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The Didactical De Morgan: a Selection of Augustus De Morgan's Thoughts on Teaching and Learning Mathematics

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The Didactical De Morgan: a Selection of Augustus De Morgan's Thoughts on Teaching and Learning Mathematics The Didactical De Morgan: a Selection of Augustus De Morgan's Thoughts on Teaching and Learning Mathematics ABRAHAM ARCAVI, MAXIM BRUCKHEIMER Less well known than Hamilton for any single achieve­ At the age of 16 he entered Trinity College, Cambridge, ment, less well known than Salmon for his thoroughness where he came under the influence of such mathemati­ in science, familiar to a much larger circle of readers cians as George Peacock He graduated in 1827 as fourth because of his wide range of interest and his skill in wrangler* He came a "mere" fourth apparently, because popularizing the science, was that eccentric but brilliant already then his interests interfered with the "drilling•· teacher, Augustus De Morgan. [I] necessary to "succeed" in the tripos. (De Morgan has left In spite of the excellence and extent of his mathematical some severe criticism of the Cambridge system **)Within writings, it is probably as a logical reformer that De a year, the University of London (what is now University Morgan will be best known to future times In this respect College) was established, and De Morgan became the first he stands alongside of his great contemporaries Hamilton professor of mathematics-at the age of 22-a post ht: and Boote, held with a break of five years, for over 30 years As a teacher of mathematics De Morgan was un­ rivalled. De Morgan's writings, however excellent, His wife, Sophia Elizabeth De Morgan [3, pp 98-99. give little idea of the perspicuity and elegance of his viva 100-10 I] quotes the following description of his teachin> voce expositions, which never failed to fix the attention of at the University as given by one of his former students all who were worthy of hearing him [2] As Professor of Pure Mathematics at University College. London, De Morgan regularly delivered four courses o1 As a popularizer of science, as a logician, or as a mathe­ lectures, each of three hours a week, and lasting through­ matician, De Morgan has his place in the history of the out the academical year. He thus lectured two hours ever~ 19th century mathematics. In this article, we would like to day to his College classes, besides giving a courst glance at a less well known aspect of his extraordinary addressed to schoolmasters in the evening during a por­ creativity: his thoughts and insights on the teaching and tion of the year His courses embraced a systematic vie" learning of mathematics It would take more serious his­ of the whole field of Pure Mathematics, from the firs1 torians than we pretend to be to present a considered view book of Euclid and Elementary Arithmetic up to tht of his writings in this field, including any influence he may Calculus of Variations from two to three years wen ordinarily spent by Mathematical students in attendanct have had on his contemporaries and the following genera­ on his lectures De Morgan was far from thinking th< tions. (It would also require access to more extensive duties of his chair adequately performed by lecturint source materials than we have available) The purpose of only At the close of every lecture in each course he gavt the present paper is to present a selection illustrating the out a number of problems and examples illustrative of th1 sharpness of his ideas, his creative insights and his wit, for subject which was then engaging the attention of the class the enjoyment of the reader. Thus we present a small His students were expected to bring these to him worke< collection, which we believe still has meaning for anyone out He then looked them over, and returned them revise< engaged in understanding the teaching and learning of before the next lecture. Each example, if rightly done, wa mathematics, even though the extracts were written 150 carefully marked with a tick, or if a mere inaccurac: years ago. The reader whose appetite has been whetted occurred in the working it was crossed out, and the prope correction inserted. If, however, a mistake of prim ipf, can find much more in the same vein in the references was committed, the words 'show me' appeared on th, cited. We precede the selection by a short description of exercise The student so summoned was expected to pre his life and work. sent himself on the platform at the close of the lecture when De Morgan would carefully go over the point witl Life and work him privately, and endeavour to clear up whatever diffi Augustus De Morgan was born in Madras, India in 1806, culty he experienced The amount of labour thus involve' where his father held a position in the army When he was seven months old his family moved back to England He attended private schools at which he acquired