Journal of the British Columbia Association of Mathematics Teachers

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Volume 26 Number 1 Fall 1984 B.C. Association of Mathematics Teachers 1984-85 Executive Committee

Past President Ian de Groot Denis M. Hamaguchi 3852 Calder Avenue 3807-22nd Avenue North Vancouver, BC V7N 3S3 Vernon, BC V1T 1H7 H: 980-6877 5: 985-5301 H: 542-8698 S: 545-0549

President, PSA Council Delegate, Membership Person and Newsletter Editor J. Brian Tetlow John Kiassen 81 High Street 4573 Woodgreen Court Victoria, BC V8Z 5C8 West Vancouver, BC V7S 2V8 H: 479-1947 S: 479-8271 H: 926-8005 S: 985-5301

Vice-President Primary Representative Garry W. Phillips Wendy Klassen 4024 West 35th Avenue 49-6880 Lucas Road Vancouver, BC V6N 21`3 Richmond, BC V7C 4T8 H: 261-4358 S: 526-3816 5: 274-9907

Secretary Intermediate Representative Nigel A. Cocking Doug Super 4-1333 Fort Street 313-2255 York Avenue Victoria, BC V8S 1Y9 Vancouver, BC V6K 105 H: 595-7716 S: 479-8271 H: 736-0960

Treasurer Post-Secondary Representative Jessie Rupp Ian de Groot - 1122 Duchess Avenue 3852 Calder Avenue West Vancouver, BC V7T 1H2 North Vancouver, BC V7N 3S3 H: 922-8315 S: 922-3931 H: 980-6877 5: 985-5301

Journal Editors Tom O'Shea NCTM Representative 249 North Sea Avenue Jim Sherrill Burnaby, BC V5B 1K6 2307 Kilmarnock Crescent H: 294-0986 North Vancouver, BC V7J 2Z3 0: 291-4453 or 291-3395 H: 985-0861 0: 228-5512 Notice to Advertisers Notice to Contributors Vector, the official journal of the British We invite contributions to Vector from all Columbia Association of Mathematics members of the mathematics education Teachers, is published three times a year: community in British Columbia. We will fall, winter, and spring. Circulation is give priority to suitable materials written approximately 550, mainly in B.C., but it by B.C. authors. In some instances, we includes mathematics educators across may publish articles written by persons Canada. outside the province if the material is of particular interest in British Columbia. Vector will accept advertising in a number of different formats. Pre-folded 21.5 x 28 Contributions may take the form of letters, cm promotional material may be included articles, book reviews, opinions, teaching as inserts at the time of mailing. Advertis- activities, and research reports. We prefer ing printed in Vector may be of various material to be typewritten and double- sizes, and all must be camera-ready. Usable spaced, with wide margins. Diagrams, if page size is 14 x 20 cm. Rates per issue are possible, should be camera-ready. We as follows: would appreciate a black-and-white Insert: $150 photograph of each author. If feasible, the Full page: $150 photo should show the author in a situa- Half page: $ 80 tion related to the content of the article. Quarter page: $ 40 Authors should also include a short statement indicating their educational position Deadlines for submitting advertising for the and the name and location of the institu- winter and spring issues are December 1, tion in which they are employed. 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 contributions 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- terrors of testing and the deficiencies of the ments on the quality of the previous issue present mathematics curriculum. of Vector, and we are pleased that the articles in that issue seemed to meet the needs Controversial issues in mathematics educa- and interests of mathematics teachers. tion are addressed by Hugh Elwood and Ed Vector has always enjoyed a good reputa- Collins. In response to the ministry's white tion, not only in this province, but across paper on curriculum, Elwood argues for the Canada. For that reason, we have decided introduction of streaming at the junior to upgrade the physical quality of the secondary level. Collins takes a hard look, journal. As a first step, we have moved to through the principal's eyes, at teacher a heavier cover, with perfect binding. We qualifications. We encourage readers to sub- hope that readers will approve of the mit opinions on these and other issues of changes. Your comments are most welcome. concern.

In this issue, we have again put together a Finally, we present reports from various variety of articles. Bonar Cow provides a sources. John Klassen and Carry Phillips delightful example of how problem-solving look at the annual conferences of the NCTM can be introduced into the elementary class- and the BCAMT. George Bluman reports the room. Walter Szetela's extension of the results of the 1984 Euclid contest, and Larry checkerboard problem is an excellent exam- Weldon summarizes events at the 1984 SFU pie of the power of - Looking Back in high school enrichment conference. problem-solving. Bernadette Harris gives many fine suggestions for using bulletin If you like the contents and appearance of boards for enrichment at the junior secon- the journal, let us know. The new binding dary level. Jim Vance's article on decimal requires that the journal be at least three multiplication is a model of a research pro- centimetres thick, so we need lots of con- ject that yields important insights into tributions. Deadlines for the winter and students' thinking in mathematics. Susan spring issues are December 1, 1984 and April Quinn's fantasy is a timely reminder of the 1, 1985.

5 AW hi AWAIF

Capilano College question and hopefully has a number of in- 2055 Purcell Way teresting applications at hand. North Vancouver, BC V7J 3H5 The minimal qualifications necessary for Dear Mr. de Groot, "Content Awareness" are the easiest to The Capilano College Mathematics Depart- address. Minimally we believe that all ment recently discussed the issue of teachers of mathematics in the public school qualifications of teachers who teach mathe- system should have Algebra 12. Further, matics in the public schools of British Algebra 11 and 12 teachers should have at Columbia. While we admit that we are not least a mathematics or physics major in their aware of all of the present requirements undergraduate degree. What formal post- necessary to teach mathematics at the secondary mathematics training a teacher of various grade levels, we were able to iden- K-10 should have is unclear, although it does tify the following three areas of critical not seem necessary (or practical) that they importance to the qualifications issue: be specialists in math.

Content Awareness The necessary qualifications for "Attitude" To gain the mathematics content familiar- and "Application" are more difficult to iden- ity, a teacher must have successfully com- tify. One step in this direction would be the pleted a number of courses beyond the requirement that all teachers of mathematics content level of the course that he/she 'is must complete at least one full year course teaching. in mathematics education. Further, mathematics teachers should be encouraged to Attitude pursue studies in mathematics-related fields One of the greatest areas of concern to our such as computing or technology. department (that many of us have observed first hand with our children in the public Hopefully these comments will be of assist- school system) is that mathematics is all too ance to your committee in the preparation often taught by a teacher who, while tech- of its report on the qualifications of teachers nically qualified to teach the subject, has of mathematics. very little enthusiasm for it. Often negative feelings toward the discipline are unwittingly Yours truly, passed on to the student. A.E.T. Bentley Application Co-ordinator of Mathematics 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

6 University of Victoria abacus does not "fit" into this part of the in- Faculty of Education structional sequence. These are points well P.O. Box 1700 made. However, one danger should perhaps Victoria, BC V8W 2Y2 be pointed out. Occasionally a teacher is May 18, 1984 tempted to represent a two-digit number (i.e., 25) on a labelled pocket chart To the Editor: The enclosed may perhaps be suitable for by using bundles of inclusion in Vector as part of the "Letters" section. ten ( S44kf4W ) and ones ( IMII Sincerely, A warning about the incorrectness of such a procedure could have been included in the Werner Liedtke article. Isn't a labelled pocket chart similar to an abacus? P.S. Keep up the excellent work! 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 Just for Fun—Or Is It? I read Clark's article. For example, many A few questions general statements are made that carry with them certain implications (too numerous to list them all!). Congratulations to you, Tom and Ian. The latest issue of Vector (Spring 1984) was a pleasant surprise. Variety and interesting "Teachers must recognize that children con- topics made for enjoyable reading. A few struct their own knowledge, rather than questions came to mind as different articles accept someone else's construction." Is this were examined. true for all subjects: reading, language, science, etc.? Is it mathematically correct to present to students a diagram of the following nature? "Children learning about numbers. . . learn by doing them. . ."; "children are given time El to think and discover. . .", "the emphasis + LI = (Dukowski article) is on process, not product." Is it implied that Li Li Li other programs lack these characteristics? Isn't a good or appropriate teaching-learning Rousseau and Owen make some excellent setting for young children described rather teaching suggestions for including physical than a book or program? devices to generate mathematical abstrac- tions. Included in these suggestions is the Isn't it true that every teacher attempts to "use of popsicle sticks (loose and bundles of apply to every subject the teaching strategies ten) together with a pocket chart to demon- and sequences described by Clark? strate graphically the grouping of tens and ones to form one ten and 'carrying' it to the Is the "book" (program) recommended as the tens column." The comment is made that an sole source for mathematics learning, or is

7 it recommended as a reference for ideas? Is BCTF Lesson Aids has suffered the same it suggested that the settings and dialogue be fate. In the rush to put all resources into an used as outlined, or are modifications possi- ill-advised war with the government, the ble? The assumption is made that there exists really important services to the classroom positive transfer from grouping in other teacher have been neglected. Let's see some bases (and learning a new language) to priorities set straight; we need more of the grouping in base ten. Is this really true? For likes of PEMC. some (most, all) children? Rick Sutcliffe I think the same book or program is used in Associate Professor some districts at three different levels (i.e., Computer Science/Math Math Their Way - K, 1 and 2). Is it possi- Trinity Western College 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.?

Box 1253 Aldergrove, B.C. HELP March 16, 1984 We would like to bind a complete set The Editor: of Vector to make a permanent I appreciate Doug Super's comment on my BCAMT collection. We are missing the fall article [Toward a Consensus on Com- following issues from the early 1970s: puter Instruction in B.C. Schools]. He should not interpret my remarks as critical of PEMC Volume Issue itself, but rather of the shortsightedness of 11 All (thought to be 2; Fall those who make funding decisions in the 1969 and Spring 1970) ministry. 12 1, 3, 4, 5 (Fall 1970 and 3 in Spring 1971) The PEMC computer support is first-rate 13 All (number unknown; (what there is of it), and I have praised Fall 1971 through Spring it highly elsewhere—see. "Quo Vadis- 1972) Software" in Call A.P.P.L.E., June 1984. 14 4 (March 1973)

However, as long as the ministry believes If anyone can locate and donate copies that supporting several assistant superin- of the above, we would be very grate- tendents per district at $100,000 p.a. is a ful. Such copies will be annotated with priority, and that the infrastructure in the doner's name, thereby ensuring Richmond and Victoria really needs all the him or her a small place in the history people it contains, genuine 'services to of mathematics education. teachers will have to be run on a shoestring.

