BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS

NEWSLETTER/JOURNAL

VOLUME 18, NUMBER 1 OCTOBER 1976

BRITISH COLUMBIA ASSOCIATION OF MATHEMATICS TEACHERS 1976-77 EXECUTIVE

PAST-PRESIDENT PRESIDENT & PSA COUNCIL DELEGATE Alan A. Taylor John C. Epp 7063 Jubilee Avenue 1612 Wilmot Place Burnaby, B.C. V5J 4B4 Victoria, B.C. V813 5S4 434-6315 (home) 592-2388 (home) 936-7205 (school) 478-5548 (school)

VICE-PRESIDENT William A. Dale 1150- 17th Street Courtenay, B.C. V9N 1Z7 338-5159 (home)

RECORDING SECRETARY Ian C. deGroot 3852 Calder Avenue North Vancouver, B.C. V7N 3S3 980-6877 (home) 987-7178 (school)

PRIMARY REPRESENTATIVE Linda Shortreid 4651 - 202nd Street Langley, B.C. V3A 5.12 530-4665 (home) 588-5918 (school)

IN-SERVICE SPECIALIST Dennis Hamaguchi 3807 - 22nd Avenue Vernon, B.C. V 1 T 1H7 542-8698 (home) 542-3361 (school)

TREASURER Grace Dilley

2210 Dauphin Place Burnaby B.C. V513 4G9

299-9680 (home) 596-0357 (school)

PUBLICATIONS CHAIRPERSON Susan J. Haberger 1390 Willow Way

Coquitlam, B.C. V3J 5M3 936-7205 (school)

INTERMEDIATE REPRESENTATIVE Ken Thompson

9352 - 119th Street Delta, B.C.V4C 6M6

433-5703 (home)

CURRICULUM CONSULTANT William J. Kokoskin

1341 Appin Road North Vancouver, B.C. V7J 2T4

988-2653 (home) 988-3161 (school)

NCTM REPRESENTATIVE SUMMER WORKSHOP 1977 Tom Howitz

Dr. Pauline Weinstein Faculty of Education, UBC

Faculty of Education, UBC

2075 Wesbrook Place

2075 Wesbrook Place Vancouver, B.C. V6T 1W5

Vancouver, B.C. V6T 1W5

325-0692 (home)

261-6803 (home) 228-5203 (UBC)

NORTHWEST NCTM CONFERENCE ORGANIZER Trevor Calkins

1623 Amphion Street Victoria, B.C. V813 4Z5

592-4463 (home) 592-1205 (school)

WIOIUIUIUIOIDIOIUIIIIDIDIUIDIDIDIOIUIUIDIDIOIUIDIDIUIDIOIUIUIOIUIU

Inside This Issue [OIOIDIDIDIOIDllhIOIUIOIUIUIUIUIDIElIDIDIIIIDIDIOIDIUIDIDIDhIllDIOIOIO

5 President's Message . John Epp

7 Your Executive Council ...........................Grace Dilley

8 New Books Across My Desk .......................Sue Haberger

10 Letter to the Editor

Elementary Teaching 11 Learning Difficulties in Math: Part II ...............Werner Liedtke

14 Performance Testing of Fraction Concepts ..........James H. Vance

& Edward W. Richmond

Secondary Teaching 20 More Activities with the Golden Ratio .................. R.F. Pea rd

22 Student Questionnaire on Individual vs. Traditional Instruction ..................A.C. Maffei

25 Metric Crossword ..............................D.W. McAdam

Computing 27 Some Hard Facts about Computer Hardware ...........Chris Weber

47 More Problems for Computer Science Students...........Doug Inglis

50 The 1976-77 Mathematics Assessment ................Jerry Mussio

53 Backward Glance: Fifth Mathematics Summer Workshop. .Doug Owens

Lesson Plans Math Match Game

Mystery Graph Activity

Shapo

Hidden Message

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President's Message EDIUIUIUIEIIDIDIDIOIOIUIUIUIUIOIUIDIDIDIDIDIUIDIIIIOIDIDIDIDIDIDIUIO

Dear Colleagues,

The new school year has begun and we're all back in the classroom and very busy. The BCAMT has hada very busy summer. The Fifth Annual Summer Workshop was well-attended and was considered a success by all participants

that I talked to.

The day before the Summer Conference some of the BCAMT Executive met with a committee of the B.C. Committee of Undergraduate Program Mathe-matics (BCCUPM). There promises to be much co-operation between the two groups to further mathematics education throughout the province. The members of BCCUPM are university and college mathematics and mathema-tics education instructors and are willing to act as resource personnel for teachers, so if you are planning a math workshop, check with your local college or university. They appear to be willing to assist.

The BCAMT is also preparing a new program, the Provincial Involvement Program (PIP). This program will endeavor to increase involvement of class-room teachers by offering math conferences and workshops at various loca-tions throughout the province and also by forming a number of local or regional chapters. In addition to the secondary math chapters, the BCAMT is encouraging the formation of elementary chapters. The workshops will pro-vide many sessions for the primary and intermediate teachers as well as for

the secondary teachers.

Discretionary days are available to schools and many are presently being used for 'administrivia.' Hopefully, the schools will declare a discretionary day if there is a math workshop within reasonable driving distance and the teachers from those schools could then attend in groups.

The BCCUPM and some of the branches of the Department of Education have indicated a desire to encourage and assist PIP.

At present the BCAMT with the Prince George Math Teachers' Association is planning a workshop in the Prince George area for this November. Hope-fully, by the time this message reaches you, the schools in the districts sur-rounding Prince George will have been contacted and sent more information.

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If your area is interested in having a conference or in starting a local math chapter, please contact me.

During the recent fire at the Campbell River Junior Secondary, the math teachers lost all of their material. If you have extra copies of worksheeIs, tests, or whatever, please send them a copy. Send them either to: Lance Klassen or to Barry Underwood, do Campbell River Senior Secondary School, 350 Dogwood Street, Campbell River, B.C. V9W 2X4. I'm sure they will be very grateful.

Respectfully yours,

John C. Epp.

[DhIUhIUlED][Dh[D1[D][U1[EI][D][U][Dh[U][Uh[D][DIEH[U][D][U1[U][D1[D][U][U][D][U]W1[UhID][U][D][D

Your Executive Council [D][D]ftIIDhEEll[U][W[D][D}[U1[D][U][DhIEIl[D1[Uh[EI][D1[D][D][U1[D][Dh[D][EI1[DIDIU][U1[DhIDl[E11[D

GRACE DILLEY

Grace Dilley has taught Grade 4 through 10 levels for many years in Quesnel, Alberni and Surrey. She is currently assigned to District Staff at School District 36 (Surrey) to assist intermediate teachers in planning and imple-menting their mathematics programs.

Grace was associated with the BCAMT as our elementary representative before being elected treasurer in May 1976. She will be looking after the association's budget for the next two years and with the planned increase in in-service activities, she will be kept busy finding the funds to pay for these

services.

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New Books Across Mq Desk by Sue Haberger

Amusements Developing Algebra Skills by A.A. Clack & Carol H. Leitch, Midwest Publications Company Inc., P.O. Box 129, Troy, Ml 48084 A collection of cross-number puzzles, connect-the-dots, coding problems, etc., with questions suitable for senior secondary algebra classes. Available in Canada from: Western Educational Activities, 10577 - 97th Street, Edmonton, Alberta T5H 21_4 Cost: about $5.25 (N. B. Western Educational's catalog is an excellent source of short reviews on math material.)

Laboratory Activities for Teachers of Secondary Mathematics by Gerald KuIn, Prindle, Weber, and Schmidt Inc., 20 Newbury Street, Boston, MA 02116

A brief general introduction to lab activities in mathematics, together with detailed outlines of 17 di fferent labs. Topics include curve fitting, constant difference, exponential functions, quadratic roots. Suitable for use in senior secondary schools and junior colleges. Cost: about $6

The Calculus with Analytic Geometry Handbook by J.R. Taylor, Taylor Associates, 59 Middlesex Turnpike, Bedford, MA 01730

A summary of the highlights of Introductory Calculus, which omits de-tailed proofs, this 54-page booklet would be of use to a college student preparing for an exam or to a teacher wishing to recall details of a seldom-used mathematical technique. Cost: about $3

Overview & Analysis of School Mathematics Grades K- 12 by the National Advisory Committee on Mathematical Education Single copies available on request from Conference Board of the Mathe-matical Sciences, 2100 Pennsylvania Avenue, N.W., Suite 832, Washing-ton, DC 20037

A comprehensive report (150 pages) published in 1975 draws input from professional groups and individuals concerned with math education. Controversial topics, such as curriculum reform, slow learners, calcula-tors, instruction methods, teacher training, and evaluation, are discussed

8

in the light of the most recent opinions and research results. The final section contains recommendations for future policy and research. Cost: Free

Math Ideas Published by the B.C. Primary Teachers' Association Classroom material suitable for duplicating, games, activities and lesson plans are included in this 34-page booklet, along with two articles on the philosophy of primary mathematics education. Contact: Shirley Thrapp, 303 - 5926 Tisdall Street, Vancouver, B.C. V52 3N3.

HAVE YOU FOUND SOME VALUABLE MATERIAL?

I should be glad to receive copies of books, or preferably short reviews of books, that B.C. teachers have found useful for the mathematics classroom.

Sue Haberger

REVISED SECONDARY CURRICULUM HELP WANTED!

The BCAMT hopes to publish booklets that will aid teachers in implementing the revised Mathematics Curriculum for Secondary Schools. If you have been among the first to use the new textbooks, please help your colleagues avoid 'learning by discovery' and duplicating any problems that you have now solved.

We welcome feedback from teachers regarding problems arising from the changeover (for example, 'general math' or prerequisites) and positive sug-gestions as to how these problems may be overcome.

