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Practical Education, Volume II
CHAPTER XIV
ON GEOGRAPHY AND CHRONOLOGYThe usual manner of teaching Geography and Chronology, may, perhaps, be necessary in public seminaries, where a number of boys are to learn the same thing at the same time; but what is learned in this manner, is not permanent; something besides merely committing names and dates to the memory, is requisite to make a useful impression upon the memory. For the truth of this observation, an appeal is made to the reader. Let him recollect, whether the Geography and Chronology which he learned whilst a boy, are what he now remembers – Whether he has not obtained his present knowledge from other sources than the tasks of early years. When business, or conversation, calls upon us to furnish facts accurate as to place and time, we retrace our former heterogeneous acquirements, and select those circumstances which are connected with our present pursuit, and thus we form, as it were, a nucleus round which other facts insensibly arrange themselves. Perhaps no two men in the world, who are well versed in these studies, connect their knowledge in the same manner. Relation to some particular country, some favourite history, some distinguished person, forms the connection which guides our recollection, and which arranges our increasing nomenclature. By attending to what passes in our own minds, we may learn an effectual method of teaching without pain, and without any extraordinary burden to the memory, all that is useful of these sciences. The details of history should be marked by a few chronological æras, and by a few general ideas of geography. When these have been once completely associated in the mind, there is little danger of their being ever disunited: the sight of any country will recall its history, and even from representations in a map, or on the globe, when the mind is wakened by any recent event, a long train of concomitant ideas will recur.
The use of technical helps to the memory, has been condemned by many, and certainly, when they are employed as artifices to supply the place of real knowledge, they are contemptible; but when they are used as indexes to facts that have been really collected in the mind; when they serve to arrange the materials of knowledge in appropriate classes, and to give a sure and rapid clue to recollection, they are of real advantage to the understanding. Indeed, they are now so common, that pretenders cannot build the slightest reputation upon their foundation. Were an orator to attempt a display of long chronological accuracy, he might be wofully confounded by his opponent's applying at the first pause,
12Elsluk he would have said!Ample materials are furnished in Gray's Memoria Technica, from which a short and useful selection may be made, according to the purposes which are in view. For children, the little ballad of the Chapter of Kings, will not be found beneath the notice of mothers who attend to education. If the technical terminations of Gray are inserted, they will never be forgotten, or may be easily recalled.13 We scarcely ever forget a ballad if the tune is popular.
For pupils at a more advanced age, it will be found advantageous to employ technical helps of a more scientific construction. Priestley's Chart of Biography may, from time to time, be hung in their view. Smaller charts, upon the same plan, might be provided with a few names as land-marks; these may be filled up by the pupil with such names as he selects from history; they may be bound in octavo, like maps, by the middle, so as to unfold both ways – Thirty-nine inches by nine will be a convenient size. Prints, maps, and medals, which are part of the constant furniture of a room, are seldom attended to by young people; but when circumstances excite an interest upon any particular subject, then is the moment to produce the symbols which record and communicate knowledge.
Mrs. Radcliffe, in her judicious and picturesque Tour through Germany, tells us, that in passing through the apartments of a palace which the archduchess Maria Christiana, the sister of the late unfortunate queen of France, had left a few hours before, she saw spread upon a table a map of all the countries then included in the seat of the war. The positions of the several corps of the allied armies were marked upon this chart with small pieces of various coloured wax. Can it be doubted, that the strong interest which this princess must have taken in the subject, would for ever impress upon her memory the geography of this part of the world?
How many people are there who have become geographers since the beginning of the present war. Even the common newspapers disseminate this species of knowledge, and those who scarcely knew the situation of Brest harbour a few years ago, have consulted the map with that eagerness which approaching danger excites; they consequently will tenaciously remember all the geographical knowledge they have thus acquired. The art of creating an interest in the study of geography, depends upon the dexterity with which passing circumstances are seized by a preceptor in conversation. What are maps or medals, statues or pictures, but technical helps to memory? If a mother possess good prints, or casts of ancient gems, let them be shown to any persons of taste and knowledge who visit her; their attention leads that of our pupils; imitation and sympathy are the parents of taste, and taste reads in the monuments of art whatever history has recorded.
In the Adele and Theodore of Madame de Silleri, a number of adventitious helps are described for teaching history and chronology. There can be no doubt that these are useful; and although such an apparatus cannot be procured by private families, fortunately the print-shops of every provincial town, and of the capital in particular, furnish even to the passenger a continual succession of instruction. Might not prints, assorted for the purposes which we have mentioned, be lent at circulating libraries?
