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ECT(DR Journal

of the British Columbia Association of Mathematics Teachers

Volume 26 Number 1 Fall 1984

B.C. Association of Mathematics Teachers 1984-85 Executive Committee

Past President Denis M. Hamaguchi 3807-22nd Avenue Vernon, BC V1T 1H7 H: 542-8698 S: 545-0549

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 V6N 21`3 H: 261-4358 S: 526-3816

Secretary Nigel A. Cocking 4-1333 Fort Street Victoria, BC V8S 1Y9 H: 595-7716 S: 479-8271

Treasurer Jessie Rupp 1122 Duchess Avenue West Vancouver, BC V7T 1H2 H: 922-8315 S: 922-3931

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

Ian de Groot 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

Primary Representative Wendy Klassen 49-6880 Lucas Road Richmond, BC V7C 4T8 5: 274-9907

Intermediate Representative Doug Super 313-2255 York Avenue Vancouver, BC V6K 105 H: 736-0960

Post-Secondary Representative Ian de Groot - 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

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, if possible, should be camera-ready. We would appreciate a black-and-white photograph of each author. If feasible, the photo should show the author in a situa-tion related to the content of the article. Authors should also include a short state-ment indicating their educational position and the name and location of the institu-tion in which they are employed.

Notice to Advertisers

Vector, the official journal of the British Columbia Association of Mathematics Teachers, is published three times a year: fall, winter, and spring. Circulation is approximately 550, 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

Deadlines for submitting advertising for the winter and spring issues are December 1, 1984 and April 1, 1985.

Inside This Issue

5 From the Editors ................................................ Tom O'Shea 6 Letters .................................................................... 9 The 1984 NCTM Annual Meeting................................. John Kiassen

11 Did You Know That ........................................... Ian de Groot

Mathematics Teaching 12 A Word About Word Problems..................................... Bonar Cow 15 Brain Teasers for Intermediate Students ....................................... 16 The Answer As a Key: Does It Shut

or Open the Door in Problem-Solving? ............................ Walter Szetela 23 Bulletin Boards: An Enrichment Tool ....................... Bernadette L. Harris 26 Diagnosing Pupil Performance in Decimal Multiplication.......... James H. Vance 33 The Terrible Demise of an Algebra Flunk-Out...................... Susan Quinn

Mathematics Issues 37 Streaming in Mathematics at the Junior Secondary Level .......... Hugh S. Elwood 40 Mathematics Teacher Qualifications: A Principal Reacts ............... Ed Collins

Miscellaneous 42 The 1984 Euclid Mathematics Contest ........................... George Bluman 47 The Third Annual SFU Mathematics Enrichment Conference ........ Larry Weldon 48 1984-85 BCAMT Executive Meeting Dates..................................... 49 The 1984 BCAMT Summer Conference ........................... Garry Phillips 50 1983-84 BCAMT Financial Statement ........................................... 51 New Books Across My Desk ........................................ Ian de Groot

90m Howitz Memorial Scholarship

The mathematics education community was saddened by the news of the death of Dr. Tom Howitz on August 8, 1984. Tom's many con-tributions to mathematics education in B.C. and beyond were featured in the last issue of Vector.

The Department of Mathematics and Science Education at UBC is creating a scholarship in the memory of Tom. If you would like to con-tribute to the fund to set up the scholarship, make a cheque out to Jim Sherrill and send it to the following address:

Jim Sherrill Faculty of Education

2125 Main Mall The University of British Columbia

Vancouver, BC V6T 1Z5

Tom O'Shea

From the Editors

Tom O'Shea and Ian de Groot

We have had a number of favorable com -ments on the quality of the previous issue of Vector, and we are pleased that the articles in that issue seemed to meet the needs and interests of mathematics teachers. Vector has always enjoyed a good reputa-tion, not only in this province, but across Canada. For that reason, we have decided to upgrade the physical quality of the journal. As a first step, we have moved to a heavier cover, with perfect binding. We hope that readers will approve of the changes. Your comments are most welcome.

In this issue, we have again put together a variety of articles. Bonar Cow provides a delightful example of how problem-solving can be introduced into the elementary class-room. Walter Szetela's extension of the checkerboard problem is an excellent exam-pie of the power of - Looking Back in problem-solving. Bernadette Harris gives many fine suggestions for using bulletin boards for enrichment at the junior secon-dary level. Jim Vance's article on decimal multiplication is a model of a research pro-ject that yields important insights into students' thinking in mathematics. Susan Quinn's fantasy is a timely reminder of the

terrors of testing and the deficiencies of the present mathematics curriculum.

Controversial issues in mathematics educa-tion are addressed by Hugh Elwood and Ed Collins. In response to the ministry's white paper on curriculum, Elwood argues for the introduction of streaming at the junior secondary level. Collins takes a hard look, through the principal's eyes, at teacher qualifications. We encourage readers to sub-mit opinions on these and other issues of concern.

Finally, we present reports from various sources. John Klassen and Carry Phillips look at the annual conferences of the NCTM and the BCAMT. George Bluman reports the results of the 1984 Euclid contest, and Larry Weldon summarizes events at the 1984 SFU high school enrichment conference.

If you like the contents and appearance of the journal, let us know. The new binding requires that the journal be at least three centimetres thick, so we need lots of con-tributions. Deadlines for the winter and spring issues are December 1, 1984 and April 1, 1985.

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AW hi AWAIF

Capilano College 2055 Purcell Way North Vancouver, BC V7J 3H5

Dear Mr. de Groot, The Capilano College Mathematics Depart-ment recently discussed the issue of qualifications of teachers who teach mathe-matics in the public schools of British Columbia. While we admit that we are not aware of all of the present requirements necessary to teach mathematics at the various grade levels, we were able to iden-tify the following three areas of critical importance to the qualifications issue:

Content Awareness To gain the mathematics content familiar-ity, a teacher must have successfully com-pleted a number of courses beyond the content level of the course that he/she 'is teaching.

Attitude One of the greatest areas of concern to our department (that many of us have observed first hand with our children in the public school system) is that mathematics is all too often taught by a teacher who, while tech-nically qualified to teach the subject, has very little enthusiasm for it. Often negative feelings toward the discipline are unwittingly passed on to the student.

Application How often have we as mathematics teachers been asked "What applications does this have?" or more bluntly "What good is this?" A good mathematics teacher anticipates this

question and hopefully has a number of in-teresting applications at hand.

The minimal qualifications necessary for "Content Awareness" are the easiest to address. Minimally we believe that all teachers of mathematics in the public school system should have Algebra 12. Further, Algebra 11 and 12 teachers should have at least a mathematics or physics major in their undergraduate degree. What formal post-secondary mathematics training a teacher of K-10 should have is unclear, although it does not seem necessary (or practical) that they be specialists in math.

The necessary qualifications for "Attitude" and "Application" are more difficult to iden-tify. One step in this direction would be the requirement that all teachers of mathematics must complete at least one full year course in mathematics education. Further, mathe-matics teachers should be encouraged to pursue studies in mathematics-related fields such as computing or technology.

Hopefully these comments will be of assist-ance to your committee in the preparation of its report on the qualifications of teachers of mathematics.

Yours truly,

A.E.T. Bentley Co-ordinator of Mathematics

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University of Victoria Faculty of Education P.O. Box 1700 Victoria, BC V8W 2Y2 May 18, 1984

To the Editor: The enclosed may perhaps be suitable for inclusion in Vector as part of the "Letters" section.

Sincerely,

Werner Liedtke

P.S. Keep up the excellent work!

Just for Fun—Or Is It?

A few questions

Congratulations to you, Tom and Ian. The latest issue of Vector (Spring 1984) was a pleasant surprise. Variety and interesting topics made for enjoyable reading. A few questions came to mind as different articles were examined.

Is it mathematically correct to present to students a diagram of the following nature?

El+ LI = (Dukowski article)

Li Li Li

Rousseau and Owen make some excellent teaching suggestions for including physical devices to generate mathematical abstrac-tions. Included in these suggestions is the "use of popsicle sticks (loose and bundles of ten) together with a pocket chart to demon-strate graphically the grouping of tens and ones to form one ten and 'carrying' it to the tens column." The comment is made that an

abacus does not "fit" into this part of the in-structional sequence. These are points well made. However, one danger should perhaps be pointed out. Occasionally a teacher is tempted to represent a two-digit number (i.e., 25) on a labelled pocket chart

by using bundles of

ten ( S44kf4W ) and ones ( IMII

A warning about the incorrectness of such a procedure could have been included in the article. Isn't a labelled pocket chart similar to an abacus?

Much has been said (at workshops) and written about Math Their Way and Mathematics: A Way of Thinking. Every time I hear or read about it, a few questions come to mind that I don't think have been answered. A few of these reappeared when I read Clark's article. For example, many general statements are made that carry with them certain implications (too numerous to list them all!).

"Teachers must recognize that children con-struct their own knowledge, rather than accept someone else's construction." Is this true for all subjects: reading, language, science, etc.?

"Children learning about numbers. . . learn by doing them. . ."; "children are given time to think and discover. . .", "the emphasis is on process, not product." Is it implied that other programs lack these characteristics? Isn't a good or appropriate teaching-learning setting for young children described rather than a book or program?

Isn't it true that every teacher attempts to apply to every subject the teaching strategies and sequences described by Clark?

Is the "book" (program) recommended as the sole source for mathematics learning, or is

7

it recommended as a reference for ideas? Is it suggested that the settings and dialogue be used as outlined, or are modifications possi-ble? The assumption is made that there exists positive transfer from grouping in other bases (and learning a new language) to grouping in base ten. Is this really true? For some (most, all) children?

I think the same book or program is used in some districts at three different levels (i.e., Math Their Way - K, 1 and 2). Is it possi-ble that there is too much repetition for some children? How does this program accom-modate the Learning Outcomes (Core) suggested in the B.C. Curriculum Guide? Has the program ever been evaluated? Is it authorized for use in B.C.?

BCTF Lesson Aids has suffered the same fate. In the rush to put all resources into an ill-advised war with the government, the really important services to the classroom teacher have been neglected. Let's see some priorities set straight; we need more of the likes of PEMC.

Rick Sutcliffe Associate Professor Computer Science/Math Trinity Western College

Box 1253 Aldergrove, B.C. March 16, 1984

The Editor: I appreciate Doug Super's comment on my fall article [Toward a Consensus on Com-puter Instruction in B.C. Schools]. He should not interpret my remarks as critical of PEMC itself, but rather of the shortsightedness of those who make funding decisions in the ministry.

The PEMC computer support is first-rate (what there is of it), and I have praised it highly elsewhere—see. "Quo Vadis-Software" in Call A.P.P.L.E., June 1984.

However, as long as the ministry believes that supporting several assistant superin-tendents per district at $100,000 p.a. is a priority, and that the infrastructure in Richmond and Victoria really needs all the people it contains, genuine 'services to teachers will have to be run on a shoestring.

HELP We would like to bind a complete set of Vector to make a permanent BCAMT collection. We are missing the following issues from the early 1970s:

Volume Issue 11 All (thought to be 2; Fall

1969 and Spring 1970) 12 1, 3, 4, 5 (Fall 1970 and

3 in Spring 1971) 13 All (number unknown;

Fall 1971 through Spring 1972)

14 4 (March 1973)

If anyone can locate and donate copies of the above, we would be very grate-ful. Such copies will be annotated with the doner's name, thereby ensuring him or her a small place in the history of mathematics education.

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The 1984 NCTM Annual Meeting John Kiassen

John Kiassen is president of the BCAMT and mathematics department head at Sutherland Secondary School in North Vancouver.

The cable cars were derailed, but that was hardly the case with the Annual Meeting held in San Francisco April 25-28. There were three thousand advance registrants, and the total registration reached five thou-sand. This was a most welcome success after the Detroit meeting in 1983. The city fathers did their utmost to welcome all teachers and even staged an earthquake, 6.2 on the Richter scale, to promote problem-solving with logarithms.

It is impossible to summarize the goals and directions of a conference that entails over six hundred sessions. Two major strands of the conference were problem-solving and technology. I would like to pass along my thoughts and reflections, hoping that it stimulates some thought or new ways of looking at old ideas.

The general feeling about the teaching of mathematics in the Untied States is that things should be changed, but there is some confusion as to the direction of this change. On the one hand, 36 state governors for-mally support school reform, but there is no accompanying infusion of funds. There is a pressing need for trained mathematics teachers, but unions balk at the proposal to offer bonuses to attract mathematics special-ists. The mathematics community advocates developing curriculum around problem-solving, but the classroom teacher must deal with the heavy influx of compulsory stand-ardized tests that reflect the learning of basic lower-level skills.

There seems to be a shortage of textbooks that deal effectively with problem-solving, although there is some consensus that this is improving. Dr. Willoughby, NCTM presi-dent, mentioned the reluctance of teachers and the community at large to embrace the available technology. He cited situations where the calculator was still considered to be taboo for general use in the classroom.

