Aids to Memory/Section IV

Next to the labour of teaching the alphabet, that of imparting the multiplication table is the most arduous. The method adopted is pretty much the same in all schools. A table-book is put into the pupil's hands, and he is told to learn one or two of the columns of figures pointed out to him. Some difficulty is experienced in accomplishing a portion of the task, and too often the first part is forgotten while the second is being learned, and even if the whole be "said," it is only by means of a consecutive recitation. For instance, ask the boy "7 times 9," and afterwards "7 times 12," and he will be puzzled. If he have a multiplication sum to do immediately after saying the table correctly, he must go through his lessons for every figure; but if the sum have to be done the next day, he will probably have forgotten his task altogether. Then, of course, he is put to it again, and feelings of aversion to figures are excited in his mind.

Lest this might be considered an exaggeration, the writer desires to state that his own practical school experience has supplied the statement. We could cite instances of boys who, with fair mental ability, have been at school two or three years, and yet could not pass a "dodging" examination on the multiplication ​table, to say nothing of the tables of money, weights, and measures. Scepticism will, perhaps, be felt when he asserts that it is possible to teach children who know the names and shape of the figures the entire multiplication table in two lessons of half-an-hour each, with little probability of their ever forgetting it.

The "seven-times table" is subjoined as an illustration. The writer is preparing for publication a book entitled "Mnemonics for Junior Scholars," in which a complete Multiplication Table will be given. The associations will be adapted to the simplest capacity, and still further assisted by poetical rhymes. Thus, to remember "7 times 2," the following couplet will be used:—

Associate according to Exercise 1. The multiplication of 7 by 10 and 11 is omitted, because too easy to require an assimilation. It will be seen that portions of other tables are learned at the same time, as 7 times 9 are equivalent to 9 times 7, and so on. A separate assimilation might be made, as "9 times 7, or Bag = 63, or Ham."

Students of arithmetic will find it advantageous to learn the extended multiplication table by this process, as much time in calculation will be saved by it. An example is given.

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Some explanations of the above will, perhaps, be acceptable. The products of the extended table never exceed three figures; therefore, in a word containing more sounds than are required, all beyond the first three should be rejected. Thus, the word delicious contains four sounds, d, l, sh, s, and the final s is not translated. It will be seen that the word key is used for the square of 7, and tomb for that of 13, as it is not necessary to repeat the factor in these cases.

Associate the number with the square, and both together with the cube. Thus, the arrow was shot into a ditch, and there struck the hare. For telling a lie, he was obliged to kneel on two nails—a very curious punishment. A great show was made of a mash in the new dish. The key was tied to the rope, and hung round the neck of the mermaid. A fee was paid for a chair at the Latin lecture. The bee engaged in a fight, and a keen battle was the result.

​Another way of learning arithmetical tables is by noticing the general results of operations. This plan is not external to our science, for the same principle pervades it—the use of the known as elucidatory of the unknown. For instance, any unit multiplied by 9 gives a product whose digits added together amount to 9, the first digit being one less than the number multiplied. 9 times 6 are 54, because one less than 6 is 5, and 4 added make 9. The product of any factor of 9 is without a remainder if divided by 9, and this process of "casting out the nines" (which should find its way into every manual of arithmetic), supplies a valuable method of setting and proving integer sums. The line of figures 7 2 1 5 6 3 8 4, if the digits are added, produce 36, and these added give 9. Whatever figures may be multiplied or divided into this line must therefore produce a result which, treated in the same way, will also give 9. Any factor of 5 yields a product equal to half of that factor multiplied by 10, or to half of the product of 10 multiplied by the entire factor. Thus, 5 times 8 are 40, because the half of 8 is 4, and 10 times 4 are 40. 5 times 9 are 45, because 10 times 9 are 90, and the half is 45. Any number of two digits multiplied by 11 is equal to the sum of the digits placed between them. 11 times 18 are 198, because 1 and 8 are 9, and that figure placed between 1 and 8 gives 198. If the sum of the digits is 10, add 1 to the first figure, and place 0 between the two figures; thus, 11 times 46 are 506. The addition of the higher figures is sometimes difficult for beginners; let it then be remembered, that to add 9 to any figure is the same as subtracting one, and then adding 10. For the addition of 8, subtract two, and for 7 three. To add 5 to any higher figure, subtract it instead, and add 10. So 5 and 6 are 11, because 5 from 6 are 1, and 10 added make 11. Nothing, surely, ought to be more acceptable in schools than the substitution of something more rational and effective than the present system of teaching tables.

​An ordinary student might as well try to get up the "London Directory," as to learn many thousands of dates by heart. Yet the feat can be performed by systematic memory, which we proceed to apply to chronology, in order to remove the difficulties attending that particular study.

These events are comparatively modern, and therefore the first figure of the date is not assimilated, being understood. Three figures therefore remain. If one word be used for the assimilation, the first three sounds represent the date; if two words be used, retain two sounds from the first and one from the second; if three words, translate the first sound of each. Should the first of two words only contain one consonant, two sounds of the second word will be required. These rules apply to all groups of three figures, and might have been adopted in Exercises 12 and 13.

It is not necessary that associations should be given for the last exercise, as they can easily be made. One exception may be found. The word "Logarithms," not being very suggestive, is assimilated as well as the date. ​A log against the shutter is the mental picture, and if the question be asked, "When were logarithms invented?" the mind reverts to the word log and thence to shutter, 614 and 1614, with marvellous quickness. The system of remembering proper names and hard English words is, therefore, identical with that for learning foreign words, and applies to lists of individuals, battles, countries, provinces, counties, towns, mountains, rivers, planets, &c. By the additional aid of the figure alphabet, we acquire the dates, areas, populations, heights, depths, lengths, distances, and every imaginable kind of statistical information.

Chronological tables of sovereigns present a difficulty which will not be met by anything in the preceding examples. For instance, in the regal table of England there are eight kings of the same name, and a mere assimilation of the name would be insufficient. We therefore furnish complete tables of English sovereigns from Egbert to Victoria, beginning with the modern table.

The first king is William I., or Wheat. W stands for William and t for I. The other Williams are respectively represented by Wine, Whim, and War. The words are immediately suggestive, as we cannot imagine any other king save the right one. The eight Henrys are remembered by Hat, Hen, Home, &c. The Edwards must be treated a little differently, as the initial letter is a vowel, and has not much combining power. We therefore use the second letter D, and proceed as before. The associations are again omitted, and, in the next exercise, no difference of type will distinguish the representative letters.

In the next section, a new principle of the Science of Memory will be developed; but, to derive the same benefit from written as from oral instruction, the preceding sections should be carefully recapitulated.