1 CHAPTER 1 BACKGROUND TO THE STUDY INTRODUCTION This is the introductory part of the study. The chapter begins with the Background to the Study, the Perceived Problem, the Diagnosis (Evidence and Causes), the Statement of the problem, the Purpose of the study, the Objective of the Study, the Research questions, the Significance of the study, and the Definition of terms. BACKGROUND TO THE STUDY Vectors, one of such aspects of mathematics is very essential due to its frequent application in our daily lives. Every now and then we either think of our movement in a specific way or direction with some specific distance to the place of our interest or vice versa. This mainly involves vectors. The systematic study and use of vectors were a 19 th and early 20 th century phenomenon. The Irish mathematician William Rowan Hamilton also contributed to the development of the concept of vectors in mathematics. Hamilton’s quaternions (the cardinal number that is the sum of three and one) were written as, q = w + ix + jy + kz, (where w, x, y, and z were real numbers). He quickly realized that his quaternions consisted of two distinct parts. The first term, which he called the scalar and the other term (x, y, z) for its three rectangular components or projections on three rectangular axes. He decided to call the trinomial expression itself, as well as the line which it represents, a vector.
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1
CHAPTER 1
BACKGROUND TO THE STUDY
INTRODUCTION
This is the introductory part of the study. The chapter begins with the Background to the
Study, the Perceived Problem, the Diagnosis (Evidence and Causes), the Statement of the
problem, the Purpose of the study, the Objective of the Study, the Research questions, the
Significance of the study, and the Definition of terms.
BACKGROUND TO THE STUDY
Vectors, one of such aspects of mathematics is very essential due to its frequent
application in our daily lives. Every now and then we either think of our movement in a
specific way or direction with some specific distance to the place of our interest or vice
versa. This mainly involves vectors.
The systematic study and use of vectors were a 19th and early 20th century phenomenon.
The Irish mathematician William Rowan Hamilton also contributed to the development
of the concept of vectors in mathematics.
Hamilton’s quaternions (the cardinal number that is the sum of three and one) were
written as, q = w + ix + jy + kz, (where w, x, y, and z were real numbers). He quickly
realized that his quaternions consisted of two distinct parts. The first term, which he
called the scalar and the other term (x, y, z) for its three rectangular components or
projections on three rectangular axes. He decided to call the trinomial expression itself, as
well as the line which it represents, a vector.
2
A vector according to the Oxford English Dictionary is a Latin word which means “one
who carries”; its’ Latin is “veho” which means “I carry”. This is the reason why we say
that a vector GH means the letter G is being carried to the letter H or one a movement
from the point G to the point H.
Dyke and Whitworth (1992), and Clarke (1974) share a common view on the definition
of a vector that, a vector is a quantity with both magnitude and direction. In other words,
a vector is a variable quantity that can be resolved into components or a straight line
segment whose length is magnitude and whose orientation in space is direction.
A statement like “a saloon car is parked outside the classroom and to locate it, move 30
meters” may provide enough information to stimulate your interest, yet not enough
information to find the car. The direction of the movement (displacement) required to
find the car has not been fully described.
On the other hand, suppose the statement is put like: “a saloon car is parked outside the
classroom; to locate it, move from the center of the classroom through the doorway 30
meters in a direction 45o to the west”, this statement provides a complete description of
the displacement vector, (that is both magnitude (30 meters) and direction (45o to the
west) relative to a reference of the starting position (the center of the classroom door).
Vector quantities are not fully described unless both magnitude and direction are listed.
Many fields of study and technological advancement like, deep-sea diving, quantity
surveyors, navigators, building contractors, aerographers, etc all apply the concept of
vector mathematics in their daily dealings. Aerographers apply vectors and resultant
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forces primarily for computing radiological fallout and drift calculations for search and
rescue operations.
Despite the integration of vectors in our daily life activities, it is a mathematical concept
that terrifies many students and even teachers alike.
