COURSE OUTLINES
EARLY GRADE
UPPER PRIMARY
JHS
COURSE CONTENTS
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HANDOUT 1
HANDOUT 2
HANDOUT 3
TEACHER ATTITUDE, BELIEFS AND VALUES ABOUT MATHEMATICS
UNIT ONE
WHY DO WE TEACH MATHEMATICS IN SCHOOL?
Lesson Objectives
By the end of this unit, it is expected of you to answer the following questions.
1. What is mathematics?
2. Explain the importance of mathematics to the upper primary teacher
3. How does mathematics relate to the society?
4. What does it mean to learn and teach mathematics?
Area of concentration
1 Definitions Mathematics
2 The relationship between mathematics and science
3 Importance of mathematics teaching and learning
4 Mathematics role in the society
5 The meaning of mathematics learning and teaching
6 Mathematical terms
7 Mathematical operations
WHAT IS MATHEMATICS?
Mathematics is one of the oldest of all field of study. It is often referred to, as used, praised, and disparaged and has long been one of the most central components of human thought.
Mathematics is a word whose meaning has varied widely from time to time and from person to person. There is no general agreement on precisely what mathematics is. Definition of mathematics is vary in relation to the type of investigation of the definer. Each definition indicates the aspect of mathematics which the investigator favors. This implies that our ideas of mathematics depend so much on our own experiences and our own knowledge of subject. Some may think of calculations involving addition, subtraction, multiplication and division. Some may want to include topics like algebra, geometry, and trigonometry. Others feel it involves logical thinking. From all these mathematics is seen as been used in finding answers to questions and problems, which arise, in every life and trades and professions.
This session discusses some definitions of mathematics.
Objectives
By the end of the session you should be able to:
a)Explain the definition of mathematics; and
b)Explain the nature of mathematics
According to the James and James Dictionary of Mathematics, ‘’Mathematics is a logical study of shapes, arrangement, quantity and many related concept’’. It is divided into three fields: algebra, analysis and geometry. No clear division can be made since branches have become thoroughly intermingled. Algebra involves numbers and their abstractions, analyses involve continuity and limits, and geometry is concerned with space and related concepts. Technically, mathematics is postulational science in which necessary conclusions are drawn from specified premises.
Mathematics is regarded as a science of quantity and space where symbols forms are used to
express them. It is a science of making general conclusion about quantity and space. Quantity
refers to arithmetic and computations. Space covers geometry, spatial relationship and
theories including the science of measurement. Dedicative science using axioms and 2
definitions and arguments. General conclusion lead to an observation about patterns,
assumptions, dedications and conclusions.
Mathematics can be seen as the servant of many field while been of great importance in its own right. Mathematics thus, is of international form. The nature of mathematics, therefore, depends on the people who do it. It is regarded as the ‘queen’ of sciences because it gives ideas of extension in science.
Morris Kline declared that mathematics is a creative or inventive process, deriving ideas and suggestions from real problems. The process is base upon intuition and contraction, with the life’s sources of this process coming from real problems. The abstractions are to be extracted from the real problems and would have a definite meaning in terms of the situation. Kline sees the physical world as the wellspring for the development for an abstract notion.
It is clear that the greatest of our mathematics creations is the material universe itself and the true nature of mathematic is, it is physical in nature. Mathematics is a symbolic representation of physical reality. Mathematics requires intelligence and the ability to learn. It is a product of intelligence creating or discovering a way to successfully represent physical reality with symbols.
Richard Skemp described mathematics as the most abstract and so the most powerful of all theatrical systems. It is therefore potentially the most useful, and scientists particular, and also economists, navigators, businessmen and engineers, it indispensable ‘tool’ (data processing tool) for their work. The main problem with mathematics lies in its great abstractness and generality achieved by successive generations of particular intelligent individuals each of whom have been abstracting from or generalizing concept of earlier generations.
