ECT(DR Journal

of the British Columbia Association of Mathematics Teachers

Volume 28 Number 2 Winter 1987

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Child-centred and activity-based, JOURNEYS IN MATH provides opportunities for students to learn through exploration, through active participation in

the learning process.

JOURNEYS IN MATH encourages them to discover, think, question, ... and "grow"!

For more information contact: JANE MILLS 926-9290

B.C. Association of Mathematics Teachers 1986-87 Executive Committee

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

Vice-President, PSA Council Delegate Carry W. Phillips 4024 West 35th Avenue Vancouver, BC V6N 2P3 H: 261-4358 S: 526-3816

Secretary Stewart Lynch 2753 St. Georges Avenue North Vancouver, BC V7N 1T8 H: 984-7206 5: 987-8141 local 303

Treasurer Grace Fraser 2210 Dauphin Place Burnaby, BC V5B 4C9 H: 299-9680 S: 596-5186

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

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

Elementary Representative Daphne Morris 430 Moss Street Victoria, BC V8V 4N4 H: 381-1971 5: 382-3212 local 299

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

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

1988 Northwest Conference Chair J. Brian Tetlow 81 High Street Victoria, BC V8Z 5C8 H: 479-1947 0: 479-8271

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

Membership Chair Richard Longman RR 4, 5135 Chute Lake Road Kelowna, BC V1Y 7R3 H: 764-7856 5: 768-7622

NCTM Representative Jim Sherrill 2307 Kilmarnock Crescent North Vancouver, BC V7J 2Z3 H: 985-0861 0: 228-5513

THE B.C. ASSOCIATION OF MATHEMATICS TEACHERS PUBLISHES VECTOR

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

2235 Burrard Street Vancouver, BC V6J 3H9

Membership rates for 1986-87 are:

BCTF members ..............................................................$25 BCTF associate members ......................................................$25 Student members (full-time university students only) .............................$10 All others (persons not teaching in B.C. public schools, e.g., publishers, suppliers).....................................................$37

Notice to Contributors

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

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

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

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

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

Deadlines for submitting advertising for the spring, and summer issues are January 21, 1987, and April 21, 1987, respectively.

Inside This Issue

4 From the Editors ................................................Tom O'Shea 5 Letters .................................................................... 7 Did You Know That ............................................Ian deGroot

Mathematics Teaching 9 Calculators are now IN and ON in B.C.

But what am I supposed to do with them ? ....................... Carry Phillips 13 Purchasing a Calculator for Primary Pupils .................... Katherine Wilson 15 The New Improved "Hand" Calculator .......................... Rudiger Krause 17 Exploring Positive and Negative Integers with Tiles .............. Thor Fridriksson 20 Formulas in Applied Mathematics................................. John Kiassen 25 Student Research as Mathematics Enrichment in Grade 10 ........... Zoe Wakelin 30 Discrete Mathematics as Enrichment Material ................. Katherine Heinrich 33 A Very Radical Complex Cross-Number Puzzle .............. Ceoffrey R. Tomlin

Mathematics Education 35 Mathematics Education in North America ..........................Alan Taylor

Mathematics and Computers 43 Remember the Function Machine ? ...........................Harold Brochmann

Miscellaneous 47 What Did You Do During Your Summer Holidays ? ................James Sherrill 49 Treasurer's Report to the Annual Meeting ..........................Crace Fraser 51 The Fifth Annual Mathematics Enrichment Conference ............Harvey Gerber 53 Report on the June 1986 Algebra 12 Provincial and

Scholarship Examinations .........................................Vic Keehn

From the Editors

Tom O'Shea

In this issue, we focus on two themes: calculators and enrichment.

Garry Phillips leads off by suggesting a number of learning outcomes teachers might consider when teaching students how to use calculators. Many teachers have enquired about what they should look for in a calculator, and Katherine Wilson recom-mends a number of useful features for teaching primary children. Rudiger Krause presents his own version of what a hand calculator should look like. Readers might want to refer to the summer 1986 issue of Vector for the original hand calculator described by Jim Sherrill.

Thor Fridriksson, who has presented work-shops, "Math: A Way of Thinking," at several BCAMT conferences, sets out a manipul,4ve model for teaching addition and suJ4j/action of integers. His procedure providsan extension of the model used in the Ministry of Education's videotape on teaching junior secondary school mathe-matics.

John Kiassen presents a sequence and set of exercises for teaching tables, graphs, and formulas. Although Klassen suggests using his sequence for general mathematics classes, the approach is applicable to all students.

Last issue, Zoe Wakelin proposed a number of curriculum units for Grade 8 enrich-ment. In this issue, she describes how her Grade 10 students carried out and presented their research on various mathematical topics.

Katherine Heinrich engages in a timely discussion of the place of discrete mathe-matics in the curriculum. I strongly concur with her recommendation of two books by A. Gardiner. Any teacher interested in teaching problem-solving at the secondary level will find them useful.

Geoff Tomlin rounds out this section with a nice self-checking activity using com-plex numbers. How about asking your students to create their own cross-number puzzles?

Alan Taylor looks at the state of mathe-matics education in Canada and the United States. His assessment shows the danger in generalizing across the 49th parallel. Nevertheless, we appear to face major challenges.

Harold Brochmann has cooled down since the last issue, and he provides a computer-ized version of the once heralded function machine.

Jim Sherrill reports on the BCAMT Summer Conference, and Grace Fraser gives us the bottom line. Harvey Gerber reports another successful SFU enrichment conference, in which Brian Thomson's presentation, "Frac-tals," was superb. Finally, Vic Keehn presents his report on students' strengths and weaknesses on the 1986 provincial algebra examination. We hope the ministry will resume publishing such reports so the results

will be available to an audience wider than Vector readers.

At the BCAMT Summer Conference, I reported that Vector has become the best journal of its kind in Canada. That has been due to the consistently high quality of material submitted to us. Keep up the good work, and let us have your ideas and reactions.

To the Editors:

Re: "To Hell with Computer-Based Instruc-tion," which appeared as an article in the fall 1986 issue of Vector.

This article seems to make the following assumptions: • that the objective of CBE in education,"

as distinct from training, is to replace teachers with machines;

• that the use of CBE will reduce the cost of education;

• that there is no good CBE;. • that CBE will not improve.

It is my opinion that these are wrong assumptions.

• CBE is most effective when it is developed, selected, and used by good teachers. Take the discipline of mathematics as an example; students have to learn many manipulative skills before they can deal with concepts. To this end, much of a math teacher's time is taken up with "training" students. CBE can be used to enable students to learn basic manipulative skills. This would allow teachers to concentrate on the more challenging task of helping the students understand the concepts—a task often

neglected by teachers who are hard-pressed to bring their students to the point where their basic math skills are adequate.

• Because CBE will become one of the tools of teaching—like books, films, and videos, and because good material does not come cheaply, I am not sure that CBE will lower the cost of education. I believe the author is correct, however, about CBE's lowering the cost of training.

• There is a lot of good training CBE and some good educational material now. Much of the better material, however, is available only via more expensive systems. GM and the U.S. military use computer-based train-ing, and one can be reasonably sure that they would not do so if they had not found them satisfactory.

• CBE is still in its infancy,hre are learn- ing to distinguish poor frorn9E materials. The ability to make such a distinction is a first step toward knowing how to create bet-ter CBE. I am optimistic about our abilities to adapt, learn, and be more creative with this new medium. The future looks exciting and challenging for those willing to explore it.

With regard to the author's remarks about simplistic simulations and instruction leading to concept formation, I do believe that even 11 simplistic simulations" can be useful teaching tools (teaching number concepts, calculus, and group theory come to mind). Furthermore, such "simplistic simulations" could be of good use to help students with concept formation. In teaching mathematics, I am sure most instructors often feel the need to draw diagrams and graphs. Simple simulations are an extension of this ap-proach. We will need to learn to be more im-aginative in our future use of computers.

I agree wholeheartedly with one of the major thoughts Brochmann expressed in his paper. Teachers need to be vigilant about how com-puters are used in education. It is our respon-sibility to ensure that we have good tools. The author makes this point.

Dr. lain Cooke, Manager, Computer-Based Education, Inter-Urban Campus, Camosun College, Victoria

UPCOMING BCAMT EXECUTIVE MEETING DATES

January 21, 1987 April 25, 1987 June 6, 1987

All meetings are held in a boardroom of the B.C. Teachers' Building on Burrard Street. BCAMT members are invited to attend. Meetings start at 09:00.

Did You Know That...?

Ian deGroot

Another bout of Rubikmania appears to be on the horizon. Erno Rubik, the Hungarian professor of architecture who invented the cube, has marked his return with a puzzle that, if anything, is more difficult to solve than the once ubiquitous cube. The new puz-zle is called Rubik's Magic and is alleged to have even more configurations than its predecessor.

The cube, as you may recall, contained 26 smaller cubes in six bright colors that rotated on horizontal and vertical axes. The trick was to unscramble the smaller cubes so that they gave the larger cube six walls of solid colors. Mathematicians calculated that the cube could be arranged in 43 quintillion posi-tions or, more accurately, 43 252 003 274 489 856 000 positions.

It is estimated that the original cube sold 100 million authorized copies, 50 million un-authorized "knockoffs," and at least 10 million books explaining how it could be solved.

It is expected that the new Rubik's Magic will be marketed around the world this fall at an estimated price of $10 (U.S.) each.

We can hardly conceal our excitement!

The bubble has burst!

Canadian universities and colleges, which two years ago were forced to limit enrolment in their computer science courses, now have trouble attracting students.

At Concordia University, in Montreal, only half of the 120 places in the first-year computer science program were filled as we went to press.

At the University of New Brunswick, in Fredericton, freshman enrolment in com-puter science has dropped by 20%.

At the University of Toronto, statistics show a 26% drop in the number of students specializing in computer science between November 1984 and November 1985.

At Ryerson Polytechnical Institute, in Toronto, applications for computer science programs are down 25%.

At the University of Waterloo, which boasts one of the country's top computer faculties, applications are lower than ever before. The university once considered ac-cepting only students with an average of 80 or 90%; now it is taking a second look at students with 75%.

Added to the mystery is the fact that while secondary school students are shunning computer science studies, employers are eager to hire bright university graduates in the field. Computer graduates command starting salaries as high as $35 000.

Some educators believe that universities may have created their own dilemma. By initially setting high enrolment standards, they may have discouraged capable students.

Other professionals wonder if the computer has become so commonplace in homes that the mystique has worn off. As words such as modems, bytes, floppy disks, and cursors have become part of everyday language, in-terest in the computer appears to have faded.

It seems that the mad rush to make second-ary school students computer-literate may have snuffed out rather than heightened their interest.

The University of Toronto has installed a huge CRAY XMP/22 supercomputer capable of some daunting feats of number-crunching.

The new machine, which will put the univer-sity in the forefront of computer research, cost $1 million just to install.

The machine is a dual-processor that can perform millions of floating decimal-point operations per second. It has the capacity to store 2 million 64-bit "words" and is so fast that it is considered to be almost interactive.

This machine can handle problems that nor-mally take superminis to their knees. For

example, a 50-hour problem on a conven-tional computer can be reduced to one hour.

The cost? I thought that you'd never ask.

With a $10-million grant from the Ontario Ministry of Technology and Trade as a downpayment, the university aims to recoup the remainder of the approximately $30-million cost through user agreements with private firms. These will permit com-panies to buy "time slices" for their work.

Some of the immediate work already lined up for the Cray:

• medical researchers at the Princess Margaret Hospital in Toronto will probe deeper into the secrets of cancer by taking readouts from their nuclear magnetic scan-ner and putting them through the machine.

• Allelix Inc. of Malton, Ontario, manufac-turer of vaccination serums and creator of the world's longest genetic strands, with more than 2 000 nucleotides in them, will use the supermachine to make its software more efficient.

The recently renovated Statue of Liberty re-quired a similar computer process to produce an analysis as a guide for U.S. reconstruc-tion engineers.

A major feature of the installation of the Cray is the freon cooling system to prevent the machine from fatally overheating and to increase its efficiency. The machine requires 1.2-million BTUs an hour to cool it. That is the equivalent to thecooling requirements of 35 homes.

MATHEMATICS TEACHING

Calculators are now IN and ON in B.C.

But what am I supposed to do with them?

Garry Phillips

Garry Phillips is vice-president of the BCAMT and vice-principal of Lord Kelvin Elementary School in New Westminster.

The recent inclusion of several intended learning outcomes and comments with regard to calculators in the British Colum-bia Mathematics Curriculum Guide 1-12 Response Draft, 1985 suggests these machines will shortly be in use in our mathematics curriculum. After reading the draft, some teachers may be feeling uncom-fortable with the lack of direction concern-ing the calculator skills necessary to satisfy these few intended learning outcomes. This article hopes to alleviate such concerns so that the integration of calculators into the elementary classroom can proceed smoothly. A short discussion on the use of the calculator within mathematics and society is provided, along with some recent develop-ments and changes in opinion with regard to calculators. Finally, this article suggests a developmental sequence for calculators that elementary teachers can use in their classrooms.