To assist our pupils in geography, we prefer a globe to common maps. Might not a cheap, portable, and convenient globe, be made of oiled silk, to be inflated by a common pair of bellows? Mathematical exactness is not requisite for our purpose, and though we could not pretend to the precision of our best globes, yet a balloon of this sort would compensate by its size and convenience for its inaccuracy. It might be hung by a line from its north pole, to a hook screwed into the horizontal architrave of a door or window; and another string from its south pole might be fastened at a proper angle to the floor, to give the requisite elevation to the axis of the globe. An idea of the different projections of the sphere, may be easily acquired from this globe in its flaccid state, and any part of it might be consulted as a map, if it were laid upon a convex board of a convenient size. Impressions from the plates which are used for common globes, might be taken to try this idea without any great trouble or expense; but we wish to employ a much larger scale, and to have them five or six feet diameter. The inside of a globe of this sort might be easily illuminated, and this would add much to the novelty and beauty of its appearance.
In the country, with the assistance of a common carpenter and plasterer, a large globe of lath and plaster may be made for the instruction and entertainment of a numerous family of children. Upon this they should leisurely delineate from time to time, by their given latitudes and longitudes, such places as they become acquainted with in reading or conversation. The capital city, for instance, of the different countries of Europe, the rivers, and the neighbouring towns, until at last the outline might be added: for the sake of convenience, the lines, &c. may be first delineated upon a piece of paper, from which they may be accurately transferred to their proper places on the globe, by the intervention of black-leaded paper, or by pricking the lines through the paper, and pouncing powdered blue through the holes upon the surface of the globe.
We enter into this detail because we are convinced, that every addition to the active manual employment of children, is of consequence, not only to their improvement, but to their happiness.
Another invention has occurred to us for teaching geography and history together. Priestley's Chart of History, though constructed with great ingenuity, does not invite the attention of young people: there is an intricacy in the detail which is not obvious at first. To remedy what appears to us a difficulty, we propose that eight and twenty, or perhaps thirty, octavo maps of the globe should be engraved; upon these should be traced, in succession, the different situations of the different countries of the world, as to power and extent, during each respective century: different colours might denote the principal divisions of the world in each of these maps; the same colour always denoting the same country, with the addition of one strong colour; red, for instance, to distinguish that country which had at each period the principal dominion. On the upper and lower margin in these maps, the names of illustrious persons might be engraven in the manner of the biographical chart; and the reigning opinions of each century should also be inserted. Thus history, chronology, and geography, would appear at once to the eye in their proper order, and regular succession, divided into centuries and periods, which easily occur to recollection.
We forbear to expatiate upon this subject, as it has not been actually submitted to experiment; carefully avoiding in the whole of this work to recommend any mode of instruction which we have not actually put in practice. For this reason, we have not spoken of the abbé Gaultier's method of teaching geography, as we have only been able to obtain accounts of it from the public papers, and from reviews; we are, however, disposed to think favourably beforehand, of any mode which unites amusement with instruction. We cannot forbear recommending, in the strongest manner, a few pages of Rollin in his "Thoughts upon Education,"14 which we think contain an excellent specimen of the manner in which a well informed preceptor might lead his pupils a geographical, historical, botanical, and physiological tour upon the artificial globe.
We conclude this chapter of hints, by repeating what we have before asserted, that though technical assistance may be of ready use to those who are really acquainted with that knowledge to which it refers, it never can supply the place of accurate information.
The causes of the rise and fall of empires, the progress of human knowledge, and the great discoveries of superior minds, are the real links which connect the chain of political knowledge.
CHAPTER XV
ON ARITHMETICThe man who is ignorant that two and two make four, is stigmatized with the character of hopeless stupidity; except, as Swift has remarked, in the arithmetic of the customs, where two and two do not always make the same sum.
We must not judge of the understanding of a child by this test, for many children of quick abilities do not immediately assent to this proposition when it is first laid before them. "Two and two make four," says the tutor. "Well, child, why do you stare so?"
The child stares because the word make is in this sentence used in a sense which is quite new to him; he knows what it is to make a bow, and to make a noise, but how this active verb is applicable in the present case, where there is no agent to perform the action, he cannot clearly comprehend. "Two and two are four," is more intelligible; but even this assertion, the child, for want of a distinct notion of the sense in which the word are is used, does not understand. "Two and two are called four," is, perhaps, the most accurate phrase a tutor can use; but even these words will convey no meaning until they have been associated with the pupil's perceptions. When he has once perceived the combination of the numbers with real objects, it will then be easy to teach him that the words are called, are, and make, in the foregoing proposition, are synonymous terms.
We have chosen the first simple instance we could recollect, to show how difficult the words we generally use in teaching arithmetic, must be to our young pupils. It would be an unprofitable task to enumerate all the puzzling technical terms which, in their earliest lessons, children are obliged to hear, without being able to understand.
