Mary Everest Boole: the first mathematical psychologist

Edmund F Robertson

University of St Andrews

1. Introduction

Mary Everest Boole was only 32 years old in 1864 when her husband, the mathematician , died leaving her with five daughters to bring up. In this paper we will look at Mary Boole's life and her contributions to mathematics teaching. She published many books on teaching in general and teaching mathematics in particular, for example Lectures on the Logic of Arithmetic (1903) and Philosophy and Fun of Algebra (1909). We give quotes from her books to illustrate her ideas which, although over 100 years old, seem surprisingly relevant today.

2. Early life

Mary Everest was the daughter of Reverend Thomas Roupell Everest (1800- 1855), Rector of Wickwar, and Mary Ryall (born 1815). Thomas Everest had one brother, George Everest (1790-1866), who was the surveyor and geographer after whom was named. Mary Ryall had a brother John Ryall who became Vice President and Professor of Greek at Queen's College, Cork. After Mary Everest married George Boole she became known as Mary Everest Boole. For simplicity, we shall refer to her as Mary Boole throughout this paper. She had a brother, George John Everest (1835-1908) who became vicar of Teynham in Kent.

Thomas Everest had studied at Pembroke College, Cambridge and was ordained in 1826. He was a priest at Bristol Cathedral before becoming rector of Wickwar, near Chipping Sodbury in Gloucestershire, in 1830. On 27 September 1830 he married Mary Ryall; their daughter Mary (the subject of this paper) was born on 11 March 1832 and their son George in 1835. In 1837 an influenza epidemic swept through England leaving Thomas Everest as an invalid. Being a strong believer in homeopathy, he went with his family to Poissy, near Paris, in France in 1837 to be treated by Samuel Hahnemann, the founder of the homoeopathic system of medicine, leaving his curate in charge of the church at Wickwar. Mary Boole was at this time five years old but she had to follow the same routine as Hahnemann devised for the whole family (as well as their servants) which included cold baths, long fast walks before breakfast and a special diet.

By 1843 Thomas Everest was well enough to return to his church at Wickwar, but the family maintained the routine devised for them by Hahnemann for eleven years. Whether Thomas Everest recovered because of Hahnemann's treatment or in spite of Hahnemann's treatment is a matter of opinion but certainly the whole family had great faith in the homoeopathic approach. While in France, Mary Boole became bilingual in French and English. In fact, in later life she considered French her native tongue. She attended school as well as being tutored by a governess. Her mother also gave her lessons which she later described as "hopelessly dreary." Later, while still in France, she was tutored by a Monsieur Deplace who impressed her with his teaching methods. When teaching her to solve arithmetic problems he would ask a series of questions, then ask her to analyse her answers so that she eventually found the solution to the original problem herself.

At age ten, when still in France, Mary Boole heard her father speaking with a friend who was visiting from England. The friend spoke about the University of Cambridge being a great centre for mathematics and spoke, in particular, about . After the friend left, Mary heard her father talking to her mother saying he didn't know what he would do with his children since his son lacked interest in education as he would expect of a girl, but his daughter had the educational interests and abilities he would expect of a boy. "If only she could go to Cambridge and study mathematics she would carry everything before her," he said, "but what could a girl do learning mathematics." Mary was shocked to learn that she couldn't go to university, that she couldn't study mathematics and that she would never meet the famous Charles Babbage.

A year later the family were back in England. Mary Boole couldn't go to university but she could learn mathematics by teaching herself from books. She met many of her father's friends and these included and Charles Babbage. This provided her with further motivation to read mathematics books. She attended school for a short time but soon was taken out of education to become an assistant to her father. She had some practical teaching experience taking the Sunday School class as well as writing experience helping her father with his sermons. In December 1843 Mary Boole's uncle, George Everest, returned from . He had undertaken much work there for the Great Trigonometrical Survey and was knowledgeable about mathematics. He also had many fascinating tales to tell of his travels in India while carrying out the surveys. George Everest spent many hours with Mary, encouraging her and telling her about his experiences. George Everest enjoyed his discussions with Mary so much that at one time he even asked her father if he couldn't adopt her, but this was not what her parents wanted.

