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ISSN 0382-0718

BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER/JOURNAL

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VOLUME 19, NUMBER 2 DECEMBER 1977

D.C. Association of Mathematics Teachers 1(07707^§

Past President John C. Epp

1612 Wilmot Place Victoria, BC V8R 5S4 h: 592-2388 s: 474-1291

President & PSA Council Delegate William A. Dale

1150 17th Street Courtenay, BC V9N 1Z7 h: 338-5159 s: 334-4438

Vice-President Dennis M. Hamaguchi 3807 22nd Avenue Vernon, BC V1T 1H7 h: 542-8698 s: 542-3361

Recording Secretary Brian Tetlow 81 High Street Victoria, BC V8Z 5C8 h: 479-1947 s: 598-3336

Treasurer Grace Dilley 2210 Dauphin Place Burnaby, BC V5B 4G9 h: 299-9680 s: 594-7548

Newsletter/Journal Editor Susan J. Haberger 1390 Willow Way Coquitlam, BC V3J 5M3 h: 939-8618 s: 936-7205

Curriculum Consultant William J. Kokoskin

1341 Appin Road North Vancouver, BC

V7J 2T4 h: 988-2653 s: 988-3161

Computer Consultant Ian C. de Groot

3852 Calder Avenue North Vancouver, BC

V7N 3S3 h: 980-6877 s: 987-7178

NCTM Consultant Tom Howitz

2285 Harrison Drive Vancouver, BC

V5P 2P7 h: 325-0692 s: 228-5203

Membership Secretary & Primary Representative

Linda P. Shortreid 4651 202nd Street

Langley, BC V3A 5J2

h: 530-4665 s: 588-5918

BCCUPM Representative Alan R. Taylor

7063 Jubilee Avenue Burnaby, BC

V3J 4B4 h: 434-6315 s: 936-7205

1977 Summer Conference Organizer Pauline Weinstein

206 - 5603 Balsam Street Vancouver, BC

V6M 4B6 h: 261-6803 s: 228-5986

2

INSIDE THIS ISSUE

5 President's Report....................................B. Dale

LETTERS 8 To Editor from David N. Ellis................................. 8 To Editor from Jerry Mussio, David Robitaille, Mary Cooper.........

10 Proposed Geometry 12 Requirements for UBC ........ Alan R. Taylor 11 Call for Nominations..............................................

MATH EDUCATION IN B.C. 12 BCAMT Exam Proposal............................Alan Taylor 17 Comments on the Mathematics Assessment Results: Problem-Solving

........................James M. Sherrill and David F. Rob itaille 32 Scitamehtam Desrever.........................Gail Spitler, UBC 49 RIPE Summaries from the Registry of Innovating Practices in Educa-

tion in British Columbia (RIPE). ....................... Julia Ellis

MATHEMATICS TEACHING 56 Fertilizer Is Important! ...................... Ian D. Beattie, UBC 60 Probability and Statistics.............................Jim Swift 63 Calculators - Boon or Bane?.........................Roger Fox 65 Using 'Real Life' Material in the Mathematics Classroom . . .Ken Keeley 69 On Areas of Rectangles.........................William J. Bruce 74 Humor in Mathematics ...........................Daniel Flegler 76 A Grade 8 Mathematics Outline ....................Bill Kokoskin

MATH EDUCATION IN CANADA 77 Canadian Mathematics Olympiad .................G.H.M. Thomas 83 What's Happening in Math Education across Canada?. . Joan Routledge

BOOKS 85 In Praise of The Calculator Workbox... Manitoba Mathematics Teacher

LESSON PLANS • Do It the Easy Way..............................Dave Morgan • Fun with Holiday Facts and Figures ....................Dave Ellis • Christmas Math Songs ............ Henry Enns and Jack Schellenberg

3

president's report bill dole

A strategy for dealing with many problems simultaneously rather than con-centrating on one major concern seems to be evolving with your executive. To make such a strategy successful demands a hard-working, talented and dedicated executive that can work in harmony. You and I are fortunate to have such a group of people working for us. Yet it is a constant source of surprise and, I must admit, irritation to learn, when speaking to members at conferences and workshops, how little is known about the work of their executive.

Part of the problem lies in the profusion of abbreviations and acronyms used to describe the various working executive committees and programs. Below is a diagram of the committees and a brief description of their function. I hope that you will gain a better understanding of how your executive repre-sents you.

B.C. Systems Ministry of Corporation PSA Education

Council BCTF 1 Universities Executive

PDAC and Colleges I

J BCCEC Curriculum

j

Revision Committee

BCCUPM _____________ BCAMT Primary

• Executive • Rep.

WSMC CAG

NCTM • • • • ( West Con- \\etc. I nce Chair-

person

Leadership Intet-me-

MCATA • diate Rep. Pubtica-

etc. PIP • tions, Vec-tor, et. al.

• Summer Workshop Chairperson

Local Chapters

5

BCCUPM British Columbia Committee on Undergraduate Pro-grams in Mathematics. An articulation committee composed of university and college mathematics and mathematics education professors. We represent the public school mathematics teachers on the committee.

BCCEC British Columbia Computers in Education Committee. A committee of persons interested in the advancement of computer education in B.C.

NCTM National Council of Teachers of Mathematics. Pub-lishers of The Mathematics Teacher and Arithmetic Teachers, yearbooks and much more. Underwriters for major conferences.

CAG Committee on Affiliated Groups to the NCTM. The BCAMT is an affiliate of the NCTM. Our representative to the CAG is Joan Routledge of Ontario. There is one representative to the CAG for all of Canada.

Canadian NCTM affiliates:

MCATA Mathematics Council of the Alberta Teachers' Associ-ation.

SMTS Saskatchewan Mathematics Teachers' Society.

MAMT Manitoba Association of Mathematics Teachers.

The BCAMT meets every two years with the above groups in a leadership conference sponsored by the Canadian representative to the CAG.

OAME Ontario Association for Mathematics Education.

QAMT Quebec Association of Mathematics Teachers. AMQ Association Mathematique du Quebec.

MTANSTU Mathematics Teachers' Association of the Nova Scotia Teachers' Union.

NBTAMC New Brunswick Teachers' Association Mathematics Council.

MCNTA Mathematics Council of the Newfoundland Teachers' Association.

PIP Provincial Involvement Program. BCAMT-initiated

6

program to establish local chapters throughout the province. It's moving along very well. East Kootenays and Langley are our latest additions - congratulations!

SUMMER Preparation is well under way - thanks to Dr. Polly

WORKSHOP Weinstein and her committee. You get a membership

No. 6 in the BCAMT included in your registration at this conference.

NORTHWEST Our Vice-President, Dennis Hamaguchi has a commit-

No. 18 tee planning an extraordinary conference for October 1979.

Plan to TURN ON IN VERNON in the fall of '79. Remember this conference is held every three years in B.C. with Washington and Oregon host in the other years. Your membership is not included with your registration at a Northwest Conference. We have an enviable record of co-operation with the WSMC, OCTM and the PSCTM.

WSMC Washington State Mathematics Council.

OCTM Oregon Council of Teachers of Mathematics.

PSCTM Puget Sound Council of Teachers of Mathematics.

PSA COUNCIL Provincial Specialist Associations' Council. The 21 PSAs of the BCTF send representativesto the council whose purpose is to work on problems common to all PSAs.

Quite an alphabet soup isn't it? 'But it does represent a hell of a lot of vol-unteër effort in the advancement of mathematics education. If you know of math teachers who aren't members of the BCAMT, encourage them to join us. We need a strong representational base to validate our efforts.

7

LETTERS

17 Dean Terrace, Edinburgh, Scotland, Great Britain.

Editor, Vector

Dear Sue:

Recently, I received the October issue of the Vector. I enjoyed receiving it and hearing the news of the B.C. Math teachers. It is now being circulated here amongst the others in the department. They will be interested to- see the notice on the Glasgow conference next summer.

I am teaching at James Gillespies High School here, on exchange for this school year. In Vancouver, I was, teaching mathematics and computer science at Sir Winston Churchill Secondary School. Although it has been an adjustment, I am enjoying it very much. I shall look forward to receiving my copy of the next Vector.

- Yours sincerely,

David N. Ellis (Dave)

Dear Editor:

We read with some concern the 'PLAP Bulletin,' which was included with the October issue of Vector, and would like to correct a number of erroneous statements which appeared in the document.

First of all, readers should keep in mind that the bulletin refers to informa-tion included in the, mathematics district assessment report for Grade 12, which was compiled by the B.C. Research Council on behalf of the Ministry of Education. This report is to be used in. conjunction with a document entitled Guidelines for interpretation: District Assessment Results, which was prepared by the Learning Assessment Branch. Also to be noted, is that the data presented in the 'PLAP Bulletin' appear to be taken from the results of a particular district - they are not overall provincial scores.

In the first example, where reference is made to Table 5 of the district report, the author infers that the only way to calculate a mean is to assign each test item an equal weight. This assumption, of course, is not true. In calculating' an overall domain score, we chose to assign equal weights to

8

objectives rather than items for the simple reason that some objectives (skills) required more items to assess. For example, even though 14 test items were used on the Grade 12 assessment to measure Knowledge of Notation and Terminology and only four items to assess Computation with important. By assigning each item an equal weight, we would have been saying that Knowledge of Notation and Terminology is more than three times as important as Computation with Fractions - a relationship which, in the Committee's view, is not appropriate. In short, the mean scores as stated in the district report are, for the above reasons, correct.

In the second example taken from Table 3 of the district report, the author calculates the standard deviation of the p-values (percentage correct for each test item). Phrased differently, the author is looking at the 'spread' of these percentage figures. This is a legitimate exercise, but unfortunately is not what was intended in our reports (see pages 7-8 of the interpretation booklet). The original standard deviations in the district reports were cal-culated on the basis of individual raw scores; this measure provides districts with an understanding of how individual student scores vary for each object-tive and domain. So once again, the original figures presented in the district report are correct.

We would also like to comment on the author's understandable concern over what difference should be considered 'significant' when comparing test scores. (Here we assume that the bulletin is referring to the difference between a provincial test score and a district score.)

First, one must make a distinction between statistical significance and edu-catiohal significance. If One is comparing population scores (as we usually are in our assessments) any numerical difference is statistically significant. But to decide whether or not this represents an educationally meaningful difference demands professional review and judgement by educators. Un-fortunately, there is no formula which provides us with clear objective directions as to what should be considered educationally significant. Readers should refer to the interpretation booklet which discusses, in detail, how a district might involve local educators and members of the public in judging the educational significance of the assessment data.

Because of the wide range of audiences reading the district reports, we chose not to present a detailed description of all of the statistical procedures used in the assessment. We have received a number of calls requesting further clarification and exemplification of our data, and have been most happy to respond to these requests. Anyone requiring more information about any aspect of the assessment program is encouraged to contact the assessment branch of the Ministry of Education.

Jerry Mussio, Director, Learning Assessment Branch David Robitaile, Chairman, Mathematics Contract Team Mary Cooper, Research Psychologist, B.C. Research Council

9

PROPOSED GEOMETRY 12 REQUIREMENTS FOR UBC

Editor's Note. The following letter was sent to the dean of science at UBC. To date no acknowledgement of receipt has been received. However, at a subsequent meeting of the BCCUPM, Alan Taylor raised the issue, and those members, present from the UBC Mathematics Department, indicated that the faculty would take a second look at the proposed requirement.

October 3, 1977

Dr. G.M. Volkoff Dean of Science University of B.C. 2075 Wesbrook Place Vancouver, BC V6T 1W5

Dear Sir:

I wish to draw your attention to an exerpt from the September 1977 issue of the UBC Newsletter. It states therein, 'It is expected that the Science Faculty will decide to require Geometry 12 for admission by September 1979.'

If your faculty implements such a policy, it will severely restrict the number of students who qualify for entry. Few schools have enough facilities to offer all courses proposed in the recent revision. A survey by the BCAMT of all secondary schools in B.C. (reported in the June 1977 issue of Vector) in-dicated that only seven out of 101 respondents planned to offer Geometry 12 this year. An additional problem with regard to this course is that no textbooks are available at present.

All schools continue to offer a good grounding in secondary school mathe-matics through the Algebra 11-12 sequence. Additional courses proposed by the revision committee are intended to supplement programs in schools where additional facilities are available. Consequently most small schools cannot offer additional courses; whereas large schools may offer additional courses only on a rotating basis.

The BCAMT is interested in fostering dialog among all levels in the educa-tional system. We invite your response and would be pleased to discuss this matter further.

Yours truly,

Alan, R. Taylor Executive member, BCAMT

10

CALL FOR NOMINATIONS

Nominations are being called for the following executive po-your past president:

Vice-President - two years Treasurer - two years

Publications Chairperson - two years

Please forward all nomina-tions by January 31, 1978 to sitions of the BCAMT,

John Epp 1612 Wilmot Place

Victoria, B.C. V8R5S4

11

MATH EDUCATION IN B.C.

bcamt exam proposal alan taylor

This report is based on the findings of a recent survey Of all schools with a Grade 12 enrollment in British Columbia. The purpose of the survey was to determine response of Algebra 12 teachers to a proposal for a province-wide examination. 'Since not all readers were polled, the proposal is presented here:

'The:executive of the B.C. Association of Mathematics Teachers has embarked on a survey of senior secondary mathematics teachers in British Columbia to determine if there is a, need to establish 'a province-wide 'diagnostic' examination for Algebra 12.

The proposal we are about to put forth for your consideration was first discussed by the executive over a year ago, and then took direction in July 1977. During the interim period, we sought approval in principle from the BCTF Executive. It, 'in turn, directed us 'to survey the wishes of all teachers of Algebra 12. In following the recommended process, we are currently assessing the needs, costs and feasibility of such an undertaking.

This proposal predates and contains objectives different from a second one that---may have been circulated to some schools. To avoid confusion, we draw your attention to the fact 'that this proposal is not associated with the other, which was initiated by one or" more members of the Math-ematics Department at UBC.

It is proposed:

That a provincial examination be. developed for the Algebra 12 course. This exam will provide a bench 'mark' for comparison purposes and also act as a diagnostic tool for 'the teacher. It is to be administered and marked internally.