8 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 There seems to be a shortage of textbooks hardly the case with the Annual Meeting that deal effectively with problem-solving, held in San Francisco April 25-28. There although there is some consensus that this were three thousand advance registrants, is improving. Dr. Willoughby, NCTM presi- and the total registration reached five thou- dent, mentioned the reluctance of teachers sand. This was a most welcome success after and the community at large to embrace the the Detroit meeting in 1983. The city fathers available technology. He cited situations did their utmost to welcome all teachers and where the calculator was still considered to even staged an earthquake, 6.2 on the be taboo for general use in the classroom. Richter scale, to promote problem-solving with logarithms. The problem-solving strand has been a highly visible one since the Agenda for It is impossible to summarize the goals and Action was presented in 1980. Based on the directions of a conference that entails over sessions I attended, it seems we are now less six hundred sessions. Two major strands of concerned about the particular definition of the conference were problem-solving and problem-solving and the uniqueness of prob- technology. I would like to pass along my lems. This emphasis has been replaced with thoughts and reflections, hoping that it a much broader approach where we talk stimulates some thought or new ways of about problem-solving in all aspects of looking at old ideas. instruction not exclusively in the section on word problems. The teacher and students The general feeling about the teaching of certainly need the time to develop problem- mathematics in the Untied States is that solving skills, but problem-solving should be things should be changed, but there is some incorporated into all mathematics instruc- confusion as to the direction of this change. tion. On the one hand, 36 state governors for- mally support school reform, but there is no Perhaps we can develop problem-solving accompanying infusion of funds. There is a skills even in our algebra units by asking pressing need for trained mathematics questions differently. For example, solve the teachers, but unions balk at the proposal to system offer bonuses to attract mathematics specialists. The mathematics community advocates x— y= 6 developing curriculum around problem- 4x - 2y = 23 solving, but the classroom teacher must deal with the heavy influx of compulsory stand- X ardized tests that reflect the learning of basic for ' 2 instead of the traditional "x" lower-level skills. and 'ay".

9 Teachers must be flexible, present different include more complex exercises, or is the strategies, and. continually keep these in alternative to use technology to perform the front of the students. Here are some ques- tasks? It certainly is a question we must con- tions raised regarding problem-solving: front. I recommend the paperback Com- 1. Once a student has written an equation puting and Mathematics, edited by James for a problem and solved the equation, is Fey, available from NCTM. It contains a he/she necessarily capable of understanding series of articles that examine the impact of or evaluating the problem? computers on algebra, geometry, calculus, 2. Is there a difference in writing an equa- etc. tion for a problem when we ask the student to translate from words to algebra or when Recent reports in the U.S.A. highlight the we talk about an equation as the expression importance of mathematics, but there is a of one thing in two different ways? concern that we are providing the same 3. Should we not be using problem-solving precalculus mathematics program for all strategies, 'such as patterning and working students. It was mentioned at a number of backward in presenting lessons even at the sessions that in a 1980 study, 25% of the most basic skill levels? Grade 12 students did not graduate, 25% entered a vocational training program, and A major reason for the re-examination of the 50% went on to junior colleges or univer- curriculum is the impact of technology. A sity. A number of speakers singled out the second reason is that an increased impor- 50% population that did not move into a tance attached to problem-solving requires university-related post-secondary program. a deletion of some topics due to the time What are we doing for this group in terms constraints. of a mathematics program? Some work is being done to develop such a program in Major curriculum-development projects are Wisconsin, and this will be published by under way at the University of Maryland (J. NCTM in the fall and in the 1985 yearbook. Fey) and the University of Chicago (Z. Usiskin). A common thread is that students may not need to spend as much time I trust there are one or more things that may developing sophisticated skills in algebra. strike a chord, for it will be very exciting in Once the student understands the basic skill the next few years to compare the changes of factoring, is it necessary to spend con- in British Columbia with those in the United siderably more time expanding this to States.

10 Did You Know That...?

Ian de Groot

The metric system is the official system of fuses or Carl Lewis wins a gold medal for measurement used by the entire world with running 100 m faster than any other human. the exception of Brunei, Burma, North and South Yeman—and our enlightened neigh- According to Time Magazine, critics like bors, the United States. Stewart Brand, creator of the Whole Earth catalogues, object to the metric system for Though Congress in 1975 legislated a the very reason that most scholars favor it: gradual and voluntary changeover in the ease of converting one unit to another— weights and measures, nothing seems harder say, kilometres to metres—by simply to do than to get Americans to adopt metric. 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 A few years ago, the metric forces thought something." they could get the U.S. to switch in a decade. Now they do not expect metric to prevail Like their colleagues in Canada and through- before the year 2000. Only when students out the world, U.S. scientists have long used who now learn some metric measurements metric measurements, and some three dozen in school reach upper-level management will states require metric instruction in the the change really occur. schools. Moreover, while football in both the U.S. and Canada still measures progress A few major manufacturers, including in yards, and Nolan Ryan's fastball blazes General Motors, John Deere, and IBM, are at 98 miles per hour, many U.S. joggers now switching rather than fighting. So are several speak knowingly of doing their weekend "10 government agencies, including NASA. Still, Ks" (for 10 kilometres). major manufacturers, such as Boeing, continue to measure in feet and inches, though Lovers of the grape originally suspected a they sell many products overseas. plot when the wine industry adopted Euro- pean measures in 1979, but they have since Even the metrically untutored do not blink learned that the newfangled litre gives them when doctors prescribe 500 mg of antibiotics 1.8 ounces more than an old-fashioned U.S. or electricians recommend 15 A (ampere) quart!

11 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 (1981) and Mathematics: A Way of Think- was beginning to wilt a little. I was teaching ing (1977). When I had finished, I selected a class of 12 Grade 5s and 20 Grade 4s and the ideas I felt would be workable in my feeling that I might make it through my first classroom and began to formulate a series year after all. That is, except for teaching of hybrid techniques. By the time I returned story-problem-solving in mathematics. to the classroom in January, I felt that I had the basis for a reasonable teaching strategy. I asked myself what had gone wrong. Hadn't I done a more than adequate job of teaching How lucky we are that children are forgiv- place value? number theory? estimation? ing! It took about two months for all 33 of operations with whole numbers? Certainly us to get over my teething problems. By I. had! Then why were my students such early March the class was beginning to solve dismal failures at solving word (or story) story problems with a high rate of success. problems? A student who scored at or above mastery level in other mathematics activities My approach took place at two levels. On was lucky to get by with 50% on word- one, there was a need to foster and, problem tests or story problems in the text- wherever possible, create, good problem- book. All my attempts to teach effective solving abilities. For example, I had worked problemsolving skills had apparently failed. with students on such skills as visualizing I decided to seek help. (drawing diagrams, sketching), estimating (predicting, checking), deleting. redundant Teachers are helpful people, but when I data or information, supplying missing in- sought advice, I found that they had similar formation, and attempting to generalize on troubles. No one seemed to have a surefire the basis of a few examples. On the other method for teaching children how to solve level, they began to write word problems for story problems. In the weeks before the each other integrating these skills, and I Christmas holidays, I tried out a number of began to use these same problems on their suggestions, but none succeeded. At the word-problem tests. same time, I also wrote to a university colleague in Victoria and requested an update To get the students into the right frame of on what research had revealed on the mind for writing and eventually solving subject. story problems, I began with manipulatives: beans, peas, kernels of corn, etc., and A package of articles came before the dust imitation money. Working in small groups, of the Christmas concert had settled. During we began to formulate simple story prob- the holidays, I sorted through the articles lems, using the manipulatives as counters. and through Mathematical Problem Solving From this very concrete level, we moved on —A Resource for Elementary Teachers to a semiabstract level and wrote down what

12 had been explained orally. Gradually the beginning of each class, the students were manipulatives disappeared, and we began to given specific instructions as to the type of work without them. 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 occasional textbook story problem, my students stuck to writing and solving their own problems. By following a prescribed problem format 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 Two students using manipulatives during the first stage most cases, these resulted from sloppy work of fraction word problems. habits rather than a basic lack of understanding (for example, a missing decimal point, regrouping mistakes, place-value errors Once the children were used to creating their created by poorly laid out work, etc.). own simple problems, I had them reserve a Second, the students were working with notebook for word problems. Initially topics that really counted, at least in their students had only to write 10 word problems eyes: foods, sports, pets, clothing, domestic of their choice. We began with addition, and situations, records, tapes, hobbies, and the then as the students became accustomed to like. writing their own questions, they were required to supply specific types of problems. There were other payoffs from student- For example, an early addition assignment generated problems, because reading, for Grade 5s might have required that: (1) writing, and mathematics skills came All questions have four addends in them. (2) together. Skills were being practised con- All addends be three- or four-digit numbers. stantly, and by placing word problems in a (3) Three of the questions involve money. separate book, the students could easily (4)One problem use metric measurement for trace their progress. In addition, the students mass. As time progressed, we moved into soon began to try stumping their classmates subtraction, multiplication, and division with what they considered their toughest problems, using the same method. At the problems.

13 This method of problem solving is carried tallied. The lowest score in the problem- out once a week in a one-hour time block, solving section was 0.2 below grade level. which is separate from the subject of mathe- Also, when we dipped into the Investigating matics. My principal, Gordon Mime, has School Mathematics text again to solve the provided four calculators, which the occasional problem, the comments ranged students share for checking answers. Once from "Mr; Gow, these are a cinch!" to a set of questions is handed in, I take the "B000ring!" notebook and check all 10 of them. Each question is given 10 marks. If you have a I now enjoy problem-solving because the class of 30 children, you can count on check- students in my class really are learning. They ing three hundred completely different ques- actually look forward to that weekly hour tions each week. Buy some extra batteries and compete with one another to create for your calculator. Story-problem tests are complex problems to challenge classmates. made up mainly from the very best questions All of us in the room have found something in the notebooks, and the students have their very worth while. names placed next to the questions they have supplied. Bibliography Baratta-Lorton, R. Mathematics: A Way of Problem-solving via student-generated and Thinking (Menlo Park: Addison Wesley, student-solved problems does work. When 1977). spring rolled around, I discovered a signifi- Vance, J. Mathematical Problem Sovling- cant increase in ability when the scores for A Resource for Elementary Teachers the Canadian Test of Basic Skills were (Victoria: Ministry of Education, 1981).

