The Parents' Review
A Monthly Magazine of Home-Training and Culture
"Education is an atmosphere, a discipline, a life."
by Mary Everest Boole.
Many of the so-called laws of number are in fact laws of the human thinking machinery. In other sciences, these laws act obscurely and can only be inferred more or less doubtfully; but all over the field of mathematics, including arithmetic, points occur where they crop to the surface and become manifest. Thoughtless persons, while recognising the necessity of obeying them in arithmetic, ignore their validity elsewhere; but there is a growing tendency among those who know better, to insist on the importance of using arithmetic as the key to all higher culture, of subordinating skill in calculation to the purpose of teaching the laws of that internal machinery by the right use of which, man draws down Truth from the Unknown. James Hinton for instance, spoke of "people wicked enough to tell a boy how to do a quadratic equation;" i.e. to sacrifice, for the sake of imparting technical knowledge, that unique opportunity for learning a special law of thought which occurs, when one is led to confront for oneself the problem of solving quadratics.
Hence those who try to learn from books how to teach arithmetic and algebra are torn between conflicting requirements, the spiritual and intellectual enthusiasts being hopelessly at issue with the "practical," who aim to teach those orderly and accurate ways of dealing with numbers so necessary for domestic and commercial purposes, with the crammers who want to make future High Wranglers, and with the specialists who would subordinate every other consideration, to prepare for event of the pupil becoming a stock broker or an actuary.
To invent any one method which shall satisfy the conflicting requirements is as impossible as it would be to give, all at once and in some one way, scientific instruction on the Physiology of muscle, hygienic practice in Gymnastics, and training for special handicrafts.
Orderly division of labour between home and school might facilitate a rational treatment of the question "How shall we teach Arithmetic?"
The principles to be grasped at starting are such as these:--
Every child's mind is in direct contact with abstract Truth, and can imbibe it by the natural use of its own powers, irrespective of statements made, or methods taught, by adults.
The Truth-absorption is the proper business of Holy-days, of quiet leisurely hours free from the stress of work.
Work is connected with and regulated by our relation to Humanity, our duties as citizens, and should be done in the most orderly and convenient way possible, irrespective of any special manner in which Abstract Truth may have revealed itself to the individual.
Methods are dictated by conventions created by experience, which we are not at liberty to ignore during work hours, though we may, and should, keep them true by comparing them, at leisure, with those abstract truths of which they should form a correct, though condensed, embodiment.
All knowledge, both of Laws of Number and of Laws of Thought is gained by conscious attention to the rationale of the processes we are going through; all skill in the use of mathematical notation depends on acquiring the habit of ignoring the meaning of the symbols and concentrating attention on using them, accurately and rapidly, according to a rule; obviously these two requirements cannot be secured at one time and by one method. [An Article by De Morgan on Arithmetical Computation, in British Almanack and Companion for 1844, shews the need for strict discipline in the regular teaching of Arithmetic, and proves (by inference) how impossible it would be to combine that discipline with the roundabout, leisurely investigation necessary to make Arithmetic fruitful for spiritual culture.] Indeed, it is almost impossible for a pupil to extract out of a mathematical operation what it has to teach about the abstract, after he has once learned to perform that operation mechanically. It is therefore desirable that a child should learn from each operation all that it can give of knowledge of Laws of Thought, before he ever sees it performed in the class-room, where skill in manipulation is being cultivated. Each operation should be introduced to the child first by a series of play-lessons, in which no eagerness is aroused, on which no examination depends, and during which he is free to soak up all it contains of Philosophic Truth, unhampered by the need for following any prescribed method.
Orderly methods can be best taught in class and by trained experts; but the preliminary play-study, on which depends the Philosophic culture, might often be more effectually superintended at home, by some one acting in concert with the school teacher. Such play-study is a more suitable form of "holiday lesson" than the tasks usually set as such.
The remainder of the paper will consist of hints for making arithmetical play-lessons fruitful for future development; it is addressed, primarily, to persons already familiar with good text books, such as those of De Morgan and Sonnenschein; and is intended not to repeat what such books contain, but to show how the carrying out of their essential purpose can be facilitated by clear division between "play-lessons" and "work."
Before the age at which arithmetic is taught at all, make the child practically familiar with the process of exchange; either by using money and getting twelve pence for a shilling, or by some game in which a red counter represents ten white ones.
Begin teaching each of the four elementary operations, by giving a few easy examples in relation to the coins or counters to which he is accustomed; do not use the written decimal notation till he understands the operation itself.
