BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER

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VOLUME 10, NUMBER 5 MAY 1969

B.C. ASSOCIATION OF MATHEMATICS TEACHERS

Executive Committee

President:

Vice-President:

Secretary: ,

Treasurer:

Publications:

Curriculum:

NCTM:

Past-President:

PETER MINICHIELLO, 2484 Aldington Drivo, Vancouver 177, B.C.

LEN GAMBLE, Clearwater

MRS, KAREN TILL, -Vancouver

GERRY NORMAN-MARTEN Buniuby

GERRY BOUMAN, 875 Camden Crescent, Richmond, B.C. ')>

LEN HAWKINS, Victoria

JIM CLARK, Burnaby,

ROY CRAVEN, Abbotsford

The B. C. Association of Mathematics Teachers publishes-Pec/or (news­letter) and Teaching Mathematics (journal). Membership in the associ­ationis $3;00 a year. Subscription rate for all publications in one year is $4.50 for those persons not eligible for BCAMT membership. Journals may be,'purchased at a single copy rute of $1.50, Please direct enquiries to the Publications Chairman.

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EDITOR 9 F R O M THE EDITOR

It waa fun putting this issuo together, There are some vory interesting items that you should road carefully, Tho president's annual roport, California textbook adop­tions, a bulletin about Departmental exams and some interesting- letters Bhould give you food for thought for a fow hours.

I am glad to sec tho discussions that havo boon started in tho Letters column, So far, I have printed the letters in full . If, as I keep hoping, quite a fow letters come in, I shall havo to odit them, But I won't do this unless it is necessary.

You should alBo havo received our second journal, Con­gratulations to E r i c MacPherson for a job well done. We are getting requests for our publications from various places throughout Canada,

N E G A T I V E F O U L - U P

Starting with the identity -1

Take the square roots of both sides: This means

Cross multiplying we get i2 = 1

from Mathematics Teacher, New Jersey,

PRESIDENT 'S R E P O R T Ji to tho

A N N U A L G E N E R A L M E E T I N G A p r i l 7, 1969 ;;

Two years ago, at tho Htart of my t e r m aa president, there was Borne doubt that this association Wjuld oxiiit long enough for mo to comploto my torm, You may recal l that at that timo PSA operations woro under review by a BCTF committee ponding a motion to the Representative Assembly In January I965 that PSA grantB bo gradually phased out, For a tima it Boomed that I should bo the Last President rather than a Past President. Actually, in view of tho BCTF commitment to curriculum, I doubt that anyone soriouBly thought that tho grants would be eliminated, with the consequent demise of the PSAs, |i

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The result of tho inquiry into PSA operations was the creation of a standing committee to maintain a l iaison betweon PSAs and the BCTF Executive Committee. The PSA Committee has made two major recommendations which wi l l havo a d i ­rect and immediate effect on our association.

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One recommendation, already accepted by the Representative Assembly, is that BCTF grants be increased, The new schedule is $3, 00 a member for each member up to 2D0 and $2, 00 for each additional membisr, The increase In revenue is timely, as you wi l l better realize when the Treasurer gives his report, ^

The other major recommendation of the PSA Committee is that membership in PSAs be open to,,all who wish to join and not restricted to BCTF members, This recommendation re­quires a change in the B C T F Constitution and the necessary amendment wi l l be presented to the delegates at the Federa­tion's Annual General Meeting this week. Other than offering the possibili ty of increased membership in our association, it is difficult to predict how an open membership policy wi l l affect our organization. I hope that an increased member­ship wi l l not only augment our financial resources, but also facilitate the exchange of ideas so necessary, but regrettably

HO lacking, to further our o b j e c t i v e of Improving tho quality of tho m a t h e m a t l c H e d u c a t i o n offered In our s c h o o l s ,

Your executive d i d not take tho throat of grant withdrawal seriously; instead, priori t ies were established and plana made for the work ahead, Two areas woro selected for con­centrated effort; namely ,„communica t ions and curr iculum.

In kooplng with the premise that our publications aro tho most valuable dividends of membership In tho association, tho Edi tor ia l Board was formed to oversee tho production of tho nowslottor and the journal. I believe the improve­ment, both in format and content, of the newsletter is read­i ly apparent, F u l l c redi t for this significant transformation must bo given to Ger ry Bouman, your Publications Chair­man and newsletter editor.

By this time you wi l l have received the second Issue of our recently-revived journal, Despite numerous difficulties and delays which have forced us repeatedly to revise our plans, I think that you wi l l agree that Teaching Mathematics is a publication in which wo can take pride. Wo were indeed for­tunate to obtain Dr . E r i c MacPherson to serve as its editor.

