1988 – 30(1)
of the British Columbia
Association of Mathematics Teachers
IN& I I
New from Nelson Canada
INTRO 11 MATHEMATICS
West Vancouver Secondary School
Intro 11 Mathematics, written expressly to meet the needs of British
Columbia Students taking Introductory Mathematics 11, will focus on the
Intended Learning Outcomes of the curriculum. Successful completion of
this course will enable students to be eligible for Mathematics 11.
Table of Contents
1. Rational Numbers
5. Geometry and Reasoning
6. Analytic Geometry
7. Linear Systems of Equations
9. Rational Expression
10. Quadratic Equations
Intro II Mathematics
Student Text 17.602341.0
F or The
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B.C. Association of Mathematics Teachers
1988-89 Executive Committee
President and Newsletter Editor
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THE B.C. ASSOCIATION OF MATHEMATICS TEACHERS
Membership may be obtained by writing to the B. C. Teachers' Federation 2235 Burrard Street
Vancouver, BC V6J 3H9
Rates for 1988-89 are as follows:
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Notice to Contributors
We invite contributions to Vector from all
members of the mathematics education
community in British Columbia. We will
give priority to suitable materials written
by B.C. authors. In some instances, we
may publish articles written by persons
outside the province if the material is. of
particular interest in British Columbia.
Contributions may take the form of letters,
articles, book reviews; opinions, teaching
activities, and research' reports. We prefer
material to be . typewritten and doublespaced, with wide margins. Diagrams
should be camera-ready. We would appreciate a black-and-white photograph of each
author. If feasible, the photo should show
the author in a situation related to the content of the article. Authors should also
include a short statement indicating their
educational position and the name and
location of the institution in which they are
Notice to Advertisers
Vector, the official journal of the British.
Columbia Association of Mathematics
Teachers, is published four. times a year:
fall, winter, spring, and summer. Circulation is approximately 600, mainly in B.C.,
but it includes mathematics educators
Vector will accept advertising in a number
of different formats. Pre-folded 21.5 x 28
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as inserts at the time of mailing. Advertising printed in Vector may , be of various
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Quarter page: $ 40
Deadline for submitting advertising for the
spring and summer issues are February 24,
1989, and April 28, 1989, respectively.
Inside This Issue
From the Editors ................................................Tom O'Shea
6 Active Learning in Mathematics ................................John V. Trivett
14 Word Problems + Calculators = Motivation + Success........Katherine Willson
16 The Adams River Sockeye Run of 1986 .....................David R. Parkinson
18 Students An Incredible Resource (2)..............................Zoe Wakelin
20 MathTalk—An Electronic Conference ............................Stewart Lynch
23 The NCTM Standards Document
Mathematics and Computers
34 Functions as Models of Reality .................................Brian Copeland
45 Secondary School Calculus .................................Harold Brochmann
54 Report on the 27th Northwest Mathematics Conference
"Mathematics for the '90s .......................................... Ian deGroot
56 The SFU Master's Program in Secondary School
Mathematics Education ..........................................Tom O'Shea
From the Editors
ell, the 27th Northwest Conference
W has come and gone. For me, it was
one of the best we have hosted in British Columbia. Facilities were good, speakers were
excellent, and the number of people attending exceeded most expectations. Congratulations to the organizing committee for a fine
job. Again, we encourage presenters to put
their ideas on paper in the form of articles
Our apologies to Jim Vance, from the University of Victoria. We inadvertently left his
name off the article in the last issue of Vector
describing the Northwest Conference.
In this issue, John Trivett begins by considering what it means to be active in learning
mathematics. John has been interested in this
area for many years, and, as usual, his perceptions are insightful and well worth considering, especially as we implement the new
Katherine Willson shares her experience in
asking Grade 1 children to create a problemsolving class booklet. It's a nice example of
how the calculator can become a natural
part of the mathematics classroom. David
Parkinson follows and shows how mathematical procedures can be found in problems
related to the local environment. Zoe
Wakelin shows how having students create
their own problems in geometry can reveal
• lot about their thinking. A side benefit is
• collection of problems that the teacher can
use later for worksheets and quizzes.
Stewart Lynch steps out of his role as coeditor and explains the new electronic conference for mathematics teachers called
MathTalk. We're both on the system, and
we welcome suggestions for future editions
of Vector. We hope to publish excerpts from
the conference to show the benefits of
The NCTM Standards document is one of
the most important developments in mathematics education in North America. It will
probably mark a turning point for the
U.S.A. in developing a national consensus
on what mathematics should be taught in
U.S. public schools. Some of us have had
a chance to respond to an initial draft, but
many math teachers are unaware of what it
is and what it means. We are reprinting a
portion of it here to give it the widest possible circulation in B.C.
In the mathematics and computers section,
Brian Copeland continues a series of articles
on using the computer to teach mathematics.
Here he examines how the computer can be
used to help understand the behavior of
functions that are particularly useful in
modelling natural phenomena. Harold
Brochmann begins a series of three articles
on teaching calculus in secondary schools.
In this issue, he argues persuasively that the
first step should be to show what calculus
can do, rather than present .- formulas to show
how to do it.
In the final section, Ian deGroot reports his
perceptions of the 27th Northwest Mathe-
matics Conference. Finally, I take a great
deal of pleasure in announcing that Simon
Fraser University has developed a Master's
Program specifically designed for secondary
mathematics teachers. We plan to begin the
program in September 1989, so if you are
thinking about graduate work, now is
the time to get started. I welcome all enquiries by mail, telephone, or electronic
I read "Calculus in the Secondary School"
in the Spring '88 edition of Vector with great
interest. Thank you for an excellent treatment of .a complex topic.
I would like to add to the debate a dimension that is usually overlooked but is relevant to B.C. teachers today.
Teachers care about the success of their
students. Many teachers say to themselves,
"All the other schools areteaching calculus.
I know because that is what the students
who were in my class last year tell me.
Therefore, if I want my students to have a
fair chance, we had better introduce a
calculus course in our school."
It is easy to explain the source of this perception, and we could even show that it is not
valid. At least we could a year or two ago..
It went something like this: "All the students
who ask questi9ns in the calculus class seem
to have had some background in it. But only
those who know enough to ask questions
speak out. The instructor responds to those
who ask and assumes that the others know
more than that. Then my ex-student gets lost
and is convinced that it is because I didn't
teach him any calculus in high school."
I used to argue that the problem was not real
and the simple solution was to keep reminding the universities that calculus was not part
of the secondary school curriculum. Now it
seems that the reality is catching up to commonly held perception. Maybe it is too late
to argue whether calculus is 'better taught
here or there. We should probably agree on
a course, teach it well, and hope the postsecondary schools adjust their courses
Active Learning in Mathematics
John V. Trivett
John Trivett is a professor emeritus
in SFU's Faculty of Education.
ere is an introduction to some of the
H important aspects of active learning in
mathematics. They apply to all students at
any grade, whether they be deemed by some
to be slow, or fast, dull or bright. The examples quoted can be multiplied almost
without limit, and they have to be for adequate classroom implementation.
In a sense every student has to discover
everything. The teachers' mere explanations,
looks, or activities do not necessarily get a
reaction by a student, let alone a hoped-for
reception. The text may be handled and so
may physical aids, but there have to be continual acts of volition by every student to
trigger some kind of reception in the student's eyes or mind.
Varying levels of reception are needed for
understanding, however. That a fraction is
conventionally written in the form -- needs
little discovery. It is seen and almost immediately accepted by the youngest of
students in school. That it represents not
only a particular element but the whole of
a particular class of ordered pairs of whole
numbers needs more and more discovery
with a different quality and quantity of
what has to go on within the person to make
it as secure as the symbol seen with the
In mathematics teaching, some have traditionally acted as though simple surface-level
discovery is all that is needed. We explain,
we do examples, we show how, and we get
the children to practise on paper. We try to
define subtleties, and when we see the pupils
failing, we tend to conclude that it is because
some are not capable. We call them inattentive, lazy, slow, and by using more simple
words, we suggest that it is they who are at
fault. Seldom do we think that we teachers
have not yet attained the highest forms of
communication and activities to induce what
is needed to have all students learn the
If, therefore, discovery is a fact in learning
processes, not just the name of a method,
we surely have to enquire more into the
ways and means by which students are compelled to take responsibility for their-own
learning, for no one ever did anyone else's.
Teachers must minimize correction from an
apparent authoritative standpoint and insist
on being an authority only where the student cannot be. That can be condensed to
just two functions: orchestrating the activities of the group and telling the children
what are the agreed conventions of symbols
and form. In all other matters, in the widening consciousness of the fundamental facts,
the pupils must be helped to discover at first
hand, and although little attention has been
paid to this under the weight of verbal and
written tables and formulae to be learnt,
every man and woman knows the effect 'of
a general understanding into which many
details fit, compared with the knowledge of
many details with no connecting substratum
Teachers need to be learning facilitators,
creators of learning environments, flexible
ones, appropriate to the tasks, to the actual
students involved and capable of minute-tominute changes. As a result of wisdom and
knowledge, using the art and craft of the
trade, the teacher of mathematics in particular initiates activities such that 'anyone
engaged in them, can't help being affected
by the concentration , noflatent
matical ideas and cocepts built in. It is
arranged deliberately that the chances are
very high that no student can avoid the
inundation and, what's more, not enjoy
The environment will include the use of
physical aids, the writing on paper, the usual
symbol work, and texts, but it will also take
into consideration' the inner consciousness of
every person in the room. No system can be
ignored. Inner systems of humans must be
recognized as essential concomitants of the
learning environment: thoughts, wishes, day
dreams, mind wanderings, feelings. These
constitute the.stuff of existence as much as,
or more than, the manipulative aids, the
books, the diagrams, and the mathematics
we aim for.
It is the teacher's responsibility to
watch the learning process and to introduce creative conflict at appropriate
Students have to do their own learning, with
their own insights, past, and present as they
see it. These are the filters through which
pass all that occurs in our math lessons. We
had better be aware that this is so.
Each students is unique and may differ much
in his or her reception of something public
from all the other receptions of the in'dividuals in the group. It is impossible that
two people receive the same thoughts,' the
same understanding,no matter how the
teacher tries. Whereas one child happily accepts a sequence, for example:
log 20 = 1.3010
log 2 = 0.3010
log .2 = 1 + 0.3010
log .20 = 2 + 0.3010
another's difficulty may be that in the third
and fourth lines there are + signs, but in the
other lines there are not. The difficulty
evaporates as soon as he or she sees that one
could write the equivalents:
log 20 = 1 + .3010
log .2 = 1 + .3010
log .02 = 2 + .3010
Now there is a balance of the forms. If the
teacher does not understand a possible
obstacle here and cannot help, the student
may well feel he or she is just incapable of
understanding where apparently the others
Another example, from elementary school
Although child D is "right," it does not
follow that we know what the child did, or
how he or she thinks. Correct answers are
frequently written for the "wrong" reasons.
What is needed is a suspended judgment, not
a stamp of approval or disapproval. If we
can find out what each child really did (and
a good way of doing this is to ask and listento replies!), this is what might emerge:
A says "9 from 8, can't, so 8 from 9 is 1
("always do what you can")
6 from 4, can't, 4 from 6 is 2
B says "9 from 8, can't, so 8 from 9 is 1
16 from 24 is 8."
9 from 8 is 1, so 6 from 4 is 2
"9 from 48 is 39, put down 9 and carry 3
3 from 20 is 17, but 1 from 17 is 7
Put down 7."
That's the kind of tiny iceberg top of what
goes on within individuals. It occurs all the
time, in all lessons, with everybody. As the
understanding that this is so grows, together
with an increase in knowledge of more and
more possibilities within this vast, commonly unexplored field, every teacher can
open new vistas in his or her approach
to teaching. The consequences are greater
self-respect and successful learning for all
Individualism also is a fact, not a method,
not an opinion. We cannot choose to treat
humans non-individually and succeed. If an
"individual approach" appears to be only a
fad, in which it is assumed that organization
is at the bottom of individualism, and if this
passes next year to some new urge that has
all students reacting the same way, people
will still go on acting individually. Even if
they are overwhelmingly herded and made
to jump through the same hoops of authority, they still won't oblige except for a short
while. Humanity's eternal quest for freedom
will be maintained despite temporary
Children and adults like playing games.
