Part 4

The mind's eye picture of the figures on the mental blackboard can be enlarged by practice so that the child can visualize problems of some complexity. This ability, of course, will come only after continued practice. Start with simple problems and increase their difficulty as the child progresses. You will be surprised to find how he will be able to retain the figures in his mind and soon will be able to work with them.

Write on the blackboard a column of figures as illustrated below. (A small one in the house is of great value in child training. A yard of blackboard cloth can be purchased and hung on the wall.) Allow the child to look at them for a few seconds and write down the result of his addition. Do not have him write the numbers as in previous exercises, for visualization, but only the total.

Now, add the first two numbers of the first example, subtract the third and add the fourth, then write the total.

In the second example let him add the first two, subtract the third and multiply by the fourth, write the answer.

These exercises of manipulation can be varied in many ways. The length of the columns can be accommodated to the ability of the child.

10 12 9 16 --

15 10 14 7 --

Learning Rules

All rules should be worked out in examples or illustrations and visually impressed upon the child's mind. One visual impression is equal to about twenty repetitions. Many times children get the idea that the problem cannot be worked unless the exact "Rule in the book" is followed. See to it that your children get a broader idea and that they understand the reason for doing a thing. The training in mathematics, that is of most value after school days are over, is, where we understand the reason and have worked out for ourselves the correct result, independent of any set rule for working the problem. When helping the child at home give him practical examples from every day life as well as those in the book.

Fractions

The first step in fractions are often confusing to children, but need not be if they have been taught to be observing and to watch for the little aids which help over the difficult places.

Nominator and Denominator are two confusing terms to many. If you will show the child that most of the fractions that he has to deal with are proper fractions, and that the Nominator, upper number, is smaller than the Denominator, lower number, and that the same relationship exists between the words.

Nominator De-nominator

The Denominator is the denomination of the fraction, the Numerator is the number of parts. Let the D of Denominator stand for Down and remember that it is Down (lower) part of the fraction.

Many scholars have difficulty in giving the correct answer to the question, What are the three kinds of fractions? The following is all that is needed to fix the answer in mind.

Give the PROPER answer. If you give the IMPROPER you will be MIXED. These capitalized words are the three kinds of fractions.

Think of a fraction as a part of a whole. When the fraction becomes a whole, or more than a whole, it is Improper. It needs to be changed to make it a unit, or a Mixed fraction, a unit and a part.

The Multiplication Tables

These are a problem which every one has to work with and because the use of them requires speed to be most valuable there must be a certain amount of repetition in learning them.

The Multiplication Game

The aim is to teach children their multiplication tables by visual repetition and at the same time to introduce the game spirit, thus to increase the interest and to prolong the period of effort without fatigue.

The child can work with these cards himself and thus by self instruction can learn this most difficult lesson of Arithmetic, and without any possibility of error, accuracy is insured.

The equipment consists of a series of eleven pieces of cardboard about 2×6 inches on which are printed in large black numbers the tables without the answers.

A series of ten odd shaped cards is then made and the digits printed on them in bright red. The following are the suggested shapes for the ten digit cards.

(Digit cards should not exceed one and one half inches in height.)

The digit cards which are the correct answer to the table printed on the larger cards are then laid in the correct position and the shapes marked out. With a sharp knife cut out the shapes a trifle larger than the marked size of the digit card. The result is a card as illustrated, with the table and two holes of irregular shape into which the digit cards with the correct answer in bright red will fit. No other card but the correct one can be put into this opening, there is never any danger of the child seeing a wrong answer to the table.

The only cards which can be fitted into this table are the two and the cipher making the correct answer 20. This card with the black 4×5= and the bright red answer 20 will make a strong impression upon the brain of the child, and by use of the strongest sense, that of sight. At the same time he can repeat the table audibly and gain the added advantage of the ear impression.

Give the child only one set at a time so that he learns one table thoroughly. When he has learned it, mix the cards and place them one at a time in front of the child and see how many correct answers he can give without fitting the cards. In cases where there is hesitation have him fit the digit cards and make sure. See to it that he is accurate and certain.

