1976 – 18(1)

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
7063 Jubilee Avenue
Burnaby, B.C. V5J 4B4
434-6315 (home)
936-7205 (school)
John C. Epp
1612 Wilmot Place
Victoria, B.C. V813 5S4
592-2388 (home)
478-5548 (school)
VICE-PRESIDENT
TREASURER
William A. Dale
1150- 17th Street
Courtenay, B.C. V9N 1Z7
338-5159 (home)
Grace Dilley
2210 Dauphin Place
Burnaby B.C. V513 4G9
299-9680 (home)
596-0357 (school)
RECORDING SECRETARY
PUBLICATIONS CHAIRPERSON
Ian C. deGroot
3852 Calder Avenue
North Vancouver, B.C. V7N 3S3
980-6877 (home)
987-7178 (school)
PRIMARY REPRESENTATIVE
Susan J. Haberger
1390 Willow Way
Coquitlam, B.C. V3J 5M3
936-7205 (school)
INTERMEDIATE REPRESENTATIVE
Linda Shortreid
4651 - 202nd Street
Langley, B.C. V3A 5.12
530-4665 (home)
588-5918 (school)
Ken Thompson
9352 - 119th Street
Delta, B.C.V4C 6M6
433-5703 (home)
IN-SERVICE SPECIALIST
CURRICULUM CONSULTANT
Dennis Hamaguchi
3807 - 22nd Avenue
Vernon, B.C. V 1 T 1H7
542-8698 (home)
542-3361 (school)
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
Faculty of Education, UBC
2075 Wesbrook Place
Vancouver, B.C. V6T 1W5
325-0692 (home)
228-5203 (UBC)
Dr. Pauline Weinstein
Faculty of Education, UBC
2075 Wesbrook Place
Vancouver, B.C. V6T 1W5
261-6803 (home)
NORTHWEST NCTM CONFERENCE ORGANIZER
Trevor Calkins
1623 Amphion Street
Victoria, B.C. V813 4Z5
592-4463 (home)
592-1205 (school)
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Inside This Issue
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5 President's Message
7
. John Epp
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
3
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President's Message
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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 Mathematics (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 mathematics 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 classroom teachers by offering math conferences and workshops at various locations 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 provide 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 surrounding Prince George will have been contacted and sent more information.
5
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 implementing 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.
7
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 ff erent 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 detailed 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 Mathematical Sciences, 2100 Pennsylvania Avenue, N.W., Suite 832, Washington, 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, calculators, 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 suggestions 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
• l3g O Willow Way
Coquitlam, B.C. V3J 5M3
9
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 Foundationsupported project, 'Electronic Hand Calculators: The Implications
for Pre-College Education,' is now available from the ERIC Information 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.
10
ELEMENTARY TEACHINQ
Learning Difficultiesin Math:
Part I I
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by Werner Liedtke, University of Victoria
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John is six and Mark is seven. In their mathematics classes, they are considered 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 suggested in Robitaille's article. He states that mathematics is 'developmental.'
New skills, ideas and concepts are, more often than not, based on previously 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 learning. 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 interview 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 interview, it is best to present the tasks in descending order of difficulty. From
our experience with young children, there seems to be no particular disadvantage when that procedure is reversed. Sometimes it may be advantageous
or just as efficient to begin an interview somewhere in the middle of a particular 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 manipulative 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, relationship.
Other common mistakes young children like John and Mark seem to make
include the following examples:
Sketch 1
o
00
000
0
00
000
O
0000
00
00
00
00
0 00
00
Sketch 2
00000.
000000
Sketch 3
2305 6
2 4 0 8 10
1230 23
1307 9
12
- They fail to realize that a number
can be represented in many different
ways. According to them, there is
nothing the same about the arrangements shown in Sketch 1.
- They find it difficult to make use
of given information. Attempts to
continue a pattern or to identify a
hidden member of an ordered sequence are unsuccessful.
- They are unable to relate symbols
or equations to their experience.
Statements like 5 + 3 and 2 x 4 may
be read correctly, but attempts to
simulate the actions associated with
the statements and attempts to identify the statements with something
from their environment seem to be
rather difficult, if not impossible,
tasks.
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 instructional 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.
13
Performance Testing of Fraction
by James H. Vance, University of Victoria Concepts
and Edward W. Richmond, Queen Charlotte, B.C.
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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 attention 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 in volvement of the subject in a problem 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 consideration.
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 provide 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 indicates 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
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 followed 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).
rIIrA
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).
