ARE BEGINNING TEACHERS OF HIGH SCHOOLMATHEMATICS ADEQUATELY TRAINED?

W. G. GINGERYGeorge Washington HigJi School, Indianapolis, Indiana

The topic assigned for this paper is manifestly a rhetoricalquestion. The answer is that beginning teachers of high schoolmathematics are not adequately trained. Such a statement how-ever is without profit unless one can tell something of therespect in which they might be better trained.

It has been only a few years since we began teaching teachersto teach. When I joined the ranks of the profession, the tech-niques of teaching were supposed to be innate. If one were ableto teach he was a born teacher. In those days the mathematicswas better taught than most any other subject. There are tworeasons for this situation. First the mathematics is an ancientsubject and the centuries have refined and organized it into anorderly discipline with simply graduated progress from the lessto the more difficult. Such a subject lends itself more readilyto the efforts of the untrained teacher. In the second place, inthose days mathematics was considered the most importantingredient of education and a pupil who could not readilymaster it was soon eliminated from the school. In that caseonly those who could easily learn the subject remained to betaught. No other subject enjoyed such distinction as did themathematics for limiting the school enrollment.

Since the training of teachers began the ease with whichmathematics was traditionally taught has been a disadvantageto it. For years it was assumed that anyone could teach algebraor geometry and so the department was largely parcelled out toteachers of other subjects who had an extra period or two. Inthe meantime other subjects, notably English and social science,have taken the key positions in the curriculum and the studentbody is no longer limited to those who can learn mathematicseasily.While these changes in the pedagogical situation have been

taking place the world in which the student is to live has beengrowing rapidly more complex and more inclusive and the majorpart of this growth has been quantitative. It takes many moreas well as more complex quantitative concepts to live success-fully now than it did a half century ago. All this has increased

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the burden on the already abbreviated courses in mathematicsand made it still less possible for the students to master it.The public now compares the pupil’s ability to manage his pres-ent complex quantitative situation with the pupil of forty orfifty years ago in a much simpler situation and blames thepresent mathematics teacher for the unfavorable impressionher pupil makes.

Actually the statement of the topic assigned for this paperaffords the high school teacher an opportunity to pay off an oldscore. I presume that the most cogent reason that may be as-signed for young mathematics teachers not being well pre-pared is the fact that much, possibly most college teaching ofmathematics is deadly dull and insignificant. The other daywhile interviewing a student still in college who will be a can-didate for a mathematics position I picked up a program of theIndiana Section of the American Mathematical Society. Muchof that program is devoted to Topology. I asked the candidateto define Topology. Like myself he had never heard of it before.But he said "I will ask my professor. He will know and he willtell me. But after he tells me I will not know, for that is the waywith everything he tells us. He makes nothing clear orinteresting."

In Indiana, I believe the Calculus is now required of all can-didates for mathematics license. It is certainly the most im-portant course in the group known as higher mathematics andit is the most powerful of all the tools in the kit of the mathe-matician. The most significant concept of all the calculus is thederivative. It is therefore a pretty important concept. Probablythe invention of the derivative should rank with the discoveryof the law of gravitation or the discovery of radium in im-portance as a contribution toward man’s understanding of andcontrol of his environment. I frequently ask young candidatesfor teaching positions to define the derivative. Very few ofthem can. To most of them, as a concept it means nothing, yetthey have ground through hundreds of problems using it, forhopelessly dull answers that likewise meant nothing to thembeyond being right or wrong.

It is not the fault of the mathematics. It is significant enough.Much of it can be shown to touch upon the most real situationsin which the student lives. It furnishes foundation material forall quantitative considerations which include the whole eco-nomic and scientific structure of modern civilization. It illus-

ARE TEACHERS ADEQUATELY TRAINED? 705

trates and supplies exercises in the fundamental concepts oflogic on which rest language and communication. The fault liesin the fact that it is too often presented as a dead, dry, abstractsubject with no* significance to the average individual.

