ECTOR Journal

of the British Columbia Association of Mathematics Teachers

Volume 27 Number 4 Summer 1986

NOW COMPLETE! from

D.C. HEATH CANADA

PROBLEM SOLVING IN THE 80'S • Levels 4-6

by A.L. Anderson

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For your FREE examination copy or more information on PROBLEM SOLVING IN THE 80'S, please contact:

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B.C. Association of Mathematics Teachers 1985-86 Executive Committee

President, PSA Council Delegate, and Newsletter Editor John Kiassen 4573 Woodgreen Court West Vancouver, BC V7S 2V8 H: 926-8005 S: 985-5301

Vice-President Garry W. Phillips 4024 West 35th Avenue Vancouver, BC VoN 2P3 H: 261-4358 S: 526-3816

Secretary Stewart Lynch 2753 St. Georges Avenue North Vancouver, BC V7N 1T8 H: 984-7206 5: 987-9341

Treasurer Grace Fraser 2210 Dauphin Place Burnaby, BC V5B 4G9 H: 299-9680 S: 588-5918

Journal Editors Tom O'Shea 249 North Sea Avenue Burnaby, BC V5B 1K6 H: 294-0986 0: 291-4453 or 291-3395

Ian deGroot 3852 Calder Avenue North Vancouver, BC V7N 3S3 H: 980-6877 5: 985-5301

Membership Person J. Brian Tetlow 81 High Street Victoria, BC V8Z 5C8 H: 479-1947 S: 479-8271

Elementary Representative Wendy Kiassen 28-9651 Dayton Road Richmond, BC V6Y 3C3 H: 277-7443 5: 274-9907

Members-at-Large Les Dukowski 3657-206A Street Langley, BC V3A 6V7 H: 530-9665 5: 856-2521

Peggy Williamson 1613-2016 Fullerton Avenue North Vancouver, BC V7P 3E6 H: 922-5984 S: 733-1195

Post-Secondary Representative Ian deGroot 3852 Calder Avenue North Vancouver, BC V7N 3S3 H: 980-6877 5: 985-5301

NCTM Representative Jim Sherrill 2307 Kilmarnock Crescent North Vancouver, BC V7J 2Z3 H: 985-0861 0: 228-5512

THE B.C. ASSOCIATION OF MATHEMATICS TEACHERS PUBLISHES VECTOR

Membership may be obtained by writing to the: B.C. Teachers' Federation

2235 Burrard Street Vancouver, BC V6J 3H9

Membership rates for 1985-86 are:

BCTF Members ..............................................................$20 BCTF Associate Members......................................................$20 Student Members (full-time university students only) .............................$ 5

All others (persons not teaching in B.C. public schools, e.g., publishers, suppliers) .....................................................$30

Notice to Contributors

We invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable materials written by B.C. authors. In some instances, we may publish articles written by persons outside the province if the material is of particular interest in British Columbia.

Contributions may take the form of letters, articles, book reviews, opinions, teaching activities, and research reports. We prefer material to be typewritten and double-spaced, with wide margins. Diagrams should be camera-ready. We would appre-ciate a black-and-white photograph of each author. If feasible, the photo should show the author in a situation related to the con-tent of the article. Authors should also include a short statement indicating their educational position and the name and location of the institution in which they are employed.

Notice to Advertisers Vector, the official journal of the British Columbia Association of Mathematics Teachers, is published four times a year: fall, winter, spring, and summer. Circula-tion is approximately 600, mainly in B.C., but it includes mathematics educators across Canada.

Vector will accept advertising in a number of different formats. Pre-folded 21.5 x 28 cm promotional material may be included as inserts at the time of mailing. Advertis-ing printed in Vector may be of various sizes, and all must be camera-ready. Usable page size is 14 x 20 cm. Rates per issue are as follows:

Insert: $150 Full page: $150 Half page: $ 80 Quarter page: $ 40

Deadline for submitting advertising for the fall issue is July 10, 1986.

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Inside This Issue

From the Editors ................................................Tom O'Shea Letters.................................................................... Did You Know That ............................................Ian deGroot

Mathematics Teaching 8 A Framework for Organizing Problem-Solving Experiences.........Daphne Morris

17 Calculators in Elementary School? ..............................Wendy Klassen 20 The "Hand" Calculator—Just for Fun .......................... James M. Sherrill 23 Homework Helper .............................................Stewart Lynch 25 The Cosine Law: An Opportunity for Exploration ................ Thomas O'Shea 30 All Parabolas Are Similar ......................................David Wheeler

Mathematics Issues 37 Summary Report on the Draft Mathematics Curriculum...........Valerie Johnson

Miscellaneous 45 The Practicum Experience ..................................Adrienne Muirhead 47 Report on the 1986 Annual Meeting of the NCTM..................John Klassen 50 Report on the Algebra 12 Provincial and Scholarship

Examinations, January 1986 .....................................Walter Hamm

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From the Editors

Tom O'Shea

So far, 1986 has been an astonishing year for mathematics education in British Columbia. In February, I returned from a two-week holiday to find a new minister of education and a new Ministry of Post-Secondary Education. Funds had been withdrawn from the BCAMT-sponsored teacher orientation to the new curriculum, and no money was available for new textbooks. At the same time, a fund for excellence in education had been created. Shortly thereafter, money was found in the fund for new textbooks in 1987. As Pogo Possum would say—rowrbazzle.

In this issue, Ian deGroot, as usual, uncovers the unusual. His story about the fellow who erred in calculating the 528th digit of p1 is one of my favorites.

In recent issues, we have featured articles on problem solving. Daphne Morris now shows how the learner, the teacher, and the sub-ject matter interact, and she draws impor-tant inferences for an optimal classroom milieu for problem solving.

Our technology section features Wendy Klassen, who argues the case for including calculators in elementary classrooms. Jim Sherrill stresses the importance of the hand

calculator. Stewart Lynch describes the highly esteemed multimedia program in

North Vancouver to help students with their mathematics and English homework.

Valerie Johnson has collated and analyzed responses to the new curriculum guide; we have condensed her two reports into a single summary report.

In the final section, Adrienne Muirhead reminds us of what it was like at the start of a new teaching career. John Kiassen reports on his impressions of the 1986 Annual Meeting of the NCTM in Washing-ton, D.C. Walter Hamm itemizes the strengths and weaknesses of student responses on the January 1986 algebra examinations.

We hope you enjoyed all four issues of Volume 27 of Vector. Ian and I have enjoyed putting each issue together, and we look for-ward to next year. Plan to attend the BCAMT summer conference in August, and write us an article based on your impressions or presentation. We wish you a happy summer holiday.

ERRATUM The previous issue of Vector showed Volume 28, Number 3. It should have read Volume 27, Number 3.

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To the Editors: I thought you might be interested in some of the comments made by respondents to the draft mathematics curriculum guide.

• Calculators promote brain corrosion. Too much stress on calculators. Leave stress to Grade 10!

• Higher (hire) some actors (re: videotapes). In-service acting for math teachers. Koko should actually eat 12/3 of the chocolate bar to illustrate the example properly.

• Take any good mathematics encyclopedia as your curriculum guide for K-12 and university. How about a curriculum guide for Grade 8 on an 8 X 11 page so I can hand it out to every student, with room for their marks record. This would also be my course outline.

• I think we'll need a campaign (or minor miracle) to get teachers to teach the cur-riculum rather than their textbook from cover to cover. It looks great! In past years, we have used the text as the curriculum. Is this to be changed?

• Surely the ministry has the expertise to select the best text for each grade. I want the ministry to be the master teacher, example, for us to follow and learn from. The best should be at the top.

• Only a master teacher can accomplish miracles of learning in a vague undelineated diversity of decentralized general individual-ization.

• It seems a bit much to expect an in-depth response on such an important matter from professional specialists for nothing and in their spare time.

• I was relieved to note that on the co-ordinating committee are two represen-tatives from UBC and one from BCIT. As a teacher, I shall bow to their wisdom.

• Mean, median, and mode for Grade 8s! It was difficult enough in teacher's training at university level.

• I hope GM9, GM10, GM11 are developed asap—last week would be fine.

• There should be some in-service education for principals, district superintendents, etc., as teachers (like sheep) respond to the shepherd.

• Math is fun—it isn't always serious or doesn't need to be.

• Could our graduates of the future be arithmetic airheads?

• Here I can't see my way out of the forest, since I'm pruning a detail from a tree.

• Great beginning. Let's not lose any steam before we get to 11/12!

Sincerely,

Wendy Swonell

Did You Know That...?

Ian deGroot

Canadian weather scientists, using a com-plicated, innovative technique, believe that shortly they will be predicting major climate changes in Canada more accurately than ever before.

The mathematical system, newly adapted to climate, is described as a "powerful statistical technique, a sophisticated technique not in use for climate anywhere else in the world."

By this summer, using complicated meteoro-logical equations and a series of predictors —factors like the surface temperature of the Pacific Ocean or the amount of snow cover in the Arctic—Canadian meteorologists in a series of experiments plan to predict on a monthly basis:

The summer temperature for Vancouver during Expo '86;

Precipitation in the drought-plagued areas Of the prairies;

Summer temperatures for Metro Toron-to, the amount of precipitation, and, more important, the number of days of rain; • Winter temperatures of Montreal and its precipitation; • Rainfall around St. John's and the autumn temperatures, to determine the problems of East Coast icing and its effect on the oil-drilling sites off Newfoundland and Nova Scotia.

Using statistics gathered at 112 places and locations across the country, meteorologists and computer technicians in the experiment now attempt to predict whether the temper-ature will be above or below normal, the same with precipitation.

Last October, the Canadian Climate Centre in North York, Ontario, predicted that temperatures in eastern Canada would be above normal, while some parts of Atlantic Canada would be below, along with the West. The prediction was correct.

It is hoped that with this new method, the accuracy of forecasts will reach 65%. The old method has been accurate 58% of the time on temperature and 46% on precipita-tion. This technique is a perfect example of how technical mathematics is being applied in society.

How do you break in a supercomputer more powerful than all the earlier versions of it combined? You have it compute p1 to 29 360 128 places.

In January, David Bailey, a computer specialist with the Ames Research Centre of the U.S. National Aeronautics and Space Administration, ran Ames's new CRAY-2

supercomputer through U.S. acceptance-period paces.

Over a 28-hour period, the CRAY-2, which is both a little bit less than a trillion times more powerful than the average personal computer, and more powerful than all the CRAY-1 computers ever sold, com-puted p1 with two different techniques developed by brothers Jonathan and Peter Borwein, Dalhousie University mathematics professors.

Computing pi to nearly 30 million places required 12 trillion arithmetic operations, providing a good test of the computer's reliability in both software and hardware.

Using the two different techniques, Bailey found that the digits of p1 matched in both cases except for the last 24 numbers. This dif-ference can be explained by the rounding-off effect.

While showing that the CRAY-2 was reliable, the technique also was the latest round in the Japanese-U.S. p1-calculating rivalry. The Japanese have reached 10 million places and are planning to push on-ward to 100 million places.

For the two Dalhousie professors, the impor-tance of the 30 million p1 computation lies in the speed of the calculating equations they have come up with. For example, to calculate pi's last number using traditional pen-and-paper multiplication techniques would have taken about 30 million steps. The Borwein techniques needed only 25 steps, and the formula underlying them could be written out on four lines of this journal.

In less practical terms, the ability to com-pute p1 marks the growth of a modern technological and scientific society.

By 1650, the number was computed to 40 places and was accurate enough to compute the circumference of the known universe down to the width of a molecule. In the 19th century, an obscure British mathematician spent 20 years computing it to 707 places, not realizing that he had made an error in the 528th digit. The first computer boosted the number to 2 000 places in 1949. This grew to 10 000 in 1958, 100 000 in 1961, 1 million in 1973, and 10 million in 1981.

The Americans have established Presidential Awards for Excellence in Science and Mathe-matics Teaching.

Fifty-four middle/junior and senior second-ary school teachers of mathematics, one from each state and eligible jurisdiction, are awarded handsome presidential citations, $5000 National Science Foundation grants for their schools, and an array of gifts and additional honors.

Nominations are made by colleagues, stu-dents, supervisors, or the general public. The National Council of Teachers of Mathe-matics co-ordinates yearly state nominations and the selections for mathematics.

In 1986, three mathematics teachers will be chosen as state-level awardees by selection committees in each state and jurisdiction (162 awardees in all); those selected will then be candidates for the Presidential Awards to be given October 21-24.

The Presidential Awards program was estab-lished to identify outstanding mathematics and science teachers who would serve as models for their colleagues. The idea behind the awards is that increased status and rewards will encourage more high-quality in-dividuals to enter teaching and remain in the profession.

