFT BEADS. 27 (1, 2) (1, 2) (1 : 2) &c. 6. (y\r ]v r 2/Yrly r j/Yr &c. ^(1,2,3) ,2,3) 10. (1, 2, 3, 4) (1, 2, 3, 4) (1, 2, 3, 4) &c. ^)Q0gX^ &c. (2, 4) (2, 4, 6) (2, 4, 6, 8) &c. 28 PRIMARY METHODS. (4,8,13 4,8,12) (5,10,15)&c. ^ (1,2, 3,4, 5 5,4,3,2,1) In the first three exercises, the members of each pair of heads differ both in color and shape, and the child recognizes them merely as "two ones, or (1, 1). It will prob- ably count, e. g., in Exercise 1 : " one green cube, one red ball ; one green cube, one red ball," etc. In the Exercises 4, 5, and 6, on the other hand, the members of each pair, though separated by color into two ones 2(1), are united by shape into one two - 1(2). Reading the fourth exercise by color, the child would say: "One red cube, one blue cube; one red ball, one blue ball," etc.; or simply: "One red, one blue," etc. Reading the same exercise by shape, it would say: " Two cubes, two balls," etc. THE SECOND-GIFT BEADS. 29 Similarly, in Exercises 7 and 8, the mem- bers of each set of three differ both in color and shape. The child "would count, while stringing 1 (in Exercise 7) : " One blue cube, one red cylinder, one yellow ball," etc. On the other hand, in Exercises 9 and 10, the shape gathers the beads into distinct sets of threes. The child would read while stringing- (Exercise 9) : " One red cube, one orange cube, one yellow cube" etc. ; or by shape alone : " One cube, two cubes, three cubes," etc. When the string is finished, it will read from its work, by shape : " Three cubes, three cylinders, three balls," etc. Similar remarks apply to Exercises 11, 12, 13, and 14. In Ex. 13, the reader will observe color waves, the color ascending in each group of five from red to yellow, and descending on the opposite side from yellow to red. In Ex. 14, the color waves of each group of five are gathered in a larger form wave ascending from the balls to the cubes, and descending on the opposite side from the cubes to the balls. A second wave may be added to this, after indicating the close of the first by means of a cube, clearly con- trasting in color with the orange balls. SO PRIMARY METHODS. In Ex. 15 counting by twos the color separates the "beads into sets of twos ; and the shape teaches the child to read suc- cessively, as it strings the beads or surveys its work : "Two, four (cubes)"; "two, four, six (balls) ; two, four, six, eight (cylinders)." Here the child may insert an orange ball, in order to mark the highest point of the number wave, and then, reversing the or- der of colors and shapes, count : " two, four, six, eight (cylinders) * two, four, six (balls) ; two, four (cubes)," etc. Ex. 16 and 17 suggest the counting by threes. In Ex. 16, the child counts: "three, six, nine (balls) ; three, six, nine (cubes) " ; and repeats this counting at pleasure. Here shape unites the threes, and color keeps them distinct. In Ex. 17, the reverse is the case ; color unites the threes, and shape keeps them distinct. Here the child counts : " three, six, nine (blue beads) ; three, six, nine (yellow beads) ; three, six, nine (red beads)," and repeats the exercise at pleasure. In Ex. 18, shape unites the fours. The exercise presents coinciding shape and color waves, the highest point being marked by & green cube. The child counts : four, eight, THE SECOND-GIFT BEADS. 31 twelve (cylinders) ; four, eight, twelve (cyl- inders). It goes without saying that the green cube, marking the highest point of the wave, is not counted ; also, that "before the exercise is repeated, an orange hall must "be strung to separate the last "blue cylinder of the first exercise from the first blue ball of the repetition. In Ex. 19, color unites the fives. The highest point of the shape wave is indi- cated by a yellow cube. The child counts : five, ten, fifteen (red beads) ; then omitting to count the yellow cube five, ten, fifteen (red beads). The repetition of the exercise would call for a green ball to separate the two exercises. Exercise 20 is an excellent counting ex- ercise. It reads both in color and shape : " One (orange ball), two (yellow cylinders), three (green cubes), four (blue cylinders), five (violet balls) " ; and from the red cube down- ward: "Five (violet balls), four (blue cylinders), three (green cubes), two (yellow cylinders), one (orange ball)." I trust that these suggestions and expla- nations will make it easy for the teacher to devise additional exercises, as they may 32 PRIMARY METHODS. be needed. Tlie exercises may "be dictated orally to the children ; or they may "be in- dicated on the "blackboard with the help of the symbols used in this book ; or the children may be left to invent form and color combinations for given number for- mulas. The last, however, should not be indulged too soon nor too frequently. CHAPTER VI. SECOND-GIFT BEADS. (Conclusion.) 'ROM the very nature of tlie opposite processes, analysis and synthesis, it follows that the same exercises will answer the purposes of "both. The inspection of the following- typical exercise with the number three will show this : (14-1-1-1) (1+2) (2+1) (3) (1+2) (2+1) (1 + 1+1) This may be read from left to right syn- thetically : 1 + 1 + 1 = 3, 1 + 2 = 3, 2 + 1 = 3, 3 = 3, 1 + 2 = 3, 2 + 1 = 3, 1 + 1 + 1 = 3. Or, it may be read, in the same direction ana- lytically : 3 = 1 + 1 + ], 3 = 1 + 2, 3 = 2 + 1, 3 = 3, 3 = 1 + 2, 3 = 2 + 1 = 3 = 1 + 1 + 1. Inasmuch as the stringing of the beads is a synthetic process, it will be necessary to begin -with the synthetic reading. As the child proceeds with the work of string- 34 PRIMARY METHODS. ing- the beads, it says in accordance with the teacher's dictation or with the number formula indicated on the blackboard : " One orange (ball), one green (ball), one violet (ball) three balls ; one red (cyl.), two green (cyl.) three cylinders ; two blue (cubes), one yellow (cube) three cubes; three violet balls," etc. When the work is clone, it surveys the string of beads, and is helped to read ana- lytically, from left to right or vice versa : " Three balls one orange, one green, one violet ; three cylinders one red, two green ; three cubes two blue, one yellow ; three violet balls," etc. Similar remarks apply to the " tables " for addition and subtraction, as well as to those for multiplication and division. The bead exercise that answers for addition is equally serviceable for subtraction ; and the exercise that teaches multiplication is equally useful for division, as will appear directly. II. SYNTHESIS AND ANALYSIS OF 2 TO 10. SECOND-GIFT BEADS. 3. "| y I v I y P) 1 (i+i+i) (1+2) (2+i) I (3) -(3) (1+2) or 3 ^YoYwYffYbYi^ I Jstifi' &c. 1 + . .A. yV ^ I ' ' (2+2) (4) ( 2 + 3 ) &c.; or, I 4 = 4(1), 4 = 1+3, 4 = 2(2), 4 = 1(4), &c. 5. (1+2+1) (1+3) (2+2) (3+1) (4) &c. ol &c. 5 (1) (1+2. +2) (1+3+1) (1+4) (2+3) (5) (3+2) (4+1) & C ; FW^^^ V o r I , 6 (1), 3 (2), 2 (3), (6), 2 (3), 3 (2), 6 (1) 9. (vXty^^^ (H-5) (3+4) (3+3) (6) (3+3) (4+2) (5+1) 36 PRIMARY METHODS. 10. y \y y \V . (3+4) (2+5) (1+6) (7) (6+1) (5+2) (4+3) 11: (33510^^ (44-4), (3+5) 13. Z> b v 171 14. o o 'J (8) (9) (10) 3 + 11- 6+ i(8)= 6 + 7 + ^(6) 4 + 13- i(9)+ll = 1(9) +11 5+ 4 = iO f (6)+ 3(4)= 8-i(15) + 4 20 1C) f (8)+ 2(6)= 4 + 4(4)-}(4) 30 JO 16+ 1(8)= f (15)-i(12) 40 60 ig For the sake of avoiding misapprehen- sion, I add the solutions of these exercises as they would appear on the children's slates : (1) (2) (3) (4) 8+5=13 11 = 1 + 10 3 (1)=: 3 12+6=2 (6) 7+5=12 11=2+ 9 3 (2)= 6 14+6=2 (6) + 2 9+5=14 11=3+ 8 3 (3)= 9 15+6=2 (6) + 3 6+5=11 11=4+ 7 3 (4)=12 18+6 = 3 (6) 12+5=17 11=5+ 6 3 (5)=15 13+6=2(6) + ! 13+5=18 11=6+ 5 3 (6)=18 16+6=2 (6) + 4 18+5=23 11=7+ 4 10+6 = 1 (6) + 4 14 + 5 = 19 11=8+ 3 20+6=3 (6) + 2 11 + 5 = 16 11=9+ 2 (5) (6) (7) (8) 1 (12)=6 f (15)= 9 12=3+9 4=11-7 t (12)=4 16+ 4=4 (4) 12=4 + 8 4=13-9 | (12)=2 (4)= 8 9+8 =17 12 = 5 + 7 4= i (8) i (12)=3 17- 8=9 12=2 (6) 4= i(12) | (12)=3 (3)= 9 16-11 = 5 12 = 3 (4) 4= t(16) t (12)= 2 6 (3) =18 12=4 (3) 4= i(20) (12)=5 (2)=10 17n-5=3(5) + 2 12 = 6(2) 82 PRIMARY METHODS. < 9 > (10) 6 +i (8)=10 6 + 7+H6) =16 i (9)+ 11=14 f (9) + ll =17 f (6)+3(4)=16 8-H15H4 - 9 f (8) + 2 (6)=18 4+4 (4)-J (4)=19 16 -hi (8)= 4 (4) | (15)-| (12) =19 CHAPTER VIII. THE COUNTING BLOCKS. tHESE are wooden one-inch "blocks, de- rived from the third gift. They are used in three sizes : (1) whole cubes (Ixlxl in.) fjjSp, (2) half cubes (lxlx|in.) gjfp, (3) quarter cubes (lxx| in.) J9. For ordinary number exercises, especially in primary work, the whole cubes are most serviceable. They are more easily handled, and because of their regular shape fit in all positions, and do not divert the atten- tion from number to form. The half and quarter cubes may be dispensed with alto- gether in number lessons; at any rate, they should not be used for this purpose until number images are well fixed by the ex- clusive use of whole cubes. The number images furnished by these blocks mark an important advance in men- tal growth with reference to number. The balls of the "numeral frame" touch only 54 PRIMARY METHODS. in one point, so that even "beads of the same color retain their individuality prom- inently in the twos, threes, fours, fives, etc., of the lessons ; the child readily recognizes in each two, three, four, five the constit- uent two, three, four, five ones. The count- ing blocks, on the other hand, coalesce quite perfectly on contact ; an entire face of one coincides with an entire face of the neighbor, and the group of blocks forms a complete, unbroken whole. The modest division lines do not force their presence on the child's attention ; still they are sufficiently clear to be readily distinguished, and to announce without difficulty the number of constituent ones in the new composite unit. Color, too, is removed so that the pupil's number conceptions are more and more freed from other phases of material existence that cling to things, and brought nearer to the pure forms of ab- stract ideas. Before the teacher attempts to guide the work of the children, she should have ex- perienced these effects of the new number forms upon her own mind. For this pur- pose she should provide herself with a THE COUNTING BLOCKS. sufficient number of these blocks (a few cents will purchase one hundred), and carry out the suggestions of the following pages for herself, extending the exercises accord- ing to her own needs. In the work of the children the suggestions already made hold good. A limited number of typical lessons in counting, analysis and synthesis of num- bers, and in the fundamental operations will, therefore, suffice to unlock the possi- bilities of this occupation. In the diagrams each square represents a cube. I. COUNTING EXERCISES. (1) The child receives ten or twelve blocks, and counts from one to ten, the successive number forms presenting the following or similar phases : QQ am, , &c. to 10. , &c. to 12. 66 PRIMARY METHODS. D. 1 1, T > 1 1 | L &c. to 10. (2) The child receives from eighteen to twenty blocks, and counts by twos, threes, fours, or fives. *nn. , &c. It will be noticed that the child thus secures a variety of mobile images of the various numbers. Thus in (a) 5 appears as 4 + 1, in (6) as 3 + 2, in (c) as 5 (1) or 1 (5) ; in (a) and (d) 6 appears as 3 (2), in (6) and (e) as 2 (3), in (r) as 5 + 1, etc. II. ANALYSIS AND SYNTHESIS OF NUMBERS. OO DOmOD OO 20), (1+1), 2, (1+1), 2(1). It will be seen that in Exercise (1), post- THE COUNTING BLOCKS. 57 tion answers a. purpose similar to that ac- complished by color on pp. 34-36, with this difference, however, that the position of each cube is in the pupil's control. In the first form, both cubes are in the same slanting- position, touching only corner to corner ; they appear as two separate ones equal in position, or as 2 (1) ; in the second form they stand in different positions, touching again only in one point ; they appear as two separate ones in different positions, or as (1 + 1) ; in the third posi- tion they have fully united, face to face, into a new composite unit ; they appear as a two, 1 (2) ; the fourth and fifth forms are repetitions of the first and second in in- verse order, forming the descending portion of a position -wave similar to the color waves of Chapter V. and VI. OOO 8 GQ, (14-1+1), (2+1), (3). In this exercise, the number three is sim- ilarly treated. We have successively, three ones in the same position, yet clearly sepa- rated, one and one and one, two and one, and one three. 58 PRIMARY METHODS. (3),(2+i)orCl-t- 2,) 3 (!}. 4. tfb (Ex. 3 and 4 read from top to bottom or vice versa. This may "be done with, all these exercises.) &c. (4), (3+ 1)., 2 (2), 4 (1). However, such exercises will prove quite cumbersome. They may be indulged for silent practice to a limited extent, but for class-instruction more efficient modes must be found. I have found it best, for this purpose, to beg-in on the analytic side of the work. Thus for Ex. 6 the child receives four fours (16 blocks) arranged at equal distances from left to right. The task is to analyze them successively in accordance with the number indications : (2 + 2) (1 + 3) (1 + 2 + 1) (4 (1)). The results THE COUNTING BLOCKS. 