PART III_The Montessori Elementary Material The Advanced Montessori Method

ARITHMETIC

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 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] 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 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 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 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.

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 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]

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 the 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 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.

[Illustration: The bead material used for addition and subtraction. Each

of the nine numbers is of different colored beads.]

[Illustration: 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.

[Illustration: The bead cube of 10; ten squares of 10; and chains of 10,

of 100, and of 1000 beads.]

[Illustration: 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. 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.

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 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.

____________________________

| MULTIPLICATION TABLE |

| |

| COMBINATION OF |

| |

| =THREE= |

| |

| WITH THE NUMBERS 1 TO 10 |

| |

| 3 × 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 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 |

|___________________________________________________________________|

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 |

___|____|____|____|____|____|____|____|____|____|____|___

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.

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 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 = | _________ |

+---------------+-----------+

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.

[Illustration: 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.

[Illustration: 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.

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 X

742.

{ 2 units { 6 units

742 = { 4 tens 356 = { 5 tens

{ 7 hundreds { 3 hundreds

Each of the first numbers is combined with the three figures of the

other number in the following manner:

u. 6 } { 12 _units_ u. 6 } { 24 _tens_

t. 5 } × u. 2 = { 10 tens t. 5 } × t. 4 = { 20 hundreds

h. 3 } { 6 hundreds h. 3 } { 12 thousands

u. 6 } { 42 _hundreds_

t. 5 } × h. 7 = { 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:

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:

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:

4 × 5 = 20 hundreds, therefore two thousands:

4 × 3 thousands = 12 thousands; so we bring forward two beads on the

thousands-wire and one on the ten-thousands-wire:

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:

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:

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 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.

[Illustration: 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 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:

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.

[Illustration: 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, 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.

[Illustration: 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 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

134185518

89457012

67092759

44728506

111821265

89457012

67092759

________________

7720914184760583

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 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.

[Illustration: 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.

[Illustration: 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 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.

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.

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

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

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 |

_______________________________________________________________

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 ||

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:

18 = 2 × 9 = °°°°°°°°°

°°°°°°°°°

= 9 × 2 = ° ° ° ° ° ° ° ° °

° ° ° ° ° ° ° ° °

[Illustration]

= 6 × 3 = ° ° ° ° ° °

° ° ° ° ° ° ° ° ° ° ° °

= 3 × 6 = °°° °°° °°°

°°° °°° °°°

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 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 = 2^{3} ×

This is the result for 60:

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2^{2} × 3 × 5

For these two numbers we get accordingly the prime factors: 2^{3} × 5;

and 2^{2} × 3 × 5. What then have the two larger numbers, 40 and 60 in

common? The 2^{2} is included in the 2^{3}; the series therefore may be

written: 2^{2} × 2 × 5; and 2^{2} × 3 × 5. The common element (the

greatest common divisor) is 2^{2} × 5 = 20. The proof consists in

dividing 60 and 40 by 20, something which will not be possible for any

number higher than 20.

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 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.

TABLE E

1 2 3 4 5 6 7 8 9 10 | 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 | 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 | 21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40 | 31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50 | 41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60 | 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 | 61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80 | 71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90 | 81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100 | 91 92 93 94 95 96 97 98 99 100

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