FOOTNOTES:

photographExercises in interpreted reading and arithmetic. (The Rivington Street Montessori School, New York.)

The fondness of children for reading and their preference for the "true" is something already demonstrated by experiments conducted elsewhere. I may refer here to the investigations on readings for children conducted by the "Education" section of the Federation for School Libraries of the province of Emilia (Italy). The questionnaire was as follows:

Do you remember what books you have read and which you liked best?How did you get them?Do you know the title of some book you would like to read?Do you prefer fairy-tales, or rather stories of true or probable facts? Why?Do you prefer sad or humorous stories?Do you like poetry?Do you like stories of travel and adventure?Do you subscribe to any weekly or monthly newspaper? If so, to which?If your mother were to offer you a choice between a subscription to a weekly or monthly and an illustrated book, which would you take? And why?

Do you remember what books you have read and which you liked best?

How did you get them?

Do you know the title of some book you would like to read?

Do you prefer fairy-tales, or rather stories of true or probable facts? Why?

Do you prefer sad or humorous stories?

Do you like poetry?

Do you like stories of travel and adventure?

Do you subscribe to any weekly or monthly newspaper? If so, to which?

If your mother were to offer you a choice between a subscription to a weekly or monthly and an illustrated book, which would you take? And why?

The answers, very carefully sifted, showed that the vast majority of children preferred readings which dealt with fact. Here are some of the reasons alleged by the children in support for their preference for "truth": "Facts teach me something; fairy-tales are too improbable; true stories don't upset my thinking; true stories teach me history; true stories always convey some good idea; fairy-stories give me desires impossible to satisfy; many good ideas come from actual experiences; fantastic tales make me think too much about supernatural things"; etc., etc. In favor of the fairy-tales we find: "They amuse me in hours free from work; I like to be in the midst of fairies and enchantments"; etc. Those who preferred sad or serious stories justified themselves as follows: "I feel that I am a better person, and realize better the wrong I do; I feel that my disposition becomes more kindly; they arouse in me feelings of kindness and pity." Many supported their preference for humorous tales on the ground that "when I read them, I am able to forget my own little troubles." In general, a great majority denied any educational value to joy and humor. In this conviction—or rather this feeling—so widely diffused among children, have we not evidence that something must be wrong in the kind of education we have been giving them?

FOOTNOTES:[5]The first readings consist of a special grammar and a dictionary.

[5]The first readings consist of a special grammar and a dictionary.

ARITHMETIC

ARITHMETICAL OPERATIONS

The children already had performed the four arithmetical operations in their simplest forms, in the "Children's Houses," the didactic material for these having consisted of the rods of the long stair which gave empirical representation of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. By means of its divisions into sections of alternating colors, red and blue, each rod represented the quantity of unity for which it stood; and so the entrance into the complex and arduous field of numbers was thus rendered easy, interesting, and attractive by the conception that collective number can be represented by asingleobject containing signs by which the relative quantity of unity can be recognized, instead of bya number of differentunits, represented by the figure in question. For instance, the fact that five may be represented by a single object with five distinct and equal parts instead of by five distinct objects which the mind must reduce to a concept of number, saves mental effort and clarifies the idea.

It was through the application of this principle by means of the rods that the children succeeded so easily in accomplishing the first arithmetical operations: 7 + 3 = 10; 2 + 8 = 10; 10 - 4 = 6; etc.

The long stair material is excellent for this purpose. But it is too limited in quantity and is too large to behandled easily and used to good advantage in meeting the demands of a room full of children who already have been initiated into arithmetic. Therefore, keeping to the same fundamental concepts, we have prepared smaller, more abundant material, and one more readily accessible to a large number of children working at the same time.

This material consists of beads strung on wires: i.e., bead bars representing respectively 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The beads are of different colors. The 10-bead bar is orange; 9, dark blue; 8, lavender; 7, white; 6, gray; 5, light blue; 4, yellow; 3, pink; 2, green; and there are separate beads for unity.[6]The beads are opalescent; and the white metal wire on which they are strung is bent at each end, holding the beads rigid and preventing them from slipping.