59 indicated in Ex. 6 may "be reached by dic- tation or "by independent study on the part of the more practical children. Bel c.< fis d.- n m n n n n n For Ex. 7 the children have received fif- teen "blocks each. The lesson is indicated on the blackboard : 1+ =5 2+ = 2(2)+ = 5 Three sets of answers are indicated in the Exercise (7). n O El m 60 PRIMARY METHODS. For Ex. 8 each, cliild lias received six "blocks. Trie lesson lias "been indicated on the blackboard : 1 + = 6 ; 2 + = 6 ; 3 + =6; 6+2= ; 6-5-3 = . The child has solved the problems successively, and has recorded the solutions on the slate, in drawing and writing, and Ex. 8 shows one of the results: D B 1 + 5 = 6, 2 + 4 = 6 9BB. = 3(2) The great value of these blocks in fac- toring deserves special notice. For this purpose it is used to great advantage even in the fifth grade. Indeed, I have found children of still higher grades derive real benefit from an occasional return to these "things," 'when problems involving factor- ing had to be solved. A few illustrations will make this clear. I have selected for this purpose the num- bers 9, 10, 15, 12, 28, 36. For the number- forms the blocks are arranged in accord- THE COUNTING BLOCKS. 61 ance with, our decimal system of notation, so that the number of tens and units may be readily distinguished ; thus : 10 15 = 10 + 5 = 10+2 28 = 2 (10) +8 36 = 3 (10) +6 The pupils should then proceed in sys- tematic order, testing the lowest factor first. Thus (9) would successively go through the following changes : a. ZK c. (10) and (15) are less refractory, yielding at once the forms 2 (5) and 3 (5), respect- ively : 62 PRIMARY METHODS, 15 = 3(5) (12) may pass through a variety of suc- cessive transformations ; in (a) 12 = 2 (6) = 2x2(3), in (6) 12 = 3 (4) = 3 x2 (2). 12 3(6) 2x2(3) 12 3(4) m en an m an m 3X2(2) The following diagrams will sufficiently illustrate the factorings of (28) and (36). THE COUNTING BLOCKS. 63 In the illustration, twenty-eight is first resolved into 2 (10) and 2 (4) =2 (14) ; then, in twice two sevens [2x2 (7)]. For thirty-six, two factorings are given on p. 64. In the first of these, "bisection ar- ranges the number in 2 (10) + 2 (5) + 2 (3) = 2 (18). These, "by a second bisection, are readily grouped, each, in 2 (9), reducing 36 to 2 x 2 (9). Lastly, the trisection of the nines yields four sets of three threes, or 2x2x3 (3). The same process is somewhat simplified in the second illustration of the reduction of thirty-six. We see successively two sets of two nines, 2x2 (9), and two sets of twice three sets of three, 2x2x3 (3). 28 2 (14) = 2X2 (7). 64 PRIMARY METHODS. 36 2 (18) = 2 X 2 (9) 2 X 2 X 3 (3) 36 &c. 2 X 2 (9) irm 2X2X3 (3). CHAPTER IX. THE COUNTING BLOCKS. (Concluded.) Q-o tN tlie use of the blocks for the funda- mental operations addition, subtraction, multiplication, division, and part-taking- the number-forms should be constructed in accordance with our decimal system of no- tation. Sixteen should appear clearly as one ten and one six; twenty-six as two tens and one six, etc. I have found it most convenient for this purpose to lay under each other as many tens (rows of ten blocks) as the number indicates, and the number of units under these. (See p. 61, where the number-forms of 9, 10, 15, 12, 28, and 36 are shown.) In the solutions of problems it is best to dispose first of the tens of the number to be added or subtracted. Thus, if 27 is to be added, add first the 20 and then the 7 ; if 39 is to be subtracted, subtract first the 30 and then the 9 ; if 4 (26) is the prob- 66 PRIMARY METHODS. lem, find 4 (20) and 4 (6), etc. This is in strict accordance with the number-names above 20 ; language says : Add twenty-seven, subtract thirty-nine, find the value of four twenty-sixes, etc. Besides this mode of pro- cedure by removing- the bulkiest portion of the task first, facilitates mental processes, and enables the pupils to ride safe between the Scylla of verbalism and the Chary bdis of slavish subjection to slate-work. In the following pages I present the suc- cessive number-forms as they occur in the solutions of a number of problems, illustrat- ing the use of the blocks in the various fun- damental operations. To these I have added a few examples of involution in connection with the finding of areas and volumes. I. ADDITION : TYPICAL PROBLEMS : 1. 8 + 5= 3. 14 + 5= 5. 14 + 7 = 2. 9 + 6= 4. 11 + 7= 6. 19 + 5 = 7. 24+15= 9. 29 + 23 = 8. 26 + 14= 10. 26 + 27= SPECIMEN SOLUTIONS : '8 + 5 = 13 THE COUNTING BLOCKS. 67 15 4- 6 = 14 + 5 = 8. 19 14 21 .19 + 24 24 15 = 68 PRIMARY METHODS. 29 + 23 = 29+20 f 3 = 52 II. SUBTRACTION ; TYPICAL PROBLEMS : 1. 16-4= 3. 28-12 = 2. 12-7- 4. 32-26 = SPECIMEN SOLUTIONS 16 4 = 5. 58-34= 6. 51-29 = 12 i itzn 12 -7 THE COUNTING BLOCKS. 69 28 _ 12 = 16 I 1 1 r 7 , T y | r ! ! ! ! ! ! 3 3 3 26 = 6 T ] T r~r~r~ T "T" T "i ....p- p-j i ...i..i._,.J j 3230- 2- <=6. III. MULTIPLICATION: SPECIMEN SOLUTIONS : 1. 4 (9) = 2 (18) = 2 (10) + 2 (8) = 36. &c. g. 3(13) = 3 (10) + 3 (3) = 39; &c. 70 PRIMARY METHODS. 8. (4) = 3 (12) = 3 (10) + 3 (2) = 36. m en m, IV. DIVISION : SPECIMEN SOLUTIONS : 1. 24 -+-3 = 8 (3). I 1 I i (The four blocks on the right are moved to the lowest row; in the second form the threes may he slightly separated from left to right.) 2. 28 -- 2 = 10 (21 + 4 (2) = 14 (2). BBBB0BBBBB m m m THE COUNTING BLOCKS. 71 V. PART-TAKING: SPECIMEN SOLUTIONS: % (25) [ (25) = 5] ^3 (5) -15. VI. INVOLUTION 2*= 4 3 ! =9 4 J =16 n r~H 5*= 25 72 PRIMARY METHODS. 6*= 36 3'= CHAPTER X. THE FOLDING SHEET. the exercises to be suggested under this head, paper sheets of various sizes and shapes may be provided. The most convenient for class use are squares of four inches to the side, equilateral triangles of four inches to the side, and circles four inches in diameter. For drawing, I have found a fair quality of unsized manilla wrapping paper quite suitable ; although, for some purposes, a good quality of un- sized printing paper may be preferred. The same sheets will answer for geometrical exercises ; nevertheless, for reasons to be stated hereafter, colored unglazed cover- paper will be found more satisfactory and not very expensive. The square and tri- angular paper can be cut cheaply on a cutting-machine by the nearest book-binder or job-printer. The circular papers will 74 PRIMARY METHODS. have to be ordered from the manufacturer, or cut to order "by the children. In folding, stress should "be laid on ac- curacy, care, and cleanliness ; the creases should be sharply defined "by drawing- the back of the thumb-nail firmly over the folds. In the dictation exercises, the pupils should not change the position of the paper unless by direction, and should not lift the paper from the table until the desired form is completed. I shall show, first, how the sheets may be prepared and used for draw- ing exercises, and then, how they may be used in form lessons. I. THE SQUAEE SHEET IN DRAWING. For an introductory exercise, the pupil should study the form features of the sheet before him ; he should count and describe the edges, corners, and angles. [" One edge in front (or below), one edge behind (or above), one edge on the right, one edge on the left." "The front (lower) edge and right edge form a corner the right front (lower) corner; the front edge and left edge form a corner the left front (upper) corner;" etc. ' The front (lower) edge and back (upper) THE FOLDING SHEET. 76 * edge run from right to left ; " etc. " The rig-lit and left edges are in the same direc- tion (from front to hack) or parallel; the front and hack edges are in the same di- rection (from right to left) or parallel." "The left edge makes a right angle "with the front edge, the left edge is perpendicular to the front edge, the front edge is perpen- dicular to the left edge, the left and front edges are perpendicular to each other," etc. "The lower and upper edges are horizontal, the right and left edges are vertical" Place the fore-finger of the left hand on the front edge, the fore-finger of the right hand on the edge parallel to this, on an edge perpendicular to it, on the opposite edge, on an adjacent edge, etc. Place the fore- fingers on the edges, forming the left front corner, the left hack corner, etc.] In all these exercises the technical terms itali- cized above should "be freely used. For the first exercises in drawing, the paper is prepared as follows, the teacher dictating: "Place the sheet before you with two edges running from right to left, and two from front to back. Place the right edge on the left edge, and crease the paper 76 PRIMARY METHODS. in tlie fold. Open the paper*. Front edge on "back edge; crease; openf. Right edge oil vertical diameter (or crease) ; crease ; open. Left edge on vertical diameter; crease; open. Front edge on horizontal diameter; crease ; open. Back edge on horizontal diam- eter; crease; open.J The thirty-five suggestive figures of the subjoined plate indicate how, "with the help of the sheet creased into sixteen square inches, as the above dictation teaches, the teacher may secure automatism in drawing- straight lines, parallel and diverging, in all directions. Similar series can easily "be * The folio-wing facts may here be fixed in lively conversa- tion -with the children: "The vertical crease passes through the middle of the paper ; it is a diameter of the square ; it cuts the paper into two equal oblongs ; each oblong is the half of the square ; the vertical diameter bisects the square ; it bisects the front edge ; it bisects the back edge ; it is parallel to the right and left edges; perpendicular to the front and back edges," etc. t Here facts like these may be brought out in conversation : "The horizontal diameter bisects the right and left edges; it bisects the vertical diameter ; the two diameters bisect each other ; the two diameters cross at the center of the square ; they divide the sheet into four equal squares," etc. $ Here the fact that the paper is divided into smaller squares may be noticed, and the squares counted. "How many rows of squares from right to left ; how many from front to back ; how many squares in each row; how many in the paper?" The number of creases, their relative directions, the angles which they form, and other things may be noticed and distinctly an- nounced by the children in full, clear sentences, until they are quite familiar with the paper and love it for the pleasure ob- tained from it. THE FOLDING SHEET. 77 constructed by the teacher for the practice of curved lines, although, for this purpose, other appliances are preferable. 78 PRIMARY METHODS. II. THE EQUILATERAL TRIANGLE IN DRAWING. Here again, for an introductory exercise, the pupil may study the form features of the triangle the position of its edges and corners, the equality of its sides and angles. The sheet may then be prepared for draw- ing, as follows : " Place the sheet "before you, one edge in front from right to left, the other two slanting backward. Place the right (slanting) edge on the left (slanting) edge, and crease the paper in the middle- Open the paper. * Front edge on left slanting edge, crease, open. Front edge on right slanting edge, crease, open." (In Fig. 1 of the plate of suggestive figures on the next page, the resulting creases are indi- cated by the lines c d, a e, 5 /.)f "Place the * Here the folio-wing facts may be brought out: "The crease lies from front to back ; it bisects the front edge ; it bisects the triangle ; it bisects the back angle ; it is perpendicular to the front edge ; it cuts the triangle into two equal, right, scalene triangles." t Here it may be noticed that the three creases (or altitudes) THE FOLDING SHEET. 79 back corner on the middle of front edge, crease, open ; right corner on the middle of left edge, crease, open ; left corner on the middle of right edge, crease, open."- (In Fig. 1, the creases are shown, "by the lines e /, e d, f d.)* intersect at a common, point, the center of the triangle, that they divide the triangle into six equal, right, scalene triangles ; that the two triangles in front (on the right, on the left) form an isosceles, obtuse triangle ; that the two triangles in the right (left, back) corner form a trapezium, etc. * Here it may be noticed that the three short creases divide the equilateral triangle into four smaller equal equilateral tri- angles ; that the short create from right to left (/ ) " The right side of the square is parallel to the left side ; the lower side is parallel to the upper side. The right side is per- pendicular to the upper (and lower) side ; the left side is perpendicular to the upper (and lower) side, etc. The opposite sides are parallel ; the opposite sides are equal. The adjacent sides are parallel (equal)." (c) "The square has four sides ; it is a quadrilateral. 