The following criteria are recommended:

1. The BCAMT and Ministry co-operate in the construction of the exam. 2. An independent research organization be employed to establish pro-

vincial norms.

12

3. The exam be self-administered and marked, with results known only to the teacher and his/her students.

4. The exam be administered approximately 3/4 of the way through the Algebra 12 course.

Deadline for Response: November 10, 1977

School

Response to Proposal:

Number of math teachers in favor.

Number of math teachers against.

Suggested Modifications:

Comments:

Send responses to: Alan Taylor Math Department Head Centennial Senior Secondary 570 Poirier Street Coquitlam, BC V3J 6A8

BACKGROUND The objectives for this exam are indicated in the opening paragraph of the proposal. At this time, however, a brief comment on each aspect of the recommendation is in order.

Approximately, the executive of the BCAMT anticipated a need two years ago, after removal of government final examinations and in light of the upcoming revision of courses, it appeared that a basis for comparison was desirable. Subsequent discussions with mathematics teachers at conferences, workshops, scholarship-marking sessions, etc., lent support to our suppo-

13

sition. However, it was felt that such a test should be an effective lesson aid for teachers. Hence, it should not only provide a basis for comparison but also identify concepts that may require modification of teaching techni-ques in subsequent course offerings. To meet the latter objectives and to avoid political use of results, the proposal includes a recommendation for internal administration and marking.

A brief comment on the criteria follows:

1. We have reason to expect that the Ministry would seriously consider such a joint venture. However, it is subject to agreement of details and BCTF approval. V

2. Provincial norms would be established externally -by an independent research organization. A cross section of schools would provide the sample space. -

3. Self-administration of the exam is necessary to assure 'corrective out-come' and to avoid political implications.

4. It is proposed the exam cover at least three-fourths of the- course to assure a substantial portion is tested and to provide adequate time for marking. (Based on survey results this criterion may be modified to include the entire course.)

The examination will be designed as a 'quality control' and 'directional' device. Teachers of Algebra 12 courses are professionals and fully capable of using such an exam as a teaching aid. It is not intended to rank teachers or schools, since no valid case has been made to show the significant dis-parities exist in school evaluation in British Columbia. Indeed, most post-secondary institutions find the secondary school letter grade to bea valid predictor of subsequent student performance. V V

SURVEY RESULTS V

In total"mathematics department heads at 188 schools were sent the pro-posal. Responses were received from 107 schools. V

Number of teachers in favor of the proposal 361 Number of teachers against the proposal 20 V

Number of abstentions 1

It is noted that the above results reflect agreement or disagreement for the proposal in principle only. Many of the responses were qualified. Since comments reflected the consensus of opinion from each mathematics de-partment, the following breakdown is based on the number of schools.

Number in favor (in principle) 103 Number against (in principle) 4

14

Number of schools with the following recommendations: The exam should be in the entire Algebra 12 course .............. .13 Similar exams should be established for other math courses aswell .................................................. 14 The exam should be externally (computer?) marked, but results given only to the respective schools .... ..... . 15 The exam should be externally marked, and results made known on either a district- or province-wide basis .................. 7

Many responses included suggestions for modification.'Almost all were sup-portive, and it would be redundant to repeat all of the comments. However, the following excerpts are listed to reflect the range.

1. We are unanimously pleased with this development. 2. After teaching any grade level after a few years, standards are appre-

ciated for reference. 3. Different schools teach the topics in different order. Perhaps con-

sideration should be given to giving the exam nearer the end of the course. 4. I see no need for two exams of this nature, so should not the BCAMT,

the Ministry, and UBC come up with one combined proposal? 5. This response is based on the guarantee of No. 3! Otherwise all op-

posed! 6. Results stress analysis of performance on test items rather than just

giving school percentile scores. 7. We felt that exams in Technical Math and Consumer Math would

help to lend more credence to these courses. 8. The exam should not force us into a formal testing timetable. The

length should be such that it could be completed in a normal class period. 9. We are strongly in favor of the proposal, but we were wondering if

there was an equitable way to give a valid exam. 10. Vector can be used as 'sounding board' re: bragging, complaining, comments on emphasis, etc. 11. Let's see the results of the whole province. 12. It should be designed by high school math teachers. 13. Exam should be given on a trial basis, and recommendations for im-provement should be sought from math teachers. 14. The corrective measures will be minimal. The benefits of this exam will not warrant the effort and expense. Maybe time and effort should be spent on developing province-wide examinations at the beginning of the Grade 8 and Grade 11 levels, so that when students arrive in those levels, they could be given an examination at the beginning of the course, and the necessary corrective measures could be made in the ensuing year. 15. We feel that there is no need for either the UBC or the BCAMT pro-posal. We feel it is unfair to impose university standards on people who are not necessarily going on to university. Are we going 'back to the good old days' or 'going forward to what used to be' or what? 16. Excellent idea. As a small school, we need this type of comparison. 17. Surely the independent agency employed to establish the norms,

15

would have the examination machine marked and then have each schools' results, together with a diagnostic analysis and a comparison with other unnamed schools, forwarded to the school. 18. We would not like to see a test with a large number of multiple choice answer questions. We prefer to see the work done to achieve an answer rather than a guess at supplied answers.

Thank you to all respondents to the proposal. It is obvious that a test sim-ilar to this one is desired by the majority of teachers. However, the actual form and administrative format are yet to be determined.

If this report is approved by the BCAMT Executive, it will form the basis for a final proposal. It, in turn, will be forwarded to the Executive of the BCTF with a request for endorsation and subsequent distribution to the Ministry. The end product of this exercise will be forwarded to members of b1.PdVI i as soon as posslole.

RESEARCH COUNCIL ON DIAGNOSTIC AND PRESCRIPTIVE MATHEMATICS

and Arizona State University present a conference on

DIAGNOSTIC AND PRESCRIPTIVE TECHNIQUES IN MATHEMATICS

April 9 - 11, 1978, Scottsdale, Arizona (just before NCTM San Diego - April 12-15)

The Department of Elementary Education of Arizona State University is hosting the annual meeting of the Research Council on Dignostic and Prescriptive Mathematics. National leaders in mathematics education (K-16) will be present-ing talks and workshops for teachers and specialists who are concerned about the difficulties students encounter in learning mathematics. A one-day mini-con-ference will be featured as an option for those interested in research relevant to diagnostic/prescriptive mathematics.

Help from national leaders for your slow students in mathematics.

For complete information clip and mail to: Dr. Jon M. Engelhardt, Conference Director Mathematics Learning Clinic Department of Elementary Education Arizona State University Tempe, Arizona 85281 or call: (602)965-3538

Name

Affiliation

Address

City State Zip

16

comments on the mathematics assessment results: problem-solving

james m. sherhH, david 1, robitaiUe, ubc Each of the three mathematics assessment tests (Grade/Year 4, Grade 8, and Grade 12) contained a number of problem-solving items. The present article contains a discussion of the results obtained on those items.

Student performance on each item was rated on a five-point scale by inter-pretation panels set up by the Ministry of Education. The three grade-level panels assigned a rating of strength, very satisfactor y, satisfactory, marginally satisfactory, or weakness to each item result as an indicator of their opinion of the performance.

The Grade/Year, 4 test measured mastery of essential mathematics concepts and skills from Grades 1 to 3. Analogously, the Grade 8 test dealt with skills and concepts from the intermediate grades, 4-7. The Grade 12 test dealt, for the most part, with content all students could be expected to have mastered upon completion of their public school education.

Grade/Year 4 Problem-Solving Results Table 1 contains the results on the 12 assessment items used to assess the problem-solving skills of Grade/Year 4 students.

For Item 42, the only problem-solving item on which the Grade/Year 4 students' performance was judged to be less than satisfactory, the students were asked to find the elapsed time between 4.25 p.m. and 5.00 p.m. Twen-ty percent of the students subtracted 425 from 500 and picked 75 minutes as their response. Many of the 51 percent of the students who 'worked the item incorrectly may have had difficulty in interpreting the notation.

TABLE 1Grade/ Year 4 Problem-Solving Results

Item No. Topic Percent Correct Panel Judgment 40 Time 77 Satisfactory 41 Multiplication 79 Satisfactory 42 Time 49 Marginally Satisfactory 43 Money 82 Satisfactory 52 Subtraction 39 Satisfactory 53 Addition 88 Strength 54 Addition/ 60 Satisfactory

Subtraction 55 Money 86 Very Satisfactory 66 Multiplication! 47 Satisfactory

Subtraction 67 Subtraction 75 Satisfactory 68 Graphs 92 Strength 69 Graphs 82 Very Satisfactory

17

I

S

• _ N __ N

N N

(I, 4-. C 0 a 0

a, .0 E

z

Teresa and Jerry played Bingo from 4.25 p.m. until 5.00 p.m. For how many minutes did they play Bingo?

Percent

95 ................LI 8

25 ................ LI 14

LI 75 ................Li 20

I don't know .......LI 8

No response 1

Figure 1: Grade/Year 4 - Item 42

The results on four of the 12 problem-solving items at the Grade/Year 4 level were rated above satisfactory. One of the items, Item 55, dealt with determining the value of some coins. Items 68 and 69 dealt with bar graphs.

Whose team came last? Oj

Sara's team.. - LI 1 Bill's team... .LI 92 Glen's team.. . LI 3 Donna's team . LI i.

I don't know . . LI 1

No response 1

How many more points did Sara's team score than Donna's team?

4 ........ ...LI 7

21 ........... LI 2

SARA BILL GLEN DONNA 3 LI g Team Captains 12 ...........LI

The bar graph shows the number of i don't know. . LI 1 points scored by four hockey teams. Use the graph to answer these ques- No response 1 tions.

Figure 2: Grade/Year 4 - Items 68-69

18

Pmv-r- ant

597 697...... 327 ..... 373 .....

LI 88

Li 4 LI 2

LI 2

Item 68, which required the students to read a graph, was answered correctly by 92 percent of the students, and this result was rated a strength. Perfor-mance on Item 68, which required the students to obtain two pieces of data from the graph and combine them to obtain the correct answer, was rated i'erv satisfac ton'.

On Monday, 185 people saw the morning whale shows and 412 people saw the afternoon whale shows. How many people saw the whale shows that day?


I don't know .......U 3

No res0002ie 1

The result on Item 53 was rated as a strength. The problem involved the operation of addition. The computation, 185 +- 412, was presented as 'a separate item in Item 26. There were three pairs of problems on the test such that one item in each pair was a computation exercise and the other was a word problem requiring the same computation. Table 2 contains the results of the three matched pairs of items.

TABLE 2

Grade/Year 4 Results Problem-Solving vs. Computation Items

Problem-Solving Computation Problem-Solving Computation Required Item No. Item No. % Correct % Correct Computation

41 16 79 86 4x8 52 33 39 87 627-500 53 26 88 92 185+412

On two of the three pairs of problems, there was very little difference be-tween the students' performance on the computational exercise and the word problem requiring the same computation. The results on the pair containing Item 52 and Item 33 do not follow the same pattern. The compu-tational exercise Item 33 was solved correctly by almost 50% more students than was Item 52.

Of the seven problem-solving items whose results were rated as satisfactory, two deserve some comment. The results on Items 52 and 66 were both

19

below the 50% mark. The problem in Item 52 required students to do a conversion from metres to centimetres before solving the problem as a first step in finding the answer. The results presented in Table 2 show that 87% of the students were able to do 'the subtraction, 627-500, involved in the problem. However, only 49 % of the students responded correctly to Item 63, which required them to convert five metres to centimetres.

Item 66, on the other hand, was simply a difficult problem. The'solution requires more than simply taking the numbers involved and combining them in one operation. Because of the complexity of the problem, the result of 47% correctwas interpreted as satisfactory.


Percent

6 ................. 8

8 ................LI 10

3 LI 14. LI 21&

I don't know .......LI 10

No response 1

Sam has 51 pop bottles and 8 cartons. Each carton holds 6 bottles.

If Sam fills all the cartons, how many bottles will be left over

Overall, the Grade/Year 4 students performed well on the problem-solving items of the mathematics assessment. Four of the 12 items had results that were rated above satisfactory, with two of the four given the highest rating. Only one item had results that were rated below satisfactory, and it was rated marginally satisfactory.

Grade 8 Problem-Solving Results The Grade 8 test contained 10 problem-solving items. The results obtained are summarized in Table 3.

Of the 10 items used to assess problem-solving at the Grade 8 level only the results of Items 37, 53, and 60 were rated as less than satisfactory. Item 60 was one of a three-item set dealing with the circle graph presented below.

Distractors B 'and D together account for 43% of the responses. The numbers involved in distractors B and D, 35 students and five students, respectively, are obtained by reading the circle graph, determining the percent of students choosing figure skating, determining the percent of students choosing gymnastics, and combining the two percents. The intermediate step of determining the number of students each percent represented was omitted. Students selecting distractor B also selected the incorrect operation.

TABLE 3

Grade 8 Problem-Soli'ing Results

Item No. Topic Percent Correct Panel Judgment 24 Multiplication 91 Strength 25 Percent 60 Satisfactory 26 Average 63 Satisfactory 27 Ratio 63 Satisfactory 36 Scale Drawing 66 Very Satisfactory 37 Area 27 Weakness 53 Geometry 60 Marginally SatisfactOry 58 Circle Graph 66 Satisfactory 59 Circle Graph 57 Satisfactory 60 Circle Graph 38 Marginally Satisfactory

The 1200 students ma secondary school were asked to name their favorite Olympic sport. The re-sults of the poll are shown in the circle graph to the right.

- Hockey

' Figure Skating

20%Gymnastics

Skiing--------_.Volley ball

\ Track and Field

Swimming

Percent

A) 420 students 7

B) 35 students 19

- C) 60 students 38 0) 5 students 24

How many more students choseE) I don't know 10

figure skating than gymnastics? No response 1

Figure 5: Grade 8 - Item 60

21

Performance on Item 36, which dealt with scale drawing, was very satis-factory, but the performance on two other items, which also dealt with geometric measurement concepts (Items 37 and 53), was disappointing. The performance was weakest on Item 37.