There are five sets of these attractive objects in each box; and so each child has at his disposal the equivalent of five sets of the long stairs used for his numerical combinations in the earliest exercise. The fact that the rods are small and so easily handled permits of their being used at the small tables.

This very simple and easily prepared material has been extraordinarily successful with children of five and a half years. They have worked with marked concentration, doing as many as sixty successive operations and filling whole copybooks within a few days' time. Special quadrille paper is used for the purpose; and the sheets are ruled in different colors: some in black, some in red, somein green, some in blue, some in pink, and some in orange. The variety of colors helps to hold the child's attention: after filling a sheet lined in red, he will enjoy filling one lined in blue, etc.

Experience has taught us to prepare a large number of the ten-bead bars; for the children will choose these from all the others, in order to count the tens in succession: 10, 20, 30, 40, etc. To this first bead material, therefore, we have added boxes filled with nothing but ten-bead bars. There are also small cards on which are written 10, 20, etc. The children put together two or more of the ten-bead bars to correspond with the number on the cards. This is an initial exercise which leads up to the multiples of 10. By superimposing these cards on that for the number 100 and that for the number 1000, such numbers as1917can be obtained.

The "bead work" became at once an established element in our method, scientifically determined as a conquest brought to maturity by the child in the very act of making it. Our success in amplifying and making more complex the early exercises with the rods has made the child's mental calculation more rapid, more certain, and more comprehensive. Mental calculation develops spontaneously, as if by a law of conservation tending to realize the "minimum of effort." Indeed, little by little the child ceases counting the beads and recognizes the numbers by their color: the dark blue he knows is 9, the yellow 4, etc. Almost without realizing it he comes now to count bycolorsinstead of byquantitiesof beads, and thus performs actual operations in mental arithmetic. As soon as the child becomes conscious of this power, he joyfully announces his transition to the higher plane, exclaiming, "I can count in my head and I can do it morequickly!" This declaration indicates that he has conquered the first bead material.

Material:I have had a chain made by joining ten ten-bead bars end to end. This is called the "hundred chain." Then, by means of short and very flexible connecting links I had ten of these "hundred chains" put together, making the "thousand chain."

These chains are of the same color as the ten-bead bars, all of them being constructed of orange-colored beads. The difference in their reciprocal length is very striking. Let us first put down a single bead; then a ten-bead bar, which is about seven centimeters long; then a hundred-bead chain, which is about seventy centimeters long; and finally the thousand-bead chain, which is about seven meters long. The great length of this thousand-bead chain leads directly to another idea of quantity; for whereas the 1, the 10, and the 100 can be placed on the table for convenient study, the entire length of the room will hardly suffice for the thousand-bead chain! The children find it necessary to go into the corridor or an adjoining room; they have to form little groups to accomplish the patient work of stretching it out into a straight line. And to examine the whole extent of this chain, they have to walk up and down its entire length. The realization they thus obtain of the relative values of quantity is in truth an event for them. For days at a time this amazing "thousand chain" claims the child's entire activity.

The flexible connections between the different hundred lengths of the thousand-bead chain permit of its being folded so that the "hundred chains" lie one next to the other, forming in their entirety a long rectangle. Thesame quantity which formerly impressed the child by its length is now, in its broad, folded form, presented as asurfacequantity.

Now all may be placed on a small table, one below the other: first the single bead, then the ten-bead bar, then the "hundred chain," and finally the broad strip of the "thousand chain."

Any teacher who has asked herself how in the world a child may be taught to express in numerical terms quantitative proportions perceived through the eye, has some idea of the problem that confronts us. However, our children set to work patiently counting bead by bead from 1 to 100. Then they gathered in two's and three's about the "thousand chain," as if to help one another in counting it, undaunted by the arduous undertaking. They counted on hundred; and after one hundred, what? One hundred one. And finally two hundred, two hundred one. One day they reached seven hundred. "I am tired," said the child. "I'll mark this place and come back tomorrow."

