UNIT 1: Introduction to the Teaching of Mathematics a) Meaning and Nature of Mathematics b) Relation of Mathematics with other school subjects (Languages, Science, Social Studies- History, Geography, Civics & Economics; Commerce, Drawing, Music) c) Values in teaching of Mathematics. NATURE OF MATHEMATICS a. Meaning: From the Greek word, which means ―inclined to learn Oxford Dictionary, ―the branch of science concerned with number, quantity and space. Locke, ― a way to settle in the mind a habit of reasoning Ancient Hindus referred to mathematics as, ―Ganita—the science of calculation. Courant and Robbins, ―Mathematics is an expression of the human mind that reflects the active will, contemplative reason and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. The following conclusions can be made related to the meaning of mathematics: It is a science of number and space Has its own language in terms of signs, symbols, terms, operations etc. Uses/Requires intuition, logic, reasoning, analysis, construction, generality and individuality. Helps in drawing conclusions and interpreting various ideas and themes. It is suited for dealing with abstract concept of any kind. Helps to solve problems of daily life. Has an aesthetic value and helps to admire the beauty of nature. b. Branches of mathematics: Pure Mathematics Applied Mathematics rithmetic Relates to a wide Algebra range of studies Geometry with a wide use. c. Nature of Mathematics: Mathematics relies both on logic and creativity: it is pursued for a variety of : Practical Purposes i.e. how mathematics applies to their work? Intrinsic Interests i.e. Essence of mathematics lies in its beauty and intellectual challenge. The nature of mathematics can be also discussed in terms of: i. Science of logical reasoning: In mathematics the results are developed through a process of reasoning. Reasoning in mathematics possesses a number of characteristics such as, Simplicity Accuracy Certainty of Results Originality Verification Conclusions follow naturally from the facts when logical reasoning is applied to the facts.
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UNIT 1: Introduction to the Teaching of Mathematics a) Meaning and Nature of Mathematics
b) Relation of Mathematics with other school subjects (Languages, Science, Social Studies-
Mind perceives knowledge as a whole. If each subject is taught as a water tight compartment,
fragments of knowledge may accumulate at one place in the form of layers. Failure in
establishing links between the layers may be an obstacle to retention of earlier knowledge.
Hence no subject can be taught in isolation. The main aim of education is the unification of
knowledge existing in the different branches of learning. The flow of knowledge should be
permeable and should be merged with all the fields of learning. In order to achieve this the
teachers while teaching, need to relate the various subjects taught in school. Thus bridging the
gap between the various school subjects .Therefore it becomes necessary to relate one subject
with another. Knowledge is useful when it can be applied to day to day life; relation of a subject
with daily life makes knowledge useful. So in order to achieve the ultimate aim of education i.e.
to achieve the development of an all round personality, is possible only through unification of
knowledge and not by teaching only a few subjects in isolation. Hence relation is the only
answer to achieve this.
We can explain the concept of relation as follows,
The term relation‘ in its simplest form means connect or to be connected.
It is the conscious effort made by teachers teaching various subjects," to show similarities or
dependence of one subject on another.
Types of Relation
1. Internal / Vertical Relation
This type of relation is between different branches (e.g. algebra, geometry, arithmetic) of a given
subject (e.g. Mathematics). Branches of a subject many times are taught by different teachers,
such that each branch is treated as a different entity. But it shouldn‘t be done so as it loses it
charm of unification of knowledge as our mind perceives, so teachers need to relate it as much as
possible.(for e.g. Teacher teaching algebra in Std VII , in the chapter Identities –the expansion of
(a+b)2
= a2 + 2ab + b
2 should relate with geometry by drawing a geometrical figure and explain.
Internal relation is necessary for continuity of knowledge and understanding of the subject. This
focuses the relevance of branches (e.g. algebra, geometry, arithmetic) of one (Mathematics)
subject and their role in providing a wholistic fund of knowledge. It makes teaching interesting
and the subject is not taught as a water tight compartment and in isolation. So it becomes easy
for students to retain and apply knowledge whenever required.