The personal, business, and corporate sec-tors of our society have been using the calculator for many years. One has to look only as far as the corner grocery store or other local business to see their use throughout society. The controversy for and against the use of calculators within the

mathematics curriculum has been a pro-tracted one that originated in the early 1970s. At that time, there was concern that the calculator was going to reshape the cur-riculum. Ten years later, Suydam (1982) reported that "not only has the calculator failed to redirect the curriculum, it has failed to enter most mathematics classrooms. In the United States, less than 20% of the elemen-tary teachers nd less than 36% of the secondary teachers have employed the calculator in mathematics instruction." In British Columbia, according to the 1985 assessment, 57% of Grade 4 teachers, 35% of Grade 7 teachers, and 15% of Grade 10 teachers do not allow students to use calculators in class (Robitaille & O'Shea, 1985, p. 227). A new attitude must emerge within the body of mathematics teachers for the implementation of calculators in our schools to have any success.

Fortunately, support for the use of calcula-tors in the mathematics curriculum has been addressed recently in two important areas. First is that of the prestigious National Coun-cil of Teachers of Mathematics. This group unanimously approved in April 1986 a replacement of original policy position statement on calculators that now ". .

recommends the integration of the calculator into the school mathematics program at all grade levels in classwork, homework, and evaluation." And further that ". . . at every grade level every student should be taught how and when to use the calculator." In a meta-analysis of contemporary research, Hembree and Dessart (1986, p. 96) concluded that "In Grades K-12 (except Grade 4), students who use calculators in concert with traditional instruction maintain their paper-and-pencil skills without apparent harm. In-deed, a use of calculators can improve the average student's basic skills with paper and pencil, both in basic operations and in prob-lem solving" and further that "Students using calculators possess a better attitude toward mathematics and an especially better self-concept in mathematics than noncalculator students. This statement applies across all grades and ability levels." These recent in-itiatives indicate strong support for the inclusion of and instruction in calculators within the mathematics curriculum.

a developmental sequence. It is hoped that after this discussion teachers and their students will be able to make more efficient use of calculators in the classroom.

A developmental sequence for the use of the calculator in the classroom should contain instruction in at least six basic areas. Pupils should understand the input, output, vocabulary, and processing functions of the calculating machine as well as being able to perform basic operations, applications, and problem solve with it. In the suggestions below, 13 intended learning outcomes have been organized into six areas of concern. They are highlighted by providing headings to indicate familiarization with input keys, calculator display, vocabulary, calculator logic, computational functions and applica-tions, and problem solving.

Need for Some Prerequisite Skills While the British Columbia Mathematics Curriculum Guide Response Draft 1-12, 1985 does not explicitly enumerate all the prerequisite skills that should accompany the use of the calculator, it is clear that some skills will be necessary to enhance efficient use of this machine. The 1985 mathematics assessment reported that 53% of Grade 4, 71% of Grade 7, and 86% of Grade 10 students own calculators. However, 91% of the Grade 4 and 72% of the Grade 7 students have never used calculators in the school. The elementary teacher is thus likely to face large numbers of pupils with access to, but not necessarily any competence in, the operation of the calculator. It is essential that teachers provide a development sequence to familiarize pupils with the machines they will be using. The remainder of this article is devoted to the advancement of just such

Suggested Developmental Sequence The pupil should be able to:

Input Keys C-i Recognize the basic numeric keys (1, 2, 3, 4, 5, 6, 7, 8, 9, 0) on a calculator and explain why these digits are necessary to pro-duce any number.

C-2 Recognize the basic operation keys on a calculator and explain their functions (ex-ample: +, —, x, -i-).

C-3 Recognize the function keys on a calculator and explain their use C/ON K (constant) OFF % (percent) Cl (clear) CE (clear entry) = (equals) M+, M-• (decimal point) (memory

[ (square root) CM, RM (clear, recall)

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Calculator Display C-4 Recognize and state the number of digits used in a calculator display and relate the number of digits available on the display to place value.

C-5 State the maximum magnitude of opera-tions that can be performed conveniently within the limits of a calculator display for • addition (for example: 7 digit + 7 digit number) • subtraction (for example: 8 digit - 8 digit number) • multiplication (for example: 7 digit X 1 digit number) • division (for example: 8 digit ± 1 digit number)

Vocabulary C-6 Identify and explain the basic calculator vocabulary

display input key output memory power source clear

Logic C-7 Recognize and discuss the type of logic used by the calculator being used.

Computational Function and Applications C-S Demonstrate the process of addition • by performing the operation on the calculator, and • by writing an expression to illustrate the operation (2+3=).

C-9 Demonstrate the process of subtraction • by performing the operation on the calculator, and • by writing an expression to illustrate the operation (3-2=).

C-10 Demonstrate the process of multipli-cation • by performing the operation on the calculator, and

• by writing an expression to illustrate the operation (2X3=).

C-li Demonstrate the process of division • by performing the operation on the calculator, and • by writing an expression to illustrate the operation (3 ­-2=).

C-12 Demonstrate the use of the calculator to • search for patterns • step count • generate multi ion tables • verify calcu tors • work with more complex computations • check the accuracy of estimations • work with repeating decimal fractions

Problem Solving C-13 Identify situations and solve problems involving • addition • subtraction • multiplication • division • a mixture of operations • practical real-life situations

The suggestions made in this article should not be considered an exhaustive list of all the operations necessary to introduce calculators into the classrooms of the province. The ideas are presented in this article in order to stimulate some thought and discussion about the implications of introducing calculators into elementary school classrooms. In addi-tion, a sequence of skills has been suggested that teachers may find useful in order to be successful in their implementation of this ex-citing new aspect of the revised Provincial Mathematics Curriculum.

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References Hembree, R. & Dessart, D.J. (March 1986). Effects of Hand-Held Calculators in PreCollege Mathematics Education: A Meta-Analysis. Journal for Research in Mathematics Education, 17 (2), 83-99.

Robitaille, D.F. & O'Shea, T. (Eds.) (1985). The 1985 British Columbia Mathematics

Assessment: General Report. Victoria, B.C.: Ministry of Education.

Suydam, M.N. (1982). The use of calculators in pre-college education: Fifth annual state-of-the-art review. Columbus, OH: Calculator Information Center. (ERIC Document Reproduction Service No. ED 220 273.)

Mathematics Teachers Recommend Calculators

The National Council of Teachers of Mathematics (NCTM) recommends • the integration of the calculator into the school mathematics program at all grade levels in classwork, homework, and evaluation; • that publishers, authors, and test writers integrate the use of the calculator into their mathematics materials at all grade levels.

These are major points voiced by the NCTM in its position statement on "Calculators in the Mathematics Classroom."

The March 1986 Journal for Research in Mathematics Education article, "Effects of Hand-Held Calculators in Precollege Mathematics Education: A Meta-Analysis," by Ray Hembree and Donald Dessart, reports that the use of calculators in concert with tradi-tional mathematics instruction apparently improves the average student's basic skills with paper and pencil both in working exercises and in problem solving. "Across all grade and ability levels, students using calculators possess a higher attitude toward mathematics and an especially better self-concept in mathematics than students not using calculators."

Results of a February 1986 membership survey indicate that a majority of mathematics teachers support a strong position by the NCTM encouraging the integration of the calculator into mathematics programs at all grade levels in classwork, homework, and evaluation. Of 1000 teachers surveyed, 75 per cent favor a strong calculator stance.

Since its Agenda for Action: Recommendations for School Mathematics of the 1980s (NCTM 1980) strongly recommended that mathematics programs must take full advan-tage of the power of calculators at all grade levels, NCTM has maintained positions stressing the importance of using calculators.

It no longer seems a question of whether calculators should be used . . . but how.

Hembree and Dessart, Journal for Research in Mathematics Education, March 1986

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Purchasing a Calculator for Primary Pupils

Katherine 011so

-'-

Katherineil.on is an assistant professor in UBC's Department o Mathematics and Science Education.

Once you ha ye made the decision to pur- chase calculators for your primary pupils you need to focus upon what type of calcula-tor to buy. After shopping around you will notice there exists a great variety of calculators from which to choose. The actual size illustration of the calculator below con-tains many features found useful for primary children.

EI.SI MATE EL- 346 SOLAR CELL CALCULATOR (9

3U

1CMJ91M

LA—

The following is a list of recommended features:

Power Supply Purchase a solar calculator. Replacing bat-teries in battery-operated calculators can lead to a great deal of frustration and main-taining them becomes costly.

Functions Any simple four function (+ - X ±) solar calculator with an eight digit display will do. Calculators with a large number of keys and functions distract young children.

Size The full-sized illustration of this Sharp solar calculator is an appropriate size for primary pupils. The numerals on the keys and on the display are large and easy to read. Beware of models with small, closely spaced keys as pupils often push an adjacent key when entering a number.

Touch Avoid calculators with extremely sensitive keys. Test a key to see that it clicks each time you touch it, or is firm to the touch. If keys are too sensitive, pupils whose fingers linger for a moment will find their calculator becomes very active indeed.

Special Keys Look for a clear key [C] or all clear key [AC] located on a corner. Many models highlight these keys in a different color. The equals key [=] in a corner location is also an asset. The fact that frequently used keys are

X;4'

13

&V a

positioned on corners aid pupils to quickly locate them.

Special Function Check to see that the calculator has a con-stant addend feature that enables you to "count" or display consecutive numbers. Enter [+] [1] [=] [=] [=] [=] on your calculator. Your calculator should display 1, 2,3,4 . . . . This feature is necessary to prac-tise counting skills or to illustrate multiplica-tion as repeated addition or division as repeated subtraction.

One calculator per student is ideal. Sharing a calculator among two students is feasible, particularly for partner activities. However, sharing a calculator among three or more students may lead to disaster.

Many models of calculators contain the features listed above. The Sharp solar calculator (EL-376) is one such model. It is available through the UBC Bookstore (phone 228-4741) for $7.56 (includes dis-count), and is protected with a plastic case.

Enjoy Bulk Rate on Feb. Issue on Using Calculators

The Arithmetic Teacher, NCTM's journal for elementary school teachers of mathe-matics, will publish in February a focus issue on using calculators in the classroom. The issue deals with the philosophy of cal-culator use and offers practical activities for use with students.

The AT Editorial Panel is making pos-sible a low prepublication price for bulk orders of this focus issue. A special price of $1 each is in effect until 15 December for orders of 50 or more copies to be sent to one address. Single copies can be ordered for $4 each.

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The New Improved "Hand" Calculator Rudiger Krause

Rudi Krause is an intermediate teacher at Pinewood School in North Delta.

I enjoyed J. Sherrill's "hand" calculator article in the summer 1986 issue of Vector. I, too, teach my pupils to use their hands to do multiplication. I must differ with Sherrill on two matters though.

First, he states, "1 have found absolutely no utilitarian value to the hand calculator. (p . 22). On the contrary, it is quite useful to some of my pupils. I have observed pupils doing operations with their fingers under their desks during a test. I have no problem with that, since I believe that the mind per-forming the mental arithmetic is more than the electro -chemical activity of the brain. Cerebellum, vocal chords, manual digits are all body parts the mind may employ to carry out its operations.

Of course, in our milieu, the pressure is on the mind to function secretly, invisibly, in-ternally, cerebrally. And, of course, inter-nalized mental operations are usually carried out much more quickly. But are speed and secrecy the only criteria we use to evaluate a mind? What about elegance or visibility, which invites participation and feedback, or the involvement of other senses, in this case, the tactile sense? Now wouldn't it be marvellous if someone discovered how to do arithmetic operations on the tongue?!

Another "use" of the hand calculator I have witnessed must be recounted anecdotally. A few days after I introduced "hand" calculators to a new class, a pupil from another division approached me after school. "Could you show me how to do that times-ing with the hands?" Somehow she had caught wind of this "neat" method.

So, for a few days, the pupil came to my room, and we programmed her hands to do multiplication. For the rest of that year, I heard, she did much better in arithmetic than she had in previous grades. The "mental block" so many children are said to have toward math was shifted enough so that light could come through. The "hand" calculator had, in fact, been pedagagically "useful" in stimulating and interesting that pupil.

The second matter on which I differ with Sherrill is in the method of using the "hand" calculator. The way I learned to do the operations seems to be significantly more elegant and easier than the method described by Sherrill. Let me describe the method I use.

To use the "hand" calculator, one holds his/her hands palms facing each other and numbers the fingers 6 to 10 beginning with the thumbs.

To indicate a number, the fingers up to and including that number touch one another. For example, to show 8, thumb, index, and middle fingers all touch.

To perform a multiplication, the touching fingers of both hands touch each other. So, to multiply, 7 times 8, left thumb and index finger and right thumb, index, and middle fingers are all touching.

I then have the children say (or think), "T for touching, T for tens." The touching fingers indicate the number in the tens place of the product.

15

The remaining, solitary fingers are then multiplied (as in Sherrill's version) to arrive at the ones' digit. (Notice that "solitary" cor-responds to "ones.") If necessary, a ten is carried over to the tens' place, as in six times seven, for example.

I find it just as easy to do the tens first, especially since we read numbers from the left (even though we usually compute them from the right). In a world where we add up numbers by (actually) adding down, it helps to be versatile.

Leaving these minor differences aside, I do agree with Sherrill that "hand" calculators are fun—fun to teach and fun to use. Even for one who has internalized the basic facts to silent, invisible perfection, it is refreshing to actually see and feel his/her mind at work. It's analogous to hooking up a TV screen to a computer to create a device with which one can monitor the otherwise invisi-ble operations of the computer.