"Seven hundred, seven hundred—Look!" cried another child. "There are seven—sevenhundreds! Yes, yes; count the chains! Seven hundred, eight hundred, nine hundred, one thousand. Signora, signora, the 'thousand chain' has ten 'hundred chains'! Look at it!" And other children, who had been working with the "hundred chain," in turn called the attention oftheircomrades: "Oh, look, look! The 'hundred chain' has ten ten-bead bars!"

Thus we realized that the numerical concept of tens, hundreds, and thousands was given by presenting these chains to the child's intelligent curiosity and by respecting the spontaneous endeavors of his free activities.

And since this was our experience with most of the children, one easily can see how simple a suggestion would be necessary if the deduction did not take place in the case of some exceptional child. In fact, to make the idea of decimal relations apparent to a child, it is sufficient to direct his attention to the material he is handling. The teacher experienced in this method knows how to wait; she realizes that the child needs to exercise his mind constantly and slowly; and if the inner maturation takes place naturally, "intuitive explosions" are bound to follow as a matter of course. The more we allow the children to follow the interests which have claimed their fixed attention, the greater will be the value of the results.

The direct assistance of the teacher, her clear and brief explanation, is, however, essential when she presents to the child another new material, which may be considered "symbolic" of the decimal relations. This material consists of two very simple bead counting-frames, similar in size and shape to the dressing-frames of the first material. They are light and easily handled and may be included in the individual possessions of each child. The frames are easily made and are inexpensive.

One frame is arranged with the longest side as base, and has four parallel metal wires, each of which is strung with ten beads. The three top wires are equidistant but the fourth is separated from the others by a greater distance, and this separation is further emphasized by a brass nail-head fixed on the left hand side of the frame. The frame is painted one color above the nail-head and another color below it; and on this side of the frame, also, numerals corresponding to each wire are marked. Thenumeral opposite the top wire is 1, the next 10, then 100, and the lowest, 1000.

We explain to the child that each bead of the first wire is assumed to stand for one, or unity, as did the separate beads they have had before; but each bead of the second wire stands for ten (or for one of the ten-bead bars); the value of each bead of the third wire is one hundred and represents the "hundred chain"; and each bead on the last wire (which is separated from the others by the brass nail-head) has the same value as a "thousand chain."[7]

At first it is not easy for the child to understand this symbolism, but it will be less difficult if he previously has worked over the chains, counting and studying them without being hurried. When the concept of the relationship between unity, tens, hundreds, and thousands has matured spontaneously, he more readily will be able to recognize and use the symbol.

Specially lined paper is designed for use with these frames. This paper is divided lengthwise into two equal parts, and on both sides of the division are vertical lines of different colors: to the right a green line, then a blue, and next a red line. These are parallel and equidistant. A vertical line of dots separates this group of three lines from another line which follows. On the first three lines from right to left are written respectively the units, tens, and hundreds; on the inner line the thousands.

The right half of the page is used entirely and exclusively to clarify this idea and to show the relationship of written numbers to the decimal symbolism of the counting-frame.

With this object in view, we first count the beads on each wire of the frame; saying for the top wire, one unit, two units, three units, four units, five units, six units, seven units, eight units, nine units, ten units. The ten units of this top wire are equal to one bead on the second wire.

The beads on the second wire are counted in the same way: one ten, two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens, ten tens. The ten ten-beads are equal to one bead on the third wire.

The beads on this third wire then are counted one by one: one hundred, two hundreds, three hundreds, four hundreds, five hundreds, six hundreds, seven hundreds, eight hundreds, nine hundreds, ten hundreds. These ten hundred-beads are equal to one of the thousand-beads.

There also are ten thousand-beads: one thousand, two thousands, three thousands, four thousands, five thousands, six thousands, seven thousands, eight thousands, nine thousands, ten thousands. The child can picture ten separate "thousand chains"; this symbol is in direct relation, therefore, to a tangible idea of quantity.