The advertising revenue that we had hoped would support a quarterly publication schedule for Teaching Mathematics has, as yet, failed to materialize, Consequently, your new exec­utive may find it necessary to l imi t publication to two 1Bsues a year, Whereas our previous journal failed for lack of pub-lishable material , Teaching Mathematics seems to have more material that it can afford to publish, This fact itself, I suppose, is a measure of success when one considers that many distinguished journals are faced with the same prob­lem.

Both E r i c MacPherson and Gerry Bouman deserve our con­gratulations and thanks for establishing our publications on a sound foundation for continued growth.

Unfortunately, our success with publications has not been matched in the area of curr iculum. Ear ly in 1967 your ex­ecutive was highly optimistic that real progress was being

miido and that a Revision Committoe. would bo formed later that yoar. Our hopes were shattered, however, with the announcement from the Department of Education that it had no plana for a new committoe.

An attempt was made by tho executive to stimulate revision by oncouraglng and offering suggestions to teachers faced with curr iculum problems, Under tho ausplcos of tho B C T F Curr icu lum Directors, the pamphlet on the articulation of Mathematics 7 and Mathematics 8 was produced and d i s t r i ­buted to a l l Bchools in addition to our membership. Meetings wore held with representatives of both the Science Teachers' and tho Shop Teachers' Associations. Discussions with tho lattor group led to the formation of an ad hoc committee of tho Curr icu lum Directors to study the General Mathematics program in relation to c r i t i c i sms of the Shop, Commerce and Homo Economics teachorB, Through lack of support, this study was not completod satisfactorily. I did, however, submit a report to tho Curr icu lum Directors containing sev­era l recommendations which were to be forwarded to the new Revision Committee upon its formation. ,(

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While the activities just mentioned were in progress, your executive continued to press the Department of Education through tho official channels to re-establish a mathematics revision committee, Final ly, about a year ago, the Depart­ment consented to form the committee, but its actual forma­tion did not occur unti l February of this year. Once again, we are optimistic that long-eought-for revisions wil lbemade.

In addition to the work on publications and curr iculum, your executive has dealt with many items of business, three of

/;which are recurrent and deserve some comment.

A s you know, this association has been involved in some way with the annual Northwest Mathematics Conferences since their inception. It was not until the 6th conference in 1967, however, that we real ly undertook our share and organized and held the conference in B . C On a number of occasions, there has been discussion with our counterpart associations in Washington and Oregon toward establishing a three-year hosting cycile for the conference. Agreement was reached

thiH year and we will next host the conference In 1970 and every third year thereafter, Planning for next year's con­ference, the 9th, to be held In Victoria , IB under way.

The Mathematics Contest has been held successfully for the third year, The contest seems to bo Increasingly popular, L i view of thia popularity, and to avoid unnecessary duplica­tion, tho universities, which sponsor tho Mathematical Association of Amer ica contost, of ft rod last year to with­draw from tho contoBt field, Thoy woro porsuaded to con­tinue sponsorship of tho M A A toBt for this year at least, pending clarification of our policy. Our position haB been that our examination attracts more students than the M A A examination because it is somewhat easier and therefore less discouraging to the candidates. However, over the past two years, the M A A tost has become easier and our test has become harder, thus weakening our argument for holding two examinations to select essentially tho same winners, It seems, therefore, that your new executive must reconsider the objectives of the contest and determine how thoy may be best achieved.

The question of affiliation with the Canadian Association of Mathematics Teachers has plagued'your executive since the topic was f irst presented and rejectee} at the 1967 Annual Goneral Meeting. The motion paBsed in 1^67, designed to prevent our affiliation, was unfortunately worded in terms of refusing financial support of the C A M T , j The interpreta­tion to be placed on the words 'financial support1 has been most troublesome. The question was finally resolved by a referendum submitted to tho membership i i i May 19,68. Since the vote was better than two-to-one in favor of aff i l ia-tion, we are now an affiliate of the Canadian Association of Mathematics Teachers. f

Your executive has dealt not only with items of direct concern to mathematics teachers, but it has also represented the association on numerous committees reflecting our interest in education as a whole. The ad hoc Committee on Electives, the ad hoc Committee on Evaluation and External Examina­tions, and the recently formed L ib ra ry Study Committee are three groups, in addition to the Curr iculum Committee and