Games are concentrated, challenging, enjoyable activities that can be given up at will;
whereas work is more unwilling, and more
pressure is accepted before it can be
Children often like to play games with
physical objects, and they have shown this
in their preschool experience. However,
games are not restricted to the use of
physical objects nor competition against
others. Word games, written games, and
games with oneself and with textbooks can
all be rewarding.
Some of the physical games are quite new
in mathematics lessons. Some teachers have
for years had children apply mathematical
and arithmetical principles to measuring,
shopping, and surveying, but nowadays
they also have for use many physical
materials that lead to awareness of mathematical principles by the very act of using
them and becoming more conscious of what
one is doing. Colored rods, attribute blocks,
multibase blocks, geometry models, geoboards, pebbles and counters,. mirrors,
cutout figures, films, one's own body and
body movements—all provide opportunities
to be taken by good teachers.
numbers, addition and multiplication, and
more of the familiar list of what we want
kids to know.
As an example, consider the use of the socalled attribute blocks, which are of different
shapes and different colors. A red triangular
block and a red square are alike because they
are both red, but they are different in shape.
When we use blocks of other shapes, other
colors, and use some with different thicknesses and even different materials, then we
have pieces for many games that demand in
the very acts of playing understandings of
differences, equivalences, intersections and
union, elements, and subsets. Such notions
are essential to a proper understanding of
By the use of games, improved mastery of
the traditional math and arithmetic can be
attained. Instead of laborious, non-related
facts that somehow have to be memorized,
the games approach involves thoroughness
and intensity, and it subtly demands every
student's ongoing dedication.
Games with physical objects are only one
type. There are many others, so here we illustrate one kind of game that seems quite
Continue this web. From any name, draw an arrow, and write another name for the
same number which comes to your mind. Have a reason why you went from one name
to the next.
7+7 - 2X7 - 1/2X4X7
(7-2)+9---(6-1)+9 - 5+9
14 - 10+4
5 X., 1 +3 X 3
Do not ally the word games with something
frivolous, a pastime, something to do "after
work." The most played games are very
demanding in energy, devotion, thought,
and time, and they increase learning and
powers. To try to meet mathematics as a
series of games is not to drag it down to
the wasteful, trivial filling of time. To
try to meet mathematics as a series of
games is to elevate certain mental activities
to the level of some of humanity's finest
activities: exploring space, serving others,
Mathematics is concerned with the dynamic
use, in the mind, of relations and relations
of relations, with some applications to social
and economic life.
Arithmetic is one of its branches having a
main emphasis on the renaming of numbers
by means of algorithms, tables, figures,
calculations, and other devices. To learn that
2 + 3=5 may only be a matter of remembering that "5" is a word that is an acceptable
substitute for the phrase "3+2," and "6" is
not the accepted word. To recognize, however, that 3+2=2+3 implies much more.
Now we get a hint of a basic principle (commutativity), and this may lead to other interests. Alternatively, it gives us a pattern
to get new names for numbers we have not
met before. (4576+687 surely must equal
687+4576 even if the former has not been
To write or say 2+3=5 is a convention, a
convenience. The symbols, as such, give no
clue to what they stand for. It is just as easy
to accept the writing 2 + 3 = 7, but historically that is not what was developed, so we
think that 2+3=5 is correct and find
2+3=7 uncomfortable. The meanings
behind the symbols, however, are not conventions. They are based on fact over which
we have no control. We can only be ignorant
of the fact. The fact is embedded in life, and
every one of us is capable of firsthand experience and enlightenment. We therefore
encourage mathematics partly to understand
and use the ideas, the concepts, the relations,
and the facts, and partly to help us
manoeuvre the old arithmetic that is still
vital and valuable.
On the whole, we do not question the kinds
of activity suggested by the use of mathematics textbooks so long as the teacher appreciates that the one printed form, the book
order, and the style of presentation is not
necessarily right for any student whatsoever.
Those mass-produced aids show the sophisticated outcome of years of thought and
centuries of argument and symbol evolution.
They must be seen as such, used for the
embodiment of desirable aims maybe, but
approached along different and unsophisticated paths. They rarely exhibit the form of
progress precisely needed by any child to
reach many of the same conclusions.
Usually in textbooks, the addition of
fractions is dealt with before the operation of multiplication.
One student pointed out that addition
-- + --
and multiplication is thus:
- 2X5 - 10
7 — 3X7
Addition, it seems, implies three multiplications and one addition. Multiplication needs only two multiplications.
On what grounds, he asked, is addition considered easier than multiplication and taught first? Had he a point?
If math is indeed concerned with relations
and activities with relations, then it is essential that students constantly have experience
that relates things for them. It is their meaning that is vitally important. Continuous
progress will be unique for each individual,
and the relations that are explored will not
by any means be confined to that portion
of the school day called the math lesson.
Relations are abstracted by all humans in all
kinds of situations. The math lesson will emphasize them and show what can be done
with them, and greater power in their use
will be gained. But the existence of relations
must be from all subjects studied, from all
life. Appreciation of integration across the
curriculum is therefore important.
Integration is also needed within the math
study itself so that growing experience is had
in relational activities that will become
second-nature and pervade all the educational scene.
Within the math study, such integrating,
creative aspects can be dealt with in many
ways. Here are two, briefly:
1. Reverse the usual task of getting an
answer to a problem. Instead, get a problem
to fit the answer, e.g., x=7. Write some
other equations of which x = 7 is the
E.g., The sum of four numbers is 2489.
What are the four numbers?
2. Rather than set word problems, give an
equation, and, from it, invent an appropriate social situation.
E.g., 3y+4=19. What situation in a
grocery store might fit this?
Across the curriculum, apart from some obvious use of number work and equations,
are rich, powerful mathematical ideas of
relations, differences, sets and subsets,
transformations, symbolisms, reversals and
repetitions to explore and use. All occur and
have their importance in social studies, art,
science, and music.
E.g., Grade 1
If children can read and write in some
form, the words "step up pat," they
can also read and write all six permütations of the words.
They can be asked to do so, with the
teacher expecting them to do it; They
can also be expected to read every one
of the six sentences and know whether
they make sense or not.
In discussing "humans," students can
be helped to list equivalences and differences of humans and see intersections of attributes. This leads to an improved awareness of racial similarities
and differences, all within the same
No mathematics lesson ever attended took
place without human beings being present.
Mathematics. may be devoid of feelings,
opinions, and other vagaries of everyday
life, but mathematics learning situations are
not, for the humans present have such attributes all the time. In the learning of the
subject, therefore, there are facts of this kind
to be taken into full consideration by the
teacher: the need for understanding, respect
for others, tolerance, frustration, tiredness,
hope, lack of communcation, annoyance,
satisfaction. Unless teachers are aware of
these as being present in every lesson, in
every human, then a large part of what is
really going on in the lesson will be completely missed, and the teachers will be
forced to resort to the traditional rationalization of "The students won't attend," or
"They are slow and incapable."
These are very real parts of everyone's life,
and we must certainly recognize their existence and the part they may be playing.
Because of such awareness and allowance,
we should help the students commune with
each other and with ourselves, since we shall
no longer depend on outward signs alone
that give little clues to the thinking and the
perceptions going on within.
Perhaps the future progress of all learning
depends solely on the harmonizing of what
has to be passed on to the new generation
within the context of the "species-specific
mode" of every learner human in this case.
It is fruitless to try to teach a dog to do what
a dog is incapable of doing. The same holds
for humans. We assume that telling is sufficient; whereas if we pause and think for a
moment, each of us has plenty of evidence
that telling is seldom sufficient for communication for anything other than
trivialities. Automatically learning by being
told is not a human mode. Humans do not
pay much attention to what others are saying. The mode surely suggests that we have
to encourage acitivities in which it does not
matter if the learners attend or not—except
to themselves, which they cannot escape.
E.g., if a Grade 5 pupil wants to add fractions thus:
-'+ -- - 7
let her and encourage her to do more like it.
When she had developed other names for
according to the same rules, she most probably will begin to sense a.contradiction with
other thoughts she has about fractions. That
contradiction within herself will provide
motivation and, at least. in this case, the
realization that that is not the way fractions
can be combined.
to act more and more asthough the learning is simple even if the organization is complex. A simple solution or a straightforward
outward appearance can invariably be
assumed to be misleading and wrong. There
have to be discovery and non-discovery;
rote learning will still play a role; there will
be partial understandings and superficial insights along a spectrum. There will not be
artificial discussion about group versus individual; for every group is one of individuals, and every individual is a member
of many-groups. The variation of possible
individual responses to every tiny situation,
the variation of movements, thoughts,
things to do, and the way for people to do
them—all contribute to a fantastic number
of alternatives that in practice, whether we
like it or not, make every lesson different,
no matter how hard we try to repeat or
Most of what goes on, most of the active
learning, will be unseen by the members of
the group. Each individual will have a concentrated view only of his/her own incoming sensations and inner thoughts. And on
top of all that, there will be the mathematics,
the activity with relations, numbers, space,
sets, operations, functions, and the arithmetic as a by-product of some of this. The
complexity is inevitably so great that it
makes one wonder how we ever got the
belief that we can determine in any but
rough form how some piece of learning
should take place... Why do teachers plan
lessons rather than prepare themselves to
meet, their students as they will be by the
time of the next lesson?
What we suggest, therefore, is a systems
approach, one that considers all the systems
active within the learning situation, outward
and inward. Much cannot be avoided, some
of it can be avoided, and some can be
initiated by . the teacher. Every learning,
situation is a complex set of circumstances;
whereas the history of education has tended
A balance is needed, not one that attempts
to remain still but one that is nevertheless
stable. Sometimes the of activity will seem
to be of.. one kind but, because of the
teacher's innate stability, and his or' her
understanding, purposes, and leadership,
there will be some vaguely perceived fulcrum
about which the kinds of activity will
oscillate. The mathematics classroom will
sometimes appear to be a hive of physical
activity. Children will be measuring,
weighing, or using blocks and ropes to
discuss sets. Others may be constructing
tetrahedrons and investigating their rotations. Computers may be in use, or some
pupils may be outside the school with angle
meters doing elementary surveying. Sets of
cubes, colored or plain, may be seen,
although the students using them may really be discovering the relations inherent in
successive cube numbers. All the fun of the
fair may seem the order of that day.
On other days, or with some children perhaps during the same time period, the
activity will be with paper and pencil, and
the appearance will be traditional.
The facts concerning the consciousness of the
different systems extant must lead to variety, for each learner, for the whole class, and
for the teacher too. Underlying all, however,
will be the commonality of an increasing
mathematical consciousness for each child
within a human community. This will result
from the effects of an environment purposely
engineered by the knowledgeable teacher in
which the learning takes place intensely, individually, integratively, and joyfully. That
is the "active learning in mathematics" to aim
Word Problems + Calculators =
Motivation + Success
Katherine Willson is an assistant professor in UBC's Department of Mathematics and Science Education.
aving children write their own word
H problems is a highly successful strategy
for problem solving. The process engages
pupils in a teaching method that éombines
reading, critical thinking, and the collection
and organization of data (Fennell &
Several years ago, toward the end of the
year, I informed my Grade 1 class that we
were going to create a problem-solving class
booklet, and each pupil would have a chance
to be an author. The children were highly
motivated, and we discussed the emphasis
on solving real problems found in the classroom, school, or home. Many teachers and
researchers are aware that students are
highly motivated when asked to solve problems that are real to them, as opposed to
contrived problems found in the textbooks.
Trading stickers and hockey cards were the
current rage, and many children were enthusiastic about writing problems that dealt
The pupils understood the meaning of addition and subtraction and had solved many
word problems throughout the year. They
had the ability to read and write numbers
to 100 and had learned basic facts to ten.
They had never created their own problems,
so I was unprepared for the results. Many
children chose numbers that would result in
calculations beyond their computational
abilities. One child wrote the following
One day I had 44 stickers. My dad
gave me 18. How many do I have
Although the student coukl read the problem
correctly and identify it as requiring addition, he could not easily solve the problem
by adding because he had not learned the addition algorithm. One solution would have
been to have him act out the problem by
using real stickers. However, many other
students had used even larger numbers,
which would have made the approach time
consuming and cumbersome. One particularly bright student had written a problem
to determine the number of children in the
school. She knew she had to add eight twodigit numbers, but was frustrated when she
could not solve the problem. When she
finally solved the problem, she was able to
read the total even though it went beyond
the Grade 1 curriculum.