The problem-solving strand has been a highly visible one since the Agenda for Action was presented in 1980. Based on the sessions I attended, it seems we are now less concerned about the particular definition of problem-solving and the uniqueness of prob-lems. This emphasis has been replaced with a much broader approach where we talk about problem-solving in all aspects of instruction not exclusively in the section on word problems. The teacher and students certainly need the time to develop problem-solving skills, but problem-solving should be incorporated into all mathematics instruc-tion.

Perhaps we can develop problem-solving skills even in our algebra units by asking questions differently. For example, solve the system

x— y= 6

4x - 2y = 23

for ' 2 X instead of the traditional "x"

and 'ay".

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Teachers must be flexible, present different strategies, and. continually keep these in front of the students. Here are some ques-tions raised regarding problem-solving: 1. Once a student has written an equation for a problem and solved the equation, is he/she necessarily capable of understanding or evaluating the problem? 2. Is there a difference in writing an equa-tion for a problem when we ask the student to translate from words to algebra or when we talk about an equation as the expression of one thing in two different ways? 3. Should we not be using problem-solving strategies, 'such as patterning and working backward in presenting lessons even at the most basic skill levels?

A major reason for the re-examination of the curriculum is the impact of technology. A second reason is that an increased impor-tance attached to problem-solving requires a deletion of some topics due to the time constraints.

Major curriculum-development projects are under way at the University of Maryland (J. Fey) and the University of Chicago (Z. Usiskin). A common thread is that students may not need to spend as much time developing sophisticated skills in algebra. Once the student understands the basic skill of factoring, is it necessary to spend con-siderably more time expanding this to

include more complex exercises, or is the alternative to use technology to perform the tasks? It certainly is a question we must con-front. I recommend the paperback Com-puting and Mathematics, edited by James Fey, available from NCTM. It contains a series of articles that examine the impact of computers on algebra, geometry, calculus, etc.

Recent reports in the U.S.A. highlight the importance of mathematics, but there is a concern that we are providing the same precalculus mathematics program for all students. It was mentioned at a number of sessions that in a 1980 study, 25% of the Grade 12 students did not graduate, 25% entered a vocational training program, and 50% went on to junior colleges or univer-sity. A number of speakers singled out the 50% population that did not move into a university-related post-secondary program. What are we doing for this group in terms of a mathematics program? Some work is being done to develop such a program in Wisconsin, and this will be published by NCTM in the fall and in the 1985 yearbook.

I trust there are one or more things that may strike a chord, for it will be very exciting in the next few years to compare the changes in British Columbia with those in the United States.

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Did You Know That...?

Ian de Groot

The metric system is the official system of measurement used by the entire world with the exception of Brunei, Burma, North and South Yeman—and our enlightened neigh-bors, the United States.

Though Congress in 1975 legislated a gradual and voluntary changeover in weights and measures, nothing seems harder to do than to get Americans to adopt metric.

A few years ago, the metric forces thought they could get the U.S. to switch in a decade. Now they do not expect metric to prevail before the year 2000. Only when students who now learn some metric measurements in school reach upper-level management will the change really occur.

A few major manufacturers, including General Motors, John Deere, and IBM, are switching rather than fighting. So are several government agencies, including NASA. Still, major manufacturers, such as Boeing, con-tinue to measure in feet and inches, though they sell many products overseas.

Even the metrically untutored do not blink when doctors prescribe 500 mg of antibiotics or electricians recommend 15 A (ampere)

fuses or Carl Lewis wins a gold medal for running 100 m faster than any other human.

According to Time Magazine, critics like Stewart Brand, creator of the Whole Earth catalogues, object to the metric system for the very reason that most scholars favor it: the ease of converting one unit to another—say, kilometres to metres—by simply multiplying or dividing by tens. Says Brand: "You can't visualize a tenth very well, but you can imagine a quarter or a half of something."

Like their colleagues in Canada and through-out the world, U.S. scientists have long used metric measurements, and some three dozen states require metric instruction in the schools. Moreover, while football in both the U.S. and Canada still measures progress in yards, and Nolan Ryan's fastball blazes at 98 miles per hour, many U.S. joggers now speak knowingly of doing their weekend "10 Ks" (for 10 kilometres).

Lovers of the grape originally suspected a plot when the wine industry adopted Euro-pean measures in 1979, but they have since learned that the newfangled litre gives them 1.8 ounces more than an old-fashioned U.S. quart!

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A Word About Word Problems Bonar Cow

Bonar Gow is a teacher at Pouce Coupe Elementary School in Peace River South.

It was late November, and my self-image was beginning to wilt a little. I was teaching a class of 12 Grade 5s and 20 Grade 4s and feeling that I might make it through my first year after all. That is, except for teaching story-problem-solving in mathematics.

I asked myself what had gone wrong. Hadn't I done a more than adequate job of teaching place value? number theory? estimation? operations with whole numbers? Certainly I. had! Then why were my students such dismal failures at solving word (or story) problems? A student who scored at or above mastery level in other mathematics activities was lucky to get by with 50% on word-problem tests or story problems in the text-book. All my attempts to teach effective problemsolving skills had apparently failed. I decided to seek help.

Teachers are helpful people, but when I sought advice, I found that they had similar troubles. No one seemed to have a surefire method for teaching children how to solve story problems. In the weeks before the Christmas holidays, I tried out a number of suggestions, but none succeeded. At the same time, I also wrote to a university col-league in Victoria and requested an update on what research had revealed on the subject.

A package of articles came before the dust of the Christmas concert had settled. During the holidays, I sorted through the articles and through Mathematical Problem Solving —A Resource for Elementary Teachers

(1981) and Mathematics: A Way of Think-ing (1977). When I had finished, I selected the ideas I felt would be workable in my classroom and began to formulate a series of hybrid techniques. By the time I returned to the classroom in January, I felt that I had the basis for a reasonable teaching strategy.

How lucky we are that children are forgiv-ing! It took about two months for all 33 of us to get over my teething problems. By early March the class was beginning to solve story problems with a high rate of success.

My approach took place at two levels. On one, there was a need to foster and, wherever possible, create, good problem-solving abilities. For example, I had worked with students on such skills as visualizing (drawing diagrams, sketching), estimating (predicting, checking), deleting. redundant data or information, supplying missing in-formation, and attempting to generalize on the basis of a few examples. On the other level, they began to write word problems for each other integrating these skills, and I began to use these same problems on their word-problem tests.

To get the students into the right frame of mind for writing and eventually solving story problems, I began with manipulatives: beans, peas, kernels of corn, etc., and imitation money. Working in small groups, we began to formulate simple story prob-lems, using the manipulatives as counters. From this very concrete level, we moved on to a semiabstract level and wrote down what

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had been explained orally. Gradually the manipulatives disappeared, and we began to work without them.

Two students using manipulatives during the first stage of fraction word problems.

Once the children were used to creating their own simple problems, I had them reserve a notebook for word problems. Initially students had only to write 10 word problems of their choice. We began with addition, and then as the students became accustomed to writing their own questions, they were re-quired to supply specific types of problems. For example, an early addition assignment for Grade 5s might have required that: (1) All questions have four addends in them. (2) All addends be three- or four-digit numbers. (3) Three of the questions involve money. (4)One problem use metric measurement for mass. As time progressed, we moved into subtraction, multiplication, and division problems, using the same method. At the

beginning of each class, the students were given specific instructions as to the type of questions they were to write for me. In this way I was able to exercise 'quality control' over what was produced.

The results of my experimenting proved interesting. With the exception of an occa-sional textbook story problem, my students stuck to writing and solving their own prob-lems. By following a prescribed problem for-mat involving five distinct steps, I noticed that the students' ability to deal with story problems improved. Every problem they handed in had to have these steps: (1) Prob-lem information, (2) A clear question, (3) An equation, (4) A correctly solved equation, (5) An answer written in a grammatically correct sentence. It soon became clear to me that creating their own problems had made it easier for the children to approach and solve the problems written by others.

Because the children were now able to put problems together, they were also able to solve problems more effectively. True, even the best students still made mistakes, but in most cases, these resulted from sloppy work habits rather than a basic lack of understand-ing (for example, a missing decimal point, regrouping mistakes, place-value errors created by poorly laid out work, etc.). Second, the students were working with topics that really counted, at least in their eyes: foods, sports, pets, clothing, domestic situations, records, tapes, hobbies, and the like.

There were other payoffs from student-generated problems, because reading, writing, and mathematics skills came together. Skills were being practised con-stantly, and by placing word problems in a separate book, the students could easily trace their progress. In addition, the students soon began to try stumping their classmates with what they considered their toughest problems.

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This method of problem solving is carried out once a week in a one-hour time block, which is separate from the subject of mathe-matics. My principal, Gordon Mime, has provided four calculators, which the students share for checking answers. Once a set of questions is handed in, I take the notebook and check all 10 of them. Each question is given 10 marks. If you have a class of 30 children, you can count on check-ing three hundred completely different ques-tions each week. Buy some extra batteries for your calculator. Story-problem tests are made up mainly from the very best questions in the notebooks, and the students have their names placed next to the questions they have supplied.

Problem-solving via student-generated and student-solved problems does work. When spring rolled around, I discovered a signifi-cant increase in ability when the scores for the Canadian Test of Basic Skills were

tallied. The lowest score in the problem-solving section was 0.2 below grade level. Also, when we dipped into the Investigating School Mathematics text again to solve the occasional problem, the comments ranged from "Mr; Gow, these are a cinch!" to "B000ring!"

I now enjoy problem-solving because the students in my class really are learning. They actually look forward to that weekly hour and compete with one another to create complex problems to challenge classmates. All of us in the room have found something very worth while.

Bibliography Baratta-Lorton, R. Mathematics: A Way of

Thinking (Menlo Park: Addison Wesley, 1977).

Vance, J. Mathematical Problem Sovling-A Resource for Elementary Teachers (Victoria: Ministry of Education, 1981).

Yes. I'd like to join the National Council of Teachers of Mathematics. I under-stand that annual dues are only $35 for all membership services, including the periodical(s) I have checked below:

. 0 ARITHMETIC TEACHER (Al) 0 MATHEMATICS TEACHER (Ml') • ' 9 issues September-May 9 issues September-May

0 Both AT and MT for an additional $13.00 0 Free Samples

Tax deductible in the U.S. as a contribution to the improvement of mathematics education. Full-time student dues are V2 regular membership dues. For mailing outside the U.S., add $5 for the first AT or MT per membership and $2.50 for each additional AT or MT. Dues support the

lbeabeuer

development, coordination, and delivery of Council services, including $13 for each subscrip- tion to the AT and MT and $2 for an NCTM NEWS BULLETIN subscription.

$ . 0 Payment to NCTM in U.S. funds enclosed 0 MasterCard 0 VISA

teacher for It.Credit Card # _________________ Expires Signature Name Address City State or Province Postal Code

M6

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Brain Teasers for Intermediate Students

Directions Answer the questions below. Use the code to help you answer the riddles. Here's how:

I am a factor of every number. Who am I? (Answer, 1) Use the code to translate your answer into a letter: 1 = U.

0=Y 2=T 4=P 6=H 8=C

1=U 3=Q 5=K 7=E 9=A

1. Add me to myself or multiply me by myself, you get the same answer. I am a factor of all even numbers. Who am I?

2. All of my multiples are the same number. Who am I?

3. Add me to myself, and you get half of what you get if you multi-ply me by myself. Who am I?

4. Add me to myself, and I double. Multiply me by myself, and I stay the same. Who am I?

5. The sum of the digits in my multiples add up to the number that I am. Who am I?

6. Add the digits in my multiples. Look for a pattern that repeats my first three multiples. Who am I?

7. and 8. Neither of us is divisible by 10, but our product is. Our sum is 11. Who are we? (List smaller number first.)

9. and 10. Our product is 56 and we differ by 1. Who are we? (Smaller number first.)

Code

What has the head of a dog, the tail of a dog, but is not a dog?

What happens to ducks when they fly upside down?

Question# 5 3 4 3 3 2

Code letter .rTIUIIIIIIIUUII

15

The Answer As a Key: Does It Shut or Open the Door

in Problem-Solving?

Walter Szetela

Walter Szetela is an associate professor in the Department of Mathematics and Science Education, UBC.

George Polya's four-step problem-solving plan has gained wide acceptance in mathe-matics education as indicated by a scan of the newer textbooks, mathematics journals, curriculum guides, etc. In Polya's fourth step, the problem solver is advised to look back after finding the answer. Yet little attention is given to the look-back step. In practice, for many people, the look-back step means mainly check the answer for reasonableness. Of course, Polya proposes much more. He suggests that one should look for another way to solve the problem or extend the problem (for example, use variations of the original problem, or generalize the problem). In other words, Polya treats the answer to a problem as an entry point to related problems. While the effort focussed on related problems may be considerable, there are satisfying rewards to the problem solver, and a much deeper understanding of the original problem is likely to develop.