What is the area of the shaded portion of this figure? Percent

A) 54 27

B) 96 28 • 8 12 J C) 120 11 I 0) 60 11

E) I don't know 21 15 .1.1 No response 1

rigure b: (irade 8 - Item 37

While the figure accompanying Item 37 was drawn to be as clear as possible, it may have confused the students. However, the fact that more students selected the area of the unshaded portion (96) as the answer than the correct answer lends support to the opinion that it was not the figure that caused the low performance. The 'I don't know' distractor was selected by more students (21 %) on Item 37 than on any other item on the test.

On the positive side, the performance on Item 24 was so high that it was judged to be a strength.

• Percent

There are 25 members in the vol- A) $ 49 2 • leyball club. If the cost for each uniform is $24, how much would B) $6000 3 it cost to buy new uniforms for C) $ 600 91 all the club members?

D) $ 96 2

E) I don't know 2

No reAnons p 1


22

The problem in Item 24 required the students to combine the only two num-bers in the problem using the proper operation. The results were very en-couraging with the 91% of the students responding correctly.

The following quote from the interpretation panel's report on the Grade 8 results serves as a good summary for student, performance in problem-solving:

In summary, it was felt that the questions and answers (distractors) were generally well constructed, and that the students' responses were satisfac-tory, given the apparent low priority accorded problem-solving. Problem-solving, not merely 'word problems,' and applications should be seen as a major focus or core of the Grades 4 through 8 curricula. Various models or strategies for problem-solving should be emphasized strongly by teacher training institutions, and through in-service programs. This area of mathe-matics should be conveyed to teachers and students alike as being central to all of mathematics.

These sentiments are also expressed in Recommendations 4 to 7 of the test results report of the mathematics assessment, which reads, 'Teachers of mathematics at all levels must emphasize problem solving. Problem solving cannot be just one unit among many; it should be given high priority as being central to all aspects of mathematics. Students must have many experiences solving multi-step problems and they should be taught to verify the reasonableness of their answers to problems.' (page. 115)

Grade 12 Problem-Solving Results The Grade 12 test contained 18 problem-solving items. The results obtained are presented in Table 4.

TABLE 4

Grade 12 Problem-Solving Results

Item No. Topic Percent Correct Panel Judgment 24 Unit Buying 65 Marginally Satisfactory 25 Credit Buying 70 Satisfactory 26 Average 89 Very Satisfactory 27 Discount 86 Very Satisfactory 33 Area 35 Weakness 36 Scale Drawing 81 Very Satisfactory 37 Area 54 Weakness 49 Surface Area 37 Weakness 53 Area of Circle 72 Very Satisfactory 58 Percent 87 Satisfactory 59 Percent 79 Satisfactory 60 Percent 66 Marginally Satisfactory 65 Similarity 63 Satisfactory 66 Theorem of Pythagoras 43 Weakness

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Item No. Topic Percent Correct Panel Judgment 69 Use Formulas 48 Weakness 70 Interpret Graphs 67 Satisfactory 71 Commission 62 Marginally Satisfactory 72 Tax Table 69 Marginally Satisfactory

Grade 12 students were divided into three groups according to their mathe-matical backgrounds: Math 12 group, those who were taking or had taken Mathematics 12; Math 11 group, those who were taking or whose highest level of mathematics course taken was Mathematics 11; Math 10 group, those who were taking Mathematics 10 or for whom this was the last mathematics course. As might be expected, the Math 12 group obtained the highest results, and the Math 10 group the lowest. The average percent correct for all problem-solving items was Math 12 - 82%, Math 11 - 59%, and Math 10 - 46%. The results in Table 4 are those for all Grade 12 students.

In commenting upon some of the problem-solving results, the interpretation panel suggested that there was room for improvement. The panel wondered if sufficient time were being spent on teaching problem solving and suggested that the performance might be improved if students studied more business and consumer mathematics topics. Finally, the panel hypothesized that students' performance might reflect inadequate training in persistence, care, and attention to detail. It may be, however, that students will improve their skills in this area if teachers succeed in bringing a little more of the real world into the mathematics classroom, thereby enlivening the discussions of topics that might otherwise be seen by the students as irrelevant and hence undeserving of persistence, care, and attention to detail.

Four of the consumer mathematics item results were rated marginally satis-factory. Item 24 required students to select the best price for an item.

Tennis balls are on sale at four sport shops. You would pay the lowest price per ball if you bought at the store which offers:

Percent

A) 8 tennis balls for $7.25 13 B) 1 dozen balls for $11.00 15 C) tennis balls for $0.95 each 4 D) 3 tennis balls for $2.70 65

E) I don't know 2

No response

SALE-TENNIS BALLS j


24

Problems of the type presented in Item 24 are encountered almost every time one goes shopping. Even though it is a multiple choice item, it is still realistic, since each choice must be interpreted in the form of a unit price. The percent correct for Item 24 for the different mathematical background groups of Grade 12 were: Math 12 - 78%, Math 11 - 60%, and Math 10 - 49%.

Item 71 contained a problem that presented the students a rate of commis-sion and the amount of commission from which they were to determine the total sales required. Only three out of five Grade 12 students responded cor-rectly. The percent correct for Item 71 for the different mathematical back-ground groups of Grade 12 were: Math 12 - 82%, Math 11 - 55%, and Math 10 40%. This same basic pattern of results with respect to mathematics background was repeated on all problem-solving items.

Percent A salesman receives 20% of the retail value of his sales as a commission. What must his total sales be if he is to earn a

A) $1200 20

commission of $60? B) $ 80 4

$ 300 62

$240 6

I don't know 5

No resnonse 2.


Item 72 deserves special mention. For this item, the students were shown a page from the Canada Income Tax Guide and asked to find the total amount of tax due on a specified taxable income. Just over 30% of the respon-dents obtained an incorrect answer; thereby giving some evidence of their inability to read a table correctly, a table many of them must have already used and which all of them will use sooner or later. A majority of the Math 10 group responded incorrectly to Item 72.

The final consumer item that had a performance rated as marginally satisfactory was Item 60. Item 60 was the same as Item 60 on the Grade 8 test and was discussed in the Grade 8 section of this paper. Their performance of both the Grade 8 and the Grade 12 groups rated as marginal/v satisfactory by their respective panels.

Of the nine items that had performances rated as being less than satisfactory, five were rated weakness. The items were Items 33, 37, 49, 66, and 69. A discussion of the results on the five items follows.

25

The perimeter of a square is 12 cm. Percent Find the area in square centimetres.

A) 48 9

B) 9 35

12 3

144 43

I don't know 9

No response 1


The problem presented in Item 33 required the students to work with a given number to find the number that would lead to the correct solution. Forty-three percent of the students treated the given number as if it were the needed number. The needed number was the length of a side of a square, but the givn number was the perimeter of the square.

The results from Item 49 show that either students did not understand the term surface area or they failed to read the problem carefully enough. As shown in Figure 11, 33% of them found the volume of the cube, not the surface area.

Percent

A) 24cm 2 37

B) 16 cm 2 12 What is the total surface area of this two centimetre cube? C) 32 cm2 8

D) 8cm 2 33

2 E) I don't know 10

2 No rosponso 1

Figure 11: Grade 12— Item 49

26

If two sides of a right triangle are 6 cm Percent and 4 cm long, find the length of the hypotenuse.

A) 6

B) 10 23

4 C) 52 6

D) -

—52— 43

6 E) I don't know 21

No roponso 1.

Figure 12: Grade 12 - Rem 66

The problem presented in Item 66 was a straightforward application of the Theorem of Pythagoras. The two numbers provided could be used directly with the formula to generate the third number, the length of the hypotenuse. Approximately one out of every four Grade 12 students simply added the two numbers. Twenty-one percent responded with the 'I don't know' dis-tractor, the highest such response on the test.

The formula to calculate simple interest is I = P r t where i is the interest, P is the princi-pal, r is the rate, and t is the time in years.

B) Find the principal, if the interest received C) after two years at an annual rate of 6% is - $oo. D)

D

$2000 9 $5000 7

$500 48 $720 17

E) I don't know 16

'o response 2


The problem presented in Item 69 required the students algebraically to manipulate the formula for simple interest, which was provided. Less than half of the students responded correctly. Seventeen percent simply multiplied the three values that were provided, treating 6% as a whole number in the process. Sixteen percent responded, 'I don't know.'

The fifth problem-solving item that had a performance rating of weakness on the Grade 12 test was Item 37. It was the same as Item 37 on the Grade 8 test and was discussed in the Grade 8 section of this article. It is interesting

27

to note that while the Grade 12 students' percentage of correct response was twice as high as that of the Grade 8 students, the relatively straightforward nature of the problem led the interpretation panel to rate the performance as a weakness at the Grade 12 level.

No problem-solving item of the Grade 12 test yielded a performance that was rated a strength. However, the performances on Items 26, 27, 36, and 53 were rated as very satisfactory.

Percent In four months, Susan spent the following amounts on records:

lstmonth .............. $17.95 2nd month ............. $22.40 3rd month.............. $ 8.25 4th month.............. $15.80

What was the average amount she spent on records per month?


A) $10.10 3 B) $64.40 4 C) $32.20 2 D) $16.10 89

E) I don't know 1

No response -

The problem presented in Item 26 required the students to find the average of four numbers. Almost 90% of the students responded correctly.

The problem presented in Item 27 was a consumer item that dealt with finding the difference in the price of a television set being offered at a dis-count at two different stores. The percent correct for Item 27 for the dif-ferent mathematical background groups of Grade 12 were: Math 12 - 94%, Math 11 —83%, and Math 10-74%.

Finding the scale distance between two points on a map given the actual dis-tance and the scale of the map was the topic of concern in Item 36. The per-formance was rated as very satisfactory. Seven percent of the students either set up the ratio incorrectly or chose the operation of multiplication instead of division. The 7% represents a small group compared to the 81% who responded correctly.

A map of B.C. is to be drawn so that 1 mm represents 5 km. If the actual distance between Vernon and Penticton is 125 km, how many millimetres apart should these two points be on the map?


Percent A)125 2 B)625 7 C) 120 2 D) 25 81 E) I don't know 7 No response -

28

The fourth problem-solving item yielding a performance rated as i'erv satis-Jactorv was Item 53, presented in Figure 16. The students were required to determine the fractional part of a circle that was shaded. Whether the stu-dents solved the problem analytically or simply 'eyeballed' the picture pro-vided is not known, but almost % of the students responded correctly. Of the seven problem-solving items dealing with geometric and measurement concepts, only the performances on Items 36 and 53 were rated as being more than satisfactory.

Percent What fractional part of the large circle is 1 shaded? A) - 10

7

E) I don't know 5

No response 1


All teachers who have dealt with word problems have probably faced the same difficulty - the students do not seem to be able to read the problems as well as they read other material. The issue of the relative importance of reading ability and computational ability in solving word problems is still unresolved. In combining the results of the B.C. mathematics and reading assessments for Grades 8 and 12 levels, a paradox was uncovered. Grades 8 and 12 females outperformed males in both computation and reading, but the males outperformed the females in solving word problems. This result parallels the result from the National Assessment of Educational Progress in the U.S. among 13 year olds. The females could read better and compute better, but not solve word problems better than the males.

The Grade 8 and 12 tests had five problem-solving items in common. Table 5 contains the results on the five common items for each of the following four groups of students: Grade 8, Math 10, Math 11, and Math 12.

29

TABLE 5

Results on Common Problem-Solving Items Grade 8 and Grade 12

Percent Correct Item No. Grade 8 Math 10 Math 11 Math 12

36 66 61 78 94 37 27 27 44 79 58 66 75 85 95 59 57 63 75 91 60 38 41 58 86

Average 51 53 68 89

The data in Table 5 show a very clear performance pattern for the common items. The Math 12 group, as expected, was far superior to all other groups. The Math 11 group was relatively equally located between the Math 10 and Math 12 groups. The Math 10 group performed considerably less well than the other two groups of Grade 12 students. What may not have been ex-pected was that the Math 10 group's performance was at the same level as that of the Grade 8 group. This same trend was evident on the approximate-ly forty items common to the two tests.

Looking back at Table 4, Items 24-27, 58-60, and 70 were classified as con-sumer items. Consumer mathematics skills are of great importance, and these results indicated that many students are completing school without having mastered the skills required to solve such problems. Some initial steps to correct this situation have already been taken (e.g., the introduction of an elective course, Consumer Mathematics, at the senior secondary level and the inclusion of a unit on consumer mathematics in the Math 10 course), and such initiatives should be endorsed by teachers. All students should have been taught the major concepts of consumer mathematics before com-pleting secondary school and, preferably, at the senior secondary level where such material is more likely to be of interest to them.

SUMMARY The Grade 4 problem-solving results were commendable. Problem-solving is the most difficult topic in the mathematics curriculum to teach, and it is at least as difficult to learn. In spite of this, no problem-solving item results were rated as weaknesses, and one was rated as a strength.

The Grade 8 results were satisfactory overall. One weakness was noted on a multistep problem dealing with the concept of area. On the other hand, a strength was noted on a problem dealing with multiplication of whole num-bers. While the Grade 8 performance was not as strong as that of Grade 4, their performance was also commendable given the difficulty of problem-

ii,]

solving tasks.

Grade 12 results for problem-solving were disappointing. Of the performances on the 18 items, none was rated as a strength, four were given a rating of rei-.v satisfactory, four were given a rating of marginally satisfactory, and five were given a rating of i'eakness. The results indicate that many stu-dents were unable to apply the computational skills they have learned to certain types of problems. This seems to be especially true in the area of geometry and measurement.

One final point must be made again - teachers and teacher educators need to stress the overriding importance of problem-solving in mathematics, and their students need to learn several strategies to use in attempting to solve problems in mathematics.