It is not from want of capacity that so many children are deficient in arithmetical skill; and it is absurd to say, "such a child has no genius for arithmetic. Such a child cannot be made to comprehend any thing about numbers." These assertions prove nothing, but that the persons who make them, are ignorant of the art of teaching. A child's seeming stupidity in learning arithmetic, may, perhaps, be a proof of intelligence and good sense. It is easy to make a boy, who does not reason, repeat by rote any technical rules which a common writing-master, with magisterial solemnity, may lay down for him; but a child who reasons, will not be thus easily managed; he stops, frowns, hesitates, questions his master, is wretched and refractory, until he can discover why he is to proceed in such and such a manner; he is not content with seeing his preceptor make figures and lines upon a slate, and perform wondrous operations with the self-complacent dexterity of a conjurer. A sensible boy is not satisfied with merely seeing the total of a given sum, or the answer to a given question, come out right; he insists upon knowing why it is right. He is not content to be led to the treasures of science blindfold; he would tear the bandage from his eyes, that he might know the way to them again.
That many children, who have been thought to be slow in learning arithmetic, have, after their escape from the hands of pedagogues, become remarkable for their quickness, is a fact sufficiently proved by experience. We shall only mention one instance, which we happened to meet with whilst we were writing this chapter. John Ludwig, a Saxon peasant, was dismissed from school when he was a child, after four years ineffectual struggle to learn the common rules of arithmetic. He had been, during this time, beaten and scolded in vain. He spent several subsequent years in common country labour, but at length some accidental circumstances excited his ambition, and he became expert in all the common rules, and mastered the rule of three and fractions, by the help of an old school book, in the course of one year. He afterwards taught himself geometry, and raised himself, by the force of his abilities and perseverance, from obscurity to fame.
We should like to see the book which helped Mr. Ludwig to conquer his difficulties. Introductions to Arithmetic are, often, calculated rather for adepts in science, than for the ignorant. We do not pretend to have discovered any shorter method than what is common, of teaching these sciences; but, in conformity with the principles which are laid down in the former part of this work, we have endeavoured to teach their rudiments without disgusting our pupils, and without habituating them to be contented with merely technical operations.
In arithmetic, as in every other branch of education, the principal object should be, to preserve the understanding from implicit belief; to invigorate its powers; to associate pleasure with literature, and to induce the laudable ambition of progressive improvement.
As soon as a child can read, he should be accustomed to count, and to have the names of numbers early connected in his mind with the combinations which they represent. For this purpose, he should be taught to add first by things, and afterwards by signs or figures. He should be taught to form combinations of things by adding them together one after another. At the same time that he acquires the names that have been given to these combinations, he should be taught the figures or symbols that represent them. For example, when it is familiar to the child, that one almond, and one almond, are called two almonds; that one almond, and two almonds, are called three almonds, and so on, he should be taught to distinguish the figures that represent these assemblages; that 3 means one and two, &c. Each operation of arithmetic should proceed in this manner, from individuals to the abstract notation of signs.
One of the earliest operations of the reasoning faculty, is abstraction; that is to say, the power of classing a number of individuals under one name. Young children call strangers either men or women; even the most ignorant savages15 have a propensity to generalize.
We may err either by accustoming our pupils too much to the consideration of tangible substances when we teach them arithmetic, or by turning their attention too much to signs. The art of forming a sound and active understanding, consists in the due mixture of facts and reflection. Dr. Reid has, in his "Essay on the Intellectual Powers of Man," page 297, pointed out, with great ingenuity, the admirable economy of nature in limiting the powers of reasoning during the first years of infancy. This is the season for cultivating the senses, and whoever, at this early age, endeavours to force the tender shoots of reason, will repent his rashness.
In the chapter "on Toys," we have recommended the use of plain, regular solids, cubes, globes, &c. made of wood, as playthings for children, instead of uncouth figures of men, women and animals. For teaching arithmetic, half inch cubes, which can be easily grasped by infant fingers, may be employed with great advantage; they can be easily arranged in various combinations; the eye can easily take in a sufficient number of them at once, and the mind is insensibly led to consider the assemblages in which they may be grouped, not only as they relate to number, but as they relate to quantity or shape; besides, the terms which are borrowed from some of these shapes, as squares, cubes, &c. will become familiar. As these children advance in arithmetic to square or cube, a number will be more intelligible to them than to a person who has been taught these words merely as the formula of certain rules. In arithmetic, the first lessons should be short and simple; two cubes placed above each other, will soon be called two; if placed in any other situations near each other, they will still be called two; but it is advantageous to accustom our little pupils to place the cubes with which they are taught in succession, either by placing them upon one another, or laying in columns upon a table, beginning to count from the cube next to them, as we cast up in addition. For this purpose, a board about six inches long, and five broad, divided into columns perpendicularly by slips of wood three eighths of an inch wide, and one eighth of an inch thick, will be found useful; and if a few cubes of colours different from those already mentioned, with numbers on their six sides, are procured, they may be of great service. Our cubes should be placed, from time to time, in a different order, or promiscuously; but when any arithmetical operations are to be performed with them, it is best to preserve the established arrangement.
