Montessori Elementary Materials - English Restoration --- ## PART III ## Chapter 15 - Arithmetic ## I **ARITHMETICAL OPERATIONS** ### Numbers: 1-10 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 a *single* object containing signs by which the relative quantity of unity can be recognized, instead of by *a number of different* units, 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 be\[206\] handled 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.[\[6\]](https://www.gutenberg.org/cache/epub/42869/pg42869-images.html#Footnote_6_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, some\[207\] in 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 as *1917* can 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 by *colors* instead of by *quantities* of 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 more\[208\] quickly!" This declaration indicates that he has conquered the first bead material. ### Tens, Hundreds, and Thousands 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. The\[209\] same quantity which formerly impressed the child by its length is now, in its broad, folded form, presented as a *surface* quantity. 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—*seven* hundreds! 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 of *their* comrades: "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. \[210\] 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. ### Counting-Frames 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. The\[211\] numeral 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."[\[7\]](https://www.gutenberg.org/cache/epub/42869/pg42869-images.html#Footnote_7_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. \[212\] 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 the\[213\] tens 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 its *position* in 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 this *position* occupied 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 at\[214\] its 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.  > **The bead material used for addition and subtraction. Each of the nine numbers is of different colored beads.**  > **Counting 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, and *in 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 numbers *on the same line* and 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.  > **The bead cube of 10; ten squares of 10; and chains of 10, of 100, and of 1000 beads.**  > **This 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.\[215\]\ \[216\] 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: [\[6\]](https://www.gutenberg.org/cache/epub/42869/pg42869-images.html#FNanchor_6_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. [\[7\]](https://www.gutenberg.org/cache/epub/42869/pg42869-images.html#FNanchor_7_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 (corresponding\[218\] to 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. 3 | Multiplication Table | | -------------------- | | COMBINATION OF<br>**THREE**<br>WITH THE NUMBERS 1 TO 10 | | 3 × 1 = ___________<br>3 × 2 = ___________<br>3 × 3 = ___________<br>3 × 4 = ___________<br>3 × 5 = ___________<br>3 × 6 = ___________<br>3 × 7 = ___________<br>3 × 8 = ___________<br>3 × 9 = ___________<br>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,\[219\] 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." \[220\] 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 Table\ PRESENTING THE COMBINATIONS OF NUMBERS IN THE\ PROGRESSIVE SERIES FROM 1 TO 10** | 1 × 1 = 1 | 2 × 1 = 2 | 3 × 1 = 3 | 4 × 1 = 4 | 5 × 1 = 5 | | --------- | --------- | --------- | --------- | --------- | | 1 × 2 = 2 | 2 × 2 = 4 | 3 × 2 = 6 | 4 × 2 = 8 | 5 × 2 = 10 | | 1 × 3 = 3 | 2 × 3 = 6 | 3 × 3 = 9 | 4 × 3 = 12 | 5 × 3 = 15 | | 1 × 4 = 4 | 2 × 4 = 8 | 3 × 4 = 12 | 4 × 4 = 16 | 5 × 4 = 20 | | 1 × 5 = 5 | 2 × 5 = 10 | 3 × 5 = 15 | 4 × 5 = 20 | 5 × 5 = 25 | | 1 × 6 = 6 | 2 × 6 = 12 | 3 × 6 = 18 | 4 × 6 = 24 | 5 × 6 = 30 | | 1 × 7 = 7 | 2 × 7 = 14 | 3 × 7 = 21 | 4 × 7 = 28 | 5 × 7 = 35 | | 1 × 8 = 8 | 2 × 8 = 16 | 3 × 8 = 24 | 4 × 8 = 32 | 5 × 8 = 40 | | 1 × 9 = 9 | 2 × 9 = 18 | 3 × 9 = 27 | 4 × 9 = 36 | 5 × 9 = 45 | | 1 × 10 = 10 | 2 × 10 = 20 | 3 × 10 = 30 | 4 × 10 = 40 | 5 × 10 = 50 |   | 6 × 1 = 6 | 7 × 1 = 7 | 8 × 1 = 8 | 9 × 7 = 9 | 10 × 1 = 10 | | --------- | --------- | --------- | --------- | ----------- | | 6 × 2 = 12 | 7 × 2 = 14 | 8 × 2 = 16 | 9 × 2 = 18 | 10 × 2 = 20 | | 6 × 3 = 18 | 7 × 3 = 21 | 8 × 3 = 24 | 9 × 3 = 27 | 10 × 3 = 30 | | 6 × 4 = 24 | 7 × 4 = 28 | 8 × 4 = 32 | 9 × 4 = 36 | 10 × 4 = 40 | | 6 × 5 = 30 | 7 × 5 = 35 | 8 × 5 = 40 | 9 × 5 = 45 | 10 × 5 = 50 | | 6 × 6 = 36 | 7 × 6 = 42 | 8 × 6 = 48 | 9 × 6 = 54 | 10 × 6 = 60 | | 6 × 7 = 42 | 7 × 7 = 49 | 8 × 7 = 56 | 9 × 7 = 63 | 10 × 7 = 70 | | 6 × 8 = 48 | 7 × 8 = 56 | 8 × 8 = 64 | 9 × 8 = 72 | 10 × 8 = 80 | | 6 × 9 = 54 | 7 × 9 = 63 | 8 × 9 = 72 | 9 × 9 = 81 | 10 × 9 = 90 | | 6 × 10 = 60 | 7 × 10 = 70 | 8 × 10 = 80 | 9 × 10 = 90 | 10 × 10 = 100 | \[221\] 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. |   |  1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |   | | - | -- | - | - | - | - | - | - | - | - | -- | - | |   |  2 |   |   |   |   |   |   |   |   |   |   | |   |  3 |   |   |   |   |   |   |   |   |   |   | |   |  4 |   |   |   |   |   |   |   |   |   |   | |   |  5 |   |   |   |   |   |   |   |   |   |   | |   |  6 |   |   |   |   |   |   |   |   |   |   | |   |  7 |   |   |   |   |   |   |   |   |   |   | |   |  8 |   |   |   |   |   |   |   |   |   |   | |   |  9 |   |   |   |   |   |   |   |   |   |   | |   | 10 |   |   |   |   |   |   |   |   |   |   | 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** |   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |   | | - | - | - | - | - | - | - | - | - | - | -- | - | |   | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |   | |   | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |   | |   | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |   | |   | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |   | |   | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |   | |   | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |   | |   | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |   | |   | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |   | |   | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |   | \[222\] 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. --- \[223\] ## III **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. | Division | Remainder | | -------- | --------- | |   | : 2 = _________ | _________ | |   | : 3 = _________ | _________ | |   | : 4 = _________ | _________ | |   | : 5 = _________ | _________ | | 27 | : 6 = _________ | _________ | |   | : 7 = _________ | _________ | |   | : 8 =    3 |    3 | |   | : 9 =    3 |   | |   | :10 =    2 |    7 | 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 10\[224\] to 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. | Division | Remainder | | -------- | --------- | | : 2 = | _________ | | : 3 = | _________ | | : 4 = | _________ | | : 5 = | _________ | | : 6 = | _________ | | : 7 = | _________ | | : 8 = | _________ | | : 9 = | _________ | | :10 = | _________ | --- \[225\] ## IV **OPERATIONS IN SEVERAL FIGURES** By this time the child can easily perform operations with numbers of two or more figures, for he possesses all the materials necessary and is already prepared to make use of them. For this work we have for the first three operations, addition, subtraction, and multiplication, a counting-frame; and for division a more complicated material which will be described later on. ### ADDITION Addition on the counting frame is a most simple operation, and therefore is very attractive. Let us take, for example, the following: 1320 +\  435\ = First we slide over the beads to represent the first number: 1 on the thousands-wire, 3 on the hundreds-wire, and 2 on the tens-wire. Then we place next to them the beads representing the second number: 4 on the hundreds-wire, 3 on the tens-wire, and 5 on the units-wire. Now there remains nothing to be done except to write the number shown by the beads in their present position: 1755. \[226\]  --- **This shows the second counting-frame used in arithmetic. The child is writing the number she has just formed on her frame. (*The Rivington Street Montessori School, New York.*)** When the problem is a more complicated one, the beads for any one wire amounting to more than 10, the solution is still very easy. In that case the entire ten beads would be returned to their original position and in their stead one corresponding bead of the next lower wire would be slipped over. Then the operation is continued. Take, for example: 390 +\ 482\ = We first place the beads representing 390: that is, 3 on the hundreds-wire and 9 on the tens-wire; or, vice versa, beginning with the units, we would first place the 9 tens and then the 3 hundreds. For the second number we place 4 beads for the hundreds and then we begin to place the 8 tens. But when we have placed only one ten, the wire is full; so the ten tens are returned to their original position and to represent them we move over another bead on the hundreds-wire; then we continue to place the beads of the tens which now, after having converted 10 of them into 1 hundred, remain but 7. Or we can begin the addition by placing the beads for the units before we place those for the hundreds; and in that case we move on the hundreds-wire first the bead representing the ten beads on the wire above, and then the 4 hundreds which must be added. Finally we write down the sum as now indicated by the position of the beads: 872. With a larger counting-frame it is possible to perform in this manner very complicated problems in addition.  > **The two little girls are working out problems in seven figures. (*The Washington Montessori School, Washington, D. C.*)** ### SUBTRACTION The counting-frame lends itself equally well to problems in subtraction. Let us take, for example, the following: 8947-\ 6735\ = We place the beads representing the first number; then from them we take the beads representing the second number The beads remaining indicate the difference between the two numbers; and this is written: 2212. Then comes the more complicated problem where it is necessary to borrow from a higher denomination. When the beads of one wire are exhausted, we move over the entire ten and take to represent them one bead from the lower wire; then we continue the subtraction. For example: 8954-\ 7593\ = We move the beads representing the first number; then we take 3 beads from the units. Now we begin to subtract the tens. We wish to take away 9 beads; but when we have moved five the wire is empty, and there are still four more to be moved. We take away one bead from the hundreds-wire and replace the entire ten on the tens-wire; and then we continue to move beads on the tens-wire until we have taken a total of nine—that is, we now move the other four. On the hundreds-wire there remain but 8 beads, and from them we take the 5, etc. Our final remainder is 1361. \[227\]\ \[228\] It is easy to see how familiar and clear to the child the technique of "borrowing" becomes. ### MULTIPLICATION When there is a number to be multiplied by more than one figure, the child not only knows the multiplication table but he easily distinguishes the units from the tens, hundreds, etc., and he is familiar with their reciprocal relations. He knows all the numbers up to a million and also their positions