Few candidates who attempted questions on vectors in the Senior Secondary School
Certificate Examination showed lack of mastery of the topic, according to the Chief
Examiner’s Report of the West African Examination Council (WAEC: 2003) on the
Senior Secondary School Certificate Examination. The chief examiner lamented on the
fact that the few students who attempted the only question on vectors could not find
vectors AB and AC and so could not calculate angle ABC of the given question.
Vector resolution has been a problem of many senior high schools and teacher training
colleges in Ghana especially the second year students of Presbyterian College of
Education, Akropong – Akuapem.
My personal interview with students during lesson periods showed that they found the
concept very difficult because only theoretical aspects are presented to them.
Again careful study of students’ mathematics exercise sheets clearly revealed that few
exercises were done. This implies that students were not practicing well enough on the
concept to acquire the needed skills.
Also a questionnaire distributed to randomly selected students from 2F class evidenced
that, they were mostly given formulas to work with.
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It is against this backdrop that I deem it expedient to look into as to why some students
have difficulties in solving vector problems especially resolution of vectors using the
triangle law and suggest ways by which students could be assisted to improve upon their
problem solving skills as far as resolution of vectors is concerned.
STATEMENT OF THE PROBLEM
I realized through my unstructured interactions with some students at Presbyterian
College of Education, Akropong - Akuapem during my one year internship programme,
that most of them see vectors as a difficult topic. From the responses of the students,
though vectors is one of the most practically applied mathematical concepts, they see it to
be very abstract since it takes only very resourceful teachers to develop teaching-learning
materials for it’s teaching. Sometimes it is left untreated due to teaching-learning
materials or difficulty in concept development, and those teachers who attempt to teach it
either give only formulas to students or overlook some aspects like resolution of vectors.
Students remember techniques without understanding why and how it works and
therefore finding it difficult to apply the techniques in solving problems in vectors.
This creates conceptual gaps in many important ideas, thus making understanding of
other related topics difficult. This in one way or the other promotes rote learning in
students.
The poor performance of students and the inability of most of the students to correctly
apply the triangle rule to resolve vectors have also necessitated this study.
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PURPOSE OF THE STUDY
This work was designed to find pedagogy to be used in teaching resolution of vectors and
to suggest suitable techniques for teachers to help make their teaching more stimulating
and effective in relation to vectors.
Another major purpose is to enable students with limited prior knowledge of using the
triangle rule to resolve vectors build on their informal knowledge to give meaning to
vectors.
RESEARCH OBJECTIVE
The main objectives of this work are:
• To enhance the use of the triangle rule in the resolution of vectors making the
learning of vectors more meaningful and applicable to solving real life problems.
• To assist students apply the triangle rule correctly to resolve vectors.
REASEARCH QUESTIONS
This research was designed to provide answers to the questions below:
• What real life situations or experiences could be associated to the topic to
make it more meaningful to aid students understanding of the resolution of
vectors?
• How can student’s knowledge of the triangle rule be improved so that they
can apply it to resolve vectors correctly?
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SIGNIFICANCE OF THE STUDY
The study is expected to help students develop a positive attitude towards the learning of
mathematics in general and vectors in particular.
To make available to teachers additional techniques of teaching vector resolution.
On the whole, this work will serve as a source of reference to others who will be
researching into the resolution of vectors and other related topics.
SCOPE OF THE STUDY
The study covers a sample of second year Diploma in Basic Education (D.B.E) students
of the Presbyterian College of Education, (P.C.E) Akropong-Akuapem.
