# The writing of Mary Everest Boole

We give below some short quotes and some long ones taken from the writings of Mary Everest Boole.

1. The following are taken from D G Tahta (ed.), Boolean Anthology: Selected Writings of Mary Boole on Mathematical Education (Association of Teachers of Mathematics, London, 1972).
1.1. Group work

The method is essentially communal not competitive. Hardly ever are there two children working on quite the same problem; therefore there can be no competition as to who answered best or quickest. At the end of the lesson, the whole class are in possession of some information which no member of it possessed before. It is not something told them by the teacher: it is often quite new to the teacher herself. The skill of the teacher is shown, not by the knowledge which she imparts, but by the manner in which she utilises the thinking power of the children for the purpose of finding out what she does not yet know.

1.2. Two elements in teaching

Suppose I am teaching, say, the process of multiplication. There are two things which the pupils can get out of my instruction: (A) skill in performing the operation of multiplication itself; and (B) a little of the power to find out for themselves how to do other arithmetical operations. Every process that I teach ought to be so taught as to add something to the pupil's chance of some day making out a rule for himself without the aid of a teacher. If we add together all the A's of a child's arithmetical career, they constitute what I called, in a former article, the body of his arithmetical knowledge; if we sum all the B's, they constitute what is called its life. The sum of the combined A elements constitutes the ability to reckon the bulk or number of dead material and to keep accounts according to any system chosen by an employer. The sum of the B elements gives the extra power of bringing one's knowledge to bear in forming a sound judgment on problems connected with living forces - e.g., on the probable behaviour of a charge of electricity under certain conditions, or the probable honesty and stability of a certain commercial enterprise. Now the A element in any mathematical lesson can be imparted while the class is alert and eager; the B element cannot be imparted except under the peculiar condition called by some mystic writers "Silence in the soul" awaiting further Light. The two states, the alert and the passive, alternate in any good educational regime; the alert phases being very much the longest, the passively recipient ones short but quite undisturbed. But under stress of competition the passive mystic phases of study are being crowded out. The reason is that England is so saturated with the spirit of advertisement that, in any given committee, the majority are almost sure to be against the teaching of anything for which there is nothing to show at the next forthcoming examination.

1.3. Pre-mathematics

Many a life of intellectual muddle and intellectual dishonesty begins at the point where some teacher explains the rule for Greatest Common Measure to a child who has not had the proper basis of sub-conscious knowledge laid in actual experiences. Therefore, if you value your child's future clearness in science, trust no teacher to tell him anything about G.C.M. or L.C.M. till you have ascertained that he is able to find, easily and accurately, by means of compasses, the longest length that will repeat exactly into each of two unequal given lengths, and the shortest length into which each of two given unequal lengths will fit.

1.4. Geometry

2. The following is taken from Mary Everest Boole, Lectures on the logic of arithmetic (Clarendon Press, Oxford, 1903).
2.1. Preface

Teachers of such subjects as Electricity complain of the difficulty of getting pupils to apply what they know of Mathematics (at what ever level) to the analysis and manipulation of real forces. It is not that the pupil does not know enough (of Arithmetic, or Algebra, or the Calculus, as the case may be), but he too often does not see, and cannot be got to see, how to apply what he knows. Some faculty has been paralysed during his school-life; he lacks something of what should constitute a living mathematical intelligence. In truth he usually lacks several things. In the first place, though he knows a good deal about antithesis of operations (e.g. he knows that subtraction is the opposite of addition, division of multiplication, movement in the direction minus $-x$ of movement in the direction $x$, and so on), he has not the habit of observing in what respects antithetic operations neutralize each other, and in what respects they are cumulative; and surely no habit is more needed than this as preparation for making calculations in electricity or mechanics. In the next place, he too often knows, about the idea of relevance, only enough to be foggy about it. The reason for this is that his study of the idea of relevance itself began where it ought to have ended; his attention was never called to it till the stage was reached when it would have been right that he should direct his action in regard to it subconsciously from long habit, leaving conscious attention free for dealing with the actual elements of some question which is difficult enough to need thinking about. He was not made to grasp the fundamental idea that a statement may be relevant to one question and irrelevant to another, till some knotty problem occurred involving consideration of which statements are relevant to the special question in hand. So he had to try to grasp at once the idea of relevance and the question, what is relevant to what, in a special problem. Such thrusting on the young brain of two difficulties of different kinds at once, is contrary to all accepted canons of Psychology. Examples are here suggested (Lesson XIII) in which there can be no doubt as to what is relevant to the question at issue; the child's attention is therefore free to focus itself on the idea that there can be facts concerning a thing which in no way concern the particular question which is just now being asked about the thing. Then again, whatever skill he may have acquired in the manipulation of those notations and formulae which he has been taught to use, he knows hardly anything about the manner in which such things come into being. Now an applier of Mathematics to real forces should be able, when occasion requires, to modify his notation, or invent a new formula, for himself. He cannot begin to learn to do this, straight away, while his mind is struggling with problems of electricity or mechanics; he should have had, from the first, the habit of seeing through formulae and notations; of watching them coming into being, of helping to construct them.
3. The following is taken from Mary Everest Boole, The preparation of the child for science (Clarendon Press, Oxford, 1904).
3.1. Teaching algebra and trigonometry

3.2. Handling objects

Each object (in school) is catalogued as intended to teach this, or to prove that, or to illustrate so-and-so; many ... seem to have no idea that it may be well to let a child have things and handle them, without anyone talking, and find out what things have to say.

3.3. Experimenting

It is desirable that children should sometimes be free to experiment under varied and accidental conditions ... and learn by making mistakes with some one about ... to whom they can apply when puzzled or discouraged.

3.4. New ideas

Receptivity cannot be generated by early teaching of a subject mixed up with the use of its appropriate technical machinery; but only by suggesting the new ideas by means of objects already familiar to the child's eye and touch.

3.5. Counting

When he begins to do sums on paper, let him still, for a considerable period, do each addition, subtraction, etc., first in counters; and then, while these are still on the table, work out the same sum on paper.

3.6. Finding results

Anything which he (the teacher) intends to prove should never be stated; children should be led up to find it out for them selves by successive questions.

3.7. Training

By training the hand to trace out Nature's actions, we train the unconscious mind to act spontaneously in accordance with Natural Law; and the unconscious mind, so trained, is the best teacher of the conscious mind.

3.8. Geometric objects

First comes the education of the senses. From the time when an infant begins to stroke the cat ... have geometric solids as ornaments or toys, so that the senses of sight and touch may actually develop in contact with true type-form.