in relation to their value. He knows from habitual practise that a unit of a higher order can be exchanged for ten of a lower order. To have the child attack this new difficulty successfully one need only tell him that each figure of the multiplier must multiply in turn each figure of the multiplicand and that the separate products are placed in columns and then added. The analytical processes hold the child's attention for a long period of time; and for this reason they have too great a formative value not to be made use of in the highest degree. They are the processes which lead to that inner maturation which gives a deeper realization of cognitions and which results in bursts of spontaneous synthesis and abstraction. The children, by rapidly graduated exercises, soon become accustomed to writing the analysis of each multiplication (according to its factors) in such a way that, once the work of arranging the material is finished, nothing is left for them to do but to perform the multiplications which they already have learned in the simple multiplication table. Here is an example of the analysis of a multiplication with three figures appearing in both the multiplicand and the multiplier: 356 × 742. \[229\] | 742 = |   | 2 units | | ----- | - | ------- | | 4 tens | | 7 hundreds |   | 356 = |   | 6 units | | ----- | - | ------- | | 5 tens | | 3 hundreds | Each of the first numbers is combined with the three figures of the other number in the following manner: | u. 6 |   |  × u. 2 =  |   | 12 *units* | | ---- | - | ---------- | - | -------- | | t. 5 | 10 tens | | h. 3 | 6 hundreds |   | u. 6 |   |  × u. 4 =  |   | 24 *units* | | ---- | - | ---------- | - | -------- | | t. 5 | 20 tens | | h. 3 | 12 hundreds |   | u. 6 |   |  × h. 7 =  |   | 42 *hundreds* | | ---- | - | ---------- | - | ----------- | | t. 5 | 35 thousands | | h. 3 | 21 tens of thousands | When this analysis is written down, the work on the counting-frames begins. Here the operations are performed in the following manner: 2 × 6 units necessitate the bringing forward of the ten beads on the first wire. However, even those do not suffice. So they are slid back and one bead on the second wire is brought forward, to represent the ten replaced, and on the first wire two beads are brought forward (12). Next we take 2 × 5 tens. There is already one bead on the tens-wire and to this should be added ten more, but instead we bring forward one bead on the hundreds-wire. At this point in the operation the beads are distributed on the wires in this manner: 2\ 1\ 1 Now comes 2 × 3 hundreds, and six beads on the corresponding wire are brought forward. When the multiplication by the units of the multiplier is finished, the beads on the frame are in the following order: \[230\] 2\ 1\ 7 We pass now to the tens: 4 × 6 = 24 tens. We must therefore bring forward four beads on the tens-wire and two on the hundreds-wire: 2\ 5\ 9 4 × 5 = 20 hundreds, therefore two thousands: 2\ 5\ 9\ 2 4 × 3 thousands = 12 thousands; so we bring forward two beads on the thousands-wire and one on the ten-thousands-wire: 2\ 5\ 9\ 4\ 1 Now we take the hundreds: 7 × 6 hundreds are 42 hundreds; therefore we slide four beads on the thousands-wire and two on the hundreds-wire. But there already were nine beads on this wire, so only one remains and the other ten give us instead another bead on the thousands-wire: 2\ 5\ \[231\]1\ 9\ 1 5 × 7 thousands = 35 thousands, which is the same as five thousands and three ten-thousands. Three beads on the fifth wire and five on the fourth are brought forward; but on the fourth wire there already were nine beads, so we leave only four, exchanging the other ten for one bead on the fifth wire: 2\ 5\ 1\ 4\ 5 Finally 7 × 3 ten-thousands = 21 ten-thousands. One bead is brought forward on the fifth wire and two on the hundred-thousands-wire. At the end of the operation the beads will be distributed as follows: | 2 | beads | on | the | first | wire | (units) | | - | ----- | -- | --- | ----- | ---- | ------- | | 5 | " | " | " | second | " | (tens) | | 1 | " | " | " | third | " | (hundreds) | | 4 | " | " | " | fourth | " | (thousands) | | 6 | " | " | " | fifth | " | (tens of thousands) | | 2 | " | " | " | sixth | " | (hundreds of thousands) | This distribution translated into figures gives the following number: 264,152. This may be written as a result right after the factors without the partial products: that is, 742 × 356 = 264,152. Although this description may sound very complicated, the exercise on the counting-frame is an easy and most interesting\[232\] arithmetic game. And this game, which contains the secret of such surprising results, not only is an exercise which makes more and more clear the decimal relations of reciprocal value and position, but also it explains the manner of procedure in abstract operations.  > **Fig. 1. The disposition of the beads for the number 49,152.** In fact, in the multiplication as commonly performed:      356 ×\      742\      712\   1424  \ 2492     \ 264152 the same operations are involved; but the figures, once written down, cannot be modified as is possible on the frame by moving the beads and substituting beads of\[233\] higher value for those of lower value when the ten beads of one wire, as a mechanical result of the structure of the frame, are all used. As multiplication is ordinarily written, such substitutions cannot be made; but the partial products must be written down in order, placed in column according to their value, and finally added. This is a much longer piece of work, because the act of writing a figure is more complicated than that of moving a bead which slides easily on the metal wire. Again, it is not so clear as the work with the beads, once the child is accustomed to handling the frame and no longer