Knowing that children are intrigued by large
numbers and that often writing problems
dealing with numbers ten or lower creates
artificial situations, I should have predicted
the outcome. No child in today's society
Would likely possess only ten or fewer
stickers. To ask that pupils limit their
problems to working with numbers ten or
lower is not only unrealistic, but also
How was the dilemma solved? We had used
calculators throughout the year to explore
counting and place-value concepts as well as
basic facts, so several pupils suggested that
we use them for problem solving. After a
quick review of adding and subtracting with
a calculator, pupils had a choice. They could
use counters (or the actual objects present
in the problem) or a calculator. All children
but one chose the calculator. Another
restriction was added. The numbers in the
problem could not total more than one hundred (in order to ensure consistency with the
curriculum). •Estimation was emphasized,
and several lessons on rounding to the
nearest ten were taught. Answers were
checked for reasonableness.
ability to read and write numbers to 100 but
were limited in their computational ability
to working with numbers to ten. With the
aid of a calculator, pupils could work with
numbers to 100 and were no longer restricted
by computation. They were motivated to
write and solve more problems than would
have been possible without a calculator.
Since problem solving success is directly
related to the number of problmes solved
correctly, the calculator is a significant asset
in the primary classroom (Suydam, 1987).
What was the final result? Once children had
access to calculators, their motivation and
productivity increased. All wrote more than
five problems, and one created twenty-three.
Solutions were written on the back of the
problem sheet and checked by a classmate.
Favorite problems were included in our class
booklet, which was frequently signed out to
be taken home and shared with parents.
Students were also enthusiastic about solving their classmates' problems, and they
solved a large number of problems.
What were the advantages of using a
calculator? Every Grade 1 pupil could add
and subtract when given a calculator. The
focus was on the problem-solving process,
not on the computation. Students had the
Bruni, J. (1982). Problem Solving for the
Primary Grades. Arithmetic Teacher, 29
Duea, J., Immerzeel, G., Ockenga, E., &
Tarr, J. (1980). Problem Solving Using the
Calculator, In Problem Solving in School
Mathematics, 1980 Yearbook of the National Council of Teachers of Mathematics. Reston, VA: NCTM.
Fennel!, F., & Ammon, R. (1985). Writing
Techniques for Problem Solvers. Arithmetic Teacher, 32 (1), 24-25.
Suydam, M. (1987). Indications from
Research on Problem Solving. In F.R.
Curcio (Ed.) Teaching and Learning—A
Problem Solving Focus, Reston, VA:
The Adams River Sockeye Run of 1986
David R. Parkinson
Dave Parkinson is a mathematics teacher in Shuswap School District.
ere is a real problem related to the
H Adams River sockeye run of 1986. It
outlines a capture-recapture method, and all
information and numbers were obtained
from a fisheries officer in September 1986.
I think the problem would be useful for
Grades 4 to 8 mathematics classes.
Background to the problem
The number of sockeye that actually spawn
in the Adams River is of concern to federal
fisheries officers because they need to predict
how many salmon will be returning four
years from now. The size of this year's run
was expected to be large because of the
number of salmon that spawned in 1982.
The number of salmon actually starting this
year's run at the mouth of the Fraser River
has been estimated at between 16 and 17
million. The estimate was made by counting
the number of fish caught in one hour by
commerical fishermen near the Fraser's
mouth. That number is then multiplied by
the number of hours that the fish are passing
Unfortunately, it is illegal to net fish for food
at the mouth of the Adams. To estimate the
number of sockeye out of that 16-17 million
that will actually spawn, fisheries officers use
a capture-tag-recover system. A seine net encircles a school of between 1200 and 1500
salmon near the mouth of the Adams, and
these are tagged with a fin marker and
released unharmed. This tagging is repeated
until between 25,000 and 30,000 salmon are
tagged. Each day between early October and
early November, officers walk the river
collecting dead salmon and throwing them
into a pile. This procedure is called dead
pitching and makes sure that no salmon are
1. How could you use this information to
estimate the number of salmon actually
spawning in the Adams River?
2. In what year will the next big run of
3. Estimate the population of untagged
Adams River sockeye actually spawning if 28,132 fish are netted, tagged, and
released, 55,141 are dead pitched, and,
of these, 556 are found to have tags on
4a. Use a calculator to find the "exact"
number of salmon that spawned. Is it
possible that this is not really the exact
4b. Explain your answer to 4a. 5c. To what place value did you express the
answer to 5b7 Is this reasonable, or
should you round it off more or make
it more accurate? Explain your answer.
5a. What fraction of the fish that entered the
Fraser actually spawned?
5b. What percent of the fish that entered the
Fraser actually spawned?
6. The number of cedar trees in a mixed
cedar-fir forest can be estimated using
another sampling procedure. Describe
how the number of trees in a 50,000 hectare forest can be estimated. Also
describe how the percentage of cedar
trees in the stand can be estimated.
MATH TECH 89
February 17, 18
August 29, 30
October 12, 13, 14
An Incredible Resource (2)
Zoe Wake/in is a mathematics teacher at Vernon Secondary School, in Vernon.
n a geometry quiz, given to two
O heterogeneously grouped classes of
Math 8 students, I included the following
question, which requires the upper-level
thinking skill of synthesis, together with
comprehension of the geometric concepts.
Below are some of the student solutions I
later used on worksheets and quizzes.
Create a question that involves alternate interior angles, supplementary angles and an
isosceles triangle to find the measure of a
particular angle in the diagram. (3 marks)Find 45
Find 4 1
Find 4 3
Find 4 6
Find 4 1
No attempt at a solution.
No property used appropriately.
Appropriate use of one property.
Appropriate use of two properties.
Three properties used, but not all
essential to the solution.
All three properties used, and all
essential to the solution.
Evaluation of this type of question is time
consuming. However, working through each
solution provides valuable information
about the student's understanding of the
geometric properties. I found the following marking scheme effective and easy to use.
National Council of Teachers of Mathematics
Leadership Is Vital to
Improving Mathematics Education
Mathematics Leaders in Secondary Schools advocates strong mathematics leadership in each school building. It is imperative that every secondary school designate
a leader, frequently called the department head. This will significantly improve
secondary mathematics instruction. This new position statement from the National
Council of Teachers of Mathematics also stresses that these leaders must provide
ongoing assistance in planning, implementing, and evaluating a comprehensive
mathematics program. They must lead and assist teachers, principals, supervisors,
and other administrators in the areas of curriculum, methodology and materials,
student assessment, professional development, and procedural duties.
Mathematics Leaders in Elementary/Middle Schools (March 1984) is a companion
piece to Mathematics Leaders in Secondary Schools. This position statement also
advocates that strong leadership in individual schools is essential for assuring a
high-quality mathematics program. These leaders must be available as a resource
person to all staff and must demonstrate knowledge, competence, and leadership
in mathematics education beyond what they may be expected to teach.
Mathematics leaders in all schools will have the responsibility for implementing
the recommendations in Curriculum and Evaluation Standards for School
Mathematics, which is scheduled for release by April 1989. Copies of both position statements are available from the NCTM Headquarters Office.
MathlalkAn Electronic Conference
Stew Lynch is vice-principal at Argyle Secondary School, in North Vancouver.
e math teachers tend to spend most
W of our time alone in our classroom
with our students. We seldom get the opportunity to discuss items concerning mathematics unless there is a major conference or
professional-day workshop. Our discussions
are limited to those within our own school
or district. With the aid of a microcomputer,
a modem, communications software, and
the telephone line, we can now participate
in a wide-ranging, intellectually stimulating,
professionally rewarding exchange with
dozens of. new friends and hundreds of colleagues every day.
This virtual community is a group of people who may or may not meet one another
face to face, and who exchange ideas
through computer bulletin boards and networks. The community. members share a
common interest; in our case, mathematics.
This network can be used to scan and gather
information on subjects that are of momentary or enduring importance, from lastminute lesson plans to long-range plans of
curriculum implementation. It is a bit like
a neighborhood pub or a coffee shop. We
can participate in hundreds of ongoing conversations, have our questions answered,
and find inspiration and support.
The communications network has several
advantages over the old-fashioned communities of place and profession. Because we
cannot see one another, we are free from
prejudices about others as we read what they
have to say: Race, gender, age, national
origin, and physical appearance are not apparent unless a person wants to make such
In a communications network, we can get
to know people then choose to meet them.
We can exchange information about one
another, share and debate our mutual interests, and sometimes become friends. In
this community, we go directly to the place
where our particular interests are being
discussed, then get acquainted with those
that share our passions. In a sense, the topic
is the address. You can't simply pick up the
phone and ask to be connected with someone who wants to talk about the new Math
9A curriculum or about a new topic in Data
Analysis; you can, however, join a computer conference on any of those topics,
then open a public or private correspondence
with the previously unknown people you
find in that conference. You will find that
your chances of making friends are
magnified by orders of magnitude over the
old methods of finding a peer group.
With so much activity going on in mathematics revision and implementation, the
BCAMT executive thought it a good idea to
have some method for mathematics educators
at all levels to communicate daily. MathTalk
is our forum for discussion.
Xchange—The Education Network
The Faculty of Education at Simon Fraser
University has developed an electronic network of teachers and students throughout
the province. The Faculty of Education
worked with SFU's Computing Centre to
design an electronic educational forum (EdForum) to serve the professional development needs of teachers in the province. Edforum carries electronic conferences dealing
with issues in many subjects. Participants
can also access the Groliers Electronic Encyclopedia and ERIC (Educational Resources
Information Centre) and open up the exciting world of on-line information for
themselves and their students.
A computer conference like MathTalk is like
any other conference. Computer conferences
are electronic forums or meeting rooms in
which participants post messages for others
to read at their convenience. A variety of
sessions are going on concurrently.
MathTalk has several active sessions where
interested participants can join in and
monitor or participate. Throughout the
year, Vector will provide its readers with excerpts from the various discussions taking
place in the conference. If a discussion does
not meet the needs of a particular individual,
that participant can begin his or her own
Xchange provides a resource guide to electronic mail, conferencing, and databases.
The Xchange materials include a tutorial and
conference directory and will be updated
regularly. Xchange is also supported by a
team of online moderators who will manage
the electronic conferences. On behalf of the
BCAMT executive, I will be moderating a
conference in Ed-forum called MathTalk that
is for all teachers and student teachers of
mathematics who want to discuss issues concerning the mathematics curriculum.
Xchange will also deliver in-service workshops, on request, to school districts and
other groups. Members of an SFU in-service
team conduct hands-on training sessions in
electronic messaging, computer conferencing, and accessing online databases for
educators wishing to extend the walls of
their classrooms and enter the rapidly
transforming world of electronic cothmunications.
Discussions on MathTalk currently include
Problems with implementation
Primary mathematics curriculum
Intermediate mathematics curriculum
Number and number operations
Math 9A and 10A
Technology and mathematics
The list can go on and on. MathTalk enables
rapid exchange of information on issues and
ideas concerning the teaching and learning
of mathematics. It is designed to promote interaction among teachers of mathematics
who are looking for information, ideas, support, and challenges for both themselves and
their students. Teachers can exchange
teaching tips, lesson plans, and personal
opinions of the latest books, materials, and
At present, access to the SFU computer
system is free. The Xchange materials,
including the resource guide and regular updates, are $25 and can be obtained from
Simon Fraser University by writing:
What You Need To Get Started
To participate in this network, you must
have access to a microcomputer that is connected to a modem. A modem is a device
that translates computer signals into sound
waves, and vice versa, so that two computers can communicate through the telephone lines. Each computer is hooked up to
its own modem. The type of computer is unimportant. Apples can talk to Commodores,
Radio Shacks, IBMs, Macintoshes, and any
other computers as long as both computers
have modems set at the same baud rate. A
1200-baud modem is recommended and can
be purchased for less than $150. Communications software is also needed to give
the commands to the modem, such as what
phone number to dial. Several modems
come with their own software, but several
integrated packages like Microsoft Works,
for the IBM and Macintosh computers, have
a communications module that will do the
SFUXchange, Faculty of Education Simon Fraser University
Burnaby, BC V6A 1S6
If you are communicating from outside the
Lower Mainland, you will connect with SFU
using the telephone number of the Datapac
Public Dial Port nearest you. Datapac is a
computer communications network that
serves most of Canada and is available in
most cities in British Columbia. Your call
from outside the Lower Mainland will be toll
free from the Datapac port to Simon Fraser
The recent Northwest NCTM Conference
was an overwhelming success. Let us keep
the exchange of ideas flowing. This is a time
of change in mathematics, and we need to
communicate. Conferences and workshops
are too few and far between. MathTalk is
your link to what is going on outside your
The NCTM Standards Document
he NCTM Commission on Standards for School Mathematics created in 1986 was charged
developing a set of standards for mathematics in an exemplary school mathematics
program and the instructional conditions necessary for that mathematics to be learned. In
addition, the commission was asked to develop standards for evaluating such a school program and evaluating of student performance in light of the curriculum standards.