So the "hand" calculator is as well a manual monitor for making mental arithmetic visible.

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Exploring Positive and Negative Integers with Tiles

Thor Fridriksson

Thor Fridriksson is a teacher at John Todd Elementary School in Kamloops.

Introduction

To understand negative numbers, pupils must have two basic concepts: 1. All numbers have many different names. Thus pupils are able to subtract -6 from 4 because another name for 4 is 4 + 6 + (-6). 2. There are only two operations in mathe-matics: moving forward through our number system and moving backward through the system. Addition is moving for-ward randomly; multiplication is moving forward by equal intervals. Subtraction is moving backward randomly; division is moving backward by equal intervals.

For this activity, tiles are used to represent images. Pupils draw a line at the middle of their working area to represent 0. Positive integers are placed above the 0 line. Negative integers are placed below this zero line. Multiplication and division are performed by repeated addition or repeated sub-traction.

Negative Numbers

Strands Arithmetic, numbers and number opera-tions, problem solving.

Materials • Tiles or bingo chips of two colors. • Two dice labelled 0, 1, 2, 3, 4, 5, and 0,

-1, -2, -3, -4, -5.

Purpose • extension of patterns • addition of negatives • subtraction of negatives

Activity Introduction Place tiles to create this pattern:

Ifl IW What comes next? Could you describe this using arithmetic words? (1, 2, 3)

Next:

1H11 What comes next? Give it arithmetic names.

Next do:

IHW(1, 2, 3, 4, 3)

Complete as:

I1 What is next?

17

H What is next?

Next do Addition

Now you are ready to add some numbers. Add 6 + 7 Add 4 + (-2)

Answer: 13 (group together and count) • place on work board • take away the tiles which cancel each

other out • read answer: 2

Using the dice marked 0, 1, 2, 3, 4, 5, and 0, -1, -2, -3, -4, -5, create some addition problems for yourself. Use the tiles to find your answer.

Could you apply arithmetic names to this? (The answer will eventually come out 1, 2, 3, 4, 3, 2, 1 and "one in the hole," "one in the red," "minus one," or "negative one.")

Students have heard about this, b.ve seen it. Point out that makes 0

^:_, ; ) one step forward and one step b.a.Ek..- = 0 change.

Therefore, we can take this number:

add to it ohe above and one below and still end up witI4. The same-Answer results if we put two on topanu two on the bottom because they cancel each other out.

Create some number statements to go along with this. For example:

4 + 1 + (-1) = 4 4 + 2 + (-2) = 4 4 + 3 + (-3) = 4

Subtraction How many different ways can we show the number 57 (Answer: lots!)

or ft or

H F We could represent this many ways:

How might we represent (-5)?

T-Iflor I I or...

Use this idea to try to find the answer to (-5) - 2.

18

We can demonstrate this by hav-ing (-5) represented like this:

Then we need to subtract 2. As we have two above the line, we can subtract the 2. We will have a result of seven tiles below the 0 line, or (-7).

Use a negatives die and a postives die, and generate some subtraction problems. If you

are stuck, then rename the numbers as we did before. Remember that every number has many names. We only need to find the one that suits us best.

The next step is to be able to do such prob-lems: -4 - (-3).

Remember the steps: • set out • do operation • rename • read.

PDK Publishes Manual for Math Program

PDK's Center for Dissemination of Innovative Programs now publishes the manuals for Mathematics Pentathlon TM , a program that brings together parents, teachers, and students to co-operate in mathematics problem solving.

This program involves a series of games for different grade levels. The students com-pete in tournaments that challenge their verbal and non-verbal mathematics problem-solving skills. The games use blocks, dice, plastic chain links, game boards, and other objects instead of a paper-and-pencil. test to teach spatial reasoning, arithmetic fundamen-tals, and other math skills. While designed to be used in special tournaments, the games also can be used by parents and teachers as enrichment activities for students.

Currently, manuals are available for three levels, including Grades K-i, 2-3, and 4-5. The manual for the fourth level, Grades 6 and 7, will be available in December. These manuals, which include the directions for the games and the official tournament rules, are available from PDK. The other materials and the procedures for organizing a tour-nament are available only from Pentathlon Institutes Inc.

The manual of directions and official tournament rules for each level cost $7 U.S. For more information about Mathematics Pentathlon TI or to order a manual, write to Phi Delta Kappa, P.O. Box 789, Bloomington, IN 47402.

19

Formulas in Applied Mathematics

John Kiassen

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

It was reassuring to see that if algebraic equations were to be included in the pro-posed Math 9A and Math 1OA courses, then considerable emphasis should be placed on their application to formulas.

The headings below and the corresponding examples provide a possible flow chart for developing the topic of formulas in Mathe-matics bA.

Tables, Graphs, Formulas It would seem that all too often formulas are considered separately and not as part of a package including tables and graphs.

Maximum Pulse Rate

Age Pulse Rate (years) (beats/mm.)

10 168 20 160 30 152 40 144 50 136 60 128 70 120 80 112

Examine the table and answer these questions: 1. Indicate the approximate pulse rates for these ages: 35, 75, 47, 14, 57. 2. Indicate the approximate age for each pulse rate: 138, 115, 140, 165.

19(

tvbc

13( E

bc

10 20 30 40 50 60 70 80

Age (years) (A)

Read the graph to answer these questions: 1. Find the maximum pulse rate for these ages: 33, 17, 23, 72. 2. Find the change in pulse rate for these ages: 10 to 30, 15 to 45, 40 to 50, 70 to 50.

20

2. The expression 7rr2 in A = 7rr2 is the same as which of the following? Choose all the correct answers.

(a) 7rXr2 (b) irXrXr (c) (7rX02

Here is a formula that allows you to calculate the maximum pulse rate for any age:

Max. Pulse Rate = 176 - (0.8 X Age) P = 176 - 0.8A

The pulse rate or "P" is the subject of the formula.

Complete the table for the formula: P = 176 - 0.8A

Age (A) 0.8A 176-0.8A = P

10 8 176-8=168 20 16 176-16 = 160 30 40

This introductory section serves to place formulas in the context of tables and graphs. This relationship should be maintained throughout the unit.

(d) 7r 2 Xr (e) (irXr)Xr (F) irrXrr

3. For F = * C + 32 if C = 25 then F equals:

(9x25) + 32 (b) 5 , 5

9X25 (c) + 32 (d) (9 X ) + 32

(e) ( - 9- x 25) + 32

4. Complete the table for the formula A = 2 and draw the graph.

S 0123 456 789

A

Evaluation of a Formula It may prove helpful in this section to state the formula in words before the algebraic form is used.

1. The formula to calculate percentage (P) of gold in jewellery is:

Percentage of gold equals the product of twenty-five and the number of carats divided by six.

= 25 X (number of carats) 6

25c 6

Evaluate for "P" when the number of carats is 12, 24, 18.

Translation into Formulas This is an opportunity to translate word statements into formulas and vice-versa.

1. Choose the expression that represents each phrase.

The difference of five time "N" and one

(a) SN—i (b) S(N—i) (c) 1—SN (d) SN+i

Five times the square of d

(a) 5 (b) (5 d)2 (c) 5d2 (d) 5+d2

2. Write a formula and evaluate for the given values.

21

(a) The number of points (P) a player scores is the sum of twice the number of field goals (C) and the number of free throws (T).

(c) Write a formula for his weight (W) after w weeks.

(d) Complete the table and draw the graph.

Find the number of points if a player scores 15 field goals and 10 free throws.

(b) For a right triangle, the square of c is the sum of the square of a and the square of b.

Find c2 if a = 9 and b = 12.

aN

Formulas and Graphs

In this section, students are expected to translate the word statement into a formula, evaluate the formula, and draw its graph.

1. A sales representative earns (E) $10 000/ year plus a 6% share of the sales (S).

(a) Write a formula for the statement. (b) Find the earnings (E) when the sales (S) are: $0, $50 000, $120 000, one million dollars. (c) Which of these graphs best represents the formula?

(a) (b) (c)

EVE EH

W 1234567

W

Changing the Subject of the Formula—Part I

A family lives in Hope, 145 km from Van-couver, and drives at an average speed of 80 km/h toward Calgary.

1 2 3 4 5 6 7 Time (t) in hours

The distance (d) from Vancouver at any time (t) is: d = 145 + 80t.

Find t when d = 300 km.

300 = 145 + SOt

700

500

o 300

100

rJ.,

300 = 145 +

2. Mr. Overwait is on a diet. His present weight (W) is 132 kg and he plans to lose 2 kg/week (w).

(a) How much weight is lost in w weeks? (b) What is his weight after three weeks?

22

155 = 80t

This approach may ease students into this important skill and strengthen their confidence.

Complete the steps and find the missing value.

(a) P =-- 70;P100

100 =- 70

e 2

=e

(b) E = $1500 + 0.065 S; E = $3200

3200 = 1500 + 0.065 S

_____ Ho.00ss

=s

(c) Choose all the correct answers.

If 80 = 25c then c equals

(1)80--!-- (2)80x

-- (3)80X 25 25

Changing the Subject of the Formula—Part II This is the section where the student generalizes the method of changing the sub-ject of a formula.

The cost (c) of renting a chainsaw for h hours is

C = $10 + 5h

Find the number of hours if the cost is $35. Use both formulas.

(1) C = 10 + 5/i (2) C-10 = h 5

35 = 10 + 35-10 = 5

[ 25] LI125

=h =

5 = /i 5 =h

Is the formula C = 10 + 5h equivalent to C—b =

h7 5

For example, make "C" the subject in F = -- C + 32

F =*C +32

F-32 =*C

(d) Wayne Gretzky is on the ice (I) 15 more minutes than twice the time (t) of a player who only kills penalties.

Write a formula and find t when I equals: 40, 35, 28, 48.

F-32

I(F_32) =LI1

23

For example, change the subject of the for-mula by following these steps. P = 15 +

(1) subtract 15

(2) divide by 4 or multiply by 3.

These subtopics provide a breakdown of the skills in developing the concept of working

with formulas. In all likelihood, it would take about 10 class periods to complete this unit.

The reversal method illustrated in the last section can prove to be quite successful. Ex-tensive use should be made of the calculator. Let us not forget the integration of tables, graphs, and formulas.

Professional Development for Teachers of Mathematics: A Handbook

Professional Development for Teachers of Mathematics: A Handbook is a new National Council of Teachers of Mathematics (NCTM) publication, co-published with the National Council of Supervisors of Mathematics (NCSM). Its precursor, the NCTM position statement 'Professional Development Programs for Teachers of Mathmatics" (April 1985), was developed by the NCTM in light of the renewed interest in the profes-sional development of mathematics teachers. The handbook was developed concurrently to provide suggestions for implementing the recommendations of the position statement, since the professional development of teachers is seen as a key to improving instruction' in mathematics.

Because ongoing professional development is necessary for educators to maintain and enhance their teaching skills and knowledge, the objective of this handbook is to pro-vide educators with practical ideas for designing and implementing professional develop-ment programs.

The handbook is available from the NCTM headquarters office for $7.50 U.S. a copy (prepaid; ISBN #0-87353-231-7).

24

Student Research as Mathematics Enrichment in Grade 10

Zoe Wakelin

Photo by Wayne Emde

Zoe Wakelin is a mathematics teacher at Vernon Secondary School in Vernon. This is the second of her two articles on enrichment.

This project was designed for the enriched Math 10 class at Vernon Secondary School to encourage the students to bring together many of the thinking, speaking, organizing, research, and independent study skills developed in earlier grades.

Each student will: • research a mathematical topic (suggested references provided). • prepare an outline for a brief—about 10-minute—presentation on the topic. • prepare any visual aids needed for the presentation. • prepare up to three questions to ask the class based on the topic. • submit outline and questions ahead of the presentation date. • make the presentation to the class.

In the preparation and presentation to the class of a selected mathematics topic, each student will practise the following skills:

Skills Independent study Each students works on his own to learn the mathematics involved in his topic. Teacher advice is available if needed.

Research Many students will search outside the in-dicated references for further clarification of a point or to extend their knowledge of the subject.

Knowledge comprehension To teach the topics to others, the student must have a good understanding of the material.

Application Each student should show some applications of the mathematics both in the presentation and in the questions set to the class at the end of the presentation.

Analysis Each student will break down the topic into component parts for easier teaching.

Synthesis Each student will create his own visual aids for the presentation and unique questions based on the topic.

Evaluation During a class discussion, the criteria to be used to evaluate the presentations will be decided. Based on these criteria, each student

25

Photo by Wayne Emcie

will contribute a mark and a written com-ment as part of the total evaluation of the presentation.

Organization As well as preparing an outline, visual aids and questions, each student must meet dead-lines for submitting the outline, making the presentation and passing on texts required by other students.

Co-operation Some topics are split into related sequential parts (compound interest, conics, bases). This will require some discussion and co-operation among students about the content of the presentation and the sharing of reference materials.

Public speaking Knowing that the mathematical content is understood, the material is well organized, visual aids and questions have been pre-pared, each student should be able to make the presentation with as much confidence as possible.

Evaluation of the Presentations

Evaluation was based on teacher assessment of the mathematical knowledge of the topic and organization of the material, and class assessment of the presentation.