An eminent English mathematician philosopher, Bertrand Russell regarded mathematics as the subject in which we never know what we are talking about nor whether what we are saying is true; We normally use undefined terms to begin explanations or discoveries e.g. point, line, etc and we try to define other objects in terms of this undefined terms and later make prepositions. Mathematics relies on logic and creativity, and it is pursued both for a variety of practical purposes and for its intrinsic interest. For some people the essence of mathematics lies in its beauty and its intellectual challenge. For others, the chief value of mathematics is how it applies to their own work. Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is a prerequisite for scientific literacy. To achieve this, students need to perceive mathematics as part of the scientific endeavour, comprehend the nature of mathematical thinking and becoming familiar with the key mathematical ideas and skills (Science for all Americans).
It is evident that mathematics is applicable to many entities. Mathematics is many-sided science. It is an outstanding achievement of human thought. It can not be uniquely defined in a few sentences or paragraphs. But the outsider can gradually develop rich senses of the nature of mathematics by examining it from various perspectives and by doing some of the things that the insiders do. It called for teachers to involve their students in doing mathematics.
Mathematics is a way of organizing our experiences of the world. It enriches our understanding and enables us to communicate and make sense of our experiences. It also gives as enjoyment. By doing mathematics we can solve a range of practical tasks and real-life problems. We use it in many areas of our lives. In mathematic we use ordinary language and special language of mathematics. We need to teach students to use both languages. We can work on problems within mathematics and we can work on the problems that use mathematics as a tool, such as a problem in science, economies, geography, etc
Mathematics can be described and explained but it can also predict what might happen. That is why mathematics is important.
Exercise 1.1
1. Discuss the assertion that mathematics is an abstract subject.
2. What is mathematics? include ideas related to:
(a ) Its definition:
Mathematics is the study of any patterns or relationships, whereas natural science is
concerned only with those patterns that are related to the observable world. Although
Mathematics began long ago in practical problems, it soon focused on abstractions from
the material world, and then on even more abstract relationships among this abstraction. In
this session, we shall learn about the common feature shared by mathematics and science.
Objectives
By the end of the session, you should be able to explain the main features shared between
mathematics and science.
(b) Is it being a creation or a discovery.
Illustrate your answer with specific mathematical ideas and symbols.
MATHEMATICS AND SCIENCE
Mathematics share many of the features of science, such as the belief in underlying order, the ideas of honesty and openness in reporting research, the importance of criticisms by colleagues in judging the value of new work, and essential role play by imagination.
Mathematics is also like science in that it incorporates both finding answers to fundamental questions and solving practical problems.
Through mathematics, people are able to think about the world of objects and happenings and to communicate those thoughts in ways that reveal unity and order. The numbers, lines, angles, shapes, dimensions, averages, probabilities, ratios, operations, cycles, correlation, etc. that make up the world of mathematics enable people to make sense of a universe that otherwise might seem to be hopelessly complicated. Mathematical patterns and relationships have been developed and refined over countries, and the process is still ongoing vigorously. Perhaps that is because today mathematics is used in more fields of endeavour than ever before it has become more essential in everyday life.
As a practical matter, mathematics is a science of patterns and order. Its domain is numbers, chance, form, algorithms and change. Mathematics relies on logic rather than on observation as it stands of truth, yet employs observation, simulation and even experimentation as the means of discovering the truth.(Mathematical Science Educational Board, 1989p. 31)
Mathematics plays a special role in education as a consequence of its universal applicability. Results of mathematics-theorem and theories-are significant and useful. Through theorems, mathematics offers science both a foundation of truth and a standard of certainty. Thus the language of mathematics for the formulation of laws physics is a wonderful gift. Mathematics has made an indelible imprint on part of modern science. Whether planned or unplanned, there is cross-fertilization between science and mathematics in problems, theories and concepts. This has never been greater than it is now. All students should have experience of discovering for themselves that an idea can be represented in different but analogous ways.
One line of research on how people learn and emphasize the helpfulness of making multiple representations of the same idea and translating from one to another. When a student can begin to represent relationships in tables in graphs and symbols and in words, one can be confident that the student has really grasped its meaning. The way students learn to make those representations and translations is to see them and practice them in contexts in which they care about what the answer is. Students engaged in this kind of activity will eventually get the idea of connectedness in mathematics.
As in other sciences, simplicity is one of the highest values in mathematics. Some mathematicians try to identify the smallest set of rules from which many other propositions can be logically derived. Theories and applications in mathematical work influence each other. Sometimes practical problems lead to the development of new mathematics theories; often mathematic developed for it own sake turns out to have new practical applications.