A draft set of standards has been prepared that reflects the output of the commission's working groups, which met several times during the summer of 1987. Membership of the working
groups represents a cross-section of classroom teachers, supervisors, teacher educators,
mathematicians, and researchers, all active in the work of the council.
The NCTM circulated a draft of the standards and asked that mathematics educators comment on the document. The comments were collated in the summer of 1988, and a final document will be presented at the 1989 Annual Meeting of the NCTM in Orlando, Florida.
The Executive of the BCAMT considers the document to be very important, and it spent a
full day at its summer retreat discussing the standards and implications for mathematics education in British Columbia. We believe that the standards will have a strong impact on curriculum and teaching in the United States, and in other educational systems. Here are some
of the questions we raised:
How realistic are the assumptions?
Are some of the beliefs just platitutdes?
Which beliefs does the BCAMT Executive support?
If there are assumptions that are not realistic in the B.C. scene, can we ignore them and
still support the standards for a specified grade level?
Even though 10 000 copies of the draft of the standards were distributed, and some reference
has been made to them in other NCTM publications, we feel that BCAMT members may
not yet be familiar with the recommendations. We cannot circulate the entire report, but we
have chosen to reprint much of the introductory section to give the flavor of what will come
to the 1989 NCTM Annual Meeting. We urge you to read this material, to consider the curriculum changes in British Columbia, and to assess whether we are in step with the NCTM,
or perhaps leading. We also hope that you will obtain the final document to see what specific
recommendations are being made at each of the primary, middle, and high school levels.
Introduction to the Standards
These standards should be viewed as one component of the mathematics education community's response to the current crisis in the teaching
and learning of mathematics and the resulting
calls for reform. The standards reflect, and are
an extension of, the community's responses to
those demands for change. Group consensus
holds that most students need to learn more, and
often different, mathematics and that the teaching
of mathematics must be significantly improved.
To provide all students an opportunity to learn
the mathematics they will need, the emphases and
topics of the present curriculum should be altered.
More importantly, methods of instruction need
to emphasize exploring, investigating, reasoning,
and communicating on the part of all students.
In particular, teachers should view their role as
guiding and helping students to develop their
mathematical knowledge and power. Thus, these
curriculum standards provide a basis for changing the intent of present courses and for developing different topics and new programs. They
characterize the spirit in which instruction should
take place. In addition, in order to provide valid
evidence that such programs actually improve
students' mathematical power, methods of gathering evidence are offered to evaluate both student
performance and program effectiveness.
THE NEED FOR STANDARDS
The driving force for the development of curriculum and evaluation standards is a vision of
the mathematics all students should have an opportunity to learn and the way in which instruction should occur. Classrooms should be places
where interesting problems are explored using important mathematical ideas. For example, in
various classrooms one could expect to see
students recording measurements of real objects,
collecting information and describing their properties using statistics, and exploring the properties of a function by examining its graph. This
vision sees students studying much of the same
mathematics currently taught but with quite a different emphasis; it also sees some mathematics
being taught which in the past has received little
emphasisin schools. Thus, the first aspect of each
standard is the mathematical content to be
This vision sees classrooms in which students are
actively engaged in making conjectures and
discussing ideas. Thus, the second aspect of each
standard specifies the expected student actions
associated with doing mathematics.
Finally, this vision sees teachers encouraging
students and probing for ideas. Hence, each
standard is then elaborated upon in a discussion
of the purpose, the emphasis, and the spirit of this
vision for instruction.
Although many readers may object to this picture of a mathematics class as too idealistic or impossible to accomplish, we are convinced otherwise. The traditions, assumptions, and constraints
underlying current educational practice must be
changed. In too many schools, teachers find it impossible to teach such mathematical topics or
create such classroom environments because of
the constraints placed on them. The mathematics
presented in many textbooks is a vast collection
of vaguely related concepts and skills which are
to be mastered in a strict order. Each page is written as if it were independent of all other pages.
The only apparent objective is that students
become competent at performing a special
algorithmic procedure which yields correct
answers on sets of stereotypical exercises. And
student work is evaluated via paper-and-pencil
tests; students' thought processes on reasoning
skills are not considered. This fragmentation, the
emphasis on paper-and-pencil procedural skills,
and a simplistic form of evaluation have effectively separated students from mathematical reality, inquiry, and intellectual growth.
Today, teachers also are constrained by directives
about chapters or pages to cover, time for instruction, and tests. They lack the necessary resources,
the time to reflect, and the opportunities to share
ideas with other teachers. Under such conditions
it is difficult to create in themselves or in their
students a sense of exploration, curiosity, or excitement. Whereas new standards alone cannot
alter these conditions, they implicitly argue for
We are convinced that many educators are eager
to reform school mathematics. During the past
decade, numerous attempts have been made to
respond to the demands for reform. For example, the majority of state legislatures have mandated that mathematics requirements for graduation be increased and that competency tests be
used for student promotion. Similarly, -new
syllabi have been or are being developed by
school districts and state departments of
education. Publishers are producing kits of
manipulatives and software and are including
more challenging problems in their texts. In addition, a number of experimental development
projects and several research studies have been
funded by the federal government and private
foundations. There is a groundswell of activity
aimed at the improvement of school mathematics.
Unfortunately, such a piecemeal approach to
reform is unlikely to have a significant effect.
Evidence from past efforts to change schools
makes it clear that a more systemic, orchestrated
approach is needed (Romberg and Price, 1981).
The NCTM Standards Project was initiated to
provide a basis for the development of school
mathematics in line with a vision which reflects
contemporary mathematics, a contemporary conception of learning and instruction, and the professionals' beliefs about what students should
Finally, we believe that NCTM is the appropriate
agency to develop and disseminate curriculum
and evaluation standards. Knowledgeable
mathematics educators, through the professional
organization which best reflects their interests and
those of their students should assume responsibility for the development of standards that closely
affect their work and the mathematical learning
of their students. For too long we have abdicated
this responsibility to others, including legislators,
administrators, and text and test publishers.
THE NCTM STANDARDS
FOR SCHOOL MATHEMATICS
The picture of school mathematics from which
these standards have been developed is based on
our concerns about:
• the mathematics that students should learn to
be productive citizens throughout their lifetimes;
• the methods of instruction which engage
students in constructive learning; and
• the expectations of our society, which make it
imperative that more students have an opportunity to learn more mathematics.
THE NEEDED CONTENT
FOR SCHOOL MATHEMATICS
Contrary to what many people believe,
mathematics is a dynamic, , growing, and changing discipline. Davis and Hersh (1981) claim that
we are now in a Golden Age of mathematical production, with over half of all mathematics invented since World War II. In fact, they argue
that "there are two inexhaustible sources of new
mathematical questions. Once source is the
development of science and technology, which
make ever new demands on mathematics for
assistance. The other source is mathematics
itself.. .each new, completed result becomes the
potential starting point for several new investigations" (p. 25). Given this explosion of knowledge,
it is impossible for any one person to know all
there is to know. For school programs we must
select carefully and emphasize fundamental
knowledge needed for contemporary mathematics. We believe that change is necessary
because many important mathematics ideas are
not reflected in current programs. These standards have been written with this goal and this concern in mind.
To illustrate the changes occuring in mathematics
and their potential impact on school programs,
we have chosen to look at changes in technology
Advances in technology
Most current mathematics programs fail to reflect
the impact of the technological revolution affecting our society; The availability of low-cost
calculators, computers, and related new
technology have already dramatically changed
the nature of business, industry, government,
sciences, and social sciences. Unfortunately, most
students are not educated to participate in this
new society. Despite the advancements that have
brought untold computational and graphical
power to our fingertips, in-school hours are spent
drilling on computational procedures in arithmetic, algebra, statistics, and even calculus. Yet
any step-by-step procedure involving the manipulation of mathematical symbols according to a
fixed set of rules can be accomplished on a
calculator or computer. Some procedures are
simple enough that they are best done mentally
or by hand; others are more complex, or take time
to work manually and should be done by
machine, as they are in the world of work.
Based on this reasoning, we have, made the
• Calculators are available to all students at all
times (a scientific calculator from Grade 7).
• A computer is available in every classroom for
demonstration purposes, and every student has
access to a computer for individual and group
The new technology not only has made calculations and graphing easier to do, it has changed
the very nature of mathematical research. Lynn
Steen, a former president of the Mathematical
Association of America, stated that "major forces
are reshaping the nature of mathematics .
Most obvious is the increasing importance of
algorithmic processes in mathematical methods"
(1986, p. 7). For example, given that computers
can perform many calculations quickly, it is no
longer adequate to consider one algorithm. Instead, important mathematical research problems
are triggered by such questions as -Can you invent an algorithm? Is this the best algorithm?
What do we mean by best? How can we prove
that it is the best? Since the computer is one force
that is changing mathematics, we have assumed
• students should learn to use the computer as
a tool for processing information and performing calculations in order to investigate and solve
Expanding use of quantitative methods
Complementing the influence of technology on
mathematics is the fact that the use and application of mathematics have dramactially expanded.
Quantitative and logical techniques have permeated almost all intellectual disciplines. Change
has been particularly great in the social sciences
and the life. sciences. The computer's ability to
process large sets of information has made quantification possible in such' areas as business,
economics, linguistics, biology, medicine, and
sociology. Furthermore, the' fundamental mathematical. ideas needed in these areas are not
necessarily those studied in the traditional
That, sequence was designed with engineering and
physical science applications in mind. Since
mathematics, is a foundation discipline for other
disciplines and grows as a result of its utility, we
have assumed that
• the curriculum for all students must provide opportunities to build mathematical models, structures, and simulations. within many disciplines.
Changes in technology, broadening of the areas
in which mathematics is applied, and growth in
mathematics itself have changed the problems of
importance to mathematics .and the 'methods
mathematicians use to investigate problems.
These changes must be echoed by changes in the
curriculum so that students have an opportunity
to learn, the skills and knowledge that are likely
to be fundamentally important in. their lives.
OF THE CURRICULUM STANDARDS
We have, used ideas about changes in mathematics
to develop curriculum standards by organizing
into three categories the content students should
have an opportunity to learn.
Initially, mathematical knowledge arose from
rudimentary ideas acquired through perceptions
of situations in the complex world around us.
Several millennia ago our ancestors planted the
first seed for this enterprise by observing some
quantitative and spatial regularities. From these
humble beginnings, mathematics has flowered
into the impressive body of knowledge we have
been fortunate enough to inherit. Thus, from its
origins, mathematics was an empirical science. Its
fundamental terms, signs, symbols, and rules are
merely abstractions and inventions created to
represent properties observed in the environment.
Thus, numbers were created to represent the
numerosity of sets of familiar objects; signs such
as "+" were invented to represent the addition
of numbers, which in turn represents the joining
of sets, and terms such as parallel and perpendicular were introduced to name spatial properties. The purpose of creating such a language is
to communicate with others; the terms of the
language become useful only when there are
shared meanings. Thus, our first category involves helping students to understand mathematics as a language [we have] created to describe
the world and a set of rules for the use of that
language. Its origins exist in the regularities of the
world in which we live. Furthermore, like any
language, mathematics grows and changes as a
i-esult of empirical investigations. Thus, all
students need to confront a rich array of common problem situations from which the empirical
language of mathematics, its notations, and its
rules can gradually be built.
A good deal of mathematical knowledge has been
created by making the empirically based language
and rules of mathematics the objects ofinvestigation. By observing properties of numbers, operations, and spatial figures, man has created another
set of terms, signs, symbols, and rules. Some are
generalizations of empirical procedures. For ex,
ample, the creation of computational algorithms
for empirical processes has made mathematics applicable to many seemingly unrelated problem
situations. In addition, no' longer bound b,: the
concrete, [we have] extended mathematics. by asking "what if. . .?" questions abstracted. from the
perceptual reality. For example, while the' 'crea tion of an equation for the distance between two
points in a plane has empirical origins, the
general ization.of this formula 'to two points in ndimensions does not. And, while multiplication
of whole numbers has empirical roots in the
grouping of objects, more abstract multiplication
algorithms do not. They grew out of observations
about properties of exponents such as a X a =
+ , and the fact that any decimal number can
be expressed as an exponent through use of
powers of ten. Thus, our second category of
mathematics emphasizes the study of abstract
systems. Hence, all students should have the opportunity to explore the properties of empirical
mathematics and to see for themselves the relationships, rules of transformations, extensions,
and structures derived from these investigations.
