BRITISH COLUMBIA ASSOCIAT ION OF MATHEMATICS TEACHERS

President; Peter Minichiello, 3118 West 10th Avenue, Vancouver 8, B, C,

NEWSLETTER

Newsletter Editor; Gerry Bouman, 875 Camden Crescent, Richmond, B. C.

Vol , 10, No. 1 September 1968

IN THIS ISSUE;" ,, Jj

9 Prom the Editor

• Reports - - Vote on CAMT

- - Journal

• Reluctant Learners

• Two Minuses Make a Plus

• Point Nine Forever

c Letters

Seventh;Northwest Math Conference

Join the BCAMT NOW and you'll get;

• A means of exchanging ideas about teaching mathematics.

o A voice tn the development of the mathematics curriculum In B. C,

• A Math contest.

• Newsletters.

• Journals. '• '". •

FROM THE EDITOR

WOW, that summer went fast. Another teaching year has started with Its ii successes and frustations. One way of preventing ulcers (so prevalent amongst teachers) is to let off steam. We are offering you a free service. It tsour PREVENT ULCERS compalgn. You can join by writing about your ups and downs in mathematics teaching. Send your articles to ^ yourself and you'll help others (me, for instance). "

DON'T BE SHY.

During the next year I hope to make some changes and Improvements to tho News' letter. So keep watching, you may be pleasantly surprised.

DON'T DELAY

JOIN YOUR MATH PSA

• REPORTS

Canadian Association of Mathematics Teachers

The vote on the Issue of participation by the BCAMT tn this organization was as follows:

.ir For - 156

Against - - 75

JOURNAL:

*V The first Journal Is going to press this month, and your copy should reach you by the end of September or early October - - If you have Joined the Math PSA, New members will receive their Journals as they Join. It Is to your advantage to Join early.

', DEADLINE: November 15.

I' < SEND IN YOUR APPLICATION FORM NOW

• RELUCTANT LEARNERS •••

The following article from the Idaho Mathematics i Newsletter has some good ideas on teaching the , ; reluctant learner. I suspect most of the General Mathematics students fall in this category. How

' do your General Math classes stack up against these criteria? Send us your evaluations.

RELUCTANT LEARNERS . From: The Idaho Mathematics Newsletter

A reluctant learner Is defined to mean a student who has developed a negative ' • attlhideJtoward learning mathematics.vAs a result, this student's mathematic performance

is at the failure or near failure level.

Wo bollovo thai a mathomatlo curriculum should provide continuous opportunity for this student to study In an appropriate mathematics program in which he Is challenged and In which he can succeed. (!

i The following suggestions aro offered for coping with tho problems of tho teaching of mathematics to such students, They have been separated Into the areas of administration and teaching, but It should bo remembered, the problems of the reluctant learner can only be solved by the |otnt efforts of competent classroom teachers, administrators and boards of education, |

ADMINISTRATIVE SUGGESTIONS

1. Each school should Identify Its reluctant foarners, preferably In the seventh and eighth grades, and provide a special program of mathematics Instruction for them, The main' goal of this special program should be the removal of the reluctant learner's negative attitude toward learning mathematics.

2. The teacher of this type of class should have freedom to structure the content of the course. He should be able to choose topics and subject material he feels will stimulate the interest and ability of the student.

A single textbook will not be suitable for this type of class. In addition, the teacher will need workbooks, single sheet materials, and supplementary texts. Also, such drill materials as flash cards, multiplication lotto and time drill sheets should be used at the discretion of the teacher.

3. The teacher should be given a reduced load to have time to prepare for this class. In order to achieve the results desired, the teacher must have ttme to find research materials in this area and to prepare or develop pro|ects.

4. The size of the class should be significantly smaller than the average in the system. We recommend a maximum of 15 students per class.

// 5. The teachers scheduled in these classes should be those who are best qualified to

teach the theory oriented mathematics courses. Specifically, this means the best teachers In mathematics departments should share in teaching these students.

6 . . A teacher's perspective can be maintained best by having his assignment Include . a variety of levels of student ability. Therefore, no teacher should be assigned more than , two such classes per.day.

.... While the competent teaching of the reluctant learner is a highly developed ski l l , . few teachers will long remain in the classroom if a major fraction of their schedule Is devoted to classes of reluctant learners.

7. These classes should provide an atmosphere which is pleasant and where there are few distractions. There should be adequate space and facilities suitable to the teaching of mathematics.

8, Every effort should be made to Increase rho prestige of teaching these classes. It Is a privilege and a challenge to toaoh ono of these classes. Only the most versatile and secure teacher should bo assigned this responsibility.

9. Parental cooperation should bo enlisted by a complete explanation of objectives of such a course. The parents should bo Informed of the needs and the limitations of those students, as well as the alms and procedures of the course by moetings, conferences, or homo visits If necessary.

t

Only when parents are Informed can they understand the reasons for such courses, and schools will be able to gain their cooporatlon In helping their children to achieve In those classes.

10. Many reluctant learners will havo emotional problems, However, there may be cases In which the'problems of a student are so severe that their presence tn a reluctant learner class can only be a disturbing factor to the other students, and their presence will Impede the class program with little real benefit to themselves.