Mathematical knowledge 5 marks of the topic

Organization of material: 5 marks outline questions visual aids

Presentation (class evaluation) 5 marks

Topics

The topics were selected for some of the following reasons: • Informal introduction of a topic taught later in the course or in a later grade (im-aginary numbers, quadratic formula, con-ics, binomial expansion). y'9 • Consumer mathematics topics not usually taught in the mainstream courses (com-pound interest, antization). • Statistics topics which are not in the cur-rent curriculum (percentile, mean, standard deviation, normal distribution). • Interesting topics for knowledge, use or pleasure (Heron's laws, Venn diagrams, probability, bases, Escher tesselations, capture-recapture method of sampling).

26

Research Topics

Topic

Imaginary numbers, and how to simplify them.

The quadratic formula: how and when to use it.

The fundamental counting pinciple and permutations.

Combinations

Probability

Heron's laws

Exterior angle of a triangle theorem

Visible horizon

Make, demonstrate and explain how a hypsometer works.

Double straightedge constructions

Escher tesselations

References

Algebra Two and Trigonometry (Dressier, Rich) Integrated Mathematics III (Keenan, Gantert)

Using Advanced Algebra (Travers, Dalton, et. a!) Modern Algebra Bk 1, Mod 6 (Dolciani, Wooton)

Mathematics, A Human Endeavour (Jacobs) Modern Algebra. and Trig. Bk 2 (Dolciani, Berman)

Mathematics, A Human Endeavour (Jacobs) Modern Algebra and Trig. Bk 2 (Dolciani, Berman)

Mathematics, A Human Endeavour (Jacobs) Mathematics 10, Addison Wesley (Kelly, Alexander)

Mathematics 11 (Del Grande) Geometry in Easy Steps (Cox)

Using Geometry (Wells, Dalton, Brunner) Integrated Mathematics II (Keenan, Dressier)

Enrichment Mathematics for the Grades (NCTM 27th Yearbook) Mathematics 11 (Del Grande)

Mathematics 11 (Del Grande)

Intermediate Mathematics 2, Teacher Ed. (Dottori, Knill)

Using Geometry (Wells, Dalton, Brunner) The Mathematics Teacher April 1974 (NCTM) Math for a Modern World 1 (Baxter, Carli, Newton)

Pages

378-383

600-604

285-287

109-112

378-501

573-581

402-410

581-587

424-441

335-342

380-381

340-341

139-140

253-256

269-272

339-340

231

243-251

200-203

299-310 335-338 342-343

27

Research Topics

References Pages

Geometry 221 (Jacobs) Mathematics, A Human Endeavour 247-249 (Jacobs)

Merrill Geometry, Teacher Ed. 458-460 (Foster, Cummins, Yunker) Geometry, Teacher Ed. 303-310 (Jurgensen, Brown, King)

Mathematics, A Human Endeavour 330-331 (Jacobs)

361 AMT Senior 299-308 (Dottori, Knill, Seymour)

AMT Senior 299-308 (Dottori, Knill, Seymour)

Paper Folding Geometry 125-129 (Johnson) 1 143-147

Math for a Modern World 2 (Baxter, Carli, Newton) Consumer and Career Mathematics

Business and Consumer Math. Teacher Ed. (Saake) AMT Senior (Dottori, Knill, Seymour)

Business and Consumer Math. Teacher/Ed. (Saake) Amf9!rtizing Loans (Teaching notes by Z. Wakelin)

Holt Math 4, Teacher Ed. (Hanwell, Bye, Griffiths) AMT: An Introduction (Dottori, Knill, Seymour)

Mathematics, A Human Endeavour (Jacobs)

Mathematics, A Human Endeavour (Jacobs)

Topic

Kaleidoscope

Explain the term locus and demonstrate several loci.

Conic sections (1) Illustrate and name the different conic sections. Give some applica-tions of each.

Conic sections (2) Define the conic sections as loci. Show how to draw each using pencil and string.

Conic sections (3) Show how to develop the conic sections using envelopes. Explain why one of the folding processes works.

Compound interest (1) Show how to calculate compound interest using the simple interest formula.

Compound interest (2) Show how to calculate compound interest using (a) tables and (b) a formula.

Am ortization

Percentiles

Mean and standard deviation

Normal distribution

210-211

100-101

178-181

366-369

340-341

34-35

526-530

526-535

28

Research Topics

Topic References Pages

Normal distribution continued AMY: An Introduction 357363 (Dottori, Knill, Seymour)

Capture-recapture sampling Math Is 3 340 technique (Ebos, Tuck)

Applications in School Math 182-185 (NCTM 1979 Yearbook) 144-145

Pascal's triangle and binomial Mathematics, A Human Endeavour 442-449 probability (Jacobs)

Binomial expansion using Pascal's Holt Algebra 2 382-384 triangle (Nichols, et. a!)

Using Advanced Algebra 414-417 (Travers, Dalton, Brunner)

Finite differences to find the th Finite Differences 31-41 term of a sequence (Seymour, Shedd)

Venn diagrams to solve counting Intermediate Mathematics 2 32-34 problems (Dottori, Knill, et. a!)

Finite Math. and it Applications 188-194 (Goldstein, Schneider)

Bases other than ten (1) Introduction to Mathematics 35-51 Writing and interpreting numerals (Brumfiel, Eicholz, Shanks) in bases other than ten. Mathematics, A Modern Approach 3-31

(Wilcox, Yarnelle)

Bases other than ten (2) Introduction to Mathematics 35-51 Addition, subtraction and multi- (Brumfiel, Eicholz, Shanks) plication in bases other than ten. Mathematics, A Modern Approach 14-43

(Wilcox, Yarnelle)

If-then statements Using Geometry 85-89 (Wells, Dalton, Brunner) Geometry 8-16 (Jacobs)

Heat loss by fat and thin people Mathematics 11 175 (Del Grande)

Cycloids Mathematics, A Human Endeavour 352-359 (Jacobs)

Stylometry Intermediate Mathematics 2 330-331 (Dottori, Knill) Applied Mathematics 208-209

• (Dottori, Knill, Seymour)

If-then statements, Euler diagrams Using Geometry 85-89 (Wells, Dalton, Brunner) Geometry 8-16 (Jacobs)

29

Discrete Mathematics as Enrichment Material

Katherine Heinrich

Katherine Heinrich is an associate professor in SFIJ's Department of Mathematics and Statistics.

During the last couple of years, there has been considerable interest in teaching discrete mathematics. Some have even gone as far as to suggest that discrete mathematics be taught in place of calculus.

Although discrete mathematics has been studied seriously by mathematicians for at least the last hundred years (making it relatively young as a mathematical subject), only now are other areas of science and mathematics realizing its value and impor-tance. One of the main reasons for this is the rapid advancement being made in computer capabilities. Computers are now able to "solve" mathematical problems that because of their size were previously infeasible. To clarify this statement, let us look at an ex-ample. One of the problems faced by large companies is to determine the weekly pro-duction levels of their various products. This level is determined by the demand for the product, the availability of raw materials, current inventory, equipment available, per-sonnel available, the fluctuating costs, the time of year, available transport, etc. All of this can be expressed mathematically as a (very) large number of linear inequalities in an even larger number of variables, and a function which must be optimized. This is a linear programming problem and, although we can give (mathematically) a good pro-cedure to solve it, we cannot carry out all of the computations without fast computers. Many of the problems studied by theoretical computer scientists are problems taken from discrete mathematics and many others are discrete mathematics problems that have arisen from the study of algorithms in

computer science (remember, the computer scientist wants to find better and more effi-cient algorithms). It is now well recognized that to be a good theoretical computer scien-tist, one should have a good solid mathe-matics background.

Discrete mathematics is a very large field employing aspects of many of the more traditional areas of mathematics, such as number theory, probability, and group theory. It also includes many new areas of mathematics. Some of these are enumera-tion, graph theory, network theory, design theory, tournament scheduling, coding theory, and optimization. So to introduce students to discrete mathematics, one can choose from a variety of topics and at a variety of levels. Many universities offer a course usually called "Finite Mathematics." This is invariably intended for students with little interest in mathematics, but who need some mathematics credits in order to satisfy a degree requirement. Such courses typically include the binomial theorem, combinations, and permutations and probability. Although such a course can be useful to many students, it is not the kind of discrete mathe-matics program that I wish to address. I am interested in using discrete mathematics as a tool to take the best students beyond the mathematics curriculum.

One of the reasons for choosing discrete mathematics as a special-topics course or as an enrichment project is that a student need not have studied a lot of mathematics previously, and whatever extra mathematics you may require can be learned as you go

30

along. Of course, one must be careful about such a statement, because it is a little misleading. While the student need not have learned a lot of other mathematics, he/she will need to have a certain level of mathematical maturity and, more impor-tant, to enjoy mathematics.

Another reason is that discrete mathematics is exciting for both the student and the teacher. Consider the following statements, all of which can be studied in discrete mathematics:

1. If n +1 numbers are chosen from the set (1, 2 .....2n), then one of them will divide another one of them evenly. 2. There are an infinite number of primes. 3. There are exactly five platonic solids. (A platonic solid is a solid object, each face of which is a regular n-gon. The platonic solids are the tetrahedron (four faces, each a triangle), the cube (six faces, each a square), the dodecahedron (twelve faces, each a pen-tagon), the octahedron (eight faces, each a triangle), and the icosahedron (twenty faces, each a triangle). 4. 1+2+3+. . .+ n3 = (1+2+3+

.+n)2 5. In any group of six people, there will always be either three people who all know each other or three people none of whom knows the other two. 6. If n married couples go to a dance, there are n! (1-1/1!+1/2!-1/3!+1/4!—. .

(-,)"n!) different ways couples can dance so that no married couple dances together.

The other reasons I would choose discrete mathematics as an enrichment tool are as follows:

1. It is possible to show the students prob-lems that are easily understood, but are as yet unsolved, thus showing them that mathematics is alive and that people (mathe-maticians) still discover new mathematics. One example is the Goldbach conjecture of

1742, which states that every even integer greater than 2 can be written as the sum of two primes. Another example is the four-color theorem that was solved only ten years ago.

2. It is possible to "keep going" on a prob-lem you thought was done. So a student is encouraged to keep asking questions like "What if . . .?" and to keep thinking and making conjectures about what he/she thinks will happen. Consider the following problem. Try to draw the pictures below without taking your pen from the page. Discover why some work and others don't. Invent an algorithm that will always yield the drawing without taking the pen off the paper, given that you know it can be done. Given any drawing, exactly how many times will you have to lift your pen?

3. Discrete mathematics is a good vehicle for introducing the students to proofs, and be-ing able to understand and construct proofs is a valuable tool for all mathematics students.

31

4. Finally, discrete mathematics gives you the opportunity to equip the student with in-valuable mathematical tools and techniques (among which I include the ability of ask questions and to think hard about problems).

References

Discrete mathematics projects In late September, we mailed from the Mathematics and Statistics Department at Simon Fraser University a package of prob-lems (mainly chosen from discrete mathe-matics) to each school in B.C. and the Yukon. We plan four such mailings during the year. Each package will consist of two parts. There is one page of challenging problems, and a larger project containing problems on a specific theme (in the first package, the theme is the pigeon-hole prin-ciple). The project is accompanied by a set of solutions and commentary for the teacher. The students are invited to submit to us their solutions to the challenging problems. Prizes and certificates will be awarded, and a com-plete set of solutions will be sent back to each student. More details are given in the letter accompanying the problems.

Books written on discrete mathematics ex-pressly for the student • Mathematical Puzzling and Discovering Mathematics: The Art of Investigation. Both are by A. Gardiner and will soon appear. They are being published by Oxford Univer-sity Press. • MATH! Encounters with High School Students and The Beauty of Doing Mathe-matics: Three Public Dialogues. Both are by S. Lang and published by Springer-Verlag. • Graphs and Their Uses, by Oystein Ore. • Groups and Their Graphs, by I. Grossman and W. Magnus.

• The Mathematics of Choice, by Ivan Niven. • Invitation to Number Theory, by Oystein Ore. • The Mathematics of Games and Gambl-ing, by Edward W. Packel.

These last five books are part of the New Mathematical Library published by the Mathematical Association of America. The books vary greatly both in quality and in ac-cessibility to the student. The MAA also has many problem books in this series.

Magazines • Crux Mat hematica, currently produced by the mathematics and statistics department at the University of Calgary. • Mathematics Magazine, a journal of col-legiate mathematics published at Santa Clara University, California is designed to enrich undergraduate study of the mathematical sciences. The aim of the magazine is to publish articles that can be used by instruc-tors as the basis for projects.

Discrete Mathematics Texts • Elements of Discrete Mathematics, by C.L. Liu. Published by McGraw-Hill. • Introduction to Combinatorial Theory, by R.C. Bose and B. Manvel. Published by John Wiley and Sons. • Graphs as Mathematical Models, by Gary Chartrand. Published by Prindle, Webef and Schmidt. • Basic Techniques of Combinatorial Theory, by Daniel I.A. Cohen. Published by John Wiley and Sons. • Combinatorial Theory: An Introduction, by Anne Penfold Street and W.D. Wallis. Published by the Charles Babbage Research Centre, Winnipeg.