To illustrate extensions and related prob-lems, we'll begin with the well-known checkerboard problem.

PROBLEM 1 How many squares are on an 8 by 8 checker-board?

SOLUTION The problem can be solved by considering a simpler case.

Note that in a 4 by 4 board, we have:

16 squares of size 1 by 1

9 squares of size 2 by 2

4 squares of size 3 by 3

1 square of size 4 by 4

....

.... NONE ....

We can extend the observations to an 8 by 8 board to obtain 64 squares of size 8 by 8, 49 of size 7 by 7, 36 of size 6 by 6, 64 squares of size 1 by 1, 49 of size 2 by 2, 36 of size 3 by 3, etc.

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A problem has been solved, and one could stop here. If you are more adventurous, and you follow Polya, you can do as Billstein (1975) and Turner (1983) have done, and solve a more general problem.

recognize these square roots as the first four triangular numbers represented below.

PROBLEM 2 How many rectangles are on an 8 by 8 checkerboard?

SOLUTION Look at a simpler case again such as a 4 by 4 board.

We record the number of rectangles of various sizes in a table.

1234

16 - 12 - 8 4

12 - 9 -

- 6

- 3

- - 8642 -

-

4 3- 2 i

- 1

The entries in the table add up to 100. Since 100 is obviously a square number, we suspect that the number of rectangles in any square board is a square number. An ex-amination of boards of sizes 1 by 1, 2 by 2, and 3 by 3 reveals that the number of rectangles is 1, 9, and 36 respectively. Evidently, square numbers do give the total number of rectangles on square checker-boards, but which square numbers? The square roots of the numbers 1, 9, 36, and 100 are 1, 3, 6, and 10 respectively. Many will

. ..

. .. S.. . S. •SS •SS• 1 3or— 6 o— 10or-

1+2 1+2+3 1+2+3+4

We are tempted to suggest that for a 5 by 5 board, the number of rectangles will be the square of the fifth triangular number or (1 + 2 + 3 + 4 + 5)2, which is 225. For the 8 by 8 board, we would use the 8th triangular number, which is 36. This would produce 1296 (362) rectangles for an 8 by 8 board.

We could stop here, but Turner takes the problem a step further:

PROBLEM 3 How many rectangles are on a rectangular frame of m by n rectangles?

SOLUTION We'll look at a simpler case, say a 2 by 3 rec-tangle, and record the number of rectangles of various sizes. See Table 1.

ENE. ..I

123

1 I 6 I I 2 I Total of 18 rectangles

23 2kfor 2 by 3 rectangle

J

Table 1

1 For example, there are 8 rectangles of

2 size 1 x 3 horizontally

3 and also 8 rectangles of size 3 by 1 verti-

4 cally.

17

Because triangular numbers were prominent in the previous problem, we might focus again on such numbers. The key numbers in this problem are 2 and 3, the dimensions of the rectangle, and 18, the total number of rectangles. The second triangular number is 3, and the third triangular number is 6. As 3 x 6 = 18, we might conjecture that the number of rectangles in a 3 by 4 rectangle is the product of the third and fourth triangular numbers, or 6 x 10 = 60. Let's ex-amine this case.

The number of rectangles of various sizes is shown in Table 2.

We could stop here, apparently having ex-hausted all variations of the checkerboard problem, but all our work has been in two dimensions. What about an extension into three dimensions? If we can work with squares, why not with cubes? Could we make discoveries and generalizations with cubes that are analogous to those with squares?

PROBLEM 4 How many cubes of all sizes are in an '8 by 8 by 8 assembly of 1 by 1 by 1 cubes?

OBSERVATION For the analogous checkboard problem, we know that the number of squares of all sizes is

12 + 22 + 32 + ... + 72 + 82.

Munn

Munn

Munn

The results in the table advance the conjec-ture that the number of rectangles in a rectangular frame is related to triangular numbers. In particular, we confirmed that in the case of a 3 by 4 rectangle, the number of rectangles is the product of the third and fourth triangular numbers, or 6 x 10. Turner indeed shows that for the general case of an m by n rectangular frame, the number of rec-tangles is the product of the mth triangular number and the nth triangular number. When the rectangular frame is square, the special case of Problem 2 arises, and the number of rectangles does become the square of the triangular number associated with the side of the given square.

If sums of squares solve the checkboard problem, might sums of cubes solve the cube problem?

CONJECTURE In an 8 by 8 by 8 assembly of cubes, the number of cubes of all sizes is 1 + 2 +

+7+8.

SOLUTION Let's look at simpler problems or more par-ticularly some extreme cases systematically.

p I ...ø I I..

Number Number Number Total of unit of cubes of cubes number

Cube size cubes of edge 2 of edge 3 of cubes

ibylbyl 1 - - 1 2by2by2 8 1 - 9 3by3by3 27 8 1 36

Table 3

18

Inspection of the table showing the first three extreme cases confirms our original conjec-ture. We would predict in a cube of edge 4 units, the number of cubes of all sizes would be 1 + 2 + 33 + 43• We also obtain a serendipitous finding when we look at the last column. In each case, the total number of cubes is a square number, in fact the squares of the first three triangular numbers. Therefore, we immediately predict that the total number of cubes in the cube of edge 4 will be the square of the fourth triangular number or 102 = 100. Investigation confirms the prediction. We obtain 64 cubes of size 1 by 1, 27 of size 2 by 2, 8 of size 3 by 3, and 1 of size 4 by 4. It is now apparent that the number of cubes in a rectangular assembly of cubes 8 units on each side is the sum of the first 8 cubic numbers. (Alterna-tively, we could express the sum as the square of the eighth triangular number, 362 or 1296 as suggested from the last column of Table 3.) More generally, we could start with a cube of edge n and note that there would be 1 cube of edge n, 8 cubes of edge (n - 1), 27 of edge (n - 2), etc.

Now that we've made a breakthrough from two dimensions into three by substituting cubes for squares, why stop? Rectangular solids in three dimensions are analogous to rectangles in two dimensions. Therefore we pose the following problem:

PROBLEM 5 How many rectangular solids of all sizes are there in a 8 by 8 by 8 assembly of unit cubes?

OBSERVATION For the analogous checkerboard problem we found that the number of rectangles of all sizes in an 8 by 8 board is (1 + 2 + 3 +

+ 7 + 8)2. (This is the square of the eighth triangular number.)

Might the number of rectangular solids in the present problem be the cube of the eighth triangular number?

CONJECTURE In an 8 by 8 by 8 assembly of unit cubes, the number of rectangular solids of all sizes is (1 + 2 + . . . + 7 + 8). (Cube of the eighth triangular number.)

SOLUTION We proceed by looking at some extreme cases. For the unit cube, the conjecture is cer-tainly true. Table 4 displays the results for a cube of edge 2.

- %dWA1 iii-Mv

Noe Dimensions of

rectangular solid Number of solids

1bylbyl 8 1by1by2PIf 4 1by2by12 4 2 by 1 by 1 2 by 2 by 2 _ 1 2by2by1 I_ j J/J 6

Total 27

TABLE 4

Thus we have 27 or 33 rectangular solids, and 3 is the second triangular number. Before we proceed with a cube of edge 3, let us note that for non-cube solids, we may obtain rectangular solids of the same size in three directions where two distinct dimen-sional lengths are indicated. Thus in the 2 by 2 by 2 case, we obtained equal numbers of rectangular solids with dimensions 1 by 1by2, lby2byl, and2bylbyl. Where it is possible to have three distinct dimen-sional lengths, we may obtain different rec-tangular solids of the same size in six ways,

19

e.g., lby2by3, lby3by2, 2bylby3, 2by3byl, 3bylby2, and3by2byl. This observation will shorten the tables and simplify the counting in more complex en-suing cases.

I..ø#J NNNO ...r

Rectangular solids in a 3 by 3 by 3 cube

Dimensions of solids

Number of solids by

type

Number of equivalent

types

Number of solids of

these types

lxlxl 27 1 27 1xlx2 18 3 54 1x1x3 9 3 27 1 x 2 x 2 12 3 •36 1 x 2 x 3 6 6 36 1 x 3 x 3 3 3 9 2 x 2 x 2 8 1 8 2 x 2 x 3 4 3 12 2 x 3 x 3 2 3 6 3 x 3 x 3 1 1 1

Total 216

TABLE 5

Note that 216 = 6 1 and that 6 is the third triangular number. At this stage, our con-viction on the correctness of the conjecture is extremely strong, but we proceed to verify that the number of rectangular solids in a 4 x 4 x 4 cube will be the square of the fourth triangular number or 10 = 1000.

Rectangular solids in a 4 by 4 by 4 cube

Number of Number of Number of Dimensions solids by equivalent solids of

of solids type types these types

ixixi 64 1 64 1x1x2 48 3 144 1x1 x3 32 3 96 1x1 x4 16 3 48 1x1x2 36 3 108 1 x 2 x 3 24 6 144 1 x 2 x 4 24 3 72 1 x 3 x 3 16 3 48 1 x 3 x 4 8 6 48 1 x 4 x 4 4 3 12 2 x 2 x 2 27 1 27 2 x 2 x 3 18 3 54 2 x 2 x 4 9 3 27 2 x 3 x 3 12 3 36 2 x 3 x 4 6 6 36 2 x 4 x 4 3 3 9 3 x 3 x 3 8 1 8 3 x 3 x 4 4 3 12 3 x 4 x 4 2 3 6 4 x 4 x 4 1 1 1

Total 1000

TABLE 6

The confirmation of the conjecture of Prob-lem 5 for n = 4 appears to be sufficient to extend the conjecture for n = 8. That is, the number of rectangular solids in an 8 by 8 by 8 assembly of unit cubes is the cube of the eighth triangular number.

Finally, we proceed to make one more ex--tension. Earlier we noted Turner's solution for the number of rectangles in an rec-tangular frame. Let's look at the analogous problem in three dimensions. We would begin with a rectangular solid that is not necessarily a cube.

PROBLEM 6 How many rectangular solids of all sizes are contained in an assembly of cubes forming a rectangular solid of dimensions 4 by 5 by 6?

20

OBSERVATION We recall that for the analogous problem in two dimensions, in a rectangular frame of size 4 by 5, the number of rectangles of all sizes is(1 +2 + 3 + 4)x(1 + 2 + 3 + 4 + 5). (Product of the fourth and fifth triangular numbers.)

CONJECTURE The number of rectangular solids in an assembly of cubes forming a 4 by 5 by 6 rec-tangular solid is the product of the fourth, fifth and sixth triangular numbers or (1 + 2 + 3 + 4)+ X5 =(1 + 2 + 3 + 4 + 5)x(1 +2+3+4+5+6).

SOLUTION We'll look at some simpler problems.

We quickly note that the product of the first, second, and third triangular numbers, 1, 3 and 6 is indeed 18. We'll look at one more simple case, a solid consisting of 12 cubes in a 2 by 2 by 3 arrangement and check to see if the total number of rectangular solids is the related product of triangular numbers 3, 3 and 6 or 54.

OEM M

mi

ME& Figure 2

Example 1 Consider a solid consisting of 6 cubes in a 1 x 2 x 3 arrangement.

IIIelpitro 9 FOM 0 d 000'A

F o WAllp-110 MMMMAJ Figure 1

Number of rectangular solids in a 1 by 2 by 3 solid

Number of Dimensions of solid such solids

ixixi 6 lxlx2 7 lxlx3 2 1 x 2 x 2 2 1 x 2 x 3 1

Total 18

TABLE 7

Number of rectangular solids in a 2 by 2 by 3 solid

Dimensions of solidNumber of such solids

ixixi 12 lxlx2 20 lxlx3 4 1 x 2 x 3 4 1 x 2 x 2 11 2 x 2 x 2 2 2 x 2 x 3 1

Total 54

TABLE 8

The two simple examples investigated seem to be by themselves sparse evidence for a claim for the truth of the conjecture, but in the light of the preceding analogous in-vestigations and results, they are reasonably convincing.

21

SUMMARY This set of investigations is an example of extension and generalization in the applica-tion of Polya's Look Back. The results demonstrate the value that can accrue from time spent on looking back, not merely to check for reasonable answers but to generate related problems. Teachers must be willing to make the effort to encourage more atten- tion to the fourth step, and not be limited to checking answers for reasonableness. Cer-tainly the fourth step with extensions and analogs of the original problem has great potential for stimulating problem-solving, consolidating understanding, and exhibiting analogies, connections, and patterns.

REFERENCES Bilistein, Richard. "Checkerboard Mathe-matics." Mathematics Teacher 68 (Decem-ber 1975): 640-646.

Turner, Sandra. "Windowpane Patterns." Arithmetic Teacher 76 (September 1983): 411-413.

