Craftsmanship In The Teaching Of Elementary Mathematics PDF
Preview Craftsmanship In The Teaching Of Elementary Mathematics
Craftsmanship in the of Teaching Elementary Mathematics BLACKIE & SON LIMITED 50OldBailey,LONDON 17 StanhopeStreet,GLASGOW BLACKIE & SON (INDIA) LIMITED WarwickHouse,FortStreet,BOMBAY BLACKIE & SON (CANADA) LIMITED TORONTO Craftsmanship in_the_ Teaching of Elementary ^ - -O - " v - Mathematics BY F. W. WESTAWAY FormerlyoneofH.M.InspectorsofSecondarySchools Authorof"ScientificMethod,itsPhilosophicalBasisanditsModesofA "ScienceTeaching: WhatitWas WhatitIs WhatitMightE "TheEndlessQuest: 3000YearsofScience"&c. BLACKIE & SON LIMITED LONDON AND GLASGOW Firstissued1931 Reprinted1934,*937 & PrintedinGreatBritainbyBlackie Son,Ltd.,Glasgow " Have some wine" the March Hare said in an encouraging tone. Alice looked round the table, but there was nothing " on it but tea. / don't see any wine" she remarked. " There isn't any'' said the March Hare. " Then it wasn't very civil of you to offer it" said Alice angrily. " You are sad," the White Knight said. " Letme sing you a song to comfortyou." " Is it very long?" Alice asked,forshe hadheard a good deal ofpoetry that day. " " It's long" said the Knight, but it's very, very beautiful. Everybody that hears me sing it either it " brings the tears into their eyes, or else " Or else what?" said Alice, for the Knight had made apause. " Or else it doesn't, you know. The song is called ' WAYS AND MEANS ', but that's only what it's called, you know!" " Well, what is the song, then?" saidAlice. " " / was coming to that," the Knight said. The songreally is 'A-SITTING ONA GATE ': andthe tune's my own invention" BY F. W. WESTAWAY SCIENCE TEACHING Whatit Was WhatitIs WhatitMightBe Second Impression. 10s.6d. net "Getthebookandreadit; itisthebestthingyet. Itispacked withpracticaladvicewhichwillalwaysbeofvalue." Journal ofEducation. "His hookwll setmanya youngteacherontheright path, and will help many an older one to raise his performance to a much higherlevelorexcellence." Nature. "Revealsoneverypagethezestfulinterestofatruecraftsmanin rteeaadchbiyngalbllehnedademdawsittehrsinafnodrmheedadgmoiosdtrseesnssees.in..s.ecTohnidsarbyooskchsohoolsu,ldanbde itisworthytobestudiedbyeveryteacherofscience. Ifitscounsels areadoptedandfollowedweshallseeagreatand beneficentchange inthepresentmethodofdealingwithscienceasafactorineducation." EducationOutlook. "Thisisaremarkablebook,criticalandstimulating,theproductof the author's long experience as teacher, headmaster, and H.M.I. h.e.lp.fcuolmgpuriedheetnostihveebiengisncnoepreaannddansoinpsrpaicrtaitciaolntthoaatll.itWwiellrebceoammmeonsdt it unreservedly to all engaged in science teaching in schools and universities." School Science Review. PREFACE When asked to write a book on the teaching of Elementary Mathematics, I felt doubtful as to the avenue by which the subject might be best approached. During the present cen- " " " " tury, the general policy and attitude to be adopted in mathematical teaching have been discussed by so many authorities that there seemed very little new to say. Finally I decided that class-room craftsmanship might be made a suitable basis of treatment. Thus the book is not intended for the experienced teacher who has already acquired skill in his art, but forthe still struggling beginner. Inthe leading schools, mathematical craftsmanship probably leaves little to be desired, but the leaven has yet to work its way into the mass. From the great variety of topics that come within the ambit of the various mathematical subjects, I have selected for treatment those which, in my experience, seem to give young teachers most difficulty. To treat all topics that come within the daily practice ofmathematical teachers is impossible; it would mean writing a dozen books rather than one. I have sometimes been asked if, as an Inspector pursuing the same daily round year after year, decade after decade, I am, when listening to lessons in mathematics, ever amused, ever really interested, ever inclined to be severely critical, ever bored. vii PREFACE viii Amused? Yes; for instance, when a young master tells his boys that mathematics is by far the most important subject they learn, inasmuch as it is the only one that leads them into " the region of pure thought ". Really interested? Yes, every day ofmy life. In the crafts- manship ofeven a beginnerthere is almost always some element of interest; in the craftsmanship of a really skilful mathe- matical teacher there is to me always a veritable joy. I never enter a classroom without hoping to find something which will make an appeal, and I am not often disappointed. Sometimes disappointed, of course; unfortunately not all mathematical teachers have come down from heaven. Severely