It should be noted, too, that some rules (or
algorithms) for transforming symbols into other
symbols are so useful that knowledge of them has
been considered essential. In fact, mastery of
many computational algorithms has been considered by many to constitute mathematical
knowledge. We do not deny that knowledge of,
in contrast to proficiency with, computational
algorithms is important, even at a time when
calculators and computers perform most computations for us, but such knowledge should grow
out of the problem situations which have given
rise to the need for such algorithms. Furthermore,
skill in procedural algorithms should not be the
backbone of elementary school mathematics.
Higher order reasoning skills
The intellectual methods that mathematicians use
when developing conjectures, reasoning about
phenomena, building abstractions, validating
assertions, and solving problems are critical to
doing mathematics. However, these intellectual
skills cut across all content networks. For example, no proposition in mathematics is considered
to be true until it has been validated. Initially,
justifications may necessarily be built upon empirical evidence since they originated in our
perceptions. However, proving an assertion by
presenting a rigorous, logical argument has
become the hallmark of abstract mathematics. For
example, no geometer who had measured the base
angles of an isosceles triangle would conclude that
they were congruent based on a demonstration,
no matter how accurate the measurements. However, such measurements may have formed the
basis of a conjecture about their congruence.
Mathematicians demand that this result be deduced from the fundamental concepts of
geometry. Thus, the discipline grows by applying mathematical methods to a wide variety of
problem situations. In this regard, to gain
mathematical power, students need to make conjectures, abstract properties and relationships
from problem situations, explain their reasoning,
follow arguments, validate assertions, and communicate results in a meaningful form.
In summary, as the curriculum standards were
being written, the mathematical content that
students should have an opportunity to learn was
foremost in our considerations.
THE NEEDED METHOD OF INSTRUCTION
The second part of each standard lists important
actions of students. These actions are expected
to result from engagement in specific instructional
activities needed to foster mathematical power.
We followed two general principles in stating
these actions: first, we have assumed that instructional activities grow out of problem situations;
note that while a specific lesson may not be problem based, a sequence of lessons should be
grounded in a problem situation. Second, since
learning occurs by construction, not absorption,
we have described some of the conditions
necessary for such learning to happen.
Traditional teaching emphases on practice in
notational transformations and solution algorithms as necessary precursors to coping with a
problem ignore the fact that knowledge emerges
from the problems. This suggests that instead of
the expectation that skill in computation should
precede word problems, experience with problems
helps develop the ability to compute. Thus, present strategies for teaching mathematics by first
teaching skills and then exposing students to
stylized application problems need to be reversed;
knowledge should emerge from experience with
A genuine problem is a situation in which, for
the individual or group concerned, one or more
appropriately analogous strategies have yet to be
developed. The situation should be complex
enough to offer challenge, but not so complex as
to be co-operatively insoluble by the group of
students. In fact, problem situations should
parallel the kind of phenomena for which mathematical structures have been typically created
(Freudenthal, 1983). Thus, pupils need to experience the phenomena for which such concepts,
structures, and ideas were created. In order to
teach the mathematizing of situations, teachers
need to create suitable contexts; the most abstract
mathematics needs the most concrete contexts.
Learning as construction
In most classrooms, the conception of learning is that students are passive absorbers of
information, storing it in easily retrievable
fragments as a result of repeated practice and reinforcement. Research findings from psychology
indicate that learning does not occur by passive
absorption (Resnick, 1986). Instead, individuals
approach each new task with prior knowledge,
assimilate new information, and construct their
own meanings. For example, before young children are taught addition and subtraction, they can
already solve most addition and subtraction problems using such routines as "counting on" and
counting back" (Romberg & Carpenter, 1986).
As instruction proceeds, children often continue
to use these routines to solve problems, in spite
of being taught more formal procedures. They
will .... accept new ideas [only] when their old
ideas do not work or are inefficient. Furthermore,
ideas are not isolated in memory but are organized and associated with the natural language that
one uses and the situations One has encountered
in the past. This constructive, as opposed to
passive, view of the learning process must be
reflected in the way mathematics is taught. Instruction based on this conception of learning is
different from that in the typical mathematics
classroom where the teacher is transmitting
lessons through exposition to a captive audience.
Instruction from this perspective should include:
• project work rather than exercises;
f group assignments as well as independent
• discussions about the origin of and relationships among concepts, rather than independence
of concepts; and
• intrinsic motivation through curiosity rather
than extrinsic reward.
IN THE CURRICULUM STANDARDS
Our ideas about problem situations and learning
are stated in the standards. They are reflected in
the verbs used to describe student actions (e.g.,
to investigate, to formulate, to find, to verify).
These verbs describe student work on different
types of problems. 'Although each problem situation is in one sense unique, we have considered
them under seven general headings. Note that,
while each category of actions is different in an
important way from the others, they are closely
Many of the activities, particularly when one is
introducing new content, involve examining or
exploring some situation. The purpose of such
activities is to abstract or to invent. To abstract
means to identify the intrinsic.form or essence,
shape or structure of a given situation. To invent
means to produce for the first time through the
use of one's imagination, ingenious thinking, or
experimentation. Pattern recognition is at the
heart of mathematical thinking and invention is
so intimately linked with abstraction as to be inseparable. Labels (terms, signs, or symbols) are
invented to represent most abstractions. An essential part of these activities is negotiation with
students about the choice and purpose of such
Activities of this type expect students to form or
to produce an image, picture, word, sentence,
chart, or table using appropriate symbols for the
referents in a problem situation. These are all part
of the process of learning to construct mathematical models of problem situations.
A transformation is an operation that involves
changing one expression into another following
one or more rules. Much textbook mathematics
involves learning how to perform transformations. Examples include computational algorithms, division of polynomials, and transforming
from one representation to another, such as from
an algebraic description of a function to a
geometric representation as a graph. Students
need to understand why such transformations are
desirable and the steps involved in performing the
changes, even though rapid proficiency may no
longer be essential.
When students have made conjectures about a
situation, they need to establish evidence that
compels acceptance of such propositions. The
kind of evidence and form of argument, as mentioned earlier, may vary but the intent.remains
constant. Students should be able to construct
cogent arguments in support of their claims..
To apply is to put to use known concepts and
procedures in new situations. Note that, in applying, one might invent. In the process of inventing, one might apply, 'and; as a precursor to inventing, one might abstract.
This category refers to true problem-solving situations; i.e., 'those in which complex situations are
presented where it is assumed that students have
acquired the necessary concepts and procedures
to find a solution but need to-use some strategies
(heuristics) to make the connections between the
given information and a method of solution.
This category of activities involves exchange of
information between individuals by use of a common symbol system. Thus, students should be
given numerous opportunities to express mathematical ideas verbally and in writing, and to comprehend and interpret ideas expressed by others.
• the ability to work with others on problems;
• the ability to see the applicability of
mathematical ideas to common and complex
• preparation for open problem situations, since
most real problems are not well formulated; and
• belief in the utility and value of mathematics.
As students' mathematical power grows and
changes, the problem situations must vary with
the maturity—both mathematical and cultural—
and experience of the students. For example, the
primary grades should emphasize the empirical
language of the mathematics of whole numbers,
common fractions, and descriptive geometry. In
the middle grades, the empirical mathematics
should be extended to other numbers and the emphasis should shift to building the abstract
language of mathematics needed for algebra and
other aspects of mathematics. High school mathematics should emphasize functions, their
representations and uses, modeling, deductive
proofs, and applications.
Notice how different this description is from that
of a student working alone on sets of well formulated exercises. While mathematics is not
taught in schools solely so students can get jobs,
their experiences should not be that dissimilar
from the expectations of employers.
In summary, in our view, instruction should be
based on problem situations. As long as the situations are familiar, conceptions are created from
objects, events, and relationships in which operations and strategies are well understood. This
forms a framework of support which can be
drawn upon in the future, when rules may well
have been forgotten but the structure of the situation remains embedded in memory, a foundation
for reconstruction. Situations should be sufficiently simple to be manageable, but sufficiently complex to provide for diversity in approach. They
should be amenable to individual, small-group,
or large-group instruction, involve a variety of
conceptual domains, and be open as to the
methods to be used.
FOR SCHOOL MATHEMATICS
In developing the overall set of standards, we
were cognizant of other evidence that changes in
school mathematics are needed. First, employers
have claimed that recent graduates have insufficient knowledge for employment. Henry Pollak
(1987), a noted industrial mathematician, recently
summarized what industry expects of its new
• the ability to set up problems with the appropriate operations;
• knowledge of a variety of techniques to approach and work on problems;
• understanding of the underlying mathematical
features of a problem;
Second, college and university officials have
documented that 15% of the current enrolment
in mathematics courses at public four-year colleges is for remedial courses, and another 37 percent is for precalculus courses (Albers, Anderson
& Loftsgaarden, 1987). Two probable reasons
may explain this situation: first, a mismatch may
exist between what is studied in school mathematics and what is needed in some college majors.
Second, there has been a change in college expectations: that is, more majors now require mathematics—a fact that many high school counsellors
have failed to recognize.
Third, the bleak national performance data cannot be ignored. For example, results from the National Assessment of Educational Progress
(NAEP) in mathematics (Carpenter, Brown,
Kouba, Lindquist, Silver & Swafford, 1978) clearly show that, although most students are
reasonably proficient in computational skills, the
majority do not understand many basic concepts
and are unable to apply the skills they have
learned in even simple problem-solving situations.
Add to this fact that, when compared with
students in other industrialized nations, particularly those in the Orient, our students do not
fare well (McKnight, et al., 1987). We expect less
of our students, they spend less time studying
mathematics, and fewer are enrolled in advanced
mathematics than are students in other countries.
Fourth, and more disturbing, is the fact that the
performance and enrolment picture is even more
bleak for women and most minorities. For example, on the average, black students complete approximately one year less high school mathematics than their white classmates (Anick,
Carpenter & Smith, 1981). Women and most
minorities are seriously . underrepresented in
careers utilizing science and technology.. Only
13% of the nation's scientists and engineers are
women and only 2% are black (National Science
Foundation, 1981). Furthermore, given that
schooling practices are inequitable, this condition
is likely to get worse. Affluent suburban school
districts already provide their students more opportunities and resources for the study of mathematics, and they are likely to be the first to react
to the current crisis and recommendations. They
are already spending more money on computers
and teacher in-service, thus widening further the
opportunity gap between affluent suburban
students and their poor, urban counterparts.
Finally, another aspect of this concern is the belief
held by many that the mastery of paper-andpencil procedural skills is necessary before one
can investigate problems. In particular, it is a
common assumption that arithmetic computations are necessary before one can study algebra
or geometry or investigate applied problems. This
assumption is not warranted. Too many students
are refused an opportunity to learn the mathematics that would make it possible for them to
be productive members of society because they
ar not proficient at skills which are now done
best on a calculator or computer. Even if there
were no technological revolution, or explosion of
mathematical knowledge, or research on learning and teaching, there is ample evidence that
school mathematics would need to be reformed.
Taken together, the evidence makes reform
We also have assumed that:
• All students should be required to study mathematics for at least 12 years (Grades K-li).
• The study of mathematics should revolve
around a core curriculum that allows all students
an opportunity to learn the important ideas and
methods of mathematics.
• Mathematics should be studied as an integrated
whole so that students understand it as a dynamic
discipline and an integral part of our culture.
• Doing mathematics involves solving problems.
• Communication is an important goal of mathematics instruction.
• Mathematics should help build students'
abilities to reason logically.
• New topics (e.g., data analysis, estimation)
must be introduced into the mainstream
• Mathematics should be taught in a natural
• Students should be encouraged to create, invent, and participate.
• Calculators and computers should be used
throughout school mathematics.