DEFINITION OF TERMS
Pedagogy - The principles and methods of instruction or the activities of educating or
instructing; activities that impart knowledge or skill
Semantics - The meaning of a word, phrase, sentence, or text
Syntax - The grammatical arrangement of words in sentences or a systematic orderly
arrangement
Quaternions - The cardinal number that is the sum of three and one
Vector Mathematics – The aspect of mathematics that deals extensively with vectors
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CHAPTER 2
REVIEW OF RELATED LITERATURE
INTRODUCTION
This chapter sought to review related literature on the topic under study. The chapter
highlights the work of other researchers whose research works has bearing on the
problem under discussion. The chapter is divided into three parts. These include:
• The Theoretical View Point
• The Empirical View Point
• Summary
The Theoretical View Point
Mathematics has evolved over the years through the use of abstraction and systematic
study of the shapes and motions of physical objects.
Abbiw, et al (1991) emphasized the fact that vectors unlike scalars have both magnitude
and direction. So if someone is asked to move from a point say A to another point B, it is
not enough to tell him the distance between A and B. He needs to know the direction of B
from A as well. The diagram below is in support of this argument: (Fig. 1)
Figure 1
N (0000)
0600
B
C
DA
5cm
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The figure (a circle) has its center A and radius 5cm. On its circumference are the points
B, C and D. They argue that the lengths are equal (i. e. |AB| = |AC| = |AD| = 5cm). The
only difference among them is the angle (bearing) from B, C, and D relative to A. It will
therefore be ambiguous to ask someone to move from one point A to another point B of
any distance without telling the angle, direction or sense of movement.
The conclusion that is drawn is that all though |AB| = |AC| = |AD| = 5cm, AB ≠ AC ≠
AD. Since their senses of movement are not the same, this presupposes that not until
students are able to make out these differences, operation on vectors would always be a
problem.
Turner (1986) uses triangle, parallelogram and ordinary quadrilateral to show the closure,
associative and commutative laws of addition of vector quantities. The introduction of
inverse of a vector and the zero vectors were defined by him. He again uses the same idea
of inverse vector to introduce subtraction of vectors.
Tallock (1970) has also given a straight forward definition of magnitude of a vector,
equal vectors and the unit vector. He expressed three essential features of a vector as
follows:
• Magnitude or length of a vector
• A direction in space of a vector
• A sense of a vector
Spiegel (1972) gives an outline of definitions which are fundamental to the analysis of
vector algebra. These were some of the definitions he gave;
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• Two vectors a and b are equal if they have the same magnitude and direction
regardless of their initial positions.
• The sum or resultant of vectors a and b is the vector c, formed by placing the
initial point of b on the terminal point of a and then pining the initial point of c to
the initial point of a and joining it to the terminal point of b. This sum is written
as a + b = c.
Sharp, (1968) defined addition of vectors using a triangle. The author carefully stated
with an appropriate illustration that AB + BC = AC so that the sum of a vector is also a
vector. He explains that to construct a vector which is the sum of two vectors, either the
triangle method or the parallelogram method may be used.
The Empirical Viewpoint
Simon Stevin of Bruges published treatments of statics (i.e. the study of bodies at rest).
He was the first Mathematician of the 16th Century to continue the work of Archimedes
in statics which he used the triangle of forces to compound two forces.
Quansah J.E. (2004) in his research work also tackles some of the factors that hinder
successful solution to vector problems. He goes on to say that the use of concrete
teaching and learning materials in interpreting the vector diagrams and graphs to illustrate
the vectors concept to students must be adopted by teachers.
Biggs and Sutton (1983) in their book “Teaching Mathematics” stated that the addition of
vectors may be defined by the Triangle laws.
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They argued that the sums of two vectors say a and b may be represented by the triangle
law by the two journeys: AB + BC = AC.
The figures below explains this argument: (Fig 2)
Figure 2
Figure 2.0
This is to show that to move from A to C, one can go through the straight line AC or
moving through line AB first and then followed by line BC to C. This implies that
AB + BC = AC (Triangle law).
Also in figure 2.0, you can go through the straight line PR (r) or through PQ (p) and then
followed by QR (q).
According to www.engin.brown.edu, (Brown University) a vector is a mathematical
object that has magnitude and direction, and satisfies the laws of vector addition. A