*1 thank Jane' Smith, graduate student in mathematics education, for her helpful suggestions.

So, you say you never heard of the COMMUTATIVE property of multiplication over addition?! It goes like this .

(a x b) + c = (a + b) x c

For example: (3x4) + 2 = 12 + 2 = 14 (5x6)+ 3 = 30 + 3 = 33

OR (3+4)x2= 7x214 (5+6)x311x333

Get the picture?

Is the property true in general? Can you find a counterexample? Is the property true for any other trio of numbers? Can you find another trio for which the property holds? Can you state the value of c in terms of a and b.

See page 5 of the May 1984 issue of the Arithmetic Teacher.

22

2,t 4 q Al

pe

I *

t

Bulletin Boards: An Enrichment Tool

Bernadette L. Harris

Bernadette Harris is a teacher in the Greater Victoria School District.

The bulletin-board space in my classroom is used to enrich my classes, which are not streamed. I have an ample number of bulletin boards, and I devote five of them to reinforcing or enriching academic topics. 1 try to change the display on one board each week, which means that many topics are displayed for five weeks, though some very useful ones are up longer.

I paper the boards in September. These backgrounds are usually a single color—some light, and others dark. I have also used checkered backgrounds by alternating two colors of standard sheets of duplicating paper. Across my largest bulletin board I have 15 cm strips of the rainbow's colors, extending halfway down, and then have finished the bottom of the board with mild pink paper. The paper is available from the art room in our school, and the cost does not come from the math budget.

It does not take long to staple up pictures and appropriate cutout letters to illustrate topics. One of the most effective topics I use is titled Reals; the letters are cut out and placed at the top. I staple this particular display during a class lecture time, having prepared the cutout letters to name the

subsets of the real-number system. I use a thin strip of paper to divide the bulletin board, labelled Reals, into two parts to be labelled rationals and irrationals; integers, wholes, and naturals are ready to be stapled onto sheets of contrasting-colored paper as subsets.

In front of a class, I stand at the bulletin board—one of the type which serves as a window-screen—and staple the material as I explain the difference between rationals and irrationals, and between integers, wholes, and naturals, so that at the end of the short talk, the display is there as a long-lasting review of the topic. It remains up longer than most other boards because it is constantly useful for many classes.

Because I teach in a Grade 8-9-10 school, I don't have the fun of introducing the students to the set of imaginary numbers. I explain to them, however, that I will leave blank the adjacent window-shade bulletin board (at least when I'm putting up the reals one), and I say that if I were teaching a class in a school that had Grade 12, I would there put up the word imaginaries in the same letter size as the reals. Having students realize they are playing with only half the

23

number system'helps me explain why we sometimes use only the set of naturals, or integers. We are using only the set of reals now. I won't reveal how to write imaginary numbers (invisible ink . . .); this mystery is a challenge to some of them, who delight in finding out from, older friends or from books, and come in and enrich us all with their discovery.

I refer to this bulletin board to explain why we use dots when graphing inequalities, with the replacement set of naturals or whole numbers or the set of integers, compared to the line for graphing with the set of real numbers.

Another bulletin board is above the chalk-board, a narrow board that runs the length of the chalkboard. There I put on the top line the cutout sentence "Have a bit of pi," and at the bottom I staple the cutout numerals 3.141592 . . . to about 30 places. It fills up that space and spills over onto the next bulletin board. Students become familiar with the number and recognize that it goes on forever. Even after that display is re-moved, every time I refer to pi everybody knows exactly what I mean when I point to that bulletin board. When something has been pinned up for five weeks, students have had enough idle time that it has become part of their memory.

After removing this pi digit display, I use a window-screen board to display a multiplic-ative series to show the students how pi may be calculated:

ir 292.4.4.6.6.. 2

3.3.5.5.797.

This fills one bulletin board, in large numerals. Students pull out their calculators and show me that yes, it does work; they do get pi.

Another presentation is an exclamation point with the words: For "!" say "factorial." On the next line, 61 = 3 . 2 . 1, and below that 9!/7! = 72.

Other displays are easy to make: Fibon-nacci's series; Pascal's triangle. Also, above the chalkboard, I quote some important mathematician to support my ideas; for ex-ample Leibnitz who, in 1698 objected to the use of "x" as a multiplication symbol say-ing he usually preferred to use a dot (Lennes, A Second Course in Algebra, 1943, p. 500). Though I prefer parentheses, I use his quote, saying that even years ago, people were find-ing that a multiplication symbol often was confused with the letter x.

To illustrate concepts such as rationalizing the denominator, I cut out digits and put them up as a display that shows

/_1 26.

I supplement this with a reproduction of a cartoon saved from a newspaper and en-larged with the aid of an overhead opaque projector. A Peanuts treasure for that par-ticular bulletin board is the little bird, Woodstock, doing the rationalization in his head after Charlie Brown has told him that Woodstock is lucky he doesn't have to attend school.

Fun boards with a math slant can often be devised.. The six, or seven Ripley's Believe It or Not paperback books are a source of ideas. for me. One display originated from the statement, in numerals and letters, 1002 pay 4180." I build this bulletin board display over several days. (Sometimes display-building is inadvertent, because I simply don't have the time to finish it at once, but this display I deliberately build slowly.) I staple the above line at the bottom of the board on the first day. Curiosity provokes many questions.

24

After a couple of days of student specula-tion, I add the next line: "Read this sentence:" More questions. Then I staple up "Zero is also pronounced ought, nought, nothing, and oh."

Short poems from mind-stretching sources such as The Space Child's Mother Goose (F. Winson, Simon & Schuster, 1958) entertain students interested in the infinity concept:

"Little Bo-Peep . . . They'll Meet in parallel space Preceding their leaders behind them."

Enlarged Peanuts and Fred Bassett cartoons, even with some words replaced, can be used as incentives by way of humor, to submit neat homework, etc.

An incomplete magic square can be put up on a bulletin board. The monthly calendar of the NCTM is another source of ideas. It is too small for students to pay much atten-tion to, I find, if it is just stapled to the board, so I pin up an enlarged version of an appropriate problem.

I vary the topics. Sometimes, I highlight the anniversary of an event. For instance, the first manned balloon flight took place 21 November 1783, and I had collected fairly large pictures of beautiful hot-air balloons from such sources as my daughters' dis-carded Owl magazines and my copies of Natural History. I put those up in November 1983 with the words "200 years ago, the 1st manned balloon flight." These magazines are also sources of nature photographs, some of which I put up with the word symmetry: an enlargement of a hexagonal honeycomb, an aerial photo of wavefronts hitting the coastline, a shell, a magnified diatom. A forest tree poster available free from the Canadian Forest Service serves as an object to measure height when I put up a bulletin

board presentation using either similar triangles or trigonometry.

M.C. Escher has provided me with an ex-cellent continuing topic. A $20 book of 20 of his prints I have broken into separate pages, and I display two or three at a time, changing them each week, varying the words below:

"M. C. Escher" "Present-day artist" "Master of tesellations"

As I remove a display, I store all the para-phernalia in a large envelope—the letters, numbers, and cutouts, with big posters folded. I sketch on the outside exactly how it will look when it is put up again (in two or three years).

No artist, I was amused (and pleased) when a teacher complimented me on my "artistic bulletin boards," I rely on tracing stencilled letter-forms that are available from teachers' supply stores, and I spend some time in front of the television cutting out these letters. My husband thinks this is my subconscious desire to be teaching at an elementary school, but I find that careful use of bulletin board space is a very effective enrichment tool at the junior secondary level.

I would welcome additional ideas. Perhaps other teachers would submit their ideas to Vector to be run as fillers, a la Reader's Digest anecdotes. Ideas would certainly help me, as once the ideas are there, a display doesn't take too long to complete.

I have thought of going to our local math teachers' meetings with some of my envelopes of completed bulletin boards and offering them for use in another school's classroom: "Here's one you could put up tomorrow." I have not yet had the courage, but this article is a start.

25

Diagnosing Pupil Performance in Decimal Multiplication

James H. Vance

Jim Vance is an associate professor in the Faculty of Education at the. University of Victoria.

Decimal multiplication is one of the key topics taught and reviewed in middle school arithmetic. Finding the product of two numbers in decimal form requires mastery of the basic facts, the ability to multiply by powers of ten and apply the algorithm for multiplying multi-digit whole numbers, and knowledge of the rule for placing the decimal point in the answer. Some examples can be computed in more than one way, depending on how the zeros in the factors are used in the algorithm. Thus decimal multiplication exercises provide many opportunities for a number of different kinds of errors. As part of a study designed to assess the ability of students in Grade 6, 7, and 8 to compute and estimate decimal products (Vance, 1983), the computational procedures used and the types of errors made by students were analyzed.

Method A 12-item test in multiplying decimals was administered to students in 10 classes in a

lower Vancouver Island middle school. The exercises were presented in horizontal form, with space provided for written computation and figuring. In seven of the questions; one factor was a multiple of 10, or both factors contained only a single non-zero digit. The other five exercises required application of the vertical multiplication algorithm.

The test papers of 150 students were chosen by stratified random sampling for analysis. At the Grade 7 and 8 levels, the samples con-sisted of 60 students, 10 boys and 10 girls at each of three achievement levels. At the Grade 6 level were 30 subjects, five in each sex and achievement category.

Results Performance on each of the 12 exercises is presented in Table 1. The group mean score for the total test, 78%, indicates that the students are reasonably proficient in decimal multiplication.

TABLE 1 Performance on Decimal-Multiplication Exercises

Item Per cent correct Item Per cent correct

61.4 x 10 87 7000 x 0.002 79 x 0.04 •1000 78 7 x 0.81 80

0.4 x 0.2 77 0.5 x 24 87 50 x 600 85 4.5 x 3.92 61 7.29 x 0.01 79 360 x 0.25 81 0.06 x 6.009 72 0.42 x 0.617 66

26

Table 2 summarizes the results by grade and achievement level. As would be expected, performance increased with both grade and achievement level. The Grade 6 students did as well as the Grade 7s and 8s on the easier items, but they had lower scores on the more

difficult exercises. Mean scores for the girls and boys were 79% and 77% respectively. The group with the lowest score was the Grade 6 low-achieving boys, who averaged 50% on the test.

TABLE 2 Performance by Grade and Achievement Levels

Achievement

Per cent correct

level Grade 6 Grade 7 Grade 8 Total*

Low 58 67 69 66 Middle 61 79 88 79 High 80 91 90 88

Total 67 79 82 78

*mean = 9.33; s.d. = 2.38; KR - 20 = 0.704

Computational Procedures Examination of students' written computa-tions revealed a variety of procedures and algorithms for multiplying two numbers. The different computational forms were organized into three main categories. A short procedure was assumed to have been used when the answer was written next to the question. given in horizontal format and the question had not been rewritten in a vertical format. Short algorithms for multiplying by a power of 10 or finding the product of two multiples of products of 10 with single non-zero digits involve "adding zeros" or "moving the decimal point." For example, in 1000 x 0.04, one moves the decimal point in 0.04 three places to the right, or in 4000, two places to the left. An intermediate form was one in which the question had been rewritten in vertical format and an efficient

algorithm used to obtain the answer. In algorithms in the long category, rows of zeros were shown as partial products, or un-necessary zeros in one or both factors were multiplied. Three types of procedures are illustrated for one of the exercises in Table 3.

For the seven items of this type, computa-tional forms considered to be long, inter-mediate, and short were used in 25%, 35%, and 40% of the exercises respectively. The frequency of use of an inefficent procedure decreased with an increase in both grade level and achievement level. Girls used a long form more often than boys—in 49% of the exercises compared to 31 %. The accur-acy rate obtained using a long form was as high as that achieved using a more efficient method in computing.

27

TABLE 3 Use of Three Types of Computational Forms for 1000 x 0.04

response number correct number of errors

40 19

Short form 40.0 40.00

5 4

- other

2 10

Total

30 10

Intermediate form Total response 1000 1000 0.04

0.04 .04 1000 40.00 40.00 40.00

number correct 19 18 1 38 number of errors 1 4 3 8

Long form Total response 1000 1000 0.04 other

0.04 .04 1000 4000 4000 000

00000 00000 0000 000000 040.00 00000

0040.00 004000 0040.00

number correct 25 13 8 3 49 number of errors 6 2 2 3 13

Error Categories - The 395 exercises in which the correct answer was not obtained were examined, and the errors analyzed. Four main cate-gories and several subcategories of errors were identified. In 24 exercises, two different types of errors were found, producing a total of 419 errors. It was observed that error type was related both to the nature of the ques-tion and to the computational form used to obtain the answer.

1. Decimal point placement errors About 40% of the errors involved misplace-ment or omission of the decimal point in the answer. When the decimal point was mis-placed, it was possible to identify a number of incorrect "rules" students had used (Table 4). In most cases, it appeared that students had some knowledge or recollection of the counting rule but were unable to apply it correctly with certain pairs of factors, par-ticularly where zeros were involved. Only seven of the 150 students made a decimal point error in calculating 4.5 x 3.92, but 25 students misplaced the decimal point in the answer to 7.29 x 0.01.