a mastery *A person placed in the first class of the mathematical tripos, which ar any of the examinations for the B. A degree with honors at Carr of Latin, Greek and Hebrew Very early in his life he bridge University developed a strong interest in mathematics, although his **See. for example, in the description of his teaching in the folio win talents were unnoticed. and (3, p 283] For the Learning of Mathematics 9, l (February, 1989) 34 Fl M Publishing Association, Montreal, Quebec, Canada was very considerable, as the number of students in lesser capacity Thus, at the same time as refusing the attendance frequently exceeded one hundred The Presidency, he proposes himself as secretary with the fundamental conceptions of each main department of following self-description: "A De Morgan-well enough Mathematics were dwelt upon and illustrated in such in his way, but cranky-for all miscellaneous work, from detail as to show that, in the judgement of the lecturer, a wax candles to Council minutes " [3, p. !54] thorough comprehension and mental assimilation of It was also a matter of principle with him not to accept great principles far outweighed in importance any mere '"honours" Thus in reply to an offer of the honorary analytical dexterity in the application of half-understood principles to particular cases. T'hus. for instance, in Trig­ degree of LL.D of the University of Edinburgh, De M or­ onometry, the wide generality of that subject, as the gan writes: "I hope I shall give no offence by very respect­ science of undulating or periodic magnitude. was brought fully declining the honour. I mean the diploma. The out and insisted on from the very first. In like manner the honour lies in the good opinion of the Senate, and that Differential Calculus was approached through a rich your communication gives me a right to say I have already conglomerate of elementary illustration, by which the earned My reason for declining the degree is my own notion of a differential coefficient was made thoroughly pecular dislike of conventional titles, which are not what intelligible before any formal definition of its meaning bridge, they seem to be" [3, p. 303] had been given The amount of time spent on any one The same principle extended even to the Royal Society lemati­ subject was regulated exclusively by the importance Thus in a lengthy and mainly critical discussion of the fourth which De Morgan held it to possess in a systematic view )ecause of Mathematical science. The claims which University or Royal Society in the Introduction to his Budget of Para­ rilling" College examinations might be supposed to have on the doxes he writes: "Whether I could have been a Fellow, I has left studies of his pupils were never allowed to influence his cannot know; as the gentleman said who was asked if he Within programme in the slightest degree He laboured to form could play the violin, I never tried Everyone who saw sound scientific Mathematicians, and, if he succeeded in iversity the three letters [F R S ] after my name would infer cer­ this, cared little whether his pupils could reproduce more tain things as to my mode of thought which would not be the first or less of their knowledge on paper in a given time On one true inference" [4, Vol I, pp 25, 28] post he occasion, when I had expressed regret that a most distin­ Similarly De Morgan never '"confessed with my lips" guished student of his had been beaten, in the Cambridge his Christian belief, ''because in my time such confession 98-99, Mathematical Tripos. by several men believed to be his eaching inferiors. De Morgan quietly remarked that he "never has always been the way up in the world" [3, p 368] A remarkable tribute to De Morgan's character is .udents. thought-likely to do himself justice in THE GREAT WRITING RACE .., All uam he held in the most sover­ given by Mary Everest Boole, wife of George Boole: "M r College, eign contempt I remember, during the last week of his Boote's work was, as he gratefully acknowledged, made )urses of course which preceded an annual College examination. possible by the generous and self-forgetting aid freely through­ his abruptly addressing his class as follows: ''·I notice that given to him by many contemporary mathematicians; and urs every many of you have left off working my examples this week. in particular by Professor De Morgan, who seemed to a course I know perfectly well what you are doing; YOU ARE take a pleasure in effacing himself to bring forward the ng a por­ CRAMMING FOR THE EXAMINATION. But I will man whom he might have been expected to feel a rival." atic view set you such a paper as shall make ALL YOUR CRAM of [5] the first no use 1p to the Augustus De Morgan died in 1871 aged 65 ~ars were De Morgan was very active on two fronts: the populariza­ .tendance tion of science on the one hand and the raising of the level nking the of scientific activity in England.
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