Information regarding Grades 9 and 10 will, it is hoped, be available this year, and a similar publication for teachers of Grades 11 and 12 for 1977.

Share your ideas - send submissions, preferably in PROBLEM-SOLUTION format to:

Ms. Sue Haberger Editor, Vector

• l3gO Willow Way Coquitlam, B.C. V3J 5M3

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Letter to the Editor

1620 Fell Avenue Burnaby, B.C.

V5B 3Z5

Dear Editor:

I was delighted to receive the May '76 issue of Vector. It is much better than most recent issues I have read.

In the 'lesson plans' section, I note that you have included 'The Five Square Puzzle.' A more comprehensive version of the puzzle with nine squares and more detailed rules and with four separate activities is available from the BCTF Lesson Aids. Authored by me and entitled 'Geometry Game,' it is in the quidance section of the Lesson Aids catalog and ahs been available since 1971.

Yours truly

R.G. Smith

NCTM NEWS

REPORT ON ELECTRONIC HAND CALCULATORS The body of the Final Report on the National Science Foundation-supported project, 'Electronic Hand Calculators: The Implications for Pre-College Education,' is now available from the ERIC Infor-mation Analysis Center for Science, Mathematics and Environmen -tal Education, 1200 Chambers Road, The Ohio State University, Columbus, OH 43212. The 350-page complete Final Report will be available later in 1976 from the ERIC Document Reproduction Service, Box 190, Arlington, VA 22210.

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ELEMENTARY TEACHINQ [D1[U1[O1[U1[D1[U1[U][U][U][U][D][D1[U][U1[U][D][D][U][D][U][UJ[U][U1[D][U1[U1[U1[U1[UJ[D][U][U][U]

Learning Difficultiesin Math:

by Werner Liedtke, University of Victoria Part I I [0][0][0110110][0110110110110][0][0110110110][0][0110110110][0110][0110110110110110110110][01101101

John is six and Mark is seven. In their mathematics classes, they are consid-ered to be slow learners. They do seem to find it difficult to keep up with the other children in their respective classrooms. They make mistakes like the following:

5 + 3 = 2 19 comes after 20 31 is read as thirteen Why do John and Mark make these mistakes? How can these children be helped?

Some readers may recognize the title and the opening paragraph as being very similar to those used by David Robitaille in an article that appeared in the May 1976 issue of Vector (p. 62-66). An excellent article-it is, and it would be nice if every teacher who works with slow learners could get a chance to read it.

My main reason for writing this is to show some support for most of the statements RObitaille made in the article. At the same time I hope that I can make a few new suggestions that may be of help to teachers who work with children in the early grades.

Mark and John are not their real names. However, children like-Mark and John can be found in many classrooms. These children tend to make a variety of mistakes, and they find it difficult to keep up with their friends. Why are they slow learners in mathematics?

Certainly two of the major reasons are related to two of the points sug-gested in Robitaille's article. He states that mathematics is 'developmental.' New skills, ideas and concepts are, more often than not, based on previous-ly learned skills, concepts and ideas. Prerequisite skills that are missed by a child can make later learning difficult. Much of the mathematics program is taught from textbook illustrations and pictures. This procedure is too abstract for many young children, especially during the early stages of learn-ing. Perhaps these two factors are the main reason why some young children have difficulty learning mathematics and why their achievement is noticeably below desired levels;

In an attempt to identify a young child's particular strengths and weaknesses, standardized diagnostic tests, checklists or individual diagnostic interviews can be used. Robitaille makes the point that diagnosis in mathematics can best be done in a one-to-one interview. This is especially true for settings that involve children in the early grades. A prerequisite for such an inter-view setting is the knowledge of skills and concepts, in order of difficulty, as they relate to a specific topic (i.e., What skills and concepts are needed for

11

the ability to count rationally?). Robitaille suggests that, during the inter-view, it is best to present the tasks in descending order of difficulty. From our experience with young children, there seems to be no particular disad-vantage when that procedure is reversed. Sometimes it may be advantageous or just as efficient to begin an interview somewhere in the middle of a par-ticular sequence.

Robitaille uses a variety of examples to illustrate the points he makes and the issues he raises. The statement that an extremely high proportion of the students referred to the Mathematics Education Diagnostic and Instructional Center have difficulties with place value is also true for many of the young children we worked with in Victoria. They were unable to use a manipula-tive aid and physically represent a number. Questions like 'which numbers come before and after a given number?' were often answered incorrectly. Frequently the numbers 12, 13.....19 were interchanged with 21, 31...... 91. Robitaille pleas with teachers to make more use of such place-value teaching aids as pocket charts, abacuses, and multibase.blocks.. Perhaps the use of abacuses and pocket charts should be delayed for some time. During the early stages of learning about place value, young children should use an aid that clearly illustrates the ten to one, or-hundred to ten to one, relation-ship.

Other common mistakes young children like John and Mark seem to make include the following examples: Sketch 1 o 0 00 - They fail to realize that a number 00 000

00 00 000 00

can be represented in many different ways. According to them, there is

O 000 nothing the same about the arrange- 0000 00 ments shown in Sketch 1.

00

- They find it difficult to make use Sketch 2 of given information. Attempts to 00000. continue a pattern or to identify a

000000 hidden member of an ordered se- quence are unsuccessful.

- They are unable to relate symbols Sketch 3 or equations to their experience. 2 3 0 5 6 Statements like 5 + 3 and 2 x 4 may

be read correctly, but attempts to 2 4 0 8 10 simulate the actions associated with

the statements and attempts to iden- 1 2 3 0 2 3 tify the statements with something 1 3 0 7 9 from their environment seem to be

rather difficult, if not impossible, tasks.

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Generally speaking, the one-to-one interview is a very rewarding experience indeed. One does get to know a child very well and most children seem to enjoy the setting and experience. Then, of course, there is always the chance of something unexpected coming up. One six-year-old boy was asked to read the statement 7 + 9. He did it correctly. He was asked to find seven counters and then nine more. As soon as he had selected the nine, and he must have anticipated the next question, he looked over his glasses and simply stated, 'Oh, my God.' It seems that in a large group, the only time childrenare talked to is when something is done incorrectly. The following experience has occurred more than once. A young child is asked to justify a correct response. For example, for 7 + 4 = 11, the question 'How do you know the answer is 11?' is asked. The forthcoming response is one of attempting to correct an apparent mistake, 'Oh, oh - it's 12.'

Contrary to some beliefs about slow learners, many young children seem to thrive in a one-to-one setting. They often display great enthusiasm, and their willingness to keep on going is, at times, a tiring experience for the person conducting the interview. The sad thing is, and the point is well made in Robitaille's article, that one cannot generalize from one child to another. A remedial program must be designed for the individual child.

NCTM NEWS

CLASSROOM USE OF HAND-HELD CALCULATORS

The November 1976 issue of the Arithmetic Teacher will focus on instruc-tional uses of hand-held calculators. Copies of this issue will be available for distribution at a special price of 50 cents each under the following conditions:

1. The minimum order is 100 copies.

2. All 100 copies must be sent to a single address.

3. Orders must be in the NCTM Headquarters office by August 31, 1976.

4. NCTM will pay the shipping charges if full payment is received with the order.

To place an order, or for further information, contact: Charles R. Hucka, Director of Publications Services, NCTM, 1906 Association Drive, Reston, Virginia 22091, or call 7031620-9840.

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Performance Testing of Fraction by James H. Vance, University of Victoria Concepts and Edward W. Richmond, Queen Charlotte, B.C. [UIUIDIDIDIuIoIoIoIonIoInIaJuIuIuIgIgIu]1 01010 I0I0I0I[II0I0I0I0I0I01

Traditionally, evaluation of learning in mathematics has been based mainly on the results of various paper-and-pencil tests. While such tests can prove useful in determining the extent to which certain goals of instruction are being met by individuals and groups, the need for other types of instruments is apparent. Performance tests in mathematics are receiving increased atten-tion as one means of gathering more information on concept formation and the achievement of process objectives (Reys and Post, 1973).

A performance test requires physical involvement of the subject in a pro-blem situation. The subject is presented with a set of concrete objects that he/she may use to respond to questions from the examiner. By referring to the material or manipulating the objects, the student arrives at a solution to the problem or demonstrates understanding of the concept under considera-tion.

The current interest in performance tests is related to the popular trend toward teaching elementary school mathematics via a laboratory approach. In math labs, students manipulate structured materials or engage in tasks involving physical objects and situations. Such activities are designed to pro-vide the learner with a background of experiences upon which abstract mathematical concepts can be built. The student is also permitted to see how mathematics is derived from the real world and that it can be applied in everyday situations. It is reasonable that evaluation of achievement under such instruction should include tasks that are similar in nature and purpose to the learning activities.

Research conducted, to date, on the effectiveness of laboratory methods in-dicates that, while students do learn new content in such settings, similar results can be obtained through nonlaboratory-based meaningful instruction. (Vance and Kieren, 1971). In a review of research comparing laboratory and demonstration methods in science teaching, Kruglak and Wall (1959) stated that paper-and-pencil tests would likely never reveal differences between the two methods, and they advocated the development of performance tests in harmony with the objectives of laboratory learning.

Two performance tests designed to measure understanding and application of fraction concepts were constructed for use in a study comparing a mani-

14

rI IrA

pulative and a nonmanipulative approach to teaching fractions at the Grade 4 level (Richmond, 1973). Each student in the experimental group first made a fraction kit that was used in the development of introductory topics. The kit consisted of a number of colored paper strips cut to varying lengths to correspond to common unit fractions. Students in the control group fol-lowed the textbook approach, which included illustrations and sketches, but no manipulation of physical materials.

The Performance Tests The Concrete Achievement Test consisted of nine items that required the student to demonstrate answers using concrete materials. For the first three test items, a set of base five blocks was used (Figure 1).

For each question, the examiner would choose two of the blocks and ask what fraction the smaller was of the larger.