Yes. I'd like to join the National Council of Teachers of Mathematics. I understand 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 development, coordination, and delivery of Council services, including $13 for each subscription 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 lbeabeuer Credit Card # ______Expires Signature teacher for It. Name Address City State or Province Postal Code M6

14 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

Code 1. Add me to myself or multiply me 5. The sum of the digits in my by myself, you get the same answer. multiples add up to the number that I am a factor of all even numbers. I am. Who am I? Who am I? 6. Add the digits in my multiples. 2. All of my multiples are the same Look for a pattern that repeats my number. Who am I? first three multiples. Who am I?

3. Add me to myself, and you get 7. and 8. Neither of us is divisible half of what you get if you multi- by 10, but our product is. Our sum ply me by myself. Who am I? is 11. Who are we? (List smaller number first.)

4. Add me to myself, and I double. 9. and 10. Our product is 56 and Multiply me by myself, and I stay we differ by 1. Who are we? the same. Who am I? (Smaller number first.)

What has the head of a dog, the tail of a dog, What happens to ducks when they fly upside but is not a dog? 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 SOLUTION plan has gained wide acceptance in mathe- The problem can be solved by considering matics education as indicated by a scan of a simpler case. the newer textbooks, mathematics journals, curriculum guides, etc. In Polya's fourth Note that in a 4 by 4 board, we have: step, the problem solver is advised to look back after finding the answer. Yet little 16 squares of size 1 by 1 attention is given to the look-back step. In practice, for many people, the look-back step means mainly check the answer for 9 squares of size 2 by 2 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 4 squares of size 3 by 3 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 1 square of size 4 by 4 considerable, there are satisfying rewards to the problem solver, and a much deeper ...... understanding of the original problem is NONE likely to develop. .... To illustrate extensions and related problems, we'll begin with the well-known checkerboard problem. We can extend the observations to an 8 by 8 board to obtain 64 squares of size 8 by 8, PROBLEM 1 49 of size 7 by 7, 36 of size 6 by 6, 64 squares How many squares are on an 8 by 8 checker- of size 1 by 1, 49 of size 2 by 2, 36 of size board? 3 by 3, etc.

A problem has been solved, and one could recognize these square roots as the first four stop here. If you are more adventurous, and triangular numbers represented below. you follow Polya, you can do as Billstein (1975) and Turner (1983) have done, and solve a more general problem...... S.. . S. •SS •SS• PROBLEM 2 1 3or— 6 o— 10or- How many rectangles are on an 8 by 8 1+2 1+2+3 1+2+3+4 checkerboard?

We are tempted to suggest that for a 5 by 5 board, the number of rectangles will be the SOLUTION square of the fifth triangular number or Look at a simpler case again (1 + 2 + 3 + 4 + 5)2, which is 225. For such as a 4 by 4 board. the 8 by 8 board, we would use the 8th triangular number, which is 36. This would We record the number of produce 1296 (362) rectangles for an 8 by 8 rectangles of various sizes in board. a table.

We could stop here, but Turner takes the 1234 problem a step further: For example, there 1 16 12 8 4 - - - - are 8 rectangles of PROBLEM 3 How many rectangles are on a rectangular 2 -12 -9 -6 3 size 1 x 3 horizontally - and also 8 rectangles frame of m by n rectangles? 3 8642 - - - - of size 3 by 1 verti- 4 4 3 2 1 cally. SOLUTION i We'll look at a simpler case, say a 2 by 3 rectangle, and record the number of rectangles of various sizes. See Table 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 ENE. square board is a square number. An ex- ..I 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. 123 Evidently, square numbers do give the total 1 I 6 I I 2 I Total of 18 rectangles number of rectangles on square checker- for 2 by 3 rectangle boards, but which square numbers? The 23 J2k square roots of the numbers 1, 9, 36, and 100 are 1, 3, 6, and 10 respectively. Many will Table 1

17 Because triangular numbers were prominent We could stop here, apparently having ex- in the previous problem, we might focus hausted all variations of the checkerboard again on such numbers. The key numbers problem, but all our work has been in two in this problem are 2 and 3, the dimensions dimensions. What about an extension into of the rectangle, and 18, the total number three dimensions? If we can work with of rectangles. The second triangular number squares, why not with cubes? Could we is 3, and the third triangular number is 6. make discoveries and generalizations with As 3 x 6 = 18, we might conjecture that the cubes that are analogous to those with number of rectangles in a 3 by 4 rectangle squares? is the product of the third and fourth triangular numbers, or 6 x 10 = 60. Let's ex- PROBLEM 4 amine this case. How many cubes of all sizes are in an'8 by 8 by 8 assembly of 1 by 1 by 1 cubes? The number of rectangles of various sizes is shown in Table 2. OBSERVATION For the analogous checkboard problem, we know that the number of squares of all sizes is

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

If sums of squares solve the checkboard problem, might sums of cubes solve the cube problem? Munn CONJECTURE Munn In an 8 by 8 by 8 assembly of cubes, the Munn number of cubes of all sizes is 1 + 2 + +7+8.

SOLUTION Let's look at simpler problems or more par- The results in the table advance the conjec- ticularly some extreme cases systematically. ture that the number of rectangles in a rectangular frame is related to triangular numbers. In particular, we confirmed that p I in the case of a 3 by 4 rectangle, the number ...ø I of rectangles is the product of the third and fourth triangular numbers, or 6 x 10. Turner I.. indeed shows that for the general case of an m by n rectangular frame, the number of rectangles is the product of the mth triangular Number Number Number Total of unit of cubes of cubes number number and the nth triangular number. Cube size cubes of edge 2 of edge 3 of cubes When the rectangular frame is square, the ibylbyl 1 - - 1 special case of Problem 2 arises, and the 2by2by2 8 1 - 9 number of rectangles does become the 3by3by3 27 8 1 36 square of the triangular number associated with the side of the given square. Table 3

18 Inspection of the table showing the first three CONJECTURE extreme cases confirms our original conjec- In an 8 by 8 by 8 assembly of unit cubes, ture. We would predict in a cube of edge 4 the number of rectangular solids of all sizes units, the number of cubes of all sizes would is (1 + 2 + . . . + 7 + 8). (Cube of the be 1 + 2 + 33 + 43• We also obtain a eighth triangular number.) serendipitous finding when we look at the last column. In each case, the total number of cubes is a square number, in fact the SOLUTION squares of the first three triangular numbers. We proceed by looking at some extreme Therefore, we immediately predict that the cases. For the unit cube, the conjecture is cer- total number of cubes in the cube of edge 4 tainly true. Table 4 displays the results for will be the square of the fourth triangular a cube of edge 2. 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, - %dWA1 and 1 of size 4 by 4. It is now apparent that iii-Mv 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 Noe square of the eighth triangular number, 362 or 1296 as suggested from the last column Dimensions of of Table 3.) More generally, we could start rectangular solid Number of solids with a cube of edge n and note that there would be 1 cube of edge n, 8 cubes of edge 1bylbyl 8 (n - 1), 27 of edge (n - 2), etc. 1by1by2PIf 4 1by2by12 4 Now that we've made a breakthrough from 2 by 1 by 1 two dimensions into three by substituting 2 by 2 by 2 _ 1 cubes for squares, why stop? Rectangular I_ j solids in three dimensions are analogous to 2by2by1 J/J 6 rectangles in two dimensions. Therefore we Total 27 pose the following problem: TABLE 4 PROBLEM 5 How many rectangular solids of all sizes are Thus we have 27 or 33 rectangular solids, there in a 8 by 8 by 8 assembly of unit cubes? and 3 is the second triangular number. Before we proceed with a cube of edge 3, let OBSERVATION us note that for non-cube solids, we may For the analogous checkerboard problem we obtain rectangular solids of the same size in found that the number of rectangles of all three directions where two distinct dimen- sizes in an 8 by 8 board is (1 + 2 + 3 + sional lengths are indicated. Thus in the 2 + 7 + 8)2. (This is the square of the by 2 by 2 case, we obtained equal numbers eighth triangular number.) of rectangular solids with dimensions 1 by 1by2, lby2byl, and2bylbyl. Where Might the number of rectangular solids in the it is possible to have three distinct dimen- present problem be the cube of the eighth sional lengths, we may obtain different rec- triangular number? tangular solids of the same size in six ways,

19 e.g., lby2by3, lby3by2, 2bylby3, Rectangular solids in a 4 by 4 by 4 cube 2by3byl, 3bylby2, and3by2byl. This observation will shorten the tables and Number of Number of Number of Dimensions solids by equivalent solids of simplify the counting in more complex en- of solids type types these types suing cases. 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 I..ø#J 1 x 3 x 4 8 6 48 NNNO 1 x 4 x 4 4 3 12 ...r 2 x 2 x 2 27 1 27 2 x 2 x 3 18 3 54 2 x 2 x 4 9 3 27 Rectangular solids in a 3 by 3 by 3 cube 2 x 3 x 3 12 3 36 2 x 3 x 4 6 6 36 Number of Number of Number of 2 x 4 x 4 3 3 9 Dimensions solids by equivalent solids of 3 x 3 x 3 8 1 8 of solids type types these types 3 x 3 x 4 4 3 12 3 x 4 x 4 2 3 6 lxlxl 27 1 27 4 x 4 x 4 1 1 1 1xlx2 18 3 54 Total 1000 1x1x3 9 3 27 1 x 2 x 2 12 3 •36 1 x 2 x 3 6 6 36 TABLE 6 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 The confirmation of the conjecture of Prob- Total 216 lem 5 for n = 4 appears to be sufficient to extend the conjecture for n = 8. That is, the TABLE 5 number of rectangular solids in an 8 by 8 by 8 assembly of unit cubes is the cube of the eighth triangular number. Note that 216 = 61 and that 6 is the third triangular number. At this stage, our con- Finally, we proceed to make one more ex- viction on the correctness of the conjecture -tension. Earlier we noted Turner's solution is extremely strong, but we proceed to verify for the number of rectangles in an rec- that the number of rectangular solids in a 4 tangular frame. Let's look at the analogous x 4 x 4 cube will be the square of the fourth problem in three dimensions. We would triangular number or 10 = 1000. 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 quickly note that the product of the first, We recall that for the analogous problem in second, and third triangular numbers, 1, 3 two dimensions, in a rectangular frame of and 6 is indeed 18. We'll look at one more size 4 by 5, the number of rectangles of all simple case, a solid consisting of 12 cubes sizes is(1 +2 + 3 + 4)x(1 + 2 + 3 + in a 2 by 2 by 3 arrangement and check to 4 + 5). (Product of the fourth and fifth see if the total number of rectangular solids triangular numbers.) is the related product of triangular numbers 3, 3 and 6 or 54.