2. External / Horizontal Relation
This type of relation is between different school subjects (e.g. Science, History, Geography ,
Economics etc. ) and a given school subject (mathematics). Different school subjects are taught
by different teachers, such that each subject is treated as a different entity. But it shouldn‘t be
done so as it loses it charm of unification of knowledge as our mind perceives, so teachers need
to relate it as much as possible.(for e.g. Teacher while solving sums based on speed and distance
in maths class should relate it with branch of science- i.e. physics, if she is teaching about life
history of great mathematicians like Euclid , Ramanujan she can relate with history, if she is
teaching a chapter based on graphs of rainfall she can relate it with geography. Etc. Thus external
relation is necessary for continuity of knowledge and understanding of the various subjects. It
makes teaching interesting and the subject is not taught as a water tight compartment and in
isolation. So it becomes easy for students to retain and apply knowledge of various subjects
whenever required. All subjects of the school curriculum contribute towards the realization of the
aims of education. Since they have the same purpose, study of one subject helps in the study of
other subjects.
This type of relation is between different subjects (e.g. Science, History, Geography , Economics
etc.) of a given subject (e.g. Mathematics).
3. Relation with Daily Life
A subject is best understood went it is applicable to daily life. So the teacher has to take care to
relate the subject she is teaching with daily life. For e.g. A mathematics teacher while teaching
simple and compound interest must pose examples from daily life transactions. Relation of a
subject with daily life is of the utmost importance in order to create interest in the subject.
Relation with daily life makes the subject relevant instead of being only theory with no practical
applications. It makes the theory more concrete and convinces the student the need for learning
the subject for practical use in his life.
This type of relation is between subjects taught in school and with daily life activities. (for e.g.
measurement of carpet area of a classroom, house , percentage of marks obtained in an
examination, profit on purchasing a product after discount, loss on selling a product etc. in daily
life can be obtained when the child uses his knowledge of a given subject (e.g. Mathematics).
Thus we can say from the above examples, that knowledge of Mathematics subject can be
applied in daily life to do various transactions and thus it is beneficial to learn mathematics and
not a waste as it helps us to do financial transactions as well as it equips the learner with
thorough knowledge so that he may not be duped or cheated by others as he himself can
calculate, measure and find percentage, profit, loss etc.
c) Values in teaching of Mathematics.
Utilitarian Value
Disciplinary Value
Intellectual Value
Cultural Value
Moral Value
Aesthetic Value
Social Value
International Value
UNIT 2: Designing Mathematics Curriculum a) Aims and Objectives of teaching Mathematics at Secondary and Higher Secondary
Levels (NCF 2009)
Secondary Levels (NCF 2009)
Students begin to perceive the structure of mathematics as a discipline.
They become familiar with the characteristics of mathematical communication:
carefully defined terms and concepts, the use of symbols to represent them,
precisely stated propositions, and proofs justifying propositions. These aspects
are developed particularly in the area of geometry.
Secondary Levels (NCF 2009)
Students develop their facility with algebra, which is important not only in the application
of mathematics, but also within mathematics in providing justifications and proofs.
At this stage, students integrate the many concepts and skills that they have learnt into a
problem-solving ability.
Secondary Levels (NCF 2009)
Mathematical modelling, data analysis and interpretation taught at this stage can
consolidate a high level of mathematical literacy.
Individual and group exploration of connections and patterns, visualisation and
generalisation, and making and proving conjectures are important at this stage, and can
be encouraged through the use of appropriate tools that include concrete models as in
Mathematics laboratories and computers.
Higher Secondary Levels (NCF 2009)
To provide students with an appreciation of the wide variety of the application of
Mathematics, and equip them with the basic tools that enable such application.
The rapid explosion of Mathematics as a discipline, and of its range of application,
favours an increase in the breadth of coverage.
Higher Secondary Levels (NCF 2009)
Such increase must be dictated by mathematical considerations of the importance of
topics to be included.
Topics that are more naturally the province of other disciplines may be left out of the
Mathematics curriculum.
The treatment of topics must have an objective, that is, the communication of
mathematical insights and concepts, which naturally arouse the interest and curiosity of
students.
b) Maxims of Teaching i. From Known to Unknown ii. From Simple to Complex iii. From Particular to General iv. From Concrete to Abstract v. From Whole to Part
Maxims of Teaching
Are the universal facts found out by the teacher on the basis of experience. They are of universal
significance and are trustworthy. The knowledge of different maxims helps the teacher to
proceed systematically. It also help to find out his/her way of teaching, especially at the early
stages of teaching.
Every teacher wants to make maximum involvement and participation of the learners in the
learning process. He sets the classroom in such a way so that it becomes attractive for them. He
uses different methods, rules, principles etc in order to make his lesson effective and purposeful.
He uses general rule or formula and applies it to particular example in order to make teaching –
learning process easy and upto the understandable level of students.