Item 3: A flat is what fraction of a large block? Students were permitted to handle or use the material in any way they wished to answer questions. For the last six items, open cardboard boxes and a supply of sugar cubes were used (Figure 2).

Q15

Item 4: Fill 'this' box one-quarter full of sugar cubes. How many cubes did you use?

Item 7: Place four sugar cubes in 'this' box. What fraction of the box is filled?

The Concrete Transfer Test consisted of eight items, two each, involving ad-dition, subtraction, multiplication and division of fractions - topics that had not yet been taught. The students were required to determine and dem-onstrate answers to the questions using fraction pieces cut from cardboard as illustrated in Figure 3.

Hi LH NH The examiner would read each question while holding up the appropriate fraction pieces. Students were to respond using the materials; in all cases, final answers were to be expressed in terms of a single fraction.

Item 1: One-half plus one-quarter Item 3: Three-quarters minus one-half Item 5: One-half of one-third Item 7: How many eighths are there in three-quarters?

The performance tests were administered directly after the completion of the three-week instruction period and again two months later as retention tests. Students were tested in groups of 15 in the school library. In addition to the examiner, five observers were present to record and evaluate student responses.

Results and Discussion Group mean scores for the two administrations of each performance test are listed in Table 1.

TABLE 1 uviau acores o" 'Erie tooncreze /cnievemen.t and Concrete Transter Tests

Test Administration Experimental Control Total Concrete Achievement Initial 2.1 2.6 2.3 (9 items) Retention 3.1 3.9 3.5

Concrete Transfer Initial 5.5 5.1. 5.3 (8 items) Retention. 6.0 5.5 5.7

None of the differences between group mean scores was found to bestatisti-cally significant. (A 38-item multiple choice paper-and-pencil test also failed to reveal any significant differences in achievement between the two treat-

16

ment groups.) Thus the experience of manipulating the paper strips during the initial work with fractions did not provide any particular advantage for the experimental group in being able to successfully apply and extend con-cepts to the physical materials used in the performance tests.

The data indicate that the Concrete Achievement Test was perhaps too diffi-cult for Grade 4 students with only three weeks of instruction in fractions. However, there was a marked improvement in performance on the second administration of the test even though there had been no school work with fractions between the two testing periods. Since this was a first experience in performance testing in mathematics for the subjects, higher scores might have resulted from a practice effect. Another possibility is that during the two-month period following the instruction, the students had become more aware of fractions and their application in real-world situations. The greatest improvement was noted in the questions that asked what fraction of a box was filled by a given number of cubes; the combined mean scores for these three items rose from 2% to 30%.

The Concrete Transfer Test that involved the four operations with fractions was easier for the students than the Concrete Achievement Test. The stu-dents were able to use the materials provided to improvise, solutions to ad-dition, subtraction, multiplication and division problems in fractions even though these skills had not been formally taught. A symbolic parallel form of this test was administered at the same time to determine the extent to which student could perform these operations without concrete materials. Combined mean scores on the concrete and symbolic forms of the test were 72% and 25% respectively, confirming that pupils can solve problems with physical materials before a symbolic procedure has been learned. As a side note, on both tests, the scores on the multiplication and division problems were approximately double those achieved on the. addition and subtraction items. S

In conclusion, the results of the performance testing provided the investi-gators with some new insights into how students are able to apply intro-ductory concepts of fractions to new problems with physical settings. While there is obviously a great deal to learn about constructing and efficiently ad-ministering performance tests of achievement in mathematics, such instru-ments appear to have potentially great value both in formative and summa-tive evaluation.

References Kruglak, H. and Wall, C. N. Laboratory performance tests for general physics.

Kalamozoo, Michigan: Western Michigan University, 1959. Reys, R. E. and Post, T. R. The mathematics laboratory: Theory to practice.

Boston: Prindle, Weber and Schmidt, Incorporated, 1973, 233-235.

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Richmond, E.W. A manipulative materials approach to teaching fractions at the Grade 4 level: A comparative study. Unpublished masters thesis, University of Victoria, 1973.

Vance, J.H. and Kieren, T.E. Laboratory settings in mathematics: What does research say to the teacher? Arithmetic Teacher, 1971, 18, 583-589.

Edward W. Richmond Supervisor of Instruction

Queen Charlotte, B.C. School District and

James H. Vance Faculty of Education University of Victoria

ANNOUNCEMENTS FROM THE DEPARTMENT OF EDUCATION

EDUCATION CIRCULARS

METRICATION The responsibility for metric conversion has been transferred to the Department of Education from the office of the Provincial Secretary effective April 1, 1976.

A Provincial Metrication Committee has been set up that represents all departments of the provincial government. This committee is responsible for making the general public aware of the changes involved in converting to the metric system and con-ducting various training programs for the public and provincial, municipal, and industrial employees.

Eugene Gosh is responsible for the administration of the program. The budget esti- mates from the Provincial Secretary's office have been transferred to the Depart-ment of Education.

Gosh reports directly to the Deputy Minister's Office through J. Phiffipson, Asso-ciate Deputy Minister, Schools.

J.L. Canty Superintendent

Administrative Services

MATHEMATICS For general information on the revised senior secondary mathematics program, ad-ministrators are referred to Instructional Services Circular 23.2.76.

As indicated in the Circular, for the 1976-77 school year, four new courses will be introduced in this revised senior secondary program: Algebra 11; Algebra 12; Con-

18.

sumer Mathematics 11; Trades Mathematics 11. The following gives information regarding the prescribed materials that will be provided to support these courses.

A. Algebra!! 1. (Each of the following titles may be ordered provided the total number of

books selected does not exceed 150% of the enrollment in Algebra 11): a. Del Grande, et a!: Mathematics For A Modern World (Gage) E b. Travers: Using Advanced Algebra (Doubleday) E

B. Algebra 12 1. (Any or all of the following titles may be ordered provided the total number of

books selected does not exceed 150% of the enrollment in Algebra 12): a. Del Grande, et a!: Mathematics For A Modern World (Gage) E b. Travers: Using Advanced Algebra (Doubleday) C. *Crosswhite, et a!: Pre Calculus Mathematics (Merrill) E

*This title has been listed particularly for use by honor students.

C. Consumer Mathematics 11 1. Bello: Contemporary Business Mathematics - Canadian Metric Edition (W.B. Saunders) A

NOTE: This Canadian metric edition, currently in production, will not be available immediately. In the interim, the present revised edition of Contemporary Business Mathematics will be provided for teacher information.

It should also be noted that copies of Business and ConsumerMathematics (Addison-Wesley), currently prescribed for Mathematics 9, 10 may be ordered for the use of students who are taking the new Consumer Mathematics 11.

(B issue)

D. Trades Mathematics 11 1. Olivo: Basic Mathematics Simplified - Canadian Metric Edition (Van Nostrand Reinhold) A

NOTE: This Canadian metric edition, currently in production, will not be available immediately. In the interim, the present unrevised edition of Basic Mathematics Simplified will be provided for teacher information:

2. (Any or all of the six modules in the following series may be ordered provided the total number of modules selected does not exceed 1 'A' issue):

Practical Problems in Mathematics Series (Van Nostrand Reinhold) E (Modules: Welders; Auto Technicians; Carpenters; Electricians; Machinists; Plumbers and PipefItters)

NOTE: These modules are available at this time in nomnetric format. When deci-sions re metric standards have been made for the industries represented, prescrip-tions will be reviewed.

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SECONDARY TEACHIN More Activities with the Colden by R.F. Peard, Windsor Secondary School, North Vancouver Ratio

[Editor's Note: Mr. Peard has sent the following three activities concerned with the idea of the Golden Ratio. They provide excellent motivation and some meaningful mathematics for classes at the secondary level.]

1. Each student is asked to draw a rectangle of any shape. and size on graph paper supplied. Each then calculates the ratio long side/short side. Pupils at this stage are familiar with the use of significant figures and are asked to cal-culate to four significant digits. The class results are summarized and a histo-gram drawn.

Ratio I Number in this ranc 1.000 - 1.500 1.501 - 1.750 1.750 - 2.000 2.001 —2.250 2.251 +

For most classes, the model interval is 1.501 - 1.750, and pupilsthen cal-culate the middle value of the most popular interval.

1.501 + 1.750 = 1.626 2

This average is more significant than finding the arithmetic mean of the class owing to the skew nature of the distribution. (The ratio has a lower limit of 1.000, but no upper limit.) This activity generally takes no more than half of a one-hour period, and pupils can go straight to the next with a minimum of instruction.

2. Pupils are asked to: a. complete the number sequence 1, 1 2, 3, 5, 8,_,_,_,_,, b. calculate the ratio of each term to its predecessor (to 3 significant fig-ures). Most pupils are able to recognize 1.62 as a limit.

3. Pupils have had previous experience at finding function rules and general graphing; they are now asked to complete the table,

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Choose suitable scales and draw a graph of the function. On the same set of axes, they then graph

I f(n)=n

1.1 2.0

2.0 2.0

and thus graphically solve the equation n 1

Again, most pupils are capable of getting a value close to 1.62. Furthermore, many are able to appreciate the significance that the ratio in each exercise is arrived at independently. From here it is relatively easy to connect the last two activities.

n

1 n1

1 n—i

'n' as the 'most popular value' of activity 2 and as the solution to exercise 3.

From here, a discussion of the Golden Ratio follows easily.

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Student Questionnaire on Individual vs Traditional Instruction by Anthony C. Maffei, Dreher High School, Columbia, South Carolina

Reprinted with permission from 'Mathematics in Michigan.'

High school students usually have a good idea of the type of instruction in which they would prefer to learn their mathematics. The best type of in-struction is one that will adequately meet the needs of each student.