Much of mathematics is done because of it intrinsic interest, without regard to its usefulness. Still, most mathematics does have applications and much work in mathematics is stimulated by applied problems. Science and technology provide a large share of such applications and stimulants. In doing their work, scientist and engineers may attempt to do some useful mathematics themselves, or many called mathematicians for help. The help may be to suggest some already completed mathematics that will suffice or to develop some new mathematics to do the job. On the one hand, there have been some remarkable cases of finding a new use for centuries-old mathematics. On the other hand, the needs of natural science or technology have often led to the formulation of new mathematics.
In school practice, science and technology should contribute to understanding the value of mathematics, and mathematics should help in doing science and engineering. The usefulness of mathematics in science and technology will be clear to students if they experience it often in simple and later sophisticated forms.
Science and technology are rich and especially important contexts in which to learn the value of mathematics and to develop mathematical problem-solving skills. But they are not the only ones. Art, music, social studies, history, physical education and sports, driver education, home economies, and other school subjects are appropriate place to learn, as use, mathematics.
Execise1.2
1. Identify and explain three features that mathematics share with other science.
2. Explain how other disciplines like music, home economies, driver education, and social sciences also create a context for students to learn the value of mathematics.
CYCLE OF INVESTIGATION
We have learnt about how mathematics share many features with science. In this session, we shall learn more about one other characteristic of mathematics as a cycle of investigation. Learning how to solve certain kinds of well-defined mathematics is important for students but does not automatically lead them to a broad understanding of how mathematical investigations are carried out. Mathematics can be characterized as a cycle of investigation that is intended to lead to the development of valid mathematical ideas. It is true that mathematical investigations involve certain processes, but the order is not fixed and the emphasis place on each process varies greatly. There are three components of the cycle –
representation, manipulation, and validation. Each of the three parts of the cycle should be studied in its own right as part of what constitutes leaning mathematics. Students should have the chance to use the entire cycle in carrying out their own mathematical investigations. The purpose of this experience is to produce not professional mathematicians but adults who are familiar with mathematical inquiry.
1. Representation.
The process of representing something by a symbol or expression is taken by many students to refer only to ‘’real things.’’ ‘Let A stand for the area of the floor in this room’’ is easier for young students to grasp than ‘Let Y equal the area of any rectangle.’’ The first students have to be convinced that substituting abstract symbols for actual quantities is worth the effort. Then they need to work their way toward the realization that using symbols to represent abstraction, also pays off in solving problems. Perhaps this means bringing students to see that in the world of mathematics numbers, shapes, operations, symbols and symbols that summarize sets of symbols are ‘’real’’ as blocks, cattle, and cedis, dollars and pounds.
2 Manipulation
As to manipulation, students should bear in mind that there if always a set of rules that must be strictly adhered to and the rules can be made up. That is where the rigor and game-playing spirit of mathematics meet. Imagine some qualities, assign them properties, select some operations, represent everything by symbols, set a problem and then, following the rules of logic that have been adopted, move the symbols around to see what solutions emerge. This process aids in finding solutions to real-life problems.
3 Validation
Validation deals with how good the solutions are. Students are used to working on mathematical problems in which the producers are predetermined and ‘’correct’’ answers are expected. But in real mathematical investigations, a good solution is one that results in new mathematical discoveries or that lead to practical outcomes in science, medicine, engineering, business, or elsewhere. Thus validation in mathematic is a matter of judgment, not authority. For practical purposes, the cycle of investigation employs heavily the use of concrete material. Concrete objects should be employed routinely to help students discover and explain symbolic relationships. A student should come to see that numbers and shapes can be used to describe many things in the world around them. Eventually, they should to realize that just as letters and words make up a language in reading and writing numbers and shapes make up a language in mathematics. The routine line use of concrete objects continues to be essential to help students connect real things and events with their abstract representations. The ability to picture and do things in their heads will be enhanced by frequent reference to real-world applications. Students should be encouraged to describe all sorts of things mathematically in terms of numbers, shapes, and operations.