86 PRIMARY METHODS. Its opposite sides are parallel ; it is a par- allelogram. Its sides are equal ; it is an equilateral parallelogram. Its angles are right ; it is a right parallelogram. It is a right, equilateral parallelogram." The children may be requested to fold the lower side on the upper one, to crease and open (Fig. 1 of the above table) : " The crease bisects the square ; it bisects the right (left) side. It is a diameter. This diameter is horizontal. It is perpendicular to the right (left) side. It is parallel to the upper (lower) side." Similarly the children may study, in Fig. 2, the oblong; in Fig. 3, the vertical diameter; in Fig. 4, another oblong; in Fig. 5, the two diameters ; in Fig. 6, one diagonal ; in Fig. 7, the right isosceles tri- angle ; in Fig. 8, both diagonals, etc. In the figures the blank spaces inclosed by clotted lines show the portion of the paper folded on the hatched part of the figure. For Fig. 9 (trapezoid), after creasing a diagonal, the lower side is folded on the diagonal. The subsequent folding of the left side on the same diagonal, gives the trapezium, Fig. 10. From this we obtain THE FOLDING SHEET. 87 Fig. 11 by folding the riglit isosceles tri- angle of Fig. 10 inward. Fig. 12 (scalene obtuse triangle) comes from Fig. 10 by folding the lower half upon the upper half of the trapezium. Fig. 13 (rhomboid) comes from Fig. 9 by folding the upper side upon the diagonal. From this Figs. 14, 15, 16, and 17 are easily derived. For Fig. 18, the child has been directed to open the last fold of Fig. 17, and to turn the lozenge over. On the reverse side, the two diagonals will be distinctly observed, and the child may be taught to see and say: "The long diagonal bisects the short one. The short diagonal bisects the long one. The two diagonals bisect each other. The short diagonal is perpendicular to the long one. The two diagonals are perpen- dicular to each other," etc. The remaining figures (19-31) suggest ex- ercises in the social synthesis of the forms obtained. Thus, for Fig. 19, four children have placed together the right isosceles tri- angles obtained by folding Fig. 7, so as to inclose a slanting hollow square. Other combinations of the same form are shown in Figs. 20 and 21. Similarly, Figs. 22, 23, 88 PRIMARY METHODS. and 25 show combinations of the trapezium (Fig. 10); Figs. 24 and 26, of the isosceles triangle (Fig. 11); and Figs. 27-31, of the lozenge (Fig. 15). The designs may he pinned to the wall or pasted on manilla paper, and thus util- ized in ornamenting the school-room and as patterns for drawing or coloring. II. THE EQUILATERAL TRIANGLE IN FORM LESSONS. After -what has been said concerning the square sheet, little need he added in ex- planation of the following cuts : THE FOLDING SHEET. 89 In Fig". 1 the cliilcl studies tlie equilateral' triangle as such and, after folding 1 the right half upon the left and opening the paper, the relation of the altitude to the triangle. In Fig. 2, he studies the three altitudes ; in Fig. 3, a right scalene triangle (half of the equilateral) ; in Fig. 4, a trapezoid ; in Fig. 5, a lozenge; in Fig. 6, a new smaller equi- lateral ; in Fig 1 . 7, after opening all the folds, the relations of parts. Figs. 8-11 show the growth of the hexag-on ; and Figs. 12-14 give some additional hints, on the basis of the net-work obtained in the last series of foldings, concerning a new line of drawing- and coloring- designs. III. THE CIRCULAR SHEET IN FORM LESSONS. 90 PRIMARY METHODS. 8. Fig. 1 represents the sheet re-opened after the creasing- of the horizontal diameter; Fig. 2, the same after the creasing of two perpendicular diameters ; Fig. 3, the semi- circle; Fig. 4, a quadrant; Fig. 5, parallel chords; Fig. 6, segments; Fig. 7, the in- scribed square. Figs. 8-12, suggest social syntheses. These suggestions must suffice. For the THE FOLDING SHEET. 91 thoughtful teaclier they will open many avenues to rich fields of investigation on the subject of form, in many grades of the school. To her, however, as in previous cases, I must leave the task of selecting and adapting the exercises and of preparing the details of each lesson. In all folding exercises it is essential for permanent success to secure neatness and promptness ; neatness first and foremost, and then promptness. K~ever, under any circum- stances, sacrifice the former to the latter. For this purpose, it is well strictly to observe a few rules, even at the risk of some ped- antry : 1. Insist that all folding and creasing be done on the table ; never allow the children to lift up the paper and do the folding and creasing in the air. 2. Insist that the paper be kept in the same position during the entire dictation, not turned so as to bring the right side in front when the direction is given to fold from right to left, etc. 3. Insist that point shall lie accurately on point and edge on edge, before any creasing is attempted. 92 PRIMARY METHODS. 4. Insist that the creasing- be clone slowly and deliberately with. 1 the back of the thumb-nail or some suitable instrument, such as a paper-knife. 5. Insist that the creasing be done thor- oughly, so that the folds may lie flat. 6. Insist that in opening the folds the child keep the paper smooth and avoid all crumpling. 7. Dictate slowly, deliberately, with ample pauses, avoiding nervous repetition. CHAPTER XII. PLASTIC CLAY. Cfci fN very many localities this material may be had for the digging-. Elsewhere it may be procured from potteries at rates varying from to 3 cents per pound. It may be broken up in a pail, moistened with water, worked into suitable consistency, wrapped in a moist cloth, covered with an oil cloth, and laid aside for use. When ready for "work the teacher may slice off pieces of suitable size with - a piece of copper wire of convenient length. Each child is furnished with a modeling board about one foot square, which will be cut to order at the planing-mill for five or six cents a piece, and with a modeling knife which may be bought at 25 cents per dozen or whittled out of soft wood by the boys. The knives should be six inches long, the 94 PRIMARY METHODS. blade three fourths to one inch wide, of a shape indicated by the following cut : In fashioning the clay, the children use only gentle pressure "with the thumbs and fingers. Violent pressure and beating are as inexpedient as they are unseemly. For smoothing, after the fingers have done their best, the blade of the knife is drawn gently over the surface with a minimum of press- ure. For trimming the edges and carving, the knife blade is used like that of an or- dinary knife. For engraving designs on. the surface of the clay, the point of the handle is used. On account of the softness of the surface, gentleness and delicacy of touch are imperative. Thus the work with clay becomes an excellent corrective of the in- jurious influences of slate-work upon the hand, as well as an excellent school of patient, thoughtful persistence. The easiest and most fertile work for primary grades, with endless resources for the exercise of inventive power and manual skill, lies in the manufacture of tiles of PLASTIC CLAY. 95 various shapes ; in succession of difficulty, square, circular, octagonal, hexagonal, tri- angular (equilateral), pentagonal. With, children who have not enjoyed the advan- tages of kindergarten training, the first few lessons should "be devoted to free random exercises or play with the clay. They may he allowed to make marbles, cakes, loaves of "bread, birds'-nests, and "birds, snakes, and hundreds of other things according to their fancy. This "will render them familiar with the plastic properties of the material, and will thoroughly arouse their interest. Dur- ing these play-lessons, too, the need of gentleness in handling the clay should be inculcated ; the children will soon learn that easy, delicate treatment yields better results. For the manufacture of the square tile, each child receives about two cubic inches of clay. By gentle pressure with the fingers, this is spread out on the modeling 'board in a flat cake, about 4 inches square, and not quite one fourth inch thick. The sur- face of this is scraped smooth with the clay knife, and the edges are trimmed so as to leave a smooth tile, four inches square. The 96 PRIMARY METHODS. teacher, in passing from child to child, giving directions, helping, and encourag- ing (or one of the children appointed for this purpose), is constantly busy picking up from the "boards scraps of clay ; this will insure neatness and respect for the material. With the help of the ruler and the pointed knife-handle, or some other suitable style, the children may then analyze the surface in a variety of "ways, reaching more or less complicated net-works for guidance in exercises of engraving, carving, or paint- ing. The following figures indicate the principal ones of these net-works. The figures on page 77, Chapter X., suggest ex- ercises for drawing or engraving. Exercises for carving and coloring will readily sug- gest themselves to a thoughtful considera- tion of the net- works. In Figs. 12, 13, and 14, the net-works appear as little picture- frames to be filled according to the child's taste 'and skill. PLASTIC CLAY. 97 10. In the manufacture of the circular tile the child proceeds as above. When the tile is ready for trimming-, he may stick a stout pin near the center of the tile, tie another pin to a piece of thread about ten inches long, pass this thread around the central pin so as to obtain a radius of four inches, and with the free pin mark the circum- ference of the circle on the tile. The ex- cess of clay is then trimmed off with the clay-knife,, and the tile is ready for work in accordance -with previous suggestions, and on the basis of net-works, indicated 1 below. 15. 16. 17. 18. 19. 98 PRIMARY METHODS. 20. The circular tiles offer convenient sur- faces for practice with dividers in the drawing- of circles and circular arcs, as in- dicated in the following cuts. One of the points of the dividers may "be protected with a small disc of card-board to keep it from penetrating- too far into the plastic clay. In the place of dividers, two pins and a thread, as above, will answer the purpose. The following figures suggest some 29. 30. 81. PLASTIC CLAY. 99 such, exercises. The various fields may be painted in different colors to make the forms more impressive and to please the children. In the making of solid forms, similar directions hold good. The pupils should, again, rely chiefly on gentle pressure with, the thumbs and fingers ; all beating and hammering should be discountenanced. For purposes of smoothing and decorating the faces of ttie solids, the same instruments and expedients are used as in the making of tiles. The fashioning of the solids offers excellent opportunities for the education of the sense of touch with reference to shape. Much is gained in this direction by re- questing the children frequently to close their eyes while going through the initial processes of coaxing the clay into the re- quired shape. This will not only add a new interest to the exercise, but will help to clear the children's notions of shape, by inducing them to concentrate their atten- tion on the sense of touch as the true shape sense. In all cases, much heed should be given to efficient means for interesting the child in the shapes under considera- 1OO PRIMARY METHODS. tion. These means are found chiefly in the decoration of the pure shapes, and in their modification and combination for imitating objects. The various shapes are "best con- sidered in the following- order : (1) Cube ; Square Prism ; Square Pyramid. (2) Cylinder; Prisms of three, six, eight, five sides. (3) Cone; Pyramids of three, six, eight, five sides. (4) Sphere; Spheroids; Tetrahedron and Octahedron. The teacher's tact and opportunities must decide to what extent each of these shapes can be used in the study of objects and of geometrical relations, as well as in drawing and coloring. In my experience I have found them of great value in all grades of the Primary and Grammar Departments. CHAPTER XIII. CUTTING AND MOUNTING. tHE materials for this occupation are the square, circular, and triangular folding- sheets and suitable paper or card-board for mounting. For the latter purpose, stout manilla wrapping- paper, cut in pieces seven to nine inches square, is quite serviceable. A pair of cheap blunt-pointed scissors, a small dish or bottle with mucilage, a small clean piece of cotton cloth, and a camel- hair brusli complete the outfit. The simplest and, for primary school use, the most efficient method of preparing- the square sheet, is indicated in the following dictation: "Lay the sheet before you, the front (nearest) side from right to left ; lay the front edge on the back edge, crease in the fold, open the paper ; the right on the left edge, crease, open ; the front edge on the middle (horizontal) crease (diameter), crease, open ; the b'ack on the horizontal 1O2 PRIMARY METHODS. diameter, crease, open ; the right on the vertical diameter, crease, open ; the left on the vertical diameter, crease, open." (These creases divide the sheet into sixteen square inches, Fig-. 