For information about the Tenth Canadian Mathematics Olympiad 1978) write to:

Professor W.J. Blundon Chairman, 1978 Olympiad Department of Mathematics

Memorial University of Newfoundland St. John's, Newfoundland, A1C 5S7

--FREE-for elementary (K-8) school teachers f

classroom tested ideas U

reproducible worksheets practical teaching activities

• from teachers for teachers in the I NEW

Arithmetic Teacher I I. I • National Council of Teachers of Mathematics

•1906 Association Drive • Reston, Virginia 22091

For a tree copy of the NEW Arithmetic Teacher to read use, and share. complete and forward this form to the Council

•tJarn

Adore,, or

I City Province Zip I Vaiiathrovqr Jun 970 Pr — — — — — — — — — — — — — — — — — — — — —

31

scitamehtam desrever goI sptIer, ubc

Perhaps it is part of the future shock syndrome, or perhaps it is because I'm getting older, but it seems to me that we in education, mathematics educa-tion, in particular, are forced to deal with an enormous number of issues related to mathematics teaching. Questions do not seem to have simple, direct solutions, and each question seems to be the concern of a multi-tude of opposing special interests, each determined to win out over the others. I took a few minutes the other day to try to identify some of the issues currently pressuring mathematics instruction. The list looked some-thing like this:

Individualization of Instruction 'Back to Basics' Impact of the Calculator/Computer 'Core' Curriculum Provincial Assessment Who Is in Control of Education? Sexism in Text Materials Diagnosis and Remediation Who Needs How Much Education? Problem Solving Values and Education Aftermaths of New Maths

I got depressed and decided that the world is in a terrible mess, floundering and directionless, and I went fishing for a few days.

While fishing, I went back to the list and wondered how much of this is simply temporary hysteria and how much will really have significant, long lasting effects upon the curriculum.

To gain some perspectives, i would like you to read a few passages I have found, and I'd like you to guess the date each was written or said.

.We have to say that the amount of mathematical discipline essential to the appropriate education of men and women as human beings, is the amount necessary to give them a fair understanding of rigor as the standard of logi-cal rectitude and therewith, if it may be, the spirit of loyalty to the ideal of excellence in the quality of thought as thought.

Cassius J. Keyser 1922

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Would it not be possible for the children in the grades to be trained in power of observation and experiment and reflection and deduction so that always their mathematics should be directly connected with matters of thoroughly concrete character?

.\Vould it not be possible to organize the algebra, geometry and physics of the secondary school into a thoroughly coherent four years' course?

E.H. Moore 1902

The number facts are the same for every child. We must explain the neces-sity of standards, and the absurdity of automatic promotion. The world does not offer a penny for a million wrong answers. The pupil's subsequent pro-fessional or vocational status depends on honest educational achievements. The average parent will understand the'cogency of such arguments. And ul-timately this insight will lead to a public demand for real educational justice to . our children and young people.

William Betz 1950

In this brief, study of the past five decades of mathematical reform we tried to give due prominence to the two major factors that really determined the course of events. The first had to do with the forward-looking plans of the pioneering mathematical leaders. While the influence of theirideas is felt to this day, it cannot be denied that these men did not sufficiently sense the coming educational revolution , under the impact of mass education. Their interest was centered almost exclusively upon academic mathematics. It was a fateful error that the role of mathematics in a system of universal educa-tion received too little attention at their hands.

William Betz 1950

These two passages provide an interesting contrast:

Thus, when the pupil comprehends clearly, by means of dots arranged in a rectangle, that three fives contain the same number of units as five threes, that is, when he sees that the commutative law is true, then it may be ex-pressed to him in the general form, A x B = B x A.

Report of the Committee on Secondary School Studies, 1893

The interpretation of multiplication of counting numbers in terms of rec-tangular arrays is quite common in contemporary programs of mathema-tics instruction for the elementary grades. There is little doubt that the emphasis upon arrays reflects the influence of curriculum innovators.

J. Fred Weaver, 1967

33

• . .The extent to which these books have been introduced into our schools during the last ten years is almost incredible. I believe that nearly one-half of the books used in our schools are from the United States.. .These books are recommended by their adaptation to elementary schools, by their style and cheapness, in comparison of school books heretofore printed in Canada ...In regard to the exclusion of American books from our schools, I have explained, as I have had opportunity, that it is not because they are foreign books simply, that they are excluded, although it is patriotic to use our own in preference to foreign publications; but because they are, with very few exceptions, anti-British, in every sense of the word. They are unlike the school books of any other enlightened people, so far as I have the means of knowing.. .And, as to the influence of such publications, I believe, though silent and unperceptible in its operations, it is more extensive and power-ful than is generally supposed. I believe such school books are one element of powerful influence against the established government of the country.

E. Ryerson 1847

One might feel that nothing has really changed in education, and that no-thing ever will. However, such is not the case.

a. We have a concept of the child that is totally different from that held even at the turn of the century. Childhood is now recognized as a Unique stage of human development. The child is not simply a miniature adult waiting to be released into the world at a specified age. This has lead to our concern about finding the types of instruction and materials appropriate for children at given stages of development.

b. More children attend school for longer periods. This has resulted in a vastly different school structure including such things as specialization of teachers, schools for certain age groups, and curriculum changes brought about by a vastly more varied school population.

c. A third major way schooling has changed, involves the impact of tech-nology upon the mathematics curriculum. The most important changes coming in education will result from technological changes in our society - particularly the technology related to the processing and transfer of in-formation.

Rank order these devices according to how much use is made of them in your school. Consider not only how much you use them but also how much they are used by members of your staff. List the most-used item first, second-most-used, and so on.

1440 Printing Press (printed material) 1600 Chalkboard 1870 Record Player

34

1887 Mimeograph 1888 Photograph 1890s Motion Picture 1900 Libraries 1920 Radio

Slide Projector 1930 Spirit Duplicator 1948 Television 1951 Computers 1960 Overhead Projector 1970 Calculator (Hand)

Compare the frequency-of-use ranking with the dates that the items became commercially available. In most cases, we will find a fair degree of corres-pondence between the two lists. The correspondence suggests that educa-tion is fairly slow about including modern technology in the enterprise. We still lecture to the students as if the textbook had not been invented. We often use the blackboard when the overhead projector or some other me-dium might be more effective, claiming, of course, that 'it just is not con-venient.' Technology has a slow, almost inconsequential, effect upon teach-ing processes. When we look at items directly related to mathematics, the case against technological impact seems stronger. Consider the slide rule and the old mechanical calculator. In the 1920s, especially with the widespread use of the slide rule during the first World War, the use of the slide rule was highly touted, particularly for junior secondary grades and above. Numerous articles, dating from 1916 to the present can be found in the Mathematics Teacher, all proclaiming the advantages of the slide rule. Here are a few quotes from these articles:

• . .It is found that attendance is better during the teaching of the slide rule than any other time during the term. A young woman who is a private secretary.. .uses the rule constantly and finds that it saves two-thirds of her time checking office work.

The slide rule introduced early.. .will prove fascinating to the students and will abridge many tedious mathematical calculations.

In the past, two things have hindered the teaching of the slide rule - ignor-ance of teachers regarding it and the expense of equipment.

Mathematics Teacher, 1916

...Very useful in computation and in checking numerical work.

Official recognition of the value of the slide rule was given last June in New York State when Niagara Falls High School asked and secured permission for its students to use the slide rule in the state regents' examination in trig-onometry.

35

...The slide rule was a very efficient calculating instrument on account of its rapidity of operation and the ease with which it could be carried in the pock-et.


Slide rule work presupposes a good understanding of decimals, so the slide rule poses a possible motivation for 'brushing up on our decimals so we can see the point.' Obviously, the slide rule is made to order for work in metric units.

A much more current advantage, however, is seen in the significant decline I have noticed in the number of careless errors. on the mathematics papers I'm marking these days.


Simply remove the words slide rule in each case and substitute calculator, and you have the kind of material now written. Do the same thing with this passage from a 1940 article exclaiming the importance of the old mechan-ical calculator.

...The question arises: should pupils be taught to perform the computations arising in the solution of problems in arithmetic by the use of machines, as they would in a similar situation in real life?

The answer to this question depends on several factors, the principal ones being the collateral educational benefits accruing from the use of such machines, and their cost.

1. They are a strong motivating force. They arouse interest in arithmetic because pupils like to use the machines, and because they relieve them of the drudgery of computation.

2. ...Before a pupil could put a problem into the machine, he had to analyze it in advance.

3. The use of machines makes easy, and gives insight into certain number relations, such as transforming fractions into decimals...


One could make a fairly strong case that the modern day hand-held calcula-tor and the computer will not have any impact upon the public schools for quite a while, at least not until you and I retire. As you read the professional journals, you can convince yourself that they may be around but that cer-tainly there is nothing to worry about. The calculator will make kids better at arithmetic - oh, we may do more with decimals and throw out fractions,

36

we may allow the calculators use during certain examinations and for home-work, but basically we will just continue as we have - teaching arithmetic, algebra, perhaps a unit on the calculator and there will be a few computer classes.

The calculator and the computer are entirely different from anything we have ever dealt with. The slide rule and the mechanical calculator were not publicly available or used. They were tools of specialized professions. The person on the street did not and does not own or use a slide rule or a mech-anical calculator. However, the person on the street does own a hand cal-culator and has learned how to use it, at least for simple calculations. What about the computer - the average person does not own a computer. Major electronic suppliers are now making minicomputer kits available for about $600. The entire minicomputer field is blossoming all over North America. Not every home is now equipped with a computer, but I'll bet that that will be the case in less than 10 years.

A second major difference between the calculator/computer and other media is the degree of control of use of the media in schools. In most cases, the teacher has almost complete control over the technology available to the student in a school setting. The teacher decides to use a motion pic-ture, an overhead projector, or whatever. Additionally, these media are generally not used for educational purposes outside of the classroom. For example, even though public educational television is. supported in the United States, the support is limited, as is the viewing audience. Commer-cial use of the media seems decidedly noneducational by intent. However, the calculator is used by students inside and outside the school setting. The advertisements for these products strongly promotes their educational use and suggests that the parents who are concerned about their child's educa-tion should definitely run out and buy their elementary and secondary school child a calculator, especially if said parents feel that they them-selves cannot help the youngster with mathematics homework.

A third way the calculator, in particular, is different from other media is that it takes relatively little sophistication to learn how to operate it and thus to be able to perform fairly sophisticated computations.

The calculator will have some kind of impact upon the mathematics curri-culum: What will be the nature of that impact? Many people are suggesting relatively small degrees of impact - trivial impact. To get a better perspec-tive on the impact of the calculator, let's look at what is now taught under the heading mathematics. Jot down the actual behaviors that students engage in while in a mathematics classroom - not what you do as the teach-er or what we assume is happening between the students' ears, but what actual overt behavior is observed. Some lists may look something like this:

• sitting still (listening) • completing textbook exercises of a routine nature

37

• answering teacher-posed questions

This list may look unappealing; let's try a different listing. List the behaviors students are required to know as evidenced by test items. I'm not sure this list looks all that good either:

• complete an algorithm • plug numbers into a formula

All too often, we teach mathematics as if it were a dead language. Those of you who took a course in Latin might think back to those classes. Mine ran something like this: first, we went over our homework, which was usually translation of a paragraph plus completion of a few questions, the teacher gave us a new rule, showed how it was applied, perhaps introduced new vocabulary, and we started the next day's homework. To study for an examination, we conjugated until it was coming out of our ears, memorized as much as possible and hoped for the best. That picture is frighteningly close to what can happen in a mathematics classroom.

We are teaching mathematics as if it were a dead language. The only prob-lem is that now a $7 machine can do it better and faster than a human being can. As long as a body of knowledge is closed - that is, routine mechanical procedures can provide the appropriate responses, for each problem there is a correct algorithm to apply and all problems come in neat, identifiable types - as long as the key is to learn the trick for the day and when to apply it - as long as a body of knowledge has those characteristics, then it is completely capable of being done by a machine, a machine that is more accurate and more efficient.

Assume that each student has a calculator/computer capable of completing the standard algorithms we now teach in the public schools - a device that can add, subtract, multiply, divide, square root, solve second-degree equa-tions using the quadratic formula, complete most trigonometric calculations, compute simple interest, plot first- and second-dregree equations, and so on. How much of what students now spend their time doing would be completed by the machine? Ninety percent of what students spend their time doing, actually doing, related to mathematics can be done by relatively unsophis-ticated calculators and computers. We are teaching Latin to every student. We teach mathematics as if mathematics had no future and no application except within the confines of absurd word problems about two planes flying in opposite directions or three bananas at 45 cents and two oranges at 67 cents.

The calculator and the computer will have a decided impact; we are at a crossroads, and critical decisions must be made. We have two alternatives. The first alternative is to continue to teach mathematics a la the Latin mode with the result that the subject will undergo continuing decreases in im-portance and time allotments with perhaps an eventual demise.

The second alternative is to consider carefully what mathematics is other than a dead language and to emphasize those aspects of mathematics that are required to deal with computers in a sophisticated way or, more impor-tant, those aspects of mathematics that are not 'do-able' by a machine.

There isn't a machine in the world that completes a piece of mathematics and says, 'Hey, that's really neat; I'd like to do some more.' (Probably kids who are trained to behave like machines don't say that either.) Mathematics has at least two important sides: content and process. Machines can handle a great deal of the content, but they cannot complete the processes mostly because the processes of mathematics are not algorithmic; there is no one guaranteed process to apply in a given situation and second, the processes are open-ended, leading who knows where.

Most often the processes of mathematics are labeled 'problem-solving.' Some 'problem-solving' can be done by machine; for example, the standard textbook story problems or the more extended government story problem called your federal income tax.

We have always paid attention to both aspects of mathematics, content and problem solving, but, in reality, content has always won out.

Those teachers who took the time to let students see patterns, discover concepts and work on 'real' problems, always felt a bit guilty in doing so and made sure that their students also knew as much content as anybody elses. We are at the stage when we can stop feeling guilty about doing neat and interesting things with kids.

I probably don't have to say all that much about what we mean by the pro-cesses of mathematics - or about what neat and interesting .things there are to do. Conferences and journals are always filled with terrific ideas. Clearly, we don't know everything about teaching problem-solving, about how to teach mathematics applications, and how to keep it interesting, but the advent of the calculator/computer allows us to have the time in our own classrooms to devote to such activity. Since it is now unecessary to spend two weeks drilling for efficie algorithm ,we

with the long multiplication algorith, we do have time to look for patterns related to multiplication.