There are three important items in the training of a mathe-matics teacher. The first is complete mastery of that part ofthe subject field to be taught. This includes an analysis of thesubject matter into small simple steps graduated as to dif-ficulty and involving only one new concept at a time. Textbook writers have done this well, but it is important that theteacher know about it and be on the alert for improvements.In this respect the mathematics is best of all subjects. It can bemastered by taking just one step at a time and these steps canall be made simple. Confusion occurs when a new concept ispresented before the previous ones are fixed or when a complexof several concepts is offered as one increment or when a pupilor class has missed a concept on which later ones depend. Ateacher should be taught to first find out, by comprehensivereview or tests, what progress the pupil has already made, thento analyze the subject matter into increments and begin toinstruct where the pupil’s competence leaves off.The second important item is a love and enthusiasm for the

subject. This should be furnished by the college professor aswell as by the high school teacher, and it can be done by ateacher who senses the real significance of its various branches.For centuries it has been an inspiration to those who under-stood and followed it, and it is not too much to say that onlya person who has caught the real thrill of power that only themathematics can give is qualified to teach it. Otherwise one’steaching is routine, wooden and meaningless like a successionof calisthenics for only the docile and stolid to follow while thepupil with creative energy goes wool gathering.One would not expect to become a good athletic coach if he

avoided all athletic functions except the routine of coachinghis team. Neither does one who does not read the literature ofthe language he teaches become a very inspiring languageteacher. Similarly it should not be expecting too much of themathematics teacher that his interest in the subject shouldextend beyond the part of the subject he teaches. He shouldhave an active interest in extending his grasp of the subjectnot because it is his duty but because of the real thrill that itfurnishes.

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The other day a young woman (I have even forgotten hername) applied for a position teaching high school mathematics.I questioned her about the calculus, the differential equationsand the theory of probability in investments and annuities. Shefinally replied, "Well, I don’t see that it makes much differenceif you are just going to teach algebra and geometry whetheryou have had any of that stuff or not." And I suspect it wouldn’tmake much difference to her pupils either. She might becomea good drill master but her students would forever be deprivedof the thrill of adventure.The third item is the establishment of a personal storehouse

of illustrative material and application. This storehouse mustbe filled by the teacher himself through excursions he may takeinto the real experiences of life. He must have shared many ofthe experiences of his pupils but he must also have becomefamiliar with a great variety of the quantitative relationshipsof our complex civilization. This background will give realityto his teaching.As previously remarked the relative importance of the

position of mathematics in secondary and higher educationhas been considerably reduced in the past few years. In thesecondary field this reduction has been chiefly in favor of theEnglish and social studies. And this change in emphasis hastaken place in the presence of the fact that our culture andour industrial and economic systems are becoming increasinglymore quantitative and more dependent upon the mathematics.The change has taken place, not because it is inherent in the

set up of our civilization but because the mathematics has beenin the hands of people who neither understood nor appreciatedits significance and could not transmit to their classes whatthey did not have.The readers of this magazine are probably representative of

the best in the country in secondary mathematics teaching.You read this and other teachers’ magazines fairly regularly.Yet those who, because of your own interest in it, regularlyread strictly mathematical writings of any grade constitute analmost negligible per cent. Number theory, mathematicalanalysis and modern geometry offer opportunities for such read-ing without entering the more highly specialized fields. Theteacher who is prompted to put forth a little effort along theselines will find the exertion stimulating.

Until secondary school mathematics is in the hands of teach-

TRISECTION OF AN ARBITRARY ANGLE 707

ers who have real mathematics scholarship enough to be able tocontinue to read the subject independently and who have beeninspired by their professors with a love and enthusiasm for thesubject that will keep them going on with it even after com-

pleting all degree and license requirements, the mathematicswill not come into its own and our cultural and economicdevelopment as a people will continue to be retarded.

TRISECTION OF AN ARBITRARY ANGLE

V. H. PAQUET, Portland, Oregon

Let OP A be the constructed half loop of a right strophoid of which 0is the double point.

Erect AB perpendicular to OA at A.Lay of the given angle BAC.Bisect angle CAO and draw the bisector AD.Construct an arc of a rectangular hyperbola passing through A in the

following manner:Let OF be any line from 0 passing through the strophoid, and let H

be its point of intersection with AD.Make angle HAG=angle AHF.The point of intersection of OF and AG is a point on the hyperbola.Through the point P, the intersection of the arc of the hyperbola with

the strophoid, draw AT.Angle CAT is ^ of the given angle BAC and is equal to 6.

ProofDraw OP, and let L be its point of intersection with AD.By known properties of the strophoid, angle OAP==90° �26..-. angle BAT =26.By construction, P being a point on the hyperbola, angle LAP--

ALP=angle OAL+6.= angle

But by construction:angle OAL=angle LAC,.’. angle ZAP-angle LAC=6=angle CAT and since angle BAT =26,

the given angle BAC ==30.Therefore angle CAT as constructed == ^ angle BAC.