Since there are basically two types of instruction, we, as mathematics educa-tors, can screen our students by means of a questionnaire before they begin their course work to determine their preference for either individual or tradi-tional instruction. Of course, such a questionnaire will be useless if we do not have provisions for implementing the results or are unable to help a stu-dent who has changed his mind once he has chosen a particular type of in-struction.

Questions 21, 22 and 23 in the questionnaire are research questions to deter-mine if the majority of responses of a student choosing a particular type of instruction can be linked with the student's self-appraisal of his ability.in mathematics course work. Such a questionnaire can also be adapted to meet other subject matters as well.

QUESTIONNAIRE Name Dear Student, The following questions will help us to find out what is the best method in which you feel you can learn your mathematics. Please answer each question as well as you can.

Place a check after each question in the box marked either 'yes' or 'no.' If it is possible, please give a reason for your response to each question. Thank you.

1. Would you prefer to learn in a traditional class where the teacher teaches the whole class at the same time? Yes 0 No 0

2. Would you prefer to learn your mathematics in an individualized class where you progress at your own rate of learning and receive individual help from the teacher when you have a problem? Yes 0 No 0

3. Do you prefer to have a teacher explain problems to the class? Yes[] No C1

22

4. Do you prefer to learn your problems on your own and get individual help from a teacher if you have trouble? Yes D No 0

5. Do you think that you would be mostly bored in a traditional class? Yes No

6. Do you think that you would be mostly bored in an individualized class? Yes No

7. Do you feel that you would learn your mathematics well in a traditional class? Yes 0 No 0

8. Do you feel that you would learn your mathematics well in an individual-ized class? Yes 0 No 0

9. Would you feel more comfortable and more secure in a traditional class? Yes No

10. Would you feel more comfortable and more secure in an individualized class? Yes 0 No 0

11. Would you prefer to be in a class where everyone is learning the same topic? Yes 0 No D

12. Would you prefer to be in a class where mostly everyone is learning dif-ferent topics? Yes 0 No 0

13. Do you feel that you are capable of learning in a traditional class? Yes No

14. Do you feel that you are capable of learning in an individualized class? Yes No

15. Would you worry in a traditional class if the teacher was teaching you too fast or too slow? Yes 0 (too fast D or too slow 0 ) No 0

16. Would you worry in an individualized class where some students were ahead of you or behind you in their work? Yes 0 (ahead 0 or behind D) No

17. Would you have the tendency to waste your time and fall behind in your work in a traditional class? Yes 0 No 0

18. Would you have the tendency to waste your time and fall behind in your work in an individualized class? Yes 0 No 0

19. Would you have the tendency to be lost and confused in a traditional class? Yes 0 No 0

20. Would you have the tendency to be lost and confused in an individual-ized class? Yes 0 No 0

Check the one (21 or 22 or 23) that best applies to you. 21. 0 Do you think mathematics is one of your best subjects? 22. 0 Do you think mathematics is one of your worst subjects? 23. 0 Do you think mathematics is neither one of your best subjects nor

one of your worst subjects?

24. What do you think are the best features of learning mathematics in a a. traditional class? b. What do you think are the worst features of such a class? 25. What do you think are the best features of learning mathematics in an a. individualized class? b. What do you think are the worst features of such a class?

23

NEWS RELEASE

FROM

SCHOOL SCIENCE AND MATHEMATICS ASSOCIATION

Career education - its implications for science and mathematics

teachers - is the theme of the special issue just published by the

School Science and Mathematics Association. The recent emphasis

in the schools on career education has stressed that teachers from

elementary school through the university level are to have exten-

sive input into the career education of their students. What input

have mathematics and science teachers had in the' implementation

of career education in their schools? Will career education be a

vehicle through which more students become interested in mathe-

matics and science? The articles in this special issue provide some

of the answers to these questions.

Some of the topic areas of individual articles include the history of

career educatiOn and the legislative and funding programs during

the last five years; how science and mathematics can be infused

with career education in the elementary schools; introducing career

education into secondary school science 'and mathematics classes;

how career education can play a larger role in the junior colleges;

how teacher education can facilitate including science and mathe-

matics in career education and a comprehensive listing of resources

for getting more ideas for infusing career education in science and

mathematics classes.

Career education is interesting and challenging. Today, mathema-

tics and science teachers should be aware of' how their disciplines

can become the first stepping stone to a life-long career. Single.

copies of this publication are available for $1.50 from the School

Science and Mathematics Association, P.O. Box 1614, Indiana

University of Pennsylvania, Indiana, PA 15701.

24

[EIIUIDhIIIOID][DIDIUh[DhIDIUl[UIUIUIOIDIU]EUID][UhIDIUIDllhIUIUIUIUIUl[DIUIEI]

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This crossword puzzle contains units from the SI, imperial system and old metric system. There are familiar units and unfamiliar units. Some you may not have heard of. It may in- dicate the confusion which exists with our existing system and the number of odd units it contains. It may even be fun. See solution on page 26. i.. I I • I MMMEEMMMMM MMEEIMEMI • a a a M 0 IIR OEM I •

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1. SI prefix for 10 2. 1000 kg 3. Surveying measure for 100 ft. 4. A unit of weight in the imperial system

equals 64.8 mg 5. Sl prefix for 106 6. SI base unit for quantity of substance 7. Common temperature unit in SI

10. Unit of energy in C.G.S. system

25

Across (continued)

20. British volume unit for 1000 ft3 21. Common unit of length imperial

system 23. Unit of energy C.G.S. system 27. Printing measure for 1/6 inch 28. Power unit, derived unit in SI 29. British force unit, commonly used

for weight of people 31. Sl prefix lo' 33. Symbol for kilowatt 34. Length unit in SI equals 1000 m 35. Force unit, imperial system, equals -

2240 pounds 36. Length measure in imperial system,

equals 45 inches 37. Mass unit in SI 38. Common unit of volume in imperial

system (1/2 of 19 down)

Down (continued)

12. British measure of 1/4 acre 14. Surveyor's measure for 5.5 yards 16. SI prefix for 10-18 18. Unit of pressure in SI 19. Common unit of volume in imperial 22. SI prefix for 103 24. Two hogsheads of wine 25. Electric capacitance, SI derived unit 26. Resistance unit in electrostatic units

(not SO 29. Old metric unit of volume, equals.

1m 3 , not used with SI 30. Force unit in SI 31. Force unit inC.G.S.System 32. , 500 sheets of paper

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26

COMPUTINQ EflhtIIDIUIOIDIDIDIUIDIOIDIUIDIUIUIDIOIDIDlEDIDIDIDIDhIIIDIOIDIUIDIUID

Some fiord Facts about , by Chris Weber Computer Hardware W][UIOIDIUIDIUIUhIIIlIIDIOIDIOIOIflIflIDhElIDIOIDIDItIIOIUIUIDIDIDl[EIIDIU

[Editor's Note: If you are considering obtaining computing facilities for your school and find your mind boggled by the vast amounts of advertising and wide selection of hardware, the following article will prove invaluable. The paper was written as an assignment for Dr. David Robitaille's course en-titled 'Computers in Education' at UBC in the Spring of 1976. Chris has graduated from UBC and moved to Alberta. We thank him for offering this information for publication and wish him good luck-in his teaching career.]

Introduction In an article entitled 'Computing on a Shoestring,' David Dempster outlined four major factors. con mrned with establishing a program for instructional use of computers These included. (1) establishing specific goals; (2) obtain-ing computing facilities; (3) deciding where to introduce computer science in the curriculum; (4) seeking approval from the authority involved.

This paper concentrates on the second factor, obtaining computer facilities. Possible options open to a school district will be presented and discussed in-cluding some Dempster did not mention. Suggestions for deciding among options will be given. The paper concludes with a district-by-district survey of computer usage in schools in British Columbia.

A comparison of the actual products has been avoided for the following reasons: (1) there is a mind-boggling range of products on the market (one survey alone compared 95 mini-computers); (2) the best choice for a school district depends to a large extent, on its own needs, and any superficial study would be of dubious value; (3) there are extensive surveys available; (4) apart from the published surveys, individual manufacturers are reluctant to quote prices directly because of the variety of possible configurations and compe-tition.

Options The options available to a school district are: 1. obtain free time on a local computer facility if possible 2. rent computer time on a local system 3. rent computer time on a large timesharing system 4. purchase a programmable calculator 5. purchase a minicomputer 0

6. rent a minicomputer Microcomputers have been excluded from the list, but their significance is discussed later in the paper.

27

Option 1 - Free Time' The obvious advantage of this option is the low cost. You may need to rent a communication device, such as a card punch, for approximately $100 a month.

The disadvantages are that (a) the languages available are not designed for student use; (b) the student involvement is limited; (c) student programs have very low priority.

Option 2 - Rented time on a local system The advantages and disadvantages are basically the same as for Option 1. You will have the additional cost of the rental fees, however.

Option 3 - Rented time on a timesharing system It will be necessary to rent a device, such as a teletypewriter and possibly a card reader, to communicate with the system. This could cost about $80 a month plus installation fees. Computer time unit cost varies with the parti-cular system, and the overallcost depends on the time used. In addition, a telephone will be needed for data transmission.

The advantages of this option are: (a) greater student involvement because of having a terminal in the classroom; (b) rapid turnaround time compared to Options 1 and 2; (c) a greater range in the languages available; (d) system support programs; (e) the capability to store student programs; (f) essentially no program size restrictions (within reason).

The disadvantages are: (a) the problem of scheduling time for each student on the available terminal(s); (b)the increased cost over Options 1 and 2.

Options 4, 5, 6 (An Overview) Options 4, 5 and 6 involve obtaining your own computing device, either through purchasing or renting.

The advantage is that you can get a system to suit your needs. Systems vary greatly in memory capacity, speed, cost, software support, available peri-pherals and expandability. The choice of device and the decision to pur-chase or rent will require careful -analysis of your own needs, as well as the available products within your price range.