has any doubt as to the position of the different values, and when it has become a sort of routine to substitute one bead of the lower wire for the ten beads of the upper wire which have been exhausted. Furthermore, it is much easier to add new products without the possibility of making a mistake. Let us go back to the point in the operation where the beads on the frame read thus: 2\ 5\ 1\ 9\ 1 and it was necessary to add 35 thousands—five beads to the thousands-wire and three beads to the ten-thousands-wire. The three beads on the fifth wire can be brought forward without any thought as to what will happen on the wire above when the five are added to the nine. Indeed, what takes place there does not make any difference, for it is not necessary that the operation on the higher wire precede that on the lower wire. \[234\]  > **Fig. 2. The disposition of the beads for the number 54,152; after adding 5 thousands to the number 49,152.** In adding the five beads to the nine beads only four remain on the fourth wire, since the other ten are substituted by a bead on the lower wire; this bead may be brought forward even after the three for the ten-thousands have been placed. By the use of the frame the child acquires remarkable dexterity and facility in calculating, and this makes his work in multiplication much more rapid. Often one child, working out an example on paper, has finished only the first partial multiplication when another child, working at the frame, has completed the problem and knows the final product. It is interesting even among adults to watch two compete in the same problem, one at the frame and the other using the ordinary method on paper. It is very interesting, also, not to work out on the frame the individual products in the sequence indicated in analyzing the factors, but to work them out by chance. Indeed,\[235\] it does not matter whether the beads are moved in the order of their alignment or at random. The beads on the ten-thousands-wire may be moved first, then the hundreds, the units, and finally the thousands. These exercises, which give such a deep understanding of the operations of arithmetic, would be impossible with the abstract operation which is performed only by means of figures. And it is evident that the exercises can be amplified to any extent as a pleasing game. ### MULTIPLYING ON RULED PAPER Take, for example, 8640 × 2531. We write the figures of the multiplicand one under the other but in their relative positions; this also can be written by filling in the vacant spaces with zeros. In this way we repeat the multiplicand as many times as there are figures in the multiplier; but instead of writing beside these figures the words units, tens, etc., we indicate this with zeros, which, for the sake of clearness, we fill in till they resemble large dots. The child already knows, from his previous exercises, that zero indicates the position of a figure and that multiplying by ten changes this position. Therefore zeros in the multiplier would cause a corresponding change of position in the figures of the multiplicand. The accompanying figure shows clearly what it is not so easy to explain in words.  **Fig. 3.** We are now ready for the usual procedure of multiplication. A child of seven years reaches this stage very easily after having done our preliminary exercises, and then it does not matter to him how many figures he has to use. Indeed, he is very fond of working with numbers of unheard of figures, as is shown in the following\[236\] example—one of the usual exercises done by the children, who of themselves choose the multiplicand and the multiplier; the teacher would never think of giving such enormous numbers. They can now perform the operation 22,364,253 × 345,234,611 22364253 ×\               345234611\ 22364253\ 22364253\ 134185518\ 89457012\ 67092759\ 44728506\ 111821265\ 89457012\ 67092759                \ 7720914184760583 \[237\] without analysis of factors and without help from the frames but by the method commonly used. This may be seen by the way in which the example is written out and then done by the child. ### LONG DIVISION Not only is it possible to perform long division with our bead material, but the work is so delightful that it becomes an arithmetical pastime especially adapted to the child's home activities. Using the beads clarifies the different steps of the operation, creating almost a *rational arithmetic* which supersedes the common empirical methods, that reduce the mechanism of abstract operations to a simple *routine*. For this reason, these pastimes prepare the way for the rational processes of mathematics which the child meets in the higher grades. The bead frame will no longer suffice here. We need the square arithmetic board used for the first partial multiplications and for short division. However, we require several such boards and an adequate provision of beads. The work is too complicated to be described clearly, but in practise it is easy and most interesting. It is sufficient here to suggest the method of procedure with the material. The units, tens, hundreds, etc., are expressed by different-colored beads: *units*, white; *tens*, green; *hundreds*, red. Then there are racks of different colors: *white* for the simple units, tens, and hundreds; *gray* for the thousands; *black* for the millions. There also are boxes, which on the outside are white, gray, or black, and on the inside white, green, or red. And for each box there is a corresponding rack containing ten tubes with ten beads in each. Suppose we must divide 87,632 by 64. Five of the\[238\] boxes are put in a row, arranged from left to right according to the value of their color, as follows: two gray boxes—one green inside and the other white—and three white boxes with the inside respectively red, green, and white. In the first box to the left we put 8 green beads; in the second box 7 white beads; in the third, 6 red beads; in the fourth 3 green beads; and in the fifth box 2 white beads. Back of each box is one of the racks with ten tubes filled with beads of corresponding colors. These beads—ten in each tube—are used in exchanging the units of a higher denomination for those of a lower.  > **The child here is solving a problem in long division. (*A Montessori School, Barcelona, Spain.*)** There are two arithmetic boards, one next to the other, placed below the row of boxes. In the one to the left, the little cardboard with the figure 6 is inserted in the slot we have described, and in the other to the right the figure 4. Now to divide 87,632 by 64, place the first two boxes at the left (containing 8 and 7 beads respectively) above the two arithmetic boards. On the first board the eight beads are arranged in rows of six, as in the more simple division. On the second board the seven beads are arranged in rows of four, corresponding to the number indicated by the red figure. The two quotients must be reduced with reference to the quotient in the first arithmetic board. All the other is considered as a remainder. The quotient in this case is 1 and the remainders are 2 on the first board and 3 on the second. When this is finished, the boxes are moved up one place and then the first box is out of the game, its place having been taken by the second box; so the gray-green box is no longer above the first board but the gray-white one instead, and above the second board we must place the box with the red beads.  > **The illustration at the top shows the square and the cube of 4 and of 5. That in the middle shows the arithmetic board being used for multiplication. In the photograph at the bottom a problem in division is being worked out on the arithmetic-board: 26 ÷ 4 = 6 and 2 remainder.** Now the beads must be adjusted. The two beads that are left over on the card marked with the number 6 are green but the box above this card is the gray-white one. We must therefore change the green beads into white beads, taking for each one of them a tube of ten white beads. The white beads which were left over on the other card must be brought to the card above which the white box is now placed. We have only to arrange the white beads now in rows of six while the other box of red beads is emptied on to the second board in rows of four, as in simple division. With the material arranged in this way according to color, we proceed to the reduction, which is done by exchanging one bead of a higher denomination for ten of a lower. Thus, for example, in the present case we have twenty-three white beads distributed on the first board in rows of six, which gives a quotient of three and a remainder of five. On the second board there are six red beads distributed in rows of four, giving a quotient of one with a remainder of two. Now the work of reduction begins. This consists in taking one by one the beads from the board to the left—in this case the white—and exchanging them for ten red beads, which in turn are placed in rows of four on the other board until the quotients on the two cards are alike. What is left over is the remainder. In this case it is necessary to change only the one white bead so as to have the other quotient reach three with a remainder of four. The same process is continued until all the boxes are used. The final remainder is the one to be written down with the quotient. The exercise requires great patience and exactness, but\[239\]\ \[240\] it is most interesting and might be called an excellent game of solitaire for children for home use. There is no intellectual fatigue but much movement and much intense attention. The quotients and remainders may be written on a prepared sheet of paper, so as to be verified by the teacher. When the child has performed many of these exercises he comes spontaneously to try to foresee the result of an operation without having to make the material exchange and arrangement of the beads; hence to shorten the mechanical process. When at length he can "see" the situation at a glance, he will be able to do the most difficult division by the ordinary processes without experiencing any fatigue, or without having been obliged to endure tiring progressive lessons and humiliating corrections. Not only will he have learned how to perform long divisions but he will have become a master of their mechanism. He will realize each step, in ways that the children of ordinary secondary schools possibly never will be able to understand, when through the usual methods of rational mathematics they approach the incomprehensible operations which they have performed for several years without considering the reasons for them. --- \[241\] ## V **EXERCISES WITH NUMBERS** ### Multiples, Prime Numbers, Factoring When the child, by the aid of all this material, has had a chance to grasp the fundamental ideas relating to the four operations and has passed on to the execution of them in the abstract, he is ready to continue on the numerical processes which will lead to a more profound study preparatory to the more complex problems that await him in the secondary schools. These studies are, however, a means of helping him to remember the things he already knows and to enlarge upon them. They come to him as a pastime, as an agreeable manner of thinking over either in school or at home the ideas which he already has gained. One of the first exercises is that of continuing the multiplication of each number by the series of 1 to 10 which was begun by the exercises on the multiplication tables. This should be done in the abstract: that is, without recourse to the material. Let us, however, set