• Success in paper-and-pencil computation need
not be a prerequisite to the study of other
GOALS FOR STUDENTS
Five general goals are reflected in the mathematics
curriculum standards across Grade K-12. These
we have labeled: (1) becoming a mathematical
problem solver, (2) learning to communicate
mathematically, (3) learning to reason mathematically, (4) valuing mathematics, and (5)
becoming confident in one's ability to do mathematics. These goals imply that students should
have numerous and various interrelated experiences which allow them to solve complex
problems; to read, write, and discuss mathematics; to conjecture, test, and build arguments
about a conjecture's validity; to value the mathematical enterprise, the mathematical habits of
mind, and the role of mathematics in human affairs; and to be encouraged to explore, guess, and
even make errors so that they gain confidence in
their actions. The opportunity for all students to
have these kinds of experiences is at the heart of
our vision of an exemplary mathematics program.
The curriculum should be permeated with these
notions to the point that they become commonplace in the lives of students. We are convinced
that if students are exposed to the kinds of experiences outlined in the standards, they will be
able to develop their mathematical power and will
become mathematically literate.
Becoming a mathematical problem solver
The development of each student's ability to solve
problems is essential if he or she is to be a productive citizen. We strongly endorse the first
recommendation of An Agenda for Action (National Council of Teachers of Mathematics, 1980):
"Problem solving must be the focus of school
mathematics" ( p. 2). The initial standard at each
of the three levels addresses this goal.
Learning to communicate mathematically
As described in the introduction, the development
of a student's power to use mathematics involves
learning the signs, symbols, and terms of mathematics. This is best accomplished in problem
situations in which the student has an opportunity
to read, write, and discuss ideas in which the use
of the language of mathematics is natural. The
importance of this goal is reflected in the second
standard at each level.
Learning to reason mathematically
Making conjectures, gathering evidence;- and
building an argument to support such notions is
fundamental to doing mathematics. In fact,
demonstration of good reasoning should be
rewarded even more than students' ability to find
corrent answers. This goal is reflected in the third
standard at each level.
Learning to value, mathematics
The mathematics curriculum should include
numerous and varied experiences related to the
cultural, historical, and scientific evolution of
mathematics so that students can appreciate the
role of mathematics in the development of our
contemporary society, and explore relationships
between mathematics and the disciplines it serves:
physical and life sciences, social sciences, and
humanities. Students should learn that mathematics is more than a collection of concepts and
skills to be mastered.
From its origins, mathematics has been simultaneously a highly esoteric and abstract discipline
and the most practical of human endeavors.
Throughout its history, practical problems and
theoretical pursuits have stimulated one another
to such an extent that it would be impossible to
disentangle them. Even today, as theoretical
mathematics has burgeoned in its diversity and
deepened in its complexity and abstraction, it has
become more concrete and vital to our technologically oriented society. Our culture has become
"mathematized." This goal's intent is to focus attention on the need for student awareness of the
interaction between mathematics and the
historical situations from which it has developed
and the impact that interaction has on our culture
and our lives. This goal is not reflected in a single
standard at each level but permeates each
The importance of coming to value mathematics
in- the manner just described has recently been
demonstrated by Allan Schoenfeld (in press).
Given nonsense problems such as 'There are 125
sheep and 5 dogs in a flock. How old is the
shepherd?", students had no trouble deciding he
was 25 because that was the most plausible
numerical combination of 125 and 5. Schoenfeld
argues that students arrived at their answers by
following a four-step procedure which is commonly taught: (1) read the problem, (2) select the
numbers and the relevant operation, (3) perform
the operation, and (4) write down the answer.
Furthermore, students have found that this
procedure usually works and is rewarded in classrooms. Students may master the formal procedures of mathematics and fail to use them
sensibly. Mastery of the procedures of typical
classroom mathematics is far different [from]
learning to think mathematically. The argument
Schoenfeld convincingly makes is that one's
knowledge is contextually bound, and that one's
understanding of a domain is shaped by the practices of those domains. To devleop mathematical
power, one cannot separate mathematics from the
cultural, historical, and scientific contexts from
which it has evolved.
One curricular difficulty is that we have paid inadequate attention to how problems are formulated for students. There is generally very little
variety in text formats, and neither the teacher
nor the students are expected to vary the bland
stereotypes so that interesting problems are posed,
discussed, extended, and solved. Cultural contexts
surely should be used for this purpose. Only by
experiencing the interactions of formal mathematics with the cultural, historical, and scientific
contexts will students come to value the discipline
as an enormous cultural achievement that is
dynamic, growing, and truly important in their
Becoming confident in one's own ability
The mathematics curriculum should include
numerous and varied experiences related to the
human aspects of the development of mathematics. This allows students to understand that
mathematics is an invention of the human mind
so that they learn to trust in their own mathematical thinking. As a result of studying mathematics, students need to view themselves as
capable of using their growing mathematical
knowledge to make sense of new problem situations in the world around them. This goal is not
reflected in any single standard but its spirit is
reflected in all the standards.
To achieve this goal, there must be a shift of
authority for the validation of conjectures, ideas,
and solutions from the teacher or the answer key
to the students' own mathematical reasoning. One
powerful way of helping students see their own
human.-capacity to do mathematics and to reason
about situations is to infuse the mathematics curriculum with examples of the present and past
struggles of humans to describe, classify, compare, contrast, measure, and relate phenomena
in the world around them. The study of mathematics msut include the development of the
mathematical enterprise, the role of mathematics
in human affairs, and the habits of mathematical
investigation. Lesson frequently should include
information about the problems, ideas, and works
that have shaped mathematics. By putting mathematics in its historical and cultural context,
students should develop an understanding of how
people discover and apply mathematics.
Another way to nurture students' confidence is
to break down the discontinuity between formal
methods and the natural strategies they develop
in their daily activities. Several recent studies have
indicated that employees show more variability
in problem-solving procedures when solving jobrelated activities than do students in mathematics
classes (see Schliemann, 1984). This research suggests that problem solving in school has to be
taught differently if it is to be useful outside the
classroom. Only through such experiences will
students come to value mathematics as an important part of their daily lives.
Finally, as Davis and Hersh (1981) have stated:
"Mathematics has been a human activity for
thousands of years. To some extent, everybody
is a mathematician and does mathematics consciously. To buy at the market, to measure a strip
of wallpaper, or to decorate a ceramic pot with
a regular pattern is doing mathematics" (p. xi).
School mathematics must enlighten all students
as to the truth of this statement.
on Mathematical Education (pp. 1-7). Boston:
McKnight, C.C., Crosswhite, F.J., Dossey, J.A.,
Kifer, E., Swafford, J.O., Travers, K.J. &
Cooney, T.J. (1987). The underachieving curriculum: Assessing U.S. school mathematics
from an international perspective. Cham-
paign, IL: Stipes.
National Council of Teachers of Mathematics.
(1980). An agenda for action: Recommendations for school mathematics of the 1980's.
Reston, VA: The Author.
National Science Foundation. (1982). Science indicators, 1982. Washington, DC: U.S.
Government Printing Office.
Resnick, L.B. (1986, April). Education and learning to think. A special report prepared for the
Commission on Behavioral and Social Sciences
and Education, National Research Council.
Romberg, T.A. & Carpenter, T.P. (1986).
Research on teaching and learning mathematics: Two disciplines of scientific inquiry.
In M.C. Wittrock (Ed.), Handbook of research
on teaching: A project of the American Educational Research Association. Third Edition
Albers, D.J., Anderson, R.D. & Loftsgaarden.
(1987). Undergraduate programs in the
mathematical and computer sciences the
1985-1986 survey. Washington, DC: Mathe-
matical Association of America.
Anick, C.M.. Carpenter, T.P. & Smith, C. (1981).
Minorities and mathematics: Results from the
National Assessment of Educational Progress.
Carpenter, T.P., Brown, C., Kouba, V., Lindquist, M.M., Silver, E.A. & Swafford, J.O.
(1987). Results from the Fourth Mathematics
Assessment of the National Assessment of
Educational Progress. Reston, VA: National
Council of Teachers of Mathematics.
Davis, P.J. & Hersh, R. (1981). The mathematical
experience. Boston: Houghton Mifflin.
Freudenthal, H. (1983). Major problems of
mathematics education. In M. Zweng, T.
Green, H. Pollack & M. Suydam (Eds.), Proceedings of the Fourth International Congress
(pp. 850-873). New York: Macmillan.
Romberg, T.A. & Price, G.G (1983). Curriculum
implementation and staff development as
cultural change. In G.A. Griffin (Ed.), Staff
development: Eighty-second yearbook of the
National Society for the Study of Education
(Part II) (pp. 154-184). Chicago: University of
Schliemann, A.D. (1984). Mathematics among
carpentry apprentices: Implications for school
teaching. In P. Damerow, M.E. Dunkley, B.V.
Nebres & B. Werry (Eds.), Mathematics for
all. Paris: UNESCO.
Schoenfeld, A.H. (in press). Problem solving in
context(s). In R. Charles & E. Silver (Eds.),
Teaching and evaluating mathematical problem solving. Reston, VA: National Council of
Teachers of Mathematics.
Steen, L.A. (1986). A time of transition: Mathematics for the middle grades. In Richard
Lodholz (Ed.), A change in emphasis (pp. 1-9).
Parkway, MO: Parkway School District.
MATHEMATICS AND COMPUTERS
Functions as Models of Reality
Brian Copeland is mathematics teacher at
Templeton Secondary School, in Vancouver.
f mathematics education is to regain a
I measure of intellectual respectability, it
must demonstrate that it is more than some
esoteric tautology. It can do this by showing the student now that it can help us to see
the world in new and useful ways. The computer seems to me an indispensable tool for
overcoming the computational difficulties in
applying mathematical functions to model
real world phenomena. This article attempts
to show how major functions can be used
to model reality.
Experiments in Functional Composition
Given two functions f and g, it is often
useful in modelling to use a combination of
the two, to get some of the behavior from
the one function and some of the behavior
from the other. The function f*g takes any
argument x and first applies g and then f.
Students can use the computer to compare
f*g to g*f and to describe the behavior of different functions in ordinary language. This
will tap into the existing knowledge base.
Most important, attention can be focussed
on mathematics as a vehicle for seeing and
describing the real world in important, quantitative ways.
Exercises like those used here can be open
ended, leaving room for considerable student exploration. The computer takes care
of calculation; the student takes care of supplying ideas for exploration, describing
results, explaining interesting features, and
suggesting possible applications. Students
can start to reinvent the mathematics they
need to learn, generating a sense of ownership, a gradual construction from hunch to
finished product and actual perception using the acquired concepts.
The examples in Figures 1 and 2 show functions that can be composed to display
various features such as changing frequency, varying amplitude, and systematic fluctuations. The computer does the low-level
computation, allowing the student to concentrate on the higher level cognitive tasks:
comprehension, application, analysis, synthesis, and evaluation.
X—AXIS:-10. . .10 STEP
X—AXIS: 0.. .50 STEP 5
Y— A XIS:
.1 STE P
The Bouncing Ball
The bouncing ball (as shown in Figures 3 and
4) needs three functions to model it:
• the sine or cosine to get up and down,
• absolute value to do away with negative
• exponential to get the damping effect.
After playing with exploratory exercises
such as those following, students could
model the behavior of specific bouncing
balls, with different frequencies, amplitudes,
and decay rates. Questions could be set
about comparisons with what would happen
on the moon and the possibility of a ball that
didn't decay but bounced higher and higher
each bounce. Preformal experimentation will
encourage student independence and confidence.
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X—AXIS: 0.. .200 STEP
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Water depth in a tidal region can be modthe definite integral being :done on the
elled with the sine function ,(see . igure 5).
Amplitude and frequency canbe modelled
and discussed in . a way that. connects with
existing knowledge., The average value of a
Variations on :the basic . exercise, might incontinuous function can, be meaningfully include changing the model to reflect deeper
vestigated here (average water de p th ovêra
water and seasonal changes ini tidal, depth.
tidal cycle). This average value can be
A discussion of tides will unpack much of
calculated as follows:
the information that 'is so. compactly stored
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Motion with Resistance
If an object is dropped from a height h above
the earth, air resistance ignored:
For simplicity, assume v,=O and use v7=g/p
v = g/p*exp(_pt) - g/p
v = a * t = 32*t
5 * a * (2 =16*(2
Velocity increases indefinitely with respect
to time, and supposedly all objects fall at the
same rate, since there is no mention of mass.