A Framework for Problem-Solving

Organizing Experiences

MATHEMATICS TEACHING

Daphne Morris4

Daphne Morris is a co-ordinator in the Ministry of Education's Curriculum Development Branch.

If elementary teachers are expected to facilitate pupils' learning problem solving as specified in the revised curriculum, they will need much more than guidelines and repack-aged collections of problems. This article reiterates the importance of problem solv-ing and provokes teachers' thoughts regard-ing the interactions of teachers and pupils as they use the resources available to them within the classroom. Through descriptions of the "commonplaces," the intents of the curriculum developers can be clarified. Fur-thermore, for those teachers who ". . . are enthusiastic and willing to teach problem solving, but.. . lack a model to work from" (Klassen, 1985), the guidelines provided here can serve as a model to enable them and their pupils to become successful problem solvers.

Background The revised curriculum for elementary school mathematics highlights the impor-tance of problem solving. It integrates application problems with concepts and skills clustered into the following five con-tent strands: Number and Number Opera-tions, Data Analysis, Geometry, Measure-ment, and Algebra. Further, the "heuristics" of problem solving and procedural processes

are described in the Problem-Solving Skills Strand. The inclusion of problem-solving ac-tivities and the focus on solving problems represent a considerable content change, but one that, in and of itself, will not ensure that pupils will be more effective problem solvers.

To increase the effectiveness of pupils as problem solvers, educators must look beyond the articles and resource books on problem solving that are currently available. Such materials are in abundance: collections of problems for teachers to use in their class-rooms. Most allude to the need to create a problem-solving atmosphere, but they fail to describe how to do that, and they assume particular attitudes in teachers and pupils. Only the subject matter is provided to guide teachers, and the other "commonplaces" (Schwab, 1973)—learners, teachers, and milieu—are seldom addressed in depth.

In The 1985 British Columbia Mathematics Assessment, Robitaille and O'Shea reported, "At the elementary level, there is a large group of teachers who have taken no mathe-

matics or mathematics education courses;" when asked how many workshops on

problem solving they had attended in the past year, about 75 % of teachers at each

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grade level said 'none,' and about 20% had attended only one," and when ". . . asked what sources they used to provide pupils with problem-solving exercises," teachers responded that "the textbook is the domi-nant source."

Elementary teachers in British Columbia are typically generalists. Lacking specialist train-ing in either mathematics or teaching mathematics, they have limited knowledge of mathematics. Their subsequent lack of confidence causes them to rely on textbooks. In his chairperson's report of a conference held in Madison, Wisconsin in 1983, Romberg (1984) cites a description of elementary classrooms as drawn from a survey (Conference Board of Mathematical Sciences, 1975) as follows:

The "median" classroom is self-contained. The mathematics period is about 43 minutes long, and about half of this time is written work. A single text is used in whole-class instruction. The text is followed closely . . . which for most students is primarily a source of problem sets.

When teachers feel anxious or uncomfort-able (or, worse yet, negative) about teaching mathematics to the point that a single text is used, whole-class instruction is the norm. The result is, as described by Romberg, situational influences that harm the develop-ment of positive pupil attitudes toward mathematics.

The Importance of Problem Solving In a general sense, problem solving encom-passes much more than school mathematics; it is ". . . vital and important in other areas of the school curriculum and in nonschool activities" (McKillip and Davis, 1980). Class-room teachers, mathematics educators, and the public believe that the ability to solve problems is essential to successful adult life. While there are some who believe that

problem-solving skills are transferable to situations beyond the mathematics class-room, most educators agree that mathe-matics ". . . can be an important context for the development and refinement of problem-solving skills" (McKillip and Davis, 1980).

When contemporary mathematics educators address problem solving, they invariably refer to the National Council of Teachers of Mathematics' Agenda for Action (1980), which recommended that problem solving be the focus for school mathematics in the 1980s. Leaders in mathematics education unanimously agree that problem solving should be integral to every student's ex-periences. Some go so far as to state that "Learning to solve problems is the principal reason for studying mathematics" (O'Daffer, 1984) and "The ultimate goal of school mathematics at all times is to develop in our students the ability to solve problems" (Lenchner, 1983).

Clearly, problem solving in the mathe-matical context lets pupils explore the "Essential characteristics of mathematics such as abstracting, inventing, proving and applying . . ." (Romberg, 1984). Through problem-solving activities, pupils experience mathematics as the mathematician does. Through problem solving, pupils use the concepts and computational skills associated with mathematics in concert with the strategies they are acquiring, and they see the reasons for doing so—learning need not occur in isolation. Problem solving is fun-damental to mathematics; therefore, it ought to be a major component of the curriculum.

Developing a Curriculum To Facilitate Problem Solving Having agreed that problem solving is fundamental to mathematics, the Elemen-tary Mathematics Curriculum Revision Committee (composed of mathematicians, educational theorists, specialists preparing pre-service and in-service teachers, and

generalist teachers who have been teaching mathematics for many years) has incor-porated problem solving and highlighted problem-solving skills in the revised curriculum. In their deliberations, the com-mittee discussed not only the fundamental aspects of the discipline and topics that con-stitute school mathematics, but also the attitudes and role of teachers, the inter-actions between teachers and children, and the resulting roles and attitudes developed in pupils. They have maintained a vision of the optimal situation—the ideal classroom environment—a place where experiences with mathematics are positive and pupils' efforts are supported and encouraged in their ever-developing understanding of concepts and facility with skills.

While the committee has discussed and embedded the commonplace aspects that in-fluence what actually occurs in classrooms as teachers begin to implement the intended curriculum, they have been mindful, too, that teachers will receive what amounts to a printed distillation of many hours of discussion and debate. The document will be interpreted by individual teachers bringing their own experiences, values, assumptions, and expertise to bear when selecting what to teach and how to teach it.

To facilitate the committee's intents, Schwab's notions of the influences that affect curriculum decisions have been super-imposed, and, for clarity, each will be discussed separately. This approach may be considered artificial and simplistic by some. However, given the problems described in the background section of this article, and a sense of the overall importance of develop-ing problem-solving abilities, a thorough examination of the variables that affect success may be the best way to understand the complexity of creating and sustaining the environment in which problem solving occurs.

As pointed out by Schwab (1973): Defensible educational thought must take account of four commonplaces of equal rank: the learner, the teacher, the milieu, and the subject matter.

Schwab states that the commonplaces must be given equal consideration; yet, while order need not imply relative importance, some organization for discussing each must be determined. Most learning resources for problem solving are structured by the following pattern: a discussion of the impor-tance of problem solving, definitions of mathematical problems, many sample prob-lems, and, finally, incidental comments regarding strategies for teaching and the role of teachers. For teachers who receive the cur-riculum guide and are expected to implement it, a more practical approach would be to begin with a vision of the successful problem solver.

The Learner The traits attributable to successful problem solvers are a function of both an attitudinal mind-set and cognitive ability. Few elemen-tary pupils are capable of acting as conscious agents in their own intellectual development, and many aspects of cognitive growth are beyond the teacher's control. Those aspects over which teachers do have control are briefly mentioned here and discussed further in the section "Teachers." The characteristics of successful problem solvers and the at-titudes necessary for successful problem solving are discussed at greater length because I believe that teachers have a great deal of control over the affective qualities of learning, and even young pupils can learn to recognize the importance their attitude has on solving problems.

Successful problem solvers must, first of all, desire a solution to a problem. Problem solvers must believe that they have the abil-ity to find a solution; they must have an "I can" attitude. They must be willing to take

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risks, beginning with attempting to solve a problem. They demonstrate perseverance: patience and persistence. Characteristically, they can recognize .when a particular ap-proach is unsatisfactory and can select alter-natives. They are self-reliant (motivation is intrinsic rather than extrinsic); they rely on their own satisfaction, not on teachers' ap-proval. Successful problem solvers can focus on processes; they use strategies selectively and are not bound by algorithmic thinking. Finally, they are effective communicators, capable of retracing their thought processes and representing their approach(es) to solu-tions in a variety of forms.

Such attitudes can be cultivated by teachers who model, them and encourage their development in their pupils. Children who discuss the problem-solving process and come to understand the importance of their attitudes will take on characteristics of suc-cessful . problem solvers.

Teachers Teachers play an essential role in facilitating problem-solving. Teachers who foster pupils' growth and development toward in-dependent problem-solving understand the dynamism and fluidity of the processes for solving problems; know the skills necessary for attempting and solving problems; and of-fer problems that will challenge, but not frustrate, their pupils.

To select and present problems that are ap-propriate to the pupils' abilities, teachers must thoroughly understand age-appropriate cognitive abilities and the degree to which such abilities can differ from individual to individual or from one group to another. Pupils who are interested in the problems and who believe they can solve them will be challenged to find solutions.

Teachers who enable problem solving en-courage their pupils to take risks, and they make it emotionally safe .to do so. These

teachers know that making mistakes is a valid aspect of the problem-solving process. They take the emphasis off finding the "right answer" on the first try and facilitate discus-sion of the strategies (both successful and un-successful) that eventually lead to satisfac-tory solutions. They encourage, comment on, and discuss the merits of patience, per-sistence, and flexibility. To assist the development of self-reliance and intrinsic motivation, teachers get pupils to justify both solutions and strategies by making comments such as, "How did you know when ...?" and "At what point in the pro-cess did you ...?" and by recognizing the achievements of individuals and groups-' Teachers model effective communication by initially outlining the steps observed and offering suggestions for other ways to repre-sent the process. As the pupils gain skill in thinking about their thinking (metacogni-tion), teachers facilitate the pupils' discus-sions to the point when, finally, they can talk about the process independently.

Teachers accept that successful problem solving does not occur as a result of memorizing a single lock-step strategy to be applied in every problematic circumstance. Teachers agree that justifiable solutions can be reached through the application of a variety of strategies (either step-after-step in sequence or unique combinations of these) and that making mistakes, "running into dead ends," making successive approxima-tions, and revising plans are valid and often necessary.

Subject Matter Experiences, selected by teachers, constitute the subject matter—problems and problem-solving strategies. O'Daffer (1984) writes, "In the elementary school a problem is a situation given to the student, usually by the teacher, where the student must use one or more strategies in order to answer a ques-tion or make a decision."

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The definitions in the literature are univer-sally consistent. A mathematics problem is a situation in which an individual or a group desires a goal, but a blockage between a given and a goal state exists, and the challenge involves mathematics. While there are a variety of types of problems (see in-set), good problems must be interesting enough to challenge learners to want to find a solution(s); a problem is not a problem for pupils who are uninterested in its solution.

Problems should be selected according to the pupils' mathematical capabilities—the chil-dren must possess the conceptual under-standings and skills to be applied to the problem. A good problem can be solved by using a variety of strategies, each valid; although some approaches may be more direct and efficient than others.

Finally, good problems can be adapted, extended, or related to other problems; problem-solving skills are strengthened as children generalize and apply to new situa-tions problem-solving strategies used suc-cessfully in the past.

Thus far, a definition of a mathematical problem, identification of problem types, and some characteristics of good problems have been provided. Problem solving, however, involves much more. It requires the use of many skills, often in unique com-binations. Using the skills to work toward solutions is strategy, and the application of a successful strategy or group of strategies is the essence of problem solving. In the revised curriculum, strategies for solving problems have been identified and grouped into three broad categories: "Understanding Problems," "Solving Problems," and "Ex-tending/Consolidating Problems." Within each category, the strategies are listed as intended learning outcomes to provide the teacher with an overview of both the struc-ture and fluidity of the problem-solving pro-cess. Sometimes, a sDlution will be reached

by a direct step-by-step approach. At other times, investigations will necessitate recon-sidering and revising plans. Post-solution analysis (individual or group) of the process used to find solutions is as critical as the solution itself.

According to O'Daffer (1984), the follow-ing constitute types of problems:

ONE-STEP Solved using single operation X, ^) Example: A jet plane has 214 seats. 167 people got on. How many seats are empty?

MULTIPLE-STEP Solved using two or more operations (+, —, x, ±) Example: An old U.S. flag had 4 rows of stars with 7 in each row and one row with only 6 stars. How many stars did it have?

PROCESS Cannot usually be solved by simply using one or more operations. Often requires other strategies. Example: 6 people each ride the roller coaster with each other person only once. How many rides?

PUZZLE Requires a flash of insight or a lucky guess. Example: Can you make 4 congruent triangles with 6 toothpicks?

APPLIED Requires organization, analysis of data, problem selection, and a variety of skills—a realistic problem. Example: How much should you charge for lemonade to make a profit at your lemonade stand?