28

TABLE 4 Examples of Decimal-Point-Placement Errors

Incorrect counting rule 3.92 Make both factors have the same number of decimal places 4.50 (2) by annexing a zero to 4.5. The answer will also have two decimal places. 19600

15680 1764.00

The zero annexed to 0.06 is used in the algorithm but is 0.009 not counted in placing the decimal point in the answer. 0.060

0000 00540

000000 0.00540

To multiply by 10, add a zero. 6.14 10

6.140

7.29

Count only non-zero decimal places (3). 0.01729

0000 00000

00.729

Count the number of digits in the factors (eight) to deter- 7000 x 0.002 = mine the number of zeros before 14. .0000000014

Incorrect application of counting rule 7.29 The four decimal places are counted from the left. 0.01

- 0072.9

The five decimal places are produced by adding zeros to 0.060.009 the right of 54.

.54000

Two decimal places are needed; add two zeros to the right 6.14 of computed numbers. - 10

614.00

2. Algorithm errors Errors involving use of an incorrect algorithm or misalignment of the partial products accounted for about 20% of the total (Table 5).

3. Calculation errors This category included basic multiplication fact mistakes (many involved 0 or 1) and carrying (renaming) errors (Table 6). Twenty-seven per cent of the errors classified were of this type.

29

TABLE 5 Examples of Algorithm Errors

Incorrect algorithm Answers to 17 x 2 and 6 x 4 are written in the same row.

Zero is added after the 6.

4 is added to 1000.

Columns misaligned Second partial product should be 24686.

Decimal point is retained (incorrectly) in partial products.

A zero is omitted in the second row because of the zero in the basic fact (30).

Extra zero is placed in third partial product.

.617

.420 .024340

6.14 x 10 = 60.14

1000 x 0.04 = 1.004

617 42

1234 2468 3702

6.14 10

0.00 6.14 6.14

600 50

000 3_000

3000

0.420 0.617 2840

04200 2520000

2.527040

4. Miscellaneous errors Three other types of incorrect answers were observed: mistakes resulting from poorly formed numerals (for example, 0 mistaken for 6); questions in which numerals had been incorrectly copied when the problem was written in vertical form (digits reversed, decimal point misplaced, zeros added or omitted), and exercises that were incomplete or not attempted.

Implications for Teaching Cox (1975) suggested that a teacher should look for three basic types of computational errors in analyzing a pupil's work. System-atic errors show a pattern of incorrect responses and occur in at least three out of five problems of a particular kind. Random errors also occur in at least three out of five problems, but no pattern is discerned. Care-less errors occur in one or two out of five problems and result from a lapse in atten-tion rather than from lack of knowledge.

30

TABLE 6 Examples of Calculation Errors

Basic fact errors 360 5x0=5

1805

7x110.81

7 5.61

9 x 6 = 56 0.009 x 0.06 = .00056

0.5 Carrying errors Carried number is added to 4 rather than to 0 (or 4 x 0 = 60 4). 1

0.002 Carried number is multiplied rather than added (or added 7000 before digit is multiplied). 0074.000

Carried numbers are not added to 7 x 4 or 7 x 0. A;

Carried number from first partial product is used in second 32 partial product. 4.5

1960 1578

Carried numbers in column addition not used. 3.92 4.5

1960 1568 16540

Carried number is give place value position in adding .42 partial product. .617

294 42

252 27714

7.00 0.4 0.420Examination of test papers produced exam- 0.81 0.2 0.617 pies of all three types of errors. One Grade - _____ 6 student (not included in the experimental 700 0.8 2940

sample) had 11 of the 12 questions wrong. 56000 04200

The error pattern is clear in the following 567.00 252000

three examples of her work. 259.140

31

It is apparent that the student knows her basic facts and how to use the multiplication algorithm, but she used an incorrect rule for placing the decimal point in the answer. Remediation would involve first having the student recognize. that her method does not produce right (reasonable) answers. The proper procedure would then be taught in a meaningful way, rather than as a rule to be memorized.

Such examples need to be discussed, and effi-cient procedures taught and practised. All questions in which one factor is a multiple of 10, or both factors contain only a single non-zero digit, should be solved directly by using rules for adding zeros and counting decimal places; the vertical algorithm should not be used. For example:

7000 x 0.02

14 - 14000 - 140.00 - 140

Random errors are more difficult to remedi-ate as no one problem can be isolated. Careless errors do not require reteaching, but students should be asked to identify their own mistakes.

The oral interview (Schoen, 1979) is an effec-tive technique for diagnosing difficulties and assessing mathematical understanding. Hav-ing a student talk out loud as he or she works through an example can help the teacher identify and isolate specific strengths and weaknesses.

Certain kinds of errors, which for an individ-ual are considered careless or random, nevertheless occur frequently for a group. Teachers should be aware of these and discuss them in class. For example, a com-mon incorrect answer for 600 x 50 was 3000. A zero was missed since the basic fact ends in a zero.

As previously noted, a large percentage of the students subjected themselves to un-necessary work and created new opportuni-ties for error by using in their algorithms a zero preceding a decimal point or zeros needlessly annexed to a factor.

2.4 3.92 0.5 4.50

120 000 000 1960

1568

17.6400

Many of the incorrect answers obtained by students were unreasonable. Consider the following examples from the study.

1000 3.92 6.14 x 10 = 6.140

0.04 4.50

0000 000

00000 19600

000000 1568000

0000.00 158.7600

Students need to be encouraged to estimate answers and to check for reasonableness of results in multiplication situations. Estima-tion is of increased importance today, since computations involving factors with two or more non-zero digits will most often be done using a calculator. Instruction in decimal multiplication must emphasize estimation strategies and provide students with a quan-tative awareness of the numbers they use.

References Cox, L.S. "Diagnosing and Remediating Systematic Errors in Addition and Subtrac-tion Computations." Arithmetic Teacher, February 1975, 151-157.

Schoen, H.L. "Using the Individual Inter-view To Assess Mathematics Learning." Arithmetic Teacher, November 1979, 34-37.

Vance, J.H. Diagnosing Pupil Performance in Decimal Multiplication: Computation and Estimation. Vancouver: Educational Research Institute of B.C., Report Number 83:13.

32

The Terrible Demise of an Algebra Flunk-Out

Susan Quinn

Susan Quinn is now enrolled in Grade 12 at McNair Senior Secondary School in Richmond. She wrote this story for a creative writing class shortly after completing Bob Campbell's Algebra 11 course, and he suggested that it would be a good article for Vector. The editors agree, and we encourage other aspiring

student authors to write for us.

The day had started out normally enough. I woke up twenty minutes late, as usual, had the proverbial bowl of soggy cornflakes for breakfast, and arrived for my first class of school ten minutes late—as usual. English was not an easy subject to start the day with, but I had managed this far into the semester; so I was doing all right. Then the bell rang. My heart stopped. My worst fears were coming true as I realized what I had to do next . . . Write an algebra exam!

I walked slowly down the hail to the math room. I sat down warily in my seat as Mr. Singleton handed out the dreaded test sheets.

For one solid hour, I sweated over this test, with visions of quadratic formulas, trinomial squares, and radical equations swimming around my head, confusing me all the more. Finally, the last inequality was solved, and I handed in my paper and left.

I opened my locker, sighing with relief, then gasped. Little graphs were sliding and

stretching all over my locker! Frantically, I tried to swipe them off my shelves, but it was too late—they were all over the place! Little graphs shrinking, flipping, always changing shape, covering everything—even my gym strip, and my peanut butter sand-wich! I groaned. Suddenly, I felt a tap on my shoulder. I whipped around, slamming the locker door amid tiny little screams. Standing directly behind me was a giant graph propped upon two unknown values, looking me straight in the eye. I gulped.

Before I could utter a sound, two ordered pairs grabbed me and shoved me into the locker. I cringed, fully expecting to feel the mini-graphs crawling all over me any second. Instead, I got the weird sensation of floating through space while sitting on a waterbed. Some strange dream . .

All of a sudden, we hit something—hard. I shut my eyes and screamed. The ordered pairs clamped my mouth shut, and a voice hissed in my ear, "Stop it, stop it!"

33

I opened my eyes and shut my mouth, then opened it again, as I stared in wonder at the world I was in. Metric rulers for trees, crumpled pieces of old worksheets as bushes, spilled Liquid Paper for rivers, eraser shav-ings for grass. What did I get myself into?

"Where am I? What am I doing here, and how did I get here?"

"This is a world of linear systems, and I am the 'prime factor.' My court and I have determined that you are a complete empty set when it comes to algebra, and you are a disgrace to Pythagoras, Euclid, and all the other math-magicians! How anyone could manage to botch a simple little operation such as complex factoring is beyond my comprehension! You must be punished for your vulgar display of incompetence. To the graphing room! You are going to pay for this!"

"No! Please, anything but the graphing room! Please! Oh no .

They dragged me down a long corridor, the walls lined with scrap paper of previous vic-tims of the graphing room: I studied them frenetically, trying to absorb everything I could in hopes of finding the answer to get me out of this hell as soon as possible.

We reached the door to the graphing room. The guard on duty took one look at the file he had on me, and burst out in laughter. "Hey, kid! Good luck! You'll sure need it! Ha, ha, ha." He opened the door and shoved me in. This was it.

The room was furnished with a cardtable and a hard wooden chair. On top of the table was a pile of paper, a brand new pink eraser, a pack of sharpened pencils, and an ancient calculator with dead batteries.

An image appeared on the far wall. "You have exactly seventy-five minutes to com-plete the exam. You may solve the questions however you wish, but you must come up with the correct solutions, then graph the last one to the nearest unit, on a sheet of blank paper. If you cannot solve it, the result will be an endless life in our world, for I will reduce you down to a lesser-than sign. You may begin."

I sank into the chair and gaped at the equa-tions that suddenly materialized in front of me. How would I ever solve them in time?

Determinedly, I gripped a pencil, and began. The exam started out easily enough, with a few simple Y = MX + B questions, and some polynomials, which I completed with no problems. The trinomials were a little more difficult, and I stumbled through the segment of "solving the difference of two squares," but I was still making good time. Then came the final section: graphs and parabolas.

I studied all the transformations disbeliev-ingly for the next fifteen minutes, mentally jotting down the easier equations, and try-ing to visualize how the parabolas would shift. "If Y = 2 X2, the vertex will move this way, and Y ½ X2 will move it this way

." I mumbled to myself, sketching out the rough form of a parabola on the tabletop. "Now, using the first equation, the whole thing should move this way. . . or maybe it moves this way . . ." It was no use, I cradled my head in my arms and started to weep.

The image appeared on the screen again. "You have only thirty-two and one half minutes left. I suggest you get cracking."

I glared at the fading face and stuck out my tongue. The calculator flew across the room, smashing against the dull brick wall. I looked at my arm in amazement.

34

The image appeared on the screen again. "You have only thirty-two and one half minutes left. I suggest you get cracking."

I smirked, bouncing slightly on the balls of my feet, full of confidence. I should have known better.

I glared at the fading face and stuck out my tongue. The calculator flew across the room, smashing against the dull brick wall. I looked at my arm in amazement.

Suddenly, I was jerked back to the present, aware that the "prime factor" was issuing a command. "Men, get the torture chamber ready. This paper is incorrect."

Once again, I picked up the pencil, still feel-ing a bit unsure of myself. I started scribbling furiously, occasionally taking abhorring glances at the hateful video screen, as all those jumbled thoughts began to sort them-selves out, piecing themselves together like a jigsaw puzzle.

"Okay I'm calm. Now, Y = (X - 5)2 moves the graph five units to the left so the vertex is at negative five on the X-axis, and if I solve this equation like this, I can move it down to negative six on the Y-axis! Bingo! I'm down to the last part; graphing it."

I glanced at the clock in the corner that hadn't been there long. Five minutes left!

I got up, stretched my stiff legs, and walked over to the remains of the shattered calcula-tor. I desperately needed a straight-edge, which the guard had conveniently neglected to supply me with to complete my graph.

Meticulously, I drew out and measured my precious graph, and had just finished when the door burst open, and two negative ex-ponents grabbed my papers and pencils. The ordered pairs were back, flanking me on either side. The "prime factor" entered the tiny cell.

"Time's up. I must congratulate you on com-pleting our 'little' exam. You are one of the elite few. Now all that is left is to mark the graph."

"What? How can that be possible? I checked everything, drew it so perfectly—"

"If you will notice, the equation of Y = (X - 5)2 moves the graph horizontally to the right, not to the left. In our books, this paper is inaccurate." He shoved it under my nose.

"Oh no! How could I have been so stupid?"