Now we must transcribe all these acts by which we have in succession counted, ten units, ten tens, ten hundreds, and ten thousands. On the first vertical line to the extreme right (the green line) we write the units, one beneath the other; on the second line (blue) we write the tens; on the third line (red) the hundreds; and, finally, on the line beyond the dots we write the thousands. There are sufficient horizontal lines for all the numbers, including one thousand.

Having reached 9, we must leave the line of the units and pass over to that of the tens; in fact, ten units make one ten. And, similarly, when we have written 9 in thetens line we must of necessity pass to the hundreds line, because ten tens equal one hundred. Finally, when 9 in the hundreds line has been written, we must pass to the thousands line for the same reason.

The units from 1 to 9 are written on the line farthest to the right; on the next line to the left are written the tens (from 1 to 9); and on the third line, the hundreds (from 1 to 9). Thus always we have the numbers 1 to 9; and it cannot be otherwise, for any more would cause the figure itself to change position. It is this fact that the child must quietly ponder over and allow to ripen in his mind.

It is the nine numbers that change position in order to form all the numbers that are possible. Therefore, it is not the number in itself but itspositionin respect to the other numbers which gives it the value now of one, now of ten, now of one hundred or one thousand. Thus we have the symbolic translation of those real values which increase in so prodigious a way and which are almost impossible for us to conceive. One line of ten thousand beads is seventy meters long! Ten such lines would be the length of a long street! Therefore we are forced to have recourse to symbols. How very important thispositionoccupied by the number becomes!

How do we indicate the position and hence the value of a certain number with reference to other numbers? As there are not always vertical lines to indicate the relative position of the figure,the requisite number of zeros are placed to the right of the figure!

The children already know, from the "Children's House," that zero has no value and that it can give no value to the figure with which it is used. It serves merely to show the position and the value of the figure written atits left. Zero does not give value to 1 and so make it become 10: the zero of the number 10 indicates that the figure 1 is not a unit but is in the next preceding position—that of the tens—and means therefore one ten and not one unit. If, for instance, 4 units followed the 1 in the tens position, then the figure 4 would be in the units place and the 1 would be in the tens position.

photographThe bead material used for addition and subtraction. Each of the nine numbers is of different colored beads.

photograph--two children sitting at tableCounting and calculating by means of the bead chains. (A Montessori School in Italy.)

The "Children's House" child already knows how to write ten and even one hundred; and it is now very easy for him to write, with the aid of zeros, andin columns, from 1 to 1000: 1, 2, 3, 4, 5, 6, 7, 8, 9; 10, 20, 30, 40, 50, 60, 70, 80, 90; 100, 200, 300, 400, 500, 600, 700, 800, 900; 1,000. When the child has learned to count well in this manner, he can easily read any number of four figures.

Let us now make up a number on the counting-frame; for example, 4827. We move four beads to the left on the thousands-wire, eight on the hundreds-wire, two on the tens-wire, and seven on the units-wire; and we read, four thousand eight hundred and twenty-seven. This number is written by placing the numberson the same lineand in the mutually relative order determined by the symbolic positions for the decimal relations, 4827.

We can do the same with the date of our present year, writing the figures on the left-hand side of the paper as indicated: 1917.

Let us compose 2049 on the symbolic number frame. Two of the thousand-beads are moved to the left, four of the ten-beads, and nine of the unit-beads. On the hundreds-wire there is nothing. Here we have a good demonstration of the function of zero, which is to occupy the places that are empty on this chart.

Similarly, to form the number 4700 on the frame, four thousand-beads are moved to the left and seven hundred-beads, the tens-wire and the units-wire remaining empty. In transcribing this number, these empty places are filled by zeros—a figure of no value in itself.

photographThe bead cube of 10; ten squares of 10; and chains of 10, of 100, and of 1000 beads.

photographThis shows the first bead frame which the child uses in his study of arithmetic. The number formed at the left on the frame is 1,111.