1.) "Fold the front edge again on the "back edge ; the left (short) edge on the right (short) edge." The paper is now ready for "work (Fig. 2). The point c indicates the center of the original large square sheet ; the dotted lines show the creases which serve as guides in cutting. In the subsequent figures (3-19), the heavy lines indicate the cuts. Thus, in Fig. 3, the left and lower margins are marked heavy. The paper is cut in these lines, and thus divided into four smaller squares of four square inches each, which may he arranged in a variety of new ways and mounted on a suitable sheet of paper or card-board, or used as rudimentary exer- cises in mensuration. In the following cut, Figs. 3-15 suggest a number of rectilinear cuts; Figs. 16-19, a few combinations of rectilinear -with cir- cular cuts; and Figs. 31-36, two cuts di- verging from the center, combined with other available cuts. CUTTING AND MOUNTING. 103 1O4 PRIMARY METHODS. Fig's. 20, 21, and 22 show three arrange- ments (or syntheses) made with the pieces resulting- from the cut (analysis) indicated in Fig. 4. For the arrangement of Fig. 23, two children who had sheets of different colors, have exchanged the squares. If one of these had a blue, and the other a yellow sheet, Fig. 23 would, then, represent a square CUTTING AND MOUNTING. 1OS made up of a "blue (yellow) cross, with a yellow (blue) square laid in eacli corner. In the figure, the difference in color is in- dicated "by the different directions of the hatching lines. The same device has "been used for a similar purpose in Figs. 25, 28, 30, 38, and 40. Fig. 24 is a re-arrangement of the pieces obtained from Fig. 11. Fig. 25 is a reconstruction of the square, after an exchange of forms among three children ; perhaps the large central square is "blue, the small triangles red, and the small squares in the corners yellow. The other two children will, then, have similar reconstructions with the same colors in different arrangements. Figs. 26, 27, and 28 are obtained from the cuts of Fig. 15 ; Figs. 29 and 30, from the cuts of Fig. 17; Figs. 37 and 38, from the cuts of Fig. 32 ; and Figs. 39 and 40, from the cuts of Fig. 36. It is scarcely needful to point out the ample opportunities which these exercises offer, for the varied and interesting use of geometric language, and for the cultivation of the esthetic sense with reference to 1O6 PRIMARY METHODS. form and color ; to speak of the certainty with which a thoughtful use of this charm- ing work will lead the child to important discoveries concerning the laws of the equality, similarity, and equivalence of fig- ures ; or to indicate how much these ex- ercises will aid the child in the work of drawing and coloring. After these explanations, very little need be added in elucidation of the subjoined * cut, suggesting the treatment of the cir- cular sheet. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. CUTTING AND MOUNTING. 107 28. Figs. 1, 2, and 3 represent the circular sheet, respectively, (1) laid oefore the pupil, (2) the lower half folded on the upper half, (3) the left half on the right half. In Figs. 4 to 27, the heavy lines indicate the cuts to he made on the quadrants. Figs. 28, 29, and 30, suggest arrangements of the pieces from the cuts of Fig. 5 ; Fig. 31 comes from Fig. 11 ; Figs. 32 and 33, from Fig. 16; Fig. 34, from Fig. 23. .For the use of the equilateral triangle (four-inch side), the follov-nng- cut will give the needed directions. 1O8 PRIMARY METHODS. CUTTING AND MOUNTING. 1O9 Figs. 1-6 indicate the successive steps in the preparation of the sheet for cutting 1 , a, 5, c mark the vertices of the triangle ; n, o, r, the middle of the sides ; ra, the cen- ter of the triangle. The dictation may take the following form : Lay the triangle before you, one side in front, from right to left (Fig. 1) ; lay the left slanting side (ca) on the right slanting side (cb), and crease the paper in the fold (en) (Fig. 2) ; lay the "back point (c) on the right point in front (ba) (Fig. 3) ; move the "back part of the paper to the right, and lay the right side from front to hack (Fig. 4) ; turn the paper over forward (Fig. 5) ; lay the left (slanting) side on the right (vertical) side (Fig. 6). The sheet is now ready ; the vertices of the triangle are all in the same point (cab), the middle points of the sides in orn, and the center of the sheet in m. Figs. 7-21 indicate simple rectilinear cuts, bisecting sides and yielding a great variety of geometrical forms for exercises similar to those suggested in the treatment of the square and circular sheets. Figs. 22-28 sug- gest a few curvilinear cuts. The subjoined foldings of the square 1 10 PRIMARY METHODS. sheet, given in kindergarten manuals, I have not found as available in primary work as those just treated, which have the advantage of greater simplicity in prepara- tion and in the forms resulting- from the cuts. i. CUTTING AND MOUNTING. Ill / V bd Fig. 1 shows the square folding sheet in position, " one corner in front and all sides slanting"; for Fig. 2 the left corner a has been placed on the right corner c, and the paper creased from d to 6; for Fig. 3 the back corner d has been placed on the front corner b, and the paper creased from m to ca ; for Fig. 4 , the corner or point d has been lifted from b and folded on ca, bringing the points cad together ; Fig. 5 shows the form turned over backward in order to bring the point b to the top ; for Fig. 6 the point b is folded on the points acb, and the paper creased from m to rost. This unites all the corners or angles of the square in the one point dacb, all the middle points of the sides of the square in the point rost, and exposes the center of the square at the point m. The folding sheet is now ready for such cuts as the teacher or pupil may devise, and all cuts will pro- duce symmetrical results. 112 PRIMARY METHODS. Fig. 7 shows the first fold of a second series starting from the position of Fig. 1 ; for Fig. 8, the left point (a) -was laid on the right (c), and the paper slightly creased at the point m ; for the fold of Fig. 9, the angle at m should span 120; in Fig. 10, the angle at m is 60; for Fig. 11, the paper is turned over from left to right; the por- tion "below the line ef is clipped off, and the form is ready for work. CHAPTER XIV. CARD-BOARD WORK. tHE cheapest and most satisfactory ma- terial for card-board work is a stout manilla board, which the printer or binder may cut of the required sizes, usually from six to twelve inches square. A pencil, a ruler, a pair of dividers, a sharp knife, some pieces of clean paper, a little mucilage, and a cutting-board, twelve inches square, com- plete the outfit. For the sake of securing a fair amount of skill in the treatment of the material and the use of the tools, the pupils 'may first practice, in a few lessons, the drawing and cutting out of given num- bers of figures of given forms and dimen- sions. Tasks like the following may be set : (1) Draw and cut out six squares, 2x2 inches. (2) Draw and cut out six equilateral tri- angles, each side two inches long. 114 PRIMARY METHODS. (3) Draw and cut out six hexagons, each side two inches long. (4) Draw and cut out six pentagons, from circles of two inches diameter. (5) Draw and cut out six two-inch squares, and "bisect them into two triangles. (6) Draw and cut out six equilateral tri- angles (two inches to the side), and trisect each into three triangles. The pieces resulting from the solutions of these tasks should toe re-arranged sym- metrically toy the pupils, and the resulting forms and angles stiidied. The following figures suggest some of these arrangements for the first two tasks : Figs. 1 and 2, for the first ; Figs. 3 and 4, for the second ; Fig. 5, for tooth comtoined. The forms thus ototainecl offer elements 3. CARD-BOARD WORK. 5. 115 4. for designs in drawing, and furnish themes for form lessons for all grades of primary and grammar schools. Next, in point of difficulty, would be a series of exercises in making hollow forms or "boxes." Some of these forms are sug- gested in the following figures, on the bases, successively, of the square, oblong, equilat- eral triangle, lozenge, and hexagon : 2. 116 PRIMARY METHODS. a w??"i The outline is first carefully drawn on the card-board ; this gives opportunity for practice in the use of instruments for geo- metrical drawing. The dotted lines (shown in a of each pair of figures) is then cut half through the paper with the help of a ruler and a sharp knife ; this enables the pupil subsequently to fold the sides of the boxes upward, as shown in & of each pair CARD-BOARD WORK. 117 of figures. Fig 1 . 2 represents the shape of small pieces of paper or " binding slips," cut of the required length for binding the faces at the edge of the boxes. The middle line represents a crease ; this is placed ex- actly over and along the edge, and each half of the ""binding slip" is firmly pasted on the two adjacent faces forming the edge. For the box (6) in Fig. 1, eight such slips of equal length would be required, four for the sides, and four for the bottom and sides ; for the box of Fig. 2 it would be necessary to prepare six slips of one length and two of doube that length, etc. The last step, the manufacture of solid forms, offers now little difficulty. Figs. 8, 9, and 10 show that by the addition of one surface the box forms 1, 3, and 4 are changed into solids, a cube and two square prisms. A little patience and care will overcome the slight difficulty of fastening the lid or sixth surface. Figs. 11, 12, and 13 show the way to the construction of the trian- gular prism, the square pyramid, and the regular tetrahedron. Figs. 14, 15, and 16 give the net-work for the regular octahe- dron, icosahedron, and dodecahedron. 118 PRIMARY METHODS. By substituting 1 , for the triangles 4 and 5 in Fig. 11, hexagons, octagons, or pentagons, and adding the required numbers of rect- angles (three, five, or two, respectively) the six-sided, eight-sided, or five-sided prisms will he obtained. By similar changes in Fig. 12, corresponding pyramids will result.; 8. 9. CARD-BOARD WORK. 119 It is evident, without further explana- tion, that these forms may "be used profit- ably in all grades of primary and grammar schools for a variety of exercises in rudi- mentary form lessons, mensuration, draw- ing, and solid geometry. It is evident, too, that "where circum- stances permit, the pupils may from the same material, with little difficulty, fashion models of pieces of furniture, buildings, and a number of objects for purposes of draw- ing; and that geometrical analysis and syn- thesis will find the material serviceable at every step. 120 PRIMARY METHODS. In all cases, it is desirable to give direc- tions in few words, setting a clearly denned task on the "basis of which the pupil may do the work independently. He should first assure himself of the correctness of his solution by carefully drawing the out- line of the cuts on a piece of paper, as in Figs. 8, 9, 10, 11, 12, 13, 14, 15, and 16. Subsequently the drawing should be trans- ferred to the manilla board, and the work may then proceed as indicated above. CHAPTER XV. FRACTION STRIPS. NY kind of strong, thin, -well calen- dered paper will answer the purposes of these strips. The children, however, will be "best pleased with strips cut from tinted cover paper or "engine-colored" paper. The paper may be bought by the quire or ream, and cut to order by the printer or binder. For use in numbers, as "fraction strips" proper, they should be one half or (later) one third inch. 'wide, and twenty or twenty- four inches long. For use in form lessons, as intertwining strips, they should be one inch wide. In forming the first definite notions of fractions, the child should have objects which he can actually break up into parts of one, in such a way that the parts can readily be re-arranged in the order of the original whole, or of new wholes dictated by the teacher or invented by the child. To 122 PRIMARY METHODS. this, in the limits of | to -j^, the fraction strips are well adapted. For a first exercise with halves, the fol- lowing is suggested : Place a (twenty-inch) strip before you from left to right. Lay the right half on the left half; crease the strip in the fold ; open the strip (Fig. 1) ; carefully tear the strip in the crease. Lay one half from left to right ; lay the other half parallel to the first one inch (two, three, etc., inches) in front ("behind) it. Lay one half from front to back ; lay the other half parallel to the first one inch (two, three, etc., inches) to the right (left) of the first ; and so on through a variety of relative positions, as indicated in Figs. 2 to 8. In each of these re-arrangements the child recognizes a new unit in which the two halves unite more or less completely into a whole. The child's attention may "be di- rected to this cautiously and without ur- gency ; in no case, however, should the growth of the ideas one whole is two halves, two halves are one whole be inter- rupted by premature formal statements or by abstract philosophizing thereon. FRACTION STRIPS. 