They dragged me, kicking and screaming, to the racks. So commenced the worst torture I have ever been through, and ever hope to go through. They made me solve quadratic equation after quadratic equation, and I fac-tored until I turned blue in the face. It was unbearable. Finally, I went over the edge.

"I don't want to conjugate your vertically replaced factoring problems, and the hori-zontal flipping vertexes are function! The ex-traneous discriminate of the slope-intercept radicand can terminate, and you can solve your Y = MX + B and transform it, for all I care!"

The "prime factor" got a steely look in his eyes, and his jaw set as he slowly raised his arm. He mumbled something unintelligible, and I felt a rumbling in my stomach. I glanced down, and froze in horror as I real-ized what was taking place. I was slowly changing into the shape of a lesser-than sign! I screamed. Sure, I wanted to lose weight and look thin, but not this thin! I blacked out.

35

I came to, feeling a gentle tap on my shoulder.

"Hey, are you all right?"

"Well young lady, from now on, I suggest you eat a well-balanced meal in the morn-ing. I will not tolerate people passing out in my classes!"

I sat up, and found myself in the medical room, surrounded by the nurse, the school principal, and Mr. Singleton.

"I know you have an extreme dislike for algebra dear, but the test wasn't so difficult that you had to faint in the hallway. Now what happened?"

"I, uh well. . . I didn't have any breakfast this morning, and I was really hungry. That's all," I said, lying through my teeth.

"Yes sir." I turned to the nurse and principal..

"May I please go back to my locker? I'm really not feeling well, and I'd like to get my things and go home." Shakily, I got up, and walked out of the room and around the corner to my locker. When I got the lock open and reached in to get my books, I froze. There were still two little graphs sliding all over my peanut butter sandwich! I grabbed my jacket and bolted out of the school. .

36

Streaming in Mathematics at the Junior Secondary Level

Hugh S. Elwood

Hugh Elwood is the mathematics department head at Edmonds Junior Secondary School in Burnaby.

The following paper criticizes the Ministry of Education's analysis of the current situa-tion regarding streaming and achievement in junior secondary school mathematics, and suggests a research-based plan for grouping students to maximize their achievement in mathematics at the junior secondary level.

On page 7 of the Secondary School Gradua-tion Requirements Discussion Paper, it is stated that:

The overall effect of this streaming is uncertain. On the one hand streaming systems that assign students to classes based on predictions of their academic performance allow the creation of a very demanding curriculum for the most able students, and standards can be seen as very high. On the other hand, curriculum expectations for other students are not as high and cor-respondingly, neither is their academic performance.

There are three inaccuracies in the above quotation. First, public schools assign students to classes not on the basis of predic-tions of academic performance, but rather on the basis of past achievement in mathe-matics. Second, all students are required by the ministry to study the core curriculum in mathematics, and all students, therefore, study the same curriculum except for those more able students who require an enriched and more rigorous mathematics curriculum.

Third, the comments on standards if they are based on the 1983 Mathematics 10 Achieve-ment Test (the reader is not given any indica-tion of the source) are very strange, because the 1983 Mathematics 10 Achievement Test was not based entirely on the government's mandatory core curriculum; therefore, how can the statement be viewed as valid? Clearly, the above quotation, which con-tains three inaccurate if not misleading statements, shows that the ministry is uncer-tain about streaming in mathematics and perhaps even misguided by its own officials.

In the next paragraph, on page 8, it is stated that:

The Ministry of Education recognizes that current practice in many schools is inconsistent with provincial policy. This paper advances the position that a common junior secondary curricu-lum that sets high expectations for all students in a core academic program and that does not disqualify, on a systematic basis, any student for any senior secondary program is the preferable path.

Both statements by the ministry are inac-curate. Provincial policy is followed by the public school system. No student is dis-qualified from any senior secondary program. All students are taught the core curriculum in mathematics; therefore, they should be eligible for any senior secondary

37

course. If they are not, then it is the fault of the Ministry of Education for not pro-viding a more enriched and rigorous Core Curriculum.

On page 8 the ministry states:

• . . it is recognized that there will be a continued need to modify courses on an individual basis to provide achiev-able goals for some students.

While this may be true for special needs students, the statement is not made in that context; therefore, one must assume that it is made in reference to the general school population. How does the ministry expect this to happen with much larger classes and dwindling availability of materials and teaching resources? Even if it were possible, would individualization, with smaller classes and more resources, affect achievement? A summary of research completed by the Association for Supervision and Curriculum Development (ASCD) of such educators as Medley, Mortimore, Rosenshine, and Edmonds has found several factors that in-fluence teacher and school effectiveness—individualization is not one of them. The ASCD report found that among other factors: (1) the -most efficient method of teaching is group instruction. (2) high teacher expectations improve achievement.

What does some of the other research say? E.G. Begle, in 1975, reviewed the research of 25 studies on ability grouping and con-cluded that homogeneous grouping based on past performance in mathematics should be used for high-ability students. In 1979, Begle further reports and confirms his findings of 1975 that high-ability students should be grouped homogeneously and that the re-maining student population when grouped

heterogeneously achieve as well even though the high-ability students are not in class with them. The latter statement is supported by the ASCD findings, which show that disad-vantaged students have higher achievement if teacher expectations of their performance is high. An efficient method of attaining this would be to group heterogeneously the low-and middle-ability students. The expecta-tions of the teacher would be higher for the lower-ability students than if they were grouped homogeneously themselves, and their achievement would be higher. The achievement of the middle-ability students would be unaffected.

One of the goals of the Ministry of Educa-tion is to promote and enhance the educa-tion of gifted students. To assist the ministry in achieving this goal, high-ability students should be grouped homogeneously based on past performance in mathematics to maxi-mize their achievement.

The Ministry White Paper states on page 8:

it is obvious that the subject of streaming at the junior secondary level requires further review.

What should be done? 1. The core curriculum for all math students at the junior secondary level should be suf-ficiently rigorous and enriched to allow all students the opportunity to enter senior math courses. 2. Lower-ability students-should be grouped with middle-ability students heterogeneously based on past performance in mathematics to maximize the achievement of the lower-ability students. There will be no effect on the achievement of middle-ability students. 3. High-ability (gifted students) should be grouped separately from other students in order to maximize their achievement and potential in mathematics.

38

References Elwood, H.S., Grouping Students for Instruction in Secondary Mathematics, May 1978.

Begle, E.G., Ability Grouping for Mathe-matics Instruction. A Review of the Empirical Literature, SMSG Paper #17, December 1975.

Begle, E.G., Critical Variables in Mathe-matics Education: Findings from a Survey of the Empirical Literature, MAA/NCTM, 1979.

Gall, M.D., "Instructional Policy, Issues in Mathematics Education." Educational Leadership, December /January 1984.

Association for Supervision and Curriculum Development, Teacher and School Effec-tiveness, videotape, March 1981.

Province of British Columbia, Ministry of Education, Secondary School Graduation Requirements, A Discussion Paper, April 1984.

UNDERGRADUATE PROGRAMMES IN THE

MATHEMATICAL SCIENCES AT

CANADIAN UNIVERSITIES

PROGRAMMES DE SCIENCES MATHEMATIQUES DU PREMIER CYCLE

DANS LES UNIVERSITES CANADIENNES

Canadian %ia:h.n,a,icalSa,uqr Sadi'iMa'hemaiiqav du Canada

The Canadian Mathematical Society has produced this 36-page booklet as an initial step toward an extensive publication that will contain complete details of Canadian university mathematics programs.

Copies of the booklet may be obtained (at cost) from: Ed Barbeau Department of Mathematics University of Toronto Toronto, ON M5S 1A1 (416) 978-5164

39

Mathematics Teacher Qualifications: A Principal Reacts

Ed Collins

Ed Collins is principal at Carson Graham Secondary School in North Vancouver.

[Editors' note: This article is a reaction to the report on the Qualifications of Teachers of Mathematics in British Columbia, which was produced by the BCAMT and circulated to all members in May 1984.]

In response to your request for reactions to the "Qualifications of Teachers of Mathe-matics in British Columbia" paper, I thought I should take the time to outline my thoughts.

First of all, I think the report makes some very significant points and raises some equally important questions. I am not con-vinced, however, that the report deals effec-tively with the full range of factors causing the problems outlined. Nor am I convinced that practical, workable, solutions are being proposed that consider the present situation as it really is and as it is really likely to be.

It is, however, not my intention to dispute the conclusions or to criticize the recommen-dations contained in the paper. I have little dubt that the authors are entirely accurate when they state (for example) that "Many teachers without sufficient mathematics skills are being asked to teach mathematics in the classrooms of this province." I also agree that "students at all levels should be taught by fully qualified teachers." What, then, is the issue?

I would like to discuss Recommendation 5a (in particular) since it is directed toward school boards and principals—the latter, my

present role. This particular recommenda-tion asks that an attempt be made to "en-sure that only persons with academic and professional training in mathematics educa-tion (be) permitted to teach mathematics."

Frankly, I think that it should be conceded that, as a general principle, an attempt is made by school principals to ensure that properly qualified teachers teach in all of the school's subjects. The question really is "why does this not always happen?" When stu-dents are not taught by subject "experts" in any particular subject, it is likely the result of one or more of the following factors: • a shortage of qualified teachers on a school's teaching staff in that particular discipline. • a dearth of qualified teachers in the school district in that discipline. • inappropriate distribution of talents and competencies within a school district. • priority being given to individual profes-sional preferences over program or curricu-lar needs of the school or districts. • the expansion or growth of a program without a commensurate growth in subject competencies; that is, hiring has not kept pace with increased clientele in a particular subject.

When the foregoing factors are considered in relation to teaching mathematics, the following conclusions are evident: • There has been a growth, or increase, in the emphasis given to mathematics and an attempt made to ensure that mathematics,

40

in one form or another, be taught to all

students for 11 of their 12 school years; that is, there is a demand for more mathematics to be taught. • Many students apparently cannot "han-dle" the academic mathematics. As a conse-quence, there is considerable movement toward "streaming," beginning at Grade 8 level or even earlier. • Most teachers who have the necessary qualifications as mathematics specialists prefer to teach the "academic" mathematics courses. • A student's academic (post-secondary) options are severely limited if he/she does not complete Algebra 11. As a result, there is much pressure to stay in "academic" math. Failure to do so generally connotes failure as a student or at least a big step in that direction.

First, I think it improbable that we will have voluntary resignations to provide jobs for those who may have superior qualifications in mathematics. That such a change can be brought about through legislation is, in my opinion, extremely unlikely.

Retraining, in turn, does not hold out too much hope to me, since local programs of this nature have not been successful. I do not anticipate a change in this, since the concept of job protection through seniority com-bined with a "willingness to try" will likely see a continuation of the present circum-stances. I think that a part of the answer might lie in a re-examination of the purposes of mathematics education: What are we educators trying to accomplish? What are our goals? For example, while there is little doubt that students who gain credit for Algebra 11 and/or Algebra 12 are compe-tent in an academic sense, it seems to me to be inherently wrong that credit in this course be indiscriminately used as a screening device for a myriad of post-secondary pro-grams. Many of these programs have little

or no connection to the study of, or prior accomplishments in, Algebra. I agree with the paper's assertion that "society of the present and future needs an ever-increasing supply of technologically trained members." I'm not convinced that this will be accom-plished through the "streaming" approach, which pits Algebra against computer studies, consumer math, trades mathematics, or accounting. This, by the way, seems to be the approach suggested by the new "white paper" on secondary school graduation requirements.

I think it can be argued that the interests of students, teachers, and society might be better served if: • Algebra were not used as an intelligence test or screening device by post-secondary institutions for the recognition of competent students. To effect this would require exten-sive lobbying with the post-secondary institutions. • Arts and science students were allowed to select any Grade 11 mathematics course to complete that requirement for graduation and college entry. • Students were to select mathematics courses on the basis of interest. Thus arts and science students selecting Algebra on this "elective" basis would likely be motivated as well as generally competent. • Students selecting the other mathematics courses were to recognize them as having equal status and value. • Teachers were more interested in teaching these other courses because of the courses' equal or equivalent status.

The improvements suggested in the paper for the teaching of mathematics can be accom-plished only through a redirection of energies on the part of the British Columbia Association of Mathematics Teachers. This extremely strong professional group could, I believe, effect some important educational gains for all students.

41

• The 1984 Euclid

Mathematics Contest

George Bluman

George Bluman is an associate professor of mathematics at UBC.

In the 1984 Euclid Mathematics Contest of the Canadian Mathematics Competition for Grade 12 students, as in previous years, the performances of B.C. students and B.C. schools ranked well ahead of those of the rest of Canada. The contest was supported nationally by the Waterloo Mathematics Foundation, the University of Waterloo, Dominion Life Assurance Company, Cana-dian Imperial Bank of Commerce, IBM Canada, Canon Canada Inc., and Mutual Life of Canada; in B.C., by the Department of Mathematics and the School of Engineer-ing at UBC.