After one table is well mastered make a similar set of cards for the next table. If you do not wish to take time to cut out the irregular shaped holes for the digit cards, the place can be blackened and the digit cards laid carefully on. The cut outs are far better and well worth the little effort necessary to make them.

For the tables up to 12's you will need the following number of digit cards; with these you will be able to work out any complete table of eleven cards. 10--1's; 8--2's; 6--3's; 6--4's; 10--5's; 4--6's; 4--7's; 5--8's; 4--9's; 16--0's.

After the child has learned two or three of the tables mix the cards, take any six and see how quickly he can fit the correct digit cards into place.

Keep him playing with these cards until he can give the correct answer to any question and give the correct table as a whole. After the tables have been learned you can make many tests of speed and competitive games with several children of the same age or school grade.

The Difficult Tables

There are certain tables which seem harder for some than the others, yet there is often a difference as to which are considered most troublesome. The 2's, 3's, 5's, 10's, and 11's are easy for all of us. The 9's are as easily learned with the aid which follows. This leaves the 4's, 6's, 7's, 8's and 12's, remaining to work on. The combinations that are new in these tables are the following; all other combinations are known from the other tables:

4 × 4 = 16 6 × 6 = 36 7 × 7 = 49* 8 × 8 = 64* 4 × 6 = 24 6 × 7 = 42* 7 × 8 = 56 8 × 12 = 96 4 × 7 = 28* 6 × 8 = 48 7 × 12 = 84 12 × 11 = 132 4 × 8 = 32 6 × 12 = 72 12 × 12 = 144 4 × 12 = 48

The first help in mastering these few necessary combinations is visualization. If you will print them in large figures and the answer in red, each table on a sheet or page by itself so that they can be handled and studied, they will form visual impressions that can be recalled with ease by almost any one. This is especially true of children at the ages when they will be learning these tables.

Repetition seems the most valuable aid, but to be most advantageously applied it should be a combination of visual and auditory repetition. Let the child look at the tables in the large form in which you have made them, while he repeats them.

Use addition and subtraction. In learning the tables there are always some which make a stronger impression and which the child will "never forget." Use these as starting points or bases of operation. For example, 4×5=20, all will recognize this at once. 4×4=16, just four less than twenty, and the subtraction will quickly give the correct answer. Also 4×6=24, or 4 more than the known point of 20. To take advantage of this it will only be necessary at first to learn 4×7=28 in order to master the entire table of 4's. The 4×4, and 4×6, would be figured from 4×5=20, and the 4×8 from the 4×7, and the 4×12, from the known 4×11=44. With these known bases to work from it is only necessary to fix the one starred combination in each table in mind indelibly at the beginning, the others will be easily figured from the known bases and will become fixtures from use.

The Table of 9's

There is a peculiar combination of figures in this table of 9's, which, if once noticed and perceived, will make this one of the easiest of the tables.

9 × 2 = 18 (1 + 8 = 9)

9 × 3 = 27 (2 + 7 = 9)

9 × 4 = 36 (3 + 6 = 9)

9 × 5 = 45 (4 + 5 = 9)

9 × 6 = 54 (5 + 4 = 9)

9 × 7 = 63 (6 + 3 = 9)

9 × 8 = 72 (7 + 2 = 9)

9 × 9 = 81 (8 + 1 = 9)

9 × 10 = 90 (9 + 0 = 9)

9 × 11 = 99 (2 9's)

9 × 12 = 108 (1 + 0 + 8 = 9)

Notice that the two digits of each answer always add up to make 9, and that each first digit of the answer is just one less than the multiple. For example, 9×5=45, the answer will begin with one less than the multiple 5, and the two digits of the answer must add to make 9, therefore it can be nothing but 4 and 5, or 45. This is true in all cases except 9×11 an already known answer, but also only 9's in this answer. This simple idea, when once understood, will master the table of 9's.