3. George Boole

In 1850 Mary visited her uncle John Ryall at Queen's College, Cork. There she met the professor of mathematics, George Boole, and despite the age difference (she was eighteen years old and he was thirty-five) they became friends. George Boole helped Mary with some of the more challenging aspects of the differential calculus and after she returned to England they corresponded frequently, much of the letters being about mathematics and science. In 1852 George Boole visited the Everest family home in Wickwar and at this time he became more formally her mathematics tutor.

On 15 June 1855 Mary's father died leaving her without means of support and George Boole proposed marriage. After a short engagement, they married on 11 September 1855 at a small ceremony in Wickwar. The lived at first in Sundays Well Road, Cork, then in Castle Road, Blackrock, Cork, before moving to Litchfield Cottage, Ballintemple, Cork. It proved a very happy marriage with five daughters: Mary Ellen Boole born in 1856, Margaret Boole born in 1858, Alicia Boole (later Alicia Boole Stott) born in 1860, Lucy Everest Boole born in 1862, and Ethel Lilian Boole born in 1864. George Boole died on 8 December 1864 with his daughter Ethel Lilian less than seven months old; she had been born on 11 May 1864. Mary Boole was only 32 years old, having five young daughters and no means of support.

4. Mary returns to England

Mary Boole moved back to England where, thanks to Frederick Denison Maurice, she was offered the position as a librarian at Queen's College, . She lived at 43 Harley Street, London and from that address she corresponded with Charles Darwin in December 1866 about the implications for his theory of evolution on religion. Mary gave a series of talks to young mothers who were worried about their religion being threatened by Darwin’s theories and used her talks as the basis for a book The message of psychic science to mothers and nurses which she completed in 1868 but it was only published in 1883. At the time of the 1871 census Mary was at the home of her unmarried uncle Robert Everest in Sunninghill, Berkshire, while her children remained at Harley Street, London. Mary's occupation is given as "Teacher of Mathematics".

Shortly after this she began working as 's secretary. James Hinton (1822-1875) was a talented surgeon who, as well as writing on his medical specialities, wrote on ethical subjects and on 'thinking'. Mary Boole described Hinton as a 'thought-artist'. James Hinton died on 16 December 1875 and after this time Mary Boole began publishing books and articles. Let us note that Mary Boole's daughter Mary Ellen Boole married James Hinton's son Charles Howard Hinton (1853-1907) in 1880. Charles Howard Hinton was a mathematician who wrote "What is the Fourth ?" but, after he was convicted of bigamy, Charles and Mary Ellen went first to Japan and then to the United States.

At the 1881 census Mary Boole was living at 103 Seymour Place, Marylebone, London with her 22 year old daughter Margaret Boole whose occupation is described as "Art Student (Painting)". In 1885 Margaret Boole married the artist Edward Ingram Taylor in Marylebone, London; their son was the mathematician Geoffrey Ingram Taylor. At the time of the 1881 census Mary's occupation is given as "Civil Service Pensioner". In 1890 Alicia Boole married Walter Stott. Alicia Boole Stott became a mathematician making contributions to 4-dimensional geometry. The word "polytope" is due to her. At the 1891 census Mary Boole was visiting her mother who was living next door to Mary's brother George Everest, vicar of Teynham, Kent. Mary was described as "living on her own means". At the 1901 census Mary Boole was living at 16 Ladbroke Road, Notting Hill, Kensington, London. She was again described as "living on her own means". She was at the same address at the 1911 census.