Exercise 1.3
1. Identify and explain the components of the cycle of investigation.
2. Why should teachers engage students in routine use of concrete materials?
ROLE OF MATHEMATICS IN THE SOCIETY
Mathematics As An Invention And As A Discovery
We have seen mathematics is science and is often characterized by a cycle of investigation in
which our student must be engaged. In this session we shall lean about the view of
mathematics as a creation of the human mind and as a discovery.
Objectives
By the end of the session you should able to:
a)
Explain why mathematics is viewed as an interventions: and
b)
Identify the aspect of mathematics that is discovered.
The basic concepts of mathematics are abstractions from experience e.g whole number and
fractions were suggested by physical counterparts. Many others are creations of the human
mind with or without partial help from experience .For example, the irrational number,√2,
was invented by mathematician to represent the hypotenuse of a right angle triangle with both
arms one unit long. Others are negative numbers, variables to represent changing physical
phenomena like temperature,!"
!#, etc. Numbers and numerals have been created or invented by
the Babylonians and Egyptians. The Romans and Mayans invented their own numerals. The
numerals vary from one group of people to another, but the structure, which shows the
relationship between these numerals is there already though, cannot be handled physically.
The structures need to be discovered and this lead to discovery nature of mathematics, but
proofs, operations, numerals etc. invented and so are creation of the human mind.6
Reality is something that can exist, it is inherent truth about existence of an object, e.g areas,
perimeter, etc all exist in reality. The human mind is to play a great role in the discovery of
all these .For example, Galois discovered the group theory; Leibnitz and others are
associated with the discovery of Calculus, Newton with mechanics etc.
One school of thought believe that mathematics exists in nature, just as certain laws of
physics exists in nature, and that mathematicians discover elements and laws of
mathematics. Other school feels that mathematics is more like the work of art, a painting
that does not exist until the artist; in this case the mathematician creates it. Mathematics
regarded as an art because we use a lot of imagination in mathematics as we do in art. We
create beauty in mathematics, using patterns like painter or a poet, but mathematical beauty
is more lasting than that of art because unlike the poet of painter, the mathematician’s
patterns are made of ideas and ideas wear less with time than words
Exercise 1.4
1.
Explain why mathematics is regarded as
a)
A creation of human mind:
b)
A discovery.
IMPORTANCE OF MATHEMATICS
1. Math is good for the brain
A study done by Dr. Tanya Evans at Stanford University proved that the students who solve
math problems in their daily life have higher logical skills than those students who don’t solve
the problems. Apart from that, the students also solve math problems for their brain exercise.
To make our bodies stay fit, we do exercise. In the same way, to stay our brain healthy and
active, we need to do brain exercise. There are plenty of ways to do brain exercise but the most
effective and robust way to do brain exercise.
2. Math helps you with your finances
Math is also helpful with your finance. With the help of math, you can easily make your
financial budget. You can calculate how much money you have and how you can spend your
money. Almost every single human in the world uses math for their finance. The salaried
person uses math to calculate their expenses and salaries. On the other hand, the businessman
uses math to calculate their profit and loss. They also use it to calculate their loans and many
more. It emerges the importance of business mathematics and also playing a crucial role in
business accounting.
3. Math makes you a better cook
Math is quite useful for the cook. Because in almost every recipe, there is a need to put the
ingredients. Such as one teaspoon turmeric, a tablespoon of garlic powder. As well as the half
cup of flour is the same thing as eight tablespoons of flour. Therefore to make anything
delicious, the cook needs to know the perfect measurements of the ingredients. Apart from that,
if the cook needs to cook food for many people. Then they need to know how much ingredients
are required to cook food for those number of people. Most of the recipes are created to serve
4 or 6 people, but only math can help you to calculate the ingredients to cook food for more
than six or more people.
4. Better problem-solving skills
Problem-solving is one of the most important factors in our life. Math is one of the most
effective ways to increase your analytical thinking. As I mentioned earlier, it also helps us to
improve our logical thinking. Both this analytical thinking and logical thinking help us to
become better problem solvers. In this way, we enhance our ability to solve the problem more 7
effectively. The more we solve mathematics problems, the better we solve the real-life
problems.