123 J Oj b.l 13. 16. PI LJ m ~j _J PRIMARY METHODS 24. 25. ill For the exercise with, one fourth, sug- gested in Figs. 9-14, attention should "be paid to the colors of the strips : the three strips, a, &, c, in Fig. 9, should "be of dif- ferent colors. These color contrasts will make the fraction contrasts more impress- ive. In Figs. 15, 16, and 17, exercises for still further contrasting halves and fourths are suggested. In Fig. 15, the equivalence of FRACTION STRIPS. 125 one (red) half and two (blue) fourths is quite prominent. Fig. 18 contrasts thirds, halves, and fourths, and Figs. 19-25 suggest re-arrange- ments of thirds. Dictations are quite helpful here in bring- ing numerical relations into prominence. Thus, for Fig. 11: "Lay two fourths par- allel from right to left, five inches apart ; lay the other two fourths between their ends, from front to back, five inches apart, making a square." This dictation contains and conveys the formula f + f = 1. The same formula lies in Fig. 14. In Figs. 12 and 13, lies the formula J + i = l. Figs. 20-25, by the similar positions (par- allel or diverging) of two thirds and the clear contrast in position of the remaining third, say clearly f + % = 1. These hints will suffice to show the edu- cational value of the fraction strip, and to enable the reader to use it effectively, in the limits indicated above, in the develop- ment of clear notions concerning the re- lations of fractions to the whole and to each other. The wide (one inch) fraction strip, pre- 126 PRIMARY METHODS. pared as an intertwining- strip, will be found very serviceable in the development of ideas of position and form. More particularly for the former, I have found it invaluable. It is prepared as follows : "Lay the strip before you from right to left (Fig. a), fold the upper third, lengthwise, on the remainder of the strip, and crease carefully in the fold (Fig. 6) ; fold the front third length- wise on the upper third and crease care- fully in the fold (Fig. c). It may be necessary to assist a number of children the first time in making these somewhat difficult long creases ; but they will find the work easy and enjoyable after this. For exercises in positions, the children, by dictation, hold the strips well stretched between their hands, " horizontal, vertical, slanting (to the right, left, back, etc.), par- allel to the front edge of the desk, perpen- FRACTION STRIPS. 127 dicular and oblique to it (or to some otlier line or plane), from North to South, from East to West," etc. The interest of exercises in relative positions will be still further increased, if children are grouped in sets of two, one of whom has a red and the part- ner a yellow strip (other color contrasts will answer the same purpose). The red (or yellow) strips are then, by dictation, held parallel, perpendicular, oblique to the yellow (or red) strips, at a variety of dis- tances, points, and angles. The subjoined figures (1-7) may suffice to indicate how the intertwining strips may be used for exercises with angles (right, ob- tuse, and acute), as well as for the making of given geometrical forms. At the points, marked m, one end of the strip is inserted between the folds of the other end. Thus Fig. 1 is a right angle which, by dictation, may be held in a variety of posi- tions, or may be united with the right angles of neighbors into a variety of forms. Or, it may be used to represent a carpen- ter's square, the letter L, a tent, etc. Fig. 2 is a square, a picture-frame, a window, an inclosed well, etc. 123 PRIMARY METHODS. 7/1 CHAPTER XVI. SPLINTS OR STICKS. QUARE match, splints, cut of the re- quired lengths, offer a remarkably cheap and serviceable material for form and number studies, as well as for drawing- ex- ercises. They can be had from dealers in lengths of one to five inches. Where this is impracticable, pieces of straw, strips of card-board, or tooth-picks, appropriately cut, will answer the purpose quite -well. In a short chapter, it is possible only to indicate the wealth and flexibility of this material in school work. For this purpose, a few typical lessons have been sketched below. The first series is devoted to lessons in the rudiments of form. 3.. ISO PRIMARY METHODS. h.H (. Jnl_ 9. 10. 11. 13. 12. SPLINTS OR STICKS. 131 15. 16. Figs. 1-4 indicate exercises for position, actual or symbolic (as in drawing- from objects). Each, child receives a supply (9-12) of splints, four inches long, with which he works, at first "by dictation, and afterward independently, as follows (Fig. 1) : "Lay six sticks near your right hand, and six sticks near your left hand. With the right hand, lay one stick (eight inches behind the edge of the desk) from right to left. With the left hand lay one stick one inch to the left of this, from front to back ; 132 PRIMARY METHODS. with, the right hand, lay a third stick from front to "back, one inch to the right of the first stick." Questions may now be asked, to fix the contrasted ideas : " How many sticks from front to hack ? from right to left ? Where is the stick from right to left ? Where are the sticks from front to back ? " The children may then be per- mitted to make a number of similar " forms " or "drawings" with the remaining sticks; and, subsequently, to imitate these on the slate, the paper, or the blackboard. Or we may dictate as follows (Fig. 2) : " I see " (the children laying on the desks the forms described) " nine inches from the desk front, near the middle of the desk, a straight line from right to left ; right and left of this I see two straight lines slant- ing inward and backward." Or (Fig. 4), using the symbolic terms, horizontal for "from right to left," vertical for "from front to back," etc. we may say: "Draw with one stick, near the middle of the desk, a vertical line ; draw right and left of the vertical two horizontal lines ; draw right and left of these lines slanting inward at the top." SPLINTS OR STICKS. 133 Similarly, Fig. 5 teach.es parallelism in different positions ; Fig-. 6, the perpendicular relation in a variety of positions ; Fig. 7, the right angle; Fig. 8 contrasts right ob- tuse and acute angles; Fig. 12 right arid acute angles; Figs. 9, 10, 11, 13 present studies of the square in varieties of shape and position ; Fig. 14 contrasts the square and lozenge; Fig. 15, the square and oblong rectangle ; Fig. 16, the lozenge and oblong rectangle. It is scarcely necessary to add that in these and similar ways the child may learn to control the form relations of all the various polygons, and to apply the knowl- edge obtained in drawing and other pur- suits. This is partly indicated in the fol- lowing nine " star forms : " 134 PRIMARY METHODS. 30, 21. SPLINTS OR STICKS. 135 Figs. 17, 18, and 19 apply the trapezium; Fig-. 20, the lozenge ; Fig. 21, the rhomboid, square, and trapezium; Fig. 22, the trape- zium, square, rhomboid, lozenge, and dodec- agon ; Fig. 23, trapezium with hexagon and dodecagon, etc. Star-forms and other symmetrical forms may thus be "drawn with the splints," and, subsequently, on slate or paper. Similar devices will help the children to gain self-confidence even in object-drawing, inasmuch as even moderate skill will find it easy to lay or "draw with splints" skel- eton sketches of houses, barns, fences, trees, pieces of furniture, etc. My " Primary Helps" contains an abundance of hints in this direction. Teachers will find match splints very useful, too, in the rudiments of arithmetic. The analysis and synthesis of numbers, ad- dition, subtraction, multiplication, division, and part-taking are performed readily in an endless variety of interesting exercises with the help of splints. For such exer- cises I have found the two-inch splint most convenient, although others may be used profitably enough. The following cut sug- 136 PRIMARY METHODS. gests a simple mode of using these for ex- ercises in the limits of 1-10. i. 2. 3. In the first series (Series 1) the number three comes to the child, successively, as three ones, two and one, one and two, one three. In the first figure of the series, the three sticks lie in the same direction at great one-inch intervals ; to the child nothing unites them, they are distinctly three ones. In the second figure of the series, the upper two sticks are more closely associated as a pair or a two "by their greater closeness. In SPLINTS OR STICKS. 137 the tliird figure, this association of the two is strengthened or intensified "by the addi- tional uniting contrast of position. In the last figure, the three sticks are closely as- sociated in a three, "one three," by the tri- angle they inclose. . Similar considerations will reveal in Series 2 the following number-readings for the little learner : 4 (1) = 4, 2 (2) = 4, 1 + 3 = 4, 2 + 2 = 4, 2 + 2 (1) = 4, 4 = 4. In Series 3 we have : 6 (1), 2 (3), 3 (2), 2 (2) + 1 (2), and 3 (2) more closely associated into a six with, the help of the inclosed triangle. In Series 4 we have : 7 (1), 2 (2) + 1 (3), or (with the help of previous knowledge), 4 + 3, 2 (2) + 1 (3), 2(3) + 1, 2(1) + 5, 4+3. By making bundles of sticks, ten in each, similar exercises may be contrived for work within even higher limits (1-100). A very profitable combination of form and number exercises is illustrated in the following figures : v A 138 PRIMARY METHODS. 19. 20. N/ A Figs. 1-5 show the four sticks, numeric- ally, as 4, 2 (2), 3 + 1, 3 + 1, 3 + 1. As sug- gestions for form concepts and drawing exercises, they represent to the child, suc- cessively, a "box or picture, a house or shed, a trowel or spade, a chair, a table. The remainder of the table treats the numbers five, six, etc., in a similar way. CHAPTER XVII. STICKS AND PEAS. 'OR this occupation the ordinary match splints may be sharpened at the ends, or a special kind of sticks about the thick- ness of stout broom-straws may be procured from a dealer in kindergarten goods. * Dried peas, soaked over night in water, serve as a cement to bind these sticks together. If the forms are to be made more permanent, the ends of the sticks may be dipped in mucilage before insertion into the peas. For advanced children, small pellets of bees-wax about the size of peas answer an excellent purpose. It will be seen from an examination of the illustrations given below that the peas represent points, and the lines mark the distances between them ; also that the sur- faces inclosed by these lines as well as the * These thin sticks come five and ten inches long, and may be cut with scissors of any required length within these limits. 14O PRIMARY METHODS. solids inclosed "by the surfaces represent comparatively pure form-concepts quite free from any admixture of materialness. The cuts show the growth and analysis of the square, the growth and study of the cube, the bisection and trisection of the right angle, 'the test of the latter with the help of the equilateral triangle and hexagon, and a few typical regular solids or crystal forms. i. 2. QC O O Q STICKS AND PEAS. 10. 11. 141 a 19. I 12 ' h 14. 'V* -01) 142 PRIMARY METHODS. For Fig. 1 a straight line the pupils hold the stick "between thumb and fore- finger of the right hand, quite near the left end of the stick, and insert the point of the latter in the pea, held "between thumb and forefinger of the left hand. The same is done on the right end of the stick, the latter being held between the thumb and forefinger of the left hand, and the pea in the right hand. For Fig. 2 two lines, forming a right angle the stick ab is held -with the left hand, the pea & being specially supported, STICKS AND PEAS. 143 and the stick ~bc is inserted with the right hand. For Fig. 3, the hands change office. For Fig. 4 the square Fig. 2 is laid to the right and in front of Fig. 3 ; the free end c of Fig. 2 is inserted in pea c of Fig. 3, and the free end a of Fig. 3 is inserted in the pea a of Fig. 2. Fig. 5 shows the first step in the anal- ysis of the square, the "drawing" of a diagonal ; Fig. 6, the separation toy the diagonal into two equal, right, isosceles triangles. In Fig. 7 the two diagonals are drawn. (For this purpose four sticks of the length of half. a diagonal may be used, all inserted in the pea at the center; or a pea may he slipped to the middle of the first whole diagonal, and the second diagonal may "be made from two halves.) In Fig. 7 the square is "broken up toy the two diagonals into four equal, right, isosceles triangles. Fig. 9 indicates a first step for analyzing the square toy two diameters. For the cutoe, two squares are made (abed in Fig. 10) ; in the two ends of one side of each of these, perpendiculars (de and cf) are inserted ; one of these forms is then in- verted, "bringing the square to the top, and 144 PRIMARY METHODS. the perpendiculars forward. ; this is lifted over and in front of trie other, and the cube (Fig 1 . 