With air resistance, the situation is different
(see equations 1 and 2 below). Some devices
even depend on the effects of air resistance
(parachute, hang glider, ski jumping). The
computer facilitates the investigation of this
type of motion under the constraint of
resistance, in which:
v = (v0 +v)*exp(_pt) -v7
= terminal velocity
p = drag coefficient
g = acceleration due to gravity
Also, the height of the object above ground
is given by:
y = y0 -
vrt + 1/p*(v0+v7)*(1_ exp(-pt))
For simplicity, use v7 =g/p, and assume
y = y 0_ g/p * t
Notice in Figure 6 that the velocity does not
increase indefinitely, but rather a terminal
velocity is attained. Experimentation should
then proceed to determine what physical and
corresponding mathematical factors will influence terminal velocity.
y=(Z32/ 1. 5)*(2. 71t(-1. 5*x) )-32/1 .5
X—AXIS: 0.. .50 STEP 5
Y — AXIS : — 260 ... 200 STEP
The height of objects in free fall can be
studied using equation (2). Obviously
feathers behave differently from cannon
balls. Examples of different free falls are plotted in Figure 7, and the student can explore
the different possibilities.
Person with parachute p=1.5, V,=21 ft/sec
Person with no parachute p=0.15, V,=213 ft/sec
Person with unbuttoned overcoat p=0.5, V,'64 ft/sec
Funi:t i cns:
y5000-213.*x+1/. 15* ('213.
X—AXIS: 0.. .40 STEP 5
Damped Oscillatory Motion
Damped oscillatory motion refers to the
progressive reduction or suppression of the
oscillation of a system. Vibrating springs and
pendulums gradually fading out are both examples of decaying periodic behavior.
Modelling periodic phenomena with
decreasing amplitudes can be accomplished
using a trigonometric function t and an exponential function exp(-bx). This is of
the form A*t(x)*exp(_bx) (see Figure 8).
Students can study the examples provided
and attempt to model an actual spring or
pendulum in oscillation, by adjusting
parameters in the general function. At this
point in their development, students should
be able to research applications of this type
of functional variation. They should also be
able to generate reverse damping and suggest applications, e.g., destructive vibration
amplification, sound theory.
Fun ';ti ':'ns:
y=4*si n (2*'-3. 14)
X—AXIS : 0.. .15 STEP
If a population reproduces itself at a rate
proportional to its size, then the population
P is given by
P(t) = P0 exp(kt),
where k is the % growth per year as a
An investigation of future world population
under the dual assumptions of uninhibited
growth and percentage increases ranging
from 0.5% to 3% is'shown in-Figure 9.
y5*2. 7lt O1O*x)
y*2. 71tC . 015*x)
y*2. 71t( .020*x)
y*2. 71i'( ..030*x)
y=*2. 71 -t .005*x)
y=5*2. 71 -r O(-.)o*x:)
X—AXIS: — 100 ... 100 STEP 10
Y—AXIS: 0. .50 STEP 5
If a population declines at a rate proportional to the present population, then
P(t) = P0exp(-kt),
where k is the % yearly decline as a decimal.
The constant k for carbon-14 is .00012. The
table of values and graph in Figure 10 can
be used to show, for example, that the half
life of the isotope is something like 5750
years and that if only 70% of an object's
C-14 remains, the object is about 3000 years
I I ri
---hX]tS: •. . . 20 1000 STED
Y -Am 1rs:
If a population is constrained by certain limits to growth such that
dP = kP(L-P),
where L is the limit and k the growth rate,
then the population is given by
P(t) - P0+exp(_Lkt)*(L_P0)
The inflection point of the curve, where the rate of change reaches a maximum and then
declines, is P = 0.5*L.
The graph and table of values in Figure 11 depict world population under the contraint of
L=30 billion. An informed discussion of the issue could be carried on using the model as
a quantitative basis.
Fun': t j csns:
,= : 5*30: / (5+2. 71 t (-30*. 001 *x) * ( 30-5))
X — AXIS: B...50 STEP 5
Figure 11 continues . .
(5*30) / '5-'-2. 71i(-3c:..o2o*:*:3o-5: :)
The computer can make the meaningful application of mathematics a reality for more
students than ever before. B.C. still lacks a
program to take advantage of the powers of
the computer to help students master the
fundamentals of quantitative literacy. This
failure remains a chronic problem in B.C.
Secondary School Calculus
Harold Brochmann is a computer consultant with the North Vancouver School District. This is the first in a series
of three articles about teaching calculus in secondary schools.
n their first year of college or university,
I many students take one or more courses
in calculus. The courses are quite difficult,
and many students do a great deal of work
to get through. Many students fail.
Getting a decent grade in first-year calculus
matters not only to the students, but also to
their secondary school mathematics teachers.
Several post-secondary institutions make a
regular practice of releasing the results of the
first-year exams, broken down by school attended. This practice, together with declining enrolment, professional pride, and plain
need to cover one's butt, prompts secondary
math teachers to go to great lengths to assure
their students' success in the year after they
part company. Enter Calculus 12.
Calculus 12 is offered as an "elective" in
many school districts, and those that don't
offer it yet are actively preparing for the inevitability. Privately, teachers will agree that
if the "competition" didn't offer such a
course, there would be little pedagogic
justification for doing so, but "under the circumstances, we have no choice."
Ah yes, calculus. I spent several years differentiating and integrating without the
slightest idea of what it was all about.
Calculus, like so many topics in mathematics, is usually presented to the student
as a ritual to be performed according to rules
laid down by person or persons unknown
for purposes also unknown, or at least not
to be revealed just yet. My Grade 9s used
to ask, "Why do we have to learn algebra
anyway?" The answer was "Until you have
learned it, you won't be able to understand
the reason for learning it. Therefore it's no
use trying to explain."
Are the reasons for doing calculus the way
we do it any better?
What is calculus anyway?
I've finally figured it out. Calculus is a collection of symbol-manipulating algorithms,
techniques, tricks, rules, and shortcuts that
allows one to find slopes of, and areas under,
curves. Do I hear any objections?
Now that we have a definition, we are left
with the questions of which should be taught
• the traditional collection of symbolmanipulating algorithms, etc.?
• the conceptual basis from which the
algorithms are derived?
• what you can do with them?
My vote is for the last of these. The first,
as far as I'm concerned, should be taught
last, and not as the core content of a first
course. The concepts of limits, like all concepts, cannot effectively be taught, but will
form in students as a result of using them in
Now the question arises as to whether it is
possible to learn about applications of
calculus without first knowing the contents
of traditional calculus courses.
Is it possible to acquire the concept of square
root and learn to solve problems involving
square roots without knowing one of the
algorithms for finding it? I say yes. Using a
calculator to solve application problems
surely leads to sound conceptual understanding. I suspect that few teachers bother with
the algorithmic treatment at all.
How many of us could calculate the sine of
74 degrees from first principles? Does this
stop us from teaching trigonometry?
How many of us who did lengthy numerical
calculations using logarithm tables can find
loge 526.7 with a pencil and paper?
Can you find 866.45 without your calculator?
So, why not teach calculus applications with
a computer—at least as a prelude to the
traditional treatment for the purpose of providing conceptual grounding? I submit that
this is a more appropriate course content
for Calculus 12 than attempting to pre-teach
the material the students will get next
This paper is not an exhaustive course
description. It illustrates a general approach
to a unit that should, in my opinion, be
taught prior to a traditional calculus
There are three articles in this Vector series:
1. Introduction to integration using the wellknown computer language BASIC.
2. Applications of integration using computer utility programs.
3. Computer-based differential calculus
What Is Numerical Integration?
This is the graph of Y = X 2over the range
0 <'X <10. Our purpose is to determine the
area between the curve and the X axis.
8 9 10
The midpoint of the rectangle is at
X = (1/210
The height of the rectangle at the midpoint is
Y = [((1/2)*10)-21/10
A = 10*[((1/2)*10)^21/10
If we make two rectangles, then their midpoints are at
'((1/2)/2). *10 and
( ( 3/2)/2)10
The, heights' are
[(( ( 1/2)/2)*10y2]/10
[(( ( 3/2)/2)*10)A2]/10
Combined area of both rectangles is
A = 10/2*f[(((1/2)/2)*10)2}/10
With three rectangles, we get midpoints at
xl = ((1/2)/3)*10,
X 2 = ( ( 3/2)/3)*10 and
X3 = ( ( 5/2)/3)*10
The heights of these rectangles are
Yl = [(((1/2)/3)*10y21/10,
Y2 = [(( ( 3/2)/3)*10)A2}/10 and
Y3 = [(( ( 5/2)/3)*10y21/10
I I I I
This gives a combined area of
A = 10/3*1[(((1/2)/3*10)^2]/10
With four rectangles, we get midpoints at
xl = ((1/2)/4)*10,
X2 = ( ( 3/2)/4)*10,
X3 = (( / 2)/4)*10 and
X4 = ((7/2)/4)'10
The heights of these rectangles are
Yl = [(((1/2)/4*10)^2]/10,
Y2 = [(( ( 3/2)/4*10)A2]/10,
Y3 = [(( ( 5/2)/4*10)^2]/10 and
Y4 = [(((7/2)/4*10)A21/10,
This gives a combined area of
A = 10/4*t[(((1/2)/4)*10y21/10
8 9 10
In general, the sum of the areas for n rectangles is
A = ( 1O/n)*t[(((1/2)/n)*10Y2]/1O
We now make three statements:
1. The sum of the areas of the rectangles we have drawn is very approximately equal
to the area under the curve.
2. The more rectangles we use, the closer the sum of the areas will be to the area under
3. If we use a sufficiently large number of rectangles, then the sum of their areas will
be close enough to the area under the curve.
How many rectangles are needed in order for
the results to be close enough?
The answer to this question is determined-by
using a computer program that finds the sum
of the areas of 10, 100, 1000, and 10 000
such rectangles. The program is written in
BASIC and looks like this:
10 A0 : N=10
20 FOR Z=1 TO 2*N1 STEP 2
30 A =A +((((2*Z1)/2)/n*1O)2)/1O
40 NEXT Z
60 PRINT A
Running this program yields an area of
33.25. Changing line 10 to N=100 gives
33.3325. Using N = 1000 gives 33.33251, and
with N=10 000 we get 33.333333.
As it happens, the area under the curve is
theoretically 33.3.... With 100 rectangles, we
get four digits of accuracy, which is adequate
for most purposes.
In this example, we found the area under the
curve Y = . Generalizing this to Y = f(X),
the sum of the rectangles that approximate
the area under the curve for the range
0< X <10 becomes
A = 10/n*tf(((1/2)/n)*10)
90 PRINT A
Line 30 in the program defines the function
used in line 60. To use the program for other
curves and/or for other ranges of X make
changes to line 20 and 30.
Also in this example, we used the range
0<X<10. Generalizing this for range
Xmin<X<Xmax, we get
Integral calculus is the branch of mathematics that deals with finding areas under
curves. What we have been doing here is
The conventional mathematical notation for
the last example is
A = ((XminXmax)/fl
Xmin) +X min )
+ f(((3/2)/n)*(X max Xmin)+Xmin)
' max Xmin)+Xmin)
± f(((.7/2)/n)*(X max Xmin)±Xmin)
f[(((2*n_1)/2)/n)* ( Xmax - Xmin) + XminIl
The final version of the computer program
which calculates the approximate area under
any curve Y=f(X), for example Y=3*Xfr52_7
over any range, for example, 5< X <17 now
10 A0 : N1000
20 MX10 : MN0
40 FOR Z=1 TO 2*N_1 STEP 2
Note that the. BASIC program and the con-:
ventional mathematical notation (CMN) are
two different ways of expressing (for all
practical intents and purposes) the same
thing. The CMN version represents the
theoretical limit to the area under the curve,
while the other actually calculates the area
to some (number of digits of) accuracy
depending on the computing facilties used,
and the precision. 5specified. The program
finds a solution quickly because it is machine
executable and it is adaptable to any function. Integration by conventional techniques
always requires a great deal of training, is
at best time-consuming, and for many functions, extremely difficult.'
Applying the Computer Program
to Familiar Examples
We will practise using our computer program to find some areas that we are already
familiar with. Consider first the area of this
The equation for the graph of a circle is
x2 + y2 =
from which we get
Consider now this quarter circle
We know that this area is 5*4 + 5*2/2 =
Substituting the values from the diagram into the computer program.
10 A0 : N1000
40 FOR Z=1 TO 2*N_1 STEP 2
70 NEXT Z
90 PRINT A
Running this program also yields 25.
The conventional mathematical notation for
what we have done is
(2/5 * x + 16/5)dx = 25
To find the area under the curve, the first
three lines of the computer program become
10 A0 : N100
30 DEFFNA(X) = SQR(52-X2)
This will yield the area of a quarter circle.