5. Every career uses math
There is no profession in the world that doesn’t use math. We know that mathematicians and
scientists rely on mathematical principles to perform their basic work. Engineers also use math
to perform their daily tasks. From blue-collar factory workers to the managerial level white
collar professionals, everyone uses math in their work. The use of math may vary for them;
likewise, the blue-collar workers use the basic arithmetic to operate efficiently in the assembly
lines. On the other hand, blue-collar professionals use advanced math to make managerial
decisions. The importance of mathematics also becomes crucial in the time of salary getting
paid to the employees.
6 Great career options
Mathematics offers a great career opportunity for students. In most careers, employers want to
hire employees who can solve complex problems. If you are good at math and have a keen
ability to solve complex problems, then you are at the top positions for applying for many jobs.
Finance analysis and cost estimation is part of every business. Therefore there are excellent
career options for the students to get math-related jobs. The importance of mathematics is not
just limited to the mathematician, even a fashion designer, a chef, a tailor, a barber all these
careers require the knowledge of mathematics.
7 Math for Fitness
Math is quite useful to stay fit and healthy. With the proper understanding of math, we can
calculate how much food we require in our daily life. How much calorie intake we expect to
stay fit. Apart from that, when it comes to food choice, we can calculate which food will give
us how much calories and fat. Thus we can make the right decision about which food we should
eat to get healthy. Besides, when it comes to gyming, we also calculate how much reps we
should do to pump our body and get it into shape. From calorie intake to calorie burning, you
can calculate almost everything with the help of math. It helps you to get better statistics of
your fitness.
8. Helps you understand the world better
Do you know that everything in nature is based on math? Even the math offers us the golden
ratio formula, which allows us to get the beauty of anything. If you want to judge the beauty
of something, then you can perform the golden ration on that thing to declare it beautiful or
not. Apart from that, you can also find the mathematical figure in the real world, likewise the
hexagonal bee combs, spider webs, triangle mountains, and many more. Every single part of
nature is based on math. You can also put the math calculation to understand nature creations.
9. Time management
Time is the key to success for everyone. Therefore we have to be more calculative for time
management. Math helps us to do better time management. The importance of math is reflected
in time management tasks. With the help of math, you can make a wise decision on how you
can spend your time effectively.
Suppose that you want to reach somewhere, and you have a few minutes, then you can calculate
the minimum time that you are going to require to reach the destination with various modes of
transportation. On the other hand, if you are doing your homework, then you can also calculate
how much time will require to finish the homework. Especially during your mathematics
exams, you can calculate the time that will take to solve the particular question. Time
management is also emerging the importance of mathematics in society.
10. To Save Money
As mentioned earlier, math helps calculate your finance. But do you know that it also helps
you to save your money? Most of the time, we spend money on unwanted stuff. Math helps us
to calculate how much money we will lose to buy that stuff. Apart from that, life is all about
risks. Almost every single person in the world takes the financial risk to become rich. But only
a few get success. Math helps you to calculate the risk before investing money in some financial scheme.
WHAT DOES IT MEAN TO ‘’DO MATHEMATICS’’?
In the previous session, we learnt mathematics as a discovery or human creation. In this session,
we shall discuss the traditional view of school mathematics and what doing mathematics really
entails. Recognize that doing mathematics is engaging in the science of pattern and order.
Discussions Questions:
Ponder over the following questions of a few minutes
a)
Distinguish between traditional and modern view of school mathematics; and
b)
How would you describe what you are doing when you are doing mathematics?
c)
Write a few sentences about what it means to know and do mathematics based on your
experiences.
Traditional View of School Mathematics
Teacher represents the sources of all that is to be known in mathematics.
Ø
Teacher review previous knowledge and move on to explain the new ideas of the lesson.
Ø
Teacher demonstrates to the student how they are to do assigned exercise.
Ø
Even with hands on activities using manipulative, teacher tell students exactly how to use
the materials in the prescribed way.
Ø
Students’ attention is mainly on the teacher’s direction and not on mathematical ideas. The
focus is on getting answers and teacher determines if an answer is correct.