CONJECTURE The number of rectangular solids in an assembly of cubes forming a 4 by 5 by 6 rectangular solid is the product of the fourth, fifth and sixth triangular numbers or (1 + mi 2 + 3 + 4)+ X5 =(1 + 2 + 3 + 4 + OEM 5)x(1 +2+3+4+5+6). MME& SOLUTION We'll look at some simpler problems. Figure 2

Example 1 Consider a solid consisting of 6 cubes in a 1 x 2 x 3 arrangement. Number of rectangular solids in a 2 by 2 by 3 solid

Number of Dimensions of solid such solids elpitroOM 09 000d F'A III ixixi 12 lxlx2 20 MMMMAJWAllp-110F o lxlx3 4 1 x 2 x 3 4 Figure 1 1 x 2 x 2 11 2 x 2 x 2 2 2 x 2 x 3 1 Total 54 Number of rectangular solids in a 1 by 2 by 3 solid TABLE 8 Number of Dimensions of solid such solids ixixi 6 lxlx2 7 The two simple examples investigated seem lxlx3 2 1 x 2 x 2 2 to be by themselves sparse evidence for a 1 x 2 x 3 1 claim for the truth of the conjecture, but in Total 18 the light of the preceding analogous investigations and results, they are reasonably TABLE 7 convincing.

21 SUMMARY REFERENCES This set of investigations is an example of Bilistein, Richard. "Checkerboard Mathe- extension and generalization in the applica- matics." Mathematics Teacher 68 (Decem- tion of Polya's Look Back. The results ber 1975): 640-646. demonstrate the value that can accrue from Turner, Sandra. "Windowpane Patterns." time spent on looking back, not merely to Arithmetic Teacher 76 (September 1983): check for reasonable answers but to generate 411-413. related problems. Teachers must be willing to make the effort to encourage more attention to the fourth step, and not be limited to checking answers for reasonableness. Cer- tainly the fourth step with extensions and *1 thank Jane' Smith, graduate student in analogs of the original problem has great mathematics education, for her helpful potential for stimulating problem-solving, suggestions. consolidating understanding, and exhibiting analogies, connections, and patterns.

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

Al Bulletin Boards: q pe An Enrichment Tool I

* t Bernadette L. Harris

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

The bulletin-board space in my classroom is subsets of the real-number system. I use a used to enrich my classes, which are not thin strip of paper to divide the bulletin streamed. I have an ample number of board, labelled Reals, into two parts to be bulletin boards, and I devote five of them labelled rationals and irrationals; integers, to reinforcing or enriching academic topics. wholes, and naturals are ready to be stapled 1 try to change the display on one board each onto sheets of contrasting-colored paper as week, which means that many topics are subsets. displayed for five weeks, though some very useful ones are up longer. In front of a class, I stand at the bulletin board—one of the type which serves as a I paper the boards in September. These window-screen—and staple the material as backgrounds are usually a single color— I explain the difference between rationals some light, and others dark. I have also used and irrationals, and between integers, checkered backgrounds by alternating two wholes, and naturals, so that at the end of colors of standard sheets of duplicating the short talk, the display is there as a long- paper. Across my largest bulletin board I lasting review of the topic. It remains up have 15 cm strips of the rainbow's colors, longer than most other boards because it is extending halfway down, and then have constantly useful for many classes. finished the bottom of the board with mild pink paper. The paper is available from the Because I teach in a Grade 8-9-10 school, I art room in our school, and the cost does not don't have the fun of introducing the come from the math budget. students to the set of imaginary numbers. I explain to them, however, that I will leave It does not take long to staple up pictures blank the adjacent window-shade bulletin and appropriate cutout letters to illustrate board (at least when I'm putting up the reals topics. One of the most effective topics I use one), and I say that if I were teaching a class is titled Reals; the letters are cut out and in a school that had Grade 12, I would there placed at the top. I staple this particular put up the word imaginaries in the same display during a class lecture time, having letter size as the reals. Having students prepared the cutout letters to name the realize they are playing with only half the

23 number system'helps me explain why we Another presentation is an exclamation sometimes use only the set of naturals, or point with the words: For "!" say "factorial." integers. We are using only the set of reals On the next line, 61 = 3.2.1, and below that now. I won't reveal how to write imaginary 9!/7! = 72. numbers (invisible ink . . .); this mystery is a challenge to some of them, who delight in Other displays are easy to make: Fibon- finding out from, older friends or from nacci's series; Pascal's triangle. Also, above books, and come in and enrich us all with the chalkboard, I quote some important their discovery. mathematician to support my ideas; for example Leibnitz who, in 1698 objected to the I refer to this bulletin board to explain why use of "x" as a multiplication symbol say- we use dots when graphing inequalities, with ing he usually preferred to use a dot (Lennes, the replacement set of naturals or whole A Second Course in Algebra, 1943, p. 500). numbers or the set of integers, compared to Though I prefer parentheses, I use his quote, the line for graphing with the set of real saying that even years ago, people were find- numbers. ing that a multiplication symbol often was confused with the letter x. Another bulletin board is above the chalkboard, a narrow board that runs the length To illustrate concepts such as rationalizing of the chalkboard. There I put on the top line the denominator, I cut out digits and put the cutout sentence "Have a bit of pi," and them up as a display that shows at the bottom I staple the cutout numerals 3.141592 . . . to about 30 places. It fills up /_1 that space and spills over onto the next 26. bulletin board. Students become familiar with the number and recognize that it goes on forever. Even after that display is re- I supplement this with a reproduction of a moved, every time I refer to pi everybody cartoon saved from a newspaper and en- knows exactly what I mean when I point to larged with the aid of an overhead opaque that bulletin board. When something has projector. A Peanuts treasure for that par- been pinned up for five weeks, students have ticular bulletin board is the little bird, had enough idle time that it has become part Woodstock, doing the rationalization in his of their memory. head after Charlie Brown has told him that Woodstock is lucky he doesn't have to attend school. After removing this pi digit display, I use a window-screen board to display a multiplic- Fun boards with a math slant can often be ative series to show the students how pi may devised.. The six, or seven Ripley's Believe It be calculated: or Not paperback books are a source of ideas. for me. One display originated from the

ir 292.4.4.6.6.. statement, in numerals and letters, 1002 pay

2 3.3.5.5.797. 4180." I build this bulletin board display over several days. (Sometimes display-building is inadvertent, because I simply don't have the This fills one bulletin board, in large time to finish it at once, but this display I numerals. Students pull out their calculators deliberately build slowly.) I staple the above and show me that yes, it does work; they line at the bottom of the board on the first do get pi. day. Curiosity provokes many questions.

24 After a couple of days of student specula- board presentation using either similar tion, I add the next line: "Read this triangles or trigonometry. sentence:" More questions. Then I staple up "Zero is also pronounced ought, nought, M.C. Escher has provided me with an ex- nothing, and oh." cellent continuing topic. A $20 book of 20 of his prints I have broken into separate Short poems from mind-stretching sources pages, and I display two or three at a time, such as The Space Child's Mother Goose (F. changing them each week, varying the Winson, Simon & Schuster, 1958) entertain words below: students interested in the infinity concept: "M. C. Escher" "Little Bo-Peep . . . They'll "Present-day artist" Meet in parallel space "Master of tesellations" Preceding their leaders behind them." As I remove a display, I store all the para- phernalia in a large envelope—the letters, Enlarged Peanuts and Fred Bassett cartoons, numbers, and cutouts, with big posters even with some words replaced, can be used folded. I sketch on the outside exactly how as incentives by way of humor, to submit it will look when it is put up again (in two neat homework, etc. or three years).

An incomplete magic square can be put up No artist, I was amused (and pleased) when on a bulletin board. The monthly calendar a teacher complimented me on my "artistic of the NCTM is another source of ideas. It bulletin boards," I rely on tracing stencilled is too small for students to pay much atten- letter-forms that are available from teachers' tion to, I find, if it is just stapled to the supply stores, and I spend some time in front board, so I pin up an enlarged version of an of the television cutting out these letters. My appropriate problem. husband thinks this is my subconscious desire to be teaching at an elementary school, but I find that careful use of bulletin I vary the topics. Sometimes, I highlight the board space is a very effective enrichment anniversary of an event. For instance, the tool at the junior secondary level. first manned balloon flight took place 21 November 1783, and I had collected fairly large pictures of beautiful hot-air balloons I would welcome additional ideas. Perhaps from such sources as my daughters' dis- other teachers would submit their ideas to carded Owl magazines and my copies of Vector to be run as fillers, a la Reader's Natural History. I put those up in November Digest anecdotes. Ideas would certainly help 1983 with the words "200 years ago, the 1st me, as once the ideas are there, a display manned balloon flight." These magazines are doesn't take too long to complete. also sources of nature photographs, some of which I put up with the word symmetry: an I have thought of going to our local math enlargement of a hexagonal honeycomb, an teachers' meetings with some of my aerial photo of wavefronts hitting the envelopes of completed bulletin boards and coastline, a shell, a magnified diatom. A offering them for use in another school's forest tree poster available free from the classroom: "Here's one you could put up Canadian Forest Service serves as an object tomorrow." I have not yet had the courage, to measure height when I put up a bulletin 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 lower Vancouver Island middle school. The topics taught and reviewed in middle school exercises were presented in horizontal form, arithmetic. Finding the product of two with space provided for written computation numbers in decimal form requires mastery and figuring. In seven of the questions; one of the basic facts, the ability to multiply by factor was a multiple of 10, or both factors powers of ten and apply the algorithm for contained only a single non-zero digit. The multiplying multi-digit whole numbers, and other five exercises required application of knowledge of the rule for placing the decimal the vertical multiplication algorithm. point in the answer. Some examples can be computed in more than one way, depending The test papers of 150 students were chosen on how the zeros in the factors are used in by stratified random sampling for analysis. the algorithm. Thus decimal multiplication At the Grade 7 and 8 levels, the samples con- exercises provide many opportunities for a sisted of 60 students, 10 boys and 10 girls number of different kinds of errors. As part at each of three achievement levels. At the of a study designed to assess the ability of Grade 6 level were 30 subjects, five in each students in Grade 6, 7, and 8 to compute and sex and achievement category. estimate decimal products (Vance, 1983), the computational procedures used and the types of errors made by students were Results analyzed. Performance on each of the 12 exercises is presented in Table 1. The group mean score Method for the total test, 78%, indicates that the A 12-item test in multiplying decimals was students are reasonably proficient in decimal administered to students in 10 classes in a 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 •1000 x 0.04 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 difficult exercises. Mean scores for the girls achievement level. As would be expected, and boys were 79% and 77% respectively. performance increased with both grade and The group with the lowest score was the achievement level. The Grade 6 students did Grade 6 low-achieving boys, who averaged as well as the Grade 7s and 8s on the easier 50% on the test. items, but they had lower scores on the more

