  Children learn to count by using objects, in the school room they count the desks, the children, the number of cards, or blocks. The first lessons are object-lessons dealing with objects which can be handled and formed into groups. Digits are symbols which represent objects, 7+3=10, is an abbreviated form for 7 (Apples) and 3 (Apples) are 10 (Apples). It is easier to teach addition and subtraction by the use of the objects to add and to take away from. The realization of the process comes by seeing the objects and the result of the change. The digits become symbols for the objects that the child has been working with. Counting boards are helpful in teaching children, for they enable you to continue the visual process. All methods of teaching through the visual processes should be continued as long as possible. The child's interest in the problem will be stimulated if he deals with objects, or things, and not with meaningless groups of figures. The problem 127+323+417= , is a meaningless one and uninteresting, but if you encourage him to think that this is the number of soldiers 127 (soldiers) + 323 (soldiers) + 417 (soldiers) = How large an army? Figuring a page of problems will be uninteresting, but if you can encourage the child to introduce the imaginary objects, it will increase his interest. Fractions are usually explained by the division of an apple or some easily divided object. Division, as a process of dividing a group of objects among a smaller group of children, is easily understood and interesting to them. Encourage your child to continue to think of the objects when dealing with fractions. Visualization Always AidsAll mental processes should take form in pictures. The adding of 4 and 7 should be seen in the mind's eye, if the problem is not written down. A parent tells the story of his difficulty with his son and this simple problem. The child got the idea fixed in his mind that 4 and 7 were 12. The father had told the boy that the answer was 11, and had the child repeat, 4 and 7 are 11, several times. But the original impression was still the stronger, and the next day, when asked by the father, "How many are 4 and 7?" the child's answer was 12. In some way this impression had become a very strong This experience is the natural result of using the stronger sense of sight in preference to the weaker one of hearing. The conscious use of the mind's eye faculty in his arithmetic lessons brought this boy from the bottom of his class up to a reasonable grade in a very short time. Do not overlook the value of visualization. It can be applied with helpful results in any lesson or problem. The Mental BlackboardThe child can easily learn to visualize his problems in mental arithmetic if he will begin while young. This is especially true if you have used the exercises for visualization given in the First Book. Those on mind's eye counting and the Number and Letter games are especially helpful. Their importance now becomes apparent, and if you have neglected them it will be well to go back and use them now. Encourage the child to see the figures in exaggerated size on an imaginary blackboard; see large white figures on the blackboard. As soon as the problem is given, let the ear impression become a mind's eye picture, as illustrated. The use of this visual method is gradually being recognized as being valuable, and will in the future come into general use. Give your children the advantage and have them use it now. Exercises in ManipulationThe 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. 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. Learning RulesAll 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, FractionsThe 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.
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. The Multiplication TablesThese 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 GameThe 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. 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 Difficult TablesThere 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:
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 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'sThere 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.
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 MeasuresSome 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 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 AnswersFamiliarity 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 Learning RulesThe 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 GeometryThe 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:
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. |
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