some limit—we will stop when each product has reached 100. In order that these series of exercises may each be in one column the first exercises will stop with 50 and another can be used for the numbers from 51 to 100. The two following tables (A and B) are the result. These are prepared in this manner in our material so that the child may compare his work with them. \[242\] **TABLE A** | 2× 1= 2 | 3× 1= 3 | 4× 1= 4 | 5× 1= 5 | 6× 1= 6 | 7× 1= 7 | 8× 1= 8 | 9× 1= 9 | 10× 1=10 | | ------- | ------- | ------- | ------- | ------- | ------- | ------- | ------- | -------- | | 2× 2= 4 | 3× 2= 6 | 4× 2= 8 | 5× 2=10 | 6× 2=12 | 7× 2=14 | 8× 2=16 | 9× 2=18 | 10× 2=20 | | 2× 3= 6 | 3× 3= 9 | 4× 3=12 | 5× 3=15 | 6× 3=18 | 7× 3=21 | 8× 3=24 | 9× 3=27 | 10× 3=30 | | 2× 4= 8 | 3× 4=12 | 4× 4=16 | 5× 4=20 | 6× 4=24 | 7× 4=28 | 8× 4=32 | 9× 4=36 | 10× 4=40 | | 2× 5=10 | 3× 5=15 | 4× 5=20 | 5× 5=25 | 6× 5=30 | 7× 5=35 | 8× 5=40 | 9× 5=45 | 10× 5=50 | | 2× 6=12 | 3× 6=18 | 4× 6=24 | 5× 6=30 | 6× 6=36 | 7× 6=42 | 8× 6=48 | | 2× 7=14 | 3× 7=21 | 4× 7=28 | 5× 7=35 | 6× 7=42 | 7× 7=49 | | 2× 8=16 | 3× 8=24 | 4× 8=32 | 5× 8=40 | 6× 8=48 | | 2× 9=18 | 3× 9=27 | 4× 9=36 | 5× 9=45 | | 2×10=20 | 3×10=30 | 4×10=40 | 5×10=50 | | 2×11=22 | 3×11=33 | 4×11=44 | | 2×12=24 | 3×12=36 | 4×12=48 | | 2×13=26 | 3×13=39 | | 2×14=28 | 3×14=42 | | 2×15=30 | 3×15=45 | | 2×16=32 | 3×16=48 | | 2×17=34 | | 2×18=36 | | 2×19=38 | | 2×20=40 | | 2×21=42 | | 2×22=44 | | 2×23=46 | | 2×24=48 | | 2×25=50 | \[243\] **TABLE B** | 2×26= 52 | 3×17=51 | 4×13= 52 | 5×11= 55 | 6× 9=54 | 7× 8=56 | 8× 7=56 | 9× 6=54 | 10× 6= 60 | | -------- | ------- | -------- | -------- | ------- | ------- | ------- | ------- | --------- | | 2×27= 54 | 3×18=54 | 4×14= 56 | 5×12= 60 | 6×10=60 | 7× 9=63 | 8× 8=64 | 9× 7=63 | 10× 7= 70 | | 2×28= 56 | 3×19=57 | 4×15= 60 | 5×13= 65 | 6×11=66 | 7×10=70 | 8× 9=72 | 9× 8=72 | 10× 8= 80 | | 2×29= 58 | 3×20=60 | 4×16= 64 | 5×14= 70 | 6×12=72 | 7×11=77 | 8×10=80 | 9× 9=81 | 10× 9= 90 | | 2×30= 60 | 3×21=63 | 4×17= 68 | 5×15= 75 | 6×13=78 | 7×12=84 | 8×11=88 | 9×10=90 | 10×10=100 | | 2×31= 62 | 3×22=66 | 4×18= 72 | 5×16= 80 | 6×14=84 | 7×13=91 | 8×12=96 | 9×11=99 | | 2×32= 64 | 3×23=69 | 4×19= 76 | 5×17= 85 | 6×15=90 | 7×14=98 | | 2×33= 66 | 3×24=72 | 4×20= 80 | 5×18= 90 | 6×16=96 | | 2×34= 68 | 3×25=75 | 4×21= 84 | 5×19= 95 | | 2×35= 70 | 3×26=78 | 4×22= 88 | 5×20=100 | | 2×36= 72 | 3×27=81 | 4×23= 92 | | 2×37= 74 | 3×28=84 | 4×24= 96 | | 2×38= 76 | 3×29=87 | 4×25=100 | | 2×39= 78 | 3×30=90 | | 2×40= 80 | 3×31=93 | | 2×41= 82 | 3×32=96 | | 2×42= 84 | 3×33=99 | | 2×43= 86 | | 2×44= 88 | | 2×45= 90 | | 2×46= 92 | | 2×47= 94 | | 2×48= 96 | | 2×49= 98 | | 2×50=100 | \[244\] **TABLE C** | 1 |                                                      | 51 |                                                      | | - | ---------------------------------------------------- | -- | ---------------------------------------------------- | | 2 | 52 | | 3 | 53 | | 4 | 54 | | 5 | 55 | | 6 | 56 | | 7 | 57 | | 8 | 58 | | 9 | 59 | | 10 | 60 | | 11 | 61 | | 12 | 62 | | 13 | 63 | | 14 | 64 | | 15 | 65 | | 16 | 66 | | 17 | 67 | | 18 | 68 | | 19 | 69 | | 20 | 70 | | 21 | 71 | | 22 | 72 | | 23 | 73 | | 24 | 74 | | 25 | 75 | | 26 | 76 | | 27 | 77 | | 28 | 78 | | 29 | 79 | | 30 | 80 | | 31 | 81 | | 32 | 82 | | 33 | 83 | | 34 | 84 | | 35 | 85 | | 36 | 86 | | 37 | 87 | | 38 | 88 | | 39 | 89 | | 40 | 90 | | 41 | 91 | | 42 | 92 | | 43 | 93 | | 44 | 94 | | 45 | 95 | | 46 | 96 | | 47 | 97 | | 48 | 98 | | 49 | 99 | | 50 | 100 | \[245\] **TABLE D** | 1 | 53 | | - | -- | | 2 | 54 = 2×27 = 3×18 = 6×9 = | | 3 | 9×6 | | 4 = 2×2 | 55 = 5×11 | | 5 | 56 = 2×28 = 4×14 = 7×8 = | | 6 = 2×3 = 3×2 | 8×7 | | 7 | 57 = 3×19 | | 8 = 2×4 = 4×2 | 58 = 2×29 | | 9 = 3×3 | 59 | | 10 = 2×5 = 5×2 | 60 = 2×30 = 3×20 = 4×15 = | | 11 | 5×12 = 6×10 = 15×4 | | 12 = 2×6 = 3×4 = 4×3 = 6×2 | 61 | | 13 | 62 = 2×31 | | 14 = 2×7 = 7×2 | 63 = 3×21 = 7×9 = 9×7 | | 15 = 3×5 = 5×3 | 64 = 2×32 = 4×16 = 8×8 | | 16 = 2×8 = 4×4 = 8×2 | 65 = 5×13 | | 17 | 66 = 2×33 = 3×22 = 6×11 | | 18 = 2×9 = 3×6 = 6×3 = 9×2 | 67 | | 19 | 68 = 2×34 = 4×17 | | 20 = 2×10 = 4×5 = 5×4 = | 69 = 3×23 | | 10×2 | 70 = 2×35 = 5×14 = 7×10 = | | 21 = 7×3 = 3×7 | 10×7 | | 22 = 2×11 | 71 | | 23 | 72 = 2×36 = 3×24 = 4×18 = | | 24 = 2×12 = 3×8 = 4×6 = | 6×12 = 8×9 = 9×8 | | 6×4 = 8×3 | 73 | | 25 = 5×5 | 74 = 2×37 | | 26 = 2×13 | 75 = 3×25 = 5×15 | | 27 = 3×9 = 9×3 | 76 = 2×38 = 4×19 | | 28 = 2×14 = 4×7 = 7×4 | 77 = 7×11 | | 29 | 78 = 2×39 = 3×26 = 6×13 | | 30 = 2×15 = 3×10 = 5×6 = | 79 | | 6×5 = 10×3 | 80 = 2×40 = 4×20 = 5×16 | | 31 | 8×10 = 10×8 | | 32 = 2×16 = 4×8 = 8×4 | 81 = 3×27 = 9×9 | | 33 = 3×11 | 82 = 2×41 | | 34 = 2×17 | 83 | | 35 = 5×7 = 7×5 | 84 = 2×42 = 3×28 = 4×21 = | | 36 = 2×18 = 3×12 = 4×9 = | 6×14 = 7×12 | | 6×6 = 9×4 | 85 = 5×17 | | 37 | 86 = 2×43 | | 38 = 2×19 | 87 = 3×29 | | 39 = 3×13 | 88 = 2×44 = 4×22 = 8×11 | | 40 = 2×20 = 4×10 = 5×8 = | 89 | | 8×5 = 10×4 | 90 = 2×45 = 3×30 = 5×18 = | | 41 | 6×15 = 9×10 = 10×9 | | 42 = 2×21 = 3×14 = 6×7 = | 91 = 7×13 | | 7×6 | 92 = 2×46 = 4×23 | | 43 | 93 = 3×31 | | 44 = 2×22 = 4×11 | 94 = 2×47 | | 45 = 3×15 = 5×9 = 9×5 | 95 = 5×19 | | 46 = 2×23 | 96 = 2×48 = 3×32 = 4×24 = | | 47 | 6×16 = 8×12 | | 48 = 2×24 = 3×16 = 4×12 = | 97 | | 6×8 = 8×6 | 98 = 2×49 = 7×14 | | 49 = 7×7 | 99 = 3×33 = 9×11 | | 50 = 2×25 = 5×10 = 10×5 | 100 = 2×50 = 4×25 = 5×20 = | | 51 = 3×17 | 10×10 | | 52 = 2×26 = 4×13 | \[246\] To read over a column of the results of each number is to learn them by heart, and it impresses upon the child's memory the series of multiples of each number from 1 to 100. With these tables a child can perform many interesting exercises. He has sheets of long narrow paper. On the left are written the series of numbers from 1 to 50 and from 51 to 100. He compares the numbers on these sheets with