12) completed. Fig. 13 shows the insertion of an axis of the cube. The diagonals in the upper and lower faces simply furnish points of support for the axis. Figs. 14-17 illustrate exercises for bi- secting the right angle. In Fig. 14, the right angle is established on both sides of the stick dc ; in Fig. 15, it is bisected on the right by stick df, on the left by stick de\ in Fig. 16, the removal of the stick dc reveals the right angle formed by the bi- secting sticks df and de; in Fig. IT, the removal of df brings out the contrast be- tween the miter (45) and the sum of the right angle and miter (135). Fig. 18 shows the trisection of the right angle on the two sides of the perpendicular dc. In Fig. 19, the removal of alternate sticks gives three angles of 60. In Fig. 20, this is verified by the insertion of sticks of the same length between the outer or free ends of the radiating sticks. If the trisection is correct, this -will give three equilateral triangles or one half of the hex- STICKS AND PEAS. 143 agon, completed in Fig. 21. Similarly treated, Fig. 15 will yield an octagon, and Fig. 18 a dodecagon. The "building of the octahedron (Fig. 22) begins with the laying of the square. On each side of this four equilateral triangles are erected, meeting in the common points o and r. Fig. 23 solves an interesting "puzzle," the "building of the octahedron in the cube ; the diagonals laid across the sides of the square merely hold the peas or "wax pellets that represent the corners of the octahedron. Fig. 24 shows the tetrahedron, "bounded "by four equilateral triangles ; the six-sided pyramid and the six-sided prism. Similarly other forms may "be treated. These skeleton models or outlines of forms may he used very profitably in the study of form and in drawing, more particularly in all kinds of perspective drawing. In the study of the propositions and problems of plane and solid geometry, as well as in the study of the rudiments of crystallography, the sticks and wax pellets furnish, even in advanced classes, a more convenient and more efficient aid than the drawing surface of paper or blackboard. 146 PRIMARY METHODS. Whenever dictation is used in these ex- ercises, it is essential for orderly success that the children should arrange their material similarly (e. g. the sticks to be used on the right side of the table or desk, and the peas in a neat paper "box or other receptacle on the left side ) ; and that all the -work be done strictly in accordance with the teacher's directions, slowly and deliberately. On the other 'hand, full free- dom in handling- the material should be granted in independent work. CHAPTER XVIII. LENTILS OR DOTS. MONG us the lentil seed is rarely planted. It is quite largely imported, however, by German grocers, who sell it at very reasonable rates. Froebel found in this flat, smooth, circular seed a most serv- iceable representative of the point or posi- tion, as I shall show directly. Other seeds have been proposed such as beans, peas, wheat, barley, etc. but had to be aban- doned as inadequate. Mrs. Hailmann, there- fore, contrived an artificial lentil seed, a circular disk, one fourth inch in diameter, cut from wood. These are furnished white, black, and in the rainbow colors, by Mr. Bradley, at low rates. For primary work, too, he furnishes similar disks cut from gummed paper, which are of great value in form and color exercises, and cost very little. The subjoined plate will suggest the manner in which this most flexible material 148 PRIMARY METHODS. may be used in lessons on form. For this purpose I have cliosen a somewhat system- atic treatment of the square, leaving- the reader to devise similar series of exercises for other forms. 1, 3. 3. 4. O O ooooo ooooo o o o ooo o ooooooooo o ooo o O O ooooo ooooo 5, 6. 7. 8. O O O 006 ooooo ooooo O O O O O o o O O O O O O o o o o ooooo O O ooo O O ooooooooo ooooo O O 000 o o O O O ooo o o o o ooooo O O ooo O O O O O ooo ooooo ooooo 9. 10. 11. 12. 00 O O O o o o o O O o o o o O O O O O ooooo o o o o ooooo O o o o o ooooo ooo o ooo O ooooo ooooo O O O ooooo O O o o O O O O O O O o o o o 13. 14. 15. 16. O O O boo ooooooooo .0 O O O O o o o O O O O ooooo o ooo ooo O o O O O O ooooo o o O O o O O O O o o o o e o o- . O G O o O O O 6 OOOOOOOOO O O ooo ooo o LENTILS OR DOTS. 149 ur. is. 19. 20. -O OOO O^"^ ^O o o ooo ooo O O O f> OQ O^o o>o o o_ o o OOO Jr o o o OO o O/-i OQO .-.Q o e / c e o g o o o o Si o ~Q 0" OQO o o ^r,OO ^0 O on o QO OXD o o o o OOO OOO ^j o ~G ooo ooo Q o OOOO Q o 21. 22. 23. o o o ooooo ooo ooo o O o o o ooo ooooooooo ooooooooo o o o ooo o ooo Q o ooooo ooo ooo CL _0 24. 25. 26. o ooooo o o O OOOOO ooooo o oo o oo ooo o ooooo o ooo oo o oo oo o oo ooo o ooo o ooooooooo o o O o o ooooo o o oo o oo o o ooo o ooo ooo ooooo ooo o ooooo ooooo o o o 27. 28. 29. o o oo n o oo ooo o u oo ooooo O ooooo o o o o o o o oo o OQ o ooo o o o o o o o o o o o o ooooooooo o o o ooooo o n o o oo o o o c o o > o 8o oo o ooooo o o o o ISO PRIMARY METHODS. 80. o o o^oo o o o ooo o o o o o ,0 ,000 oo o 9P o o o 81. o o o o o o o o o o o o o o o o o o o o o o o o o o o o 00 o o o o o o o o ,0 O O o go '0 O O ^0 o o o o o o o o OO o 33. 34. ooo ooo ooooooooooo ooooo o On O O ^O O OOOOOOO O W rt ^.^ ^U Q Q o o ooooo o o ooo o o ooo o o o boo ooc5 o o o o o o 35. 36. OOOOO OOOOO OOOn Q o o o o o cr o o^t OOOOOO On OOO o o o o ooooooooooo o o ooooooooooo O O O O O O O^ nn O 00 O O 0,0 O ooooo ooooo ^^ o ooo o o o o o O .-.OOOr, O CkOX) ^ o " ooo 87. 000 o o O oo o Oo ooO o0 / 00 o o o o o ooo o o o o o o o ooo o o o o n ooooo ooooo ox Xo O 00 OOO OO ooo~ o o ooo o o o o nO O OO u o 39. o 00. OQ 00' CQ0 O O o o o o o o o 00^0 O^ CO o o oo 40. ooooooo o o o o o o o o o ooooo o o o o o o o o o o ooooo o o o o o o o o o ooooooo o o 41. o OO o o . o o / 3 O O 1 o o s O O 'ocPA o o O O o o 00, o O LENTILS OR DOTS. 151 42. 43. 44. O^OA ^O^ 00 00 00 00 o o oo o c> _ o o On 00 oO OQOOO O O OOOQp O O O 6o o o n^> cPoo o o ooSb oooo O O O U O O O O O 00 O 00 o 0f0q-.o u o o o c oq-,o o oo o oo o Rpo o xox o o ooooo o .0 OT50 U O O 00d0 O 00 o CO o O o o o U o o o n o o o o o oo o oo 0,0 ,0 O OoO Q O O C OOOO O .G O ^O ODOOO O O OOOOO O O O oo oooo oo O^o OO^OOOn 00 O O A Cs ?5 O i c^ 1 5= a 9 e ? r3 CB 5s o i. ? s o -- ceive the inexperienced, while adding materially to the value of the work as a reliable index to the character and institutions, as well as the historv of the Roman people. 35 THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. HISTORY Continued. Hanna's Bible History. The only compendium of Bible narrative which affords a connected and chronological view of the important events there recorded, divested of all superfluous detail Summary of History; American, French, and English. A well-proportioned outline of leading events, condensing the substance of the more extensive text-books in common use into a series of statements so brief, that every word may be committed to memory, and yet so comprehensive that it presents an accurate though general view of the whole continuous life of nations. Marsh's Ecclesiastical History. Affording the History of the Church in all ages, with accounts of the pagan world during the biblical periods, and the character, rise, and progress of all religions, as well as the various sects of the worshippers of Christ. The work is entirely non-sectarian, though strictly catholic. A separate volume contains carefully prepared questions for class use. Mill's History of the Ancient Hebrews. With valuable Chronological Charts, prepared by Professor Edwards of N. Y. This is a succinct account of the chosen people of God to the time of the destruction of Jerusalem. Complete in one volume. Topical History Chart Book. By Miss Ida P. Whitcomb. To be used in connection with any History, Ancient or Modern, instead of the ordinary blank book for summary. It embodies the names of contemporary rulers from the earliest to the present time, with blanks under each, in which the pupil may write the summary of the life of the ruler. Oilman's First Steps in General History. A "suggestive outline" of rare compactness. Each country is treated by itself, and the United States receive special attention. Frequent maps, contemporary events in tables, references to standard works for fuller details, and a minute Index constitute the " Illustrative Apparatus." From no other work that we know of can so succinct a view of the world's history be obtained. Considering the necessary limitation of space, the style is surprisingly vivid, and at times even ornate. In all respects a charming, though not the less practical, text-book. Baker's Brief History of Texas. Dimitry's History of Louisana. Alison's Napoleon First. The history of Europe from 1788 to 1815. By Archibald Alison. Abridged by Edward S. Gould. One vol., 8vo, with appendix, questions, and maps. 550 pages. Lord's Points of. History. The salient points in the history of the world arranged catechetically for class use or for review and examination of teacher or pupil. By John Lord, LL.D. 12mo, 300 pages. Carrington's Battle Maps and Charts of the American Revolution. Topographical Maps and Chronological Charts of every battle, with 3 steel portrait* of Washington. 8vo, cloth. Condit's History of the English Bible. For theological and historical students this book has an intrinsic value. It gives the history of all the English translations down to the present time, together with a careful review of their influence upon English literature and language. 36 THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. DRAWING. BARNES'S POPULAR DRAWING SERIES. Based upon the experience of the most successful teachers of drawing in the United States. The Primary Course, consisting of a manual, ten cards, and three primary AtMTing hooks, A, B, and C. Intermediate Course. Four numbers aiid a manual. Advanced Course. Four numbers and a manual Instrumental Course. Four numbers and a manual. The Intermediate, Advanced, and Instrumental Courses are furnished either in book or card form at the same prices. The books contain the usual blanks, with the unusual advantage of opening from the pupil, placing the copy directly in front and above the blank, thus occupying but little desk-room. The cards are in the end more econom- ical than the books, if used in connection with the patent blank folios that accompany this series. The card* are arranged to be bound (or tied) in the folios and removed at pleasure. The pupil at the end of each number has a complete book, containing only his own work, while the copies are preserved and inserted in another folio ready for use in the next class. Patent Blank Folios. No. 1. Adapted to Intermediate Course. No. 2. Adapted to Advanced and Instrumental Courses. ADVANTAGES OF THIS SERIES. The Plan and Arrangement. The examples are so arranged that teachers and pupils can see, at a glance, how they are to be treated and where they are to be copied. In this system, copying and designing do not receive all the attention. The plan is broader in its aims, dealing with drawing as a branch of common-school instruction, and giving it a wide educational value. Correct Methods. In this system the pupil is led to rely upon himself, and not upon delusive mechanical aids, as printed guide-marks, &c. One of the principal objects of any good course in freehand drawing is to educate the eye to estimate location, form, and size. A system which weakens the motive or re- moves the necessity of thinking is false in theory and ruinous in practice. The object should be to educate, not cram ; to develop the intelligence, not teach tricks. Artistic Effect. The beauty of the examples is not destroyed by crowding the pages with useless and badly printed text The Manuals contain all necessary instruction. Stages of Development. Many of the examples are accompanied by diagrams, showing the dilt'erent stages of development. Lithographed Examples. The examples are printed in imitation of pencil drawing (not in hard, black lines) that the pupil's work may resemble them. One Term's Work. Each book contains what can be accomplished in an average term, and no more. Thus a pupil finishes one book before beginning another. Quality not Quantity. Success in drawing depends upon the amount of thought exercised by the pupil, and not upon the large njimber of examples drawn. Designing. Elementary design is more skilfully taught in this system than by any other. In addition to the instruction given in the books, the pupil will find printed on the insides of the covers a variety of beautiful patterns. Enlargement and Reduction. The practice of enlarging and reducing from copies is not commenced until the pupil is well advanced in the course and therefore better able to cope with this difficult feature in drawing. Natural Forms. This is the only course that gives at convenient intervals easy and progressive exercises in the drawing of natural forms. Economy. By the patent binding described above, the copies need not be thrown aside when a book is filled out, but are preserved in perfect condition for future use. The blank books, only, will have to be purchased after the first introduction, thus effect- ing a saving of more than half in