Ø
Student emerge with the view that mathematics is a series of arbitrary rule handed
down by the teacher, who in turn get it from a ‘’reliable source’. Student role is largely
passive. They accept what they are told and attempted to master each new rule. The danger
is student always have to check the teacher for correctness of answers.
As students progress, many refuse to attempt a problem that has not first been explained by
the teacher. Saying ‘’you haven’t shown us how to do this.’’ Student accept that every problem
has a predetermined solution and only one way to solve any problem and that the teacher most
show the way first. This is describe as ‘’follow-the –rules, computation-dominated, answer
oriented view of mathematics’’ it distorts what mathematic is really about. This tradition
system rewards the leaning of rules but offer little opportunity actually to do mathematic.
Students are involved mainly in listening, copying, memorizing and drills.
Modern View of Mathematics
Mathematics is a science of pattern and order. This definition challenges the popular social
view of mathematics as a discipline dominated by computation and rules without reasons.
Science is a process of figuring things out or making sense of things. It beings with problematic
situations. Mathematics is a science of things that have a pattern of regularity and logical order.
Finding and exploring this regularity or order and then and making sense of it is what is doing
mathematics is all about. Even every young pupil can and should be involve in the science of
pattern and order. Notice that (i) 6+7, 5+8 and 4+9 are the same, why? What is pattern? Is
there any relationship?9
(ii) When two odd numbers are multiplied the product is odd but when they added or subtracted
the result is even why? Any logic behind it?
Pattern is not just in numbers, but in everything in the world in which we live. Pattern and order
are in buildings, in music, in commerce, in science, medicine manufacturing and sociology.
Mathematics discovers this order, makes sense of it, and uses it in multitude of fascinating
ways improving our lives and expanding our knowledge. School mathematics must help
students with this processing of discovery. Students should be able to count accurately, know
the basic fact of operations, compute whole numbers, fractions and decimals using efficient
methods, finding gradients, do differentiation and integration, state geometric ratios, etc. But
repetitive drill of the bits pieces is not ‘doing mathematics’ and will never result in
understanding. There is a time for drill and practice but should never come before
understanding. Drill may produce short term result on traditional test, but long term effects
have produce citizen happy to admit they can’t do mathematics. More mastery of skills is not
doing mathematics. Doing mathematics is engaging science of pattern and order
Most students think of mathematics as an old dull subject that was invented hundreds or
thousands of years ago. But mathematics is alive and constantly changing. In fact, outstanding
problems in mathematics are quit recently solved and others are yet to be solved. The common
believe is that mathematics is to be pursued only in a clear-cut, logical fashion. This belief is
perpetuated by the way mathematics is presented in most textbooks (and so how it is taught).
It is often reduce to a series of definitions, method to solve various type of problems and
theorems. These theorems are justified by means of proofs and deductive reasoning. Proofs
give mathematics its strength but the power of imagination is as important as the power of
deductive reasoning.
Textbook rarely show the long history of development of a concept or any of the blind alleys
that were taken. The mathematician seeks out relationships in single cases, looks for patterns,
and only then tries to generalize. It is usually not until much later that the generalization is
proved and finds it way into the textbook. It is therefore important for students to:
1. Understand in what sense that mathematics is the study of pattern and relationship
2. Become familiar with some of it patterns and relationship
3. Learn to use them in daily life
The latter two of these general goals should be sought in parallel rather than sequentially. For
the most part, learning mathematics in abstract before seeking to use it has not proven to be
effective. Thus, teachers should arrange instruction so that student encounter any given
mathematical pattern or relationship in many different contexts before, during, and after its
introduction in mathematics itself. Learning skills and remembering fact in mathematics are
important but they are only means to an end. Fact and skills are not important in themselves.
They are important when need them to solve a problem. Students will remember fact and skills
easily when they use them to solve real problems. As well as using mathematics to solve real
life problems, student should also be taught about the different part of mathematics, and how
they fit together. Mathematics can be taught using step-by-step approach to a topic but it is
important to show that many topics are linked and that mathematics is done all over the world.
Exercise 1.5
1. State two disadvantages of employing the traditional approach to teaching mathematics.
2. Why should mathematics teachers engage their students in the science of pattern and
order?
3. Explain two reasons why mathematics is important.
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