These settled principles, tenets, working rules or general truths through which teaching becomes
interesting, easy and effective are called the maxims of teaching. They have universal
significance. Every person who is expected to enter into the teaching profession have to
familiarize himself with the maxims of teaching. Their knowledge helps him to proceed
systematically.
The different maxims of teaching are briefly explained below. The teacher should always
proceed keeping them in view.
The different maxims of teaching are briefly explained below.
When a child enters into school, he possess some knowledge and it is the duty of teacher to
enlarge his previous knowledge. Whatever he possesses should be linked with the new
knowledge. If we link new knowledge with the old knowledge our teaching becomes clearer and
more definite.
This maxim facilitates the learning process and economises the efforts of the teacher and the
taught.
This way of teaching helps the learners to understand things fully. This way the teaching
becomes definite, clearer and more fruitful.
This maxim is based on the assumption that the student knows something. We are to increase his
knowledge and widen his outlook. We have to interpret all new knowledge‘ in terms of the old.
It is said that old knowledge serves as a hook on which the new one can be hung. Known is
trustworthy and unknown cannot be trusted. So while teaching we should proceed from known
and go towards unknown. For example, while teaching any lesson, the teacher can link the
previous experiences of the child with the new lesson that is to be taught.
2. From simple to complex:- The main objective of teacher is to teach and the learners objective is to learn something. In this
process of teaching and learning, simple or easy things should be first presented to the students
and gradually he should proceed towards complex or difficult things. The presentation of simple
material makes the learners interested, confident and feel encouraged. As they will show interest
towards the simple material, they becomes receptive to the complex matter. On the other hand, if
complex matter is presented first, the learner becomes upset, feel bored and finds himself in a
challenging situation.
For example in mathematics we first present the idea of +, - , x and then division.
When the child gets admitted to 9th and 10th class we introduce algebra, surds, trigonometry,
geometry etc. As he proceeds further he becomes familiar with the complex material like
matrices, integration, differentiation etc. In this way a learner shows interest by proceeding from
simple mathematics to complex one. But if we reverse the situation, he will find himself in a
challenging situation and will leave his studies due to complexity of matter. Simplicity or
complexity of the subject matter should be determined according to the view point of the
learners. It makes learning convenient and interesting for the students.
Class-room teaching is formal where the teacher tries to teach and the students try to learn
things. In this process of teaching-learning, the teacher should see that simple things are
presented first to the students. That way they will start taking interest. Once they become
interested, thou gradually complex type of things can also be learnt by them. By learning simple
things, they feel encouraged and they also gain confidence. On this basis, they become further
receptive to the complex matter. On the other hand, if complex types of things are presented to
the learner first, he becomes upset, feels bored and finds himself in a challenging situation lot
which he is not yet ready being immature and unripe.
Gradually more difficult items of learning may be presented to the students. It will smoothen
teaching being done by the teacher and make learning convenient and interesting for the
students.
3. From concrete to abstract:-
Concrete things are solid things and they can be visualized but abstract things are only
imaginative things. The child understands more easily when taught through their senses and
never forgets that material. On the other hand if abstract things or ideas are presented, they forget
it soon. As Froebel said, ―Our lessons ought to start in the concrete and end in the abstract‖. For
example when we teach the solar system, we first visualize the sun through our senses and gives
the concept of eight planets, galaxies, meteorites etc. Through this process, the learners
understand the materials more easily. Some power of imagination also develops in them .But if
we reverse the situation, it will become difficult for learners to understand anything. Another
example, when we teach counting to the students we should first take the help of concrete objects
like beads, stones etc. and then proceed to digits and numbers.
Concrete things are solid things and they can be touched with five senses. But abstract things can
only be imagined. So it is rather difficult to teach the children about abstract things. The students
are likely to forget them soon. On the other hand, if we teach the students with the help of
concrete objects, they will never forget the subject matter.
For example when we teach counting to the students we should first examine concrete nouns
like, laptop, book, Pen etc. and then proceed to digits and numbers. The stars, the moon, the sun
etc. being taught first whereas the abstract thing:, like planet, satellites etc. should be taught
afterwards.
5. From particular to general:- A teacher should always proceed from particular to general statements. General facts, principles
and ideas are difficult to understand and hence the teacher should always first present particular
things and then lead to general things.
While teaching, the teacher should first of all take particular statements and then on the basis of
particular cases, generalization should be made.
6. From Whole to Parts: This maxim is the offshoot of gestalt theory of learning whose main emphasis was to perceive
things or objects as whole and not in the form of parts. Whole is more understandable,
motivating and effective than the parts. In teaching, the teacher should first give a synoptic view
of lesson and then analyze it into different parts.