TABLE 2 Performance by Grade and Achievement Levels

Per cent correct Achievement 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- algorithm used to obtain the answer. In tions revealed a variety of procedures and algorithms in the long category, rows of algorithms for multiplying two numbers. zeros were shown as partial products, or un- The different computational forms were necessary zeros in one or both factors were organized into three main categories. A short multiplied. Three types of procedures are procedure was assumed to have been used illustrated for one of the exercises in Table 3. when the answer was written next to the question. given in horizontal format and the For the seven items of this type, computa- question had not been rewritten in a vertical tional forms considered to be long, inter- format. Short algorithms for multiplying by mediate, and short were used in 25%, 35%, a power of 10 or finding the product of two and 40% of the exercises respectively. The multiples of products of 10 with single non- frequency of use of an inefficent procedure zero digits involve "adding zeros" or decreased with an increase in both grade "moving the decimal point." For example, level and achievement level. Girls used a in 1000 x 0.04, one moves the decimal point long form more often than boys—in 49% of in 0.04 three places to the right, or in 4000, the exercises compared to 31 %. The accur- two places to the left. An intermediate form acy rate obtained using a long form was as was one in which the question had been high as that achieved using a more efficient rewritten in vertical format and an efficient method in computing.

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

Short form - Total response 40 40.0 40.00 other number correct 19 5 4 2 30 number of errors 10 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

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- Error Categories - placed, it was possible to identify a number The 395 exercises in which the correct of incorrect "rules" students had used (Table answer was not obtained were examined, 4). In most cases, it appeared that students and the errors analyzed. Four main cate- had some knowledge or recollection of the gories and several subcategories of errors counting rule but were unable to apply it were identified. In 24 exercises, two different correctly with certain pairs of factors, par- types of errors were found, producing a total ticularly where zeros were involved. Only of 419 errors. It was observed that error type seven of the 150 students made a decimal was related both to the nature of the ques- point error in calculating 4.5 x 3.92, but 25 tion and to the computational form used to students misplaced the decimal point in the obtain the answer. 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 19600 two decimal places. 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). 0.01 729 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

0.06 The five decimal places are produced by adding zeros to 0.009 the right of 54. .54000

Two decimal places are needed; add two zeros to the right 6.14 of computed numbers. - 10 614.00

3. Calculation errors 2. Algorithm errors This category included basic multiplication Errors involving use of an incorrect fact mistakes (many involved 0 or 1) and algorithm or misalignment of the partial carrying (renaming) errors (Table 6). products accounted for about 20% of the Twenty-seven per cent of the errors classified total (Table 5). were of this type.

29 TABLE 5 Examples of Algorithm Errors

.617 Incorrect algorithm .420 Answers to 17 x 2 and 6 x 4 are written in the same row. .024340

Zero is added after the 6. 6.14 x 10 = 60.14

4 is added to 1000. 1000 x 0.04 = 1.004 617 Columns misaligned 42 Second partial product should be 24686. 1234 2468 3702 6.14 Decimal point is retained (incorrectly) in partial products. 10 0.00 6.14 6.14

A zero is omitted in the second row because of the zero in 600 the basic fact (30). 50 000 3_000 3000 Extra zero is placed in third partial product. 0.420 0.617 2840 04200 2520000 2.527040

Implications for Teaching Cox (1975) suggested that a teacher should 4. Miscellaneous errors look for three basic types of computational Three other types of incorrect answers were errors in analyzing a pupil's work. System- observed: mistakes resulting from poorly atic errors show a pattern of incorrect formed numerals (for example, 0 mistaken responses and occur in at least three out of for 6); questions in which numerals had been five problems of a particular kind. Random incorrectly copied when the problem was errors also occur in at least three out of five written in vertical form (digits reversed, problems, but no pattern is discerned. Care- decimal point misplaced, zeros added or less errors occur in one or two out of five omitted), and exercises that were incomplete problems and result from a lapse in atten- or not attempted. tion rather than from lack of knowledge.

30 TABLE 6 Examples of Calculation Errors

Basic fact errors 360 5x0=5 1805 0.81 7x11 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 0.420 Examination 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

It is apparent that the student knows her Such examples need to be discussed, and effi- basic facts and how to use the multiplication cient procedures taught and practised. All algorithm, but she used an incorrect rule for questions in which one factor is a multiple placing the decimal point in the answer. of 10, or both factors contain only a single Remediation would involve first having the non-zero digit, should be solved directly by student recognize. that her method does not using rules for adding zeros and counting produce right (reasonable) answers. The decimal places; the vertical algorithm should proper procedure would then be taught in not be used. For example: a meaningful way, rather than as a rule to 7000 x 0.02 be memorized. 14 - 14000 - 140.00 - 140 Random errors are more difficult to remedi- Many of the incorrect answers obtained by ate as no one problem can be isolated. students were unreasonable. Consider the Careless errors do not require reteaching, following examples from the study. but students should be asked to identify their own mistakes. 1000 3.92 6.14 x 10 = 6.140 0.04 4.50 The oral interview (Schoen, 1979) is an effec- 0000 000 tive technique for diagnosing difficulties and 00000 19600 assessing mathematical understanding. Hav- 000000 1568000 ing a student talk out loud as he or she works through an example can help the teacher 0000.00 158.7600 identify and isolate specific strengths and weaknesses. Students need to be encouraged to estimate answers and to check for reasonableness of Certain kinds of errors, which for an individ- results in multiplication situations. Estima- ual are considered careless or random, tion is of increased importance today, since nevertheless occur frequently for a group. computations involving factors with two or Teachers should be aware of these and more non-zero digits will most often be done discuss them in class. For example, a com- using a calculator. Instruction in decimal mon incorrect answer for 600 x 50 was 3000. multiplication must emphasize estimation A zero was missed since the basic fact ends strategies and provide students with a quan- in a zero. tative awareness of the numbers they use.

As previously noted, a large percentage of References the students subjected themselves to un- Cox, L.S. "Diagnosing and Remediating necessary work and created new opportuni- Systematic Errors in Addition and Subtrac- ties for error by using in their algorithms a tion Computations." Arithmetic Teacher, zero preceding a decimal point or zeros February 1975, 151-157. needlessly annexed to a factor. Schoen, H.L. "Using the Individual Inter- view To Assess Mathematics Learning." 2.4 3.92 Arithmetic Teacher, November 1979, 0.5 4.50 34-37. Vance, J.H. Diagnosing Pupil Performance 120 000 in Decimal Multiplication: Computation 000 1960 and Estimation. Vancouver: Educational 1568 Research Institute of B.C., Report Number 17.6400 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. stretching all over my locker! Frantically, I I woke up twenty minutes late, as usual, had tried to swipe them off my shelves, but it the proverbial bowl of soggy cornflakes for was too late—they were all over the place! breakfast, and arrived for my first class of Little graphs shrinking, flipping, always school ten minutes late—as usual. English changing shape, covering everything—even was not an easy subject to start the day with, my gym strip, and my peanut butter sand- but I had managed this far into the semester; wich! I groaned. Suddenly, I felt a tap on so I was doing all right. Then the bell rang. my shoulder. I whipped around, slamming My heart stopped. My worst fears were the locker door amid tiny little screams. coming true as I realized what I had to do Standing directly behind me was a giant next . . . Write an algebra exam! graph propped upon two unknown values, looking me straight in the eye. I gulped. I walked slowly down the hail to the math room. I sat down warily in my seat as Mr. Before I could utter a sound, two ordered Singleton handed out the dreaded test sheets. pairs grabbed me and shoved me into the locker. I cringed, fully expecting to feel the For one solid hour, I sweated over this test, mini-graphs crawling all over me any with visions of quadratic formulas, trinomial second. Instead, I got the weird sensation of squares, and radical equations swimming floating through space while sitting on a around my head, confusing me all the more. waterbed. Some strange dream . . Finally, the last inequality was solved, and I handed in my paper and left. All of a sudden, we hit something—hard. I shut my eyes and screamed. The ordered I opened my locker, sighing with relief, then pairs clamped my mouth shut, and a voice gasped. Little graphs were sliding and hissed in my ear, "Stop it, stop it!"

33 I opened my eyes and shut my mouth, then An image appeared on the far wall. "You opened it again, as I stared in wonder at the have exactly seventy-five minutes to com- world I was in. Metric rulers for trees, plete the exam. You may solve the questions crumpled pieces of old worksheets as bushes, however you wish, but you must come up spilled Liquid Paper for rivers, eraser shav- with the correct solutions, then graph the ings for grass. What did I get myself into? 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 "Where am I? What am I doing here, and reduce you down to a lesser-than sign. You how did I get here?" may begin."