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TABLE 1

TABLE 2

STRAND: Problem-Solving Skills STRAND: Problem-Solving Skills TOPIC: Understanding Problems TOPIC: Solving Problems

Intended Learning Outcomes

The student should be able to:

• P-i ask the kinds of questions needed to clarify the problem

• P-2 restate and demonstrate an understanding of a problem by (a) using objects (b) drawing a picture (c) rephrasing (d) writing open sentences (if

appropriate)

• P-3 identify key words

• P-4 identify needed, missing and/or extraneous information

• P-5 determine whether a problem can be • solved

• P-6 recognize sub-problems

• P-7 give alternate interpretation(s) (multiple word meanings, point of view, validity of data)

• P-8 analyze a given value as being a reasonable estimate/ solution

Intended Learning Outcomes

The student should be able to:

• P-9 use one or more of the following strategies (a) act out problems (b) draw a diagram or make a

model (c) classify and order information (d) look for a pattern (e) make a list, table or graph (f) guess and check (g) design and use a simpler case (h) break problems into parts (i) work backwards (j) write and solve number

sentences

• P-10 estimate solutions to help solve problems

• P-li use a calculator to help solve problems

• P-12 formulate and revise plans as necessary

• P-13 solve a variety of problems such as (a) application problems, puzzle

problems, open-ended problems,

(b) those not requiring computation (c) those requiring more than one

application of an operation or multiple operations

The Milieu Given that the three preceding common-places are operational as described, the environment that enables and supports the process of problem solving is very different from the "median" classroom described earlier. In contrast, the pupils solving prob-lems are pursuing meaningful mathematical

experiences. They may or may not be using a textbook as the source of problems. The most noticeable aspects of a series of mathematics classes will be excitement, in-creased time for discussion—with a corres-ponding decrease in written work—and departure from repetition and routine.

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TABLE 3

STRAND: Problem-Solving Skills TOPIC: Consolidating/ Extending Problems

Intended Learning Outcomes

The student should be able to:

• P-14 state a solution

• P-15 determine the reasonableness of a solution

• P-16 identify an alternative strategy to solve a problem

• P-17 check the solution by using strategies such as (a) relating it to an estimate (b) using a different strategy (c) checking the steps, using a

calculator

• P-18 explain the solution by reviewing the information, question, strategy and process

• P-19 consider the possibility of other solutions

• P-20 make predictions and generalizations

• P-21, recognize problems similar to those solved

• P-22 create problems (a) given information, a picture, a

number sentence, data, (b) by varying a given problem (c) that can be solved using a given

strategy

Initially, teachers may find it easier to in-troduce problem solving and to model effec-tive problem-solving behavior by presenting a single problem to the whole class or the same problem to a number of groups of children. To facilitate this purpose, an open-ended lesson plan is provided below.

Sample Lesson Plan • for Problem Solving*

Problem

1. Understanding the problem (Read, discuss, and ask questions for understanding.)

2. Solving the problem (Discuss possible strategies; observe and question pupils to facilitate solutions. . . .)

3. Consolidating the problem (Pupils check and give answer; discuss solu-tion(s), strategies, and features of the prob-lem....)

4. Extending the problem (Recall related problems, invent similar problems, create new problems . . .

* Adapted from O'Daffer (1984)

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The observer of the mathematics class will see the teacher present a problem to the class, either orally or in written form. The problem will be discussed at length; teacher and pupils may be asking questions to clarify understanding of what is expected. Once the teacher is assured that the pupils understand the problem, a variety of possible strategies will be discussed. The teacher accepts all sug-gestions as possible; learning what doesn't work is as valuable as discovering the most direct and effective strategy. The group(s) can then begin to work toward a solution, by applying the selected strategy to the problem.

Once the pupils have entered this phase of the process, the observer will note that pupils are talking with one another; they are asking questions and making suggestions. The teacher moves from group to group, re-questing that the pupils describe what they are doing and asking them to justify their course ofaction. Pupils unfamiliar with the process may need the teacher to provide a reflective articulation of what is observed. Here is an example of the dialogue between teacher and pupils:

T: I can see that you are trying a number of different answers and checking to see if each one solves the problem. That's called "Guess and Check." Who gave these answers?

(Two pupils respond)

T: What happened when you tried them as solutions?

P: David's worked; John's answer didn't fit.

T: Tell me why one worked and the other didn't.

(Discussion)

T: Not everyone has made a suggestion. Alice, can you think of a number that we could try?

P: 7.

T: All right. Let's try 7. Hmm. That one won't work, but look! We're getting closer. Thanks, Alice—you got us closer to the answer.

T: I'm going to work with another group. When your group has a solution you're satisfied with, give me a signal, and we'll discuss your answers.

The teacher has described and identified the strategy and participated in the problem-solving process with the pupils in the group. The teacher has modelled appropriate behavior, encouraged all group members to participate, and challenged the pupils to con-tinue until they have reached a satisfactory solution. The focus has been taken off in-dividuals, and responsibility for solutions, directed to the group. Thus, individual anxiety is reduced; co-operation and com-munication are encouraged. A mistake has been recognized as useful in finding the solu-tion, and the individual's contribution has been acknowledged.

In the problem-solving classroom, pupils are given time to discuss and solve problems. They are guided to describe strategies and are enabled to recognize alternative pro-cedures and solutions. The teacher and pupils are completely engaged in their in-teractions, and the process could be des-cribed as "guided search and shared discovery." Both teacher and pupils use probing questions, and children are en-couraged to find validity in their own resources, thus moving ever closer to becom-ing independent problem solvers.

15

Conclusion Problem solving is not intended to be a com-plete program for school mathematics. Nor should it be considered "supplementary" for all or "enrichment" for a few. Problem solv-ing is central to the study of mathematics, and it should be integrated into the school mathematics curriculum. By considering the four commonplaces, attempting problem solving, practising the strategies, and using the process, teachers and pupils can develop positive attitudes and begin to internalize those skills and procedures that result in in-creased intellectual autonomy.

References Greenes, et al. (1980). Teacher's commen-

tary: techniques of problem solving (Problem Deck A). U.S.A.: Dale Seymour Publications, 1-6.

Kelly, W.P., & Tomhave, W.K. (1985, Jan-uary). A study of math anxiety/math avoidance in preservice elementary teachers. Arithmetic Teacher, 51-53.

Klassen, W. (1985). Problem-solving cur-riculum for the elementary grades. Vector, 26, (3) 11-15.

Labinowicz, E. (1985). Learning from children: new beginnings for teaching numerical thinking. Don Mills, Ontario: Addison-Wesley Publishing Company.

Lenchner, C. (1983). Creative problem solv-ing in school mathematics. Toronto: Houghton Mifflin Company.

McKillip, W.D., & Davis, E.J. (1980). Mathematics instruction: early childhood. Morriston, New Jersey: Silver Burdett Company.

O'Daffer, P.C. (1984, April 26-29). Ideas for teaching problem solving. Handout pre-pared for the 1984 National Council of Teachers of Mathematics Annual Confer-ence. San Francisco.

Robitaille, D.F., & O'Shea, T .J. (1985). The 1985 British Columbia mathematics assessment. British Columbia: Ministry of Education.

Romberg, T.A. (1984, June). School mathematics: options for the 1990's. Volume 1, chairman's report of a con-ference, Madison, Wisconsin, December 5-8, 1983. Washington, D.C.: U.S. Government Printing Office.

Schwab, J.J. (1973, August). The practical 3: translation into curriculum. School Review, 501-522.

Szetela, W. (1982). Problem solving activities teacher's resource book 6. Markham, Ontario: Houghton Mifflin Canada Limited.

16

Calculators in Elementary School?

Wendy Kiassen

Wendy Kiassen is a teacher at Kingswood Elementary School in Richmond.

Whenever I think about calculators in the elementary school, one particular comment always comes to mind:

A century from now people will gasp when they learn that six to nine years were spent getting good at paper-and-pencil arithmetic. They will ask: Didn't they have calculators then?

(Usiskin, 1983, p. 2)

That certainly makes me think about the role of calculators in elementary schools today. And just what is that role? In the following, I give a brief rationale regarding the use of calculators in the elementary school, and outline some of the common fears.

Rationale Most children already have access to a calculator. First of all, a calculator is a tool to be used with all mathematics topics. Its use is neither a curriculum strand nor a self-contained unit. The calculator is the most common calculating tool in our society, and the majority of students in this province have access to a calculator at home. The recent B.C. Mathematics Assessment reveals that 49% of Grade 4 pupils and 68% of Grade 7 pupils in the province have access to a

calculator at home. At the Grade 4 level, this represents an increase of 10 percentage points from 1981, and at the Grade 7 level, an increase of 32 percentage points. So, it seems that children are using, or at least have the potential to use, calculators with or without specific instruction in schools. We should therefore teach children to use calculators intelligently, starting in elemen-tary school.

The use of calculators will enable us to re-examine and reorder the current mathe-matics curriculum. The availability of calculators requires a re-examination of the computational skills needed by every citizen. Some computa-tional skills will no longer be important; whereas others will become more important. For example, there should be a decreased emphasis on long division and an increased emphasis on estimation. Mathematics in-struction in the elementary schools needs to emphasize the skills required for efficient and effective use of calculators.

Calculators improve attitude and moti-vation. Perhaps most important, calculators im-prove pupils' attitudes and motivations. As with anything new, elementary-school-age children will inevitably be initially interested

17

in calculators. Their interest can be main-tained if they are given interesting things to do with calculators. Pupils will remain inter-ested given more interesting problems. And given a calculator, pupils can be given more interesting problems. In addition to an im -provement in general attitude and motiva-tion, more specific benefits in the following areas accrue: • All pupils can succeed. • All pupils can add, subtract, multiply, and divide with a calculator. • Individual differences in the classroom are reduced: all children can now do the basics, while more capable pupils can go on to ex-tending problems, generalizing, looking for patterns, solving problems in different ways, etc. • Discipline problems resulting from frustra- tion and/or ignorance lessen. • Every pupil can participate in problem solving when using a calculator.

Common Fears Children will become dependent on cal-culators. Yes, since they will, with or without us, we must teach the intelligent use of calculators, beginning in elementary school. Pupils won't take long to recognize and judge for them-selves the arithmetic that is easier done in their heads than with a calculator. A simple demonstration involves a test of 10 to 20 basic arithmetic facts. First, pupils should be timed doing the questions in their heads and marking down their answers with a pencil. Second, they should be timed doing the same questions, but keystroking each number into a calculator and again marking down their answers with a pencil. The latter test gener-ally averages more than twice as long to do. It becomes obvious that in certain situations, mental calculations are much more efficient than calculators.

Children will not learn to think. Calculators themselves do not think. They only follow the thinking of people. Children have to choose the necessary numbers, operations, and order of operations needed to solve a problem. Furthermore, allowing unrestricted use of calculators for problem solving and applications focusses instruction and learning on higher thinking skills and decision-making. Obviously, the higher the grade, the more complex the arithmetic. By alleviating the tedium of complex computa-tion, calculators facilitate the introduction of new topics, such as real-world applica-tions, investigations of numerical patterns, and analyses of statistical data.

Children will not learn the basics. Extensive research on calculator use in classrooms is reassuring. When calculators are used appropriately, basic computational skills (with paper and pencil) are maintained, and problem-solving skills improve. In the past decade, close to 200 studies have been undertaken, at least 75 of which explored whether calculators harm achievement. The answer is a resounding no. In all but a few instances, achievement scores were as high, or higher, when calculators were used for in-struction (but not on the test) (Suydam, 1982).

Moreover, calculators not only can help as a computational tool, but also can help teachers develop concepts about numbers and counting, the four arithmetic opera-tions, decimals, and estimation.

Summary The calculator has the potential to become a valuable instructional tool in the elemen-tary school for computational skill develop-ment and problem solving. A century from now, when people ask, "Didn't they have calculators then?", what would the answer be?

18

References Bell, M.S. (1979). Calculators in elementary

schools? Some tentative guidelines and questions based on classroom experience, Calculators, Reston, VA: NCTM.

Math Implementation Committee. (1985). Calculators in elementary schools, Rich-mond: SD 38.

Suydam, M.N. (1982). The use of calcula-tors in pre-college education, Columbus, Ohio: Calculator Information Center, ERIC: ED 220-273.

Usiskin, Z.P. (1983, May). Arithmetic in a calculator age Arithmetic Teacher, 30, 2.