the usual cost of drawing-books. Manuals for Teachers. The Manuals accompanying this series contain practical instructions for conducting drawing in the class-room, with definite directions for draw- ing each of the examples in the books, instructions for designing, model and object drawing, drawing from natural forms, &c. 38 THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. FAMILIAR SCIENCE. Norton & Porter's First Book of Science. Sets firth the principles of Natural Philosophy, Astronomy, Chemistry, Physiology, and Geology, on the catechetical plan for primary classes an I beginners. Chambers's Treasury of Knowledge. Progressive lessons upon first, common things which lie most immediately around us, and first attract the attention of the young mind ; second, common objects from the, (mineral, animal, and vegetable kingdoms, manufactured articles, and miscellaneous substances ; third, a systematic view of nature under the various sciences. May be used as a reader or text-book. Monteith's Easy Lessons in Popular Science. This book combines within its covers more attractive features for the study of science by children than any other book published. It is a reading book, spelling book, com- position book, drawing book, geography, history, book on botany, zoology, agricul- ture, manufactures, commerce, and natural philosophy. All these subjects are presented in a simple and effective style, such as would be adopted by a good teacher on an excursion with a class. The class are supposed to be taking excursions, \7ith the help of a large pictorial chart of geography, which can be suspended before them in the school-room. A key of the chart is inserted in every copy of the book. With tlm book the science of common or familiar things car; be taught to beginners. NATURAL PHILOSOPHY. Norton's First Book in Natural Philosophy. Peck's Elements of Mechanics. A suitable introduction to Bartlett's higher treatises on Mechanical Philosophy, and adequate in itself for a complete academical course. Bartlett's Analytical Mechanics. Bartlett's Acoustics and Optics. A complete system of Collegiate Philosophy, by Prof. W. H. C. Bartlett, of West Point Military Academy. Steele's Physics. Peck's Ganot. GEOLOGY. Page's Elements of Geology. A volume of Chambers's Educational Course. Practical, simple, and eminently calculated to make the study interesting. Steele's Geology. CHEMISTRY. Porter's First Book of Chemistry. Porter's Principles of Chemistry. The above are widely known as the productions of one of the most eminent scientific men of America. The extreme simplicity in the method of presenting the science, while exhaustively treated, has excited universal commendation. Gregory's Chemistry (Organic and Inorganic). 2 vols. The science exhaustively treated. For colleges and medical students. Steele's Chemistry. 47 THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. NATURAL SCIENCE Continued. ASTRONOMY. Peck's Popular Astronomy. By Win. G. Peck, Ph.D., LL.D., Professor of Mathematics, Mechanics, and Astron- omy in Columbia College. 12mo. Cloth. 330 pages. Professor Peck has here produced a scientific work in brief form for colleges, acade- mies, and high schools. Teachers who do not want an elementary work like Steele's . Astronomy, for instance will find what they want in this book. Its discussion of the Stars, Solar System, Earth, Moon, Sun and Planets, Eclipses, Tides, Calendars, Planets and Satellites. Comets and Meteors, &c., is full and satisfactory. The illustrations are numerous and very carefully engraved, so the student can gain an accurate comprehen- sion of the things represented. Professor Peck is wonderfully clear and concise in his style of writing, and there is nothing redundant or obscure in this work. It is intended for popular as well as class nse, and accordingly avoids too great attention to mathe- matical processes, which are introduced in smaller type than the regular text. For higher schools this astronomy is undoubtedly the best text-book yet published. Willard's School Astronomy. By means of clear and attractive illustrations, addressing the eye in many cases by analogies, careful definitions of all necessary technical terms, a careful avoidance of verbiage and unimportant matter, particular attention to analysis, and a general adop- tion of the simplest methods, Mrs. Willard has made the best and most attractive elementary Astronomy extant. Mclntyre's Astronomy and the Globes. A complete treatise for intermediate classes. Highly approved. Bartlett's Spherical Astronomy. The West Point Course, for advanced classes, with applications to the current wants of Navigation, Geography, and Chronology. NATURAL HISTORY. Carll's Child's Book of Natural History. Illustrating the animal, vegetable, and mineral kingdoms, with application to the arts. For beginners. Beautifully and copiously illustrated. Anatomical Technology. Wilder & Gage. As applied to the domestic cat. For the use of students of medicine. ZOOLOGY. Chambers's Elements of Zoology. A complete and comprehensive system of Zoology, adapted for academic instruction, presenting a systematic view of the animal kingdom as a portion of external nature. ROADS AND RAILROADS. Gillespie's Roads and Railroads. Tenth Edition. Edited by Cady Staley, A.M., C.E. 464 pages. 12mo. Cloth. This book has long been and still is the standard manual of the principles and prac- tice of Road-making, comprising the location, construction, and Improvement of roads ii-cimmiin, macadam, paved, plank, &c.) and railroads. It was compiled by Win. Gillespie, LL.D., C.E., of Union College. 49 THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. AESTHETICS. Huntington's Manual of the Fine Arts., A view of the rise and progress of art in different countries, a brief account of the most eminent masters of art, and an analysis of the principles ot art. It is complete in itself, or may precede to advantage the critical work of Lord Kames. Boyd's Kames's Elements of Criticism. The best edition of this standard work ; without the study of which none may be considered proficient in the science of the perceptions. No other study can be pursued with so marked an effect upon the taste and refinement of the pupil ELOCUTION. Watson's Practical Elocution. A scientific presentment of accepted principles of elocutionary drill, with black- board diagrams and full collection of examples for class drill Cloth. 90 pages, liimo. Taverner Graham's Reasonable Elocution. Based upon the belief that true elocution is the right interpretation of thought, and guiding the student to an intelligent appreciation, instead of a merely mechanical knowledge, of its rules. Zachos's Analytic Elocution. All departments of elocution such as the analysis of the voice and the sentence, phonology, rhythm, expression, gesture, &c. are here arranged for instruction in classes, illustrated by copious examples. SPEAKERS. Northend's Little Orator. Northend's Child's Speaker. Two little works of the same grade but different selections, containing simple and attractive pieces for children under twelve years of age. Northend's Young Declaimer. Northend's National Orator. Two volumes of prose, poetry, and dialogue, adapted to intermediate and grammar classes respectively. Northend's Entertaining Dialogues. Extracts eminently adapted to cultivate the dramatic faculties, at well as entertain. Oakey's Dialogues and Conversations. For school exercises and exhibitions, combining useful instruction. James's Southern Selections, for Reading and Oratory Embracing exclusively Southern literature. Swett's Common School Speaker. Raymond's Patriotic Speaker. A superb compilation of modern eloquence and poetry, with original dramatic exercis. Ksariy every eminent modern orator is represented. 52 THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. MIND. Mohan's Intellectual Philosophy. The subject exhaustively considered. The author has evinced learning, candor, and independent thinking. Mahan's Science of Logic. A profound analysis of the laws of thought. The system possesses the merit of being intelligible and self-consistent. In addition to the author's carefully elaborated views, it embraces results attained by the ablest minds of Great Britain, Germany, and France, in this department Boyd's Elements of Logic. A systematic and philosophic condensation of the subject, fortified with additions from Watts, Abercrombie, Whately, &c. Watts on the Mind. Edited by Stephen N. Fellows. The " Improvement of the Mind," by Isaac Watts, is designed as a guide for the attainment of useful knowledge. As a text-book it is unparalleled ; and the discipline it affords cannot be too highly esteemed by the educator. MORALS. Peabody's Moral Philosophy. A short course, by the Professor of Christian Morals, Harvard University, for the Freshman class and for high schools. Butler's Analogy. Hobart's Analysis. Edited by Prof. Charles E. West, of Brooklyn Heights Seminary. 228 pages. 16mo. Cloth. Alden's Text-Book of Ethics. For young pupils. To aid in systematizing the ethical teachings of the Bible, and point out the coincidences between the instructions of the sacred volume and the sound conclusions of reason. Smith's Elements of Moral Philosophy. 140 pages. 12mo. Cloth. By Win. Austin Smith, A.M., Ph.D., Professor of Moral Philosophy in the Columbia (Tenn.) Athenseum. This is an excellent book for the use of academies and schools. It is prepared to meet the wants of a much larger public than has heretofore been reached by works of this class. The subject is presented in clear and simple language, and will be found adapted to the comprehension of young pupils, at a time when they particularly need an insight into the laws which govern the moral world. Janet's Elements of Morals. By Paul Janet. Translated by Mrs. Prof. Corson, of Cornell University. The Elements of Morals is one of a series of works chiefly devoted to Ethics, and treats of practical, rather than theoretical morality. Mr. Janet is too well known that it be necessary to call attention to his excellence as a moral writer, and it will be sufficient to say that what particularly recommends the Elements of Morals to educators and students in general is the admirable adap- tation of the book to college and school purposes. Besides the systematic and scholarly arrangement of its parts, it contains series of examples and illustrations anecdotic, historical gathered with rare impartiality from both ancient and modern writers, and which impart a peculiar life and interest to the subject. Another featnra of the work is its sound religious basis. Mr. Janet is above all a religious moralist THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. FRENCH. Worman's First French Book. On same plan as the German and Spanish. The scholar reads and speaks from the first hour understandingly and accurately. 83 pages. Worman's Second French Book. Continues the work of the First Book, and is a valuable Elementary French Reader. 96 pages. Worman's Le Questionnaire. Exercises on the First French Book. 98 pages. Cloth. Worman's Grammaire Frangaise. Written in simple French, but based on English analogy. It therefore dwells upon the Essentials, especially those which point out the variations of the French from the student's vernacular. 184 pp. Worman's Teacher's Hand-Book. Or Key to the Grammaire Francaise. Worman's French Echo. This is not a mass of meaningless and parrot-like phrases thrown together for a tourist's use, to bewilder him when in the presence of a Frenchman. The " Echo de Paris " is a strictly progressive conversational book, beginning with sim- ple phrases and leading by frequent repetition to a mastery of the idioms and of the every-day language used in business, on travel, at a hotel, in the chit-chat of society. It presupposes an elementary knowledge of the language, such as may be acquired from the First French Book by Professor Worman, and furnishes a runniny French text, allowing the learner of course to find the meaning of the words (in the appended Vocabulary), and forcing him, by the absence of English in the text, to think in French. ' CHER MONSIEUR WORMAN, Vous me demandez mem opinion sur votre " Echo de Paris" et quel usage j'en fais. Je ne saurais mieux vous repondre qu'en repro- duisant une lettre que j'ecrivais derniere- mcnt a un collegue qui etait, me disait-il, " bien fatigue d HRgirl A little girl. Here is a girl. Here is alittle girl. THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. COPY-BOOKS, &c. Barnes's New National System of Penmanship. This new series of Copy-books, iii six numbers, is based upon many years' experience. The publishers think these books are the best ever made in this country, for the following reasons : 1. They contain a. thoroughly Practical System of Penmanship, which, if once well learned in school, will not prove too difficult of execution for business purposes or private correspondence. 