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The Parents' Review

A Monthly Magazine of Home-Training and Culture

Edited by Charlotte Mason.

"Education is an atmosphere, a discipline, a life."
Home Algebra and Geometry.

by Mary Everest Boole
Volume 3, 1892/93, pgs. 854-857

Parents who know ever so little of Algebra and Geometry might forestall many difficulties by familiarizing their child, in play, with certain elementary ideas, months, or even years, before he has to deal with them in connection with the notion of "work." I refer to such ideas as the following:--

If we know a statement must be true about all numbers whatever, it is always legitimate, and often convenient, to express it in a way which does not specify about what number we make it. Such statements are--
a added to b = b added to a;
a multiplied by b = b multiplied by a;
P + a - b = P - b + a.
whatever numbers or quantities are represented by a, b and P. In the last instance the child should be led to think of P as a stock in hand; b a sum or amount taken away from that stock; a a sum of amount added to the stock. Point out that, though the process represented by P + a - b is different to that represented by P - b + a, yet we can know beforehand that the ultimate result must be the same.

Shew that if a = b,
Then a + c = b + c
a - c = b - c
ac = bc must be true, whatever be the values of a, b, and c, whether numbers or cyphers; whereas if ac = bc, a must be = b, provided c be a number, but need not be so when c is not a number, but zero (the negation of number). This curious distinction is of fundamental importance to get understood and soaked in before the time when it has to be used in actual work; for it lies at the root of all true comprehension, both of quadratic equations and of the laws of thought, and is too startling to be used efficiently as a working proposition, till it has had time to soak into the consciousness.

Be careful to ascertain whether the child clearly realizes the distinction between zero as a factor, and zero written merely to indicate that something else is a multiple of ten; between the act of multiplying or dividing by zero, and the act of writing or striking out a zero as a short way of multiplying or dividing by ten. Confusion on such points, once fixed by working in the dark for results, seems hardly ever to be quite cleared away by any subsequent explanations.

Beware of "telling a child how to do a quadratic equation." When he is doing simple equations for "work," give an easy quadratic, e.g.
x2 + 6x = 16,
and let him grope about at it till he has given it up in despair, and feels sure it cannot be solved by any direct process. Then set him successively these questions:--
"Square x + 3;"
"Square the equation x + 3 = 5;"
"Find x in the equation x2 + 6x + 9 = 25;"
"Find x in the equation x2 + 6x = 16;"
taking care that they all remain uneffaced, for reference. Let the pupil ponder over this sequence of operations, this solution of the unsolvable by inverse inference, till he has soaked in the sense of a new power, of a hitherto unknown intellectual tool.

Before the child has to work at geometry, see the he realizes:-- That there are in nature no lines, except the edges of solids, nor surfaces, except the sides of solids. That natural movements and growths take place usually in curves, not straight lines. That axes, diameters and co-ordinates are not part of any natural figure, but measuring rods introduced for man's convenience. Connect in the pupil's mind these devices with that earlier invention, mentioned in the former paper, of breaking big numbers into tens and hundreds; explain that man extracts no more real instruction out of such devices than a monkey gets by breaking things in bits, unless he completes the cycle of study by the transcendental (or supra-simian) act of synthesis. The parent will translate these words into such as she finds suit the child; the essential thing is for the parent herself to grasp the idea clearly. If she does not, she should get herself taught to do so by some competent person.

In connection with this idea of synthesis, a child should be taught that if he wants to learn anything about an object by breaking it open, he should not break it till he has taken a good look at it in its original state, so as to be able to reconstruct it in his mind as a whole.

Explain that an ellipse is (not exactly the path of any planet, but) the one among the simpler curves which is most like a planet path, and therefore important to understand.

Then, shew that we cannot draw an ellipse without putting two pins in the two foci; yet nature makes the planet go round with only one sun in one focus, the other being left empty. This is an excellent typical instance of the relation between nature's facts and our devices for analyzing those facts.

Set the child to draw ellipses with the pins nearer and nearer together, till they come into one hole; then with the pins further and further apart. Tell him that the resulting circle and straight line are called the limiting forms of the ellipse. Call his attention to the difference between a dead straight line (i.e., a bit cut from a longer line), and a living straight line (i.e., the limiting form of an ellipse, the path traced by a vibrating pencil point).

Shew that, though the distance of the pencil from any one focus is constantly varying, yet the sum of the distances from the two foci is always the same in the same ellipse.

Shew that if a stone has been whirling in a sling, it flies off straight from the point where it escapes from the sling. Explain that, though a planet never does escape the pull of the sun's attraction, yet men have imagined the line it would follow if it did escape; and have agreed to call that line the tangent to the actual path. Shew that the sling itself represents the invisible line of pull of the sun on the planet. Ask the child to draw the pull-line, and tangent. If he draws the former to the centre of the ellipse, do not correct him, but lead him, by questions, to see that it should go to the focus instead.

Connect early in his mind the name Finite Differences, with the idea of men trying to investigate a difficult curve by dividing it into convenient bits; the word Differential with the idea of drawing the attention off from the bits themselves, and studying the successive direction which the curve takes at the points of junction; and the word Integration with the final act of synthesis, by which we recover the idea of the curve from the study of successive directions. This will diminish that bewilderment of the imagination which constitutes the main difficulty in grasping the ideas of the Differential and Integral Calculus.

And before the time when he has to deal with problems in maxima and minima, let his finger-tips, and his whole nervous system have been long in tune with the main idea, by the habit of playing with the Indian toy called Bandelore; make him notice the momentary stillness of the disk at certain points. The first time a lad comes upon such an equation as dx = o in a text book, it should strike him, not as a mere formula invented by tutors for the puzzlement of the boy-mind, but as a neat expression of something which, ten years ago, his bandalore used to do every time it reached the highest or lowest point of its string.

Professor de Morgan [Ram Chundra, Maxima and Minima, edited by De Morgan.] pointed out that Hindoos are able to solve, without reference to the calculus, problems in maxima and minima, which the European mind had found unsolvable, except by the aid of that artificial intellectual apparatus. I attribute this, in great part, to the fact that Indian toys train the nervous system, through the fingers, in tune with the subtle play of nature's forces. Play can teach no actual mathematics; mathematics can only be learned by severe study, and should be tested by rigorous examinations, conducted by specialists. But such simple precautions as those indicated above might so much to make the future study a living reality, an incessant revelation, to prevent the formation of those vicious thought-habits which are at once the despair of all true mathematical teachers, and the disgrace of the colleges that turn out so many sham ones; and to promote (that most important of all educational desiderata) harmony of feeling between the respective devotees of inspired idealism and of exact science.


Note.--Those who wish to pursue the subject further may refer to Hinton, Life and Letters, chapters xiv and xv; Gratry Logique, vol. ii (Douniol Paris): Babbage, The Influence of Signs (Transactions Cambridge Philosophical Society, 1826); Boole, Laws of Thought; Logic taught by Love (Edwards, High Street, Mary-le-bone.) Symbolical Methods of Study (Edwards).