These books were all written for a second-year university mathematics course; some of the problems are extremely difficult.

32

A Very Radical Complex Cross-Number Puzzle

Geoffrey R. Tomlin

Geoffrey Tom/in is a mathematics teacher at Burnaby South Senior Secondary School.

Here is a very radical, very complex, cross-number puzzle that is a real i-opener. Photocopy, if you wish, the next page and hand out to your senior algebra students. (Beware: if you are not a member of BCAMT, the next page is programmed to automatically , self-destruct upon contact with a copier, all before your very i's.)

If you have not covered rationalization of a binomial denominator with your students, substitute the following clues:

4 across: (3 + 20(1 - i)

18 down: (-17 + 9J )(2 + V5)

Solution Across Down 1.3+5i 15.41 1.3+15i 4.5-i 16.7.J 2.5-Ji 6. 2ñ + /3 i 17. 5-,F3 3. 312-i 7. + Si 18. 128 4. sJ + 2J5 9. 5 + 1 19. + 11 5. -i

10. -21-.15 21. 1 + F7 6. 25 12. 17,[5- 22. 22 + 8. + 18J7 13. 27i 23. J7 + lii 10. -71

14. 8i 24. V5- i

11.hi 12. 18J3 + 21 13. 2V3 + 5V7 15. 42 + 16. 7'.J

18. 11 + \/ 20. 121

33

A Very Radical Complex Cross-Number Puzzle

Across 1. (4 + i)(1 + 1) 4. 4-61

1-i

6.J+V-3 7. (iä + i)(1 + J2i) 9. The complex conjugate of 4 across

10. -\ö - .Jii - -

12. 40 + -..J 5.

13. A pure imaginary number with an ab-solute value of 27.

14.

15. 140 - 911

16.

Write answers in the form a + bi. Simplify radicals.

• MEN • . I.... • MEMO 0 -1 E MEN - I... • lilt.

• I.. I.. lU No .l.• 1ll 1111 I 'ill I• I lull MENEM No

17. \/ 18. 2 19. 18 down, backwards. 21. (-2 + \f )(3 + fi) 22. (5 + 2J)(8 - 3.1)

23. + J-121

24.

Down

1. (3 + 2i)(3 + 31) 2. 7 across X 5 down.

3. \/-18

4. 6 + \/ +

5. i3

6. (24 across)4

8. 10 + 63 T-I 10.

11..•

54 2 12. --- ----\I

13. V2_4 + 14 + \/j

NF2 V-7

15. (2 + 3J 0(3 - i)

16. Jö +

18. 17 - 2-J

20. 112

34

MATHEMATICS EDUCATION

Mathematics Education in North America

Alan Taylor

Alan Taylor is director of instruction for the Coquitlam School District.

Introduction As we proceed into the latter half of the 1980s, it is timely to pause and reflect on the extent to which some of the recommenda-tions contained in the National Council of Teachers of Mathematics' Agenda for Action have been implemented. Since I view public perceptions of mathematics education to be extremely important, the focus for this paper is on Recommendation #8, which reads as follows:

Public support for mathematics in-struction must be raised to a level com-mensurate with the importance to mathematical understanding to in-dividuals and society. (NCTM, p. 26)

While all of the eight recommendations con-tained in the-Agenda for Action are impor-tant, I believe that Recommendation #8 pro-vides the key for the rest. Solutions to issues related to the other seven recommendations, for example, cannot be achieved solely within the educational community. They re-quire the active participation and support of parental and societal groups.

To address this topic in an orderly manner, I plan to first deal with the need for public support, as articulated at the beginning of this decade in the Agenda for Action. Second, I will include a few words on government support of education during the 1980s, with reference to both the United States and Canada. From this setting, I then plan to review public perceptions of education—which, in effect, act as drivers for the extent of government support it

receives. Following this dialogue, the paper deals with the following question, as an in-dicator of success in this area: Is there, or is there not, a crisis in mathematics educa-tion? The balance of the paper views needs and impacts, including demands to be faced in subsequent years and plans to deal with them. Since much of the data available in Canada does not "break out" discipline areas, such as mathematics, a number of references deal with education in general.

The need for public support Public support is essential to the promotion and success of mathematics education. It brings with it direction for planning, credibility and respect, and resources necessary for the implementation and maintenance of programs. This support, however, is assured only through mean-ingful involvement of a number of different participant groups.

The council's publication refers to a number of unnecessary obstacles to the effective functioning of teachers and students in a true teaching /learning interaction, due to a lack of public support. These include unproduc-tive record keeping, unmotivated and un-disciplined students, lack of parental support, ambivalence in government regula-tions, shifting societal priorities, and a lack of school and home agreement on out-of-school study assignments.

To address this situation, it is essential that a common goal is developed, shared by both

35

the professional community and society, to bring all citizens to the full realization of their mathematical capacity. This is a com-plex task requiring the commitment and co-operation of all segments of society. In response to this need, the Agenda for Action listed the following recommendations:

• Society must provide incentives that will attract and retain competent, fully prepared qualified mathematics teachers. • Parents, teachers and school ad-ministrators must establish new and higher standards of co-operation and teamwork toward the common goal of educating each student to his or her highest potential. • Goverment at all levels should oper-ate to facilitate, not dictate, the attain-ment of goals agreed on co-operatively by the public's representatives and the professionals.

What are the public's perceptions?

In forming opinions on the credibility of the educational system, the public relies on two major sources of information: through direct interaction with the school and by news reported in the media. The first source, which falls within our immediate domain is by far the most accurate and effective of the two. The media, on the other hand, may provide inaccurate information through weighted emphases and failure to report in-formation within a contextual framework. It is our responsibility to improve the lines of communication between school and com-munity, and to provide the media with in-formation set within an interpretive frame-work.

Often events far removed from education can generate the greatest change. For exam-ple, we saw ripples from the launching of Sputnik in the 1950s wash to shore, in great haste, the "New Mathematics" program.

A concern of simila magnitude resulted in the late 1970s when Issac Wirszup, in a com-parative study of mathematics education between the U.S.S.R. and the United States, reported that enrolment levels and the con-tent of mathematics programs in the U.S. were sadly lacking when compared to Russia. His report, which received little coverage in Canada, generated headlines in the United States, causing President Jimmy Carter to term it a "national crisis." Subse-quently, large sums of money were made available for mathematics programs and research. In spite of this infusion of resources, however, public perception on the state of mathematics education in the United States remained on the downside. This was due, in large measure, to evidence reported from a number of sources. For example, trends toward lower scores on standardized tests, such as the Scholastic Aptitude Tests (SAT) and the National Assessment of Educational Progress (NAEP), supported that viewpoint.

In Canada, the state of education is viewed on a more global basis, in which disciplines such as mathematics are usually not singled out. This is due partly to provincial autonomy in education gained under terms of the British North America Act, and also to the lower profile given to national defence—an area in which mathematics and science are major players.

The "back-to-basics" movement remains alive and well in both the United States and Canada, but does not generate the same fer-vor in this country. This is likely due to a greater degree of confidence in the education system held by the Canadian public. For ex-ample, results of a p011 on public perceptions of the school system conducted in 1984 by the Canadian Educational Association, as well as a number of findings reported in the provincial report entitled "Lets Talk About Schools" (1986), provide evidence to challenge the following myths:

36

• respect for the importance of educa-tion has been lost. • confidence in schools no longer exists. • standards of education have dropped.

Evidence related to each of these myths is as follows:

MYTH 1 The public has lost traditional respect for the importance of education.

• 78% of the public indicated that schools are extremely important for future success (only 3% chose not im-portant or had no opinion).

MYTH 2 The Canadian public has lost confidence in schools.

• 48% of the public graded schools as "A" or "B" (61% of parents). • Only 3% graded schools as "Fail."

MYTH 3 Standards of education have dropped.

• Achievement change data in the 1981 and 1985 assessments of mathe-matics in British Columbia show either improvement or no difference across domains. • Norming studies for the Canadian Test of Basic Skills and the Canadian achievement tests show higher Cana-dian results on those items which are common to their American counter-parts (e.g., the Iowa tests of basic skills and the California achievement tests). • Comparisons with NAEP and GED test results show significantly higher Canadian results than those in the United States. • Performance of B.C. and Ontario schools compared favorably to other countries in the International Study of Mathematics (1981).

In the same poll, Canadians were asked to rate eight public institutions according to the degree of confidence they have in them. Responses were given on a five-point Likert-type scale ranging from "no confidence" to "a great deal of confidence." Figure 1 shows results where the two positive responses in the scale are combined.

Public Perceptions of Institutions

Rank % Confidence 1. Schools 75.5 2. Local government 70.8 3. The church 70.7 4. Federal government 67.6 5. The courts 66.9 6. Provincial government 63.1 7. Big business 59.4 8. Labor unions 46.0

FIGURE 1

What government support has been forthcoming? It appears that crises generate action and the resources needed to correct them, whereas satisfaction without clearly defined expecta-tions, yields only complacency. For exam-ple, after the American public was informed of the evidence provided by Wirszup, declin-ing test scores, and a number of other con-cerns expressed in the publication entitled "A Nation at Risk," the issues were translated into the following sequence of events: from a beginning in 1981, when Reagan termin-ated all National Science Foundation (NSF) programs at the pre-college level, congress awarded the foundation $15 million in 1982-83, $55 million in 1984, and $500 million in 1984-85.

In Canada, we tend to be too complacent. As a result, the B.C. model of "leaner is

37

meaner" is fast becoming the norm. While efficiency and ability-to-pay are important considerations, it is important that high ex-pectations, teacher morale, and incentives remain integral components of the system. Illustrations of funding levels for education and changes between 1984 and 1985 are shown by province in Figures 2 and 3.

Share of Total Provincial—Local Government Spending for Elementary and Secondary

Education 1984

Newfoundland 16.19 Prince Edward Island 14.58 Nova Scotia 16.86 New Brunswick 15.59 Quebec 11.50 Ontario 16.50 Manitoba 13.33 Saskatchewan 14.91 Alberta 14.17 British Columbia 10.84 Yukon Territory 17.46 Northwest Territories 14.76

Canada . 13.95 Source: Statistics Canada, Consolidated Government Finance. 1984, unpublished data.

FIGURE 2

School Funding Compared: Cross-Canada Highlights

/, Increase in Government Funding of Public Schools in 1985 over 1984 Levels

Nfld. 1.9

P.E.I. -I 6.0

N.S. . 49

N.B. 37

Que. 3,5

Ont. 4.0

Man. 2.0

Sask. 10.0

Alta. 2.0

B.C. (2.4)

-2 0 2 4 6 8 10 % Increase

Sanr,'e CTF Otob.' 85 Economic Se,'i,.' Bdh'ti,,

FIGURE 3

These data paint a rather bleak picture for British Columbia. For example, education's share of government spending in 1984 ranked last among the provinces, and it had the dubious distinction of being the only province in which funding levels for educa-tion between 1984 and 1985 decreased.

A major task we now face is to convince the government to allocate needed resources before a crisis stage. is reached. To ac-complish this objective a highly complex process is required.

Is there a crisis in mathematics education?

Perceptions on the state of mathematics education vary among constitutent groups. To address this issue, I suggest we look at two major indicators. First, the question of standards, and second, the supply of and de-mand for mathematics teachers.

Standards and Comparisons The question of standards is often raised, but seldom dealt with through hard evidence. To address this question, both expectations and comparisons need to be considered.

Expectations, which are subjective in nature, are determined through a process where con-sensus on levels of student performance are reached by a representative group of educators and members of the community. Program level assessments in several prov-inces include this process in the inter-pretation of student achievement results, and evidence suggests that, for the most part, ex-pectations are met.

Comparisons, on the other hand, are a more objective measure. Student performance can be compared with other jurisdictions or else over periods of time. Evidence of these com-parisons points to positive results. For ex-ample, jurisdictional comparisons between

38

the United States and Canada in GED (General Equivalency Development), CTBS (based on the Iowa tests of basic skills), and the CAT (based on the California achieve-ment tests) show significantly higher results by Canadian students over their American counterparts. Favorable comparisons also exist between British Columbia and Ontario, and a number of other countries who par-ticipated in the International Assessment of Mathematics (1981-82).

Data measuring change over time, however, are not as readily available. To make com-parisons which are meaningful in this area, objectives measured must be of the same level of importance in each of the curricula of the day, student retention rates must be examined and test items must be equivalent. Taking these considerations into account, much of the change data among the 1977, 1981, and 1985 provincial assessments of mathematics in British Columbia show an improving trend in student performance.

Information based on the issue of standards does not point to a crisis in mathematics education in Canada. Teacher supply, the second indicator for review, is addressed next.

Supply of Mathematics Teachers Evidence in the United States'suggests that the supply of mathematics teachers is a prob-lem of alarming magnitude. The following examples, reported by Norrie, et. a! (1984), illustrate this concern:

• California graduated 114 students in math education, but Los Angeles dis-trict alone needed twice that number. • The University of Nebraska received requests for 464 new mathematics teachers, but graduated only 18. • 32 275 students in New York State were taught mathematics by non-mathematics teachers.