When the child fully understands this process he makes up many exercises of his own accord and with the greatest interest. He moves beads to the left at random, on one or on all of the wires, then interprets and writes the number on the sheets of paper purposely prepared for this. When he has comprehended the position of the figures and performed operations with numbers of several figures he has mastered the process. The child need only be left to his auto-exercises here in order to attain perfection.

Very soon he will ask to go beyond the thousands. For this there is another frame, with seven wires representing respectively units, tens, and hundreds; units, tens and hundreds of the thousands; and a million.

This frame is the same size as the other one but in this the shorter side is used as the base and there are seven wires instead of four. The right-hand side is marked by three different colors according to the groups of wires. The units, tens, and hundreds wires are separated from the three thousands wires by a brass tack, and these in turn are separated in the same manner from the million wire.

The transition from one frame to the other furnishes much interest but no difficulty. Children will need very few explanations and will try by themselves to understand as much as possible. The large numbers are the most interesting to them, therefore the easiest. Soon their copybooks are full of the most marvelous numbers; they have now become dealers in millions.

For this frame also there is specially prepared paper.On the right-hand side the child writes the numbers corresponding to the frame, counting from one to a million: 1, 2, 3, 4, 5, 6, 7, 8, 9; 10, 20, 30, 40, 50, 60, 70, 80, 90; 100, 200, 300, 400, 500, 600, 700, 800, 900; 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000; 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000; 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000; 1,000,000.

After this the child, moving the beads to the left on one or more of the wires, tries to read and then to write on the left half of the paper the numbers resulting from these haphazard experiments. For example, on the counting-frame he may have the number 6,206,818, and on the paper the numbers 1,111,111; 8,640,850; 1,500,000; 3,780,000; 5,840,714; 720,000; 500,000; 430,000; 35,840; 80,724; 15,229; 1,240.

When we come to add and subtract numbers of several figures and to write the results in column, the facility resulting from this preparation is something astonishing.

FOOTNOTES:[6]At the present time, because of the difficulty of getting beads of certain colors, owing to war conditions, the following colors have been approved by Dr. Montessori to replace those originally used: 10 bead bar, gold; 9, dark blue; 8, white; 7, light green; 6, light blue; 5, yellow; 4, pink; 3, green; 2, yellow-green; 1, gold. These same colors are retained for the bead squares and the bead cubes. They will be supplied by The House of Childhood, 16 Horatio Street, New York.[7]It would, perhaps, be better in this first counting-frame to have the beads not only of different colors, but of different sizes, according to the value of the wires, as was suggested to me by a Portuguese professor who had been taking my course.

[6]At the present time, because of the difficulty of getting beads of certain colors, owing to war conditions, the following colors have been approved by Dr. Montessori to replace those originally used: 10 bead bar, gold; 9, dark blue; 8, white; 7, light green; 6, light blue; 5, yellow; 4, pink; 3, green; 2, yellow-green; 1, gold. These same colors are retained for the bead squares and the bead cubes. They will be supplied by The House of Childhood, 16 Horatio Street, New York.

[7]It would, perhaps, be better in this first counting-frame to have the beads not only of different colors, but of different sizes, according to the value of the wires, as was suggested to me by a Portuguese professor who had been taking my course.

THE MULTIPLICATION TABLE

Material:The material for the multiplication table is in several parts. There is a square cardboard with a hundred sockets or indentures (ten rows, ten in a row), and into each of these indentures may be placed a bead. At the top of the square and corresponding to each vertical line of indentures are printed the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. At the left is an opening into which may be slipped a small piece of cardboard upon which are printed in red the numbers from 1 to 10. This cardboard serves as the multiplicand; and it can be changed, for there are ten of these slips, bearing the ten different numbers. In the upper left-hand corner is a small indenture for a little red marker, but this detail is merely secondary. This arithmetic board is a white square with a red border; and with it comes an attractive box containing a hundred loose beads.