Careers in Statistics

Statistics is closely allied with both mathematics and computing, but distinct and requiring special skills. Applications of statistics are routinely made in a great variety of fields—in the design of weather modification experiments, for exam-ple, and in the testing of new drugs like polio vaccine, in the exploration for new energy sources, in monitoring for pollution, in analyzing courtroom evidence, in estimating the abundance of wildlife and fish populations. Of great current interest is quality control in manufacturing, which is intrinsically a statistical problem. In consequence, a shortage of statisticians exists at all levels in North America.

Preparation for a career in statistics begins at the college or university either at the undergraduate or at the graduate level. Some programs of study would emphasize theory possibly with concentrations in computing or mathematics. Others would be heavily concerned with applications.

At the University of British Columbia, a new Department of Statistics has been established, and it currently offers graduate study in the statistical sciences. While its proposed undergraduate program in statistics will not be implemented until possibly 1987/88, students interested in pursuing statistics at that level may select a statistics option from the programs offered by the Department of Mathematics. For further information, contact Professor J.V. Zidek, head, Department of Statistics, the University of British Columbia, 2021 West Mall, Vancouver, BC V6T 1W5.

19

20

FIGURE 1: 9 X 7 FIGURE 2(a): 9 X 3

if JI

gHThe "Hand" Calculator

—J ust for Fun

James M. Sherrill

Jim Sherrill is a professor in UBC's Department of Mathematics and Science Education.

In this "day of modern technology" (a term that has been used every day since the Indus-trial Revolution), we are bombarded with in-formation concerning calculators and com-puters. Let's give ourselves and our students a break. I want to discuss the "Hand" calculator. It does multiplication only. I want to talk about its capability to com-pute the "large" multiplication basic facts (from 6 X 6 to 9 X 9; it will even do 10 X 10).

Probably very few of you reading these words haven't seen the method of doing the 9s table on your hands. One simply num-bers the fingers 1-10 from left to right. If one wants to multiply 9 X 7, for example,

one puts down the seventh finger (see Figure 1).

One now counts how many fingers are up to the right of the down finger (Fingers 8, 9, and 10—so 3), which is the units value, and counts how many fingers are up to the left of the down finger (Fingers 1, 2, 3, 4, 5, and 6—so 6), which is the tens value. In the example, the answer is 63 (9 X 7 = 63).

Figure 2 shows some more examples. Stu-dents enjoy such techniques just for fun. The Hand calculator has more steps in its multip-lication algorithm, but why not; it will do any multiplication fact from 6 X 6 to 9 X 9.

FIGURE 2(b): 9 X 9

To use the Hand calculator, one holds one's hands palm up in front and numbers the fingers 6 to 10 from left to right on each hand as in Figure 3.

'1 \ffI cm'

FIGURE 3: The Hand Calculator

Four steps are required to use the Hand calculator to compute a large multiplication fact. Let me demonstrate with the computa-tion of 7 X 8.

Step 1: Clear memory. This is done by shak-ing both hands vigorously.

Step 2: Remembering the numbering scheme in Figure 3, put DOWN all the left-hand fingers up to and including , the 7th finger; put DOWN all the right-hand fingers up to and including the 8th finger. The result of this step appears in Figure 4.

Step 3: Always compute the units value first. Multiply the number of UP fingers on the

£1

FIGURE 4: 7 X 8

left hand (3) by the number of UP fingers on the right hand (2). In the example 3 X 2 = 6. If you get a two-digit product, carry the 1 in the tens place over to Step 4.

Step 4: Compute the tens value by adding the number of DOWN fingers on the left hand (2) to the number of DOWN fingers on the right hand (3). In the example 2 + 3 = 5. If carrying occurred in Step 3, you'd add 1 to the sum computed in this step.

In the example 7 X 8, the units are 6, and the tens are 5, so the answer is 56. Are you confused or simply don't believe? Let's do an example that requires carrying. Let's do 6 X 7.

Step 1: Clear memory.

Step 2: Put down finger 6 on the left hand and fingers 6 and 7 on the right hand (see Figure 5).

FIGURE 5: 6 X 7

21

Step 3: Units value. Four fingers are up on the left hand, and three fingers are up on the right hand. Since 4 X 3 = 12, the units are 2, and we will have to add 1 to the sum com-puted in Step 4.

Step 4: Tens value. One finger is down on the left hand, and two fingers are down on the right hand. Since 1 + 2 = 3 and we add 1 from Step 3 and 3 + 1 = 4, the tens are 4. Voila! 6 X 7 = 42.

(a) 9 X 7 = n (c) 72 = n

(b)8X6=n (d)lOXlOn

ANSWERS: OOt (P) 6 7P () 8T' (q) £9 (e)

FIGURE 6 Homework

Figure 6 shows several more examples. The After an exhaustive review of the research answers are given upside down. Try to do literature, I have found absolutely no utili-the problems without looking at the tarian value to the Hand calculator or this answers. article. Both are Just for Fun!

r

leachers of

22

HOMEWORK

Stewart Lynch is a mathematics teacher at Carson Graham Secondary School, in North Vancouver.

Homework Helper uses telephone and televi-sion to help North and West Vancouver students in Grades 6 to 10 with their mathematics and English homework.

A team of regular classroom teachers employed by North Vancouver School District (44) meets in the studios of Shaw Cable 10 each Wednesday evening. Students from across the North Shore are invited to telephone the teachers for help with their homework. Most students' questions are answered on the phone. Selected questions—those that are thought to be of interest to other students and those that have a strong visual component—are answered over tele-vision, with the telephone conversation be-tween teacher and student being carried live over Cable 10.

Phone calls are handled from 18:30 to 20:30 each Wednesday evening. The television seg-ment of Homework Helper runs from 19:00 to 20:00. Though the service is aimed directly at mathematics and English students in Grades 6 to 10, students—and adults—can call in with questions relating to other subjects and other grade levels.

Though the students may be the main bene-ficiaries, Homework Helper also reinforces

instruction given by regular classroom teachers. The service complements the help provided by parents, and the general public gains an opportunity to see current methods of instruction, to realize the complexity of today's curricula, and to recognize the dili-gence of North Shore students and teachers.

The response has been overwhelming. The program has been on the air since February 26, 1986, and calls related to mathematics have outnumbered English by two to one. We field an average 80 calls each night, so we barely have enough time to catch our breath.

Before the first show, all of us in the project were nervous about appearing on live tele-vision. Two simulated live rehearsals familiarized us with talking to the correct camera and watching for cues from the director. One of the difficult aspects was try-ing to cram all of our board work into a small enough area to be picked by the camera for a close-up.

Though all of us are experienced teachers, we felt like student teachers on practicum. No matter how much producer Leslie Payne told us how easy it was, there just seemed to be too many things to remember.

23

In spite of all our worries, the first show was successful, and our nervousness was not evi-dent. Student, staff, and public response has been very encouraging, and some of us are even being recognized in shopping malls as "TV stars."

Performing on television does come a little easier with experience, but with live TV, one really has to be ready for anything. For in-stance, what do you do at five seconds before air time when the student you are planning to interact with over the phone extension to the studio gets disconnected? Having just finished your mini-lesson, with nothing else planned, what do you do when the director signals that you still have three more minutes to go in your segment? My

greatest fear was realized on the third show when I was well into introducing my topic and trying to have a conversation with my student. I was informed that my microphone was not operational and that no one had heard a world I'd said.

Fortunately all of us are experienced teachers and are able to cope with those unexpected little things that continually crop up.

Homework Helper was a pilot project running 13 weeks—until May 28. A science segment will likely be included in the pro-gram next year. The program may be ex-tended to two nights a week, and additional teachers will be included in the service.

24

The Cosine Law: An Opportunity for Exploration

Thomas O'Shea

Tom O'Shea is an associate professor in SFU's Faculty of Education.

At a recent meeting of the BCAMT Execu-tive Committee, one of my colleagues took the faculties of education in British Colum-bia to task. He criticized faculty members for not giving student teachers sufficient knowledge about mathematics and methods of teaching mathematics in the secondary school. In a follow-up letter, he said, "I believe that courses must be offered in the various faculties of mathematics education that deal with various techniques of intro-ducing curriculum topics. Students need to know how to effectively introduce topics such as logarithms, cosine law, ellipses, in-verse functions, absolute value functions." During our meeting, he said, "They don't even know the cosine law, much less teach it."

I, of course, took this as a personal insult and, because I was at that time teaching a small group of pre-service secondary teachers. I decided to develop a teaching sequence by which my students would learn the cosine law and which also could serve as a model for them to use in their own teaching. Having taught secondary school mathematics for 11 years, I checked back to see how the texts of my time dealt with the topic. In my first year of teaching, we used A First Course in Trigonometry (Oliver, Winters, & Campbell, 1941), which developed the Law of Cosines as follows:

Sec. 6. The Law of Cosines. C C

/X C N_X X-C A C

Solution: Draw CD -LAB and let AD=x. Then DB=c—x in the first figure and (x—c) in the second figure. In both As, h=b'—z5. In the acute A, 10a1 —(c—x) 1, while in the obtuse A,

h2—c0—(x—c)1

Hence in both figures, I&'=o—(c—x)'

or b1_xI=ae_cs+2cx_xi

or b1+c2—a1—Ecx. b5+c'—&

2c

Now, cos A

cos Ab+ca which is Lite Law of Cosines. It is 2 b sometimes written in the form, &=b2 +c2_ 2bc cos A.

FIGURE 1

That was also the text from which I was taught as a student in Grade 12, and I re-called that this proof for the cosine law was not entirely enlightening. I had always felt that the authors had pulled a rabbit out of a hat, that the procedure was contrived to prove a point. When I look at it now, it seems straightforward, mathematically sound, and compelling. But when I was a student, I thought it a lot of jiggery-pokery.

I remembered that Dolciani used a very dif-ferent approach, one that built on a logical sequence, so I checked Modern Algebra and Trigonometry (Doliciani, Berman, &

25

Wooton, 1963) and found on page 436 the following demonstration:

11-8 The Law of Cosines

The formula c' - a t + b' is true for every right triangle (Figure 11-6). The distance formula (page 423) enables you to derive a similar formula true for every triangle.

Given any triangle ABC, choose a coordinate Figure 11-6 system with origin at C and angle C in standard position (Figure 11-7). Then, '

vertex B is a units from the origin and has a position angle of 0u; vertex A isb units from the origin and has angle C as position angle; while the distance be-tween A and B is c.

Figure 11-7 c) .. "

Applying the distance formula (page 423). you find

c' - (AB)' - at + b' - 2abcos(C O°)

c' - at + b - 2abcosC. -

FIGURE 2

Because I did not recall the distance formula listed, I checked page 423 and found:

'This leads to the formula for the cosine of the difference of two angles:

cos(a - 13) = cos a cos 3 + sin a sin 0.

This result now enables you to write the formula in the last line of

page 420 in the simpler form(PQ)' - p' + q 3 - 2pq cos(a - 13) for the distance between any points P and Q.

FIGURE 3

What next? I turned to page 420, of course. This is what it said:

By following the steps above, you can derive a formula for the.squarc

of the distance between any points-. P and Q in terms of their respective

position angles, a and 13, and dis-

tances from the origin, p and q "N., (Figure 11-2).

1. P: x, - pcosa;y 1 - p sin a

Q: x, - q con 13; y, - q sin ft Figure 11-2

2. (PQ)' - (x - x,)' + (Ya - - (p cos a - q cos 13)3 + (p sin a - q sin 13)5

- p' cost a - 2pq Cos a Cos ft + q' cos t $

+p'sin'a - 2pq sin a sin ft + q 2 sin* - p'(cos' a + sin' a) + q'(cos' ft + sin' 13)

- 2pq(cos a cos ft + sin a sin 13) - p'I + q' I - 2pq(cosa Cos ft + sin a sin ft) (PQ)' - Ps + q' - 2pq(cos a cos p + sin slap)

FIGURE 4

In despair, I apologized to all the students to whom I had subjected this monstrosity over the years. How could new math, which was to have created enlightenment, have been perverted to such an extent?

Finally, I consulted a more recently author-ized textbook in B.C., Pre-Calculus Mathematics (Crosswhite, Hawkinson, & Sachs, 1976). The authors did not attempt any justification and simply stated the three forms of the law of cosines. Which is worse, an overly academic argument or a "here it is, kids" approach? Surely there must be an acceptable compromise.

What could I do with my own students? In class, I had continually emphasized the im-portance of hands-on activity and discovery, particularly at the secondary school level. From deep in the recesses of my mind, I recalled teaching in Malaysia from a text that had an interesting British perspective on geometry, to wit, we were required to teach it. I recalled doing something with triangle inequalities where sometimes c2 was greater than or less than a2 + b2 , and I started to experiment with triangles of various con-figurations. I also thought about the approach Bill Davidson had used in his master's thesis to teach the Pythagorean theorem (see Vector, Summer 1985, for an example of Davidson's procedure). As a result, I devised a teaching sequence that in-corporated most of the components impor-tant for learning.