2. They have been prepared in the most careful manner, without regard to expense by experts in the business ; and the copies are infinitely superior to all others in the mar.cet, not only in beauty and grace of style, but also in accuracy of outline and proportion. 3. Pupils who use these books as directed will write in a free, graceful, rapid man ner, and not in the slow, cramped, and crabbed style so common in many schools. 4. The classification of capitals is wonderfully simplified. Eleven letters are formed on one general plan ; ten on another, and the rest on a third. The number of elements is reduced to h've. Other systems have from seven to thirteen. 5. The gradation is perfectly simple. Only familiar words, and those easy of forma- tion, are used, and not such unusual enigmatical words as "zeugma," " unjue.sne " " xylus," " tenafly," ''quinque," " minetic," " xuthus," and the like, that have long been the bane of both teachers and pupils. 6. No time has been lost by placing before the pupil disconnected words, unmean- ing phrases, and stale aphorisms ; but the complete sentence has been used early in the series, that the pupil, while writing, might be expressing thoughts, rather than simply copying abstract words. 7. The business forms are elaborately engraved on steel and printed on patent safety- tint paper. They are exactly like the checks, notes, drafts, receipts, etc., used in business, and the learning how to fill them out will serve as an admirable introduction to the counting-room. 8. The whole series for ungraded schools is comprised in six books ; but for the benefit of the large, graded schools in both city and country, there are six additional books, of smaller size, to meet the demands of a still closer gradation. Ward's Letter-Writing and Business Forms. For Schools and Academies. In Four Numbers. No. 1, Letters and Bills. No. 2, Letters, Receipts, Accounts, etc. No. 3, Letters, Notes, Drafts, etc. No. 4, General Practice. 48 pages each. Price 15 cents each. Intended to supply more practical work in the school-room. Letter-writing, Bills, Receipts, Accounts, Checks, Notes, and all Commercial forms are given for practice. Beers's System of Progressive Penmanship. This "round hand " system of Penmanship, in twelve numbers, commends itself by its simplicity and thoroughness. The first four numbers are primary books. Nos. 5 to 7, advanced books for boys. Nos. 8 to 10, advanced books tor girls. Nos. 11 and 12, ornamental penmanship. These books are printed from steel plates (engraved by McLees), and are unexcelled in mechanical execution. Beers's Slated Copy Slips. Slate exercises, familiarizing beginners with the form of the letters, the motions of the hand and arm, etc. , etc. These copy slips, 32 in number, supply all the copies found in a complete series of writing-books, at a trifling cost. Fulton & Eastman's Chirographic Charts To embellish the school-room walls, and furnish class exercise in the elements of Penmanship. Payson's Copy-Book Cover. Protects every p;if;e except the one in use, and furnishes " lines " with proper slope *r the penman, under. Patented. 65 THE: NATIONAL SERIES OF STANDARD MISCELLANY. LIBRARY AND MISCELLANEOUS PUBLICATIONS. TEACHERS' WORKING LIBRARY. Object Lessons. Welch. This is a complete exposition of the popular modern system of "object-teaching," for teachers of primary classes. Theory and Practice of Teaching. Page. This volume has, without doubt, been read by two hundred thousand teachers, and its popularity remains undiminished, large editions being exhausted yearly. It was the pioneer, as it is now the patriarch, of professional works for teachers. The Graded School. Wells. The proper way to organize graded schools is here illustrated. The author has availed himself of the best elements of the several systems prevalent in Boston, New York, Philadelphia, Cincinnati, St. Louis, and other cities. The Normal. Holbrook. Carries a working school on its visit to teachers, showing the most approved methods of teaching all the common branches, including the technicalities, explanations, demon- strations, and definitions introductory and peculiar to each branch. School Management. Holbrook. Treating of the teacher's qualifications ; how to overcome difficulties in self and others ; organization ; discipline ; methods of inciting diligence and order ; strategy in management ; object-teaching. The Teachers' Institute. Fowle. This is a volume of suggestions inspired by the author's experience at institutes, in the instruction of young teachers. A thousand points of interest to this class are most satisfactorily dealt with. Schools and Schoolmasters. Dickens. Appropriate selections from the writings of the great novelist. The Metric System. Davies. Considered with reference to its general introduction, and embracing the views of John Quiney Adams and Sir John Herschel. The Student ; The Educator. Phelps. 2 vols. The Discipline of Life. Phelps. The authoress of these works is one of the most distinguished writers on education, and they cannot fail to prove a valuable addition to the School and Teachers' Libraries, being in a high degree both interesting and instructive. Law of Public Schools. Burke. By Finley Burke, Counsellor-at-Law. A new volume in " Barnes's Teachers' Library Series." 12mo, cloth. "Mr. Burke has given us the latest expositions of the law on this highly im- portant subject. I shall cordially com- mend his treatise." THEODORE DWIGHT, LL.D. From the HON. JOSEPH M. BECK, Judge of Supreme Court, Iowa. " I have examined with considerable Wire the manuscript of ' A Treatise on the Law of Public Schools. ' by Finley Burke, Esq., of Council Bluffs. In my opinion, the work will be of great value to school teachers and school officers, and to law- yers. The subjects treated of are thought- fully considered and thoroughly examined, and correctly and systematically arranged. The style is perspicuous. The legal doc- trines of the work, so far as I have been 66 THE NATIONAL SERIES OF STANDARD MISCELLANY. MISCELLANEOUS PUBLICATIONS Continued. able to consider them, are sound. I have examined quite a number of the authori- ties cited ; they sustain the rules an- nounced in the text. Mr. Burke is an able and industrious member of the bar of the Supreme Court of this State, and has a high standing in the profession of the law." " I fully concur in the opinion of Judge Beck, above expressed." JOHN F. DIL- LON. New York, May, 1880. Sioux CITY, IOWA, May, 1880. I have examined the manuscript of Finley Burke, Esq. , and find a full citation of all the cases and decisions pertaining to the school law, occurring in the courts of the United States. This volume contains valuable and important information con- cerning school law, which has never before been accessible to either teacher or school officer. A. ARMSTRONG, Suft. Schools, Sioux City, fotca. DES MOINES, May 15, 1880. The examination of " A Treatise on the Law of Public Schools," prepared by Fin- ley Burke, Esq. , of Council Blufl's, has fiven me much pleasure. So far as I now, there is no work of similar charac- ter now in existence. I think such a work will be exceedingly useful to lawyers, school officers, and teachers, and I hope that it may find its way into their hands. G. W. VON COELLV, Supt. Public Inst. for Iowa. Teachers' Handbook. Phelps. By William F. Phelps, Principal of Minnesota State Normal School. Embracing the objects, history, organization, and management of teachers' institutes, followed by methods of teaching, in detail, for all the fundamental branches. Every young teacher, very practical teacher, every experienced teacher even, needs this book. This is the key-note of the present excel- lent volume. In view of the supreme importance of the teacher's calling, Mr. prepare the child for the discipline of life. Phelps has presented an elaborate system Th country schoolmaster, accordingly, of instruction in the elements of learning, From the New Fork Tribune. "The discipline of the school should prepare the child for the discipline of life. Th country schoolmaster, accordingly, holds a position of vital interest to the destiny of the republic, and should neg- lect no means for the wise and efficient discharge of his significant functions. with a complete detail of methods and processes, illustrated with an abundance of practical examples and enforced by judicious councils." Topical Course of Study. Stone. This volume is a compilation from the courses of study of our most successful public schools, and the best thought of leading educators. The pupil is enabled to make full use of any and all text-books bearing on the given topics, and is incited to use all other information within his reach. American Education. Mansfield. A treatise on the principles and elements of education, as practised in this country, With ideas towards distinctive republican and Christian education. American Institutions. De Tocqueville. A valuable index to the genius of our Government. Universal Education^ Mayhew. The subject is approached with the clear, keen perception of one who has observed its necessity, and realized its feasibility and expediency alike. The redeeming and elevating power of improved common schools constitutes the inspiration of the volume. Oral Training Lessons. Barnard. The object of this very useful work is to furnish material for instructors to impart orally to their classes, in branches not usually taught in common schools, embracing a. 1 ' departments of natural science and much general knowledge. Lectures on Natural History. Chadbourne. Affording many themes for oral instruction in this interesting scisnc*, especially in schools where it is not pursued as a class exercise. 67 X THE NATIONAL SERIES OF STANDARD MISCELLANY. MISCELLANEOUS PUBLICATIONS Continued. Outlines of Mathematical Science. Davies. A manual suggesting the best methods of presenting mathematical instruction on the part of the teacher, with that comprehensive view of the whole which is necessary to the intelligent treatment of a part, iu science. Nature and Utility of Mathematics. Davies. An elaborate and lucid exposition of the principles which lie at the foundation of pure mathematics, with a highly ingenious application of their results to the develop- ment of the essential idea of the different branches of the science. Mathematical Dictionary. Davies and Peck. This cyclopaedia of mathematical science defines, with completeness, precision, and accuracy, every technical term ; thus constituting a popular treatise on each branch, and a general view of the whole subject. The Popular Educator. Barnes. In seven volumes, containing interesting and profitable educational miscellany. Liberal Education of Women. Orton. Treats of " the demand and the method ; " being a compilation of the best and most advanced thought on this subject, by the leading writers and educators in England and America. Edited by a professor in Vassar College. Education Abroad. Northrop. A thorough discussion of the advantages and disadvantages of sending American children to Europe to be educated ; also, papers on legal prevention of illiteracy, study, and health, labor as an educator, and other kindred subjects. The Teacher and the Parent. Northend. A treatise upon common-school education, designed to lead teachers to view their calling in its true light, and to stimulate them to fidelity. The Teachers' Assistant. Northend. A natural continuation of the author's previous work, more directly calculated for daily use iu the administration of school discipline and instruction. School Government. Jewell. Full of advanced ideas on the subject which its title indicates. The criticisms upon current theories of punishment and schemes of administration have excited general attention and comment. Grammatical Diagrams. Jewell. The diagram system of teaching grammar explained, defended, and improved. The curious in literature, the searcher for truth, those interested in new inventions, as well as the disciples of Professor Clark, who would see their favorite theory fairly treated, all want this book. There are many who would like to be made familiar with this system before risking its use in a class. The opportunity is here afforded. The Complete Examiner. Stone. Consists of a series of questions on every English branch of school and academic instruction, with reference to a given page or article of leading text-books where the answer may be found in full. Prepared to aid teachers in securing certificates, pupils in preparing for promotion, and teachers in selecting review questions. How Not to Teach. Griffin. This book meets a want universally felt among young teachers who have their expe- rience in teaching to learn. It undertakes to point out the many natural mistakes into which teachers, unconsciously or otherwise, fall, and warns the reader against dangers that beset the path of every conscientious teacher. It tells tlie reader, also, the proper and acceptable way to teach, illustrating the author's ideas by some practice-lessons in arithmetic (after Griibe). gg THE NATIONAL SERIES OF STANDARD MISCELLANY, MISCELLANEOUS PUBLICATIONS C