The disadvantages include: (a) program size restrictions based on memory size; (b) the responsibility of keeping the system operative; (c) the expense and (d) the time required for the initial planning.

'The discussion of options 1, 2 and 3 is really a summary of Démpstér's comments. This paper concentrates on options 4, 5 and 6.

continued on page 37

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Option 4 - Programmable calculator The characteristics of the calculators being considered are: (1) they are pro-grammable; (2) they are interactive, and they offer a wide range of peripheral devices. Calculator programming languages are of two types; keyboard and algebraic.

A keyboard language calculator has the following characteristics: (1) there is one key for each complete operation; (2) they have 12-digit precision; (3) they can store approximately 100, 12-digit numbers; (4) they can store from 500 to 3000 program steps.

The advantages of this type are that they are very easy to operate and, conse-quently, easy to program. They may also be used for a wide range of calcula-tions. The disadvantage is that the assembler-style language is cumbersome for complicated problems.

In contrast, algebraic languages are similar to well-known computer langu-ages. You can enter expressions that can be executed immediately or stored. In the case of erroneous expressions, error diagnostics are provided. Algebraic-language calculators are most like a small computer. They sell for less than $6,000, can store approximately 200 numbers and allow 1400 characters of program .2

Options 5 and 6 - Minicomputers. A computer will be considered a minicomputer if (1) it is usable with 4096 words of memory for less than $10,000 and (2) if it can be used for a variety of applications (i.e., not dedicated). The trends in minicomputers for 1971 to 1974 have included: (1) lower cost; (2) higher system performance; (3) in-creased use of microprogramming techniques; (4) increased use of bus struc-tures that allow interface with more peripherals, such as terminals; (5) in-creasingly complex software, and (6) the use of semiconductor memories and microprocessors.

The cost of minicomputers has decreased at a rate of 20 percent per year since 1971, but is now generally leveling off. The bottom price will likely be in the $2,000 - $3,000 range. Internal memory is faster and cheaper because of integrated semiconductor memories. Microprocessor chips are used to a large degree in the CPU. This has led to the use of microprogramming tech- niques, which, .together with bus structures, have led to greater flexibility and usefulness. Powerful macro-level languages can be tailor-.made. The bus structures have allowed for more on-line terminals. There is a wider range of lower cost auxiliary storage devices such as tape cassettes, single or dual plat-ter discs, floppy discs and diskettes. Manufacturers. have also concentrated

2 Asmus, P., 'Calculators vs Minis,' Datamation, April 1972, p. 55

37

on more powerful and flexible system software. .Most systems provide con-versational BASIC and batch Fortran. Other compilers may be available. The operating system is a multiterminal time-sharing executive system.

Choosing between a Calculator and Minicomputer Clearly programmable calculators overlap with the lower end of the mini-computer spectrum. The , difference is a question of dedication -calculators are dedicated to calculating and problem-solving, while computers are cap-able of any data-processing task.

The advantages of a calculator are: (1) they are self-contained, and conse-quently there is no problem concerning the configuration; (2) they have a brief set of operating and programming instructions, so they are easy to op-erate and program; (3) they are easy to implement, and (4) they are relatively powerful. The disadvantages are: (1) the limited amount of storage restricts the program size; (2) they have a slower data rate (ranging from a few micro-seconds to several hundred milliseconds); (3) they have a specific set of peripherals; (4) it is not possible to manipulate character'strings; (5) they only accept decimal numbers.

The choice between a calculator and a minicomputer depends on the need for generality, the user's knowledge of computers and hardware, and the cost. Generality implies knowing more about the device. Following are some guidelines for choosing between a calculator and minicomputer.3

1. If the calculation is standardized, a dedicated mini with specialized 1/0 can be cheaper and more efficient. (Although dedicated to calculating, a calculator is really general. purpose within this domain.) 2. For a moderate to light amount of programming by a few with limited background, a keyboard style' calculator is the best choice. 3. For a heavy amount of calculating and programming, choose an alge-braic style calculator. 4. If any one of the following factors is important, select a minicomputer. a. memory size in excess of limited calculator storage b. availability of a specific programming language c. ability to put any number into system . d. ability to handle numeric and non-numeric input e. ability to interface with any type of peripheral f. ability to alter the configuration (i.e., expand ibility and flexibility) g. throughput speed

It is not straightforward to compare the number of registers (memory size) of calculators and minicomputers. For keyboard language calculators, stor-

3

38

age for 1500 program steps and 100 data items corresponds roughly to .1000, 16-bit .words. For algebraic language calculators, 400 registers of storage cor-responds to approximately 1600, 16-bit words.

In evaluating calculators, one is interested in how much calculation is per-formed by each program step as well as the functions available. In April 1972, the cost of a calculator was generally lower than the cost of a mini-computer. For example, a keyboard calculator with storage for 300 pro-gram steps and 50 data items cost less than $3,000. (Algebraic calculators were more expensive.) Since then, the price of minicomputers has decreased so that now there is wide range of products in the $3,000 to $9,000 range. If you are faced with the decision of what computing device to obtain, it would be well worth-while to investigate the programmable calculator mar-ket to see if there has been a corresponding trend.

Choosing a minicomputer 'Buying in today's rapidly expanding minicomputer market poses a consider-able challenge. There is a proliferation of hardware and software, and myriad specs and performance capabilities are offered by suppliers.' 4 The correct choice will require a thorough knowledge of your needs followed by a care-ful analysis of the computing devices within your price range. Once your needs have been determined, the following steps should be taken.

Step 1 Obtain a current survey periodically. Data processing magazines publish sur-veys of the minicomputer market. (Developing your own would be a time-consuming task.) These provide a good overview of the products available. Two such surveys are Hobbs and McLaughlin, 'Minicomputing Survey,' Datamation, July 1974, p. 50 and 'How To Keep Pace with Minicomputer

Innovation,' Survey 1975, Canadian Datasystems, July 1975, p. 37.

The first survey compares 44 machines from 23 manufacturers. The memory capacities range from 4k to 128k in steps of 4, 8 and 16k. The languages available are conversational BASIC, Fortran, and ALGOL. The cost for a 4k system is in the range $3,000 to $9,000.

The second survey compares 97 computers from 31 manufacturers. The prices range from $3,500 to $500,000, thus accounting for the larger num-ber. Nevertheless, there are many in the $3,500 to $30,000 range. (The $3,500 machine is an Interdata 7/16 with 8k memory and Fortran, BASIC, and Assembler.) . . .

Many factors are considered under the general headingsof memory, pro-

4 'How To Keep Pace with Minicomputing Innovations,' Survey, '75 Cana-dian Datasystems, July 1975, p. 37

39

cessor, registers, arithmetic functions, available peripherals, available soft-ware, and pricing.

Step . Seek Advice. The second step is to consult with someone knowledgeable in purchasing computer hardware. He/she can provide help in deciphering the specifications presented in the surveys as well as warn you of possible pro-blems.

Step Contact manufacturers. Once you have restricted your search, you should contact the vendors involved. They will demonstrate their products and pre-sent the system that best matches your needs, then negotiate a price. Re-member, the field is very competitive.

Step Check reputation. Try to obtain the comments of someone who has exper-ience with the proposed system. You should take into account the reli-ability of the equipment and the availability of service. Each year some manufacturers drop out of the market, while others join. Therefore, repu-tation and stability will be a concern.

Choosing between renting and owning Assuming that a product and configuration has been selected, the decision must be made to rent from the manufacturer, lease from a third, party, or purchase. Purchasing can be done directly or through an installment plan. The difference is the interest cost paid to the manufacturer.

D.H. Brandon has written an article, 'Computer Acquisition Method Analy-sis,' 5 in. which he presents the major factors involved, the principal cost factors under each scheme, and two methods of comparing the schemes. The following is a summary of that article.

The major factors in the decision include economic considerations, financial leverage (i.e., withholding payments to ensure prompt service) and obsele-scence. Obsolescence is subdivided into physical, technical and economic components.

The major concern is the economic factor. Brandon analyzes it in two ways - the breakeven method (the time value of expense is not included) and the discounted cash flow method. Under the breakeven method, he predicts the breakeven point of purchase over rental as approximately three years, while

Brandon, D.H., 'Computer Acquisition Method Analysis,' Datamation September 1972, p. 76

40

the breakeven point of purchase over lease is about six years. By the dis-counted cash flow method, the first breakeven point is five years, and the second breakeven point is seven and one-half years.

With regard to physical obsolescence, machines are designed to last 12-16 years, which far exceeds the breakeven points. A manufacturer plans on covering the cost of a machine in five years (the monthly , rental is about 1/50 the purchase price).

In conclusion, Brandon suggests purchasing is preferable to leasing, while leasing is preferable to renting from the manufacturer.

Role of Microprocessors and Microcomputers No larger than 1 % inch square, they contain all the essential elements of a central processor, including the control logic, instruction decoding, and arithmetic processing circuitry.. To be useful, the microprocessor chip or chips are combined with memory and I/O integrated circuit chips to form a 'microcomputer,' a machine almost as powerful as minicomputer which usually fills no more than a single printed circuit board and sells for less than $1,000.6

The above is a very impressive description. Microcomputers are extremely compact, inexpensive and highly reliable. Are these an option'for school dis-tricts?

At present, microcomputers are very much in the design stage. They are re-stricted currently by, lack of software support and incomplete sets of inter-face chips. They use assembly language predominantly and are primarily for dedicated applications.