The exercise which is done with this material is very simple. Suppose that 6 is to be multiplied by the numbers in turn from 1 to 10: 6 × 1; 6 × 2; 6 × 3; 6 × 4; 6 × 5; 6 × 6; 6 × 7; 6 × 8; 6 × 9; 6 × 10. Opposite the sixth horizontal line of indentures, in the small opening at the left is slipped the card bearing the number 6. In multiplying the 6 by 1, the child performs two operations: first, he puts the red marker above the printed 1 at the top of the board, and then he puts six beads (correspondingto the number 6) in a vertical column underneath the number 1. To multiply 6 by 2, he places the red marker over the printed 2, and adds six more beads, placed in a column under number 2. Similarly, multiplying 6 by 3, the red marker must be placed over the 3, and six more beads added in a vertical line under that number. In this manner he proceeds up to 6 × 10.

The shifting of the little red marker serves to indicate the multiplier and requires constant attention on the part of the child and great exactness in his work.

Multiplication TableCOMBINATION OFTHREEWITH THE NUMBERS 1 TO 103 × 1 = ___________3 × 2 = ___________3 × 3 = ___________3 × 4 = ___________3 × 5 = ___________3 × 6 = ___________3 × 7 = ___________3 × 8 = ___________3 × 9 = ___________3 × 10 = ___________

While the child is doing these operations he is writing down the results. For this purpose there is specially prepared paper with an attractive heading which the child can place at the right of his multiplication board. There are ten sets of this paper in a series and ten series in a set,making a hundred sheets with each set of multiplication material. The accompanying cut shows a sheet prepared for the multiplication of number 3.

Everything is ready on the printed sheet; the child has only to write the results which he obtains by adding the beads in columns of three each. If he makes no error he will write: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

In this way he will work out and write down the whole series from 1 to 10; and as there are ten copies of each sheet, he can repeat each exercise ten times.

Thus the child learns by memory each of these multiplications. And we find that he helps himself to memorize even in other ways. He walks up and down holding the multiplication sheet, which he looks at from time to time. It is a sheet which he himself has filled, and he may be memorizing seven times six, forty-two; seven times seven, forty-nine; seven times eight, fifty-six, etc.

This material for the multiplication table is one of the most interesting to the children. They fill six or seven sets, one after the other, and work for days and weeks on this one exercise. Almost all of them ask to take it home with them. With us, the first time the material was presented a small uprising took place, for they all wished to carry it away with them. As this was not permitted the children implored their mothers to buy it for them, and it was with difficulty that we made them understand that it was not on the market and therefore could not be purchased. But the children could not give up the idea. One older girl headed the rebellion. "The Dottoressa wants to try an experiment with us," she said. "Well, let's tell her that unless she gives us the material for the multiplication table we won't come to school any more."

This threat in itself was impolite, and yet it was interesting; for the multiplication table, the bug-bear of all children, had become so attractive and tempting a thing that it had made wolves out of my lambs!

When the children have repeatedly filled a whole series of these blanks, with the aid of the material, they are given a test-card by means of which they may compare their work for verification, and see whether they have made any errors in their multiplication. Table by table, number by number, they do the work of comparing each result with the number which corresponds to it in each one of the ten columns. When this has been done carefully, the children possess their own series, the accuracy of which they are able to guarantee themselves.