When next we met as a class, I gave each stu-dent a ruler, a set square, a protractor, and some squared paper. I asked each person to draw carefully a triangle ABC in which AC = 4 cm, BC = 8 cm, and angle C unique to each student. There were six students in the group that day, and I assigned each an angle: 20°, 40°, 60°, 80°, 100°, or 120°. Then I asked each student to draw a perpen-dicular from A to CB at point D, and then

26

Then I asked the students to determine the area of each square, and I entered their results in a table on the overhead projector. We obtained the following results:

to draw a solid square on each of AB, BD, DC, and CA, and a dashed square on BC.

The diagram produced by students assigned an acute angle was as follows:

I I I I

a

FIGURE 5

The diagram produced by students assigned an obtuse angle was as follows:

n

FIGURE 6

b a CArea of the square on AC AB BD CD

4 8 200 16 20 14 18

4 8 400 16 30 9 25

4 8 600 16 46 4 36 4 8 800 16 66 1 50 4 8 100 0 16 90 1 78

4 8 120 0 16 110 4 100

Some interesting things occurred. Two stu-dents had problems drawing a square that did not have horizontal or vertical sides, such as the square on AB. Although several students were mathematics majors, it was clear that no one had been asked to draw anything accurately for a long time, probably because a sketch usually would have sufficed for mathematical demonstra-tions. Students had no difficulty determining the area of the square o .n AC because the length of AC was given as, 4 cm. However, some students had problems determining the area of the other squares because they wanted to calculate the area from the given information. It took them a while to note that they could measure the side of the square and do a simple calculation. Finally, they were not sure whether it was all right to use a calculator. As we struggled with these problems, I thought if university graduates had such difficulties, how much more serious must the problem be at second-ary school?

After we completed the task, I posed a number of questions. What can you say about the area of the square on AC? What about that on AB? On BD? How large can the squares on AB and BD get? How small? How might the change in area be related to

27

the size of angle C? Tell me about the behavior of the square on DC. The questions established the dynamic relationships among the four squares as angle C varied.

Then I asked students to look at the values we had obtained as shown in the table and to suggest a possible mathematical relation-ship among the four areas. Of course, the students found it much more difficult to focus on four variables at once rather than on one at a time. I needed to insert some prompts at strategic points, but finally one of the students suggested the key relation-ship: AC' is approximately equal to AB + CD' - BD'. We checked it out for the six cases and found perfect agreement in the first case and a maximum of 3 units "error" in the fifth case. A discussion ensued as to whether we should accept the relationship as demon-strated. Why didn't the values agree exactly? Answer: because they do only when pre-sented in textbooks. Moreover, the person who drew the fifth triangle admitted not tak-ing the instructions very seriously, and all agreed they could have been more careful in drawing and measuring.

I added convenient labels for the sides and recast the relationship in a mode that I felt more comfortable with, that is, c2 + m2 = b2 + n2 . I then asked how we could devise a formula for the area of the square on AB knowing only the values of a, b, and c.

As the first step, we began with a formula for c2 in terms of b, n, and rn as follows:

c2 =b2 +n2 —m 2 (1)

To focus more clearly on the geometric inter-pretation of n 2 - m 2 , I isolated the square on BC and "shifted" m 2 to a new location where it cut out a portion of n 2 , as shown in the following diagram.

mD B I . /^ ^^/

C

n2_m2

_

FIGURE 7

Now n2—rn2=a2—rnn—rnn—m2—rn2

= a2 - 2mn - 2m2 = a 2 - 2m(n + rn)

But n + m = a

Therefore n 2 - = a 2 - 2rna

Substituting this into equation (1) led to:

c2 = b2 + a2 - 2ma

And, from the original diagram, m = b cos C.

Therefore c 2 = b 2 + a2 - 2b cos C a

or c2 = a 2 + b2 - 2ab cos C QED

One student exclaimed: "Oh! It's the cosine law."

Discussion For several reasons, I believe the approach described above is superior to any textbook approach.

The activity is student-centred. Each student contributes information necessary to see the whole picture developing. Students do

28

things, rather than just copy down the teacher's notes or the textbook demonstra-tion. They use instruments to construct figures; they measure angles and segments; they calculate areas; they investigate possi-ble relationships; and, most important, they talk mathematics.

The procedure also integrates ideas in mathe-matics. Students use geometry, trigonom-etry, algebra, and arithmetic. Moreover, the true notion of a function underlies the pro-cess, that is, c2 = f(C). Too often, "laws" such as the cosine law and sine law are seen only as formulae from which to determine the specific measure of an angle or side in a given triangle. Such an approach is essen-tially anti-algebraic because the dynamic relationship among elements is obscured.

The question really becomes Why do we teach the cosine law at all? Nobody uses it for computations. In the old days, when logarithms were used to determine missing parts of triangles, say in surveying, the most appropriate formula was

"A—B\ (a —b \ (A+B\ tan 2 ) = "a +b ) tan 2 )

This results in a value for A - B, and know-ing that A + B = 180 - C, the two equa-tions can be solved for A and B.

Now, perhaps the best application of the cosine law is as an exercise in programming computer solutions for triangles. But what is the mathematical value to students in learning the cosine law? It must lie in the opportunity that is presented for students to

explore, speculate, and validate hypotheses; that is, the mathematical process, not the product, should be valued.

Finally, the method presented here may be used as a source of further questions at a high cognitive level. What happens when angle C is a right angle? Why wasn't the choice of dimensions 4 cm, 8 cm, and 600 a good one? Would the same formula develop if the perpendicular had been drawn from B to AC? Could you arrive at the same formula by holding one side and the angle constant while varying the length of the other side? Can you see the present pro-cedure hidden in the method used by Oliver, Winters, & Campbell?

So there's my sequence, colleagues. We took an hour to work our way through it. Perhaps it could be done over two periods in secondary school. Too much time, you might say, just to develop the cosine law. Perhaps, if your aim is to have students memorize the cosine law. But consider all the other learning going on. If you do try it, please let me know your reaction and your students' comments.

References Crosswhite, F.J., Hawkinson, L.D., &

Sachs, L. (1976). Pre-calculus mathematics. Agincourt, Ont.: Charles E. Merrill.

Dolciani, M.P., Berman, S.L., & Wooton, W. (1963). Modern algebra and trigonom-etry. Boston: Houghton Mifflin.

Oliver, W.J., Winters, P.F., & Campbell, J.E. (1941). A first course in trigonometry. Toronto: School Aids and Text Book Pub-lishing.

29

All Parabolas Are Similar

David Wheeler

David Wheeler is a professor in Concordia University's Mathematics Department, and editor of For the Learning of Mathematics.

Jake Penner's letter in Vector 27 (2) prompted me to write. I'm no expert, and there is nothing original in what follows, but I, like Jake and is colleagues, find the conic sections vastly more entertaining than many textbook treatments suggest. So I have put together a few miscellaneous items hoping that some of them will be fresh to some readers.

A little history The great Greek geometer Apollonius (third century B.C.) classified the conic sections by considering a general circular cone and slic-ing it by different planes. He was the first known person to consider a complete "double" infinite cone—so he was aware that a hyperbola has two branches—and he gave the sections the names we still use. But more than a hundred years earlier, Meneachmus identified three different types of section of a (single) cone sliced by a plane perpen -dicular to an edge (generator). The three types corresponded to conditions on the angle at the vertex of the cone: less than a right angle (making the cross-section an ellipse), equal to a right angle (parabola), and greater than a right angle (one branch of a hyperbola).

The focus-directrix-eccentricity relation was made explicit by Pappus (third century A.D.), though it was probably known earlier. Diodes, a contemporary of Apollo-nius, proved the "burning glass" property of the parabola: rays parallel to the axis

of a paraboloid are reflected through the focus.

The first major applications of the conic sec-tions were made by Kepler (1571-1661), who identified the orbit of Mars as an ellipse with the sun at a focus; by Galileo (1564-1642), who showed that the path of a projectile (neglecting air resistance) is a parabola; and by Gregory (1638-1675), who designed the first reflecting telescope, later constructed by Newton.

New directions to the study of the conics were initiated by Desargues (1591-1661), Pascal, etc.; who identified them as projec-tions of a circle, and by Descartes (1596-1658), Fermat, etc., who developed the methods of analytic geometry in which the conics appear as the plane curves correspond-ing to algebraic equations of the second degree in two variables. Both approaches brought powerful and unifying techniques to bear, and it seems unfortunate that most elementary teaching of the conics in schools and colleges has exclusively adopted the analytic algebraic approach and totally neglected the intuitive geometric approach.

The study of the conics and the quadrics (their three-dimensional counterparts) con-tinued sporadically for another 200 years. Too much happened during this period to be noted here, even in summary, but one further item may be added because of its relevance to the origins of the conics as sec-tions of a cone.

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FIGURE 1

The beautiful result published in 1822 by the Belgian mathematician Dandelin showed that a sphere inscribed in a cone so that it touches a plane that sections the cone actually touches the plane at a focus of the conic section and, further, that the directrix corresponding to this focus is the line of in-tersection of the section plane and the plane through the circle of contact of the cone and the sphere. This result has such a flavor of Greek geometry that it is difficult to believe the Greeks didn't know it.

Penner's questions All squares are similar, all right isosceles triangles are similar, all regular hexagons

are similar, all circles are similar . . . all parabolas are similar. But to know whether the last properly belongs to the list requires deciding what we are going to mean by say-ing two unbounded figures are similar. In the case of bounded figures, we have a percep-tual criterion: "They are exactly the same shape." We can't use this naive test for un-bounded figures, since, for example, one parabola can be "blunt" at the vertex and another, "pointed."

An alternative approach is to think of similarity in terms of projection—in the sense of a movie projector sending a similar image of certain marks on celluloid onto a screen. Out of this mental model, we can abstract the key ideas of magnification and scale factor relative to a "centre" (the lamp filament). This idea gives a test we can apply to bounded or unbounded figures. If we have two parabolas in the same plane, we can shift them around until the vertices and axes coincide. With the usual co-ordinate conventions, we can then give them the equations y k 1x2 and y = k2 X2.

FIGURE 2

It is easy to show that if a variable ray from O intersects the parabolas at P 1 and P2.

then OP, :0P 1 = k1 :k2 . Since the cor-respondence between points of the two

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parabolas is one-to-one, and since k 1 :k2 is independent of the position of the ray, we can say that the parabola y = k2

X2 is a magnification with respect to 0 of y = k1x2

by the scale factor k 1 /k2 . Hence, the two parabolas (and so all parabolas) are similar.

The same approach can be used to show that not all ellipses are similar, nor are all hyper-bolas, but I find equally convincing an argu-ment based on the observation that one can identify a subset of congruent figures from a set of similar figures by specifying the linear measure of just one associated quan-tity. For example, all circles with radius 5 cm are congruent, all regular hexagons with side 8 cm are congruent, and so on. In the same way, a subset of congruent parabolas can be identified by specifying, say, the measure of the latus rectum (which is equivalent to specifying the a in y2 = 4ax. In the case of ellipses, two linear measures must be given to ensure congruence: the lengths of the major and minor axes, say. So not all ellipses are similar. And ditto for hyperbolas.

Appollonius proved (in his definition of similarity, more complicated than the one given here) that parallel sections of a cone always produce similar figures. I think this result is intuitive—at least I find it so when I visualize parallel elliptical sections. Visualizing parallel hyperbolic sections isn't so easy, for the reason brought out in Penner's question about the symmetry of the hyperbola: a plane that produces a hyper-bolic cross-section doesn't necessarily cut the two nappes of the cone at equal distances from the vertex. So there is no overall sym-metry of the total configuration about a plane through the vertex of the cone (con-trary to what one would perhaps expect). I know no simple way to make the symmetry of the hyperbola obvious at a visual level. The symmetry has to be established by argu-ment, not by inspection.

The climax of Book VI of Apollonius's study of the conics is the theorem that it is possi-ble to cut from any given cone a section con-gruent to any given conic. So it doesn't "matter" what shape the cone is: any double cone (provided it is not degenerate) can be sectioned to give every ellipse, every parabola, and every hyperbola.

Whether these remarks answer some of Jake Penner's questions is for him to say. That his questions were about the intuitive geometry of the conic sections triggered my responding. I think that part of a teacher's job is to help students clarify, sharpen, and educate their intuition. How can we help them do that if we always try to replace in-tuition by something else? Some of the following items can work on intuition, too, though they leave the cone behind.