5. Mary Boole's publications

Let us list here some of Mary Boole's publications. These include The message of psychic science to mothers and nurses mentioned above and: Symbolic Methods of Study (K. Paul, Trench & Company, 1884); Logic taught by love (Alfred Mudge & Son, 1889); Mathematical psychology of Gratry and Boole: Translated from the Language of the Higher Calculus Into that of Elementary Geometry (Swan Sonnenschein & Company, 1897); Boole's Psychology as a Factor in Education (Benham & Company, 1902); The cultivation of the mathematical imagination (Colchester, 1902); Lectures on the Logic of Arithmetic (Clarendon Press, 1903); The Preparation of the Child for Science (Clarendon Press, 1904); Mistletoe and Olive. An Introduction for Children to the Life of Revelation, Etc. [In Prose and Verse.] (London, 1908); The Message of Psychic Science to the World (C.W. Daniel, 1908); The Message of Psychic Science to Mothers and Nurses (C.W. Daniel, 1908); Miss Education and Her Garden: A Short Summary of the Educational Blunders of Half a Century - dedicated (without Permission) to the Educational Authorities of Great Britain (C.W. Daniel Company, 1908); Philosophy and Fun of Algebra (C.W. Daniel, 1909); A Woodworker and a Tentmaker (C. W. Daniel, 1909); Suggestions for Increasing Ethical Stability (C.W. Daniel, 1909); Some Master Keys of the Science of Notation: A sequel to "Philosophy and Fun of Algebra" (C.W. Daniel, 1911); The Forging of Passion Into Power (M. Kennerley, 1911); What One might say to a Schoolboy (C.W. Daniel, 1921); At the foot of the Cotswolds (C.W. Daniel Company, 1923); The Psychologic Aspect of Imperialism: A Letter to Dr Bose (C.W. Daniel, 1931).

6. Teaching algebra and trigonometry

Let us look at Mary Boole's ideas about teaching algebra and trigonometry. We give a substantial quote from her book The preparation of the child for science (Clarendon Press, Oxford, 1904):

The mathematical formulae which some people speak of contemptuously as "dry" are in reality as beautiful as microscopes, or any other well-adjusted and well-finished machinery intended to extend the scope of men's powers; and as for mathematical ideas themselves, they are as grand as any expressed in poetry. Comparatively few people get any real enjoyment out of either; and the reason obviously is that their faculties have been stamped into confusion by a method which I can only compare to that of making a child use a telescope before it can see properly with its eyes; using a complicated machine for extending and refining certain work before he has learned to do, without it, the simpler kinds of that work. To make clearer what it is that is wrong, let us think of an orchard, at harvest time, which contains, besides the trees and fruits, the natural human limbs with which man picks the fruits he can reach; the stools, steps, and ladders of various lengths, which are an extension of his own legs and by means of which he rises to the level of the fruits which he cannot naturally reach; the sickles or fruit-scissors which improve on the action of the hand and enable us to bring down the choicest fruit without risk of spoiling its delicate bloom; the room fitted with shelves on which the fruit is stored for future use; and the baskets in which it is temporarily packed for safe conveyance to the store- house. It is no exaggeration to say that all these various items have analogues in science, especially in mathematics. A mathematical textbook contains truths valuable in themselves because throwing light on the nature of human thought; other truths valuable for use in commerce or physical science; natural processes of reasoning by which the student can gain direct perception of some of these truths; artificial devices for arriving at others: refinements of various sorts; and formulae and tables in which truths are classified, stored for easy access, and preserved in the memory. It is pitiable to see how far are many students, even advanced ones, from any clear realization which of these various things is which. Many have no conception of the difference between direct and inverse in mathematics. They make no clear distinction between the truth itself, the ladder of devices by which they reached it, and the formulae in which they stored it. And can we wonder at this? No human being, I suppose, ever attempted to teach a child to climb a ladder, to use a fruit-sickle, or to store fruits on shelves, in the same summer in which it first was able to stand on its legs and grasp a low- growing apple, in which it first experienced the delight of eating fruit. It would probably be an under-statement of the case to affirm that a mode of treatment analogous to this has been inflicted on ninety-nine per cent of the young people who have learned Algebra and Trigonometry even within the last ten years. The methods of doing this stupid thing have no doubt improved enormously since the early days of De Morgan; all praise is due to the patient workers who have done so much to improve them; but the stupid thing is still done; and the parents and the public still insist on its being done. The questions so often put by parents, 'At what age do you think my child had better begin Algebra?' (or Trigonometry) and, 'Can you recommend me to a good teacher?' really mean something analogous to this:- 'I intend to keep my child ignorant of all experiences concerning fruit, and all processes connected with it, till he is old enough to begin receiving straight away, in one continuous series of lessons, information, conveyed by verbal explanations, about how to stand on one's legs, how to climb ladders, how to use sickles, how fruits taste, their hygienic and economic value, their botanical classification, and the best means of preserving them. At what age do you consider this series of lessons should begin, and whom do you recommend me to employ to give it?' The only answer one could make to such a question would be that there is no age at which any such course should begin, and no person who ought to be asked to give it. The question asked by a parent should be, 'At what age would you recommend me to let my child begin learning such portions of Algebra (or Trigonometry) as can only be learned by the aid of complicated devices invented, centuries after the science itself was an actual working possession of our race, for the sake of projecting its action into fields which would be inaccessible to it if only natural and simple tools were used?' The answer should be, 'When the process of learning by the more direct means has become so familiar as to be performed sub-consciously.'