"This is a world of linear systems, and I am I sank into the chair and gaped at the equa- the 'prime factor.' My court and I have tions that suddenly materialized in front of determined that you are a complete empty me. How would I ever solve them in time? set when it comes to algebra, and you are a disgrace to Pythagoras, Euclid, and all the Determinedly, I gripped a pencil, and began. other math-magicians! How anyone could The exam started out easily enough, with a manage to botch a simple little operation few simple Y = MX + B questions, and such as complex factoring is beyond my some polynomials, which I completed with comprehension! You must be punished for no problems. The trinomials were a little your vulgar display of incompetence. To the more difficult, and I stumbled through the graphing room! You are going to pay for segment of "solving the difference of two this!" squares," but I was still making good time. Then came the final section: graphs and parabolas. "No! Please, anything but the graphing room! Please! Oh no . I studied all the transformations disbeliev- ingly for the next fifteen minutes, mentally They dragged me down a long corridor, the jotting down the easier equations, and try- walls lined with scrap paper of previous vic- ing to visualize how the parabolas would tims of the graphing room: I studied them shift. "If Y = 2 X2, the vertex will move this frenetically, trying to absorb everything I way, and Y ½ X2 will move it this way could in hopes of finding the answer to get ." I mumbled to myself, sketching out the me out of this hell as soon as possible. rough form of a parabola on the tabletop. "Now, using the first equation, the whole thing should move this way. . . or maybe We reached the door to the graphing room. it moves this way . . ." It was no use, I The guard on duty took one look at the file cradled my head in my arms and started to he had on me, and burst out in laughter. weep. "Hey, kid! Good luck! You'll sure need it! Ha, ha, ha." He opened the door and shoved The image appeared on the screen again. me in. This was it. "You have only thirty-two and one half minutes left. I suggest you get cracking." The room was furnished with a cardtable and a hard wooden chair. On top of the I glared at the fading face and stuck out my table was a pile of paper, a brand new pink tongue. The calculator flew across the room, eraser, a pack of sharpened pencils, and an smashing against the dull brick wall. I ancient calculator with dead batteries. looked at my arm in amazement.

34 The image appeared on the screen again. I smirked, bouncing slightly on the balls of "You have only thirty-two and one half my feet, full of confidence. I should have minutes left. I suggest you get cracking." known better.

I glared at the fading face and stuck out my Suddenly, I was jerked back to the present, tongue. The calculator flew across the room, aware that the "prime factor" was issuing a smashing against the dull brick wall. I command. "Men, get the torture chamber looked at my arm in amazement. ready. This paper is incorrect."

Once again, I picked up the pencil, still feel- "What? How can that be possible? I checked ing a bit unsure of myself. I started scribbling everything, drew it so perfectly—" furiously, occasionally taking abhorring glances at the hateful video screen, as all "If you will notice, the equation of Y = those jumbled thoughts began to sort them- (X - 5)2 moves the graph horizontally to the selves out, piecing themselves together like right, not to the left. In our books, this paper a jigsaw puzzle. is inaccurate." He shoved it under my nose.

"Oh no! How could I have been so stupid?" "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 They dragged me, kicking and screaming, to this equation like this, I can move it down the racks. So commenced the worst torture to negative six on the Y-axis! Bingo! I'm I have ever been through, and ever hope to down to the last part; graphing it." go through. They made me solve quadratic equation after quadratic equation, and I fac- I glanced at the clock in the corner that tored until I turned blue in the face. It was hadn't been there long. Five minutes left! unbearable. Finally, I went over the edge.

I got up, stretched my stiff legs, and walked over to the remains of the shattered calcula- "I don't want to conjugate your vertically tor. I desperately needed a straight-edge, replaced factoring problems, and the hori- which the guard had conveniently neglected zontal flipping vertexes are function! The ex- to supply me with to complete my graph. traneous discriminate of the slope-intercept radicand can terminate, and you can solve your Y = MX + B and transform it, for all Meticulously, I drew out and measured my I care!" precious graph, and had just finished when the door burst open, and two negative ex- The "prime factor" got a steely look in his ponents grabbed my papers and pencils. The eyes, and his jaw set as he slowly raised his ordered pairs were back, flanking me on arm. He mumbled something unintelligible, either side. The "prime factor" entered the and I felt a rumbling in my stomach. I tiny cell. glanced down, and froze in horror as I realized what was taking place. I was slowly "Time's up. I must congratulate you on com- changing into the shape of a lesser-than sign! pleting our 'little' exam. You are one of the I screamed. Sure, I wanted to lose weight elite few. Now all that is left is to mark the and look thin, but not this thin! I blacked graph." out.

35 I came to, feeling a gentle tap on my "Well young lady, from now on, I suggest shoulder. you eat a well-balanced meal in the morning. I will not tolerate people passing out in "Hey, are you all right?" my classes!"

I sat up, and found myself in the medical "Yes sir." I turned to the nurse and principal.. room, surrounded by the nurse, the school principal, and Mr. Singleton. "May I please go back to my locker? I'm really not feeling well, and I'd like to get my "I know you have an extreme dislike for things and go home." Shakily, I got up, and algebra dear, but the test wasn't so difficult walked out of the room and around the that you had to faint in the hallway. Now corner to my locker. When I got the lock what happened?" open and reached in to get my books, I froze. There were still two little graphs "I, uh well. . . I didn't have any breakfast sliding all over my peanut butter sandwich! this morning, and I was really hungry. I grabbed my jacket and bolted out of the That's all," I said, lying through my teeth. 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 Third, the comments on standards if they are of Education's analysis of the current situa- based on the 1983 Mathematics 10 Achieve- tion regarding streaming and achievement in ment Test (the reader is not given any indica- junior secondary school mathematics, and tion of the source) are very strange, because suggests a research-based plan for grouping the 1983 Mathematics 10 Achievement Test students to maximize their achievement in was not based entirely on the government's mathematics at the junior secondary level. mandatory core curriculum; therefore, how can the statement be viewed as valid? On page 7 of the Secondary School Gradua- Clearly, the above quotation, which con- tion Requirements Discussion Paper, it is tains three inaccurate if not misleading stated that: statements, shows that the ministry is uncertain about streaming in mathematics and The overall effect of this streaming is perhaps even misguided by its own officials. uncertain. On the one hand streaming systems that assign students to classes In the next paragraph, on page 8, it is stated based on predictions of their academic that: performance allow the creation of a very demanding curriculum for the The Ministry of Education recognizes most able students, and standards can that current practice in many schools be seen as very high. On the other is inconsistent with provincial policy. hand, curriculum expectations for This paper advances the position that other students are not as high and cor- a common junior secondary curricu- respondingly, neither is their academic lum that sets high expectations for all performance. students in a core academic program and that does not disqualify, on a There are three inaccuracies in the above systematic basis, any student for any quotation. First, public schools assign senior secondary program is the students to classes not on the basis of predic- preferable path. tions of academic performance, but rather on the basis of past achievement in mathe- Both statements by the ministry are inac- matics. Second, all students are required by curate. Provincial policy is followed by the the ministry to study the core curriculum in public school system. No student is dis- mathematics, and all students, therefore, qualified from any senior secondary study the same curriculum except for those program. All students are taught the core more able students who require an enriched curriculum in mathematics; therefore, they and more rigorous mathematics curriculum. should be eligible for any senior secondary

37 course. If they are not, then it is the fault heterogeneously achieve as well even though of the Ministry of Education for not pro- the high-ability students are not in class with viding a more enriched and rigorous Core them. The latter statement is supported by Curriculum. the ASCD findings, which show that disad- vantaged students have higher achievement if teacher expectations of their performance On page 8 the ministry states: is high. An efficient method of attaining this would be to group heterogeneously the low- • . . it is recognized that there will be and middle-ability students. The expecta- a continued need to modify courses on tions of the teacher would be higher for the an individual basis to provide achiev- lower-ability students than if they were able goals for some students. grouped homogeneously themselves, and their achievement would be higher. The achievement of the middle-ability students While this may be true for special needs would be unaffected. students, the statement is not made in that context; therefore, one must assume that it is made in reference to the general school One of the goals of the Ministry of Educa- population. How does the ministry expect tion is to promote and enhance the educa- this to happen with much larger classes and tion of gifted students. To assist the ministry dwindling availability of materials and in achieving this goal, high-ability students teaching resources? Even if it were possible, should be grouped homogeneously based on would individualization, with smaller classes past performance in mathematics to maxi- and more resources, affect achievement? A mize their achievement. summary of research completed by the Association for Supervision and Curriculum Development (ASCD) of such educators as The Ministry White Paper states on page 8: Medley, Mortimore, Rosenshine, and Edmonds has found several factors that in- it is obvious that the subject of fluence teacher and school effectiveness— streaming at the junior secondary level individualization is not one of them. The requires further review. ASCD report found that among other factors: What should be done? (1) the -most efficient method of teaching is 1. The core curriculum for all math students group instruction. at the junior secondary level should be suf- (2) high teacher expectations improve ficiently rigorous and enriched to allow all achievement. students the opportunity to enter senior math courses. What does some of the other research say? 2. Lower-ability students-should be grouped E.G. Begle, in 1975, reviewed the research with middle-ability students heterogeneously of 25 studies on ability grouping and con- based on past performance in mathematics cluded that homogeneous grouping based on to maximize the achievement of the lower- past performance in mathematics should be ability students. There will be no effect on used for high-ability students. In 1979, Begle the achievement of middle-ability students. further reports and confirms his findings of 3. High-ability (gifted students) should be 1975 that high-ability students should be grouped separately from other students in grouped homogeneously and that the re- order to maximize their achievement and maining student population when grouped potential in mathematics.

38 References Elwood, H.S., Grouping Students for Gall, M.D., "Instructional Policy, Issues in Instruction in Secondary Mathematics, Mathematics Education." Educational May 1978. Leadership, December /January 1984. Begle, E.G., Ability Grouping for Mathe- Association for Supervision and Curriculum matics Instruction. A Review of the Development, Teacher and School Effec- Empirical Literature, SMSG Paper #17, tiveness, videotape, March 1981. December 1975. Province of British Columbia, Ministry of Begle, E.G., Critical Variables in Mathe- Education, Secondary School Graduation matics Education: Findings from a Survey Requirements, A Discussion Paper, April of the Empirical Literature, MAA/NCTM, 1984. 1979.