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the PSA Officers' Committee, in which memberB of tho ex-cut ivo have been active. A I B O , WO havo maintained a l iaison with the Br i t i sh Columbia Committee on the Undergraduate P rogram in Mathomatics. The B C C U P M is an Informal group representing tho four universities and the currently established junior colleges,

It has been my privilege to servo as your president for tho past two years. Although there have boon some hectic times, I have onjoyod my work and it has boon very rewarding to me, My job was mado easier because of tho support of my follow executive members, a support which was gonoroualy given and for which I now givo my sincere thanks,

REPORTS 1 R E P O R T S C

£p Executive Meeting, March 15. Q The following reports were received! j H a. Curr iculum: Tho now Revision Committee has been i% instructed to examine the curr iculum for K - 12 and j P make recommendations about texts, course content, *_ philosophy, evaluation and so on.

b. N C T M : J i m C ' l r k reported that sales are breaking even, but little interest has been shown by individual teachers, The university professors seem to account for most of the Bales.

c. Canadian Association of Math Teachers -Roy Craven reported on the Ottawa Meeting of March 7-8.

"- C A M T w i l l publish two newsletters yearly. The first one was available at the A G M , The next one w i l l be mailed to the membership with our own news­letter. - C A M T planB a conference on Mathematics and Educational Television in the spring of 1970. It is hoped that a l l provincial departments of education wi l l participate.

Annual, tumoral Mooting - A p r i l 7.

ThlB moating took tho following actional - Acloptotl a foo incroaao of $1. 00 for 1969-70. A look at tho proposed budget wil l indicate the reason why this was nocosBary.

Income Foes 470 members at 3. 00 1410. 00 B C T F Grant 250 at 3.00 750.00

220 at 2.00 440,00 $2600.00

Expenses Journals 2 at 550.00 1100.00 Newsletters 6 at 130. 00 780.00 Executive Meetings 8 at 100. 00 800.00 Annual General Meeting 100,00 Dolegato to C A M T 200. 00 Math Contest 100.00 M i s c . (N. W, conference, mai l ,

stamps, etc.) 100.00

$3,180.00

- Revised the constitution to allow for mai l ballot and sev­era l other items, A new, constitution wi l l be mailed to each member with the next newsletter. - Elected those teachers to serve as your Executive for 1969-1970 or 1971:

President: Len Gamble, Clearwater Vice -Pres : Menno Wiebe, Ladner Treasurer: Mike Bakery Surrey Secretary: Isabel Lea sk, North Vancouver i Publications: John Turnbull, Richmond

E L E M E N T A R Y ' On the following page you w i l l be able to read the textbook adoptions now in force in California. How many of these have you seen? i ;

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T E X T B O O K A D O P T I O N S A N NO U N C E D

At tho February 13 mooting of the State Board of Education, the recommendations, of the Curr icu lum Commission on adoption of mathomatics textbooks K-8 for September, 1970, woro approved, Tho feeling of tho Board waB that thoy would proSB for an increase in tho budget allowed to implement tho adoption, Those books selected are!

BASIC T E X T B O O K S 1-6 Slow , Addison-Wosioy, Elementary School

, * Mathomatics 1-6 Average-if .as t , , , Houghton Miff l in , Modern School

If | \ Mathematics: Structure and UBC 7-8 Slow . , ,V(, . , , Science Research AB&oclatos ,

Mathematics, Structures and Skil ls 7-8 A v e r a g e . , Addison-WoBloy, School Mathematics 7-8 Fast , , Houghton Miff l in , Modern School

>' Mathematics: Structure and Method 8 Accelerated . . . . Houghton Miff l in , Modern School

! Mathematics: Algebra 1

I K I N D E R G A R T E N Ginn,' Kindermath K Holt, Rinehart, Winston, Developing Number Experiences,

lj T E A C H E R S ' M A N U A L S

Wiley (manuals), I Do, and I Understand Beginnings Mathematics BeginB! 1 Pic to r i a l Representation

Wir tz -Bote l , Developing Insights into Elementary Mathe­matics: Operations with Whole Number a 1

S U P P L E M E N T A R Y 1-6 Encyclopedia Britannica, Math Workshop 3-6 Frankl in Publications, Learning About Measurement

Learn to Fold - Fold to Learn Mirror Magic Paper and Penc i l Geometry

,, Mathematics Around the Clock

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Making and Uoinfj Nomographs and Graphs Patterns and Puit/.loB.l.n Mathematics

4-6 Harcourt, Brace and World, Elomontarv Mathomatlcs Enrichmont.