Rather than compete with minicomputers, microprocessors and large-scale integrated standardized products will be used increasingly in their design. Indeed, the increase in performance and the decrease in minicomputer cost have been due largely to the advance in semiconductor technology. 'Engi-neers are becoming familiar with this new technology and are applying it with increasing confidence in new hardward.'7

Survey of Computer Science Programs in British Columbia The purpose of the survey is twofold; first, to aid future computer science teachers in locating districts with programs and, second, to enable districts

6 Theis, D.J., 'Microprocessor and Microcomputer Survey,' Datamation, December 1974, p. 90 'Microprocessors: Getting Ready for a Larger Role in Data Processing,'

Canadian Datasystems, July 1975, p. 24

-

41

TABLE A Summary of B.C. School Districts with Computer Science Program (1976)

School No. of District Location Schools Computers Status Comments

11 Trail 1 proposed 1976/77 2. 24 Kamloops 1 IBM 370/155 time

rented 3. 34 Abbotsford 1 HP 9830A owned 1st yr. of program

will be expanded 4. 35 Langley 1 HP 9830A approved in process of pur-

chasing computer 5. 36 Surrey 5 HP 2000F time

rented 6. 37 Delta 3 PDP 8F owned

PDP8E owned PDP8E owned

7. 38 Richmond 2 HP 2000E leased 1st yr. of program will be expended

8. 39 Vancouver 16 HP 2000F owned 9.* 40 New Westminster considering for 76/77

10. 41 Burnaby 1 IBM 370/155 time rented

11. 43 Coquitlam 1 Wang owned part of Bus. Ed. Dept. 12. 44 North Vancouver 2 HP 9830A leased 13. 45 West Vancouver 2 HP 2115A owned

HP 9138 owned 14. 52 Prince Rupert 1 Hewlett owned

Packard 15. 61 Greater Victoria 9 IBM 370/145 time

rented PDP 1140 time

rented IBM 370/155 time

rented 16. 62 Sooke 2 time

rented 17. 63 Saanich 1 IBM 370/145 time

rented 18 . * 65 Cowichan considering for 76/77

80 Kitimat 2 Digital Classic 8 owned proposed for 76/77 20.** 85 Vancouver Is. N. considering for 76/77 21. 88 Terrace 4 PDP8E owned 22. 89 Shuswap 1 HP 9830 owned

*considering for 1976/77

proposed for 1976/77

42

contemplating a program to benefit from the experiences of others.

Of the 75 school districts in B.C., 16 have a computer program: two will be starting a program in September 1976, three are considering a program for 1976/77 and one has a small computer associated with the business educa-tion department (see Table A).

Table B indicates the various policies used for obtaining computing facilities.

TABLE B Computer Policies Employed

Policy No. of Districts

computer owned 13 computer leased 3 time rented 8

Of the 24 computers utilized, more than half are owned. Renting time is-the next popular option, while leasing is third.

Table C indicates the use of peripheral devices.

TABLE C Peripheral Devices Used

Device Available . No. of Devices

interactive terminal . 16 keypunch . 5 card reader 10 printer 9 deck tape 1 paper tape reader and punch 1

The results indicate that, an interactive terminal and card reader are perfectly adequate for conducting a computer course. School District 89 (Salmon Arm) reported that 'the interactive terminal along with the card reader has been satisfactory for instruction in classes of up to 24 students at one time.' The small number of keypunches used as compared to card readers implies that mark sense cards are employed extensively.

It should be pointed out that the peripheral devices are not always located in the school, although this is often the case.

Table D indicates the programming languages available.

43

TABLE D Lanauaoes Available

Language No. of Districts BASIC 13 - FORTRAN 9 ASSEMBLER 4 PLI 2 ALGOL 2 FOCAL 2 SNOBOL 1 COBOL 1 APL -. 1 MACHINE 1 TURING 1

BASIC is almost universally used. Fortran is very popular,. with-an appreci-able number of districts providing ASSEMBLER. Although a district has ac-cess to a language, this does not necessarily imply it is being used in com-puter instruction. If time is rented on a large system, many languages may be available.

Summary

There are several options available to a school district seeking computer facilities: 1. obtain free time on a local facility where possible 2. rent computer time on a local system 3. rent computer time on a large timesharing system 4. purchase a programmable calculator 5. purchase a minicomputer 6. rent a minicomputer

The choice made depends largely on the needs of a district and the funds available. This article has attempted to summarize the issues involved. The advantages and disadvantages of each option have been presented. Generally, the higher the number of the option, the more costly it is, but the closer it will fit your needs.

In selecting a minicomputer, you must first determine your needs. The next step is to obtain a current survey of the available products.

In comparing purchasing to renting, purchasing will be cheaper over the long term, but requires added responsibility. The breakeven point is approxi-mately five years for purchasing over renting and approximately seven and one-half years for purchasing over third-party leasing.

44

The significance of microprocessor technology is the impact it is having on the performance and cost of minicomputers.

The paper concludes with the results of a survey of computer science pro-grams in British Columbia. Of the 75 districts, 16 have a computer science program while five more are considering implementing one in the near future. Consequently, 25 percent of the school districts are involved in computer science education.

Glossary

Central Processing Unit, CPU, Central Processor The unit in any digital computer system that co-ordinates and controls the activities of all the other units and performs the logical and arithme-tic processes to be applied to data.

Hardware Physical equipment as opposed to a program (i.e., software) or a method of use: for example, card readers, line printers, card punches, magnetic tape drives, central processing units, input/output channels etc.

Input/Output, I/O A term used to describe anything (e.g., equipment, data, programs, etc.) involved in communication with a computer.

k In computer terminology, the symbol 'k' usually represents two to the tenth power; i.e., 1k = 2' ° = 1024

Peripheral Unit A machine that can be operated under computer control. Peripheral equipment includes input devices (e.g., card readers, magnetic tape), output devices (e.g., hue printers, plotters), and storage devices (e.g., disks, data cells, drums).

Software Programs, as opposed to hardware.

Time sharing A method of operating a computer system so that the time of the central processor is shared among a number of users by giving short bursts of activity to each in turn. In suitable circumstances, each user can operate as if the whole system were dedicated to him/her.

45

Bibliography

Asmus, P., 'Calculators vs Minis,' Datamation, April 1972, P. 55

Berger, A. and Leigh, J.J. (ed.), UBC Glossary, Computing Center, Univer-sity of British Columbia, December 1973.

Brandon, D.H., 'Computer Acquisition Method Analysis,' Datamation, September 1972, p. 76

Hobbs and McLaughlin, 'Minicomputer Survey,' Datamation, July 1974, p.50

Sharp, D.E., 'Small Business Computers, Survey '75,' Canadian Datasystems, July 1975, p. 28

Theis, D.J., 'Microprocessor and Microcomputer Survey,' Datamation, December 1974, p. 90

'How To Keep Pace with Minicomputer Innovations, Survey '75,' Canadian Datasystems, July 1975, p. 37

'Microprocessors: Getting Ready for a Larger Role in Data Processing,' Canadian Datasystems, July 1975, p; 24

'WANG Seeks out 1st Time Users for New Systems,' Canadian Datasystems, May 1975

Dempster, David, 'Computing on a Shoestring' (see D. Robitaille, Faculty of Education, University of B.C.)

46

[DIDIIIIDIDIDIDIDIDIDIUIOIUIDIUIDIOIDIDIDIOIDIDIDIUIUIOIDIDIOIOIEIIU

More Problems forCom-0--o-ter Science Students

by Doug Inglis, Caledonia Senior Secondary School, Terrace, B.C. [UIDIDIOIDIDIUIDIDIOIDIDIOIOIOIDIDIUIDIDIDIEIIDIOIUIIIIUIDIDIOIDIDID [Editor's Note: These problems are taken from a longer paper entitled 'Mathematics and the Computer.' This is the third set of problems published from the paper.]

SEQUENCES, SERIES, LIMITS Infinite processes and their study represent one of the crowning achieve-ments of work in mathematics. Many of these ideas are particularly weII suited to computer investigation. The use of the computer helps to develop a detailed picture of finite sequences and provide an excellent background for the understanding of infinite processes. This section will provide guidance for this study of sequences.

Note: It is assumed that a student understands the meaning of sequence, terms, finite and infinite before beginning these exercises. It is also assumed that he/she knows how to program a recursive process and a formula process for generating the terms of a sequence.

Exercise 1: A famous mathematical problem from medieval Italy describes the population growth of a hypothetical group of rabbits. Starting in their second month of life, a pair of these rabbits can produce one new pair per month. If there is only one pair in the first month, the following pattern emerges:

Month No. of Pairs 1 I 1 2 I-.. 1 3 I II 2 4II1'i^

11I II 3

5 lV 5 6 V Iiii IV 8

VII V VIII

In this sequence, each term after the second is the sum of the two preceding terms. This gives us the recursive formula

a11 • a2 = 1 aa_1+an_2 when n> 2

Write your own computer program to calculate the first 20 terms of this sequence, known as the Fibonacci sequence.

47

Exercise 2: The Fibonacci sequence has an interesting property concerning the ratios of consecutive terms. Modify your program in order to calculate the ratios of consecutive terms, that is the ratio a n / a n....l; This ratio was known as the Golden Ratio to the ancient Greeks. What information can you find concerning the Golden Ratio?

Note: This simple extension of the Fibonacci sequence program will show students that you often have to 'fiddle' with a sequence to ascertain some of its properties. This can be very important in the development of a student's ability to analyze sequences.

Exercise 3: You have been offered two jobs. The first one pays $1,000 per day with a $1,000 raise per day. The second one pays 1 cent per day with the pay doubling each day. Since each job lasts only 25 days, you want to know which job will be paying you more on the day that you finish. To help you with. your decision, write a computer program to calculate the daily pay for each job.

Exercise 4: Probably a question of more interest to you is which job pro- vides the greater total wages. Modify your program to find this information.

Note: These two programs should give the student a good, intuitive under-standing of the difference between an arithmetic and geometric sequence. The formulas for each of these sequences could then be formally developed in the normal manner. To assist in the understanding of these formulas, the following exercise could then be assigned.