UNDERGRADUATE PROGRAMMES IN THE MATHEMATICAL SCIENCES AT CANADIAN UNIVERSITIES PROGRAMMES DE SCIENCES MATHEMATIQUES DU PREMIER CYCLE DANS LES UNIVERSITES CANADIENNES

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

Canadian %ia:h.n,a,icalSa,uqr University of Toronto Sadi'iMa'hemaiiqav du Canada 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 present role. This particular recommenda- report on the Qualifications of Teachers of tion asks that an attempt be made to "en- Mathematics in British Columbia, which sure that only persons with academic and was produced by the BCAMT and circulated professional training in mathematics educa- to all members in May 1984.] tion (be) permitted to teach mathematics."

In response to your request for reactions to Frankly, I think that it should be conceded the "Qualifications of Teachers of Mathe- that, as a general principle, an attempt is matics in British Columbia" paper, I thought made by school principals to ensure that I should take the time to outline my properly qualified teachers teach in all of the thoughts. school's subjects. The question really is "why does this not always happen?" When stu- First of all, I think the report makes some dents are not taught by subject "experts" in very significant points and raises some any particular subject, it is likely the result equally important questions. I am not con- of one or more of the following factors: vinced, however, that the report deals effec- • a shortage of qualified teachers on a tively with the full range of factors causing school's teaching staff in that particular the problems outlined. Nor am I convinced discipline. that practical, workable, solutions are being • a dearth of qualified teachers in the school proposed that consider the present situation district in that discipline. as it really is and as it is really likely to be. • inappropriate distribution of talents and competencies within a school district. It is, however, not my intention to dispute • priority being given to individual profes- the conclusions or to criticize the recommen- sional preferences over program or curricu- dations contained in the paper. I have little lar needs of the school or districts. dubt that the authors are entirely accurate • the expansion or growth of a program when they state (for example) that "Many without a commensurate growth in subject teachers without sufficient mathematics competencies; that is, hiring has not kept skills are being asked to teach mathematics pace with increased clientele in a particular in the classrooms of this province." I also subject. agree that "students at all levels should be taught by fully qualified teachers." What, When the foregoing factors are considered then, is the issue? in relation to teaching mathematics, the following conclusions are evident: I would like to discuss Recommendation 5a • There has been a growth, or increase, in (in particular) since it is directed toward the emphasis given to mathematics and an school boards and principals—the latter, my attempt made to ensure that mathematics,

40 in one form or another, be taught to all or no connection to the study of, or prior students for 11 of their 12 school years; that accomplishments in, Algebra. I agree with is, there is a demand for more mathematics the paper's assertion that "society of the to be taught. present and future needs an ever-increasing • Many students apparently cannot "han- supply of technologically trained members." dle" the academic mathematics. As a conse- I'm not convinced that this will be accom- quence, there is considerable movement plished through the "streaming" approach, toward "streaming," beginning at Grade 8 which pits Algebra against computer studies, level or even earlier. consumer math, trades mathematics, or • Most teachers who have the necessary accounting. This, by the way, seems to be qualifications as mathematics specialists the approach suggested by the new "white prefer to teach the "academic" mathematics paper" on secondary school graduation courses. requirements. • A student's academic (post-secondary) options are severely limited if he/she does not complete Algebra 11. As a result, there I think it can be argued that the interests of is much pressure to stay in "academic" math. students, teachers, and society might be Failure to do so generally connotes failure better served if: as a student or at least a big step in that • Algebra were not used as an intelligence direction. test or screening device by post-secondary institutions for the recognition of competent students. To effect this would require exten- First, I think it improbable that we will have sive lobbying with the post-secondary voluntary resignations to provide jobs for institutions. those who may have superior qualifications • Arts and science students were allowed to in mathematics. That such a change can be select any Grade 11 mathematics course to brought about through legislation is, in my complete that requirement for graduation opinion, extremely unlikely. and college entry. • Students were to select mathematics Retraining, in turn, does not hold out too courses on the basis of interest. Thus arts much hope to me, since local programs of and science students selecting Algebra on this nature have not been successful. I do not this "elective" basis would likely be anticipate a change in this, since the concept motivated as well as generally competent. of job protection through seniority com- • Students selecting the other mathematics bined with a "willingness to try" will likely courses were to recognize them as having see a continuation of the present circum- equal status and value. stances. I think that a part of the answer • Teachers were more interested in teaching might lie in a re-examination of the purposes these other courses because of the courses' of mathematics education: What are we equal or equivalent status. educators trying to accomplish? What are our goals? For example, while there is little The improvements suggested in the paper for doubt that students who gain credit for the teaching of mathematics can be accom- Algebra 11 and/or Algebra 12 are compe- plished only through a redirection of tent in an academic sense, it seems to me to energies on the part of the British Columbia be inherently wrong that credit in this course Association of Mathematics Teachers. This be indiscriminately used as a screening extremely strong professional group could, device for a myriad of post-secondary pro- I believe, effect some important educational grams. Many of these programs have little 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 TABLE 2 the Canadian Mathematics Competition for School Enrolment Grade 12 students, as in previous years, the performances of B.C. students and B.C. B.C. percentage schools ranked well ahead of those of the rest B.C. Canada of Canada total of Canada. The contest was supported 1980 42 440 10% nationally by the Waterloo Mathematics 1981 52 513 10% 1982 54 591 9% Foundation, the University of Waterloo, 1983 74 661 11% Dominion Life Assurance Company, Cana- 1984 98 765 13% dian Imperial Bank of Commerce, IBM Canada, Canon Canada Inc., and Mutual Life of Canada; in B.C., by the Department TABLE 3 of Mathematics and the School of Engineer- Percentage of Students from B.C. on the ing at UBC. Canadian Student Honor Roll of 100

By any criterion (e.g., Tables 3 to 6), B.C. 1980 14% results continue to be superior. (Keep in 1981 14% mind that as of the first quarter of 1984, B.C. 1982 22% had 11% of Canada's population.) 1983 19% 1984 21%

TABLE 1 Student Enrolment TABLE 4 Percentage of Schools from B.C. on the Canadian School Honor Roll of 50 B.C. percentage B.C. Canada of Canada total 1980 291 3178 9% 1980 18% 1981 337 3596 9% 1981 18% 1982 552 4667 12% 1982 30% 1983 743 5301 14% 1983 22% 1984 1097 6649 16% 1984 28%

42 TABLE 5 Cory Ristock, Mathematics student, UBC. Median School Team Score on the Euclid Al Sarna, University Hill Secondary, Contest (Maximum Possible Score is 300) Vancouver. Jack Schellenberg, John Oliver Secondary, B.C. Canada Vancouver. 1980 117 95 John Stigant, 1981 141 122 Hillside Secondary, West 1982 - 178 138 Vancouver. 1983 152 130 John Turnbull, McNair Senior Secondary, 1984 173 140 Richmond.

TABLE 6 The following results are based on the mark- Year-by-Year Ranking of B.C. Students ing at UBC. Year-by-year a student placing in the top 10% in Among the 39 schools with 10 or more B.C., placed nationally in the top: entries, the highest ranking schools (based on the results of the 10 highest scoring 1980 7% students) were: 1981 8% 1. St. Michael's, Victoria (78114) 1982 6% 2. Churchill, Vancouver (710) 1983 7% 1984 6% 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) For B.C. schools, the written part of the 10. Burnaby North, Burnaby (5501%) Euclid Contest was marked by a team 11. Penticton (548) including: 12. South Delta, Delta (5391%) Dr. Andrew Adler, Department of Mathe- matics, UBC Among the 77 schools with five or more Dominic Alvaro, Argyle Secondary, North entries, the highest ranking schools (based Vancouver. on the results of the five highest scoring Leo Boissy, Churchill Secondary, Van- students) were: couver. 1. St. Michael's (420) Dr. John Coury, Department of Mathe- 2. St. George's, Vancouver (400) matics, UBC. 3. Churchill (378Y4) Sharon Cutcliffe, Burnaby North Senior 4. L.B. Pearson, Victoria (3771/2) Secondary, Burnaby. 5. Handsworth (356) Ian de Groot, Sutherland Secondary, North 6. Nanaimo (351Y2) Vancouver. 7. Sentinel (3473/4) Larry Dubois, Centennial School, Coquit- 8. Port Moody (3461%) lam. 9. Argyle (344Y4) Dr. Harvey Gerber, Department of Mathe- 10. Steveston (3431%) matics, SFU. 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 CONNECT program for bright B.C. high top-ranking student (on the written part of school students, June 24-29, 1984, with all the contest) in each school with five or more fees covered, and free accommodation and entries received book prizes (141 students). transportation (25 students). 2. A school ranking in the top 10 in B.C. 5. The Euclid Contest was an important (based on 3, 5, or 10 entries) received a book criterion in selecting the 26 winners of the prize (16 schools). UBC entrance scholarships, each worth at 3. Students scoring well in the contest were least $10,000 (over four years) provided a invited to the honors calculus courses student maintains a first-class standing in a Mathematics 120/121 at UBC (312 students). full program of study. Of the 26 winners, 4. Grade 10 or 11 students scoring well in 17 wrote the Euclid Contest and all 17 had the contest were encouraged to study high scores resulting in congratulatory calculus and were invited to the UBC 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 Sutherland Burns Lake Dawson Creek Hope Maple Ridge Windsor Lakes District South Peace Hope Garibaldi Delta McKenzie Kamloops Penticton Delta McKenzie Penticton Clearbrook North Delta Kamloops 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 University Hill Spectrum Princeton Vancouver Vancouver College Crofton House Sidney Vancouver Technical West Vancouver Parkland - Gladstone Windermere Quesnel John Oliver Hillside Surrey Winston Churchill Sentinel Correlieu Killarney 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 5. B.V. Gnedenko. "Mathematical 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 of a set of numbers. The general strategy for participated, the Mathematics Enrichment the algorithms he discussed was called conference that took place at Simon Fraser "divide and conquer," the idea being this summer was a great success. The con- analagous to the well-known trick of finding ference combined lectures, problem sessions, a word in a dictionary by repeatedly asking social events, and tours over a three-day which half of a section of the dictionary the period. About thirty SFU personnel, faculty, word is in. A highlight of Professor Babai's staff, and senior students, were involved. lectures was his collection of charming Twenty school teachers and 190 students cartoons featuring Merlin the magician and from 75 schools participated in the con- King Arthur, and a recipe for Irish stew! ference. The students were selected by their schools, and while a majority were from the The opening talk by Professor Len Berggren Fraser Valley schools, many students were focussed on the Babylonian origins of our from all over B.C., and even one was from arithmetic system, including the explanation Whitehorse. of why our time and geographic co-ordinate systems use base 60 arithmetic instead of. the The academic part of the conference decimal system. - focussed on six lectures. The keynote speaker was Professor Laszlo Babai from Professor Tiko. Kameda of the SFU depart- Eotvos University in Hungary. In his first ment of Computing Science discussed the talk, Professor Babai discussed a problem on rationale of coding electronic signals to allow round-table seating satisfying constraints automatic error detection. It was nice to see regarding which pairs of diners are allowed the practical application of the binary to sit beside each other. Professor Babai system which has received so much em- showed that this problem is related to a phasis since the advent of electronic number of other famous problems, including computers. the four-color problem, the travelling sales- man problem, and class scheduling prob- "The impossibility of complete chaos" was lems. He showed clearly that no computer the comforting title of Professor Tom was ever going to be able to solve these Brown's talk. If one puts a long sequence of problems by brute force, and that more colored beads on a string, and if there are clever mathematical strategies would need a finite number of colors and the string is in- to be invented. finitely long, then certain patterns have to appear over the whole string. Apparently Professor Babai's second talk concerned these patterns are the domain of a branch algorithms for such tasks as finding a median 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 DATES TO of a traffic queue: the model is complex to KEEP IN MIND study analytically, but it can be studied empirically as if it were a real-world object subjected to scientific investigation. The NCTM 63rd Annual Meeting technique of Monte Carlo simulation was San Antonio, Texas outlined for this purpose. APRIL 17-20, 1985 Most, but not all, of the students were able to follow these lectures, with their sophis- 24th Northwest Mathematics ticated material. The conference is aimed at Conference gifted students with a particular facility in Richmond, B.C. mathematics. OCTOBER 10-12, 1985 Problem sessions were of several kinds, including 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. UPCOMING Social events included an evening barbeque and soccer game, a teacher's lunch, and three BCAMT EXECUTIVE well-attended buffet lunches. Billets for out- MEETING DATES of-town students—provided by in-town participants as well as SFU faculty, staff, and September 29, 1984 students—provided memorable experiences December 1, 1984 for many participants. February 2, 1985 A reunion of participants will be held this April 13, 1985 fall, and a continuation of the conferences June 8, 1985 is planned for 1985 and beyond. Informa- tion will be distributed to the schools. Harvey Gerber, of the SFU Department of All meetings are held in a board- Mathematics and Statistics, will organize the events. room of the B.C. Teachers' Building on Burrard Street. [Editor's note: The six lectures were video- taped during the conference. These will be BCAMT members are invited to available for use by the schools. Contact the attend. Meetings start at 09:30. Department of Mathematics and Statistics at SFU for further information.]