1-2 Holt, Rlnehart, Winston, Elomontarv Matho matte B : ;( Patterns and Structure, Ejcrciclos

4-8 McGraw-Hill, Programmed Math K - 6 Singer, Sotfl and lumbers 7-8 Addison-WQBIOV. S Exploration in Mathematics: Student

Discussion Guido J 7-8 Franklin Publications, F rom Fingers to Computers

Probabili ty: Thci'Styience of Chance Mathematics; Manx's Key to Progross

7-8 Harcourt, Brace ixiid World, Learning to Compute 7 Harper and Row, ||Strotchers and Shrlnkors 8 Harper and Row, ['Motion Geometry: SlideB, F l ips , and

Turns: Book 1 j j;

7-8 Houghton Mifflin, '!Experiments in Mathematics j, 7-8 Silver Burdott, Foundations of School Mathematics 7-8 D. C, Heath, Mathematics, Modern Concepts and Ski l ls 7-8 Holt, Rinehart arid Winston, Exploring Modern

Mathematics I if

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T W E L V E Y E A R S IN M A T H E M A T I C S : PAST AND ^FUTURE

-from Texa a Ma thematic si1 Teache r B

During the Dallas meeting of the National Council Of Teach­ers of Mathematics, held in conjunction with the N E A last July, Dr . W. Eugene Ferguson, Newton High School, Newtonville, .Massachusetts, spoke on 'Twelve Yea'rs of Change in Mathematics Education, ' and John Wagner, Michigan State University, East Lansing, Michigan, spoke on 'The Next Twelve Years of Change in Mathematics Edu­cation. ' VT' ! | I ' . • ;|t

(Dr. Ferguson: I \

More than a dozen groups have participated in the rjevision of and improvement of mathematics education in the last twelve years. He said that each one has made a tremendous

contribution to tho change H . The HO organization groups aro: UnivorBity of Illinois Committee on School Mathematics, College Entrance Examination Board, Madison Project, Universi ty of Maryland Mathematics Project, B a l l State Program,/] Boston College Mathomatlcs Institute, h ;hool of Mathematics Study Group, Educational. Services, Greater Cleveland Mathematics Program, Minnesota Mathematics and Science Teaching Project, and Stanford Program in Computer Assisted' Instruction,

Tho advanced placement program has established tho fact that many students and schools are ready for calculuB in high school, Many topicB have been introduced and spiralod through tho twelve grades, Beginnings of algebra and geom­etry are found in the grades K - 6 , Topics now finding wide acceptance that wore introduced during this period are: prob­ability, statistics, matr ix algebra, calculus, elementary functions, beginnings of modorn algebra, vectors, and use of automatic digital computers in th<5 high school. Geometry now includes plan/^' geometry, solid geometry, and analytic geometry, Sots, \velations, and functions are emphasized; they cause traditional topics to be presented in new methods,

John Wagner: „ \

'In any specified act of teaching,- a new environment is created ,from familiar objects and events; in responding to the changed environment, the variety or learners gain capacities not achieved through pr ior experience, ' He predicted these ex­tensions and changes: j 1. Improved rationalization of mathematics - - better sym­bolism, sequencing topicB, .sharper definitions, overall de­sign of total; subject. 2. Introduction of new and/or more sophisticated mathema­t ics of al l ' levels with rationalization in terms of child develop­ment and psychological research - - statistics and probability, l inear algebra, matrices, computer algorithms. 3. Improvement of basic mathematics - - genuinely job-oriented. mathema (tfcs. 4. Increased individualization of instruction - - laboratory experiences. .. 5. Synthesis of arithmetic, algebra, geometry.

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6, Incraaaod understanding of tho interaction botwoon mathomatlcB and reality; application of roal mathomatics to tho roal world,

7, Cur r icu lum planning that omphasiaos problem finding rather than simply problem solving. 8, 'Human being is tho master of mathematics. ' 9, Moro attention on tho 'roading-and-writing' aspects of

mathematics . ji 10, Somo attention to team teaching and interrelation of mathematics and other subjects; creation of now environ­ment in classroom, 11, Computer assisted instruction - - programmed mater i ­als — task analysis, 12, New approaches'to teachor training fusion of content and method,';

CUSP S E E K S 200 MATHS AND S C I E N C E T E A C H E R S • W ^ » W W W ^ « ^ » ^ ^ W M M » _ J _ M _ _ W » M I _ _ M M M M M _ M » « _ _ M M ^ _ _ I » M W M « _ M M

; CUSO this year has received requests for some 200 mathe­matics and science teachers to work for two-year assign­ments in ,40 developing nations of Afr ica , Asia , tho Caribbean and La t in Amer ica ,

;.'An independent, non-profit Canudian organization, CUSO recruits , selects, trains and sends professionally arid

,. technically qualified personnel abroad in response to specif­ic requests from governments and agencies overseas, It is a technical assistance manpower program; not a relief, religious or emergency aid scheme.