Q
15
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 addition, subtraction, multiplication and division of fractions - topics that
had not yet been taught. The students were required to determine and demonstrate 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
(9 items)
Concrete Transfer
(8 items)
Initial
Retention
Initial
Retention.
2.1
3.1
2.6
3.9
2.3
3.5
5.5
6.0
5.1.
5.5
5.3
5.7
None of the differences between group mean scores was found to bestatistically significant. (A 38-item multiple choice paper-and-pencil test also failed
to reveal any significant differences in achievement between the two treat16
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 concepts to the physical materials used in the performance tests.
The data indicate that the Concrete Achievement Test was perhaps too difficult 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 students were able to use the materials provided to improvise, solutions to addition, 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 investigators with some new insights into how students are able to apply introductory concepts of fractions to new problems with physical settings. While
there is obviously a great deal to learn about constructing and efficiently administering performance tests of achievement in mathematics, such instruments appear to have potentially great value both in formative and summative 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.
17
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 conducting various training programs for the public and provincial, municipal, and
industrial employees.
Eugene Gosh is responsible for the administration of the program. The budget estimates from the Provincial Secretary's office have been transferred to the Department of Education.
Gosh reports directly to the Deputy Minister's Office through J. Phiffipson, Associate Deputy Minister, Schools.
J.L. Canty
Superintendent
Administrative Services
MATHEMATICS
For general information on the revised senior secondary mathematics program, administrators 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; Con18.
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):
E
a. Del Grande, et a!: Mathematics For A Modern World (Gage)
E
b. Travers: Using Advanced Algebra (Doubleday)
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):
E
a. Del Grande, et a!: Mathematics For A Modern World (Gage)
b. Travers: Using Advanced Algebra (Doubleday)
E
C. *Crosswhite, et a!: Pre Calculus Mathematics (Merrill)
*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 decisions re metric standards have been made for the industries represented, prescriptions will be reviewed.
19
SECONDARY TEACHIN
More Activities with the Colden
by R.F. Peard, Windsor Secondary School, North Vancouve r
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 calculate to four significant digits. The class results are summarized and a histogram drawn.
Ratio
1.000 - 1.500
1.501 - 1.750
1.750 - 2.000
2.001 —2.250
2.251 +
I
Number in this ranc
For most classes, the model interval is 1.501 - 1.750, and pupilsthen calculate 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 figures). 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,
20
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
1
n1
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.
21
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 instruction is one that will adequately meet the needs of each student.
Since there are basically two types of instruction, we, as mathematics educators, can screen our students by means of a questionnaire before they begin
their course work to determine their preference for either individual or traditional 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 student who has changed his mind once he has chosen a particular type of instruction.
Questions 21, 22 and 23 in the questionnaire are research questions to determine 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 individualized 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 different 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 individualized 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 extensive 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 mathematics 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 mathematics 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, mathematics 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]
Metric Square
by D.W. McAdam, P.Eng.
contributed by Jane Srivastava
[IlIIDIDIOIUhEiIDIDIDIUIDIUIDIEIIUIOIDIEIIUIUIUIDIDIOIUIUIDIUIDIDIDIDID]
Reprinted from the B.C. Professional Engineer
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 indicate 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..
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1. SI base unit of length
8. Power unit in imperial system
9. SI base unit for luminous intensity
11. Area unit, equals 100 m 2 (not recommended for Canadian use)
13. Same as 10 down
15. SI prefix for 10
17. Force unit in imperial system, used
by civil engineers
SI prefix for 10
1000 kg
Surveying measure for 100 ft.
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
1.
2.
3.
4.
25
Across (continued)
Down (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)
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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COMPUTINQ
EflhtIIDIUIOIDIDIDIUIDIOIDIUIDIUIUIDIOIDIDlEDIDIDIDIDhIIIDIOIDIUIDIUID
Some fiord Facts about ,
Computer Hardware
by Chris Weber
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 entitled '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) obtaining 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 including 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 competition.
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
6. rent a minicomputer
Microcomputers have been excluded from the list, but their significance is
discussed later in the paper.
27
0
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 particular 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 peripherals and expandability. The choice of device and the decision to purchase 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
28
page 37
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Option 4 - Programmable calculator
The characteristics of the calculators being considered are: (1) they are programmable; (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, consequently, easy to program. They may also be used for a wide range of calculations. The disadvantage is that the assembler-style language is cumbersome
for complicated problems.
In contrast, algebraic languages are similar to well-known computer languages. 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) increased use of microprogramming techniques; (4) increased use of bus structures that allow interface with more peripherals, such as terminals; (5) increasingly 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 techniques, 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 platter 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 conversational 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 minicomputer spectrum. The , difference is a question of dedication -calculators
are dedicated to calculating and problem-solving, while computers are capable of any data-processing task.