On the surface, Canadian data convey a

somewhat different picture since the total number of teachers exceeds the number of job openings. However, due to cutbacks in hiring and the re-assignment of non-specialists to teach mathematics, increased numbers of teachers trained in subjects other than mathematics are teaching the subject. For example, in the 1985 British Columbia Assessment of Mathematics, it was reported that 40% of teachers at the Grade 10 level completed fewer mathematics courses than the minimum number required for a major. Further evidence showed that 18, 33, and 24 per cent of teachers at Grades 4, 7, and 10 respectively, completed no ' mathematics methods courses at all.

A crisis in the supply of mathematics teachers currently exists in the U.S. and is close at hand in Canada. To deal with this issue, we must determine our needs, allocate resources to train teachers and provide in-centives to attract and keep them.

What are the future needs and how should we address them?

At this point, I should tread rather softly as speculation enters the picture. For example, a measure of the difficulty I face in predict-ing the future can be gained through a review of my poor record in playing the stock market or guessing Loto 649 numbers. The difficulty of prediction was expressed by two notable personalities: Mike Harcourt, former mayor of Vancouver, and the Chinese leader Mao Tse Tung. The former was quoted as saying "The future is not what it used to be," whereas the latter shared the following perceptive observation, "The problem with predicting the future is thatit requires too much speculation."

A graphic display of projected increased needs for elementary and secondary teachers in the 'United States until the year 1993 is shown in Figure 4. This trend will likely be similar in Canada.

39

Projected Demand for Additional Teachers to 1993

:ntar

U

Secondary

C

0

25

0U I U I I I I I I I

1980 1983 1988 1993 U.S. L'poot..'ot of Ei,otio,: t,',,t,o,,aI C,t', I,, Education Statistic,

FIGURE 4

The graph shows a decline in need for teachers at the secondary level until 1988. During this period, mathematics classes will, in many cases, be taught by teachers assigned from other subject areas. This is.a time when massive in-service and re-training is needed. Beginning in 1988, an accelerated need for mathematics teachers will develop. These teachers are currently entering our faculties of education. The time to attract and train these teachers is now. Equally important is a need to retain both current teachers of mathematics and new graduates once they arrive in the classroom. To accomplish this objective, we need to implement what I term the "3 Rs"—Recognition, Reward, and Remuneration.

Recognition involves an overt expression of appreciation for a job well done. We often fall short in this category and take for granted excellent work done by our col-leagues. This recognition could take the form

of reporting staff accomplishments in school newsletters or formal recognition of teachers who are doing excellent work by the profes-sional associations they belong to.

In the reward category, we should rid our-selves of the notion that the best way to reward teachers is through their promotion to administration—thereby removing them from the activity they do best. There are other ways to reward excellence in the class-room. Examples of a highly prestigious award which goes a long way to improve the morale of recipients as well as their col-leagues, are the presidential awards for ex-cellence in science and mathematics in the United States, established in 1983. In this program, one mathematics and one science teacher from each state is brought to Washington for award ceremonies, as well as to receive gifts for their schools. Canada would do well to establish a similar program for recognition for excellence in teaching.

40

Remuneration is usually based on formal training and experience. In the current system, little or no attention is paid to how well a teacher does the job, or how much the prerequisite skills are in demand in the marketplace. For example, comparisons of starting salaries in technical occupations and professions, which require mathematics and science training, show that teaching salaries are lagging behind. It is contended that this variance is a major reason for the shortage of mathematics teachers in the United States. To address this concern, a number of dif-ferent possibilities should be explored. Among these are the master teacher concept, merit pay, performance contracting, the voucher system, incentive plans, and demand-supply indices. All of these concepts have both advantages and disadvantages, discussion of which is beyond the scope of this paper. However, a brief description of each follows.

The master teacher concept involves iden-tification of teachers who are superior in their field to work with others in the role of helping teachers. The master teacher is assigned different duties and receives better compensation. Although this concept pre-sents a significant problem in the selection and evaluation of teachers in this category, it is growing in popularity in the United States. Willoughby (1986) reported that 14 states currently have this plan in place and 23 others are considering it.

Merit pay is an issue which has been debated extensively. In the October 1984 issue of Phi Delta Kappan, results of a Gallup poll were reported in which 32% of classroom teachers and 76% of the general public supported the concept.

Performance contracting involves payment based on results. The amount of remunera-tion is a function of student growth. This concept was piloted in Texarkana in the late 1960s and met with mixed findings.

The voucher system was first introduced in Ontario in the late 1800s. Since that time, it has been piloted in a number of school districts throughout the United States. In this plan, students are provided vouchers, or purchase requisitions, payable to the schools in which they enrol. A pilot on this concept was conducted in Seattle schoollin the late 1970s, and there has been talk of it recently by some politicians in British Columbia.

Both the incentive plan and demand-supply indices relate to the need for and the availability of teachers in a specific discipline. For example, due to a shortage of mathematics teachers, some jurisdictions in the United States, such as the Houston (Texas) Independent School System, offer a higher salary scale to attract teachers of this subject. Demand-supply indices provide a more precise method of determining the need for teachers of specific disciplines. An index for each subject is determined by dividing the number of positions required by the number of qualified people teaching it or seeking positions. Lichtenberg, in an article in the May 1985 Mathematics Teacher, sug-gests that this index be a third variable, along with training and experience, for use in the determination of salary.

To meet these future needs, we need assur-ance of the resources to provide them. I predict, however, that there will be greater competition for these resources than currently exists, and our major challenge will be to earn education's rightful share.

What is our major challenge of the future?

By the end of this century, the proportion of parents in the community will decline, while that of citizens over age 65 will in-crease substantially. Consequently, there will be fewer taxpayers and more users of social services. We will be in the fight of our

41

lives to maintain education's share of a dwindling pot.

In order to maintain our share of resources, we need to keep our house in order. We must articulate our goals in such a way that the public understands and accepts them, and so that they will be above political tampering. Parents, the business community and other professionals must have a sense of owner-ship of the educational system and fully understand the benefits to be acccrued from it. We must be accountable for what we do and have the data to support our position.

To articulate our goals, we should begin with a mission statement which provides a sense of purpose and a basis from which to develop guidelines. To operationalize this statement, we need to clearly identify the goals, content emphases and process-affective objectives of the curriculum.

Parents and the community must develop a sense of ownership of the system and be in-volved in meaningful ways at all stages of its development. This involvement can be fostered through invitations to parents and the community to visit classrooms, articula-tion with organizations (such as Rotary, PTA, medical and law associations, etc.), and the use of "twinning arrangements" where schools are "adopted" by businesses.

Accountability is a must. Ways to improve supervision and evaluation should be developed. A first step in this direction is to list all outcomes of the educational system and determine corresponding indicators of quality. Once these indicators are identified, we would attempt to measure them in either concrete or perceptual terms. Results should then be interpreted and reported. Through this process, educators and society in general will gain a meaningful knowledge of the strengths and weaknesses of the system, determine direction for the allocation of resources and monitor change over time.

In this paper, I have discussed the need for public support of education, suggested several indicators of it, and attempted to predict future trends and needs. In assessing the impact of Recommendation #8, I con-clude that little progress has been made since 1980. The need to address this recommen-dation, however, has become increasingly important as the decade progresses.

Through this discussion, I hope to have put to rest, for the short term, at least, one of the following three greatest myths:

• Your cheque is in the mail. • I will respect you in the morning. • The Canadian education system

lacks credibility.

References Horizon Research & Evaluation Affiliates, Education in British Columbia, Myths and Facts. Victoria: Ministry of Education, March 1985.

Ministry of Education, Let's Talk About Schools—Summary and Highlights. Victoria: Ministry of Education, 1986.

Mort, Janet N. School Successes: An unsung song. BCPVPA Bulletin, February 1986.

National Council of Teachers of Mathematics, An Agenda for Action. Palo Alto: Dale Seymour Publications, 1980.

Norrie, Alexander L. & Kirkwood, Kristion. Report on teachers of mathematics shortage, Ontario 1983-84. QAME/A OEM. March 1984, 43-45.

Paul Clyde. Mathematics teachers—we need you! The Clearing House, April 1983, 344-344.

Robitaille'

David F. & O'Shea, Thomas J. (Eds.) British Columbia Mathematics Assessment 1985 General Report, Victoria: Queen's Printer, 1985.

Say, Elaine. A shortage of mathematics teachers in Houston, Mathematics Teacher, December 1983, 644-645.

Wirszup, Isaac. Education and national survival: Confronting the mathematics and science crisis in American schools. Educational Leadership, December 1983/January 1984, 5-11.

Wood, Fred, H. Hold the line for quality in mathematics and science teachers, Educational Leadership, December 1983/January 1984, 43-44.

42

MATHEMATICS AND COMPUTERS

Remember the Function Machine?

Harold Brochmann

Harold Brochmann is a North Vancouver computing teacher on a leave of absence in Thailand.

The function machine is a hypothetical "black box" which we feed with rumbers. For each number input, the function machine outputs another number related to the first according to some rule. If, for ex-ample, the function machine contains the rule Y = 2x + 3, and we provide it with "5," we get "13" in return. After several such trials, the student is expected to infer the rule the machine contains.

One problem with how we have used the function machine in our classrooms is that the common function rules have very little motivational value because activities are not usually placed in context of applications. The function machine has tended to be perceived by students as just another piece of mathematics esoterica with no obvious usefulness.

Consider a really versatile function machine that can accept complex inputs like the following: AREA SQUARE SIDE 5 AREA RECTANGLE CIRCUMFERENCE 45 WIDTH 3

AREA POLYGON 5 12 AREA POLYGON "ABCDE CURRENT VOLTS 12 RESISTANCE 6 AMOUNT DEPOSITS 25 RATE 8 TIME 12

This proposed function machine would be a real black box with keys for inputs and a display for outputs, and it would contain a

range of function rules applicable to disciplines from statistics to engineering and chemistry. It would automatically select the appropriate rule on the basis of the input syntax.

If such a machine were readily available in the classroom, could it be useful in teaching mathematics? Would such a machine allow us to extend the scope of mathematics in-struction, and allow us to illustrate that mathematics really is an integral part of vir-tually all human activity? If we had such a machine, how would we use it?

Possible activity: This is a schematic of an electric electric cir-cuit (diagram). Electric "pressure" is measured in VOLTS. The rate at which the current flows is measured in AMPERE. Please read pages 45-46 of Understanding Science.

Enter CURRENT [VOLTS 10 RESISTANCE 21 into the function machine. The result is the number of ampere flowing in a 2 ohm resistor with 10 volts applied.

Make a table of values for current and voltage, keeping resistance constant at 2 ohm.

Make another table of values obtained for current and voltage; this time keeping the resistance at 6 ohms.

43

Write a statement which describes what happens to the value for area as the number of sides increases.

Prepare a properly labelled set of co-ordinate axes and plot two graphs from your results.

Give an equation which defines the relation-ship between current, voltage, and resist-ance. Substitute several value from the results you obtained to verify that the equa-tion is valid.

Write a sentence which describes the function.

Note: This activity cannot be done with real voltmeters and ammeters in junior second-ary, because of a variety of practical reasons: the observed results would simply not fit the "correct equation," there would be problems with students reading the meters, it would take so long that the students would lose track of what they were doing, etc. If you are going to try this sort of thing efficiently, you pretty well have to use a function machine.

Possible activity: A square is a polygon with four sides. Sup-posing a square had a perimeter of 100 cm, then each side would be (100/4) = 25 cm long and the area would be 25 X 25 = 625 cm squared.

Enter AREA POLY 4 (100/4) into the func-tion machine. You get 625.

To find the area of a triangle with a perimeter of 100 cm, you enter AREA POLY 3 (100/3).

Make a table of values which compares the areas of polygons of perimeter 100 cm with number of sides in the range 3-20. Prepare a labelled graph of the results.

Make another table which compares areas of polygons with number of sides in the range 100-500. Prepare a labelled graph of the results.

Predict the area of a polygon of perimeter 100 cm with an infinite number of sides.

Enter AREA CIRCLE CIRCUMFERENCE 100 into the function machine.

Note: The calculations for finding the area of so many polygons are tedius, and also too complex for junior secondary students. If you are going to do this sort of thing, you pretty well have to use a real function machine.

Is it always necessary that students be able to write a specific equation for a function? Could it be that a verbal description of the resulting graph is in some circumstances of more or equal educational value?

For example, must applications of the areas of polygons be left until the student is capable of proving as well as performing the algorithm by which the area is found?

Those teachers who would like to experi-ment with a function machine of the type I describe will be glad to know that I have one which I will gladly share. As you might have guessed, it is simply a suitable Logo work-space. At the moment, my workspace con-tains only area-related functions, but I have plans for unlimited expansion!

Obviously, it would also be possible to write a function machine program in BASIC. Those who have read my previous contribu-tions to Vector will know that I am a Logophile, and I would state without fear of contradiction that Logo lends itself to this sort of thing much better, because list pro-cessing is a far better tool for input parsing than BASIC string functions.

In 1984, the National Council of Teachers of Mathematics produced a booklet titled

44

"Computing and Mathematics." The lead article suggests that we may have been teaching mathematics in an inappropriate se-quence. In general, mathematics instruction proceeds from the general to the specific, from principles to applications, from algorithms to "problems." The availability of computers facilitates an inversion of this order.