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 further attempt was made to feature Cana- was held at Palmer Junior Secondary School dian authors. Marshall Bye, Les Dukowski, on August 28 and 29, 1984. This year's event Frank Ebos, Olive Fullerton, Heather was admirably hosted by the teachers in Kelleher and Shawn McPhail added yet Richmond under the direction of Conference another dimension to an already impressive Chairperson Mary Stewart (Errington Ele- program. mentary School). Other highlights of the workshop: The workshop was sited in the lower main- • Becky Mills (Ministry of Education) in- land again this year, and more than 200 paid troduced the Elementary and Secondary registrants attended. In all, nearly 300 Mathematics Revision Committee members. mathematicians attended the workshop in- The subsequent update and question period cluding registrants, speakers, publishers and were very informative. committee members. The Richmond site was • Twenty publishers, commercial compan- very popular with teachers from around the ies, and computer corporations provided an province. In fact 35% of the registrants were impressive array of print and supplemental our colleagues who travelled from areas such materials for review. as Victoria, Salmon Arm, Alexis Creek, Pen- • The BCAMT Annual General Meeting ticton, Prince Rupert, Osoyoos, Prince was held. George, Kitimat, and Burns Lake. The Summer Workshop is primarily for ex- The Summer Workshop program commit- changing new ideas and preparation for the tee continued the tradition of providing a new school year. However, social events are program with a wide range of topics relating not neglected. Lunches were provided at the to all areas of mathematics. Milt McClaren school both afternoons for a small charge. (SFU) gave a thought-provoking address, Registrants used their lunch hour to meet "Mathematics, Computers and Human new people from throughout the province as Thought," to keynote the start of the well as to renew old friendships. Free coffee workshop. The next two days were filled and donuts were provided by the publishers with 82 sessions. These sessions were throughout the conference, so coffee breaks devoted to problem-solving, computers, were enjoyable. Two textbook publishers, calculators, Math Their Way, math games Houghton Mifflin and Nelson, provided and activities, practical math programs, con- hospitality suites on the first night of the ics, logarithms, and estimation. This lists workshop, and a closing wine-and-cheese only a small sample of the available sessions. event hosted by the BCAMT workshop Of special interest were the sessions espe- committee was a resounding success. cially tailored to an administrative view- point: "Mathematics Scope-Sequence" and There have already been numerous favor- "Trends in Mathematics Education." A able comments about the workshop from

49 many people. I would like to take this shop Committee for their co-operation and opportunity to express our appreciation hard work in organizing the conference. from both the registrants and the BCAMT executive to School District #38 (Richmond), P.S. See you all at next year's 24th North- Palmer Junior Secondary School administra- West Mathematics Conference, October 10, tion and staff, and the 11th Summer Work- 11, 12, 1985 at the Richmond Inn.

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 fourth chapter is more concerned with what 3. The Mathematical Education of Primaiy- actually happens in the classroom. School Teachers. Edited by Robert Morris. Published in 1984 by the United Nations The chapter on computers and calculators Educational, Scientific, and Cultural is likely to prove as controversial to the Organization, 7 Place de Fontenoy, 75700 reader as it is stimulating. It sees the greatest Paris, France. potential of these aids as that of a tool for exploring mathematical ideas, capable of ex- This third volume of Studies in Mathematics tending enormously the range and scope of Education has been prepared as a part of mathematical activity in school. UNESCO's program for improving mathematics instruction through the production of I think that all mathematics educators, resource materials for those responsible for primary and secondary, will find this study mathematics teaching. It examines the interesting, thought-provoking, and cer- responsibility of primary-school teachers for tainly useful. the mathematics component of the curriculum and the implications thereof of teacher education. Dottori, D., C. Knill, and J. Stewart. FMT Senior: The Ryerson Mathematics Pro- In the present volume, Robert Morris, the gram. McGraw-Hill Ryerson. editor, has unified contributions from 16 countries while maintaining the individual This text is prescribed in several provinces, style of each author. including Alberta and Ontario, for the senior mathematics course; in addition, I use it As with the earlier volumes in this series, often as a source of many good applied contributions were solicited from far and problems in Algebra 12 classes. wide. The views expressed are from Africa, Asia, the Caribbean, Europe, Latin America, I like this text for several reasons: North America and the South Pacific. The • The problems throughout the text are contributions combine the views of teachers, realistic and topical. teacher educators, curriculum developers, • The content is ideally suited for the and research workers. They make a rich current B.C. Grade 12 curriculum. mixture; yet, surprisingly perhaps, they • The polynomial chapter has as its major reflect a remarkable agreement about what objective, the graphing of polynomial and primary mathematics should be. rational functions. • The exponential and logarithmic functions The study is organized such that the first two section contains many realistic applied ex- chapters are general—the responsibilities of ponential and logarithmic questions. The the teacher of primary mathematics. The authors do not waste time in using logs for third chapter discusses the environment as calculations or in using terms such as a source of problems, and, in contrast, the characteristic and mantissa.

51 • The section in which calculus is introduced I've taught a 10-period unit developed using is logically developed and uses a geometric other sources. All other aspects of the approach to introduce the concept of Algebra 12 course are dealt with more than derivative. adequately by the text. The layout provides • The text is compact, with no unnecessary for good readability, and the presentations information or pictures. are clear. Oral exercises are a good base for • The final chapter concerns problem- discussion in each section, leading smoothly solving, and a unique collection of problems into the A, B, C, sets of the written exercises. from many different areas of mathematics Appropriate questions at each level of dif- is presented. ficulty are provided in sufficient numbers. The student text has answers for odd- The publishers tell me that the latest edition numbered questions. of this text is due very early in the new year. I.look forward to this event with keen antici- The teacher's edition contains many helpful pation. features: teaching suggestions for each section, extra illustrative examples, suggested time schedules for assignments, and suggested assignments. These have been Algebra and Trigonometry—Book 2— developed with considerable expertise and Structure and Method, new edition. Do!- are particularly useful to someone using the ciani, Sorgenfrey, Brown, Kane, Houghton text for the first time. Answers to all ques- Mifflin, 1982. tions appear with thequestions or in the margins. Quizzes are provided: at regular in- Reviewed by Don Gordon, Seycove Second- tervals. Of course, reviews and practice tests ary. School, North Vancouver. are given at the end of each chapter in the / student text, and the teacher's edition con- I have used a few sections from Chapters 6 tains chapter tests, cumulative reviews and and 7 and almost all of Chapters 8-14 in topical reviews. Over 500 questions and teaching my Algebra 12 classes during the answers are provided. last two years at Seycove Secondary School in North Vancouver. I recommend the text In summary, I have thoroughly enjoyed very highly; I think it is one of the very best using this text, and I think that it would I've used for any secondary math course. satisfy the needs of Algebra 12 students having a wide range of ability in mathematics, The only supplementation I've found neces- and that it provides excellent guidance and sary is in "graphing techniques," in which resource materials for the teacher.

Y84-0068 October 1984

52 utfe Get it all together with Houghton Mpin Mathematics.

00 I

CO C,

C) C, Houghton Mifflin 00 150 SteekaseRd. West, Toronto, ONT L3J? 1B2 / Teachers' ft.HELPLINE Discuss any personal or professional problem. A caring, well-informed colleague will listen to your concerns. Your conversation will be completely confidential and you will remain anonymous. 112-800-663-9166 TolIfr.- or 7314120. Vancouvr arsa

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