These students usually need tho help of a trained psychologist, and the teacher cannot help to resolve this student's problems without the assistance of a trained psychologist.

11. A flexible schedule should be maintained so that students who do Exceptionally well In these classes can be transferred to regular classes, while students falling far behind In the regular classes may be placed In a class for reluctant learners for a period of time.

TEACHER SUGGESTIONS

1. The teacher should be free to structure course content and should use many different teaching materials and techniques. One text will not be sufficient for the < flexibility that makes such a course a success.

Reluctant learners have a short attention span and therefore need a variety of learning experiences. Several different types of activities must be planned for each class period; These may include work with concrete objects, oral group drill sessions, practice with number games such as number lotto and'multiplication quiz me, class work sheets, chalkboard work, and a short time to work on a daily homework assignment. Vj

2. It is important for the teacher to provide students with small successes dally. This may mean covering topics that^are less mathematically sophisticated than anticipated. Each success improves the student's opinion of himself and increases his motivation.

3. ' in order to develop a positive attitude toward mathematics, fun Upraise, encourage­ment, and good grades In recognition of honest achievement must be provided.

>-:•:' ; / , • : • ) ' • • ' • . . " ; V : : . ' ".. ' - V •''[''•'" :)•.'':•'' ' ft

4. The teacher must be enthusiastic about(,his role as a teacher of such a class. He is teaching students as well as content. The student should be accepted and liked as he is so he can like himself.

• • • TURN THE AIR-CONDITIONER DOWN or TWO MINUSES MAKE A PLUS • • •

by Louts R. McCroory San Marino High School

A hearer Is a plus agent, for It raises temperature. An alr-condltloner Is a minus agent, for It lowers temperature. An Increase tn power of either one Is a plus, a docroaso In power Is a minus. Your students will understand this table,

Up (+) |, Down (-)

Heater (+) + -AlrCondlrlonor (-) - +

Certainly, turning an alr-condltloner down tends to Increase the temperature, a plus result, Also a decrease, a minus operation, in unemployment, also a minus situation, makes for a plus for the economy. \

Examples such as these make 'minus times a minus lo a plus' a very sensible conclusion. *

• '•.'.•.•.*<>.;• >. • •• Froms The Common Denominator

„ ••• POINT NINE FOREVER? • o o John O'Connor

, Mission San Jose High School Fremont Unified School District i

• ^ • ' v : : ' - " " r ^ " Sooner or later a mathematics student at ho high school level encounters a set of

problems of which the following is typical:

Express . 121212 . . . (written .12) cii the quotient of two integers. The usual . device is employed:

Let X = .12 ' Then 100X * 12 + . 12 or 100X = 12 +,X

99X = 12 Hence .12= 12/99

Sometimes the problem crops up: Express .9 in non-decimal form. Many students rebel when, after applying the above procedure, which they held suspect all along, the 1

answer 'one' appears. This situation affords a perfect opportunity to bring forth some unifying concepts which seldom fail to generate curiosity and genuine interest. For one thing the student can be made aware of the fact that the usual definition given for a rational number (a number which can be expressed in the form p/q where p and q are integers, q 7* 0) is equivalent to that which states that a rational number is an infinite periodic decimal (which includes the so-called terminating decimals - - which rebeuts

5

zoros), And Hits fact can bo demonstrated to him. Geometric series, Jjie density property of real numbers, and Indirect proof all can bo brought to bear on tho .9 problem. Hore aro some alternate solutions to the problem.

1. The sum of an tnftnlto geometric series Is glvenjpy tho formula S « a/(l-r) whore a Is the first term and r Is tho common ratio. ,"9= .9 + .09+ .009 . . . whore a = , 9 and r « . 1. Then by the formula . 9 « 1.i This solution may not gain much wider acceptance than tho original solution since tho derivation usually given for the formula depends on limits. However, one studont moro or less bridged this gap by observing that S - a + ar + ar 2 + , , . can bo written S = a + r (a + ar + a r 2 * , . . ) so that S » a + rS. Solving this equation for S yields the formula, Note that this ts a valid derivation for values of r for which S Is a number, I.e., for /r/<r 1.

2. The density property for real numbers assurosjjs that between any two real numbers there ts at least one other real number. If .9 Is not equal to 1, then what number lies between them? We also postulate a one-to-one correspondence between the real numbers and the points on a line. So, looking at the problem from a geometric point of view the question becomes: If .9 and 1 are the coordinates of distinct points A and B, then what ts the coordinate of a point between them? (The student Is hard pressed to give an answer but may present some misconceptions such as .9 having a last digit, Infinity being a number, the existence of a potnt 'next to' A and the like,) Along these lines we may generate an Indirect proof as follows: Suppose . 9 Is not equal to 1 . T h e n let k be the coordinate of the potnt M , the midpoint of segment AB. We are assured of every segmont having a midpoint}\and Its coordinate as given by the midpoint formula ts: k = (1 +,,5} / 2 => 1.999 . . . /2 = .9*, the coordinate of A . Hence, segment AB does not have a midpoint, clearly a contradiction, Therefore, .7= 1,

3. A plausibility argument: Division yields 1/9= . 1 , 2/9= .2 , . . . 7/9 - . 7, 8/9 = .8 . Then . 9 ought to equal 9/9 or 1.