Teachers, assigned to established positions in secondary school systems, are paid by the host country employer at approximately counterpart, not Canadian, salaries, CUSO, provides health and life insurance and can defer Canada Student Loans by paying interest during term of service overseas. ! . : • ...

M i. Applicants must be,Canadian citizens or landed immigrants of at least two years' residence in Canada. Mar r ied couples are considered if botfrhusband and wife are qualified i a r

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available assignments, Couples with ono child can bo con-sidorod in special cases provided child is under school ago at timo of posting and wi l l remain so during assignment, There is no upper ago l imit , but good health is ossontial.

A l l outgoing personnel attend a six- to oight-wook orionta-tion and training course in July and August covering language and area studios before flying out to aonignmont in Septem­ber,

Those interested should apply to their nearest CUSO local committee located at 80 universities, colleges, institutes of technology and community colleges across Canada, CUSO Information, 151 Slater Street, Ottawa 4, Ontario, w i l l send committee addresses on request,

N C T M //'»

The National Council of Teachers of M a t h e m a t i c a l approach­ing its Golden Jubilee, In 1970, half a centuryjwill have passed since a few dedicated teachers organtnedthe N C T M ,

What haB the Council accomplished in these 50 years? What should it plan to do in the next 50?

In 1920, 100 courageous and forward-looking teachers met in Cleveland to found the N C T M , They quickly persuaded some 3, 000 colleagues to join them.

• ; • ' ,'/ •

During the next 30 years, an elite corps of teachers of mathe­mat ics , never more than 10, 000, worked patiently and assidu­ously to keep alive the light of their aspiration: constant improvement in the teaching of mathematics.

Fifteen years ago, membership was s t i l l under 10, 000. Ten years ago, membership was close to 20, 000. Today our membership and subscription l is t numbers more' than 80, 000.

Why this meteoric r i se? Essential ly because schools and teachers ,°aw the challenge in the increasing need for mathe-

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matlcs in the modern world. Teachers of mathematics, through tho N C T M , havo actod together to moot that chal­lenge. Thoy needed help, and the Council haB boon providing that help - - through its conferences and conventions, Its periodicals and othor publications, its films, and its many committees,

Whoro do wo go from horo? During tho noxt fow years, certainly we shall have to place emphasis on the following!

By 1970, membership and subscriptions may pass the 100, 000 mark. We should not enlist now members merely for tho Bake of impressive statistics. Rather, wo must recognize that, with some 200, 000 persons teaching mathematics at tho high school level and above, and possibly 1, 000, 000 at the elementary school level, communications between the Council and the profession at large wil l be far moro effective i f we enrol more of the Be teachers.

Notable improvements have been made in curr icula for sec­ondary schools. Improvements in instructional materials have been stimulated by the experimental work and texts of such groups as SMSG. The commercial ly produced textbooks of today are better than those of the past. In the elementary grades, however, we s t i l l face the question of how to make a significant impression oil the curriculum, How can we communicate with more and more elementary school teach-e r s?

Max Beberman has called the new mathematics 'suburban mathematics. ' It is a fact that suburban and private schools have more readily introduced the newer proposals and cur­r i cu la and more will ingly implemented them, by, supplying mater ials and by providing opportunities for teachers to re­view and strengthen their preparation. But what of the spe­c i a l problems of schools in the great cities? How can the N C T M study these problems and help teachers to meet them?

It has often been charged that the 'new curr icula ' are for the top 5%, or 10% or 20%^of the students. If this is true, and if the assumptions behind the new programs are correct, how do we work toward the goal of good, necessary, and

useful mathematics for aU tho children?

Over the past 50 years tho N C T M has mado a strong and valuable contribution to education. In tho yoars to como, wo hope to continuo and strengthen that contribution - - b y reaching moro teachers, by searching out and exploring their problems, and by employing tho full resources of tho Council to help solve them, This can be accomplished only with your help.