By any criterion (e.g., Tables 3 to 6), B.C. results continue to be superior. (Keep in mind that as of the first quarter of 1984, B.C. had 11% of Canada's population.)

TABLE 1 Student Enrolment

B.C. percentage B.C. Canada of Canada total

1980 291 3178 9% 1981 337 3596 9% 1982 552 4667 12% 1983 743 5301 14% 1984 1097 6649 16%

TABLE 2 School Enrolment

B.C. percentage B.C. Canada of Canada total

1980 42 440 10% 1981 52 513 10% 1982 54 591 9% 1983 74 661 11% 1984 98 765 13%

TABLE 3 Percentage of Students from B.C. on the

Canadian Student Honor Roll of 100

1980 14% 1981 14% 1982 22% 1983 19% 1984 21%

TABLE 4 Percentage of Schools from B.C. on the

Canadian School Honor Roll of 50

1980 18% 1981 18% 1982 30% 1983 22% 1984 28%

42

TABLE 5 Median School Team Score on the Euclid Contest (Maximum Possible Score is 300)

B.C. Canada 1980 117 95 1981 141 122 1982 - 178 138 1983 152 130 1984 173 140

TABLE 6 Year-by-Year Ranking of B.C. Students

Year-by-year a student placing in the top 10% in B.C., placed nationally in the top:

1980 7% 1981 8% 1982 6% 1983 7% 1984 6%

For B.C. schools, the written part of the Euclid Contest was marked by a team including: Dr. Andrew Adler, Department of Mathe-

matics, UBC Dominic Alvaro, Argyle Secondary, North Vancouver.

Leo Boissy, Churchill Secondary, Van-couver.

Dr. John Coury, Department of Mathe-matics, UBC.

Sharon Cutcliffe, Burnaby North Senior Secondary, Burnaby.

Ian de Groot, Sutherland Secondary, North Vancouver.

Larry Dubois, Centennial School, Coquit-lam.

Dr. Harvey Gerber, Department of Mathe- matics, SFU.

Cory Ristock, Mathematics student, UBC. Al Sarna, University Hill Secondary,

Vancouver. Jack Schellenberg, John Oliver Secondary,

Vancouver. John Stigant, Hillside Secondary, West

Vancouver. John Turnbull, McNair Senior Secondary,

Richmond.

The following results are based on the mark-ing at UBC. Among the 39 schools with 10 or more entries, the highest ranking schools (based on the results of the 10 highest scoring students) were: 1. St. Michael's, Victoria (78114) 2. Churchill, Vancouver (710) 3.Handsworth, North Vancouver (6441/2) 4. Argyle, North Vancouver (634) 5. Carson Graham, North Vancouver

(613) 6. Steveston, Richmond (603) 7. Killarney, Vancouver (600N) 8. John Oliver, Vancouver (580) 9. Sentinel, West Vancouver (574)

10. Burnaby North, Burnaby (5501%) 11. Penticton (548) 12. South Delta, Delta (5391%)

Among the 77 schools with five or more entries, the highest ranking schools (based on the results of the five highest scoring students) were: 1. St. Michael's (420) 2. St. George's, Vancouver (400) 3. Churchill (378Y4) 4. L.B. Pearson, Victoria (3771/2) 5. Handsworth (356) 6. Nanaimo (351Y2) 7. Sentinel (3473/4) 8. Port Moody (3461%) 9. Argyle (344Y4)

10. Steveston (3431%) University Hill, Vancouver (3431%)

43

Among the 92. schools with three or more 6. Handsworth (2281/2) entries, the highest ranking schools (based 7. Sentinel (2251%) on the results of the three highest scoring 8. Nanaimo (2243/4) students) were: S University Hill (2243/4) 1. St. Michael's (260) 10. Argyle (2201/2) 2. St. George's (251 3/4) 11. Port Moody (2191/2) 3. L.B. Pearson (237) 12. Steveston (2183/4) 4. Burnaby Central, Burnaby (2301%) 13. Burnaby North (2161%) 5. Churchill (229 3/4) 14. Centennial, Coquitlam ' (216)

The top 50 students in B.C. (in the case of equal scores, the order is according to the higher written score) are:

1. Huang, Peter D.W. Poppy Langley [93] 2 . Yen, Lily Burnaby Central Burnaby [93] 3. Feir, Bryan St. Michael's Victoria [92] (Grade 10) 4. Westwick, Paul St. George's Vancouver [89¼] 5. Palmer, Aaron Burnaby North Burnaby [87] 6. McCorquodale, Peter Handsworth North Van. [86Y4] (Grade 11) 7. Tang, Kunikyo St. Michael's Victoria [85] 8. Turner, Geoffrey L.B. Pearson Victoria [85] 9. Ma, Alex St. Michael's Victoria [83] - Marziali, Andre Argyle North Van. [83]

11. Wong, Norman Nanaimo Nanaimo [83] - Mounthanivong, Bounsou Sardis Sardis [83] (Age 19)

13. Neher, Darwin University Hill Vancouver [82¼] 14. Cheung, Johnson St. George's Vancouver [811%] - Maskall, Douglas St. George's Vancouver [81¼]

16. Quon, Kim St. Michael's Victoria [81] 17. Andrews, Barry Centennial Coquitlam [79] - Kasapi, Steven St. Michael's Victoria [79] (Grade 11)

19. Obura, Yumi L.B. Pearson Victoria [79] 20. Balzer, Michael Alberni Port Alberni [78] 21. Blumenfeld, Aaron St. George's Vancouver [77¼] 22. Yang, Joseph Churchill Vancouver [77] (Grade 11) 23. Chang, Francis Sentinel West Van. [77] 24. Watt, Byron Handsworth North Van. [77] (Grade 11) 25. Tsai, David , Churchill Vancouver [761/2] 26. Hsiung, Robin Churchill Vancouver [76¼] 27. Rabbani, Firouzeh West Van. West Van. [76¼] 28. Rabbani, Farhang West Van. West Van. [76¼] - To, Mary Norkam Kamloops [76¼]

30. Ho, Tommy Churchill Vancouver [76] 31. Denny, Kevin Steveston Richmond [76] - Wong, Hubert Quesnel Quesnel [76]

33. Lau, Benjamin Steveston Richmond [751/2] 34. Chin, Ronald Nanaimo Nanaimo [75¼] 35. White, Murray Brentwood College Mill Bay [75] 36. Lee, David Port Moody Port Moody [75] 37. Koga, Yotto University Hill Vancouver [74¼] (Grade 11) 38. Wai, Robert Sentinel West Van. [74¼] - Wong, Fred Vancouver Tech. Vancouver [74%] (Grade 11)

40. Steere, Jonathan Vernon Vernon [74] 41. Behrouzan, Mansour Sentinel West Van. [74]

44

42. Lo, Terence Sutherland North Van. [74] 43. Booth, Paul St. Michael's Victoria [731/4] 44. Chan, Michael L.B. Pearson Victoria [73] 45. Ho, Francis Richmond Richmond [73] 46. Miura, Brian Churchill Vancouver [73] 47. Lai, Winnie Churchill Vancouver [73] 48. Sclater, Rod Port Moody Port Moody [73] 49. Haydar, Lily Burnaby Central Burnaby [73] 50. Sturve-Dencher, Goesta St. Michael's Victoria [73] (Grade 11)

The following accolades resulted from outstanding performance in the 1984 Euclid Contest -1. The top 10% of all B.C. entrants and the top-ranking student (on the written part of the contest) in each school with five or more entries received book prizes (141 students). 2. A school ranking in the top 10 in B.C. (based on 3, 5, or 10 entries) received a book prize (16 schools). 3. Students scoring well in the contest were invited to the honors calculus courses Mathematics 120/121 at UBC (312 students). 4. Grade 10 or 11 students scoring well in the contest were encouraged to study calculus and were invited to the UBC

CONNECT program for bright B.C. high school students, June 24-29, 1984, with all fees covered, and free accommodation and transportation (25 students). 5. The Euclid Contest was an important criterion in selecting the 26 winners of the UBC entrance scholarships, each worth at least $10,000 (over four years) provided a student maintains a first-class standing in a full program of study. Of the 26 winners, 17 wrote the Euclid Contest and all 17 had high scores resulting in congratulatory letters.

List of Participating B.C. and Yukon Schools in the 1984 Euclid Contest for Grade 12

Abbotsford Coquitlam Esquimalt Mt. Boucherie Nelson Abbotsford Centennial Esquimalt Spring Valley L.V. Rogers

Port Coquitlam Armstrong Port Moody Golden Langley New Westminster Pleasant Valley Golden Aldergrove New Westminster

Courtenay D.W. Poppy North Vancouver Burnaby G.P. Vanier Grand Forks Langley Argyle Alpha Grand Forks Mountain Carson Graham Burnaby Central Cranbrook Handsworth Burnaby North Mt. Baker Hazelton Lumby St. Thomas Aquinas Cariboo Hill Haze l ton Charles Bloom -,eycove

Burns Lake Dawson Creek Hope Maple RidgeSutherlandWindsor

Lakes District South Peace Hope Garibaldi

Delta Delta Kamloops

McKenzie McKenzie

Penticton Clearbrook North Delta Kamloops Penticton M.E.I. Seaquam Norkam Mill Bay Port Alberni W.J. Mouat South Delta Westsyde Brentwood College Alberni

Comox Duncan Kelowna Nanaimo Prince George Highland Queen Margaret's Kelowna Nanaimo Duchess Park

45

Prince Rupert Salmon Arm Trail St. George's L.B. Pearson Prince Rupert Salmon Arm J. L. Crowe Templeton Reynolds

Sardis Tupper St. Michael's Princeton Sardis Vancouver University Hill Spectrum Princeton

Crofton House Vancouver College Sidney

- Gladstone Vancouver Technical West Vancouver Parkland

Quesnel John Oliver Windermere Winston Churchill

Hillside Correlieu Surrey Killarney Sentinel

York House Quesnel Frank Hurt Little Flower West Vancouver

Fraser Valley Academy Christian Lord Byng Vernon Winfield

Richmond Semiahmoo Magee Vernon George Elliot McNair Notre Dame Richmond Terrace Point Grey Victoria Yukon Steveston Caledonia Prince of Wales Claremont Del Van Gorder

Source Materials for Secondary School Mathematics Problems

1. Australian Mathematics Competition 14. G. Polya and J. Kilpatrick. "The Stanford (Canberra College of Advanced Education, University Competitive Examination in P.O. Box 1, Belconnen, A.C.T. 2616 Mathematics," AMM, 627-640, June-July Australia). 1973.

2. 1001 Problems in High School Mathematics: 15. "ICMI Report on Mathematical Contests in Book I, II, and III. (Canadian Mathematical Secondary Education," Educational Studies Society, 577 King Edward Avenue, Ottawa, in Mathematics (ESM) 2, 80-114, 1969. Ontario KiN 6N5). 16. F. Swetz and A. Chi. "Mathematics Entrance

3. "Mathematics Prize Competition for Los Examinations in Chinese Institutions of Angeles High Schools," Mathematics Higher Education," ESM, 14, 39-54, 1983. Magazine, 77-82, November-December 17. I. Wirszup. "The School Mathematics Circle 1955. and Olympiads at Moscow State Universi-

4. T. Rado. "On Mathematical Life in ty," Mathematics Teacher (MT), 194-210, Hungary," American Mathematical Monthly April 1963. (AMM), 85-90, February 1932. 18. D.W. Stover. "Pretesting for the College

"Mathematical 5. B.V. Gnedenko. Education in Boards," MT, 537-541, November 1969. the U.S.S.R.," AMM, 389-408, June-July 1957. 19. C. Jones, M. Rowen, and H. Taylor, "An

6. R.C. Buck. "A Look at Mathematical Corn- Overview of the Mathematics Achievement petitions," AMM, 201-212, March 1959. Tests Offered in the Admissions Testing Pro-

7. J. Aczel. "A Look at Mathematical Competi- gram of the College Entrance Examination tions in Hungary," AMM, 435-437, May Board," MT, 197-208, March 1977. 1960. 20. J. Goldberg and F. Swetz. "Mathematics

8. J. de Francis. "Mathematical Competitions Education in the Soviet Union," MT. in China," AMM, 756-762, October 1960. 210-218, March 1977.

9. M.N. Bleicher. "Searching for Mathematical 21. S. Conrad, "The Widening Circle of Mathe-Talent in Wisconsin," AMM, 412-416, April matics Competitions," MT, 442-447, May 1965. 1977.

10. J.R. Smart. "Searching for Mathematical 22. J. Becker and K. Hsi. "Mathematical Olym-Talent in Wisconsin II," AMM, 401-407, piad Competitions in China," MT, 421-433, April 1966. September 1981.

11. B. Kneale, "A Mathematics Competition in 23. J. Becker, "1979 National Middle School California," AMM, 1006-1010, November Mathematics Olympiads in China," MT, 1966. 161-169, February 1982.