4. The 'Clincher': Rare ts the student who ts not convinced that 1/3 = .333 . , . Then multiplication of both members of the above equation by 3 yields the desired result. 1/9=. I l l . . . can also be employed In a similar fashion.

5. 'Proof by long division: This one really begs the question but is bound to get 100% attention. Let's divide 1 by 1.

• • . 9 9 ? , , , i y I.OOO . . .

10

PPp.-y:p^:\--M:x:y-:'y \ -$-r--y-[ •;• . • :••:. ': ::• .

-r-: 'i\-Zy ZCi b Z - • • " \ ' ' ' :'; : ". If that doesn't satisfy 'em try 2/2. From: ACME - - spring 1968

j ••• LETTERS • • • ii ii

Tho following letter was rocolvod during tho summer.

Do you have any comment?

B . C . Association of Mathematics Toachers Newsletter Editors

Dear Mr. Bou'man:

This past term (1967 - 68) I have been auditing at U Vie their new first year Mathematics courso, Math 130 uses as a text Motsei Calculus. It Is a very rigorous text and students found It very difftcuIt. Students who'got A grades In Math 12 were falling Math,130, Nlnety*nlne percent of Math 12 was valueless as far as Math 130 was concerned.

If my experience this past winter would bo of any Interest to the Revision Committee I would be most happy to pass It along.

This coming term U Vic is offering two other first year Mathematics courses along with their present Math 130. ..,..„•..••••••••..•.•..<*.,;>..

i .1 find the Newsletter most helpful and look forwardto the Journal.

• Yours sincerely,

(Mrs.) Christian B. Reld.

MAKE YOUR PLANS NOW TO ATTEND • • • THE SEVENTH NORTHWEST MATHEMATICS • • •

CONFERENCE IN TACOMA • NEXT OCTOBER

Save tho weekend of October 4th and 5th for the Seventh Annual Northwost Mathe­matics Conference In Tacoma on the campus of Pacific Lutheran University, |

The conference offers an outstanding program featuring well-known and nationally recognized leaders In the field of mathematics education, Between tho opening general session and the closing banquet, there will bo something for everyone. Elementary school teachers will hear Robert Wlrtz, James Smart and Eugene Smith. Secondary speakers Include John Egsgard, Jack Forbes, Boyd Henry and Patrick Suppes. Howard Eves will speak at the banquet.

Numerous section meetings will direct their appeal to Individual groups. In addition to elementary and secondary sections, a number of Junior college and administrative sections have been scheduled.

Glance over this list of topics;

Computer-Based Instruction Algorithms of Arithmetic The Mathematics Laboratory Measurement Demonstration Classes Special Problems for Supervisors of A Plan for the Low-Achiever in Mathematics 1 Mathematics New Teaching Materials • The Slow Learner Geometry, K-12 , J Occupational Mathematics Topics in Algebra

Are you Interested in any of these topics? These and many more will be presented at the Seventh Northwest Conference tn Tacoma on October 4th and 5th. You can't afford to miss this conference I

Complete the attached form and mall as directed,

Sponsoring Organizations!

Washington State Mathematics Council (Host) Puget Sound Council of Teachers of Mathematics Oregon Council of Teachers of Mathematics British Columbia Association of Mathematics Teachers National Council of Teachers of Mathematics <{)

General Chairman of the Conference:

Alan O. Martin ' P . O . Box 1357 Tacoma, Washington 98401

8

ADVANCE REGISTRATION FORM

Registrations will be receipted at tho Conference Registration desk. Foos must accompany each registration, Make cheque payablo to the Seventh Northwest Mathematics Conference,

NAME.

CITY_

ADDRESS.

.(PROV.)_

FEESt

Registration $7,00 ($5.00 for pre-roglstratlon) No. .

Banquet $5.50 , , , . > • > , > • • • >•<••• • No. .

Luncheons $2.00 - - Washington - - Sherwood Inn No. .

$ 2 . 0 0 O r e g o n B r a d ' s Restaurant. . . . . . No._

$2.00 - - British Columbia - - Barb's Restaurant. No._

MEMBERSHIP:

Washington State Mathematics Council - - $2.00 . . . . . . N o u

Puget Sound Council of Teachers of Mathematics - - $1.00 . No. .

Oregon Council of Teachers of Mathematics - - $2.00. . . . No. .

British Columbia Association of Mathematics Teachers - -(For B. C. Teachers only) $3.00 . . . . No. .

, Amount $„

.Amount $.

.Amount $_

.Amount $„

.Amount $.

.Amount $.

, Amount $_

Amount $_

.Amount $.

TOTAL

MAJOR ASSIGNMENT: Elem.Q] Jr. H i g h Q Sr. H i g h • Adm. -Sup .Q CoW.^J

MAIL TO: Mr. John S. Marvin, 2915"North 27th Street, Tacoma, Washington 98407

•I'd

X-68-417/mo