Multiplication TablePRESENTING THE COMBINATIONS OF NUMBERS IN THEPROGRESSIVE SERIES FROM 1 TO 10

1 × 1 = 12 × 1 = 23 × 1 = 34 × 1 = 45 × 1 = 51 × 2 = 22 × 2 = 43 × 2 = 64 × 2 = 85 × 2 = 101 × 3 = 32 × 3 = 63 × 3 = 94 × 3 = 125 × 3 = 151 × 4 = 42 × 4 = 83 × 4 = 124 × 4 = 165 × 4 = 201 × 5 = 52 × 5 = 103 × 5 = 154 × 5 = 205 × 5 = 251 × 6 = 62 × 6 = 123 × 6 = 184 × 6 = 245 × 6 = 301 × 7 = 72 × 7 = 143 × 7 = 214 × 7 = 285 × 7 = 351 × 8 = 82 × 8 = 163 × 8 = 244 × 8 = 325 × 8 = 401 × 9 = 92 × 9 = 183 × 9 = 274 × 9 = 365 × 9 = 451 × 10 = 102 × 10 = 203 × 10 = 304 × 10 = 405 × 10 = 50

6 × 1 = 67 × 1 = 78 × 1 = 89 × 7 = 910 × 1 = 106 × 2 = 127 × 2 = 148 × 2 = 169 × 2 = 1810 × 2 = 206 × 3 = 187 × 3 = 218 × 3 = 249 × 3 = 2710 × 3 = 306 × 4 = 247 × 4 = 288 × 4 = 329 × 4 = 3610 × 4 = 406 × 5 = 307 × 5 = 358 × 5 = 409 × 5 = 4510 × 5 = 506 × 6 = 367 × 6 = 428 × 6 = 489 × 6 = 5410 × 6 = 606 × 7 = 427 × 7 = 498 × 7 = 569 × 7 = 6310 × 7 = 706 × 8 = 487 × 8 = 568 × 8 = 649 × 8 = 7210 × 8 = 806 × 9 = 547 × 9 = 638 × 9 = 729 × 9 = 8110 × 9 = 906 × 10 = 607 × 10 = 708 × 10 = 809 × 10 = 9010 × 10 = 100

The children should write down on the following form, in the separate columns, their verified results: under the 2, the column of the 2's; under the 3, the column of the 3's; under the 4, the column of the 4's, etc.

123456789102345678910

Then they get the following table, which is identical with the test cards included in the material. It is a summary of the multiplication table—the famous Pythagorean table.

The Multiplication Table

1234567891024681012141618203691215182124273048121620242832364051015202530354045506121824303642485460714212835424956637081624324048566472809182736455463728190102030405060708090100

The child has built up his multiplication table by a long series of processes each incomplete in itself. It will now be easy to teach him to read it as a "multiplication table," for he already knows it by memory. Indeed, he will be able to fill the blanks from memory, the only difficulty being the recognition of the square in which he must write the number, which must correspond both to the multiplicand and to the multiplier.

We offer ten of these blank forms in our material. When the child, left free to work as long as he wishes on these exercises, has finished them all, he has certainly learned the multiplication table.

DIVISION

Material:The same material may be used for division, except the blanks, which are somewhat different.

Take any number of beads from the box and count them. Let us suppose that we have twenty-seven. This number is written in the vacant space at the left-hand side of the division blank.

DivisionRemainder: 2 = __________________: 3 = __________________: 4 = __________________: 5 = __________________27: 6 = __________________: 7 = __________________: 8 = 33: 9 = 3:10 = 27

Then taking the box of beads and the arithmetic board with the hundred indentures we proceed to the operation.

Let us first divide 27 by 10. We place ten beads in a vertical line under the 1; then in the next row ten more beads under the 2. The beads, however, are not sufficient to fill the row under the 3. Now on the paper prepared for division we write 2 on a line with the 10to the left of the vertical line, and to the right of the same vertical line we write the remainder 7.

To divide 27 by 9, nine beads are counted out in the first row, then nine in the second row under the 2, and still another nine under the 3. There are no beads left over. So the figure 3 is written after the equal-sign (=) on a line with 9.

To divide 27 by 8 we count out eight beads, place them in a row under the 1, and then fill like rows under the 2 and the 3; in the fourth row there are only three beads. They are the remainder. And so on.

A package of one hundred division blanks comes in an attractive dark green cover tied with a silk ribbon. The multiplication blanks, with their tables for comparison and summary tables, come in a parchment envelope tied with leather strings.

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