Chapter XII: Part III: Arithmetic (2)

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.

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.

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.

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 = ° ° ° ° ° ° ° ° °
° ° ° ° ° ° ° ° °

= 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} × 5.

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
________________________________|________________________________
|
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
________________________________|________________________________
|
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
________________________________|________________________________
|
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
________________________________|________________________________

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100
______________________________

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 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: 5^{2} = 25; 7^{2} = 49; 10^{2} = 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 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.

The quantity is always the same: four times the square of four. 4 × 4 × 4 = 4^{2} × 4 = 4^{3}.

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: 4^{2} × 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 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 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.