It is actually the reverse of the maxim ―analyses to synthesis‖.
In teaching, the teacher should try to acquaint the child with the whole lesson first and then the
different portions of it may be analyzed and studied intensively. This principle holds good while
teaching a thing to the small children. At the early stages, the child loves to speak full sentences
because in daily life situations, full sentences are used. The child should be given a full sentence.
Then he may have full familiarity with the different words contained in that sentence. Later he
may have the knowledge of words. Then he will have the knowledge of different letters forming
the words.
It will help the teacher to teach better and the learners to learn things conveniently.
c) Concentric and Topical Approach of Curriculum Construction
Concentric Approach
This is a system of organising a course rather than a method of teaching. It is, therefore,
better to call it concentric system or approach. It implies widening of knowledge just as
concentric circles go on extending and widening. It is a system of arrangement of subject
matter. In this method the study of the topic is spread over a number of years. It is based on
the principle that subject cannot be given an exhaustive treatment at the first stage. To begin
with, a simple pre-sentation of the subject is given and further knowledge is imparted in
following years. Thus beginning from a nucleus the circles of knowledge go on widening
year after year and hence the name concentric method.
Procedure
A topic is divided into a number of portions which are then allotted to different classes. The
criterion for allotment of a particular portion of the course to a particular class is the
difficulty of portion and power of comprehension of students in the age group. Thus it is
mainly concerned with year to year teaching but its influence can also be exercised in day-to-
day teaching Knowledge being given today
Merits of Concentric Method (i) This method of organisation of subject matter is decidedly superior to that in which one
topic is taken up in particular class and an effort is made to deal with all aspects of the topic
in that particular class.
(ii) It provides a framework from course which is of real value to students.
(iii) The system is most successful when the teaching is in hand of one teacher because then
he can preserve continuity in the teaching and keeps his expanding circle concentric.
(iv) It provides opportunity for revision of work already covered in a previous class and
carrying out new work.
(v) It enables the teacher to cover a portion according to receptivity of learner.
(vi) Since the same topic is learnt over many years so its impressions are more lasting.
(vii) It does not allow teaching to become dull because every year a new interest can be given
to the topic. Every year there are new problems to solve and new difficulties to overcome.
Drawbacks For the success of this approach we require really capable teacher. If a teacher becomes over
ambitious and exhausts all the possible interesting illustrations in there introductory year then the
subject loses its power of freshness and appeal and nothing is left to create interest in the topic in subsequent years. In case the topic is too short or too long then also the method is not found to be useful. A too long portion makes the topic dull and a too short portion fails to leave any permanent and lasting impression on the mind of the pupil.
EXAMPLE: TOPIC: INDICES Std. Content
VI Expression, Meaning, Base, Power, Reading, Finding Value, Indices of
–ve numbers
VII Introduction, Revision of Rules,expanded form of numbers
VIII Revision of all rules,laws of indices,fractional indices
IX Used to introduce Surds
X Trignometry, Solving sums
Concentric approach is really a very good approach to be adopted. Teachers have to be careful so
that portion is neither too long nor too short. In every consecutive year positive points can be
added, if same teacher, teaches the particular content every year, thus teacher can start with full
interest and vigor every year and will bring all possible and interesting illustrations in class.
C) TOPICAL APPROACH OF ORGANISING CURRICULUM
Topical arrangement means that a topic should be finishes entirely at one stage. It takes the topic
as a unit. Topical arrangement requires that easy and difficult portions of a topic should be dealt
with one stage only which is psychological. In this system the topic which is dealt with earlier
receives no attention later and so there is every likelihood of its being forgotten. The main defect
in the topical method is that it introduces in the curriculum a largeness of irrelevant material for
which the pupil finds no time and no immediate need or the use of which cannot be appreciated
by the pupil at the stage.
They are introduced with a view to make the teaching of the topic complete and through. Hence
topical method demands that a topic once taken should be finished in its entirely. This is not
more useful for lower classes.
MEANING:
A topic is taken with a unified whole in itself & it is unbreakable unit.
TOPICS which cannot be analysed into smaller unit are best thought though topical approach.
Principle of Topical Approach:
Take any topic, do not leave it half-done.
Finish the entire topic before starting the next topic.
Steps:
1.Systematic arrangement of subject matter.
2.Syllabus is also suitable organized.
3.Topic is chosen & is taught at stretch.