the same numbers in the tables, series by series, and writes down the different factors which he thus finds; for example, 6 = 2 × 3; 8 = 2 × 4; 10 = 2 × 5. Then finding the same number in the second column and the other columns his result will read, 6 = 2 × 3 = 3 × 2; 18 = 2 × 9 = 3 × 6 = 6 × 3 = 9 × 2. In this comparison the child will find that some numbers cannot be resolved into factors and their line is blank. By this means he gets his first intuition of prime numbers (Tables C and D). When the child has filled in this work from 1 to 50 and from 51 to 100 and has reduced the numbers to factors and prime numbers he may pass on to some exercises with the beads. The children now meditate, using the material, on the results that they have obtained by comparing these tables. Let us consider, for example, 6 = 2 × 3 = 3 × 2. The child takes six beads, and first makes two groups of three beads and then three groups of two.  And so on for each number he chooses. For example:  \[247\]  The child will try in every way to make other combinations and he will try also to divide the prime numbers into factors. This intelligent and pleasing game makes clear to the child the "divisibility" of numbers. The work that he does in getting these factors by multiplication is really a way of dividing the numbers. For example, he has divided 18 into 2 equal groups, 9 equal groups, 6 equal groups, and 3 equal groups. Previously he has divided 6 into 2 equal groups and then into 3 equal groups. Therefore when it is a question of multiplying the two factors there is no difference in the result whether he multiplies 2 by 3 or 3 by 2; for the inverted order of the factors does not change the product. But in division the object is to arrange the number in equal parts and any modification in this equal distribution of objects changes the character of the grouping. Each separate combination is a different way of dividing the number. The idea of division is made very clear to the child's mind: 6 ÷ 3 = 2, means that the 6 can be divided into three groups, each of which has two units or objects; and 6 ÷ 2 = 3, means that the 6 also can be divided into but two equal groups, each group made up of three units or objects. The relations between multiplication and division are very evident since we started with 6 = 3 × 2; 6 = 2 × 3. This brings out the fact that multiplication may be used to prove division; and it prepares the child to understand the practical steps taken in division. Then some day when he has to do an example in long division, he will\[248\] find no difficulty with the mental calculation required to determine whether the dividend, or a part of it, is divisible by the divisor. This is not the usual preparation for division, though memorizing the multiplication table is indeed used as a preparation for multiplication. From the above exercises (Table D) others might be derived involving further analysis of the same numbers. For example, one of the possible factor groups for the number 40 is 2 × 20. But 20 = 2 × 10; and 10 = 2 × 5. Bringing together the smaller figures into which the larger numbers have been broken, we get 40 = 2 × 2 × 2 × 5; in other words 40 = 23 × 5. This is the result for 60: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 22 × 3 × 5 For these two numbers we get accordingly the prime factors: 23 × 5; and 22 × 3 × 5. What then have the two larger numbers, 40 and 60 in common? The 22 is included in the 23; the series therefore may be written: 22 × 2 × 5; and 22 × 3 × 5. The common element (the greatest common divisor) is 22 × 5 = 20. The proof consists in dividing 60 and 40 by 20, something which will not be possible for any number higher than 20. **TABLE E** |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 | |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 | |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 | |  1    2   3    4   5   6   7  8   9  10<br>11 12 13 14 15 16 17 18 19 20<br>21 22 23 24 25 26 27 28 29 30<br>31 32 33 34 35 36 37 38 39 40<br>41 42 43 44 45 46 47 48 49 50<br>51 52 53 54 55 56 57 58 59 60<br>61 62 63 64 65 66 67 68 69 70<br>71 72 73 74 75 76 77 78 79 80<br>81 82 83 84 85 86 87 88 89 90<br>91 92 93 94 95 96 97 98 99 100 | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | We have test sheets where the numbers from 1 to 100 are arranged in rows of 10, forming a square. Here the child's exercise consists in underlining, in different squares, the multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10. The numbers so underlined stand out like a design in such a way that the child easily can study and compare the tables. For instance, in the square where he underlines the multiples of 2 all the even numbers in the vertical columns are marked; in the multiple of 4 we have the same linear\[249\] grouping—a vertical line—but the numbers marked are alternate numbers; in 6 the same vertical grouping continues, but one number is marked and two are skipped; and again in the multiples of 8 the same design is repeated with the difference that every fourth number is underlined. On the square marked off for the multiples of 3 the numbers marked form oblique lines running from right to left and all the numbers in these oblique lines are underlined. In the multiples of 6 the design is the same but only the alternating numbers are underlined. The 6 therefore, partakes of the type of the 2 and of the 3; and both of these are indeed its factors. --- \[251\] ## VI **SQUARE AND CUBE OF NUMBERS** Let us take two of the two-bead bars (green) which were used in counting in the first bead exercises. Here, however, these form part of another series of beads. Along with these two bars there is a small chain:  By joining two like bars, the chains represent 2 × 2. There is another