- Dr , Julius I-I. Hlavaty, N C T M Presidont,

2, Name-of-Site Meeting: The Name-of-Site Meeting this summor w i l l bo held in Bi l l ings , Montana. Date: August 18-20, Speakors include: Jack E , Forbes, Calvin T, Long,

R, B , Davis, David A , Page, Howard F . Fehr, Glenadine Gibb, M , L'. Koedy, J . D. Hancock, D. L . Po^,,ln, Eugene P . Smith, Irvine Brune, and many others,

3. New publications of the N C T M :

T H I R T I E T H Y E A R B O O K \\ Jr

M O R E TOPICS IN M A T H E M A T I C S F O R E L E M E N T A R Y SCHOOL T E A C H E R S .

; ( • )

ORDER YOURS NOW F R O M JIM C L A R K , 21054 Cla rk Avenue, R. R. #3, Langley, B. C.

D E P A R T M E N T A L E X A M I N A T I O N S

The following bulletin about changes in Departmental exams, was received.. We reprint it here for your information.

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ChangoB In Departmental Examination Policy for Grade XII

Tho attention of School Pr inc ipals and toachors Is drawn to tho following changoB In Departmental Examination policies and procedures, These should bo read in conjunction with Administrative Ci rcu lar 5.2.68 which rcmalnB in effect for tho year 1969 except as noted below.

1, For a varioty of reasons it has become desirable to make tho regular Department of Education examinations at tho Grado XII level available more ofton than onco a year. (By regular examinations are meant those which must be written by candidates who are not recommended.) This means that marking procedures must bo simplified in order to mako pos­sible earl ier release of Statements of Standing, It has there­fore boon decided to mako these Grado XII regular Dopart-

>-mental Examinations, commencing Juno 1969, exclusively objective in format.

Although it is realized that not a l l aspects of a candidate's work can be examined in this manner, "it is nevertheless possible to test certain essentials adequately. The school assessment, which counts for 50% of a non-recommended candidate's final mark in an examinable course, must there­fore place greater emphasis on other vi ta l elements such as essay writing, oral exposition, laboratory work, and other aspects which are best evaluated in a more subjective man­ner. Because the two types of evaluation wi l l be'to some degree measuring different things it is not expected that they w i l l correspond although the degree of difference at the pass level in an individual case should not be great,

2, Effective June 1969, a scholarship candidate from an accredited Vchool who is eligible for recommendation stand­ing in a Grade XII examinable course w i l l no longer be re­quired to write the regular Departmental Examination should he wish to offer that course for scholarship purposes. A candidate's scholarship average would be based on his per­formance in the best two scholarship Departmental Examina­tions and wi l l not include any mark made on the regular examinations should these also have to be written. (Amends items 11 and 12, Administrative Ci rcu la r 5.2.68.)

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It should bo rcali/.od that tho scholarship examination is designed to rank tho bost students In a course. It Is compe­titive by nature and, if it is to perform its task, must d i s t r i ­bute tho candidates. Therefore, it should offer a degree of difficulty to even tho very capable student, Principals should onsuro that students electing to write for scholarship are aware of this fact and that thoy do not fool unduly discouraged becauso thoy find tho examination hard, As scholarship stu­dents thoy should rocognizo that this is a special examination designed to test thorn and that it is not juBt an achievement tost for tho average student.

Scholarship examinations wil l continue to include both objec­tive and subjective items but with a preponderance of tho latter, The proportions of each type of item wi l l vary some­what among courses and from year to yoar in a particular course,

3, Previous pol icy has been to require a l l students to write the regular Departmental Examination in any examinable course having a total annual enrollment of fewer than 200 pupils, Commencing in June 1969> capable students regis­tered in Lat in 12, German 12 or Spanish 12 may be granted recommendation standing by accredited schools providing they satisfy the usual conditions pertaining to recommenda­tion,

Students who do not qualify for recommendation in one of these courses w i l l continue to write the applicable regular Depart­mental Examination. (Amends item 4, Administrative C i r - ... cular 5. 2, 68.)

4, Commencing in 1970, Grade XII regular Departmental Examinations w i l l be available twice annually, in January and in June. Unt i l further notice, Grade XII scholarship Departmental Examinations wi l l be available in June only.

5, Grade XIII Departmental Examinations may be written as before in June and/or August. There is no change at this time in examination policy where Grade XIII is concerned.

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6, Dotailcd information on modified examination procedures wi l l bo provided to schools and candidates by tho Division of Examinations. Scholarship regulations wi l l also bo d is t r i ­buted at a lator dato.

F , P , Lov l r s , Superintendent of Education.