String constructions The well-known "gardener's method" of con-structing an ellipse with two fixed stakes and a fixed length of rope needn't be given here, but comparable constructions for the other conics are not so familiar. Kepler worried away at this problem, and in 1604, he gave his solutions. His paper contains some other discoveries and nice insights: for example, the parabola can be conceived as having one focus "at infinity."

The following diagrams suggest a pair of possible string methods, though others ex-ist. (None is easy to make work well.)

B

A

FIGURE 3

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FIGURE 4

In each figure, BPT is a taut non-elastic string. Its length is less than AB in the first case and equal to AB in the second. T is a fixed point (nail/drawing pin), and P is a moving pencil point. In the first figure, AB is a rigid rod (e.g., a ruler) fixed at A but free to rotate. In the second figure, the T-square must slide freely against the edge of the drawing board. The addition of a ring or sleeve that will hold P against the rod or the T-square improves the chances of suc-cessful drawing.

The ellipse as a glissette and a roulette A rigid rod slides with its ends on two perpendicular tracks. A general point P of the rod will describe an ellipse. (Why?)

FIGURE 5

Consider the effect of various positions of P. What happens if P is at the centre of AB? at A? Consider symmetrical positions of P (at the same distance from opposite ends of the rod). Suppose P could be fixed somehow on the extension of AB while AB continues to slide in the same way as before. Does P still trace an ellipse? (Yes.)

The system described is the basis of the ellipsograph or Archimedean trammel for drawing ellipses.

What is the envelope of the sliding rod; that is, what curve is touched tangentially by every possible position of AB? (An astroid.) The envelope touches each member of the family of ellipses generated by the interior points of AB.

Will P trace an ellipse if the tracks are not at right angles? (Intuitively unlikely, but actually "yes.")

Form a crossed parallelogram with two pairs of loosely jointed rods or strips.

AFIGURE 6

Hold the side AB fixed, and move CD about. Then the intersection point X describes (part of) an ellipse. (Why?)

Its foci are A and B. (Why?)

Since X traces an ellipse, with A and B as foci, by symmetry, it must at the same time trace an ellipse with C and D as foci. So the

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motion of the crossed parallelogram can be seen as implying a motion in which one ellipse rolls on an equal fixed ellipse.

A circle rolls inside a circle of twice the diametef. A point fixed on the circumference of the rolling circle traces a line segment: a diameter of the large circle. (Why?) What does a point fixed in the interior of the roll-ing circle trace? (How do you know?)

Envelopes The first two constructions develop the central conics (ellipse and hyperbola) from a circle, and the parabola from a straight line. It seems highly appropriate that the conics, intimately related to circles and straight lines in the context of the cone, can also be wedded to them in the plane.

In the first type of construction, the conics are generated as "pedals" of a circle or a line. Take a circle C and a point P inside it. Draw a ray from P crossing the circumference of the circle. At the point of intersection, draw a line at right angles to the ray. Repeat many times with different rays. The construction can be carried out with the help of a set-square, placing the right-angled vertex on the circumference of C while one of the short sides touches the point P, and drawing along the other short side. The set of drawn lines envelopes an ellipse.

FIGURE 7

Repeating the construction with P outside C gives a set of lines enveloping a hyperbola. To obtain the parabola, replace C by a straight line 1 (a circle of infinite radius, perhaps).

The second construction is a variant of the first. Draw the circle C (or the line 1), mark the point P on a sheet of waxed paper and fold the paper so that the point P falls over a point of the circumference (or the line). Crease the fold well. Repeat many times, folding so that P falls over more or less equally spaced points on the circumference of C (or the line 1).

A parabola can also be produced as an envelope starting from two line segments having a common endpoint. Divide each segment into the same number of equal parts, and number the division points on the two segments, one in the reverse direction from the other. Join pairs of division points bearing the same number.

FIGURE 8

This construction can readily be cross-stitched on card with colored thread, and in this form, it is sometimes found in books for elementary school children as an example of curve-stitching (an activity invented by Mary Boole, wife of the famous logician).

In spite of a hint of asymptotes, the stitches envelope a parabola, not a hyperbola. Try-ing to prove this is an interesting exercise.

34

A

Another generation Given a circle C and a point P inside it, the centre of a variable circle touching C and passing through P traces an ellipse. (Why?) As a good visual exercise, try to see in the mind's eye the ellipse being generated. A film of the movement is available: see "A few resources" at the end of this article.

FIGURE 9

The point P is one focus, and the centre 0 of C is the other. Consider the effect on the ellipse of changing the position of P while leaving C fixed.

If, on the other hand, P and the point A of C nearest to P are held fixed and the circle C is gradually blown up as a balloon is, the ellipse gets longer and longer as 0 recedes from A until (we can imagine) 0 eventually "goes to infinity." C is now a circle of in-finite radius (so all that can be seen of it is a straight line I through A perpendicular to AP) and the trace of the centre of the variable circle (which now touches I while passing through P) is a parabola.

And what about a hyperbola? Well, the cen-tre of a circle of infinite radius is just as legitimately to the left of A as to the right of it, so 0 can now approach A from the left. As it does, the line 1 curves again, con-cave on the left now as it bends to become

part of the circumference of the circle C cen-tre 0. But P is now outside C, and the path of the centre of the variable circle touching C and passing through P is a hyperbola.

Can both branches of the hyperbola be generated by this means? (Yes.)

A few resources Lockwood, E.H. (1961). A book of curves.

Cambridge University Press. Yates, R. C. (1974). Curves and their proper-

ties. Classics in Mathematics Series, 4, National Council of Teachers of Mathematics. (First published in 1952. The NCTM book is a republication of the second 1959 edition.)

The above are both excellent sources of information about plane curves. Principal properties are presented, but not proved; various construction methods are illustrated. Lockwood includes many practical sugges-tions for drawing exercises.

Also see Encyclopaedia Britannica (14th edi-tion) under "Curves, special." Some texts on engineering graphics may be worth examin-ing for drawing methods.

Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination. Chelsea Publishing Co.

Whicher, 0. (1971). Projective geometry. Rudolf Steiner Press.

The first of the above books is a classic: the fruit of a lecture series by the first author and an exhibition organized by the second. Too comprehensive and condensed to be al-together easy to read, it is marvellous for reminding readers of the scope of geometry and of the power of geometrical intuition. The second book applies a strong pedagog-ical approach to the task of "pulling down" projective geometry from the abstract regions it normally occupies. Readers who

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can surrender to the author's style, or at least suspend judgment about it, will find the sub-ject matter presented with considerable skills and imagination. The illustrations make one's fingers itch to pick up a straight edge and pencil and experiment for oneself.

Coolidge, J.L. (1968). The history of the conic sections and quadric surfaces. Dover Publications. (A reprint of the original 1945 publication by Oxford University Press.)

The above book probably contains more in-formation than most readers will need or want, but it demonstrates the' persistent attraction that the conic sections have had for mathematicians ever since they were first discovered.

Gattegno, C. Common generation of conics. Animated Geometry New Series, No. 3, 16 mm, color. Educational Solutions Inc., 95 University Place, New York, NY 10003-4555. (Based on the original b/w film designed and animated by J.L. Nicolet.)

Fletcher, T.J. Four-line conics. 16 mm, color. National Film Board of Canada.

The first of the above two films shows how the conics emerge as the paths traced by the centre of a variable circle constrained to touch a circle (or a line) and pass through a specified point. See "Another generation" earlier in this article. The second film ex-plores poles and polars with respect to the family of conics touching four given lines. (A conic is determined by five distinct points on it or by five distinct lines that it touches. If one of these constraints is dropped, an in-finite number of conics that satisfy the rest can be found. They constitute a "family with one degree of freedom.") Fletcher's film shows how the medium can be used to com-press and co-ordinate a large number of "theorems" into a comparatively short visual experience.

Whether the prodigious capabilities of the computer will lead anyone to develop im-aginative new approaches to the conics as the film medium has done is an open question.

36

Summary Report on the Draft Mathematics Curriculum

Valerie Johnson

MATHEMATICS ISSUES

Valerie Johnson is a research co-ordinator in the Ministry of Education's Curriculum Development Branch. This summary is an edited version of two separate reports

she prepared for the Mathematics Revision Committee.

Introduction In early fall 1985, a draft of the Mathematics Curriculum Guide 1-12 was circulated throughout the province. The document was distributed to all schools (including indepen-dent schools), district offices, and post-secondary institutions, along with question-naires soliciting responses from B.C. educators.

The following report represents a summary of the data accumulated through the questionnaire.

General Information A total of 806 questionnaires or submissions related to the elementary grades were com-pleted and returned, representing input from 1142 individuals. Of those questionnaires, 77 were returned by groups representing schools, districts, and committees. Coquit-lam, District 43, for example, compiled data representing 60 individuals in the district.

A total of 212 questionnaires or submis-sions related to the secondary grades were

completed and returned, representing input from 403 individuals. Of those responses, 39 were submitted by groups representing schools, districts, and committees. North Vancouver, District 44, for example, submit-ted a report representing input from 30 teachers.

Where actual numbers of teachers partici-pating were not given, a weighting, based on the information provided (for example, the response represents the primary teachers in a small school) was assigned. The weighting became particularly important when it was used to determine the response to the major changes as indicated in the opinion scale.

Information identifying the respondents was optional, and many questionnaires were received with no personal information in-cluded. Where the identification was pro-vided, the information is reported here. Distribution by grade level as indicated in Table 1 was relatively even, with each grade well represented.

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TABLE 1 Distribution of Respondents by Grade

Grade Number of teachers Grade Number of Teachers

K 20 Administrators, learning 1 174 resource teachers, others 8 2 199 8 284 3 174 9 298 4 145 10 316 5 150 11 264 6 155 12 226 7 138 Administrators 23

TABLE 2 Comments by Elementary Respondents on Format of Draft

Category Individual Group Comment Responses Responses

No comment 321 16

No change 164 20

Scope and sequence Delineate by grade (grade-specific wall charts, checklists, one-

page overview). 70 15 Set up as for Grade 8. 26 7 Include Kindergarten. 7 Show scope and sequence for K-8 and 7-12. 3 2 Indicate more clearly where skill or concept is meant to be

introduced, mastered, or maintained (reviewed) perhaps by codes or colors. 17 7

Outline areas as core, optional, or enrichment. 6 2 Only outline strands, not individual ILOs in detail for all

grades. 1 Show scope and sequence by topic. 1 Provide flowchart to show skill development and prerequisites. 2 Reference scope and sequence to ILOs. 2

Language Include glossary. 9 9 Simplify language. 12 2

Examples Include more examples (at least one per ILO). 34 7 Explain use of examples (meant to be mastered? by all?). 2 Include examples for lower limit as well. 2

Text Text selection must effect a match with curriculum including

philosophy, problem-solving approach, etc. 24 15 Reference text to guide. 2 1

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Format of Draft Two items in the questionnaire focussed at-tention on the format of the Response Draft. There were open-ended questions, requiring some manual analysis of the comments made. In order to do this, the following pro-cedure was employed: • Comments were recorded on separate sheets of paper.

• Where comments were the same or essen-tially the same, the numbers of occurrences were tabulated. • Through a process of synthesis, categories of comments were generated.

These comments are reported in Tables 2 and 3, by categories, with an attempt to show the numbers of occurrences.

TABLE 3 Comments by Secondary Respondents on Format of Draft

Frequency of Frequency of Category Individual Group Comment Response Response

No comment 97 14

No change 23 8

Time, order, expectations Recommend time allocations. 9 4 Include expected level of mastery. 5 1 Suggest order for covering topics. 4 Generally, increase time allotted to mathematics. 4

Examples Provide more than one example. 5 1 Include answers to examples. 1 1 Examples should represent average difficulty. 2 Clarify use of examples, i.e., are they intended to

be mastered? 2

Textbooks Find one to match curriculum. 6 6 Reference more than one text. 4 2 Provide answers in text. 2 Ensure accuracy of reference. 4 Provide separate workbook. 2 There should be a teacher's guide with answers to

student text. 1 Publish guide as a text, adding more examples. 1

Scope and sequence Include a one-page overview for each grade. 4 Provide a wall chart for each grade. 1 Flowchart each strand through grades. 1 Provide overview charts for K-8 and 7-12. 1

General Provide index, summary, table of contents. 2 1 Identify software. 1 Incorporate cross references if present outcome is prerequisite for a subsequent grade. 1

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By far the greatest occurrence was indicated by the "no change" comment. Elaboration on this point included words of praise similar to the following examples: "well organized," "clear and simplified," "such an improve-ment." If one can assume that the "no com-ment" is also an indication that respondents like the draft as is, then it appears that there is consensus. There are some recommenda-tions in terms of additions, which may make the document even more useful.