12. D.W. Crowe. "Searching for Mathematical 24. J. Becker and K. Hsi. "Mathematical Educa-Talent in Wisconsin, III," AMM, 855-858, tion in China—Directions in the 'New September 1967. Period'," International Journal of Mathe-

13. F. Swetz. "The Chinese Mathematical Olym- matical Education in Science and Tech-piads," AMM, 899-904, October 1972. nology, 11 (2), 151-162, 1980.

46

The Third Annual Mathematics Enrichment Conference

Simon Fraser University June 24-26, 1984

Larry Weldon

Larry Weldon is an associate professor in the mathematics department at SFU.

According to the teachers and students who participated, the Mathematics Enrichment conference that took place at Simon Fraser this summer was a great success. The con-ference combined lectures, problem sessions, social events, and tours over a three-day period. About thirty SFU personnel, faculty, staff, and senior students, were involved. Twenty school teachers and 190 students from 75 schools participated in the con-ference. The students were selected by their schools, and while a majority were from the Fraser Valley schools, many students were from all over B.C., and even one was from Whitehorse.

The academic part of the conference focussed on six lectures. The keynote speaker was Professor Laszlo Babai from Eotvos University in Hungary. In his first talk, Professor Babai discussed a problem on round-table seating satisfying constraints regarding which pairs of diners are allowed to sit beside each other. Professor Babai showed that this problem is related to a number of other famous problems, including the four-color problem, the travelling sales-man problem, and class scheduling prob-lems. He showed clearly that no computer was ever going to be able to solve these problems by brute force, and that more clever mathematical strategies would need to be invented.

Professor Babai's second talk concerned algorithms for such tasks as finding a median

of a set of numbers. The general strategy for the algorithms he discussed was called "divide and conquer," the idea being analagous to the well-known trick of finding a word in a dictionary by repeatedly asking which half of a section of the dictionary the word is in. A highlight of Professor Babai's lectures was his collection of charming cartoons featuring Merlin the magician and King Arthur, and a recipe for Irish stew!

The opening talk by Professor Len Berggren focussed on the Babylonian origins of our arithmetic system, including the explanation of why our time and geographic co-ordinate systems use base 60 arithmetic instead of. the decimal system. -

Professor Tiko. Kameda of the SFU depart-ment of Computing Science discussed the rationale of coding electronic signals to allow automatic error detection. It was nice to see the practical application of the binary system which has received so much em-phasis since the advent of electronic computers.

"The impossibility of complete chaos" was the comforting title of Professor Tom Brown's talk. If one puts a long sequence of colored beads on a string, and if there are a finite number of colors and the string is in-finitely long, then certain patterns have to appear over the whole string. Apparently these patterns are the domain of a branch of combinatorics called Ramsey theory.

47

The last talk was given by the conference co-ordinator, Professor Larry Weldon. He described some simple models including one of a traffic queue: the model is complex to study analytically, but it can be studied empirically as if it were a real-world object subjected to scientific investigation. The technique of Monte Carlo simulation was outlined for this purpose.

Most, but not all, of the students were able to follow these lectures, with their sophis-ticated material. The conference is aimed at gifted students with a particular facility in mathematics.

Problem sessions were of several kinds, in-cluding small-group competitions, open-ended discussions, and in one case, an animated film of geometic phenomena. Students were asked to recall what they could from that film.

Tours of the departments of Computing Science, Physics, and Chemistry were provided.

Social events included an evening barbeque and soccer game, a teacher's lunch, and three well-attended buffet lunches. Billets for out-of-town students—provided by in-town par-ticipants as well as SFU faculty, staff, and students—provided memorable experiences for many participants.

A reunion of participants will be held this fall, and a continuation of the conferences is planned for 1985 and beyond. Informa-tion will be distributed to the schools. Harvey Gerber, of the SFU Department of Mathematics and Statistics, will organize the events.

[Editor's note: The six lectures were video-taped during the conference. These will be available for use by the schools. Contact the Department of Mathematics and Statistics at SFU for further information.]

DATES TO KEEP IN MIND

NCTM 63rd Annual Meeting San Antonio, Texas

APRIL 17-20, 1985

24th Northwest Mathematics Conference

Richmond, B.C. OCTOBER 10-12, 1985

UPCOMING

BCAMT EXECUTIVE MEETING DATES

September 29, 1984 December 1, 1984 February 2, 1985 April 13, 1985 June 8, 1985

All meetings are held in a board-room of the B.C. Teachers' Building on Burrard Street.

BCAMT members are invited to attend. Meetings start at 09:30.

48

BCAMT 11th Summer Workshop

Carry Phillips

Garry Phillips, vice-president of BCAMT, teaches at Lord Kelvin Elementary School in New Westminster.

The 11th Mathematics Summer Workshop was held at Palmer Junior Secondary School on August 28 and 29, 1984. This year's event was admirably hosted by the teachers in Richmond under the direction of Conference Chairperson Mary Stewart (Errington Ele-mentary School).

The workshop was sited in the lower main-land again this year, and more than 200 paid registrants attended. In all, nearly 300 mathematicians attended the workshop in- cluding registrants, speakers, publishers and committee members. The Richmond site was very popular with teachers from around the province. In fact 35% of the registrants were our colleagues who travelled from areas such as Victoria, Salmon Arm, Alexis Creek, Pen-ticton, Prince Rupert, Osoyoos, Prince George, Kitimat, and Burns Lake.

The Summer Workshop program commit-tee continued the tradition of providing a program with a wide range of topics relating to all areas of mathematics. Milt McClaren (SFU) gave a thought-provoking address, "Mathematics, Computers and Human Thought," to keynote the start of the workshop. The next two days were filled with 82 sessions. These sessions were devoted to problem-solving, computers, calculators, Math Their Way, math games and activities, practical math programs, con-ics, logarithms, and estimation. This lists only a small sample of the available sessions. Of special interest were the sessions espe-cially tailored to an administrative view-point: "Mathematics Scope-Sequence" and "Trends in Mathematics Education." A

further attempt was made to feature Cana-dian authors. Marshall Bye, Les Dukowski, Frank Ebos, Olive Fullerton, Heather Kelleher and Shawn McPhail added yet another dimension to an already impressive program.

Other highlights of the workshop: • Becky Mills (Ministry of Education) in-troduced the Elementary and Secondary Mathematics Revision Committee members. The subsequent update and question period were very informative. • Twenty publishers, commercial compan-ies, and computer corporations provided an impressive array of print and supplemental materials for review. • The BCAMT Annual General Meeting was held.

The Summer Workshop is primarily for ex-changing new ideas and preparation for the new school year. However, social events are not neglected. Lunches were provided at the school both afternoons for a small charge. Registrants used their lunch hour to meet new people from throughout the province as well as to renew old friendships. Free coffee and donuts were provided by the publishers throughout the conference, so coffee breaks were enjoyable. Two textbook publishers, Houghton Mifflin and Nelson, provided hospitality suites on the first night of the workshop, and a closing wine-and-cheese event hosted by the BCAMT workshop committee was a resounding success.

There have already been numerous favor-able comments about the workshop from

49

shop Committee for their co-operation and hard work in organizing the conference.

P.S. See you all at next year's 24th North-West Mathematics Conference, October 10, 11, 12, 1985 at the Richmond Inn.

many people. I would like to take this opportunity to express our appreciation from both the registrants and the BCAMT executive to School District #38 (Richmond), Palmer Junior Secondary School administra-tion and staff, and the 11th Summer Work-

B.C. ASSOCIATION OF MATHEMATICS TEACHERS Statement of Receipts and Disbursements (Note 1)

for the year ended June 30, 1984

Balance (Deficit) July 1, 1983 $(2,094.62)

Receipts BCTF Grant $3,465.00 Membership Fees (Note 2) 7,671.00 NCTM Refund 1,815.96 Canadian Math Teacher 1,861.20 Publications Sales 218.00 Summer Workshop Income 3,184.84 18,216.00

Disbursements Executive Meetings $ 889.50 Subcommittee Meetings 281.68 Publications—Journals 5,388.22 Publications—Newsletter 1,257.61 Publications—Other 3,523.87 Conferences and In-Service 550.66 Affiliation Fees and Meetings 3,686.24 Operating Expenses 102.13 Other Projects 373.84 16,053.75

Balance, June 30, 1984 $ 67.63

Notes: 1. This statement reflects only funds held by the B.C. Teachers' Federation on behalf of the B.C. Association of Mathematics Teachers. 2. Unearned fees as of June 30, 1984—$3,994.12.

50

New Books Across My Desk

Ian de Groot

Studies in Mathematics Education, Volume 3. The Mathematical Education of Primaiy-School Teachers. Edited by Robert Morris. Published in 1984 by the United Nations Educational, Scientific, and Cultural Organization, 7 Place de Fontenoy, 75700 Paris, France.

This third volume of Studies in Mathematics Education has been prepared as a part of UNESCO's program for improving mathe-matics instruction through the production of resource materials for those responsible for mathematics teaching. It examines the responsibility of primary-school teachers for the mathematics component of the curricu-lum and the implications thereof of teacher education.

In the present volume, Robert Morris, the editor, has unified contributions from 16 countries while maintaining the individual style of each author.

As with the earlier volumes in this series, contributions were solicited from far and wide. The views expressed are from Africa, Asia, the Caribbean, Europe, Latin America, North America and the South Pacific. The contributions combine the views of teachers, teacher educators, curriculum developers, and research workers. They make a rich mixture; yet, surprisingly perhaps, they reflect a remarkable agreement about what primary mathematics should be.

The study is organized such that the first two chapters are general—the responsibilities of the teacher of primary mathematics. The third chapter discusses the environment as a source of problems, and, in contrast, the

fourth chapter is more concerned with what actually happens in the classroom.

The chapter on computers and calculators is likely to prove as controversial to the reader as it is stimulating. It sees the greatest potential of these aids as that of a tool for exploring mathematical ideas, capable of ex-tending enormously the range and scope of mathematical activity in school.

I think that all mathematics educators, primary and secondary, will find this study interesting, thought-provoking, and cer-tainly useful.

Dottori, D., C. Knill, and J. Stewart. FMT Senior: The Ryerson Mathematics Pro-gram. McGraw-Hill Ryerson.

This text is prescribed in several provinces, including Alberta and Ontario, for the senior mathematics course; in addition, I use it often as a source of many good applied problems in Algebra 12 classes.

I like this text for several reasons: • The problems throughout the text are realistic and topical. • The content is ideally suited for the current B.C. Grade 12 curriculum. • The polynomial chapter has as its major objective, the graphing of polynomial and rational functions. • The exponential and logarithmic functions section contains many realistic applied ex-ponential and logarithmic questions. The authors do not waste time in using logs for calculations or in using terms such as characteristic and mantissa.

51

• The section in which calculus is introduced is logically developed and uses a geometric approach to introduce the concept of derivative. • The text is compact, with no unnecessary information or pictures. • The final chapter concerns problem-solving, and a unique collection of problems from many different areas of mathematics is presented.

The publishers tell me that the latest edition of this text is due very early in the new year. I.look forward to this event with keen antici-pation.

Algebra and Trigonometry—Book 2—Structure and Method, new edition. Do!-ciani, Sorgenfrey, Brown, Kane, Houghton Mifflin, 1982.

Reviewed by Don Gordon, Seycove Second-ary. School, North Vancouver.

/I have used a few sections from Chapters 6 and 7 and almost all of Chapters 8-14 in teaching my Algebra 12 classes during the last two years at Seycove Secondary School in North Vancouver. I recommend the text very highly; I think it is one of the very best I've used for any secondary math course.

The only supplementation I've found neces-sary is in "graphing techniques," in which

I've taught a 10-period unit developed using other sources. All other aspects of the Algebra 12 course are dealt with more than adequately by the text. The layout provides for good readability, and the presentations are clear. Oral exercises are a good base for discussion in each section, leading smoothly into the A, B, C, sets of the written exercises. Appropriate questions at each level of dif-ficulty are provided in sufficient numbers. The student text has answers for odd-numbered questions.

The teacher's edition contains many helpful features: teaching suggestions for each sec-tion, extra illustrative examples, suggested time schedules for assignments, and sug-gested assignments. These have been developed with considerable expertise and are particularly useful to someone using the text for the first time. Answers to all ques-tions appear with thequestions or in the margins. Quizzes are provided: at regular in-tervals. Of course, reviews and practice tests are given at the end of each chapter in the student text, and the teacher's edition con-tains chapter tests, cumulative reviews and topical reviews. Over 500 questions and answers are provided.

In summary, I have thoroughly enjoyed using this text, and I think that it would satisfy the needs of Algebra 12 students hav-ing a wide range of ability in mathematics, and that it provides excellent guidance and resource materials for the teacher.

Y84-0068 October 1984

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