ThankB to M r , Mandoli for sending mo a series of reports of tho Seventh Northwest Math Conference. I have had those reports in my possession for some time, but I havo not had enough space to print them,

Here is tho first one, Perhaps another one wi l l follow in the next newsletter,

Advanced Intuitive Geometry for Grades 7, 8 and 9 - Dr . D, Prekogos

Dr . Prekeges discussed some of the ideas about geometry taught at the junior high level . He believes geometry should be taught intuitively, and not with the rigor of proof that is now required,

His presentation began with the question: 1 What do we mean by mathematical r igor ? 1 - - 'to sanction or to legitimize the intuit ive. ' Thus, he concludes, it seems logical to de­velop the intuitive before the r igor. The rigor should be left to the higher levels of the study of mathematics.

Dr . Prekeges argues that an informal heuristic approach should be used in the study of geometry at this level; that is , a discovery pattern approach which wi l l tend to lead the stu­dent to investigate further on his own.

His examples included:

1. Find the area of a simple closed curve.

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Make use of a grid and count tho number- of square units within tho curve only (LOWER BOUND), and those units which aro within and touch tho curvo ( U P P E R BOUND).

Tho results may bo: L . B: 14 aq. unite) /y V G 1 . a go = 24 sq U. B: 34 sq. units)' units

T r y with a smaller scaled grid: L , B.) A v o r a H O - 2 4 , 25u2 U. B.) B

Further approximation to an Average = 24. 375U2

Now using a s imilar technique approximate the area of a c i r ­cle, (Check annwor with the formula A =J\ r^, and approxi­mate T ) Through systematic approximations of the area of a c i rc le , a value for IT can be obtained,

2, Given:

A B

Measure the angles, then add. m L A C D + m L D C B = 180

Bisect the angles. What can we say about m L E C F ?

What is the sum of the angles of a / \ ? - tear off the angles of a A and place vertices at a common point - ZL L B = 180

(Exterior Angle Thm.)

m L a•+ m L b +m L c = 180 _ _ _ _ _ m L c + m L d = 180 •Then m l_a + m L b = ? (mLd)

Draw diagonals to a SQUARE. Write down a l l the properties, (ex. diagonals bisect each other, etc.)

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Now draw a roc tannic with diagonals, r - 1 What propert ies -st i l l hold?

Draw Paral le logram

Aro tho L s within congruent?

Trapezoid:

What properties s t i l l hold true? (Ls are equal)

Quadrilateral!

4.

A l l above properties lost.

Square - connect mid-points do s imi lar to: rectangle

parallelogram trapezoid quadrilateral

Now conjecture the results!

- mid-points of any quadrilateral form a parallelogram,

5. Finding the square root.

a. flS

J

19

,

b. 72209 W in Ho botwocn 40 and 50, If inaldo square region has area o l 1600, outer region has area of 2209 - 1600 = 609,

a + b + c = 609 sq, units

<< 40 -> 40

Outer region is about 80 units long. Therefore it is about 7 units wide. Thus a good approximation for y^209

(Check 47 2 a 2209)

c. ytOOO lies between 30 and 40 Outer region a 100 u 2

I

80 J609

40 + 7 a 47

3b 30

900

width?

Outer region is about 60 u. long Width is about 1 unit,

61 x l a 61 61 x 2 = 124

Thus width is between 1 and 2 units, Therefore /1000 is between 31 and 32. Further approximation:

60 1

JlOO

1000 - 961 = 39

31

31

961

31

Outer region is about 62 u. long, Width about . 6 units

62. 6 x . 6 = 37. 56 ' 62. 6 x . 7 a 43. 82

62 J39T0

Width is between . 6 and . 7

20

. A

Approximately /1000 s 31 f .6 3 31,6 Using closer approximation of 31.6

1000 - 998. 56 = 1. 44 31 31. 6

.6 \

T H ! " 3T7o ' . 02

Outer region length is 63. 2 units 63. 2 ) 1. 44 63,22 x , 02 = 1. 2644 63,22 x . 03 = 1. 8966

Width is between . 02 and . 03

Approximate 71000 = 31,62

(Check 31.622 = 999.8244)

t e r m s £ Sir :

T jL R. J . Cu r r ' s letter on the 'Allendoerfer method1 of pro-Jf viding identities demonstrates just one-more instance in

f which our emphasis upon mathematical formalism may set up barr iers to mathematical concept;growth rather

ft than enhancing this growth. I think it incorrect to say . Q that teachers are not acquainted with this method. Every 9 . time I teach t r ig I encounter this method; the students

themselves supply it. I think the question boils down to: 'What is our objective?' If our objective is to prove an identity, it is inescapable that we must agree with Curr , If our objective is to teach formalism, that is another matter, and should open the question of the validity of formal ism in the high Bchool curr iculum.