Exercise 4a: Write a program that will analyze the first four terms of a se-quence and print 'ARITHMETIC SEQUENCE. DIFFERENCE IS 'or 'GEOMETRIC SEQUENCE. RATIO IS 'or 'NEITHER ARITHMETIC NOR GEOMETRIC SEQUENCE.'

Exercise 5: For each of the following sequences, have the computer calculate and print the first 10 terms, then the 50th, 100th, 150th, . . . and 1000th terms. a. (3n+2) / (4n+1) b (3n+2) / (5n)

c. (5n2 +3) / (2n 2 +1) d. 1/n

Each of these sequences is said to be 'convergent' or to have a 'limit.' In other words, as n gets very large, the values of the terms get very close to a specific value. The previous sequences for which you wrote programs are not convergent, because they never get close to any number; the values just get larger and larger. .

48

Exercise 6: A frog is trying to jump the length of a 2-foot log. He success-fully jumps 1 foot, 1/2 foot, 1/4 foot, 1/8 foot and so on, each time cover-ing half the remaining distance. Will he ever reach the end of the log? The frog's jumps can be described by an 'infinite series,' which is simply the in-dicated sum of the terms in the sequence

1 + 112 + 1/4 + 1/8 +... Since the sequence involved is geometric, the series is an 'infinite geometric series.' Write a computer program to try to evaluate the sum of this infinite series. Use the geámetric series formula

S1 ' (1_(1/2)n)

1.-1/2 and evaluate it for n = 10, 20, 30, . . . 100.

Exercise 7: In attempting to evaluate an infinite geometric series, the value of the term r in the formula is very important. Why? Write a computer pro-gram to evaluate r'1 for the f011owing ratios of an infinite geometric series with a first term of 1, using n = 10, 20, . . ., 100.

r = (.5, .2 5, .9, 2, 1.5, 1.1,-.86, .98, 1, 1.01, —.5, —2, —.8, —.6, —1.1) When can a value be found for an infinite geometric series? Why?

Note: Whendeveloping theorems regarding infinite geometric series, students have difficulty realizing that lim r 0 when In < 1.

n+co, The preceding pair of programs will help students to learn this concept. They will also help in learning that all-the terms of a series are often not necessary in order to calculate the sum.

Supplementary Problems 1. Given a set of scores, write a program to calculate the mean and the standard deviation. 2. Find the median of a set of scores. 3. Given two sets of scores from thö same sample, write a computer pro-gram to determine if there is a relation between the two variables by calcu-lating the correlation coefficient. 4. Find the resultant of two vectors. 5. Write a program to plot the graph of the sine (or cosine) function. 6. Write a general function tabulator to write a table of values for any function of the form y = f(x) / g(x), given the first value of x, the change in x, and the final value of x. 7. Evaluate 11 . 1 + 1 +

1 1x2 1x2x3 1x2x3x4 for 1, 2, 3, . .., 10 terms.

8. Find the roots of a fifth degree equation with integer coefficients using the factor theorem. 9. Given the co-ordinates of n points on the Cartesian plane, find the equa-tion of the best linear approximation for these points.

49

[UIDIDIUIUIDIDIDh[U1[UIUIDIUIUIUIUIUIUIUIU][E1]WIflI[IIUIUIUIDIUID!UIUIO

The 1976 77 Mathematics Assessment

by Jerry Mussio, Acting Director, Learning Assessment Branch, Department of Education - [DIDIUIDIOIDIUIUIDIOIUIDIDIDIUIUIDIDIOIUIUIUIIIIIDIEIIOIDIUIUIUIUIDIU

Early this spring, the department announced a long range assessment plan for the province (see table). This plan, which covers a five-year period and in-cludes mathematics every second year, was prepared after two years of plan-ning and discussion with a number of groups in the province..

Why an assessment program? Or more to the point, why should tests be ad-ministered on a province-wide basis? The central principle underlying the provincial assessment program is that if. decisions are to be made about edu-cation, to be effective, they should be based on an understanding of what our youth are learning and what their needs are.

How is this information to be used? The assessment program is being imple-mented to gather information for the following purposes:

• To assist curriculum developers at the provincial and local levels in the process of improving curriculum and developing suitable resource materials.

• To provide directions for change in teacher education and professional development.

• To inform , the public of the strengths and weaknesses of the public school system. .

• To provide information that can be used in the allocation of resources at provincial and local levels. . .

• To provide directions for educational research.

(It is important to note that the program is not designed to report informa-tion on individual pupils. The smallest reporting unit will be a school sum-mary which will be provided on request.)

ASSESSMENT TIMETABLE' Smallest Publication

Content Area Grade Reporting Unit of Report 1975-76 Reading 4 district summer 1976 (Pilot Study) Writing 8,12 province

1976-77 Functional Skills 1 4, 8, 12 school2 summer 1977 (Reading, Math...)

Social Studies/ 4, 8, 12 province Citzenship

1977-78 Functional Skills 2 4,8, 12 province summer 1978 (Writing...)

Physical Sciences 4,8,12 school2

1978-79 Functional Skills 1 4, 8, 12 school2 summer 1979 (Reading, Math...)

Career and 8,12 province Occupational Development

1979.80 Functional Skills 2 A8,12 province summer 1980 (Writing...)

Recreational and 4,8, 12 province Health Education

'Note that Functional Skills (Reading, Mathematics, Writing,...) are assessed over a two-year cycle; if possible, other related skills such as speaking and listening will be assessed under the heading of Functional Skills. Content areas such as Social Studies, Physical Sciences or even the Fine Arts will be assessed over a five or six-year cycle and may not be assessed in conventional fashion (instruments other than paper and pencil tests may be used).

2 Optional reporting unit; results provided at district request. Provincial and district re-sults will automatically be made available to the board.

Organization of the 1976/77 Mathematics Assessment During the first phase of the study, which is now in progress, a contract team consisting of university faculty members and classroom teachers, has been involved in the generation or selection of possible goals and objectives and involved in the identification of sample test items. This work is being done in collaboration with a management committee consisting of teachers, uni-versity faculty members, a school trustee and a representative from the de-partment. The management committee was organized to provide overall guidance to the assessment and to serve as a sounding board for the contract team before a more extensive external review of any draft materials is con-ducted.

51

During the months of October/early November, we hope to meet math educators and members of the public to obtain feedback on the following questions:

- Are the goals/objectives proposed in the Math Assessment realistic and meaningful? Do they represent skills that all or most students should acquire?

- Do the sample test items do an adequate job in measuring the respective objectives?

In March of 1977, and following pre-testing of the instruments, we plan to administer a series of tests to Grades 4, 8 and 12 students. During this stage we hope to collect other useful sources of information. For example, we would like to continue the practice, which was started in the pilot study in the language arts, of asking teachers what they think of the books that are being prescribed - Are the books being used effectively? What are their strengths and weaknesses? In the pilot study, we found that there appear to be a number of textbooks not being used - this information will have impor-tant implications in terms of redirecting money to more suitable instruction-al resources or providing more in-service help in assisting teachers to use these materials more effectively. Information on university training could be collected at this time as well - information that will assist planners of teach-er education programs.

After all tests and questionnaires have been scored, and the goals and objec-tives reviewed, we plan to involve mathematics educators and members of the public in the interpretation of results. 'If 87% of the Grade 4 student population can successfully add two simple whole numbers, does this meet our expectations as professionals and members of the public?' We hope to obtain the answers to these types of questions from mathematics educators and members of the public. A report of the assessment results is planned for September 1977.

The success of the Mathematics Assessment will depend on the co-operation and direct involvement of mathematics educators throughout British Colum-bia. If you are interested in getting involved, please write to:

Jerry Mussio Assessment Branch

Department of Education Victoria, B.C.

52

[DIU1[D1[UIWI[D][U][U1[D1[U1[U1[U][U][U][D][Dh[D1[O][U][DI[U][D][U1[D1[U][U][U][DID1[U][U][U][U

Backward Qlance:

5th Mathematics Summer Workshop

by Doug Owens, University of British Columbia [U][fl][U][Dh[D][U1[U1[D1[U][D][D][DhIU][D1[U1[U1[U][U][D][U][D][D][D][][U][D1[D1[UIU][D][U]ID][D

It is rewarding to note that the mathematics summer workshop was again a com- plete success. Success is mea-sured in terms of good atten-dance, excellent sessions and general enthusiasm of the participants. Many partici-pants made a special effort to relate positive reactions to the workshop, and there were a remarkably small num-ber of negative comments.

There were 271 preregistrations and 106 on-site registrations. With 70 pro-gram participants, 10 committee members, and over 20 publishers and sup-pliers, attendance was about 475. This is down from over 500 of one year ago. The total of 377 registrations in 1976 compares to 430 registrations of 1975. We had predicted that the numbers would level off at some point. Since each year prior to this showed an increase over the previous year, it is not surprising that the number of registrations is down slightly.

With the Northwest Mathematics Conference being held in Victoria on October 29 and 30, we prediced that fewer Vancouver Island residents would attend the summer workshop. However, that was not the case. Considering incomplete data from 1975, registration decrease appears to be evenly spread throughout the province. Secondary teachers seemed to turn out in record numbers again this year. There was a slight decrease in percentages of ele-mentary teachers - especially from the kindergarten and intermediate grades.

The general session was opened with a welcome and remarks by Superinten-dent Rod Wickstrom of North Vancouver School District. Then Dean Eric MacPherson of the University of Manitoba's Faculty of Education gave an inspiring keynote address on curriculum development. He made predictions,

53

presented challenges and suggested strategies for making curriculum changes.

Following the general ses-sion, a panel composed of Dave Robitaille, Jim Sherrill,

11 SP u/' Heather Kelleher, John f...,t...1Lt. Klassen, Jerry Mussio and

fr'4' ob Aitken gave--in-forma -

tion and answered questions concerning the upcoming B.C. Mathematics Assess-ment. More than 100 per-sons from all levels attended the session.

Discussing...