Major Changes The response draft outlined 16 major changes in the revised curriculum. Respondents were asked to indicate their reaction to each change on a Likert-type scale, with five positions ranging from strongly disagree (1) to strongly agree (5). Changes outlined were under the following headings: 1. Curriculum Guide, 2. Early Number Concepts, 3. Conceptual Develop-ment, 4. Enrichment, 5. Problem Solving, 6.

Geometry, 7. Data Anaylsis, 8. Mastery of Complex Numerical Algorithms, 9. Com-mon and Decimal Fractions, 10(a). Mathe-matics 9/10, 10(b). General Mathematics 9/10, 11. Mathematics 11, 12. Introductory Mathematics 11, 13. Senior Courses, 14. Calculators, 15. Computers, and 16. Resource Books.

For each major change outlined, responses were recorded to determine frequencies of indications by position on the scale. This information is illustrated by graph in Figure 1 for elementary respondents and Figure 2 for secondary respondents. The item numbers (16 changes) are indicated on the horizontal axis. In each case, the extreme left position illustrates the "No Response" for the item. The numbers of responses in each position of the five-point scale or a "no response" are recorded for each by a bar corresponding to numbers on the vertical axis.

40

800

700

600

500

400

300

200

100

Curriculum Early Concept Enrichment Problem. Geometry Data Mastery of Fractions Guide

Number Development Solving Analysis Algorithms Concept

800

700

600

500

400

300

200

100

Mathematics General Mathematics Introductory Senior Calculators Computers Resource 10 Mathematics 10 11 Mathematics 11 Courses Guides

U No response El Strongly disagree n Disagree 0 0/A Agree Strongly agree

FIGURE 1 Opinions Expressed on the Major Changes (Elementary Respondents)

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240

210

180

150

120

90

60

30

0

240

210

180

150

120

90

60

30

Number of Responses (N = 384) I

Curriculum Early Concept Enrichment Problem- Geometry Data Mastery of Fractions Guide Number Development Solving Analysis Algorithms

Concept

Mathematics General Mathematics Introductory Senior Calculators Computers Resource 10 Mathematics 10 11 Mathematics 11 Courses Guides

• No response E3 Strongly disagree Disagree 0 0/A D Agree Strongly agree

FIGURE 2 Opinions Expressed on the Major Changes (Secondary Respondents)

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Concerns While the overall response to the draft was positive, some general concerns were raised. Concerns expressed by elementary respondents are summarized in Table 4, and those by secondary respondents, in Table 5.

TABLE 4 General Concerns about the Mathematics Curriculum (1-7)

(Note: Positive comments are NOT included here.)

Category Comments

MANIPULATIVES If the curriculum is to rely on the use of manipulatives, they must be supplied in schools.

Teachers should not be asked to "scrounge" as was men-tioned on the videotape.

Advice must be given on the best process to follow on bridg-ing the gap from manipulative to abstract work.

IMPLEMENTATION Much in-service education will need to be provided to teachers if implementation is to be successful.

Introducing new areas such as data analysis, problem solv-ing, geometry, and calculators means that extensive in-service education must be provided.

Resource materials for problem solving, geometry, and calculators must be provided.

The program should be reassessed after one year of implementation.

EXPECTATIONS Grade 3 has too much content. Too much is expected at primary level. The heavy content will mean less opportunity to work on

building a positive attitude and using a problem-solving approach.

Multiplication /division skill expectations are too easy for intermediate.

Content is too watered down; expectations could be higher. The additional expectations indicate that time allocations for

math should be re-examined.

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TABLE 5 General Concerns about the Mathematics Curriculum (7-12)

(Note: Positive comments are NOT included here.)

CommentFrequency

by individualsFrequency by groups

Too much content. 20 9 Need teacher in-service. 5 4 Too rigorous. 6 2 Develop a General Math 8. 8 2 Include a Consumer Math strand. 3 Develop a General Math 12. 2 Address metrication. 0 1 Include word problems. 3 1 Need provincial examinations. 4 Include more algebra in Grades 9 and 10. 1 A second evaluation and possible additional revision

will need to be considered after a year of using the program. 1 1

Topics: New topics are at the expense of basic principles and

problem solving. 1 New topics are superfluous to the evolution of mathe-

matical concepts. 1 Appears to be a collection of isolated topics. 1

Approach is not one that leads to understanding, but is indoctrination. 1

Include: —rational expressions before Grade 8 1 —ratio and proportion 1 —SAS, ASA, etc., in Grade 8 not Grade 9 1 —variety of uses of rational numbers 1 —graphing ordered pairs and simple linear equations in Grade 8 1 —work on radicals in Grade 9 1

Calculus should not be introduced. It is too little to help in post-secondary and will result in poor coverage of traditional topics. 1

Conclusions The data clearly indicate that respondents are generally happy with the work of the committee to date. Although there are sugges-tions for change, teachers see the revised cur-riculum, as outlined, as a vast improvement.

The data formed the basis for a number of recommendations the curriculum re-vision committees subsequently considered during deliberations about appropriate revisions.

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MISCELLANEOUS

The Practicum Experience

Adrienne Muirhead

Adrienne Muirhead is a fifth-year education student at the University of British Columbia.

I'm standing in front of the class, just before the period begins. The students are waiting quietly, and my sponsor teacher is sitting at the back of the room, pen poised and ready. The bell rings, and everyone is watching me. Suddenly I realize that I cannot remember what I was supposed to teach. To admit this to my sponsor teacher, however, would be admitting a lack of preparation. So I stand there in front of the class without doing or saying anything. They all keep silently waiting. I discover, with relief, that I have been dreaming.

I have these nightmares before, during, and after each practicum, and I have learned that such dreams are common among student teachers. Fortunately, I have never en-countered real situations like them. Instead, my practica have been interesting, exciting, and rewarding.

I transferred into the Education Faculty at the University of British Columbia in my third year, with the goal of becoming a mathematics teacher. My first practicurn was in May of that year, at Stanley Humphries Secondary School in Castlegar. I was in-formed that I was to start teaching a Grade 9 class on the second day. I was so nervous that I could neither eat nor sleep. The next day, I managed, with shaking hands and a

monotone voice, to deliver my lesson. I put only one student to sleep. Fortunately, I had two excellent, patient sponsor teachers, Harry Plotnikoff and Peter Young, who taught me the value of using questioning to increase class involvement. Using such techniques also prevented students' falling asleep.

I was gradually introduced to more and more classes, and by the end of that first week, I had stopped shaking. As time passed, my confidence grew, and my inflec-tion, questioning techniques, and lesson planning began to improve. My classes were not discipline problems, so I could concen-trate on learning teaching strategies.

Not until my second practicum, in Pentic-ton a year later, did I encounter discipline problems. By then, my inflection was ex-cellent and my questioning techniques were improving. No one fell asleep in my first class in Penticton. Instead, one of my Grade 8 classes tested me. Unfortunately, I had not been taught any strategies for handling discipline problems. Mr. Bobbitt, my spon-sor teacher, pointed out that my starting lessons without having the class's attention encouraged the students to ignore me. I worked on this for the three weeks, but my lack of sternness occasionally stood in my

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way. Even the students, who evaluated me on my last day, noted that I should be "meaner." My experience with that class taught me the importance of setting reasonable rules and enforcing them con-sistently, with consequences.

One of my most positive experiences also occurred in Penticton. A Grade 10 class was responsive and seemed really to enjoy geometry. I have yet to find a class that shows as much interest in mathematics. On my last day, the boys came to shake my hand and thank me; the girls gave me a card they had all signed. Moments like that make teaching rewarding.

After the two practica, I had learned a lot about teaching. I had learned the importance of pacing lessons, of classroom management, and of questioning techiques. I also realized that I had much more to learn. During my third and fourth practica, at Sutherland Secondary School in North Vancouver, my skills improved further. Under the supervi-sion of John Klassen and Ian deGroot, I learned to teach with several texts rather than to follow just one. I explored many ways to present material to facilitate students' comprehension.

At Sutherland, I was introduced to the use of the overhead projector. Initially I had

some reservations, but I found that the con-tinual eye contact I had with the class was beneficial. I was aware of what was happen-ing in the class, with both discipline and comprehension.

Another thing I learned at Sutherland was the value of continuing my own education. I was able to attend two professional days. On the first one, I was introduced to the pro-posed new British Columbia mathematics curriculum guide. -On the second profes-sional day, I attended the VANSTAT Statistics Workshop at Capilano College, where I learned novel techniques for teaching data analysis to secondary school students. The speakers were entertaining, and the wine-and-cheese party that followed gave me the.opportunity to speak to math teachers from all over the province.

I hope this overview of my thoughts about student teaching and the relating of my experiences will encourage other student teachers as they prepare to stand in front of a class for the first time. I am heading into my last practicum with great confidence in my ability and a feeling of excitement. I will sleep well and eat heartily this time, but I know that I will be nervous at the beginning of my first class on my first morning, if only for a few moments. No amount of practice could change that.

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NC

Report on the 1986 Annual Meeting of the NCTM

"Better Teaching, Better Mathematics: The Perfect Mix for '86"

John Kiassen

John Klassen is president of the British Columbia Association of Mathematics Teachers.

Washington, D.C., hosted a meeting of more than six thousand mathematics teachers. The capital was the perfect lady, complemented by blossoming cherry trees and glorious sunshine. The temperature climbed to 30°C, and I can only give the highest grades to the many sights of Washington. It is indeed a great city, and its citizens are justifiably proud.

I was reminded once again of the Canadian element in the NCTM family. The American and Canadian flags both graced the podium at the major presentations. The Canadian caucus on Wednesday was an opportunity to meet with our provincial counterparts across the country.

One general concern was the introduction of province-wide examinations. Few provinces will escape this experience over the next few years. The caucus would like to see a Canada-wide recognition of excellent mathe-matics teachers similar to the Presidential Awards in the United States. We will try to enlist the help of the provincial ministers in this matter.

The theme for the annual meeting was a departure from those of previous meetings. The concerted effort to remind all par-ticipants of the value and characteristics of good teaching was refreshing. Many of us have had direct or indirect exposure to the research on effective teaching. A number of sessions reviewed the findings, but it was valuable to discuss them in the context of teaching mathematics.

"A teacher who can be replaced by a com-puter should be." Joe Crosswhite, the past president of NCTM, expressed these sentiments in his address. The sensitivity of a teacher cannot be replaced by a machine, and education is not an assembly where production is measured by standardized tests. Rather we should more often ask students the Why and How questions, i.e., the process questions, not just the product questions. This thought was echoed by a number of speakers.

The results • of the Second International Study continue to be discussed. Dr. David

47

Robitaille reported on the British Columbia data in the spring issue of Vector. One general response seems to be a proposal for a more rigorous curriculum. Zalman Usiskin's comment was: "By the end of Grade 6, all students should have their arithmetic skills down pat."

His subsequent suggestion is to introduce a heavy algebra component in Grade 8.

With regard to curriculum development, I sense that a unified mathematics program, an integration of geometry, measurement, statistics, arithmetic, and algebra, will be a serious consideration for many states over the next few years. New York State has already adopted the program for the present model of Algebra I. Geometry and Algebra II will not accommodate, such additional topics as statistics (see Soundoff in Mathematics Teacher, April 1986). Zalman Usiskin, at the University of Chicago, has certainly moved away from the traditional model.

A number of speakers cited the research that showed an overwhelming tendency for classroom teachers to review homework and follow that with assignment of seat work. Where does the teaching and development of new concepts fit into such lessons? There apparently was very little evidence of actual teaching. This is disturbing news, especially since about half the class time should be devoted to the delivery of new material. A second component of good teaching high-lighted throughout was continuous review. Again this is not a revelation, but when I feel pressed for time I give review short shrift. Regular weekly attention to reviewing previous concepts is most important, and perhaps a designated day or time slot would help us to remember.

"Not good manners to have the radical in the denominator." The above quote reminded me of the British

Columbia curriculum revision and the sug-gested provincial changes. We seem to be moving away from the insistence on radicals as demonstrated in the Grade 12 examina-tions.

A research session on teaching secondary mathematics examined the understanding of the variable in equation solving. Engineer-ing students were asked to write an equation for this statement: There are six times as many students as professors. Only 63% of the students gave a correct response. The analysis indicated that the respondents tied the variable to the student and not the number of students, and their equation was 6S = P.

The same session compared the teaching of algebraic equations by the reversal method and the compensation method.