To find the area of the complete circle, we
change line 80 to read
80 A=A*(MXMN)/N * 4
Running the program produces 78.5484.
This compares to
* 52 = 78.5398.
The slight discrepancies observed are related
to compromises made in the computer's
what we have done is
The conventional mathematics notation for
Next, we find the area bounded by a SINE CURVE
10 A0 N100
20 MX2*ir : MNO
30 DEFFNA(X) = SIN(X)
the program produces an area of zero. This is because the area below the curve is considered
we get 2.
The conventional mathematics notation for what we have done is
(sin x)dx = 2
On the right is the graph plotted over the
range -2<X<'2 and 0< Y < 100.
Using the computer program, we find that
the numerical solution is 0.111111....
The purpose of integration is to find the area under a curve. Although this area can be expressed symbolically, it is the numerical value which is needed in applications—and it can
be found quickly to any needed degree of precision using a simple, virtually intuitive,
"homemade" computer program.
It is always desirable to have available the graphs of the functions we integrate. In the first
three examples, we saw, the graphs of the functions were familiar, and plotting the graphs
is quite straightforward. Plotting the graph of the last function involves some pretty tedious
calculations. In the next article of this series, we have a look at another "homemade" computer program that not only performs the calculations, but also draws the graphs for us. We
also look at a commercial program that determines appropriate "best fit" functions from raw
Report on the 27th Northwest
• Mathematics Conference
"Mathematics for the '90s"
Ian deGroot is president of the British Columbia Association of Mathematics Teachers; he teaches in North Vancouver.
he Empress Hotel in Victoria was a
T gracious hostess to more than two thousand mathematics educators from Oregon,
Washington, and British Columbia. It was
great to renew friendships with our colleagues and to welcome distinguished
visitors such as Shirley Frye, president of the
National Council of Teachers of
The organizing committee is to be commended for the superb program and smooth
organization. Everything went off without
a hitch. We tip our hats to Bob Belcher,
Trevor Calkins, Brent Carberry, Dana
Close, John Clemens, Joan Foster, Liz
Hamblett, Lyle Melnyck, Betty Milne,
Daphne Morris, Wendy Swonnell, Brian
Tetlow, and Jim Vance.
A standing-room-only audience in the
Empress Ballroom greeted keynote speaker
Frank Ebos as he provided his insights into
the teaching and learning process. I knew
then that it was going to be the best
Friday's sessions continued in the same
positive manner. Lola May, no stranger to
B.C. math teachers, was welcomed by a
packed audience in Newcombe Auditorium.
She held the crowd in her hands with her
particular brand of energy and charm as she
talked. about calculator 'and computer
Richard Skemp, of Coventry, England,
presented his view on the teaching of
mathematics, and Shirley Frye, of the
NCTM, enlightened us with her plan for the
NCTM standards for school mathematics.
That document will certainly, change
mathematics instruction in North America..
I believe that we will be hearing about and
experiencing the objectives of these standards over the next few years.
Brendan Kelly was his usual dynamic self,
with his "New School Mathematics Programs for the 1990s." Frank Ebos was inventive when he had to be as he made his second
contribution with "Motivation in Mathematics."
No Northwest conference would be complete without the presence of Father Stanley
Bezuska, of Boston College. His topic,
"Mathematics for the Countdown to the
Twenty-First Century," was the icing on the
cake for many of tHe participants. Harold
Jacobs, the teacher's teacher, presented
"Mathematical Snapshots of 1988" to
another responsive audience, focussing on
the conference theme of Mathematics for the
We should not forget the usual array of
talented local presenters: Jim Sherrill, John
Kiassen, Tom O'Shea, Zoe Wakelin, Becky
Matthews, Grace Fraser, Don Vance, Ivan
Johnson, Jim Nakamoto, Dennis Hamaguchi, Les Dukowski, Bill Kokoskin, Jim
Mennie, Bill Toth, Doug Owens, Mark
Mahovlich, Tom Schroeder, David Robitáille, Craig Newell, -Alan Taylor, Gary
Phillips; David Parkinson, Jim Swift, and
Walter Szetela: Have I missed anyone?
These dedicated- mathematics educators have
shared their ideas and expertise with us for
many conferences and have helped to make
Our British Columbia organization one of the
best in Canada.
I was particularly impressed with the two
sessions that I attended on the graphing
calculators. Jim McManüs and Bob Hill, of
Albany, Oregon gave us hands-on experience On the use of the HP 28S symbol
manipulator, a truly incredible instrument;
I call mine DEBE, an acronym for does
everything but eat. In the second session that
I attended, Duane Olson, of Lacey,
Washington, showed how the CASIO 7000
g graphing calculator will eventually be a
necessity in every senior mathematics class.
I was delighted with the level of debate
at the BCAMT annual general meeting,
which was the final session on Friday afternoon. We then all adjourned to sample
the hospitality of various publishers'
Saturday morning, it was my turn to present, and I chose the topic "Motivation, the
Key to Success in Applied Math."
Dr. Martin Collis,, -of the University of Victoria, closed the conference with his fitting
talk on the unchanging qualities a teacher
needs-to teach in a changing world.
I left Victoria with many memories of the
conference: I won't forget Eldon Egbers, the
supervisor of mathematics from Washington
State, who told me that he -has attended all
27 Northwest conferences! I won't forget the
participation of the publishers and their appropriate displays. I will -not forget the
warmth and enthusiasm of all of the participants. I can hardly wait for the Seattle
conference in October 1989.
The SFU Master's Program in
Secondary School Mathematics Education
Tom O'Shea is an associate professor in SFU's Faculty of Education.
t the present time, there appears to be
A renewed interest among secondary
mathematics teachers in a graduate program
structured to meet their needs. Secondary
mathematics teachers are facing a revised
curriculum that will be fully in place in 1990.
This curriculum will contain a renewed emphasis on geometry, a new strand on probability and statistics, and a new unit on
calculus. Our graduate program is designed
to help teachers develop insights into the
nature of mathematics and its place in the
school curriculum. Teachers will also
become familiar with research on how
secondary students learn mathematics, and
current ideas on how best to teach the subject. This is an opportune time for teachers
to examine curriculum changes and upgrade
The theme of the graduate program emphasizes the human aspects of mathematics:
the role of mathematics in society and the
natural development of mathematics as a
growing, changing, entity. Developments in
the school mathematics curriculum, and in
pedagogy, will be related to historical,
cultural, and psychological forces operating
within society. The goal is to produce
teachers who have a broad understanding of
mathematics and mathematics education,
and who will be qualified to deal with rapid
curriculum change in the next several
decades. To this end, the Faculty of Education and the Mathematics Department at
Simon Fraser University have collaborated
to develop five courses specifically designed
for secondary mathematics teachers.
Course Structure of the Program
Students require at least 23 credits and a
thesis for the Master of Science (Education)
degree. Equal numbers of courses will be
taken in the Mathematics Department and
in the Faculty of Education. The Mathematics Department has developed three new
courses for the program, and two new
courses have been developed in the Faculty
of Education. All courses are four credits.
The program has been designed so that
teachers can take one course each semester
at the same time as they continue their normal teaching career. Classes will be held
from 16:30 to 20:30 one day a week for 13
weeks each semester. Classes will be held at
SFU's Burnaby campus or perhaps at the
Downtown campus, depending on where the
majority of students live. At the present
time, the program is feasible only for those
students within commuting distance of these
Courses offered by the Mathematics
1. MATH 601-4 Foundations of Mathematics
"Crises in mathematics, their historical
and philosophical background, and their
resolution." In this course, all necessary
mathematics will be taught as part of the
course. The intent is to show mathematics
in the making rather than as a finished product. Various critical periods in the development of mathematics will be examined. The
emphasis will be on the mathematical problem of the time and how the problem was
2. MATH 602-4 Geometry
"Euclidean and non-Euclidean geometries.
Klein's Erlanger program." In this course,
students will look at the development of
geometry to the present time. Emphasis will
be placed on how geometry was interpreted
at various times in history, including the influence of Euclidean geometry on philosophy,
and the crisis precipitated by the discovery
of non-Euclidean geometry. Modern geometrical treatment will include transformations of the plane.
the teacher." Students will examine implications for instruction of the ideas of various
mathematics educators and school of
thought, for example, Dienes, Gattegno,
Skemp, and the constructivist school. In
part, the course will focus on teaching
geometry, anticipating the content of the
geometry course in mathematics, and on applications and problem solving reflecting the
content of the mathematical modelling
3. One elective in the Faculty of Education.
3. MATH 603-4 Mathematical Modelling
"Introduction to mathematical modelling
using algebraic and geometric techniques,
along with techniques using calculus." This
course is designed to give students experience
in creating and fitting mathematical models
to real-world problems. It is based on recommendations of the MAA's committee on the
Undergraduate Program in Mathematics,
and includes modelling using the computer
Courses offered by the Faculty of Education
1. EDUC 846-4 Foundations of Mathematics
"An examination of historical, cultural,
and psychological forces shaping the secondary school mathematics curriculum. Current
developments in mathematics curriculum
and in mathematics education research." The
emphasis will be on the historical underpinning of the curriculum and the cyclical
nature of reform in mathematics education.
The course will have a structure similar to
the one on the foundations of mathematics
and will focus on critical periods in the
development of the school mathematics
2. EDUC 647-4 Teaching and Learning
"The theory and practice of mathematics
teaching at the secondary level. Emphasis on
the nature of the learner and the function of
The program is intended to meet the needs
of practising secondary mathematics
teachers. The mathematical content is
designed to increase students' breadth of
understanding, and will not require students
to recall details of their undergraduate
mathematics courses. We recognize that
teachersinterested in this program may have
completed their university work some time
ago, and the courses are designed with this
In addition to the standard requirements for
entry to the SFU graduate program, the
following prerequisites are normally
• sufficient mathematics background for
certification to teach secondary school
• mathematics teaching experience in
• an undergraduate methods of teaching
mathematics course, for example, SFU's
EDUC 475-4 Designs for Learning:
Timetable for Implementation
The program is designed for a cohort of
students. The entire program is a cooperative venture between mathematics and
education, and each course is designed to
complement the other courses. The initial intake will be in the fall of 1989, with a second
intake planned for two years after that time.
This schedule may change, subject to
MATH 601-4: Foundations of Mathematics
supervision of students' theses. Students will
be encouraged to undertake work in which
they can apply the ideas from the courses to
The deadline for application to enter in fall
1989, is March 1, 1989.
For further information, please contact:
Dr. Tom O'Shea
Faculty of Education
Simon Fraser University
Burnaby, BC V5A 1S6
EDUC 847-4: Teaching and Learning
MATH 602-4: Geometry
EDUC 846-4: Foundations of Mathematics
MATH 603-4: Mathematical Modeling
Dr. Harvey Gerber
Department of Mathematics and Statistics Simon Fraser University
Burnaby, BC V5A 1S6
Education: One elective course
For application forms, please write directly
It is expected that students will take one year
to complete a thesis. Members of the Facul ty of Education and the Mathematics
Department will share responsibility for
Graduate Programs Office Faculty of Education
Simon Fraser University
Burnaby, BC V5A 1S6
National Council of Teachers of Mathematics_______
Whereas, mathematical literacy is essential for citizens to function effectively in society; and,
Whereas, mathematics is used every day—both in the home and in the
Whereas, the language and processes of mathematics are basic to all other
Whereas, our expanding technologically based society demands increased
awareness and competence in mathematics; and,
Whereas, school curricula in mathematics provide the foundation for
meeting the above needs;
Now, therefore, I, Shirley M. Frye, President of the National Council of
Teachers of Mathematics, do hereby proclaim the month of
April 1989 as
Mathematics Education Month
To be observed in schools and communities in recognizing the increased
importance of mathematics in our lives.
In witness thereof, I have hereunto set my hand and caused the corporate
seal of the National Council of Teachers of Mathematics to be
affixed on this 1st day of September 1988.
You can count on it
A special new video designed to encourage students to get excited about math.
This fast-paced video shows students enthusiastically dealing
with everyday math problems and having lots of fun at it. The
video can be purchased by schools and school groups for the
cost of duplication—just $8.75 pet copy.
If you would like to order this video, please write:
VCA Teletronics, 1801 Royal Lane, Suite 1010
Dallas, TX 75229. Or call: 1-800-TI-CARES
© Copyiight 1988, TI
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