One cube and one other, are called two.
Two what?
Two cubes.
One glass, and one glass, are called two glasses. One raisin, and one raisin, are called two raisins, &c. One cube, and one glass, are called what? Two things or two.
By a process of this sort, the meaning of the abstract term two may be taught. A child will perceive the word two, means the same as the words one and one; and when we say one and one are called two, unless he is prejudiced by something else that is said to him, he will understand nothing more than that there are two names for the same thing.
"One, and one, and one, are called three," is the same as saying "that three is the name for one, and one, and one." "Two and one are three," is also the same as saying "that three is the name of two and one." Three is also the name of one and two; the word three has, therefore, three meanings; it means one, and one, and one; also, two and one; also, one and two. He will see that any two of the cubes may be put together, as it were, in one parcel, and that this parcel may be called two; and he will also see that this parcel, when joined to another single cube, will make three, and that the sum will be the same, whether the single cube, or the two cubes, be named first.
In a similar manner, the combinations which form four, may be considered. One, and one, and one, and one, are four.
One and three are four.
Two and two are four.
Three and one are four.
All these assertions mean the same thing, and the term four is equally applicable to each of them; when, therefore, we say that two and two are four, the child may be easily led to perceive, and indeed to see, that it means the same thing as saying one two, and one two, which is the same thing as saying two two's, or saying the word two two times. Our pupil should be suffered to rest here, and we should not, at present, attempt to lead him further towards that compendious method of addition which we call multiplication; but the foundation is laid by giving him this view of the relation between two and two in forming four.
There is an enumeration in the note16 of the different combinations which compose the rest of the Arabic notation, which consists only of nine characters.
Before we proceed to the number ten, or to the new series of numeration which succeeds to it, we should make our pupils perfectly masters of the combinations which we have mentioned, both in the direct order in which they are arranged, and in various modes of succession; by these means, not only the addition, but the subtraction, of numbers as far as nine, will be perfectly familiar to them.
It has been observed before, that counting by realities, and by signs, should be taught at the same time, so that the ear, the eye, and the mind, should keep pace with one another; and that technical habits should be acquired without injury to the understanding. If a child begins between four and five years of age, he may be allowed half a year for this essential, preliminary step in arithmetic; four or five minutes application every day, will be sufficient to teach him not only the relations of the first decade in numeration, but also how to write figures with accuracy and expedition.
The next step, is, by far the most difficult in the science of arithmetic; in treatises upon the subject, it is concisely passed over under the title of Numeration; but it requires no small degree of care to make it intelligible to children, and we therefore recommend, that, besides direct instruction upon the subject, the child should be led, by degrees, to understand the nature of classification in general. Botany and natural history, though they are not pursued as sciences, are, notwithstanding, the daily occupation and amusement of children, and they supply constant examples of classification. In conversation, these may be familiarly pointed out; a grove, a flock, &c. are constantly before the eyes of our pupil, and he comprehends as well as we do what is meant by two groves, two flocks, &c. The trees that form the grove are each of them individuals; but let their numbers be what they may when they are considered as a grove, the grove is but one, and may be thought of and spoken of distinctly, without any relation to the number of single trees which it contains. From these, and similar observations, a child may be led to consider ten as the name for a whole, an integer; a one, which may be represented by the figure (1): this same figure may also stand for a hundred, or a thousand, as he will readily perceive hereafter. Indeed, the term one hundred will become familiar to him in conversation long before he comprehends that the word ten is used as an aggregate term, like a dozen, or a thousand. We do not use the word ten as the French do une dizaine; ten does not, therefore, present the idea of an integer till we learn arithmetic. This is a defect in our language, which has arisen from the use of duodecimal numeration; the analogies existing between the names of other numbers in progression, is broken by the terms eleven and twelve. Thirteen, fourteen, &c. are so obviously compounded of three and ten, and four and ten, as to strike the ears of children immediately, and when they advance as far as twenty, they readily perceive that a new series of units begins, and proceeds to thirty, and that thirty, forty, &c. mean three tens, four tens, &c. In pointing out these analogies to children, they become interested and attentive, they show that species of pleasure which arises from the perception of aptitude, or of truth. It can scarcely be denied that such a pleasure exists independently of every view of utility and fame; and when we can once excite this feeling in the minds of our young pupils at any period of their education, we may be certain of success.