I don't believe that there is nothing new under the sun. It is true that many things are not new; many of the pressures upon the mathematics curriculum come and go, fade in and then fade out. Most of these pressures are related to shifting political, social and economic pressures. But changes brought about by technology are neither reversible nor avoidable. We cannot pre-tend that the steam engine was not invented, and society cannot withdraw to a pre-steam-engine style of life. The same is true for the classroom. This year, we may be 'Cored' and 'PLAPed,' but five years from now, it may all be forgotten. We cannot, however, go back to using fingers for reckoning, nor can we assume that paper-and-pencil calculation is the goal of

39

mathematics; it can no longer be so. We are at an important point for the curriculum. We have the opportunity to rethink mathematics, to make it a growing, alive activity, engaged in by human beings - with other human beings, all enjoying the process. It's an opportunity to share with the stu-dents some of the joy we feel about what is neat about mathematics and not worry that they can't divide 7 into 987654321 in three seconds. It is a time for back to basics, but there are more important basics than 4 x 5 or (a+ b)2

UBC ANNOUNCES WINTER COURSES

Education 580, Section 161 (1.5): The B.C. Mathematics Assessment. An in-depth examination of aspects of the 1977 B.C. Mathematics Assessment: objectives, item construction, test results and teacher ques-tionnaire results. Participants will investigate an aspect of the assess-ment results. Instructor: D. Robitaille Time: Thursdays, 4.30 p.m. to 7.00 p.m., January 5 - April 6 Place: Scarfe, Room 1214

Education 580, Section 162 (1.5): Computers in Secondary Education An examination of applications of computers in schools: goals of teaching computing studies, instructional applications of computers, supportive applications of computers, and current issues. Participants will have hands-on use of several computer systems. Instructor: M. Westrom Time: Mondays, 4.30 p.m. to 7.00 p.m., January 9 - April 3 Place: Scarfe, Room 1226

Education 485 (1.5): Mathematics History for Teachers A study of the historical development of selected topics from the mathematics curriculum of elementary and junior secondary schools. Topics will include systems of numeration, methods of evaluating, and measurement systems. Instructor: W. Szetela Time: Tuesdays, 4.30 p.m. to 7.00 p.m., Janaury 3 - April 4 Place: Scarfe, Room 1211

Education 373 (1.5): Geometry for Elementary Teachers Topics in geometry related to the geometrical content of the elemen-tary school curriculum. Prerequisite Education 370 or equivalent. Instructor: C. Chandler Time: Mondays and Wednesdays, 4.30 to 6.00 p.m., January 4 - April 5 Place: Scarfe, Room 1214

For further information consult Extra-Sessional Winter Session 1977-78 UBC Credit Courses or phone 228-5203.

40

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ripe summaries from the registry of innovating practices in education in

british columbia (ripe) contributed by Julia ellis, communications

co-ordinator, eribc To enable RIPE users quickly to identify projects of interest, summaries such as this one have been prepared for all categories contained in RIPE. Originally, RIPE contained only the one-page project descriptions provided by contributors. As many categories became very bulky, the projects within them were grouped according to common approaches used. These approaches are described in the summary, and contact people are listed accordingly. If you would like the original one-page project descriptions, please indicate the category and sub-heading(s) of interest. There is no charge for this ser-vice. We hope that you find the RIPE information useful and that you will take the time to contribute to RIPE as well.

Write to.. Educational Research Institute of British Columbia Second Floor, 1237 Burrard Street, Vancouver, BC V6Z 1Z6

or call: 688-8574

ELEMENTARY - CURRICULUM

Mathematics 1. Individual and/or Non-Graded Progress 2. Teaching By Means of Manipulative Materials

1. Individual and/or Non-Graded Progress Programs are designed to allow for each child to learn basic arithmetic con-cepts at his own speed.

Example By means of careful planning and a great deal of preprogram preparation, students are given an arithmetic program which teaches all the basic arith-metic skills in small steps to allow for easy student assimilation. Careful attention is given to pre-testing and post-testing at all levels to assure ef-ficient learning and to guard against skill/knowledge gaps for the slower students and boredom for the quicker ones.

2. Teaching By Means of Manipulative Materials Programs where arithmetic skills are developed by a variety of activities and projects.

49

Example A corner of the classroom, a section of the hallway, or some other area within the school building is equipped with apparatus which allows the children to learn basic arithmetic concepts. Teaching involvement is flex-ible since most of the program is carried out on an individual or small group basis. Extensive pre-planning is needed to ensure continuity of learning but this is compensated for by high pupil interest, involvement and commit-ment to task. -

ELEMENTARY - CURRICULUM

Mathematics

1. Individual and/or Non-Graded Progress Distqr Method Armstrong Elementary School R.R. No. 3, Armstrong, BC J. Stephenson, Principal

ICPS - Individualized Computational Skills Champlain Heights Community School 6955 Frontenac St., Vancouver, BC - Mr. J.A. Downs, Principal

Edwin S. Richards Elementary School 33419 Cherry Avenue, Mission, BC V2V 2S4 David Bird, Principal

Organic Day Lord Beaconsfield Elementary School 3663 Penticton St., Vancouver, BC V5M 3C9 Shirley Sulentich and Marilyn Northrop, Teachers

Individualized Programs in Math and Language Sahtlam Annex Old Lake Cowichan Rd., R.R. No. 2, Duncan, BC H. David Preece, Head Teacher

Bayview Arithmetic Program Bayview Elementary School 140 View St., Nanaimo, BC Mrs. L. Addison and Mrs. M. Prescott, Teachers

Individualized Reading and Math Program Vanier Elementary School 1600 King Albert Ave., Coquitlam, BC Ms. J. Mitchell, Teacher

50

Learning by Doing Shaughnessy Elementary School 4250 Marguerite St., Vancouver, BC Belinda Puilvaw, Teacher

2. Teaching By Means of Manipulative Materials jllat/, Room Actii'itv Approach to Priinw MathMacquinna Elementary School 3881 Bruce St., Port Alberni, BC V9Y 11.16 Lorna Rankin, Teacher; Mr. M. Lessard, Principal

Rocketry/The Solar System and Space Travel Okanagan Falls Elementary School P.O. Box 6, Okanagan Falls, BC VOH 1 RO Mr. S. Hancheroff, Science Teacher

Mat/i Program Walter Lee Elementary School 949 Ash St., Richmond, BC Mary E. Dean, Teacher

MacCorkindale Math Project MacCork indale Elementary School 6100 Battison St., Vancouver, BC Mr. B. MacFadden, Teacher Co-ordinator

Metric Work Stations - 1975 Parkside Centennial School General Delivery, Aldergrove, BC VOX lAO I. Redekopp, Head Teacher

Curriculum Extension Program for Academically Gifted Students Southiands Elementary School 5351 Camosun St., Vancouver, BC Mrs. J. Stancombe, Teacher, Enrichment Program

SECONDARY - CURRICULUM

Mathematics 1. Computer Math 2. Metrics 3. Advanced Teacher Training 4. Alternative Programs - Providing Choices 5. Upgrading

1. Computer Math To introduce students to computer language and programming.

51

Example A program that teaches computer programming and data processing. It begins with instruction in computer language and basic operating procedures and culminates in actual student use of a complete mini-computer system.

2. Metrics To study the metric system of measurement.

Example A course that teaches all facets of metric measurement - linear, area, mass, volume, time, speed - in preparation for conversion to this system in Canada.

3. Advanced Teacher Training To provide practising teachers with the opportunity to update their know-ledge of mathematics and mathematics education.

Example A program that operates in the late afternoon and evening so that working teachers are free to attend. It is designed to prepare teachers to assume leadership roles in their school or district.

4. Alternative Programs - Providing Choices Designed to more adequately meet the needs and interests of the individual student.

Example A program where students are allowed to choose from a variety of courses the one that they feel they are most interested in and/or can handle best. Through the use of a wide variety of materials, including many texts, kits, work cards, etc., depending on the course chosen, the student works at a level appropriate to his ability.

5. Upgrading Programs to assist students who lack competence in mathematics.

Example A program where older students serve as tutors to younger pupils who need help with mathematics. The senior students prepare lessons, teach them on a one-to-one basis, and report to the classroom teacher regularly.

SECONDARY - CURRICULUM

Mathematics

1. Computer Math Vancouver School Board Computer Use John Oliver High School 530 E. 41st Ave., Vancouver, BC Miss P. Dyer and Miss B. Breen, Staff Assistants

52

Computing and C'oinputer Sciences Burnaby North Secondary School 751 Hammarskjold St., Burnaby, BC Gopaulsingh Karam, Teacher

Computer Math Delta Senior Secondary School 4615-51st St., Delta, BC Mr. C. Hardy, Teacher

Computer Technology for the WHOLE Sc/zoo! Hillside Secondary School 2295 Queens Avenue, West Vancouver, BC William P. Goddard, Co-ordinator of Data Processing

Computer Program Caledonia Senior Secondary School 3605 Munroe, Terrace, BC Mr. J.E. Bastin, Principal

Computer Science 8 Williams Lake Junior Secondary School 640 Carson Dr., Williams Lake, BC V2G 1T3 Mr. F. Kika

Computer Usage Steveston Senior Secondary School 1044 No. 2 Rd., Richmond, BC Mr. Richards, Principal.

Computer Science 1] Steveston Senior Secondary School 1044 No. 2 Rd., Richmond, BC R. Gregory, Chairman, Computer Co-ordinating Counsellor

Computer Math Claremont Secondary School 4980 Wesley Rd., Victoria, BC John Ellis, Teacher

Computer Science Dunsmuir Secondary School 3341 Painter Rd., Victoria, BC Barry W. Arnsdorf, Head, Mathematics Department

2. Metrics Metrication Ann Stevenson Junior Secondary School

53

1250 Western Dr., Williams Lake, BC V2G 1H7 Mr. H. Beausoleil and Mr. N. Weber

3. Advanced Teacher Training Emphasis on Secondary School Mathematics Faculty of Education Simon Fraser University, Burnaby, BC V5A 1S6 G.R. Eastwood, Professor, Director of Graduate Programs

4. Alternative Programs - Providing Choices General Maths, 8, 9, 10 Continous Program Brookswood Junior Secondary School 20962-37A Ave., Langley, BC Mr. Don Neumann, Principal

General Math 10 and]] Southern Okanagan Secondary School Box 990, Oliver, BC VOH iTO Bob Fleming, Mathematics Department Head

Phasing System for English and Maths Richmond Secondary School 717 Foster Rd., Richmond, BC Mr. H. Lindsay, Principal

Learning Activity Packages - Mathematics Grades 8-12 Handsworth Secondary School 1044 Edgewood Rd., North Vancouver, BC Isabel C. Leask, Department Head

Junior High Math A.D. Rundle Junior Secondary School Box 250, Chilliwack, BC Mr. Alf Bakken, Teacher

5. Upgrading Remedial Math (Junior Secondary) John McInnis Secondary School 3400 Westwood Dr., Prince George, BC Rosemary Allen, Teacher

Tutorial Prograinnie Pitt Meadows Secondary School 116 B Ave., Pitt Meadows, BC Mrs. I. Krickan, Teacher

54

SECONDARY - INSTRUCTIONAL ORGANIZATION AND TECHNIQUES

CAI - Computer Assisted Instruction To provide additional educational technology.

Example A working CAI package using a mini-computer, used by teachers and stu-dents.

Computer Applications in Instruction Camosun College 1950 Lansdowne Rd., Victoria, BC J. Diemer, Director, Science and Technology

BCAMT and MALASPINA COLLEGE

present

a workshop on the teaching of

PROBABILITY AND STATISTICS

at Malaspina College, Nanaimo

on February 24-25, 1978

Registration: $25 (before January 13, $30 after January 13). Send to: Jim Swift, R.R. 3, Site E, Nanaimo, BC V9R 5K3.

55

MATHEMATICS TEACHING

fertilizer is important! ion d, beottie, ubc

It is somewhat paradoxical, but at a time of increasing concern for mastery of basic arithmetic skills, there is growing recognition that the able child is a neglected child. The basic program in the elementary school, it is generally accepted, includes optional topics intended to be challenging and interesting. This is known as enrichment.

Actually, these optional topics have one or more of the following character-istics. They are often busywork, similar to the material in the regular pro-gram, perhaps a little more difficult. They are often unrelated to the material just covered, apparently drawn randomly from some reservoir of mathema-tical tidbits. They are generally of the pencil-and-paper variety and provide little opportunity for creativity. Surely this is not enrichment.

This discrepancy between the intent and the application of enrichment activities can be rectified, but the meaning of the term enrichment must be clarified and expanded. Webster's Third New International Dictionary includes the following definitions of the verb enrich. While only the last is intended to refer to a course of study, the others are of considerable in-terest and can broaden and give direction to our concept of enrichment.

1. To increase the intellectual or spiritual riches of. 2. To fill with things of value; add to the valuable contents of. 3. To improve with additions. 4. To supply with ornament. 5. To make more productive; to fertilize. 6. To expand a course of study by increasing the variety of subjects as

well as the depth of materials.

Consideration of these definitions suggests that, if an activity is to be truly enrichment, it should perform at least one of the following functions.

1. It should improve the program. Examples of improvement include the provision of additional insights into a topic just covered and the extension of ideas previously learned. Open-ended investigations, the solution of non-routine problems and the use of alternative algorithms or problem-solving methods also fall into this category. Note that the additions should be val-uable mathematically, add another dimension to the program and increase its worth.

2. It should ornament the program. How easy it is to get bogged down in the daily grind and omit activities that, from a mathematical standpoint,

56

may not be particularly valuable in themselves, but that could add interest and provide a change of pace and motivation. This is done as a matter of course in other subject areas, but many teachers are simply unaware of the possibilities in mathematics. A wide selection of published materials provides this type of activity, and yet school and class libraries include only a meagre selection.

3. It should fertilize the program. At first glance, fertilize seems a strong word to use in this context, but it is an evocative word, and the gardener in me gets really excited by it. It is appropriate. Just as fertilizer is the stimu-lus for plant growth, a mathematical activity can stimulate mathematical growth in two ways. Those participating in the activity may be stimulated by this exposure to new and interesting ideas. Open-ended activities provide opportunities for a child to send out new 'shoots,' to explore a new idea, to formulate and express a solution to a problem and to be creative. Growth may also occur in those who observe the results of such activities. Displays, projects and demonstrations will involve other children in class.