combination of these same objects—the two bars are joined together not in a chain but in the form of a square:  They represent the same thing: that is to say, as numbers they are 2 × 2; but they differ in position—one has the form of a line, the other of a square. It can be seen from this that if as many bars as there are beads on a bar are placed side by side they form a square. In the series in fact we offer squares of 3 × 3 pink beads; 4 × 4 yellow beads; 5 × 5 pale blue beads; 6 × 6 gray beads; 7 × 7 white beads; 8 × 8 lavender beads; 9 × 9 dark blue beads; and 10 × 10 orange beads; thus reproducing the same colors as were used at the beginning in counting. For every number there are as many bars as there are beads for the number, 3 bars for the 3, 4 for the 4, etc.; in addition there is a chain consisting of an equal number of bars, 3 × 3; 4 × 4; and, as we have seen, there is a square containing another equal quantity. The child not only can count the beads of the chains\[252\] and squares, but he can reproduce them by placing the corresponding single bars either in a horizontal line or laying them side by side in the shape of a square. The number repeated as many times as the unit it contains is really the multiplication of the number by itself. For example, taking the small square of four the child can count four beads on each side; multiplying 4 by 4 we have the number of beads in the square, 16. Multiplying one side by itself (squaring one side) we have the area of the little square. This can be continued for 5, 8, 9, etc. The square of 10 has ten beads on each side. Multiplying 10 by 10, in other words, "squaring" one side we get the entire number of beads forming the area of the square: 100. However, it is not the form alone which gives these results; for if the ten bars which formed the square are placed end to end in a horizontal line, we get the "hundred chain." This can be done with each square; the chain 5 × 5, like the square 5 × 5, contains the same number of beads, 25. We teach the child to write the numbers with symbol for the square: 52 = 25; 72 = 49; 102 = 100, etc. Our material here is manufactured with reference to the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10. It is "offered" to the child, beginning with the smaller numbers. Given the material and freedom, the idea will come of itself and the child will "work" it into his consciousness on them. In this same period we take up also the cubes of the numbers, and there is a similar material for this: that is, the chain of the cube of the number is made up of chains of the square of that number joined by several links which permit of its being folded. There are as many squares for a number as there are units in that number—four\[253\] squares for number 4, six squares for 6, ten squares for 10—and a cube of the beads is formed by placing the necessary number of squares one on top of the other.  Let us consider the cube of four. There is a chain formed by four chains each representing the square of four. They are joined by small links so that the chain can be rolled up lengthwise. The chain of the cube, when thus rolled, gives four squares similar to the separate squares which, when drawn out again, for a straight line.  > **Fig. 5.—This shows only part of the entire chain for 43.** The quantity is always the same: four times the square of four. 4 × 4 × 4 = 42 × 4 = 43. The cube of four comes with the material; but it can be reproduced by placing four loose squares one on top of the other. Looking at this cube we see that it has all its edges of four. Multiplying the area of a square by the number of units contained in the side gives the volume of the cube: 42 × 4. In this way the child receives his first intuitions of the processes necessary for finding a surface and volume. With this material we should not try to teach a great\[254\] deal but should leave the child free to ponder over his own observations—observing, experimenting, and meditating upon the easily handled and attractive material. **\*             \*             \*** Little by little we shall see the slates and copybooks filled with exercises of numbers raised to the square or cube independently of the rich series of objects which the material itself offers the child. In his exercises with the square and cube of the numbers he easily will discover that to multiply by ten it suffices to change the position of the figures—that is to say, to add a zero. Multiplying unity by ten gives 10; ten multiplied by ten is equal to 100; one hundred multiplied by ten is equal to 1,000, etc. Before arriving at this point the child will often either have discovered this fact for himself or have learned it by observing his companions. Some of the fundamental ideas acquired only through laborious lessons by our common school methods are here learned intuitively, naturally, and spontaneously. An interesting study which completes that already made with the "hundred chain" and the "thousand chain" is the comparison of the respective square chain and cube chain. Such differing relations showing the increasing length are most illustrative and make a marked impression upon the child. Furthermore, they prepare for knowledge that is to be used later. Some day when the child hears of "geometric progressions" or "linear squares" he will understand immediately and clearly. It is interesting to build a small tower with the bead cubes. Though it will resemble the pink tower, this tower, which seems to be built of jewels, gives a profound notion\[255\] of the relations of quantity. By this time these cubes are no longer recognized superficially through sensorial impressions, but their minutest details are known to the child through the progressively intelligent work which they have occasioned.