4.No break or gap is given in topic
Example:
1)In 5th standard: Perimeter
2)In 7th standard:
(a)Time & Work
(b)Discount, commission & Rebate.
3)In 9th standard:
(a)Sets,
(b)Surds,
(c)Logarithms.
(d)Computing.
MERITS:
Students keeps Complete Concentration on particular topic.
Teachers give best effort to impart information, knowledge & illustration of topic.
Students attention is not diverted.
Students ability, creativity, capacity is directed only to particular topic.
Students get concrete & thorough knowledge.
It illustrates advantage of co-relation.
DEMERITS:
It is an unpsychological approach: Difficulty level of student is not considered.
All student will not be able to receive & understand complicated parts of topic.
Teachers get bored of teaching same topic.
Interest of student will wane.
Since topic is not carried to next consecutive year, knowledge gained is forgotten by student.
Topical approach is really a very good approach. Teachers will have to select a topic, which can
be delivered to its fullest in the same consecutive years .It is one of effective way of imparting
information. If topic is kept a center, co-relation is also taken care.
UNIT 3: Pedagogical Analysis a) Content Analysis
b) Instructional Objectives
c) Instructional Strategies
• Concept Pedagogical analysis- Meaning and Process
• What is Content analysis and how it is done in Mathematics
• What are the Instructional Objectives for Mathematics
• What are the different Instructional Strategies that can be used in Mathematics
The word Pedagogy was derived from the Greek words, ‗paid‘ meaning child and ‗agogus‘
meaning leader of.
It refers to passive methods of teaching-learning.
It maintained that the students are empty vessels and the teacher can pour knowledge into them.
This approach to learning was called Pedagogy. Thus in pedagogy, the concern is transmitting
the content and the teacher alone takes all decisions about learning.
Today, however, the term 'pedagogy' has taken a new meaning. Thus the New Meaning of
Pedagogy is facilitating the learner in mastering the content.
Hence Pedagogy means the art and the science of teaching orLearner focussed education for
people of all ages.
PedagogicalAnalysis
Why to Teach: Aims/ Learning Outcomes
What to Teach: Content/Concepts
How to Teach: Approaches to teaching & learning
• In pedagogy, development is based upon a content plan:
What content needs to be covered?
How can this content be organized into manageable units or modules?
How can this content be transmitted in a logical sequence?
What would be the most effective method for transmitting this content (media)?
• Use of the word ‗linear‘ (giving example of non-linear).
• General Form of linear equation
• New word ‗Solution‘
• Domain of the variable
• Word or Story Problems classified as problems on numbers, problems on age.
• Graded presentation of problems with respect to difficulty level
• Standard X
Content
• Simultaneous equations:
• Solving by Graphical Method.
• Solving by Algebraic Method.
• Solution Set
• Word problems
• Graded Presentation
• Quadratic Equations:
• General form
• Identification
• Solving: factorization method
Method of completing the square
• Formula
• Equations reducible to quadratic
• Word problems
• Graded Presentation
• Unit - Equations involving one variable Std. VII
• Learning outcomes
After completing this unit the students will be able to:
1. Define an equation .
2. Determine the value of the variable which satisfies the given equation.
3. Use the addition, subtraction, multiplication, and division properties of equalities to solve
equations.
4. Translate the word problem into mathematical language (equation form).
5. Solve practical problems in arithmetic and geometry.
6. Verify the answer.
The students will be sensitized to and appreciate the underlying core elements.
Core Elements in this unit
Following core elements are depicted in number of word problems.
Secularism, (Names in problems)
Equality of sexes, (Names and activities)
Protection of environment, (such as planting of trees)
Removal of social barriers (different sections of society)
Observance of the small family norms, (two children a boy and a girl)
Inculcation of scientific temper.
Logical thinking – Framing equation
Solving step by step
Reasoning - Step and Reason side by side
Each step has a reason
Confidence – Verification of answer
All knowledge is one -Solving Practical
(Arithmetic and Geometry) problems using
equations (Algebra)
Methods to be followed
Inquiry
Problem Solving
Dramatization
Simulation
Framing word problems from given equations (always vice versa is done)
Evaluation questions
Why is this topic important?
How can this topic be used in different fields?
What if we modify this topic?
UNIT 4: TEACHING OF MATHEMATICS
a) Teaching Concepts (Concept Development Design) b) Teaching Generalizations (Inductive Deductive) c) Teaching Problem Solving (Problem Solving) d) Teaching Constructions (Lecture cum Demonstration) e) Teaching Proofs (Analytical Synthetic)