There wasn't very much local reaction at the time. Maybe we weren't ready to hear such heresy. Perhaps the climate has changed since then. Perhaps the function machine I have described could make such an inver-sion possible. I urge everyone to get a copy of the above mentioned booklet. It contains much that needs to be thought about—and discussed in the pages of Vector.

UBC Hosts the Largest Open House Ever

In March 1987, UBC will open its doors to the people of British Columbia, hosting the largest Open House ever held on campus

For three days from March 6 to March 8 UBC will be open from 10:00 to 17:00, providing an exciting free program of special events activities displays shows and lectures All twelve faculties and specialized campus attractions will be open showing some of their innovative people and ideas. The open house program will include:

• A show of magic chemistry tricks • A poetry workshop given by the legendary Earle Briney, himself a UBC graduate • Concerts given by members of the music department • The Physics Olympiad for secondary school physicists • A lecture by Dr. David Suzuki on the role of the humanities in a high tech world • Free entry to the Museum of Anthropology • Free swimming at the UBC pool • Tours of TRIUMF (Tr-University Meson Facility) * A Triathalon event * Major sporting events—maybe rugby, volleyball, field hockey * Special children's activity area * Displays and demonstrations by UBC activity clubs * Skate-board demonstrations * International refreshments * Career information * An aqua show in the UBC pool * Hot air balloon rides UBC faculty will be on hand to answer questions about courses in their disciplines, career

paths, and opportunities. Further information will be coming to you regularly, as planning for the open house pro-

gresses. Meantime, put March 6-8, 1987 in your calendar. We'll look forward to seeing you at UBC.

For further information, please contact Elaine Stevens, Community Relations, 228-2183.

45

8-11 APRIL

65TH ANNUAL MEETING

Learning, Teaching, and Learn* ing Teaching

TM

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

MISCELLANEOUS

What Did You Do During Your Summer Holidays?

James Sherrill

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

About 200 teachers of mathematics spent part of their summer on the Simon Fraser University campus attending the BCAMT Summer Mathematics Conference, and they were certainly rewarded for their time and effort.

The conference was opened by Dr. Dan Birch, Vice-President for Academic Affairs at the University of British Columbia. The title of his presentation was "The Geography of the Mind and the Arithmetic of Intelli-gence." He brought us up to date on many of the things happening in the area of measuring intelligence. He was able to make use of metaphors from both his academic area and ours. I'm sure what most of us will remember, however, were the excellent anecdotes he had of teachers he has known.

After the opening session, we had the op-portunity to attend any four sessions out of the 32 offered on the first day of the con-ference. As usual, it was very difficult to select only four of the sessions. Eight of the sessions were directly related to the proposed new curriculum with emphases on the higher visibility to be given to problem solving.

Since many people feel that whatever ap-pears in the mathematics curriculum is going

to be tested, Walter Szetela discussed how to evaluate students in the area of problem solving. Janet Elliot gave a two-session presentation on "Teaching Problem Solving —A Problem?" Other sessions dealing with the proposed new mathematics curriculum included Bill Kokoskin from Handsworth Secondary School in North Vancouver discussing "Introductory Algebra 11—The New Bridging Course," and John Klassen, the president of the BCAMT, discussing "A Revised Curriculum and Materials for General Math 10."

The first day of the conference ended at 16:00. It was clear by.the end of the first day that the conference had managed to over-come the enormous obstacles of magnificant weather and Expo. One could tell that people were pleased with what they were getting out of the conference and would be back for the second (and final) day.

The second day also began with the entire group of teachers gathered in the Images Theatre for a general session. The general session speaker on the second day was Brendan Kelly who is known to almost everyone who has made any study of the current mathematics textbooks. Brendan has a doctorate in mathematics, is working on

47

a doctorate in computing studies, and has already mastered the art of the non-belligerent use of ethnic dialects. Brendan's general session presentation was "Old-Fashion and New-Fangled Mathematics." Brendan also gave a senior secondary presentation later in the day entitled "Ten Mathematical Gems for Your Students." I would be hard pressed to pick which of his two presentations I enjoyed more, both were excellent.

The second day was also loaded with ses-sions dealing with topics which will be given much greater visibility in the proposed new curriculum. The new curriculum, however, is not the only issue of interest to BCAMT members. There was a packed house for the panel discussion on "Provincial Examina-tions in Mathematics: Questions and Answers." The panel consisted of John Klassen, representing the BCAMT; Stew Lynch, who served as a marker for one of the examinations; and Alan Frisk from the Ministry of Education. Alan is a secondary mathematics teacher seconded to the Ministry of Education, and you could tell he had some of the same concerns as other mathematics teachers. The audience

participated fully as did the panel. The interplay between asking and answering questions was frank and profitable for those in attendance.

At the end of the conference, it had to be agreed that there was something there for everyone in attendance. The sessions ranged from primary graphing to computer net-working, from daily drill to mu math. My own personal highlight of the conference, however, occurred in the penultimate session —the BCAMT Annual Meeting. At that ses-sion, the very first BCAMT outstanding mathematics teacher award was given. The recipient was Dominic Alvaro of Argyle Secondary School in North Vancouver. Dominic has been active in the teaching of mathematics in B.C. for many years. He has been a leader in the local PSA as well as ser-ving as vice-president of the BCAMT. He is a mathematics teacher par excellence!

I hope you had a great summer, but if you missed the Summer Mathematics Confer-ence, then you missed a great deal. It was packed with excellent sessions, excellent speakers, and excellent teachers in atten-dance. See you there in 1987!

48

Treasurer's Report to the Annual Meeting Grace Fraser

INCOME Deficit from 1985 ($1219.20) BCTF grant 5 196.00 Membership fees 9667.00 Conference income 33 069.04

Total Income $46 712.84

EXPENSESExecutive meetings $1023.89 PSA Council 8.40 Subcommittees 97.35 Vector 8817.30 Newsletter 3249.40 Other publications 5044.86 Conferences and in-service 1 240.43 NCTM affiliation fees and meetings 256.63 Operating expenses 210.84 Other projects 2 142.99 Treasury bills 20 285.00

Total Expenses $42 377.09

Balance as of June 30, 1986 $4 335.75

49

A CHANGE IN EMPHASIS The consensus of reports from groups such as the American Association for the Ad-

vancement of Science (AAAS); the College Entrance Examination Board (CEEB); the Conference Board of Mathematical Sciences (CDMS); the National Assessment of Educa-tion Progress (NAEP); the National Council of Supervisors of Mathematics (NCSM); the National Council of Teachers of Mathematics (NCTM); the National Science Foun-dation (NSF); and others, is that we need a change of emphasis in the content and delivery of mathematics. The topics that are consistently stressed in these reports are:

estimation /mental spatial thinking problem solving computers /calculators probability/ statistics

number theory geometry/ measurement

Although several of the professional groups mentioned above have established some definitions and indicated direction for teachers, the effort to date has been fragmented by efforts with supplemental materials, and suggested schemes for curriculum enhancement.

Papers were solicited from the listed educators to provide direction on these topics to teachers, curriculum developers, and textbook publishers. This collection of papers is an effort to address each topic as it would (could/should) appear in textbooks for students moving from arithmetic to algebra. The "experts" contributing to this docu- ment are considered to be knowledgable in their respective topics by their previous study, research, and experiences.

Position Papers on Mathematics Curriculum in Grades 7 and 8 Mathematics Content Lynne Steen St. Olaf College St. Olaf, Minnesota

Estimation/ Mental Computation Robert Reys, Barbara Rey University of Missouri-Columbia Columbia, Missouri

Geometry Mary Lindquist Columbus College Columbus, Georgia

Probability and. Statistics Albert Shulte Pontiac Public Schools Pontiac, Michigan

Spatial Visualization Grayson Wheatley Purdue University Lafayette, Indiana

Technology Robert Kansky Texas A&M University Bryan-College Station, Texas

Problem Solving Number Theory Development Lesson Jesse Rudnick, Steven Krulik John Dossey Thomas Good Temple University Illinois State University Univer. of Missouri-Columbia Philadelphia, Pennsylvania Bloomington, Illinois Columbia, Missouri

Project Consultant Editor, Project Director Douglas Grouws Richard Lodholz University of Missouri-Columbia Parkway School District Columbia, Missouri St. Louis, Missouri

For a copy of this document, please contact: Richard Lodholz

Instructional Service Center 12657 Fee Feed Road

Creve Coeur, Missouri 63146

50

The Fifth Annual Mathematics Enrichment Conference

Simon Fraser University June 23-25, 1986

Harvey Gerber

Harvey Gerber is an assistant professor in SFU's mathematics department.

With the inclusion of four students and one teacher from Bellingham, Washington, we can truly say that our enrichment conference is now international in scope. The confer-ence combined lectures, problem sessions, social events, tours, and contests. There were 145 students and 20 teachers from all over B.C. and from Bellingham.

Talks for students and teachers This year, we had two invited speakers: Dr. Mary Williams of the Institute for Marine Dynamics, National Research Council, and Professor Evelyn Nelson of the Depart-ment of Mathematics, McMaster University. Mary Williams gave an interesting talk con-cerning the difference between textbook problems and the problems applied mathe-maticians face. Specifically, she spoke about the mathematics involved in the construction of ships and platforms used in the Arctic.

Professor Nelson gave two talks. Her first was "The Story of the Infinite Hotel." Many deep and difficult topics in set theory, in-cluding the continuum hypothesis, were woven in a delightful and entertaining man-ner. Dr. Nelson also spoke on "Women in Mathematics from Antiquity to the Present." Here she traced the accomplishments that women made in mathematics and the diffi-culties that they encountered.

"Modelling the Flights of Homing Pigeons" was presented by Professor Michael Stephens of our department. Professor Stephens used a classic result to describe the flights of home and migratory birds. The talk was an excellent presentation of what statisticians do, and how they do it.

Professor Brian Thomson used computer graphics to speak about "Fractals." The sub-ject, which is currently in fashion, was presented at the right level for the Grade 11 students attending. The secondary school teachers, along with our own students and faculty, also learned more about the subject.

The final talk in this series was "The Uses of Paradox—Godel's Theorem" given by Dr. Malgorzata Dubiel. Dr. Dubiel explained the statement and proof of Godel's incomplete-ness theorem in a clear and simple manner. This theorem is one of the most beautiful and profound results of the 20th century.

Talks for teachers Professor Tom O'Shea of the Faculty of Education at SFU not only helped to organ-ize the teacher's part of the conference, but he also gave an interesting talk on "The Challenge of the Unknown," in which he showed and analyzed some new and exciting audiovisual materials.

51

One of the three guest speakers for this part of the conference was Professor Lars Jensson of the education faculty at the University of Manitoba. Professor Jansson's talk "Things are Seldom What They Seem . . . Or Para-doxes, Anomolies, and Discrepant Events" was warmly received.

Professor Leon Bowden, formally of the mathematics department at the University of Victoria, spoke on "Heuristics and the Scien-tific Attitude." Professor Bowden was qualified to speak on this topic; he was a stu-dent and a colleague of one of the foremost exponents of heuristics, Professor Polya.

The last talk in this series, "Implementing a Contest Program in the Schools," was given by Mr. Craig Newell, who is head of the mathematics department at St. George's Secondary School. Mr. Newell described the program that St. George's uses to achieve ex-cellent results in contests. We also discussed how SFU uses contest scores to award scholarships in mathematics and the sciences.

Other activities Each day, students attended problem ses-sions. On the average, 12 students were in each group. For the first time, undergraduate students on NSERC scholarship conducted the sessions. Without a doubt, based on stu-dent attendance and reaction, these sessions were the best in the five years of the con-ference. The undergraduate students did a

superb job. In the problem sessions, very few hints were given to the carefully graded problems—the students were encouraged to solve as many of them as they could.

A highlight of the conference for many of the students was the barbecue on Tuesday night. The teacher's lunch at the university club was an enjoyable social event for all of us.

Professors Allen Friedman and Alan Mekiar constructed difficult problems for book prizes. These problems were marked by Alan Meklar and seven book prizes were awarded.

For the first time, we had a mathematics relay contest. This idea, suggested by one of our undergraduate students, Lily Yen, proved to be exciting and entertaining. Book prizes were given to the winning team. In-cidently, Lily Yen also constructed the full set of relay problems.

Tours of the departments of chemistry, com-puting science, and engineering were pro-vided. This year, instead of a tour of physics, Professor Leigh Palmer gave a physics lecture.

A reunion of participants will be held sometime in November at SFU. This event will allow students to renew their friend-ships. There will be a lunch, one short enter-taining mathematics lecture, and time to talk. No registration fee is charged for the reunion.

52

Report on the June 1986 Algebra 12 Provincial and Scholarship Examinations

Vic Keehn

Vic Keehn is Mathematics Department Head at Summerland Secondary School.

As always, the examination marking session offered algebra teachers an excellent oppor-tunity for in-service with exchange of ideas and resource materials. An added bonus was provided when Mark Mahovolich spent two hours talking with us about the curriculum revision, and answering questions from the committee. Special thanks to Mark for spending that time with us.

As with other provincial algebra examina-tions, the teachers of the marking commit-tee have produced a summary of how well the students did on the questions marked by the committee. This summary outlines the strengths and weaknesses of the students on each question.

As a teacher of Algebra 12, this summary has been valuable to me as it underlines the areas of the curriculum which need to be stressed as well as areas where student achievement has improved.