With these criteria in mind, consider the following activities as appropriate for enrichment:

1. Researching topics of special interest; e.g., the history of numbers, dif-ferent numeration systems, nonstandard algorithms. 2. Recreation - puzzles, games, etc. 3. Collections - scrapbooks, exhibits. 4. Learning about calculators and computers. 5. Investigating to discover patterns and rules. 6. Investigating geometric relationships through such activities as paper folding and curve stitching. 7. Forming a mathematics club. 8. Collecting and analyzing data. 9. Devising strategies for solving problems or playing games.

Here is an example of an enrichment activity I used successfully with Grades 6 and 7. It can be used as shown, but check that each part has been under-stood and completed successfully before allowing the next part to be at-tempted. Some persistence is required in part two. Encourage the children to share their ideas. Work through it yourself before using it. You'll like it. Some notes and suggestions follow the activity.

BRACELETS

1. There is a well-known sequence of numbers that starts with 1, 1 and goes like this: 1, 1,2,3,5,8, 13, 21, . . .

What will the next number be?

How did you get that answer?

57

How is each number in the sequence obtained?

Does the sequence stop, or go on forever?

2. Write down the sequence and continue it, but write down only the ones digit of each number that you get. This means that the numbers in the se-quence above would look like this: 1, 1, 2, 3, 5, 8, 3, 1,

Does the sequence still go on forever?

Does it start to repeat?

After how many numbers däes it start to repeat?

Look at your sequence up to the point it starts to repeat. If you write them in a circle and draw arrows from one number to the next, you can make a 'bracelet.'

What number patterns can you see in the bracelet?

For example, are there as many odd numbers as there are even ones?

Does every number occur equally often?

Make a list of your observations.

3. Remember we started with the number pair: 1, 1. Start with another pair, say 2, 6, or 3, 5, etc., and use the same rule to make new numbers, and again write down only the last digit.

How many different bracelets can you make? (Remember the bracelet is continuous and can have different starting points.) Are the bracelets all the same length?

Draw the bracelets.

How can you be sure you have found them all?

What number patterns can you see in these bracelets?

Make a bulletin board display on this for your class.

Notes: This is the well-known Fibonacci sequence. Each term is found by adding the previous two. The sequence never ends.

Writing down only the first digit, you get: 11235831459437077415617853819099875279 6516730336954932572910/11 2, etc.

58

The sequence begins to repeat after 60 terms.

Some patterns: Every fifth digit is either a 5 or a 0. There are twice as many odd numbers as even ones. A number occurs either four times or eight times in the sequence.

Four other bracelets can be found:

055 2684 134718976329 02246066280886404482

A table can be constructed to record the pairs found, and thus verify that all possible pairs have been checked. , If you start with any consecutive num-bers on the 60-term bracelet and follow the same rules, the same bracelet will be found.

Number pairs accounted for by the 60 Bracelet

0 1 2 3 4 5 6 7. 8 9

0 x x x x

1 x x x x x x x x

2 , x x x x

3 x x x x x x x x

4 x x x x

5 x x x x x x x x

6 x . x x x

7 x x x x x x x x

8 x x x x

9 x x x x x x x x

59

probability and statistics corner jm swift, nanaimo senior secondary

The Statistics and Probability workshop mentioned in the previous issue of Vector has been arranged for Friday and Saturday, February 24-25 at Malas-pina College, Nanaimo. The workshop, NANSTAT'78, will focus attention on the Probability and Statistics 12 course. We hope to draw on the exper-ience of the NCTM/ASA committee on probability and statistics. The key-note speaker will be Mr. Richard S. Pietrs. He was one of the editors of the series 'Statistics by Example' and has had many years experience in teach ing statistics at the high school level. This committee was responsible for the production of the series 'Statistics by Example' (Addison-Wesley) and the book Statistics - A Guide to the Unknown (Holden-Day).

The aim of the NANSTAT workshops is to encourage teachers to offer the probability and statistics course. Further details will be found in the adver-tisement, elsewhere in this issue of Vector.

Because of the unavailability of the prescribed textbook, the PS12 course at Nanaimo District Senior Secondary School had to be built around units developed for the mathematics program at A.Y. Jackson Secondary School in Ontario. The theme running through the course is that this is essentially a practical subject, and must draw on the experience of the students them-selves. A brief outline of the units of the course follows. More detailed ac-counts of individual chapters may appear in futures issues of Vector.

SAMPLES AND POPULATIONS - AN INTRODUCTION TO STATISTICAL INFERENCE

Statistics Is Chapter 1 This chapter examines some situations that arise when numerical informa-tion is collected. The framework for the entire course is outlined, based on two concepts: sample and population. Students are introduced, to the Data File, which is an important feature of the course. Through critical examination of information collected in this file, students learn the valuable habit of close investigation of numerical data.

Information from Samples Chapter 2 Here we see at work, a fundamental idea and technique of statistics: data obtained from a sample can be used to provide information about the parent population from which the sample has been taken. This technique is illus-trated by the capture/recapture techniques used in biology. This consti-tutes an intuitive approach to the binomial model.

60

An Interlude with APL Chapter 3 The language of mathematics known as APL was developed in the 1960s to handle ideas that arose with the processing of data by computers. It has become a language that can be understood by a computer and used to com-municate mathematical ideas very effectively.

Some of the basics of this language are taught in this chapter. The language is used to illuminate ideas and calculations. APL also becomes a highly useful tool for students to use when they have access to computers.

Probability - Counting Outcomes Chapter 4 In chapter 2, a 'measurement' was given to the word unlikely, and so 'pro-bability' was introduced. Now we examine such things as die throwing, card drawing and coin tossing in which all the outcomes are equally likely, and students extend their knowledge of probability. The essential skills of efficient methods of counting are seen to be very important. The grid-path model is also introduced.

Expectation and Achievement Chapter 5 A knowledge of probability tells one what to expect, but only experience will show what one gets, and there is a difference. As a mathematician, one is not really content until one has measured the difference!! Then one tries to discover how big, how small, this difference is unlikely to be. This is an intuitive introduction to the chi-squared distribution.

Up the Eucalyptus Tree Chapter 6 In Chapter 4, students find the probabilities of outcomes that are equally likely and so begin a study of probability. They look at combinations of such events and extend, even further, their knowledge of probability.

Samples and Their Statistics Chapter 7 Data are almost always obtained from measurements of a sample. Such mea-surements, called statistics, are of many kinds, averages and measures of spread. This chapter examines such. statistics and the methods of their cal-culation. The experience gained, in this chapter, of the distribution of sample means will be used in late work on the normal distribution.

The Binomial Model Chapter 8 A set of samples provides a distribution of statistics. The set of samples of marked and unmarked beads, examined in chapter 2 can be modeled using the paths on a grid used in chapter 3. A model, known as the Binomial Model, is developed in this chapter, and used to introduce the distribution described in chapter 9.

61

The Normal Distribution Chapter 9 This distribution is one of the most widely used distributions in statistics. Its shape is accurately tabulated, and the use of the standard normal dis-tribution table is a feature of this chapter.

Bivariate Distribution Chapter 10 Data has been used to demonstrate apparent connections between pairs of such variables as fluoridation and cancer, transcendental meditation and smoking. In this chapter, we examine correlation using scatter diagrams.

The Investigation Chapter 11 The conclusion of this course is the application of the knowledge acquired to data collected. This project is a very important aspect of learning about statistics.

INTERNATIONAL MATHEMATICS EDUCATIONAL RESEARCH Glasgow, Scotland

The Groupe International de Recherche en Pedagogie de la Mathema-tique will hold its seventh meeting in Glasgow, Scotland, July 15-23, 1978 'You who teach mathematics - who are you?' will be the theme of the meeting. Additional information may be obtained from Edward C. Martin, 18 Cairnview Road, Milton of Campsie, Glasgow, G65 8131, Scotland.

Ne

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---' - --

62

calculators, boon or bane? roger fox, prince george

Calculators came upon the education scene just a few years ago. What boons they are to the mundane, repetitive calculations so often encountered in laboratory work! What a delight in accuracy and speed' A lot of laboratory work has become a pleasure, especially in physics and chemistry. The price has been dropping rapidly, putting sophisticated instruments within the reach of many.

Much the same was said about slide rules when in their heyday. However, the controversy over slide rules now applies to hand calculators. Should we or should we not allow their use in certain circumstances? To what extent should their use in schools be tolerated? Doesn't a student who has exper-tise and facility with a calculator pose some deep educational problems? Will he/she be able to add, subtract, etc., without one? Will the art of arithme-tic be forever lost once the calculator holds sway? What fond memories this brings to mind, especially when thinking back to the times of the slide rule.

Despite the reservations expressed about slide rule use, it was accepted and permitted on examinations and on other hallowed events. Nobody has ever claimed since that time, that the slide rule was a factor in the demise of the arithmetic ability of students or (heaven help us) their ability to think. The slide rule did become exactly what it always was - a tool. It made demands on a student's knowledge of mathematics such that it became quickly and efficiently useless when used without a sound mathematical grounding. But one interesting fact to note was that despite the widespread acceptance of the slide rule, no school provided them for the student, and no govern-ment agency, or the like, provided them for students writing their examina-tions. Slide rules became quite sophisticated too. Remember the log-log, the trig-log, folded scales, and those that were a metre long.

Now we have the calculators. More sophisticated, more versatile, more efficient, speedier and far more attractive, and we have the same contro-versy. Listening to the discussions and arguments pro and con - careful listening - will show very minor deviations from those used some twenty years ago.

A few facts remain, the most significant of which is 'garbage in - garbage out.' A student of any age must be able to tell the calculator what to do. Unless the instructions are correct and comply with arithmetic principles, the result is useless. Consider the most sophisticated programmable cal-culator. To program such a monster to do a students' bidding with any problem in physics, he/she must have a fundamental understanding of the nature of the problem. Sceptics should try to write a program to appreciate the statement.

63

Let us consider a few more basic arguments such as the ability to multiply two members. The calculator will always get it correct if the instructions are correct; students do get such problems wrong - they have done so for ages. What is far more important is whether or not a student is aware of the feasibility of an answer - does the answer make sense in terms of the pro-blem at hand? A common experience comes to mind - a Grade 8 class, generally competent in arithmetic is calculating the volume of some small tin cans. A tin is filled with water, and the volume is measured. It is then dimen-sionally measured, and the volume is calculated. The results are recorded:

Volume of-water 110 ml Volume by calculation = 126 1.9835 ml

'Which is right?' one asks. The reply, inevitably; seems to be either 'I don't know' or 'The first one.' 'Then what about the second one?' one asks. 'Well, that is the way it worked out,' is the prompt answer. And so begins the dia-log.

The important thing here, is the ability of students to recognize the nature of an answer in a given situation. The problem that was apparent. with slide rules, which just gave an answer, will be the same with calculators. Thus it could be argued that the calculator necessitates that a student think - an acceptable educational criterion. Justice in calculator use is a major concern. With the variety of calculators available, and the financial disparities among students, how can assurance be given that all students have equal oppor-tunity? Here the parallel drawn with slide rule does not fit. The argument that all students should have access to calculators is not significant; as this was not true of slide rules. But the greater degree of sophistication in calcu-lators does pose a problem. This problem could be dealt with by placing restrictions on calculator use, even calculator type. Such restrictions may be in the form of prohibiting the use of prepared programs, declaring when a calculator has been used in work submitted, selected use by specifying questions or areas in which calculators may be used. A sensible approach to this problem is really all that is required.

The problem of calculator use in classrooms and on examinations should be approached by reviewing attitudes toward the slide rule and moving forward from that point. The controversy mustnot get bogged down with issues and opinions that have already been resolved.

The positive attributes of calculator use must be considered first and they must direct policy development. All science and mathematics teachers should work to promote a district policy and then insist that the Ministry of Education adopt a sensible attitude toward the use of calculators in schools.

64

using 'real life' material in the mathe-matics classroom, ideas for junior

secondary slow learners ken keeley (mission junior secondary school)

1. A train schedule offers a wealth of math exercises.

a. Discussion of a 24-hour clock. b. Time zones in Canada. c. How will a delay in a train's starting day affect arrival dates at other

towns? d. Distances between towns. e. Comparison of rates for different types of accommodation.

2. Catalog shopping

a. Have some of the students draw a floor plan of a house (to scale). House plan books are available at most bookstores. You may have to adapt one of these plans.

b. Let the students list furniture for the house. c. Sears stores will give you some catalogs. Find the prices of the items

using the catalog. d. Once the students have the idea, copy an order form. Repeat the above

procedure. Have students fill in an order form - catalog number, unit price, color, size, etc.

3. Math in sports

a. Scoring leaders in the Canadian Football League. b. Standing of National Hockey League teams. c. Bowling scores.

4. Divide the class into groups that are even in ability. Name the teams - Fractured Fractions, Mad Multipliers, Mighty Mathematicians, etc. Points for work can be awarded each team. There are some good books available explaining games, for example, Games for Learning Mathema-tics available from J. Eston Walch. At the end of a month, reward theteam with the most points by taking them out to lunch.