This quote presents a common theme in much of Mary Boole's view of the educational process. She believed that a child should become familiar with mathematics from a young age, not through formal teaching, but rather through handling object such as spheres, cubes, cones, etc. She believed in suggesting new ideas to children by means of objects already familiar to them through sight and touch. The ideas which then became part of their unconscious mind of a child would act as a teacher to their conscious mind. When moving to more advanced mathematical results, the teacher should never state them but should ask the child questions until they find the result for themselves. Here she was very much advocating the method used by her tutor Monsieur Deplace when she was growing up in France.

7. Applying mathematics

Let us now quote from the Preface of Mary Boole's book Lectures on the logic of arithmetic (Clarendon Press, Oxford, 1903):

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 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.

Let us first note that Mary Boole's says that one problem is that children don't know how the mathematical ideas they are being taught came into existence. This surely is a plea to teach the history of mathematics, and I must totally agree with this sentiment. Again she says that children are not taught relevance. We teach students integration by parts, then give a series of examples on integration by parts. Perhaps, if we follow her advice, we should give students a list of examples and ask, "Which of these should be tackled using integration by parts?"

8. Two elements in teaching

The quote below by Mary Boole, from D G Tahta (ed.), Boolean Anthology: Selected Writings of Mary Boole on Mathematical Education (Association of Teachers of Mathematics, London, 1972), repeats again some of her ideas about teaching that we have already encountered:

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.

9. Conclusion

To appreciate fully Mary Boole's contribution it is important to understand how children were taught in school in her day. Rigid military discipline saw children learning by rote, sitting 6 or 7 hours a day on a bench in a classroom, and beginning again to a task of 2 or 3 hours at night. We have moved far since those times and teaching today has certainly moved a long way in the direction Mary Boole advocated. Do we still have further to go? One final thought. Mary Boole was taking a psychological approach long before Sigmund Freud; she was born 24 years before Freud. Her writings, put into the context of the time, are remarkably far sighted and she deserves more credit that she has received. I am, however, encouraged that on International Women's Day, Thursday 8 March 2018, Cambridge University Press circulated a list of 17 leading women mathematicians with short biographies. Mary Everest Boole was on that list.

10. References

1. A little bit of history. Famous Mathematicians - Mary Boole (1832-1916), Primary Magazine Issue 21 (National Centre for Excellence in the Teaching of Mathematics, 2010).

2. M Boole, Lectures on the logic of arithmetic (Clarendon Press, Oxford, 1903)

3. M Boole, The preparation of the child for science (Clarendon Press, Oxford, 1904)

4. E M Cobham, Mary Everest Boole: A Memoir with Some Letters (Ashington, 1951).

5. E M Cobham (ed.), Mary Everest Boole, Collected Works (4 Volumes) (London, 1931).

6. R S Creese, Boole, Mary (1832-1916), Dictionary of National Biography (Oxford, 2004)

7. S Innes, Mary Boole and curve stitching: a look into heaven, Endeavour 28 (2004), 36-38.

8. G Kennedy, The Booles and the Hintons (Atrium Press, 2016).

9. L M Laita, Boolean algebra and its extra-logic sources: the testimony of Mary Everest Boole, History and Philosophy of Logic 1 (1980), 37-60.

10. K D A Michalowicz, Mary Everest Boole (1832-1916): An erstwhile pedagogist for contemporary times, Vita Mathematica: Historical Research and Integration with Teaching (1996), 291-298.

11. A Pinch, Thinking about Other People in Nineteenth-Century British Writing (Cambridge University Press, Cambridge, 2010).

12. D G Tahta (ed.), Boolean Anthology: Selected Writings of Mary Boole on Mathematical Education (Association of Teachers of Mathematics, London, 1972).

13. K A Valente, Giving wings to logic: Mary Everest Boole's propagation and fulfilment of a legacy, The British Journal for the History of Science 43 (1) (2010), 49-74.