The marking key used by the marking com-mittee will be published by the Ministry of Education this year. Copies of the booklet containing the keys for the January and June examinations will be sent to school districts and schools. Additional copies will be avail-able at cost through the Publications Service Branch of the ministry.

The following is a list of the strengths and weaknesses on each of the open-ended questions.

Algebra 12 Provincial Examination, June 1986

Part B: Written Response Questions

X 12 27 n

y 12 8 3

1. The above table shows that x varies in-versely with the square of y. Find "n."

Strengths: • This question was generally well done.

Weaknesses: • Many students used direct instead of in-verse variation. • Some used "square root" instead of "square."

53

2. Solve the equation 2x 3 + 4x2 + 4x + 8 = 0 over the complex numbers.

Strengths: • Many students obtained one integral root by synthetic substitution. • Many students obtained the depressed equation. • When taking the square root, students in-dicated both the positive and negative roots. • Students were able to solve the equation by factoring.

Weaknesses: • Many students made the error

X' + 2 = 0 x2 = 2

• Many students did not know the dif-ference between roots and factors. • Some students inserted an extra root (con-jugate. of -2 is +2. • Many students did not understand the definition of complex number (omitted -2 from the solution set).

3. Give the equation, in standard form, for the parabola shown in the above graph.

Strengths: • Students generally recognized that the ver-tical parabola was of the form y = ax2. • Most students translated the vertex correctly. • Most students recognized that the coeffi-cient must be negative.

Weaknesses: • Many students were unable to calculate the coefficient. • Many students assumed the focus was at (3, 0).

• Students demonstrated a poor under-standing of the relation between co-ordinates on a graph and the values of x and y in an equation.

4. Solve for x over the complex numbers.

X' - ix + 2 = 0

Strengths: • Many students were able to factor, and quickly arrived at the answer.

Weaknesses: • Many students stated the quadratic for-mula incorrectly or substituted incorrectly into the formula.

54

• Some found the roots -1, 2i, but gave ±1, ±21 as solutions. • A few gave the factors rather than the roots. • Many tried synthetic division with integral divisors. • Many students wrote (_j)2 = 1.

• Many students did not simplify j±3j

and lost ½ mark.

5. If log 5 = 0.6990, solve the following equation for x to the nearest thousandth.

(10x3 )(103x) = 5

Strengths: • Students generally understood the relation-ship between exponential and logarithmic form. • Students generally recognized the approach that must be taken to obtain a solution.

Weaknesses: • The majority of students did not use parentheses when expressing log 104x3 as (4x-3) log 10. • Many students confused log 10 and log 10 (x —3). •' Some •students had difficulty dividing 16990 by 4 and rounding correctly.

• A number of students arrived at the cor-rect answer x = 0.925 and proceeded to find its antilog.

6. Prove the following identity.

1—tan 2 x = cos 2x 1—tan 2 x

Strengths: • Students exhibited a variety of approaches leading to a finished proof.

Weaknesses: • Many students exhibited poor form in developing a proof. The layout of sequen-tial steps was not orderly. Unacceptable ab-breviations were used (e.g., S2 for sin 2X, sin for sin 2X).

• Many weaknesses in operative skills (e.g., improper cancellation, poor use of brackets, inability to handle complex fractions.

7. A vertical flagpole stands on a slope that is inclined at an angle of 100 with the horizontal. When the angle of elevation of the sun measures 25° with the horizontal, the shadow of the flagpole down the slope is 40 m in length. How tall is the flagpole, to the nearest tenth of a metre?

55

Algebra 12 Scholarship Exam, June 1986

1. Determine the remainder when 3x25 + 2x'° - 5 is divided by x + 1.

Strengths: • The few students who were able to do this question were concise, neat, and accurate.

Weaknesses: • Most students made interpretive errors: assumed the flagpole was perpendicular to the slope, placed the 40 m incorrectly, com-bined the 25° and 100 angles to make a 35° angle. • Many students attempted this problem with right angle trigonometry rather than the sine law. • Many students rounded too early or too indiscriminately throughout the question.

Strengths: • Those who used direct substitution in-variably obtained the correct answer.

Weaknesses: • Most students chose the more difficult synthetic substitution method and many made placement errors. • Many students gave the remainder as:

-6 x+1

8. A hot-air balloon rises 100 m the first minute after it is released, and in each minute after the first it rises 7% less than in the previous minute. Ignoring all other factors, calculate the maximum height that the balloon will reach. Round the answer to the nearest metre.

Strengths: • Generally done. Approximately 4

1 stu- dents received full marks. • Most students understood that this was an infinite geometric series and were able to find the first term.

Weaknesses: • Many students incorrectly interpreted the phrase "7% less" to arrive at ratios of 1.07, 0.97, 0.7, or thought the balloon went down instead of up. • Over 10% of the students attempted to find the answer by brute force (adding con-secutive terms). Several calculated over 60 terms. • Many students demonstrated poor arith-metic skills, especially in decimal division and rounding.

2. Solve the inequality 2 - 2 log x>0.

Strengths: • Most students recognized the need to change from logarithmic to exponential form.

Weaknesses: • Many students made careless transposi-tion errors. • Very few students recognized the lower restriction x>0 (approximately 3% of the students).

3. Solve for "fl". 3n(n-2)! = (n-3)!

Strengths: Approximately 50% of the students

received 21/2 or 3 marks for this question.

Weaknesses: • The most common error was the inclusion of n = -5 in the answer. • Many students lacked familiarity with fac-torial expressions involving variables (e.g., (n-2)! was expressed as (n-2)(n-1) (n). .

56

4. Determine a polynomial equation of lowest degree with integral coefficients and having roots 4- and 2i. Write the equation in simplified form using decreasing powers of the variable.

Strengths: • Good understanding of the complex con-jugates theorem.

Weaknesses: • Some difficulty with reading the question, particularly "integral coefficients" and "equation." • Many students did not equate to zero. Others expressed the answer as a function f(x) = 3x • Some students had difficulty multiplying by i.

5. Jack started a method of saving in which he saved x dollars more. each month than in the previous month. At the end of 10 months, he had saved a total of $560, and at the end of 20 months, he had saved a total of $1920. If he saved y dollars during the first month, determine the values of x and y.

Strengths: • Good knowledge of simultaneous equa-tions.

Weaknesses: • Some students read $560 as the 10th term rather than the sum of the first 10 terms. • Nearly half the students had no response or got zero marks for this question.

6. Solve the following equation for 0 where 0°<_ 0 < 360 0 . cot 0 tan 20 = 3.

Strengths: • Good substitution from double to single angle identity.

Weaknesses: - • Inability to arrive at an equation with a single trigonometric function. Those who converted to sine and cosine generally got lost. • Many students who arrived at tan 0

= 4 omitted the negative root. • Poor cancelling of fractions.

7. Give the equation, in standard form, of the circle that intersects the y-axis at (0, 2) and (0, 8), and that .is tangent to the x-axis at (4, 0).

Strengths: • Students knew the standard form of the equation of a circle.

Weaknesses: • Many students were unable to draw the diagram properly, especially the point of tangency (4, 0). • Most students did not recognize or recall that the perpendicular bisectors of chords pass through, the centre of, a circle. • Those students who chose to use the distance formula, or a system of three equa-tions, or the general form of the conic, fre-quently bogged down in the algebra.

8. A conic has the equation 4x 2 + 9y2 - 16x + 18y - 11 = 0.

(a) Change the equation to standard form. (b) Draw the graph of the conic on the grid

provided. Although points on the graph do not have to be labelled, the graph should have its correct centre and should have the correct shape.

Strengths: • Generally well done. Over 40% received full marks.

57

Weaknesses: • Some difficulty in completing the square and in adding the equivalent amount to the other side. • Many students had difficulty reading the scale on the graph.

9. A 15-metre ladder and a 13-metre ladder are leaning against a building. The bottom of the longer ladder is 4 metres father, on level ground, from the base of the building than is the bottom of the shorter ladder. Both ladders reach the same distance up the vertical wall of the building Find the exact measure of this vertical distance up the wall.

Strengths: • Generally well done. Approximately 55% received full marks. • Diagram was sketched well. • Students used both algebraic and trigonometric solutions effectively.

Weaknesses: • Some students had difficulty with the ap-plication of the Pythagorean theorm. • Some students misinterpreted the phrase "4 metres farther."

10. Determine the value of k so that the following system has one solution over the real numbers.

x—y1 x - 2 =' k - y2

Strengths: • Most students were able to substitute in the linear quadratic system.

Weaknesses: • Many students made algebraic errors (e. g., k - (x-1) 2 = k - x2 - 2x + 1). • Very few students understood that the discriminant must be equated to zero to get a single root.

11. Given the function y = 3 sine' x - 3 cos2 x + 1.

(a) Express the function in the form

y = a cos bx + c

(b) Sketch one complete period of the func-tion on the grid provided and clearly label the co-ordinates of 3 points on the curve.

Strengths: • Good simplification using identities. • Students who answered 11(a) correctly gave the correct amplitude and period.

Weaknesses: • Students often equated sin 2X - cos2x with Cos 2x - sin 2x and missed the negative sign. • Students demonstrated weakness in writing the function in the required form for 11(a). • Students frequently positioned and labelled points incorrectly in 11(b).

12. The lengths of two sides of a parallel-ogram are 5 m and 8 m. The length of the longer diagonal is 11 m. Find the length of the shorter diagonal. Give the answer to one decimal place or in simplest radical form:

Strengths: • Student demonstrated good and diverse problem solving skills (11 different correct solutions were given). • Good knowledge of cosine law. • Most students understood that the dia-gonals of a parallelogram bisected each other.

Weaknesses: • Some students assumed that the diagonals of a parallelogram were perpendicular or that they bisected the angles. • A surprising number of students had dif-ficulty drawing the parallelogram properly.

58

13. Find the value of "a" and "rn" if the above graph has an equation of the form y = (a)(2mx).

Strengths: • Good substitution from graph co-ordinates into the equation. • Good understanding of exponents and logarithms as used in solving equations.

Weaknesses: • Some errors with fractional exponents. • Many of those using log equations got bogged down. • Frequent errors:

If y = a(2-x) then y = rnx log 2a

or log ab = ( log a)(log b)

Y86-0125 December 1986

59

AREYOUR COFFEE BREAKS AS MUCH FUN AS YOUR MATH CLASSES? They can be with this colorful MathMug from the National Council of: Teachers of Mathe-matics. Delightfully decorated with problems, equa-tions, and formulas, the 9-ounce NCTM MathMug is calcu-lated to stretch your mind as well as your coffee break. Makes a great gift!

rPlease rush to: Ti I Name I

Address_________________________

State or Province_________________ Zip or Postal Code_______________

mugs at $7.00 each: $-__.

•1 Handling Charge: $__ TotaI:$

I (Discount: Buy five mugs, get one free)

Io Please send information on other special member products

0 Please send information on NCTM 0 Payment to NCTM in U.S. funds enclosed

i 0 MasterCard 0 VISA Credit Card # Expires_ Signature

All orders must be prepaid. Prices subject to change without notice. VA residents add 4% sales tax.

I J I I I National Council of Teachers of Mathematics

1906 Association Drive Reston, Virginia 22091

Telephone: (703) 620-9840 CompuServe: 75445, 1161

MCI Mail: 266-2653

L__ The Source: STJ228 __J

2.00

NOMINATION FORM

BCAMT Teacher Awards

These awards are presented annually to mathematics classroom teachers who are active BCAMT members. Present members of the BCAMT executive are not eligible. One secondary teacher and one elementary teacher will be selected.

The undersigned wishes to nominate

who is a member in good standing and teacher at

to be a recipient of the BCAMT Teaching

Award.

Please provide information about the nominee under each of the following. Please try to be specific as possible and include dates where applicable or possible.

A. Excellence as a classroom teacher (rapport with students and staff, classroom methods, curriculum methods).

B. Professional leadership (sharing of ideas, teaching resources, and in-formation through seminars, workshops, conferences).

C. Other outstanding efforts (committee work, marking scholarship ex-ams, etc.).

Nominated by

School address

This nomination form with any supporting documents should be put into an envelope and clearly marked BCAMT Teaching Award, and mailed by MARCH 13, 1986 to:

Carry Phillips 4024 West 35th Avenue

Vancouver, BC V6N 2P3

B.C. Association of Mathematics Teachers MEMBERSHIP FORM Fl 1-25/Rev. December 1986

To join, here is all you have to do.

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

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

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

Home address I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

Postal code l I I I I I I I

Name & address school/institution/business

School district no.

Type of membership

Full(a) fl BCTF associate member(b) 0 Non-BCTF member(c)

Student(d) J BCTF honorary associate member(e) fl BCTF honorary life member(f) fl

Teaching or interest level (check one only)

Kindergarten(1) LI

Primary(2) J Intermediate(3) 0 Junior secondary(4) D

Senior secondary (5)

J College/University(6) 0 Elementary(7)

Secondary(8) LI

All(9) 0

BCTF members—$25

Student—$1 0

Non-BCTF members—$37

Total fee enclosed Cheque fl

Money order 0

PSA 50

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EDUCATIONAL COMPUTER

A New Star in our Math Classrooms

For more information, contact

Ron Blache-Fraser - Reg Nordman Phone 688-2431