The advantages of a calculator are: (1) they are self-contained, and consequently there is no problem concerning the configuration; (2) they have a
brief set of operating and programming instructions, so they are easy to operate 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 microseconds to several hundred milliseconds); (3) they have a specific set of peri
pherals; (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 algebraic 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, stor3
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 corresponds to approximately 1600, 16-bit words.
In evaluating calculators, one is interested in how much calculation is performed 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 minicomputer. For example, a keyboard calculator with storage for 300 program 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 market to see if there has been a corresponding trend.
Choosing a minicomputer
'Buying in today's rapidly expanding minicomputer market poses a considerable 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 careful 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 surveys of the minicomputer market. (Developing your own would be a timeconsuming 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 number. 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, pro4 'How To Keep Pace with Minicomputing Innovations,' Survey, '75 Canadian Datasystems, July 1975, p. 37
39
cessor, registers, arithmetic functions, available peripherals, available software, 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 problems.
Step
Contact manufacturers. Once you have restricted your search, you should
contact the vendors involved. They will demonstrate their products and present the system that best matches your needs, then negotiate a price. Remember, the field is very competitive.
Step
Check reputation. Try to obtain the comments of someone who has experience with the proposed system. You should take into account the reliability of the equipment and the availability of service. Each year some
manufacturers drop out of the market, while others join. Therefore, reputation 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 Analysis,' 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 obselescence. 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,'
September 1972, p. 76
40
Datamation
the breakeven point of purchase over lease is about six years. By the discounted 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 districts?
At present, microcomputers are very much in the design stage. They are re- stricted currently by, lack of software support and incomplete sets of interface 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. 'Engineers 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
District Location
11
No. of
Schools
Computers
Status
Comments
2.
24
Trail
Kamloops
1
1
IBM 370/155
time
rented
3.
34
Abbotsford
1
HP 9830A
owned
4.
35
Langley
1
HP 9830A
5.
36
Surrey
5
HP 2000F
approved in process of purchasing computer
time
rented
6.
37
Delta
3
PDP 8F
PDP8E
PDP8E
owned
owned
owned
7.
38
Richmond
2
HP 2000E
leased
8.
9.*
39
40
Vancouver
16
HP 2000F
owned
1
IBM 370/155
time
rented
10.
41
New Westminster
Burnaby
proposed 1976/77
11.
43
Coquitlam
44
North Vancouver
1
2
Wang
HP 9830A
owned
leased
13.
45
West Vancouver
2
owned
owned
14.
52
Prince Rupert
1
15.
61
Greater Victoria
9
HP 2115A
HP 9138
Hewlett
Packard
IBM 370/145
PDP 1140
IBM 370/155
time
rented
time
rented
time
rented
time
rented
time
rented
62
Sooke
2
17.
63
Saanich
1
IBM 370/145
18 . *
65
80
Cowichan
Kitimat
2
Digital Classic 8 owned
21.
85
88
Vancouver Is. N.
Terrace
4
PDP8E
owned
22.
89
Shuswap
1
HP 9830
owned
proposed for 1976/77
42
part of Bus. Ed. Dept.
owned
16.
*considering for 1976/77
1st yr. of program
will be expended
considering for 76/77
12.
20.**
1st yr. of program
will be expanded
considering for 76/77
proposed for 76/77
considering for 76/77
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 education department (see Table A).
Table B indicates the various policies used for obtaining computing facilities.
TABLE B
Computer Policies Employed
Policy
No. of Districts
13
3
8
computer owned
computer leased
time rented
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
interactive terminal
keypunch
card reader
printer
deck tape
paper tape reader and punch
.
No. of Devices
.
.
16
5
10
9
1
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
BASIC
FORTRAN
ASSEMBLER
PLI
ALGOL
FOCAL
SNOBOL
COBOL
APL
MACHINE
TURING
-.
No. of Districts
13
9
4
2
2
2
1
1
1
1
1
BASIC is almost universally used. Fortran is very popular,. with-an appreciable number of districts providing ASSEMBLER. Although a district has access to a language, this does not necessarily imply it is being used in computer 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 approximately 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 programs 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 arithmetic 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, University 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 achievements 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
I
1
I-..
2
II
I
3
II
4II1'i^ 11I
lV
5
VII
V Iiii IV
6
V
No. of Pairs
1
1
2
3
5
VIII
8
In this sequence, each term after the second is the sum of the two preceding
terms. This gives us the recursive formula
a11
•a 2 = 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 provides the greater total wages. Modify your program to find this information.