1a ^'

Jo Routledge of Aurora, Ontario is the Canadian Regional Representative , V on the Committee on Affiliated Groups of the NCTM. We were happy to have Joan attend the workshop, and her sessions on developing basic skills in the primary grades were greatly appreciated. Further sessions on developing computational skills at the elementary and secondary levels were given by W.A. Gar.neau. Dave Robitaille informed participants of types of computa-tional errors to be expected of students in Grades 4 to 8.

Margaret Stroyan gave a presentation on informal testing in the primary classroom. The presentation was followed by a make-and-take workshop on the same topic. Bill Bober's make-and-take workshops at both primary and Displaying.... intermediate levels were

well-received. Bober's work -shops contained ideas and games.

Primary teachers looking - for good ideas turned out in

droves to sessions on activi -ties and math centers given by Ann Warrender and Sheila Donnelly, Lynn Mat-thews, Linda O'Reilly and Lorna Rankin. It seems that

primary teachers remain interested in ideas for using multiple texts as evi-denced by the number attending Ozan McSweeny's session on that topic. Diane Brow gave a session in primary and one in intermediate on supplement-ing Investigating School Mathematics to meet the needs of all children. Ses-sions on using Mathways as a resource in the primary grades were given by Barbara Colbert of Calgary.

54

Various approaches to numeration concepts were developed by Werner Liedtke for primary teachers. Kay McKinnon stressed the importance of

Listening... multiplication and division concepts as she presented various suggestions for devel-oping understanding and skills. Ralph Gardner gave a comprehensive session on teaching division of whole numbers at the intermediate level.

Since more geometry is in the elementary curriculum, we were fortunate to have

Betty Huff giving two sessions on geometry activities for primary children. At the intermediate level, Grace Dilley presented a wealth of ideas on teach-ing symmetry and related topics. Jill Glaridge shared her ideas for teaching primary children measurement using metric units, Walter Szetela gave an in-teresting session on graphical presentation of measurement data at the inter-mediate level.

Primary teachers are interested in using manipulative materials to develop concepts. Sessions on this topic by Jean Aston for primary grades and Alice Ross for kindergarten were enthusiastically recieved. Other Kindergarten-Grade 1 workshops were given by Belinda Putnam on her individualized math program and by Bev Nikiforuk and Judy MacDonald on beginning work with Cuisenaire. Joy Ruffki presented methods for using Cuisenaire rods to teach understanding of fractions in the intermediate grades.

Relaxing.... Ian Beattie presented a vari-ety of types and uses of games for primary grades. Sessions on games and activi-ties for the intermediate grades were given by Ray Melendez-Duke and Bill Biles. Workshops on a vari-ety of other topics were held for intermediate tea-chers. Peter Makeiv and Kenneth Woodcock present-ed sessions on probability and using a mini-unit approach to teaching mathematics, respectively. Ses-sions on using volunteers and on using competition for motivational pur-

55

poses were given by Dorthea Lock and Doug Forbes, respectively.

A new topic in the program this year is the use of hand-held calculators. Jim Vance gave a session on using calculators as an instructional aid at the inter-- mediate level. Allen Neufeld, co-author of Project Mathematics, included use of calculators in his sessions on computation and problem-solving at the intermediate level. Tom Howitz and Walter Szetela presented two workshops on using calculators at the junior and senior secondary levels.

In a two-hour session followed by a one-hour session, Gail Spitler discussed teaching for problem-solving at the secondary level. Jim Bourdon gave a ses-sion on problem-solving in intermediate grades.

Revision-related topics were the order of the day for secondary teachers. Bill Dale described an individualized approach using the modern algebra modules

• for Grades 9 and 10. The forum, an open discussion led by Hugh Elwood was a new kind of session. Three persons were prepared to speak briefly on how their schools had handled the Grades 9 and 10 revision over the last year. When the floor was opened, several other people volunteered to report on their successes and concerns with the new programs. We also learned that, because the new courses were on a permissive basis, and because texts were

• not available in September 1975, many schools had adopted a wait-and-see strategy and will be beginning the new courses this year. Later in the work-shop, Brian Tetlow gave a session on curriculum development in Mathematics 9 and 10 in Victoria schools. Neil Baer . related his experiences of giving courses in occupational and general mathematics and using resources like Career Mathematics.

Robert Peard described various approaches to integrating math with other subjects at Grade 8. Harold Brochmann gave some novel ideas on using maps in the mathematics classroom. Anita LoSasso's math lab sessions provided a source of ideas for activities for the junior secondary classroom.

Bill Kokoskin gave two sessions on the status of revision at the senior secon-dary level. He shared a wealth of information on availability, status, and ordering of textbooks as well as other pertinent details. Alan Taylor gave a repeat session in which he discussed plans to implement the new mathematcis 11-12 program and described his experience in piloting Using Advanced Algebra. Ray Mickelson outlined several locally-developed courses for Grade 12 and described the procedure for obtaining approval of locally-developed courses.

John Del Grande, co-author of Mathematics for a Modern World, gave a re-peat session on proving theorems using physics concepts. In a session en-titled 'Today's Mathematics for Tomorrow's World,' Betty Kennedy expressed

56

the need for building mathematical models of real-world applications. John Hazell convinced those in attendance that logarithms aren't deadwood, as he gave many applications and uses of logarithms other than for calculation. John Taylor showed a variety of applications and natural occurances of the Fibonacci numbers, which can provide considerable enrichment and motiva-tion at the secondary levek/ohn Trivett, Sandy Dawson, Jim McDowell and Barry MacFadden had a new approach to their workshops. They showed a film in joint session entitled '9 x 4 = 37' and then broke into small discussion groups of grade levels.

There was considerable interest in the sessions on using computers in the secondary school. Ron HarrQ gave his views on why and where computing fits into the school curriculum. James Nakamoto, Wayne Gatley, Dave Ellis -and Jack Schellenbe.!9 gave several enrichment topics that can be facilitated by computer. Participants in Ian deGroot's session learned programming in BASIC.

The Fifth Mathematics Summer Workshop would never have been a success without the organizational efforts of the Workshop Committee. Thanks go out to Joanne Shutek for her assistance in locating program persons for the elementary sessions. Les Humphries, in charge of publicity, got publicity out well in advance. Ken Silen, secretary-treasurer, could be depended upon to keep accurate minutes of planning committee meetings ., and he managed the finances admirably. Gary Phillips can be thanked for obtaining the freebees that went into making an attractive registration packet. Florine Carlson did an excellent job of organizing presiders for 90 workshop sessions. Since Linda Shortreid l . ikes getting mail, she did a superb job as registrar and in running the registration desk. Commercial exhibitors will join in expressing thanks to Heather Kelleher for so capabably managing the displays. Thanks go out to Bob Campbell, Diana Mumford and Grace Dilley for the displays in the Ideas Room. Ian deGroot and Ken . Mayson as site co-chairpersons did an admirable job of such details as moving furniture, making coffee, and arranging for audio-visual equipment as requested by each speaker. I would like to acknowledge the tremendous co-operation on the part of the Carson Graham School administration, teaching staff, and nonteaching staff.

As outgoing Summer Workshop chairperson, I express sincere thanks to those who presented workshop sessions, served on the Workshop Committee, or otherwise made a contribution to a successful endeavor. Many people gave a lot of time and energy to the task. I am confident that Polly Weinstein as chairperson of the Sixth Mathematics Summer Workshop will be as fortunate in locating those who are willing to help.

57

IN$ERVICE MONEY

by Dennis Hamaguchi

The Professional Development Advisory Committee of the BCTF adopted new guidelines that could have a tremendous impact on you and I, the class-room teacher. This dramatic breakthrough will allow many valuable in-service programs that are pertinent to an individual to blossom and provide professional growth.

The new in-service grants are earmarked to promote local school in-service programs. This will allow different schools in a district to participate dis-tinctly in a series of sequential workshops. Teachers will have the opportun-ity for professional input into their unique product.

The grant regulations read as follows: '14.C.20 (a) To become eligible for a BCTF grant, applicants must submit a plan for any proposed activity to the PDAC co-ordinator or the BCTF Professional Development officer BEFORE the program is held. The following details should be included: 1. explicit objectives for the project, 2. anticipated activities, 3. names of resource people, 4. budget breakdown. (b) Approval forms for use in submitting the foregoing, and planning guides, are available from the PDAC co-ordinator and/or the BCTF Professional Development office. Assistance yv-ith planning is available upon request. (c) After consideration of the approval form, a reply will be sent to the ap-plicant indicating whether or not the project will be eligible for a grant from the BCTF. (d) After the workshop, seminar or conference, a request for a grant (in terms of the actual revenue and expenses) should be made on the application form designed for this purpose.'

Executive Committee Minutes, May 14 and 15, 1976

58

MATHEMATICS

3rd Annual Conference Wednesday to Friday

June 8-10, 1977 Approx. Cost: $75

Distribution: This announcement has been distributed across Canada to educational jurisdictions at all levels

PROGI

^SPE^CIAL^^ EDUCATION

(EDEXS) jointly with the Faculty of Education

AMS FOR EDUCATORS

NREADING

Couse for Leaden Monday, Tuesday, February 14-15,1977 Approx. Cost: $55

Course for Leaden

Wednesday to Friday, May 11-13, 1977 Approx. Cost: $75

Symposia Series Fridays: October/76 to June/77 Fees: $50 (series of 8)

$8 (per symposium)

10th Annual Conference Wednesday to Friday, February 16-18,1977 Approx. Cost: $85

Symposia Series Fridays: October/76 to May/77 Fees: $40 (series of 6)

$8 (per symposium)

For further information and/or brochure/ application forms, phone, write or visit:

The Centre for Continuing Education York University 4700 Keele Street Downsview, Ontario M3J 2R6 (416)667-2502

PSA76-89 59

published by B.C. Teachers' Federation