Be sure that the children realize that 7×4 in the tables of 7's are the same in value as 4×7, so that the answer to 7×4 becomes familiar with learning the table of 4's. Ask the question both ways 7×4 and 4×7.

The Tables of Weights and Measures

Some of these we learn easily and always retain; some always seem confusing. These can be mastered by the use of the Number Code and the Visual picture combined. Some examples follow:

24 sheets = 1 quire, and 20 quires = one ream. The picture of Two Dozen Squires in a Nice Room, will fix these figures and terms in mind. Two Dozen is 24, Squires is a reminder for Quires. Nice is 20 (2 is N and 0 is C) and room a reminder for Ream.

16-1/2 Feet = 1 Rod, 320 Rods = 1 Mile. Picture a Dish and a Half balanced on a Rod. Dish is your code word for 16 (1 is D and 6 is sh) and the Half Dish makes 16-1/2 Feet on (in) a Rod. Next--Many's the Rod in a Mile. Many's is 320 or the number of rods in a mile.

30-1/4 Sq. Yards = 1 Sq. Rod. Picture--MISTER takes a yard stick and measures off a Sq. Rod. Mister is 3-0-1-4, or 30-1/4.

160 Sq. Rods = 1 Acre. Picture--See a pile of Dishes out in the Acre being broken up by a rod. Dishes is 160 the number of Sq. Rods in an Acre.

640 Acres in a Sq. Mile. Picture--Take the Shears and cut up the mile into squares. Shears is 640, the number of Acres in a Sq. Mile.

792 Inches--1 Link. Picture--792 is Cabin, see the link hanging on the side of the cabin.

4 Rods = 1 Chain. Picture--See 4 Rods wrapped around with a chain. 80 chains = 1 mile. Your Code Word for 80 is Vase; put a chain around it and drag it a mile.

A few picture associations like these will help in fixing the difficult points in mind. Associations which you make yourself will help you most. Be sure to repeat them at intervals; make them permanent.

Pictures for Answers

Familiarity with the Number Code given in the book on Memory, will aid the child in keeping the result of a problem. The numbers of the answer can quickly take the form of an object which can be translated again into the correct numbers. Many children will not be able to hold the visual picture of the digits for any length of time. There is considerable difference in the ability to hold the visual picture of the digit 127. Many children, and adults, will be far more accurate and remember longer if they see a TANK, which is easily translated by the Code into 127, when the answer is wanted.

Learning Rules

The exaggerated example illustrating the rule to be learned, will make its meaning clear and thus make the problem of learning it many times simpler than if it is learned as a group of words, the meaning of which is not always well understood. It is always best to understand the rule first and learn it afterwards. Use the suggestion given for learning verbatim and the exaggerated example as given in the suggestions in spelling. After you understand the rule it will not be difficult to memorize.

Visualizing Geometry

The Theorem in geometry should have the visual process applied to it in the same manner. Make a strong picture of the figure which illustrates it. For example:

=The square on the hypotenuse of a right angle is equal to the sum of the square on the other two sides.=

To visualize the figure, as illustrated, will aid in fixing this Theorem in mind. Do the same with others. Another example of emphasizing the important lines as in the Theorem:

Two rectangles are to each other as the products of their bases by their altitudes.

In the illustration below the bases and altitudes are emphasized to remind you of the fact that they are the factors to be dealt with. Notice that in the first pages of the Geometry all simple figures are illustrated as explained or defined. Learn to visualize the problem with your book closed, work until you can see it clearly, and you will understand it better.

AIDS IN STUDYING CHEMISTRY

Experiments in Chemistry are its most interesting phase. Let its problems take form in your visual mind and you will add to the enjoyment and also the ease of your understanding.

A teacher of this subject, after appreciating the value and ease of visualization, worked out picture combinations of atoms which helped him greatly. He could see the two atoms of Hydrogen floating through the air and combining with the atom of Oxygen and could see the result of the combination.