CANADIAN PACIFIC RAILWAY - ACROSS CANADA SERVICES

Time Train 41 Example Miles Daily Zone Example Train 42 2050 Thursday 0 Dp Saint John, NB Ar AT Tuesday 0945 0605 Friday 372 Sherbrooke, PG ET Monday 2240 0835 Friday 479 Ar Montreal, PG Dp ET Monday 1950

65

Time

Example Miles Daily Zone Example

The Canadian The Canadian

No.1 No.2

1115 Friday 0 Dp Montreal, P0 Ar ET Monday 2005

1121 2 Westmount 1958

1127 5 Montreal West 1950

1133 9 Dorval, PQ 1910

1232 57 Vankleek Hill 1845

1335 109 Ottawa, Ontario 1755

1420 141 Carleton Place 1655

1455 164 Arnprior 1615

1520 182 Renfrew 1551

1614 217 Pembroke 1500

1655 239 Chalk River 1430

1935 v 356 North Bay 1150

2115 Friday 435 Ar Sudbury Dp Monday 1000

No. 11 No.12

1530 Friday 0 Dp Toronto, ON Ar Monday 1610

1540 5 West Toronto -' 1550

1840 131 Mac Tier 1305

1909 4 155 Parry Sound '4' 1225

2135 Friday 260 Ar Sudbury Dp Monday 1010

No.1 No.2

2225 Friday 435 Dp Sudbury Ar Monday 0920

2320 469 Cartier 0835

0250 Saturday 606 Chapleau 0515

0610 736 White River Monday 0140

0915 854 Schreiber Sunday 2225

1255 987 Thunder Bay ET 1910

1453 1134 Ignace CT 1500

1600 Saturday 1197 Dryden CT Sunday 1341

1752 1280 Kenora,ON 1158

2045 1406 Winnipeg, MB 0920

2143 1461 Portage-la-Prairie 0738

2328 Saturday 1539 Brandon 0616

0017 Sunday 1586 Virden, MB 0513

0200 1670 Broadview, Saskatchewan 0346

0355 1763 Regina 0145

0500 1805 Moose Jaw Sunday 0055

0710 1915 Swift Current CT Saturday 2243

0902 2063 Medicine Hat, AB MT 1850

1255 2239 Calgary 1535

1510 2320 Banff 1300

1555 2355 Lake Louise 1205

1715 2375 Field, BC 1110

1830 2410 Golden MT 0940

2055 2501 Revelstoke PT 0540

66

Time Example Miles Daily Zone Example

2238 Sunday 2564 Salmon Arm 0325 0045 Monday 2629 Kamloops Saturday 0130 0440 2751 North Bend Friday 2140 0633 2810 Agassiz 1935 0711 2838 Mission City 1855 0747 2863 Coquitlam 1817

(New Westminster) 0825 Monday 2880 Ar Vancouver, BC Dp PT Friday 1745

Club Upper Lower Mini- Standard Dayniter Car Berth Berth Roomette Bedroom Bedroom

Campellton and Levis 4.00 11.00 18.00 23.00 46.00 50.00 Montreal 4.00 12.00 20.00 25.00 50.00 56.00

Edmonton and Toronto 11.00 32.00 53.00 68.00 136.00 148.00 Vancouver 5.00 14.00 24.00 30.00 60.00 67.00

Halifax and Montreal 6.00 15.00 26.00 33.00 66.00 73.00 Vancouver 19.00 53.00 90.00 113.00 226.00 252.00

Hamilton and Ottawa 11.00 12.00 15.00 23.00 34.00 34.00:

Louis and Gaspe 12.00 20.00 25.00 50.00 56.00

Montreal and Edmonton 11.00 32.00 53.00 68.00 136.00 148.00 Gaspe 14.00 24.00 30.00 60.00 67.00 Moncton 5.00 14.00 24.00 30.00 60.00 67.00 Toronto 4.00 12.00 15.00 23.00 34.00 34.00 Vancouver 13.00 38.00 64.00 80.00 160.00 179.00 Winnipeg 7.00 22.00 36.00 45.00 90.00 101.00

Toronto and North Bay 4.00 11.00 18.00 23.00 46.00 50.00 Saskatoon 9.00 28.00 46.00 59.00 118.00 129.00 Vancouver 13.00 38.00 64.00 80.00 160.00 179.00 Winnipeg 7.00 22.00 36.00 45.00 90.00 101.00

Winnipeg and Edmonton 5.00 15.00 26.00 33.00 66.00 73.00 Saskatoon 4.00 12.00 20.00 25.00 50.00 56.00 Vancouver 8.00 24.00 40.00 50.00 100.00 112.00

67

SAMPLE EXERCISES

Editor's Note: Mr. Keeley sent extensive questions and we include a sample of each type.

The 24-Hour Clock What are these times on a 24-hour clock?

6.00 a.m

10.40 p.m

What are these times on an 'ordinary' clock?

2006 ___________________ 0948

Time Zones Students should have a map showing Canadian time zones.

7.00 a.m Mountain Time is ________ Atlantic Time.

2.45 p.m. in Winnipeg is in Ottawa.

Distance How far is it between:

Sudbury and Winnipeg?

Time Start in Montreal on Wednesday. What day are you in:

Thunder Bay ___________ ? Golden

Fares

Between Upper Berth Lower Berth Save Toronto and Winnipeg Montreal and Edmonton

68

"A b 'B

on areas of rectangles inscribed in a sequence of triangles william j . bruce

(university of alberta, edmonton)

Consider a sequence of triangles drawn with a common base of constant length b units and of constant altitude a units. In Figure 1, we identify, from left to right, five different types of triangles as follows: Scalene triangle ABC, Isosceles triangle ABCi with ACt = BCi, Isosceles triangle ABC2 with AC2 = AB, Right triangle ABC3, Scalene triangle ABC4 with angle ABC4 >900.

Relative to the aforementioned sequence of triangles, inscribed rectangles will be located such that one side of each will lie on the base AB. Two dif-ferent problems associated with these rectangles will be examined. One of these is to determine the dimensions of the rectangles of maximum area for all possible triangles as vertex C moves from the extreme left to the extreme right. The other problem is to find a function to represent the maximum areas of the rectangles inscribed in the sequence of triangles. It is conven-ient to discuss the former problem in two parts.

C l C2 C3

Figure 1

First we consider the usual procedure, namely, that based on the drawing shown in Figure 2. Here scalene triangle ABC has been selected, but the proof applies identically to triangles ABCi, ABC2, and ABC3. Triangle ABC4 will be treated separately: Let the dimensions of the inscribed rectangle be w units and h units as shown. If K units is the area of the rec-tangle, it follows

-

A' I I

b-

Figure 2 K=wh

69

A

a

R D a cot ci

b —

Since triangles ABC and MNC are similar, we have that

wa — h b a

or

a

Use this to replace w in K = wh and obtain

K =--(ah - h2)

Complete the square for the quadratic function oh the right to get

K=_[(h_a/2)2 _a2/4]

Now K will be a maximum when (h - a/2) 2 is least, that is, when h = a/2.

It follows that, when h a/2, w = b/2. These are the dimensions of the rec-tangle of maximum area.

It should. be noted in the above proof that, since a and b are fixed values and since this proof applies to all of triangles ABC, ABCi, ABC2 and ABC3, the rectangle of maximum area inscribed in all of these triangles is unique. Students of calculus can use differentiation of K with respect to h to arrive at the same result.

Consider. triangle ABC4 as shown in Figure 3. Note that one end of the base of the rectangle is taken at B and that one vertex does not lie on the triangle.

Ct,

Figure 3

The area of the rectangular region is given by K = wh.

Since triangles AIDE and ARC4 are similar, we have from Figure 3 that

h a b — w b+a cot o

70

orh=

a (b — w) b + a cot

Substitute this for h in K = wh to obtain

K= a (bw—w2) b + a cot

Complete the square for the quadratic to get

a K=— b + a cot cc [(w—b/2)2—b2/4]

K will be a maximum when w = b/2, the same value found for the other

triangles. However, when w = b/2, we obtain h = a (b/2) for tn-b + a cot cc

angle ABC4 and this is not the same as for the other triangles unless we have CC = 900. Differentiation of K with respect to w can be used to obtain the same result.

Note that as cc tends to 0, cot cc gets larger and larger. But a and b are fixed values. Thus h tends to 0 as cc tends to 0. Since w = b/2 is constant, it fol-lows that, as cc tends to 0, the area of the largest rectangle decreases. Hence, for triangles ABC4 , the rectangle of maximum area is not unique.

Let S, herein called the maximum area function, be the function that re-presents the maximum areas of the rectangles inscribed in the sequence of triangles. Consider triangle ABC placed on a Cartesian grid as shown in Figure 4.

Y

Figure 4

As x varies, all of the types of triangles previously discussed are formed. We have shown that, for all triangles considered, the width of the rectangle of maximum area is given by w = b/2. It also follows from our discussion that, for this maximum, h a/2, -b/2 x b12, so that

S (x) = ab/4, -b/2 x b/2.

71

For x > b/2 (or x <-b/2), the situation is quite different, because h changes. Since triangles AOE and ARC are similar, we get

= a x>b/2

b/2 x+b/2 that is,

h = ab , x>b/2 2x + b

We now have

S(x)=wh=.ab2 x > b/2 2b+4x

Similarily, since in this problem symmetry exists about x = 0, we obtain

S(x) = a b 2 x <- -b/2 2b - 4x

Therefore, the maximum area function for all cases is defined by

x<-b12 2b-4x

S(x) ab-b/2xbI2 4

ab2 x>b/2 2b + 4x

We note that the first part of this function represents a portion of an hyper-bola with vertical asymptote given by x = b/2 and horizontal asymptote represented by S 0. The second part represents a horizontal line segment, whereas, the third part defines a portion of an hyperbola with vertical as-ymptote given by x = -b/2 and horizontal asymptote again given by S = 0. Figure 5 is a sketch of the graph of the maximum area function S(x).

S

ab/4

/ TI-

-b/2

b/2

X

0

Figure 5

72

This graph shows that, as the vertex C of the triangle ABC moves from the extreme left to the right, the maximum area of the inscribed rectangle in-creases until the triangle becomes a right triangle at x = -b/2, at which time the maximum becomes abI4. This maximum remains fixed until the triangle becomes a right triangle again at x = b/2 after which the maximum continues to decrease and tends to zero as x becomes larger and larger. With a and b fixed, each choice for x specifies the triangle being considered and determines the maximum area of the rectangle.

ThEEfl I for secondary (7-14) school teachers

classroom tested ideas I I reproducible worksheets I practical teaching activities I I from teachers for teachers in the

Matliumatics Teachif I I I I I National Council of Teachers of Mathematics I 1906 Association Drive • Reston, Virginia 22091 I

For a free copy of the Mathematics Teacher to read, use, and share, I I complete and forward this form to the Council.

I I Name i

u Address

I State or City Province Zip I

Valid through June 1978 Pr

— — — — — — — — — — — — — — — — — — — — — —

73

humor in mathematics doniel fiegler

Reprinted with permission from SUMMATION: Association of Teachers of Mathematics of New York City

The general impression which the layman has of mathematics teachers is that of the eccentric professor who is slightly out of touch with reality. A rather typical caricature is found in the film Willy Wonka and the Choc-olate Factory, in which a. mathematics teacher is shown harassing his stu-dents with an incomprehensible explanation of per cent problems. More recently, Woody Allen has included a brief portrait of a mathematics teacher in his latest movie, Annie Hall. Allen presents-us with a mathematics teacher scribbling numbers on a blackboard without any logic to their appearance. While both of these films present mathematics teachers in a humorous light, it is the very seriousness with which the teachers take their work which leads to the humorous response on the part of the audience.

I have no intention of suggesting that mathematics is not serious business; however, what I do suggest is that most mathematics teachers do not utilize the possibilities for humor which exist in their lessons. All of us have suf-fered through lessons on algebraic word problems white our classes have shown little enthusiasm for solving these problems. Consider the following problem:

A vat can be filled by one pipe in 4 minutes and by a second pipe in 5 min-utes. The drain can empty the vat in 3 minutes. If the vat is empty, how long will it take to fill the vat if both pipes are turned on and if the drain is left open?

Should we really be surprised if most students fail to get overly excited about the solution to this type of problem? Is there any reason why anyone should be curious about the length of time it would take to fill an anony-mous vat? However, here is an alternate version of the same problem:

Secret Agent 008 is captured by the diabolical Doctor Yes. Imprisoned in Doctor Yes's torture chamber, Agent 008 is told by Doctor Yes that poison gas will be filling the room from two different pipes. The first pipe will fill the room with poison gas in 4 minutes, and the second pipe will fill the room with poison gas in 5 minutes. Secret Agent 008 discovers the exhaust fan which empties the room of poison gas in 3 minutes. If both pipes are acti-vated and the exhaust fan is turned on, how long will it take for the room to completely fill with poison gas?

74

There are few students who can resist the excitement of trying to solve this revised version.

Here is another example of a typical problem:

If x varies directly as the square of y and if x = 75 when y = 5, find x when y = 8. The following restatement of the questions. would surely generate greater student interest:

The IQ of a martian varies directly with the square of the number of ten-tacles which it possesses.. If a martian with 5 tentacles has an IQ of 75,' then what is the IQ of a martian with 8 tentacles?

When I first used this problem in class 12 years ago, the existence of some unusual life-form on Mars was considered much more likely than it'is'today. Luckily for us mathematics teachers, we have a fairly large number of planets on which we may place our many-tentacled monsters.

As a final example of humor in mathematics, let us look at an elementary trigonometry question:

A person stands 500 feet from the base of, a building 1000 feet high. What is the angle of elevation from.the person to the top of the building?

Here is the version I prefer to use (while instructing my class to temporarily make the assumption,, teáhnically incorrect, that the path of a bullet is a straight line):

Fay Wray is held captive by King Kong, at the top of the Empire State Building. If a sharpshooter stands 500 feet from the base of the Empire State Building and if King Kong is at a height of 1000 feet, at what angle must the sharpshooter aim to shoot King Kong?

Unfortunately, Dino DeLaurentiis has caused this version of the problem to become slightly obsolete. However, I prefer, the original King Kong to his version, so I refuse to change heroines or buildings.

Paraphrasing Jerome Bruner, I am tempted to say that any mathematical topic can be presented in a humorous fashion. My hesitation in making such a statement is a result of the fear that someone will challenge me to find a humorous approach to some esoteric topic and I shall be unable to satisfy the request. But I do believe that there are many topics in the high school mathematics curriculum which are suited to humorous treat-ment. The creative mathematics teacher must search for these topics and then apply his individual sense of humor to restructure them in a humor-ous fashion. One caveat: Be prepared for moans and groans from students who do not 'appreciate' your humor. I would rather have my classes laughing at me in this context than laughing at some caricature of mathematics teachers in the movies.

75

a grade 8 mathematics outline bill kokoskn

The mathematics teachers of North Vancouver were unhappy with the Grade 8 curriculum outline. Through their association, the North Vancouver Mathematics Teachers' Association (NVMTA), and a secondary curriculum committee, set up by the assistant superintendent of program and develop-ment, three teachers were given the task of writing an outline to meet the needs of School District 44.