Reversal5 x-2----- = 2 3

What must I subtract from 5 to leave 2?

x-2 3

What divided by 3 gives 3 as an answer?

x-2 9

What number take away 2 leaves 97

x=11

Compensation If you act on one side, compensate by doing the same to the other side.

5—=2

15 - (x-2) = 6

15 - x + 2 = 6

17 - x = 6

-x = -11

X = 11

48

At the concrete operational stage for a stu-dent, using the reversal method has an advantage over using the compensation technique. Once the student is at the formal operational stage, there is no advantage to either method. These ideas on research in teaching secondary mathematics are from the book Research within reach: secondary school mathematics (NCTM). A second book distributed by NCTM proved to be very popular: Classroom ideas from research in secondary school mathematics.

Bii[i Red r

A spoon of red solution is poured into the blue beaker. Then a spoon of the blue-red solution is poured into the red solution.

Is there more Blue in the Red or moreRed in the Blue?

Blue (4 spoons) (1) 4B (2) 4B + 1R (3) 4B + 1R - -- (4B + 1R)

3 + B +*R Red (4 spoons) (1) 4R (2) 3R (3) 3R + *(4B + 1R)

3 * R +*B

The colored solutions are effective on the overhead projector.

Max Sobel, former president of NCTM, reviewed with us the past trends of teaching mathematics and future possibilities. He reminded us of the new mathematics in the '60s, back to the basics in the '70s, and prob-lem solving in the '80s. We must remember that a balance should be struck among the various influences and that an overzealous response can do serious damage. Sobel's thought for the '90s is that we should teach mathematics better: use the technology to teach problem solving, examine the research on effective instruction, adjust the cur-riculum to include such topics as statistics, and, above all, clearly model the joy of teaching mathematics.

He gave us this problem from a Grade 3 textbook:

3 0 4 - 5 7 0 3 -k 2 5 0 1 - 5 4 0 6 - ? 8 0 3 - 7

The conference was a tremendous success, and I thank all of you for supporting my attending this excellent meeting. Do make an effort to go to the 65th Annual Meeting in Anaheim, California, April 8-11, 1987.

49

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50

and Scholarship Examinations, Report on the Algebra 12 Provincial

January 1986 Walter Hamm

Walter Hamm is a mathematics teacher at Windsor Secondary School, in North Vancouver, and he chaired the Marking Committee

of the January 1986 algebra exams.

A recent edition of Vector (27, 2) contained an excellent report on the marking of the June 1985 Provincial Algebra 12 exams. The report gave general comments on the mark-ing session, as well as detailed comments on student responses to open-ended questions.

Many Algebra 12 teachers are finding the report valuable, particularly while prepar-ing their students for the June final. In response to requests for a similar coverage of the January 1986 exams, I offer the following summary of students' specific strengths and weaknesses on open-ended questions.

Algebra 12 Provincial Examination, January 1986 Part B: Written Response Questions

1. Determine "m" so that x-2 is a factor of 2x + 3x2 - mx + 10.

Strengths: • Correct use of +2 as divisor in syn-thetic division (or in substitution). • Correct equating of remainder to zero.

Weaknesses: • Many students solved for -m, getting -19 for the answer.

2. Prove the following identity: 2 cot 20 = cot 0 - tan 0

Strengths: • Correct substitution from double-angle to single-angle identity. • Correct manipulation of trigonometric fractions;

e.g., cos0 - sinO - cos20—sin20 sinO cos0 sin0 cosO

Weaknesses: • Poor organization and layout of work. • Negligence of use of brackets. • Omission of angle symbol. • A few students attempted to "cross-multiply."

3. Determine the real value for x and y, given that

____ = x + yi (i =

51

Strengths: • Most students recognized need for com-plex conjugate for simplification of left side.

Weaknesses: • Many students did not comprehend that the definition of equality in the set of complex numbers was being tested;

i.e., If a + bi = x + yi, then a=x and b=y

4. On the grid provided, sketch a graph of the following equation.

16x2 = 144 - 9y2

Strengths: • Most students recognized the equation as that of an ellipse. Since no detail was asked for, markers gave full marks for any reasonable "sketch" having correct intercepts and shape.

5. Find two numbers such that their product is 2 and the sum of their reciprocals is

Strengths: Setting up a system of equations.

Weaknesses: • Weak in routine algebraic skills, most notably, in converting, the system of equations into a quadratic equation;

1 e.g., 1 11 x y 6

xy=2—x=2—y or

6. A plane headed 600 east of north at 650 km/h encounters a wind of 80 km/h blowing from the east. The plane's resul-tant speed is slowed to 582 km/h due to the effect of the wind. Determine the resultant direction of the plane to the

nearest degree. Include a clearly labelled diagram.

Strengths: • Ability to use Law of Sines or Law of Cosine. • Ability to use calculator for inverse trig functions. • Vector addition was good.

Weaknesses: • Weak and varied use of vector nota-tion. • Some East-West directional co- nfusion. • Inability to recognize the supplemen-tary (obtuse) angle where using the Law of Sines with side 650. • Inability to make a "clearly labelled vector diagram" approximately to scale. (One full mark was awarded for a dia-gram showing angles, sides, and vector arrow heads.)

7. The number of bacteria in a growing col-ony is a function of time. If a colony begins with n bacteria, and y represents the number that will be present after time t, then

y = n(10)"

where the constant k depends on the type of bacteria.

For a certain strain of bacteria, k = 0.32 when t is measured in hours. How long will it take 10 bacteria, of this strain, to increase to 1000 bacteria?

Strengths: • Most students were able to substitute correctly into formula. • Those who solved by logarithms showed good manipulative ability with logarithms. • Those who were able to express both sides to the same base were able to equate exponents correctly.

52

Weaknesses: • The most frequent error, by far, was to multiply the bases rather than add the exponents.

10 X lO' = 100032t

8. Find the sum of all the integers between 10 and 650 that are multiples of 4.

Strengths: • No real strengths demonstrated. Ap-proximately half the students were able to find a 1 = 12 and a = 648.

Weaknesses: • Many students were confused by the word multiple and chose a geometric se-quence instead of arithmetic. • Students did not use "multiples of 4," and frequently chose a 1 = 10 and a,, = 650. This, incidentally, led to self-correcting errors, making marking difficult.

Algebra 12 Scholarship Examination, January 1986 1. The point (m, -3m) is the midpoint of

the line segment joining P 1 (4, 7) and P2 (x, y). Express x and y in terms of m.

Strengths: • Most students knew the midpoint formula.

Weaknesses: • Many students did not understand the instruction "express x and y in terms of m. • Numerous computational errors in

solving equations X = m for x, and

= -3m for y.

2. Give the middle term in the expansion Of (m2 - i.)s. Simplify your answer.

Strengths: • Most students recognized the fifth term as the middle term. • Most students were able to select a correct formula and substitute the cor-rect values.

Weaknesses: • Numerous errors in exponent com-putation and simplification. Quite a few students added or subtracted the factors

(m2 )4 and (_)4•

3. If F. atm = 3, solve for a.

Strengths: • Those who recognized this as an infinite geometric series selected the cor-rect formula.

Weaknesses: • Quite a few students did not under-stand how to apply sigma notation. They wrote a' = 3, a2 = 3, a3 = 3, instead of &+a2 +a3 + . . . = 3.

4. Given that i is a root of the equation 3f - 5x3 + x2 - 5x - 2 = 0, find the other roots.

Comment: • A fairly easy question. If any weakness was evident, it was in finding the depressed equation.

5. John, Jan, and Larry went to the store to purchase pencils, notebooks, and erasers. John paid $6.05 for 3 pencils, 5 notebooks, and 2 erasers. Jan paid $2.60 for 4 pencils, 2 notebooks, and 1 eraser.

53

Larry paid $4.40 for 6 pencils, 3 note-books, and 4 erasers. Find the cost of one notebook only.

Comment: • Also a fairly easy question, with most students finding three equations. Many computational erorrs in solving the system.

• Of those who attempted it, the ma- jority placed the 2 on the wrong side of the distance formula. • Many made computational errors in completing the square and in simplifi-cation.

9. Determine the numerical value of x1

given that

6. Evaluate the following, leaving the (log46)(log5x)(10965) = 3 answer in simplest form.

4 3

1096k +F4 10963j k=2 j=2

Comment: • Some difficulty in handling the two concepts series and logarithms. After the expansion, some students found the log (base 6) of each term instead of simpli-fying first, then finding the log.

7. Give the polynomial function of least degree with integral co-efficients whose graph is given above. Show the function in descending powers of the variable.

Comment: • Many students did not know the change of base formula. Those who got past the first step showed good skills in log and exponential manipulation.

10. Solve for 0, where 00 < 0 < 360°. cos 20 = cos 0 - 1

Strengths: • Good understanding of replacing cos 20 by a single angle identity. • Good manipulative ability in solving quadratic equations (surprizingly few divided by cosO).

Comment:11. Two vertices of an isosceles triangle are

• Using the three zeros, most students the points (0,0) and (2,4). The base of

expressed a polynomial of degree three, this triangle measures Find the

but totally ignored the given point (1, measure to the nearest degree of one of

-4) for finding the constant factor. The the base angles of the triangle.

majority did not express the answer as a function, = . . . Strengths:

• Good understanding of distance for-mula and cosine law.

8. A conic consists of all points (x,y) which Weaknesses: are twice as far from (0,2) as from (3,2). • Many students drew sloppy or map-Give the standard form equation for this propriate diagrams, not showing clearly conic, which angle was to be found.

Weaknesses: 12. Write an equation for the hyperbola • A high percentage (approximately which passes through (2,3) and has 80%) of students either did not attempt asymptotes with the following this question, or tried and got zero, equations.

54

y= 1 --x

and y=+x+2

Strengths: • Students were able to plot asymptotes. • Students were able to recognize equa-tion of hyperbola.

Weaknesses: • Poor recognition of translation. • Poor application of a and b values in hyperbola equation. Many found the correct ratio a:b = 2:1, but did not find the actual values, a=4, b=2.

13. Two swimmers sight the top of a 60 m lighthouse. The angle of elevation to the top from the first swimmer is 45°, while from the second swimmer the angle of

elevation to the top is 30°. The base of the lighthouse is.at sea level. The angle between the lines of sight from the base of the lighthouse to the two swimmers is 150°. How far apart are the swim-mers? Give the answer in simplest radical form or to the nearest metre.

Strengths: • Angles of elevation well understood. • Good use of trig laws for right angle trigonometry.

Weaknesses: • Poor recognition of the three-dimensional aspect of the question. • Most common error was putting the 150° angle between the lines of sight from the top of the lighthouse instead of the base.

55

PLAN TO ATTEND

The BCAMT Summer Conference '86

August 26 and 27 Simon Fraser University

KEYNOTE SPEAKERS: Dr. Daniel Birch, vice-presient, UBC

Dr. Brendan Kelly, author and mathematics consultant, Ontario

Registration includes a one-year membership in the B.C. Associa-tion of Mathematics Teachers ($25), whose services include Vector and newsletters. (Current memberships are extended a year.)

Early registration (up to July 1) - $50

Registration after (July 1) - $60

Copy and mall to: Ian deGroot

3852 Calder Avenue North Vancouver, BC V7N 3S3

Name

Address

Phone - SIN

(Make cheques payable to 24th Summer Workshop)

Y86-0056 56

May 1986 utfe

B.C. Association of Mathematics Teachers MEMBERSHIP FORM F11-25/Rev. May 1985

To join, here is all you have to do.

Mail to: B.C. Teachers' Federation 105-2235 Burrard Street Vancouver, BC V6J 31-19

Print your name, address, etc., below. Enclose your cheque or money order (do not mall cash), made payable to the B.C. Teachers' Federation.

Social Insurance Number I I I I I I I I I I Mr., Mrs., Miss, Dr., Ms.I I I I I Surname I I I I I I I I I I I I I I I I I I I I I I Maiden/former name I I I I I I I I I I I I I I I I I I I I I I Given name I I I I I I I 1 1 I I Initial I.._..j

Home address I I I I I I I I I I I I I I I. I I I I I I I

liii I 11111 IPostal code l 1111111 Name & address school/institution/business

School district no. I I I

Type of membership

Full(a) [J BCTF associate member(b) [J Non-BCTF member(c)El

Student(d) El BCTF honorary associate member(e) [J BCTF honorary. life member(f) El

Teaching or interest level (check one only)

Kindergarten(1) Primary(2) 0 Junior secondary(4) Senior secondary(S)

Elementary(7) Secondary(8)

BCTF members—$20 Student—$5

Total fee enclosed Cheque [J

lntermediate(3) 0 J College/University(6) 0

All(9) LI

Non-BCTF members—$30

Money order El

PSA 50