Using the Initialing Idea

The ideas which have been given in this and the preceding books can be applied in many ways to the problems of any subject. There is no attempt on the part of the author to work out all applications, but merely to suggest a few possible ones and leave the rest to the student. Each will think of different methods, and those aids which each one works out for himself will be most valuable and most easily recalled.

The Elemental Substances

The six elemental substances of the organic world are: Carbon, Hydrogen, Nitrogen, Oxygen, Phosphorus, and Sulphur, which can be easily remembered by the following: The Organic World--Can Have No Other Principal Story.

The initials of each of the words following "World" stand for one of the elements.

Chemical Formulae

Remembering of chemical formulae can be simplified very greatly by reducing the formulae to an idea using the initialing plan; for example, the formula for Wood Alcohol is CH3OH. This formula in itself has no meaning, and is difficult to carry in mind. By using the initials you can easily make some ideas which will represent this formula and help you to remember it, as for example: CAT HAD ham ON HAND. In this formula the C of Cat stands for Carbon, the H of Had for Hydrogen; Ham being a Code Word for 3 is indicative of 3 atoms of Hydrogen, and the O. H. is represented by the O and H of On Hand.

The formula for Glycerine is C3H5(OH)3, and can be remembered by the following idea: COME HEEL O HAM. In this example notice that the first letter of the word initials the substance and the last letter the number of atoms by the number code. As COME: C for Carbon, and M for 3. HEEL: H for Hydrogen, and L for 5. O for Oxygen. HAM: H for Hydrogen, and M for radicle 3 times. Use whichever method suits you best.

The formula for Carbolic Acid, C6H5OH, or CASH HAUL O, HAY. The formula for Benzine, C6H6, or CASH HASH.

Hardness of Substances

It is often valuable to know the degree of hardness of different substances, and these can easily be remembered by the following list. In degree of relative hardness the list is as follows, the hardest coming first.

Diamond Corundum Topaz Quartz Iridium Apatite Bell Metal Boric Acid Rock Salt Kaolin

Take Kaolin as a basis. The number opposite each substance in the following list indicates its comparative degree of hardness in relation to Kaolin:

1 Kaolin TIE See tie on Kaolin

2 Rock Salt SNOW Poured over Rock Salt

3 Boric Acid HOME Built of Boric Acid

4 Bell Metal WIRE Swinging a Bell

5 Apatite WHEEL With a big appetite for running

6 Iridium SASH Irritating the wearer

7 Quartz EGG Quartz taken from an Egg

8 Topaz IVY To pass the Ivy

9 Corundum WHIP Made Cora run

10 Diamond TOES Set with Diamonds

In the list you have ten substances. Kaolin, the base, is 1, Rock Salt is 2, which indicates that Rock Salt is twice as hard as Kaolin. Iridium is 6, and six times as hard as Kaolin. Diamond is 10, which means that it is ten times as hard as Kaolin.

Learning this list by picturing reminders with the word of the Code list will enable you to easily recall these ten substances and the degree of hardness compared with Kaolin.

Atomic Weight Table

Some students of Chemistry have learned the entire list of elements and their atomic weights. The following are a few examples of how the list can be arranged and learned. It will be excellent practice for you to use this method and make a list of your own.

Element Code No. Reminder Wt. Code Word

1 Carbon TIE Carbine 12.005 Tin Sizzle 2 Hydrogen SNOW Hydrant 1.008 The Saucy Foe 3 Nitrogen HOME Night 14.01 Deer Sat 4 Oxygen WIRE Ox 16.0 Dash 5 Sulphur WHEEL Sulphur 32.06 Money Sash 6 Phosphorus SASH Fuss for us 31.04 Mad Sir 7 Sodium EGG Soda 23.0 Nome 8 Potassium IVY Pot 39.1 Mop It 9 Calcium WHIP Calsomine 40.07 Horse Sack 10 Iron TOES I Run 55.84 Lily Fire 11 Arsenic DOT Arson 74.96 Gray Page 12 Gold TOWN Gold 197.2 Dipping In

Learning Foreign Vocabularies

The principle of using a reminder can be applied with advantage in learning a foreign language. The majority may learn foreign words more easily and permanently by the Reminder Link. In this case the reminder is the connecting link between the English word and the foreign word. Those who usually learn foreign words only by laborious repetition will find a saving of time in learning by the reminder link.