critical? Yes, occasionally, more especially at the rather slavish adoption of certain doubtful forms of traditional procedure. For instance, a teacher may include in the work ofthe bottom " Set " ofa Form the Italian method of division, well knowing that two-thirds of the boys will thenceforth always get their sums wrong. Another teacher may adopt " standard form ", not because he has examined it and found " it to be good, but because everybody does it nowadays ". Instead of saying, " I thought we had to do these things," why do not mathematical teachers hold fast to the faith which is really in them? If their faith, theirfaith, includes the Italian method, standard form, and the score of other doubtful ex- pedients that spread like measles from school to school, I have nothing more to say. Bored? Yes, though not often. The petrifying stuff often doled out to Sixth Form specialists, the everlasting Series and Progressions,theolddodgesanddevicesandbookwork"proofs" ad nauseam in preparation for scholarship examinations, all this is virtuallythesame nowas fortyyears ago. True, teachers PREFACE ix are not much to blame for this. Boys have to be prepared for the scholarship examinations, and according to prescription. But that does not soothe an Inspector who has to listen to the same thing year after year, and I admit that, with Sixth Form work, sometimes I am almost bored to tears. If I had to pick out those topics which in the classroom make the strongest appeal to me, I should include (i) Arith- metic to six-, seven-, and eight-year-olds, when well taught; (ii) Beginners' geometry; (iii) Upper Fourth and Lower Fifth work when the rather more advanced topics in algebra, geo- metry, trigonometry,and mechanics are being taken forthe first time (not the Upper Fifth and its revision work); (iv) Upper Sixth work when examinations are over and the chief mathe- matical master really has a chance to show himself as a master of his craft. Sixth and Upper Fifth Form work often savours too much of the examination room to be greatly interesting: everything is excluded that does not pay. But inasmuch as examination success is a question of bread and cheese to the boy, the teacher is really on the horns of a dilemma, and very naturally he prefers to transfix himself on that horn that brings him the less pain. Why, of all the subjects taught, is mathematics the least popular in girls' schools? and whyjs [t_the one subject in which the man in the street feels no personal interest? It is not because mathematics is difficult to teach. My own opinion is that it improbably the easiest ofall subjects to teach. When it is taught by well-qualified mathematicians, and when those mathematicians are skilled in their teaching craft, suc- cess seems always to follow as a matter of course, in girls' schools equally with boys'. The failure to make any head- way, even under the best conditions, on the part of a small x PREFACE proportion of boys and a rather larger proportion of girls is probably due to a natural incapacity for the subject. Had I my own way, I would debar any teacher from teaching even elementary mathematics who had not taken a strong.-doge^of the calculus, and covered a fairly extensive field of advanced work generally. It is idle to expect a mathematical teacher to handle even elementary mathematics properly unless he has begjx.through the mathematical mill. And yet I have heard a Headmaster say, " He can take the Lower Form mathe- matics all right; he is one of my useful men: he took a Third in History." As long as University Scholarships are what they are, so long will Sixth Form specialists* work proceed on present lines. But one purpose of the book is to plead for considera- tion of the many neglected byways in mathematics and for their inclusion in a course for all Sixth Form boys; suggestions to this end are made in some of the later chapters. We want a far greater number of ordinary pupils to become mathe- matically interested, interested in such a way that the interest will be permanent; and we want them to learn to think mathe- matically, if only in a very moderate degree. Why do ordinary pupils shrivel up when they find a mathematician in their midst? It is simply that they are afraid ofhis cold logic. There is, in fact, a curious popular prejudice against mathe- maticians-a-s.a,class. It probably arises from the fact that we *~~_-.._.~ - * j arenotanationofclearthinkers,andwe dislikethefewamongst us who are. Foreigners at least the French, the Germans, and the Italians are mathematically much keener than we are. They seem to become immediately interested in a topic with mathematical associations, whereas we turn away from it, disinclined to take part in a discussion demanding
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