Note: These two programs should give the student a good, intuitive understanding 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 sequence 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. (5n 2 +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 successfully jumps 1 foot, 1/2 foot, 1/4 foot, 1/8 foot and so on, each time covering 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 indicated 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
1 ' (1_(1/2)n)
S
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 program 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
lim
r
0 when In < 1.
have difficulty realizing that
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 program to determine if there is a relation between the two variables by calculating 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 equation 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 includes mathematics every second year, was prepared after two years of planning and discussion with a number of groups in the province..
Why an assessment program? Or more to the point, why should tests be administered on a province-wide basis? The central principle underlying the
provincial assessment program is that if. decisions are to be made about education, 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 implemented 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 information on individual pupils. The smallest reporting unit will be a school summary which will be provided on request.)
ASSESSMENT TIMETABLE'
Smallest
Reporting Unit
Grade
Content Area
Publication
of Report
1975-76
(Pilot Study)
Reading
Writing
4
8,12
district
province
summer 1976
1976-77
Functional Skills 1
(Reading, Math...)
4, 8, 12
school2
summer 1977
Social Studies/
Citzenship
4, 8, 12
province
Functional Skills 2
(Writing...)
4,8, 12
province
Physical Sciences
4,8,12
school2
Functional Skills 1
(Reading, Math...)
4, 8, 12
school2
Career and
Occupational
Development
8,12
province
Functional Skills 2
(Writing...)
A8,12
province
Recreational and
Health Education
4,8, 12
province
1977-78
1978-79
1979.80
summer 1978
summer 1979
summer 1980
'Note that Functional Skills (Reading, Mathematics, Writing,...) are assessed over a twoyear 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 results 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, university faculty members, a school trustee and a representative from the department. 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 conducted.
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 important implications in terms of redirecting money to more suitable instructional 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 teacher education programs.
After all tests and questionnaires have been scored, and the goals and objectives 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 Columbia. 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 complete success. Success is measured in terms of good attendance, excellent sessions and
general enthusiasm of the
participants. Many participants made a special effort
to relate positive reactions to
the workshop, and there
were a remarkably small number of negative comments.
There were 271 preregistrations and 106 on-site registrations. With 70 program participants, 10 committee members, and over 20 publishers and suppliers, 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 elementary teachers - especially from the kindergarten and intermediate grades.
The general session was opened with a welcome and remarks by Superintendent 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.
Discussing...
Following the general session, a panel composed of
Dave Robitaille, Jim Sherrill,
SP u/' Heather Kelleher, John
Klassen, Jerry Mussio and
f...,t...1Lt.
fr'4' ob Aitken gave--in-forma tion and answered questions
concerning the upcoming
B.C. Mathematics Assessment. More than 100 persons from all levels attended
the session.
11
1,a
' ^
V
Jo Routledge of Aurora, Ontario is the Canadian Regional Representative 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 computational 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 Matthews, Linda O'Reilly and
Lorna Rankin. It seems that
primary teachers remain interested in ideas for using multiple texts as evidenced by the number attending Ozan McSweeny's session on that topic.
Diane Brow gave a session in primary and one in intermediate on supplementing Investigating School Mathematics to meet the needs of all children. Sessions 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 developing 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 teaching symmetry and related topics. Jill Glaridge shared her ideas for teaching
primary children measurement using metric units, Walter Szetela gave an interesting session on graphical presentation of measurement data at the intermediate 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 KindergartenGrade 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 variety of types and uses of
games for primary grades.
Sessions on games and activities for the intermediate
grades were given by Ray
Melendez-Duke and Bill
Biles. Workshops on a variety of other topics were
held for intermediate teachers. Peter Makeiv and
Kenneth Woodcock presented sessions on probability
and using a mini-unit approach to teaching mathematics, respectively. Sessions on using volunteers and on using competition for motivational pur55
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 intermediate 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 session 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 workshop, 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 secondary 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 repeat session on proving theorems using physics concepts. In a session entitled '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 motivation 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 classroom teacher. This dramatic breakthrough will allow many valuable inservice 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 distinctly in a series of sequential workshops. Teachers will have the opportunity 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 applicant 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
PROGI AMS FOR EDUCATORS
^SPE^
CIAL^^ NREADING
EDUCATION
Couse for Leaden
Monday, Tuesday, February 14-15,1977
Approx. Cost: $55
(EDEXS) jointly with the
Faculty of Education
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)
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
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)
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
59
published by
B.C. Teachers' Federation