I personally feel that formalism is great - but for about 5% of our students. For the majority, to teach formalism before useful concept mastery i s a gross e r ror on the part of anyone claiming the title of teacher. Yet we do it a l l the time, and our curr iculum seems to require it.

21

Our emphasis oti'jitho verbalisation of the commutative, asBociative and distributive principles provided an example of this e r ror in. more elementary work, yet you cannot add 56 + 23 'I- 79 without correctly using al l throo. We seem to bo aiming at a situation in which to correctly vorbali/.o the principle constitutes, a licence to perforin tho act, no others aro to bo permitted to try,

An example of tho same error In more advanced work, guar­anteed to turn away from mathomatics about half of UIOBO

successful in Grade J.2, is tho use of Rolle 's theorem to prove the 'mean value thoorom,' whoa tho general statement of tho latter is simply a reconstruction oi: the definition of the derivative for any function you may choose, but not yet taken in the l imi t ,

I hearti ly agree with M r , C u r r in tho particular area ho cites, It is , however, only a small part of the question of correctly balancing formal i sm on the one hand and useful concept mas­tery on the other, This imbalance is obvious in our cur r icu­lum and has the effect of excluding too many from mathema­t ics , ,. ;J

7 I)

Bruce Ewen, / ; v Carson Graham Sr. Socondary.

March 10, 1969

Dear M r . Bouman:

It was good to see your reprint of,,1 Reluctant Learners ' in the September 1968 Newsletter /arid I thought youjhiight be interested in a copy of what seems to be the original put out by the Michigan Mathematic Cur r icu lum Committee in A p r i l I967. Good ideas seem never, to die but keep on!being re­printed. ' u

/ T h e letter from M r . Bobbitt in the October 1968 Newsletter "' <Veems amusing to a teacher new to B . C . What is the real

complaint? Did the instructions in 1967 or 1968 state or

22

imply that oach student was expected to complete tho exam­ination? It is unusual horo to give 'too long' a paper in tho hope that oach student can find something In which ho can display a dogroo of competence? (In this day of tho com­puter, such 'too long' tests followed by an Itom analysis might furnish valuable feedback to tho teacherB, )

Having taught in a school where 60% was tho magic number relative to passing, and in another whore 50% filled that role, this wri ter cannot got upset about 35% being tho number used for a part icular examination, Indeed, 35%, 50%, X % of what, tested by whom?

How about adapting the oncloBod application form for tho Associat ion of Teachers of Mathematics for the last page of a future newsletter, please? A fair number of U. S, and Canadian teachoi'B, mostly in the east, belong to A T M and enjoy its journal Mathematics Teaching.

Yours truly,

R, Longman

Edi tor ' s Note:

Thank you, M r , Longman, for bringing this to my attention, Although we are not able to print an application form for A T M , we shall include a description and the address of A T M .

. G. B.

THE ASSOCIATION OF T E A C H E R S OF M A T H E M A T I C S

The Associat ion of Teachers of Mathematics was formed in 1952 with the a im of improving the teaching of mathematics at a l l levels . Over 6, 000 teachers of mathematics in a l l types of schools and colleges in various parts of the world are now members of the Association.

The annual subscription for ordinary members (home and overseas) is thirty shillings (four dollars) and for student

23

members is ton ahillinga (ovoraoao ono pound). Ordinary subscription's are payable on 1st January each year. Mom-bo rs joining after October 1st in any yoar aro regarded as having subscribed for the subsequent calendar yoar. Stu­dont subscription runs from l s | September to 31st August,

A l l members receive free copies of Mathematics Teaching which is published four times a year, in February, May, September and November,

. . . 1

Other activities of tho Association include tho publication of 'Mathematics Teaching Pamphlets' (occasional publications on mathematical topics and teaching aids), and tho ' A T M Supplement, 1 the organization of conferences for teachers of mathematics in conjunction with Loca l Education Authorities, University Institutes of Education, etc.,. regular branch . meetings for members of the Association in various parts of the country, the initiation of study projects, tho production of mathematical f i lms and other aids, etc.

There are branches of the Association throughout.the country; a l ist of Branch Secretaries appears regularly in Mathematics Teaching.

The Association is concerned with the teaching of mathema­tics at a l l levels f rom the infant school to Universi ty,

If you are interested, write to:

Associat ion of Teachers of Mathematics, Vine Street Chambers, Nelson, Lancashire, England. 1 ;

24 X69-276: l f