The outline they came up with is divided into sections as follows:

I. Intended learning outcomes. A. Mastery level. B. Instructional level. C. Enrichment level. II. Teaching strategies. Ill. Resources/references/supplies/equipment. IV. Evaluation.

The intended learning outcomes were selected on the basis of the provincial guide, teacher experiences, public desires, and what is known about child development. For each intended learning outcome, a specific example is given of the level of difficulty expected a. mastery; b. instructional; c. en-richment. The suggested teaching strategies help , to achieve the intended learning outcomes. The resources/reference/supplies/equipment section describes materials available to teachers for a particular teaching strategy. The evaluation section suggests other processes that teachers might use to determine the degree to which expectations are being met, and to pro-vide teachers with feedback on their teaching strategies.

At the end of the outline, books, charts, materials, games and filmstrips/ transparencies are listed in an order form. The school board is now sup-plying the teachers with some of the material, and it is hoped that all of the material needed to support the program will be available in two years.

The school board provided funds for development by hiring three teachers to work on it. Since it is only a first draft, it is hoped that teachers will im-prove it after they have used it for a few years.

If you are interested in looking at what we've come up with in North Van-couver, please write for a copy of 'Mathematics Year Eight' to:

Dr. Leo Marshall Assistant Superintendent Program and Development School District No. 44 721 Chesterfield Avenue North Vancouver, BC V7M 2M5

76

MATH EDUCATION IN CANADA

canadian mathematics olympiad contributed by g. h m. thomas, chairman

977 olympiad committee 1. If f(x) = x 2 + x, prove that the equation 4f (a) = f(b) has no solutions in positive integers a and b.

2. Let 0 be the center of a circle and A a fixed interior point of the circle different from 0. Determine all points P on the circumference of the circle such that the angle OPA is a maximum.

3. N is an integer whose representation in base b is 777. Find the smallest positive integer b for which N is the fourth power of an integer.

4. Let p(x) = ax n + a n 1 x - +. . . +ax + a0

andq(x)bxm+b_1xm_l+...+b1x+b0

be two polynomials with integer coefficients. Suppose that all the coef-ficients of the product p(x) q(x) are even, but not all of them are divisible by 4. Show that one of p(x) and q(x) has all even coefficientsand the other has at least one odd coefficient.

5. A right circular cone of base radius 1 cm and slant height 3 cm-is given. P is a point on the circumference of the base and the shortest path from P around the cone and back to P is drawn (see diagram). What is the minimum distance from the vertex V to this path?

3cm

/

P 1

77

I. IC) 10 IJ

6. Let 0 < u < 1 and define u 1 = 1 + u, u2 =1. + u, U1

U n+1 _+u,n1.

Show that U n > 1 for all values of n = 1,2,3.....

7. A rectangular city is exactly m blocks long and n blocks wide (see dia-gram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day, but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number f(m, n) of different paths she can take to work satisfies f(m,n) 2mn

m blocks

ninth canadian mathematics olympiad 1977 solution key

1. Suppose a and b are two positive integers such that 4f (a) = f(b); i.e., 4a2 + 4a = b2 + b.

Solution i

4a 2 - b2 + 4a - b = 0 (2a - b) (2a + b) + (2a - b) = —2a (I) (2a—b)(2a+b+ 1)=-2a<0 :.b>2a

But then the absolute value of the left-hand side of (I) is> 4a + 1, contra-diction. Alternatively, from (I), we get 2a - = —2a

2a + b + 1

But this is a contradiction, since 2a - b is an integer and —1 <_-2a <0 2a + b + 1

Solution ii

Regard the equation as a quadratic in a

78

.. a = -4 ±/l6 -f 16b 2 + 16b

a = —1 ±,/b2+ b + 1

The right hand side of this equation will be an integer only if b 2 + b + 1 is the square of an odd integer.

But b 2 <b2 + b + 1 <(b + 1)2, so this is impossible.

2. Solution i

By the Sine Law sin OPA = sin PAO = sin AOP OA OP AP

OA Therefore sin OPA = -- sin PAO.

OP OA and OP are constant, and 0 <angle OPA <900.

Angle OPA is a maximum when sin OPA is maximum and this occurs when sin PAO = 1; i.e., angle PAO = 900.

Angle OPA is a maximum when AP is .1. to OA.

Solution ii

Extend P0 to a diameter POQ and extend PA to meet the circumference at B. Join QB. LQBA = 90° since POQ is a diameter.

LOPA is maximum when LPQB is minimum.

This occurs when AB has minimum length of all the chords through A. This occurs when OA I PB (this lemma should be proved).

Solution iii

Consider the circle C 2 through 0, P and A. Angle OPA is an angle at the circumference of C 2 determined by the segment OA. This angle will be a maximum when C 2 is tangent to the given circle at P. Then OP is a diameter Of C 2 and so angle PAO = 90°.

79

Solution iv

From the point of view of an observer at P, as P moves around the circum-ference, A appears to move in a circle C 3 with center 0 and radius OA. The angle OPA will be a maximum when PA is tangent to the circumference Of C3. This occurs when LPAO = 900.

3. Suppose N = 777 in base b representation and N x 4 for some integer x. Then 7b 2 + 7b + 7 = 7(b2 +b+1)=x4

7 divides x 4 50 7 divides x since 7 is a prime. Thus x = 7k for some integer k. 7(b2+b+1)=74k4 b 2 + b + 1 = 7 3 k 4

The smallest possible value of b will occur when k is smallest. Try k 1. Then b2 + b + 1 = 7 3 = 343 b2+b-342=0 (b - 18) (b + 19) = 0 :.b=18

4. First we establish s n

Since p(x)q(x)=(

m+n =0 p(x)q(x) = Zckxk =

k=0

ome notation.

a i x i ) ( bxJ)we can write

Cm+nXn+...+CkXk+...+Clx+cO

where

k C k = atbk -t = a0bk + a l bk - 1 . + ak - 1 b 1 + a kb0 = ab

t=0 i + j = k

For example, c0a0b0

c1=a0b1+a1b0

c2 = a0b2 + a 1 b 1 + a2 b2, etc.

(some of the terms atbk - t will be 0 when k is large).

a. First we show that either p(x) or q(x) must have at least one odd co-efficient. Otherwise a t and bL. - are both even for all values of t and k and thus afbk - = (2d)(2e) = 4de br some integers d and e, 50 4 divides ck for each k, contrary to assumption.

80

.(p= ) Pi P2 (=P

h /3

p1

b. Suppose both p(x) and q(x) have at least one odd coefficient. Suppose r and s are the smallest integers such that ar and b5 are both odd. Consider:

Cr + - r S i+j=r+s i

In this sum, apart from the term a r b , every term a 1 b1 involves a subscript i < r or j < s. Since a i is even if i Z!r and b is even if j < s, these terms ab will be even. However arbs is odd since both ar and b5 are odd.

c + s is odd, contrary to assumption. .. either p(x) or q(x) has only even coefficients.

5. If the curved surface of the cone is cut along the segment VP, it can be considered as the sector of a circle of radius VP = 3 and arc length = (270 (1) since the radius of the base of the cone equals 1.

The shortest path around the cone thus corresponds to the shortest distance from P 1 to P2 in the sector, namely the chord P1 P2, and the shortest dis-tance to this path is the segment h = VA in the following diagram.

VA

In general for a sector in a circleL

27r 27rR

so in this case we have 2 20 2ir (3) or 07r/3

_:cOS (7

..h=(3)(Y2)=f

6. Proof by Mathematical Induction

Solution i For n = 1, u 1 = 1 + u> 1 since u >0.

1 1 1 lfn=2,u2—+u=----+u

+u+u2 u2 =1+ >1.

u 1 1+u 1+u 1+u

—u Also note that

1 = 1 = > 1 - u. That is, u 1 < U i 1+u 1+u 2 1—u

81

Inductive assumption

** l<u < 1 1 u , 1nk

Consider n = k + 1. Since u k < 1—u ,weobtain

Uk+1=+U>(l—U)+U=l. uk

Since uk> l , 1 < l SOU k+1 <1 +U< 1

Uk 1—u

Therefore ** holds for n = k + 1 if it holds for n = k. Therefore it is true for every positive integer n.

Solution ii

Inductive assumption 1 <U n 1 + U.1 1

True for n = 1, 2. Assume true for 1 n k. Then <1 1 1. k

<1+u I U U k+1

Also u' _ 1+u U1

7. Every allowable path separates the set of all city squares into two dis-joint subsets (one of which may be empty; e.g., for the path going straight up and then straight across). e.g.

a.. LI..

Therefore the number N of allowable paths is less than or equal to the num-ber of subsets of the set of all city squares. Since there are m n squares, the number of such subsets equals 2mu1 The number N is actually always strictly less than 2mn if both m and n are greater than 1, since some sub-sets cannot be obtained in this way, even for m = n = 2. e.g.

However, if n = 1, it can be proved that we always have equality, that is, N = 2m in this case.

Note: The contestants submitted a variety of correct solutions for some of the problems, especially problems 1, 2 and 6. In some cases, these solutions displayed a great deal of ingenuity.

82

what's happening in math education across canada? j routledge (canada nctm rep)

bhtish columbia BCAMT - Bill Dale, President BCAMT held another very successful Summer Workshop. I was able to accept its invitation to attend, and enjoyed renewing friendships and making new ones. BCAMT is already making plans for the 18th Northwest Mathema-tics Conference to be held in October 1979 at Vernon. No wonder its con-ferences are tops!

alberta MCATA - K. Allen Neufeld, President This council held its 17th Annual Conference and Business Meeting in Red Deer on October 14 and 15. It was a large, enthusiastic, stimulating, success-ful conference, and a privilege to attend. A special thanks to Eric Mãcpher-son who 'filled in' for the special speaker Bob Wirtz, when he forgot to come. Also high praises to the planning committee who kept their collective cool when what all planning committees have nightmares about really happened! MCATA is planning more miniconferences for this year, and is already think-ing about the Name-of-Site to be held in Calgary in October 1979.

saskatchewan SMTS - Gerald Goski, President Because of a commitment to speak at a conference the same weekend at Saskatchewan's Sciematics Conference, I will be unable to bring greetings to them from NCTM. It appears to be an excellent program, and I am sure that it will be a very successful conference.

manitoba MAMT - Allan Wells, President Manitoba is planning the St. Benedicts Conference to be held in May 1978. The committee on the use of hand calculators has prepared a very interesting report which is being discussed by the association.

ontario OAME - Neil J. Williamson, President The Ontario association has just held another very successful Leadership Conference. Two chapters have held fall conferences - SEYMA (Scar-borough East York Mathematics Association) and Quinte-St. Lawrence. FWTAO (Federation of Women Teachers) is holding a conference in Toronto

83

at which both OAME and NCTM have been invited to set up membership tables. This conference, for elementary teachers, will deal with new curri-culum guidelines - Formative Years (K-6) and Intermediate (7-10). Plans are well-advanced for the annual spring conference, in Hamilton this year (May 1978).

quebec QAMT AMQ - Michael Cassidy, President No news from Quebec this fall, but I know that there is a lot going on, and I will catch you up on it in my winter newsletter. We are still hoping to hold a leadership meeting for eastern Canada in Montreal in February. More about that shortly.

nova scofia MTANSTU - Sr. Agnes Cordeau, President I was able to attend the Nova Scotia math teachers annual conference in Halifax on September 23 and 24. The theme was Math Update, and it was an excellent conference. Thanks for including me in your program.

new brunswick NBTAMC - Diane Manning, President The New Brunswick math teachers held their annual conference in Sack-ville last spring. It was my privilege to take part in the program. The program was excellent and the hospitality very warm in spite of the weather! It was my first visit to New Brunswick, but it will not be my last. I made many new friends and am looking forward to going back.

newfoundbnd MCNTA - Roy Pitcher, President As we expected; the St. John's Name-of-Site meeting was a roaring success. An excellent conference with a real Newfie flavor. The hospitality was out-standing. Most of the mainlanders were 'screeched in' and are now staunch and loyal Newfoundlanders. The attendance was higher than expected (not by me!) inspite of the ferry difficulties.

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BOOKS

in praise of the calculator workbox Reprinted from the Manitoba Mathematics Teacher

Any teachers who believe the calculator has a p lace within the math pro-gram, The Calculator Workbox is an invaluable aid. Designed as a program-med learning lab, it contains five major sections of assignment cards com-plete with answers and teacher's guide.

Topics covered within the sections are: 'The Calculator,' 'Numbers,' 'Mea-surement,' 'Real Life Math' and 'Extra Points.'

'The Calculator' coincides with the teaching of both estimation and the order of operations. It affords a comprehensive overview of the use and fallibility of the calculator and the necessity for estimating an unexpect-ed answer.

Sections two and three, dealing with numbers and measurement, can be referred to at any point throughout the year to reinforce both skill areas.

'Real Life Math' complements any consumer math or business education program and is particularly useful in any problem-solving situation.

'Extra Points' provides interest and fun through the use of games and puz-zles worked out on the calculator.

The set has various teaching applications. It can be drawn upon for variety in worksheet material as a supplement to the regular work and presented to the entire class. It can also be used in activity centers in a less formal way, and it lends itself well to self-programmed study as either an endea-vour in enrichment or remediation.

The Calculator Workbox can be purchased through Addison-Wesley Pub-lishers for a relatively low price. The program it provides can only enhance the teaching of math in light of the present popularity of the calculator.

PSA77-1 04 December 1977 85

THE B.C. ASSOCIATION OF MATHEMATICS TEACHERS PUBLISHES VECTOR

(combined newsletter/journal).

Membership may be obtained by writing to the:

B.C. Teachers' Federation 105 - 2235 Burrard Street Vancouver, BC V6J 3H9

Membership rates for 1977-78 are:

BCTF Members............................................$10

BCTF Associate Members....................................$10

Student Members (full-time university students only)................$3

all others (persons not teaching in B.C. public schools, e.g., publishers, suppliers). . . $15

printed by B.C. Teachers' Federation

issn 0382-0718 british columbia association … › ... › vector › 192-december-1977.pdfd.c....

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