When you use notation, shew him that ten was chosen as our "carrying" standard, because savages counted on their fingers; make him realise early, that ten has no special value as a standard except what is given to it by the conformation of man.
Beware of writing, in play-lessons, anything which does not represent some process actually going on in the child's mind. E.g. It is natural to a child to count the more valuable coins or counters before the less valuable ones; allow him to do additions in that order, till he discovers the inconvenience of doing so. The first few examples of each operation should involve no "carrying" and therefore no inconvenience from beginning at the "wrong" end. When he begins to "carry," let him still work in the wrong order, and correct his results. If it does not soon occur to him to spare himself this trouble, you may suggest it to him; but for some time after you have suggested it, make him do each sum in both ways, the clumsy and the convenient way, and become accustomed to see the identity of results.
When he can do an easy addition, of about three columns and three rows, slowly, but without effort, beginning indifferently at either end, and can explain the rationale of each process, addition may pass to the stage of "work." Subtraction should them be taken up for play-lesson, the same principle being observed as in play addition.
If the play is being thoroughly well done at home, then the class teacher need never tolerate any clumsy, roundabout ways, any scribbling of needless figures, any sort of untidyness; he should insist on the most rapid, orderly and convenient methods.
No child should use a multiplication table until he has made one. Rule wide squares, and write the heading numbers. Give the child, as addition sums, two 2's to add together, then two 3's, two 4's, &c. Let him write the result of each of these additions into its proper square of the table. He may take many weeks or months to complete it. He may begin to learn by heart the easy rows before he has filled in the more difficult ones. Let him learn from the copy which he has himself make. Meantime he can be doing play-multiplication sums. When he comes to multiply by more than one digit, let him write all the zero's in full, till he has satisfied himself that leaving them out can make no difference to the result provided the figures are kept in their right places. Multiplication should not be done as work till his is satisfied of this, and can do a sum either with or without the zeros.
As soon as multiplication passes into work, no needless zeros should be permitted, nor any stopping to reckon; the work should be done by sheer effort of mechanical memory. He should occasionally revise or recreate his table in "play," but, during "work," should take its truth for granted.
By the time that he begins play division, he will probably have reflected that it is useless to begin at the end of greatest values, and will try to divide, beginning with the smaller value. Let him do that till he finds,--probably to his great astonishment,--that, in division, this is the inconvenient way.
When he can easily do a short division by one small digit, let him do one on the top of the slate and leave it; prepare the rest of the slate as for long division, and set the same sum over again in long division form. Tell him to write down all the steps (multiplication and subtraction) by which he got his successive remainder. Repeat this process during several weeks, or even months: do not set any division by more than one digit till the child quite realizes that long and short division are identically the same process; that the former helps memory when the divisor is large, but gives needless trouble in writing when it is small. This will forestall the difficulty which many, even clever, children experience in understanding long division. Remember too that no time is wasted which serves to impress on the child's consciousness that abstract truth has a sanctity and authority of its own, independent of special method; and that our choice between methods, equally valid abstractly, is to be decided by human convenience.
On the same principle let proportion be done at first by the longest, most roundabout gropings after successive steps of reasoning; let this be continued for some time before a shorter method is suggested, and do not let proportion pass into work till the child has done many easy examples by both long and short method.
Form early the habit of designating by a big "ANS." the final answer to the question originally asked. Treat this clear distinguishing of the final from all partial results as an essential part of the solution.
Give occasionally, as multiplication sums, questions about the number of days in so many years. Accustom the child to picture, in imagination, the days passing steadily on from one new year or birthday to another; shew that there comes no special stop or break at the end of ten days or a hundred days; that we are obliged to do our sums as if days were grouped in tens and hundreds. Point out that, though we perform our investigations on fragments of a large number, because, our faculties being limited, we cannot multiply it as a whole, yet we write the word "Answer" opposite to no partial result, but only to that expression which indicates the summary or synthesis of our various partial investigations. This forms a good preparation for understanding that doctrine, which is at once the basis of Mathematical Philosophy and its crowning glory; viz: the doctrine that man is related by his analytic faculties to the monkeys who investigate by breaking things in pieces, and, by his synthetic faculty, to that Unity in Whom are reunited all that has been separated; that the normal use of man's thinking organ consists in perfect cyclic alternation of Analysis and Synthesis; that the dominant revealing point is the point of completed synthesis; that man falls into error when he vitiates the cycle; but while he keeps faithful to it his mind is constantly inspired with Truth.
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