Spanish words:

English Link Spanish

cold freeze frio drink beer beber written inscribed escrito sing cantata cantar full complete completo sweet delicious dulce window ventilate ventana keep guard guardar sell vend vender

Latin Vocabularies

Latin is the base from which most modern languages are derived, and you will find in English a very large proportion of the words taken directly from the Latin source. This makes the learning of Latin Vocabularies simpler than any other.

In a great many cases the word is a direct derivative and needs no reminder or intermediate step; for example, the Latin word ANIMAL is the same as in English, although pronounced differently; or Latin: ORNAMENTUM, and the English ORNAMENT. Be resourceful, draw upon your imagination. Note the following suggestions:

English Reminder Latin

boyish Puerile puer crown coronation corona free liberate liber land terrace terra dog canine canis think cogitate cogito mind mental mentis (gen.) running current curro pleasing gratifying gratus soldiers militia milites (pl.) teaching doctrine doceo more majority maior unending perpetual perpetus shortness brevity brevis time temporary tempora (pl.) faith fidelity fides

German Vocabularies become very much less difficult if you search for an intermediate step or reminder:

English Link German

fork gobble gabel coffee-pot coffee-can kaffekanne amusing comical komisch ancient old timer alterthuemlich easy light leicht meat flesh fleisch writing scribed schreiben gloves hand shoe handschuh quilt bed cover betdecke walking going gehen stove oven ofen flowers blooms blumen

Studying Music

The visual memory is the best memory for music. Many of the better musicians who learn music readily and remember it well have the visual memory. They can see the page, the bar, and the notes in the mind's eye. This ability can be developed in the child by the use of the exercises for visualization given in the first book. When the child begins to study music give part of the time to practice of visualizing and memorizing music.

First, teach him to visualize a perfect clef. Draw imperfect ones on paper or slate and have the child tell what is the matter with them. Draw different notes and have him become thoroughly familiar with them by reproducing them. Have him draw the whole, half, quarter, and eighth notes, etc.

Teach the child the division of time by grouping the notes with reference to beats. Write a line of notes and have him divide them into groups of whole note value. Then indicate a certain time to be followed and have him divide other rows of notes into bars in accordance with the time indicated.

Teach the child the different rests by the same visual process. Have him write bars of music using the different rests and completing the bar of given time by filling in with the proper notes. Teach the use of sharps and flats and the difference in signatures by the same visual process. Let all practice be simple in the beginning and increase in complexity as he grows older.

Teach the child to combine the use of the eye and ear in musical practice. Have him transfer ear impressions to visual ones by seeing the notes on a staff as he hears the tone. Write a few bars of a familiar tune and have the child tell what it is.

Another application of the visual memory is to look carefully at the staff, then close the eyes and see it in the mind's eye, then look back and correct and improve the picture. Another plan is to see the staff exaggerated in size, covering the entire wall of the room. This exaggerated picture can be colored according to the above suggestions.

It will be helpful to take the piece which is to be memorized, and after fixing the picture of it in mind write it upon a blank staff. Keep improving this written copy of music, writing only that part of the score which is seen clearly. These methods will help to improve the visual ability to carry a picture of the page; and continuous practice with them will help in improving the ability to memorize in this way.

Be systematic in all your efforts. It is best first to memorize the words, then the air, then the technical part. A thorough understanding of the composition and its general plan will be of assistance.

In learning songs apply the principles given in the second book, and learn the words thoroughly. This will enable you to devote all of your time and attention to the technical part of the music. When you do not know the words thoroughly your attention is divided between learning the technical part of the music and recalling the words. Better master one thing at a time and do that well.

Speaking in Public